Systematic Approaches to Predictive Computational

APPROVED:

Angela K. Wilson, Major Professor Weston T. Borden, Committee Member Thomas R. Cundari, Committee Member Martin Schwartz, Committee Member Michael G. Richmond, Chair of the Chemistry

Department Michael Monticino, Interim Dean of the Robert

B. Toulouse School of Graduate Studies

SYSTEMATIC APPROACHES TO PREDICTIVE COMPUTATIONAL CHEMISTRY

USING THE CORRELATION CONSISTENT BASIS SETS

Brian P. Prascher, B.S.

Dissertation Prepared for the Degree of

DOCTOR OF PHILOSOPHY

UNIVERSITY OF NORTH TEXAS

May 2009

Prascher, Brian P., Systematic Approaches to Predictive Computational

Chemistry using the Correlation Consistent Basis Sets

The development of the correlation consistent basis sets, cc-pVnZ (where n = D,

T, Q, etc.) have allowed for the systematic elucidation of the intrinsic accuracy of ab

initio quantum chemical methods. In density functional theory (DFT), where the cc-pVnZ

basis sets are not necessarily optimal in their current form, the elucidation of the

intrinsic accuracy of DFT methods cannot always be accomplished. This dissertation

outlines investigations into the basis set requirements for DFT and how the intrinsic

accuracy of DFT methods may be determined with a prescription involving recontraction

of the cc-pVnZ basis sets for specific density functionals. Next, the development and

benchmarks of a set of cc-pVnZ basis sets designed for the s-block atoms lithium,

beryllium, sodium, and magnesium are presented. Computed atomic and molecular

properties agree well with reliable experimental data, demonstrating the accuracy of

these new s-block basis sets. In addition to the development of cc-pVnZ basis sets, the

development of a new, efficient formulism of the correlation consistent Composite

Approach (ccCA) using the resolution of the identity (RI) approximation is employed.

The new formulism, denoted ‘RI-ccCA,’ has marked efficiency in terms of computational

time and storage, compared with the ccCA formulism, without the introduction of

significant error. Finally, this dissertation reports three separate investigations of the

properties of FOOF-like, germanium arsenide, and silicon hydride/halide molecules

using high accuracy ab initio methods and the cc-pVnZ basis sets.

. Doctor of Philosophy (Physical

Chemistry), May 2009, 275 pp., 49 tables, 28 illustrations, references, 307 titles.

ii

Copyright 2009

by

Brian P. Prascher

iii

ACKNOWLEDGEMENTS

I would like to acknowledge Jesus Christ as my source of strength in all things. (“I have

strength for all things in Christ who empowers me,” Philippians 4:13) Further, the love, support,

and encouragement of my wife and best friend, Laura; my beautiful daughters, Emma and

Charlotte; my Mom and Dad; and Lew and Diane, has been paramount for me during my

graduate career.

I wish to thank my colleagues during my time in the Wilson Group (in order of

appearance): Xuelin Wang, Scott Yockel, James Seals, Ray Bell, Ben Mintz, Pankaj Sinha, John

Determan, Nathan DeYonker, Sammer Tekarli, Gavin Williams, Brent Wilson, Gbenga Oyedepo,

Kameron Jorgensen, and Marie Majkut, for constantly challenging me with questions about

computational chemistry and, in so doing, kept my knowledge of the subject up to par. In

particular, I want to thank and acknowledge Brent Wilson and Gavin Williams for their

contributions to projects discussed in this dissertation. Many thanks and acknowledgements

also go to the numerous undergraduates that served in the Wilson Group, particularly Rebecca

Lucente‐Schultz, Arjun Kavi, and Jeremy Lai for their contributions to work presented herein.

I also want to thank the professors of the Chemistry Department, who challenged and

molded me into the academic that I am today. Finally, I would like to give a heart‐felt thanks to

Angela Wilson, my academic mother and mentor, whose open‐door, yet hands‐off approach

gave me room to spread my wings.

iv

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS ................................................................................................................... iii

LIST OF TABLES ............................................................................................................................... vii

LIST OF ILLUSTRATIONS .................................................................................................................. xi

Chapter

1. INTRODUCTION ................................................................................................................... 1

2. COMPUTATIONAL QUANTUM CHEMISTRY ........................................................................ 4

2.1 The Schrödinger Equation ....................................................................................... 4 2.2 Ab initio Methodology ............................................................................................ 8

2.2.1 The Hartree‐Fock Approximation ............................................................. 10 2.2.2 Configuration Interaction .......................................................................... 16 2.2.3 Coupled Cluster Theory ............................................................................. 20 2.2.4 Many‐Body Perturbation Theory .............................................................. 24

2.3 Density Functional Theory .................................................................................... 27 2.3.1 The Hohenberg‐Kohn Theorems ............................................................... 30 2.3.2 The Kohn‐Sham Method ........................................................................... 33

2.4 Basis Set Theory .................................................................................................... 35 2.5 Model Chemistries ................................................................................................ 40

3. CORRELATION CONSISTENT BASIS SETS ........................................................................... 43

3.1 Introduction .......................................................................................................... 43 3.2 Valence and Tight d Basis Sets .............................................................................. 45 3.3 Augmented Basis Sets ........................................................................................... 50 3.4 Core‐Valence Basis Sets ........................................................................................ 50 3.5 Scalar Relativistic Basis Sets .................................................................................. 51

4. SYSTEMATIC TRUNCATION OF THE CORRELATION CONSISTENT BASIS SETS IN DENSITY FUNCTIONAL THEORY CALCULATIONS ............................................................................. 53

4.1 Introduction .......................................................................................................... 53 4.2 Computational Methodology ................................................................................ 56 4.3 Results and Discussion .......................................................................................... 58

4.3.1 Atoms ........................................................................................................ 58 4.3.2 Homonuclear Diatomics ............................................................................ 60

v

4.3.3 CH4, SiH4, NH3, PH3, H2O, and H2S ............................................................. 63 4.3.4 Computational Time Savings ..................................................................... 65

4.4 Conclusions ........................................................................................................... 66

5. SYSTEMATIC RECONTRACTION OF THE CORRELATION CONSISTENT BASIS SETS FOR DENSITY FUNCTIONAL THEORY CALCULATIONS ............................................................... 82

5.1 Introduction .......................................................................................................... 82 5.2 Computational Methodology ................................................................................ 85 5.3 Results and Discussion .......................................................................................... 87

5.3.1 Atoms ........................................................................................................ 87 5.3.2 Molecules .................................................................................................. 89 5.3.3 Kohn‐Sham Limits ...................................................................................... 92 5.3.4 Basis Set Superposition Error .................................................................... 94 5.3.5 Diffuse Functions ....................................................................................... 95

5.4 Conclusions ........................................................................................................... 96

6. THE DEVELOPMENT OF S‐BLOCK CORRELATION CONSISTENT BASIS SETS .................... 116

6.1 Introduction ........................................................................................................ 116 6.2 Computational Methodology .............................................................................. 118 6.3 Basis Set Construction ........................................................................................ 120

6.3.1 Tight d Functions for Inner Valence Correlation ..................................... 120 6.3.2 Core‐Valence Functions for Sub‐Valence Correlation ............................ 121 6.3.3 Recontracted Basis Sets for Scalar Relativistic Computations ................ 123

6.4 Benchmark Computations .................................................................................. 125 6.4.1 Ionization Potentials and Electron Affinities ........................................... 125 6.4.2 Optimized Geometries and Vibrational Frequencies .............................. 128 6.4.3 Thermochemistry .................................................................................... 129

6.5 Conclusions ......................................................................................................... 130

7. THE RESOLUTION OF THE IDENTITY APPROXIMATION APPLIED TO THE CORRELATION CONSISTENT COMPOSITE APPROACH ............................................................................ 137

7.1 Introduction ........................................................................................................ 137 7.2 Computational Methodology .............................................................................. 142 7.3 Results and Discussion ........................................................................................ 144

7.3.1 RI‐ccCA Implementation ......................................................................... 144 7.3.2 Auxiliary Basis Sets for RI‐ccCA ............................................................... 145 7.3.3 Energetic Properties ................................................................................ 146 7.3.4 Computational Cost................................................................................. 147

7.4 Conclusions ......................................................................................................... 152

8. A SYSTEMATIC INVESTIGATION OF DIHALOGEN‐μ‐DICHALCOGENIDES ......................... 168

vi


8.3.1 Stability of X2A2 Compounds ................................................................... 173 8.3.2 XAAX Structures ...................................................................................... 174 8.3.3 Vibrational Frequency Analyses .............................................................. 176 8.3.4 Anomeric Effects ..................................................................................... 177

8.4 Conclusions ......................................................................................................... 180

9. A SYSTEMATIC INVESTIGATION OF GERMANIUM ARSENIDES ....................................... 188


9.3.1 Optimized Structures .............................................................................. 192 9.3.2 Classical Barriers to Isomerization .......................................................... 195 9.3.3 Thermochemistry and Relative Stabilities .............................................. 197

9.4 Conclusions ......................................................................................................... 199

10. A SYSTEMATIC INVESTIGATION OF SILICON HYDRIDES AND HALIDES ........................... 218

10.1 Introduction ........................................................................................................ 218 10.2 Computational Methodology .............................................................................. 220 10.3 Optimized Structures .......................................................................................... 221

10.3.1 SiH, SiF, and SiCl ...................................................................................... 221 10.3.2 SiH2, SiF2, SiCl2, SiHF, and SiHCl ............................................................... 222 10.3.3 SiH3, SiF3, SiCl3, SiH2F, SiH2Cl, SiHF2, and SiHCl2 ...................................... 223 10.3.4 SiH4, SiF4, SiCl4, SiH3F, SiH3Cl, SiH2F2, SiH2Cl2, SiHF3, and SiHCl3 .............. 224

10.4 Thermochemistry ................................................................................................ 226 10.4.2 Atomization Energies and Enthalpies of Formation ............................... 227 10.4.3 Dissociation Reaction Enthalpies ............................................................ 233

10.5 Conclusions ......................................................................................................... 234

11. CONCLUDING REMARKS ................................................................................................. 257

REFERENCES ................................................................................................................................ 259

vii

LIST OF TABLES

Table 3.1 The composition of the correlation consistent basis sets for the first three rows of the main group. ................................................................................................ 45

Table 4.1 Total energies (Eh) of the carbon and oxygen atoms computed with full and truncated basis sets; energy differences are listed in mEh. .................................. 69

Table 4.2 Ionization potentials (eV) of the carbon and oxygen atoms computed with full and truncated basis sets; relative differences are listed below the full basis set values. ................................................................................................................... 71

Table 4.3 Electron affinities (eV) of the carbon and oxygen atoms computed with full, truncated, and augmented truncated basis sets (aug); relative differences are listed below the full basis set values. ................................................................... 72

Table 4.4 Total energies (Eh) of the C2 and O2 molecules computed with full and truncated basis sets; energy differences are listed in mEh. ................................................... 74

Table 4.5 BLYP vertical ionization potentials (eV), electron affinities (eV), atomization energies (kcal/mol), and optimized bond lengths (Å) of the C2 and O2 molecules computed with full and truncated basis sets; electron affinities include diffuse functions and relative energy differences are listed below the full basis set values. ................................................................................................................... 76

Table 4.6 B3LYP atomization energies (kcal/mol) and optimized bond lengths (Å) for CH4 and SiH4 (both

1A1); relative differences are listed below the full basis set values................................................................................................................................ 78

Table 4.7 B3LYP optimized geometries (Å and degrees) for NH3 and PH3 (both 1A1); relative

differences are listed below the full basis set values. .......................................... 79

Table 4.8 B3LYP optimized geometries (Å and degrees) for H2O and H2S (both 1A1); relative

differences are listed below the full basis set values. .......................................... 80

Table 4.9 Percent CPU time and average time saved computing B3LYP single‐point energies with truncated basis sets, relative to the full basis sets. ....................... 81

Table 5.1 A comparison of the BLYP and B3LYP total energies (Eh) of the first row atoms (including hydrogen) computed with the original and recontracted cc‐pVnZ basis sets; relative differences to the original basis sets are shown in mEh under [rc].............................................................................................................................. 101

viii

Table 5.2 A comparison of the BLYP and B3LYP ionization potentials and electron affinities (eV) of the first row atoms (including hydrogen) computed with the original and recontracted cc‐pVnZ basis sets; relative differences to the original basis sets are shown under [rc]. ................................................................................................ 102

Table 5.3 A comparison of the BLYP and B3LYP total energies (Eh), ionization potentials, and electron affinities (eV) of the first row dimers (including hydrogen) computed with the original and recontracted cc‐pVnZ basis sets; relative differences to the original basis sets are shown in mEh (total energies) and eV under [rc]. ........................................................................................................... 103

Table 5.4 The BLYP and B3LYP optimized geometries (Å and degrees) of several first row molecules computed with the original and recontracted basis sets; relative differences to the original basis set are shown under [rc]. ................................ 105

Table 5.5 The BLYP and B3LYP atomization energies (kcal/mol) of several first row molecules computed with the original and recontracted basis sets; relative differences to the original basis set are shown under [rc]. ................................ 107

Table 5.6 Two‐ and three‐point extrapolated BLYP and B3LYP Kohn‐Sham limits of atomization energies (kcal/mol) computed with the original and recontracted basis sets. Non‐monotonic behavior is denoted by ‘‐‐‐’. .................................... 110

Table 5.7 The BLYP and B3LYP atomization energies (kcal/mol) and their extrapolated Kohn‐Sham limits of four molecules corrected for basis set superposition error (BSSE). ................................................................................................................. 113

Table 5.8 A comparison of BLYP and B3LYP atomization energies (kcal/mol) and their extrapolated Kohn‐Sham values using the aug‐cc‐pVnZ, aug‐cc‐pVnZ[rc], and aug‐cc‐pVnZ[rc] basis sets corrected for basis set superposition error (BSSE). ........ 114

Table 6.1 The composition of the correlation consistent basis sets for s‐block atoms. .... 119

Table 6.2 CCSD(T) ionization potentials and electron affinities of Li, Be, Na, and Mg; the tight d correlation consistent basis sets for Na and Mg have been used. ......... 131

Table 6.3 CCSD(T) optimized geometries (Re, Å) of some s‐block molecules computed with various families of correlation consistent basis sets. ......................................... 132

Table 6.4 CCSD(T) harmonic vibrational frequencies (Å) of some s‐block molecules computed with various families of correlation consistent basis sets................. 134

Table 6.5 CCSD(T) standard state enthalpies of formation at 298.15 K (kcal/mol) for some s‐block molecules computed with various families of correlation consistent basis sets; tight d functions have been used for the Na and Mg atoms. .................... 135

ix

Table 7.1 Differences between ccCA and RI‐ccCA total energies (mEh), relative CPU times, and relative disk space using different combinations of RI‐SCF and RI‐MP2 auxiliary basis sets.a ............................................................................................ 154

Table 7.2 Total energies (Eh) computed with ccCA, RI‐ccCA, and RI‐ccCA+L; differences (mEh) are relative to the ccCA energies. ............................................................. 155

Table 7.3 Enthalpies of formation (ΔHf°) at 298 K (kcal/mol) computed with ccCA, RI‐ccCA, and RI‐ccCA+L. ..................................................................................................... 159

Table 7.4 CPU times and disk space usage of RI‐ccCA and RI‐ccCA+L relative to ccCA...... 164

Table 8.1 Stabilities (kcal/mol) of the different isomers of X2A2 molecules relative to the gauche isomer, XAAX, computed with B3LYP and CCSD(T). ............................... 182

Table 8.2 B3LYP and CCSD(T) optimized geometries (Å and degrees) of the fluorine species using various correlation consistent basis sets. ................................................. 183

Table 8.3 B3LYP and CCSD(T) optimized geometries (Å and degrees) of the chlorine species using various correlation consistent basis sets. ................................................. 184

Table 8.4 B3LYP and CCSD(T) optimized geometries (Å and degrees) of the bromine species using various correlation consistent basis sets. ................................................. 185

Table 8.5 B3LYP and CCSD(T) harmonic vibrational frequencies (cm‐1) computed with the aug‐cc‐pVTZ basis set. ......................................................................................... 186

Table 8.6 A comparison of CCSD(T) harmonic and anharmonic vibrational frequencies (cm‐1) of FSSF computed with the cc‐pVDZ basis set. ......................................... 187

Table 9.1 Non‐relativistic and spin‐orbit (SO) total energies (Eh) of some main group atoms computed at the MCSCF/cc‐pVTZ level of theory; SO corrections are given in kcal/mol. ............................................................................................................. 202

Table 9.2 A comparison of the CCSD(T) optimized geometries (Å and degrees) for the L‐GeAs and GeAs‐L isomers; previously reported geometries are included for comparison. ........................................................................................................ 203

Table 9.3 A comparison of the B3LYP optimized geometries (Å and degrees) for the L‐GeAs and GeAs‐L isomers; previously reported geometries are included for comparison. ........................................................................................................ 205

Table 9.4 CCSD(T) and B3LYP classical reaction barriers (kcal/mol) for the forward (f) and reverse (r) reactions; previously reported reaction barriers are included for comparison. ........................................................................................................ 207

x

Table 9.5 Non‐relativistic (NR), Cowan‐Griffin (CG), and Douglas‐Kroll (DK) enthalpies of formation at 298 K (kcal/mol) of the L‐GeAs isomers computed with CCSD(T) and B3LYP; spin‐orbit effects have been included in the relativistic enthalpies. ...... 209

Table 9.6 Non‐relativistic (NR), Cowan‐Griffin (CG), and Douglas‐Kroll (DK) enthalpies of formation at 298 K (kcal/mol) of the GeAs‐L isomers computed with CCSD(T) and B3LYP; spin‐orbit effects have been included in the relativistic enthalpies. ...... 212

Table 9.7 CCSD(T) and B3LYP enthalpies of isomerization (kcal/mol) from L‐GeAs to GeAs‐L at 298 K. Various basis set levels and CBS limits are represented as well as non‐relativistic (NR) and relativistic energies using the Cowan‐Griffin (CG) and Douglas‐Kroll (DK) approach. .............................................................................. 215

Table 10.1 CCSD(T) optimized geometries (Å) of SiH, SiF, and SiCl. ..................................... 239

Table 10.2 CCSD(T) optimized geometries (Å and degrees) of SiH2, SiF2, SiCl2, SiHF, and SiHCl.............................................................................................................................. 240

Table 10.3 CCSD(T) optimized geometries (Å and degrees) of SiH3, SiF3, SiCl3, SiH2F, SiH2Cl, SiHF2, and SiHCl2. ................................................................................................ 242

Table 10.4 CCSD(T) optimized geometries (Å and degrees) of SiH4, SiF4, SiCl4, SiHF3, SiH3F, SiHCl3, SiH3Cl, SiH2F2, and SiH2Cl2. ....................................................................... 244

Table 10.5 CCSD(T) and ccCA atomization energies (kcal/mol) and extrapolated energies; atomic spin‐orbit corrections have been applied to the CCSD(T) results. ......... 246

Table 10.6 CCSD(T) and ccCA enthalpies of formation at 298 K (kcal/mol) and extrapolated energies; atomic spin‐orbit corrections have been applied to the CCSD(T) results.............................................................................................................................. 249

Table 10.7 CCSD(T) and ccCA enthalpies of dissociation at 298.15 K (kcal/mol) of the silicon hydrides, computed with the aug‐cc‐pV(n+d)Z basis sets; the mean absolute deviation (MAD) is relative to the experimental values. .................................... 252

Table 10.8 CCSD(T) and ccCA enthalpies of dissociation at 298.15 K (kcal/mol) of the silicon fluorides, computed with the aug‐cc‐pV(n+d)Z basis sets; the mean absolute deviation (MAD) is relative to the experimental values. .................................... 253

Table 10.9 CCSD(T) and ccCA enthalpies of dissociation at 298.15 K (kcal/mol) of the silicon chlorides, computed with the aug‐cc‐pV(n+d)Z basis sets; the mean absolute deviation (MAD) is relative to the experimental values. .................................... 255

xi

LIST OF ILLUSTRATIONS

Figure 2.1 Representations of restricted closed‐shell (I), restricted open‐shell (II), and unrestricted (III) wavefunctions. .......................................................................... 14

Figure 2.2 A comparison of the dissociation of H2 using a single determinant (SCF) and a two‐determinant (MCSCF) wavefunction. ............................................................ 16

Figure 2.3 Examples of singly‐, doubly‐, and triply‐excited configurations of the HF reference wavefunction. ....................................................................................... 18

Figure 2.4 Plots comparing the behavior of a single STO to a single GTO (left) and that of a single STO with a linear combination (LC) of three GTOs (right). The units are arbitrary. ............................................................................................................... 38

Figure 2.5 A conceptual view of basis set size versus level of sophistication in correlated ab initio methods. The number of determinants with respect to excitation level (n), number of electrons (N), and basis set size (K) is given by the formula on the lower axis. ............................................................................................................. 41

Figure 4.1 BLYP ionization potentials of the oxygen atom computed with truncated basis sets (hash marks) and plotted against the full basis set values. The inset shows more detail at the quadruple‐ζ level. Points denoted ‘l > 3’ are the truncated basis sets with only s, p, d, and f basis functions. ................................................. 61

Figure 4.2 BLYP ionization potentials of the O2 molecule (3Σg) computed with truncated basis sets (hash marks) and plotted against the full basis set values. Points denoted ‘l > 3’ are the truncated basis sets with only s, p, d, and f basis functions................................................................................................................................ 62

Figure 4.3 Comparisons of CPU time savings (closed circles), ionization potentials (triangles), electron affinities (upside‐down triangles), atomization energies (open squares), and total energies (closed squares) between the full and truncated cc‐pV5Z basis set using B3LYP. Truncation levels: ‘0’ is the full basis, ‘1’ is cc‐pV5Z(‐1h ; 1g), ‘2’ is cc‐pV5Z(‐1h1g ; 1g1f), etc. .......................................... 68

Figure 5.1 The BLYP potential energy curve of O2 (3Σg) computed with the original and

recontracted correlation consistent basis sets. .................................................... 91

Figure 5.2 A comparison of the BLYP atomization energies computed with the original and recontracted correlation consistent basis sets. .................................................... 92

xii

Figure 5.3 A comparison of BLYP atomization energies computed with the original and recontracted correlation consistent basis sets, and the recontracted basis sets with basis set superposition error (BSSE) removed. ............................................ 99

Figure 5.4 A comparison of BLYP atomization energies computed with the pc‐n, aug‐cc‐pVnZ, and aug‐cc‐pVnZ[rc] basis sets with basis set superposition error (BSSE) removed. .................................................................................................. 100

Figure 6.1 The incremental CISD correlation energy lowering of the Mg atom due to each polarization function. .......................................................................................... 121

Figure 6.2 Comparisons of the Na atom valence (cc‐pVnZ) exponents with the core‐valence (cc‐pCVnZ) and weighted core‐valence (cc‐pwCVnZ) exponents; the exponents are grouped by angular momentum. .................................................................. 122

Figure 6.3 The spacing of the d exponents in the cc‐pVnZ basis sets compared with those in the cc‐pV(n+d)Z basis sets. ................................................................................. 123

Figure 6.4 CCSD(T) valence correlation energies recovered by the cc‐pVnZ (V/V), cc‐pCVnZ (V/CV), and cc‐pwCVnZ (V/wCV) basis sets. ....................................................... 124

Figure 6.5 CCSD(T) core‐valence plus valence correlation energies recovered by the cc‐pVnZ (CV/V), cc‐pCVnZ (CV/CV), and cc‐pwCVnZ (CV/wCV); the 1s orbitals of Na and Mg were kept frozen in each calculation. ........................................................... 126

Figure 6.6 Comparisons of the scalar relativistic corrections to the atomic Hartree‐Fock energy using the original contractions (diamonds) and the Douglas‐Kroll contractions (squares) with increasing basis set size. ........................................ 127

Figure 7.1 CPU times of RI‐ccCA (top) and RI‐ccCA+L (bottom) versus ccCA; each data point represents a molecule in the test set. ................................................................ 148

Figure 7.2 Disk space usage of RI‐ccCA (top) and RI‐ccCA+L (bottom) versus ccCA; each data point represents a molecule in the test set. ....................................................... 150

Figure 7.3 Correlation between molecular size and average CPU time and disk space saved using RI‐ccCA and RI‐ccCA+L, relative to ccCA. ................................................... 151

Figure 8.1 The orbital representation of anomeric delocalization in XAAX systems: the lone pair on a bridging A atom delocalizes into the adjacent σ*(AX) orbital. ............ 170

Figure 8.2 The lowest unoccupied CISD natural orbitals of FSSF: both contour plots are F‐S σ* orbitals (14A, left; 13B, right). The perspective is down the z‐axis (iso‐contour value: 0.095). ...................................................................................................... 178

xiii

Figure 8.3 The molecules FSSF (left), ClSSCl (center), and BrSSBr (right), and the lone pair orbitals responsible for anomeric delocalization. The prespective is down the z‐axis (iso‐contour values: 0.06, 0.03, and 0.03, respectively). ............................. 179

Figure 9.1 A schematic of the forward (f) and reverse (r) reaction coordinates of L migration from the germanium atom to the arsenic atom. ................................................ 195

Figure 9.2 A side‐by‐side comparison of non‐relativistic and Douglas‐Kroll (DK) relativistic CCSD(T) and B3LYP relative energies from mixed Gaussian/exponential extrapolations. .................................................................................................... 201

Figure 10.1 Plots comparing the CCSD(T) optimized geometries of SiH, SiF, and SiCl using four different families of correlation consistent basis sets. ....................................... 237

Figure 10.2 Plots comparing the CCSD(T) enthalpies of formation at 298 K of the silicon fluorides using four families of correlation consistent basis sets. ..................... 238

1

CHAPTER 1

INTRODUCTION

The practice of chemistry, in one form or another, has been around for millennia, but

only in the past two centuries has the practice of chemistry evolved from archaic alchemy to a

pure and applied science. Modern chemistry is able to probe the structure, microscopic

properties, and reactions of atoms and molecules through systematic laboratory methods that

include synthetic techniques, such as directly reacting a substance with another substance in a

controlled environment; instrumental techniques, including the myriad of spectroscopic

techniques; and, more recently, computational techniques. With the advent of quantum

mechanics in the early 20th century, the way in which chemical properties traditionally are

elucidated changed dramatically. Schrödinger’s wave mechanics, simultaneously introduced

alongside Heisenberg’s matrix mechanics, offered ways of calculating both static and dynamic

properties of quantized systems. Following the construction of the first electronic computer,

chemists and physicists began to submit atomic and molecular properties for calculation using

Schrödinger’s and Dirac’s mathematics. The amount of information acquired about chemical

properties during the early years of quantum mechanics was not substantial, limited primarily

by the computational expense of both the underlying mathematics and the physical

computational resources (i.e. storage, time, and software).

The general organization of this dissertation is such that systematically approaching a

problem with computational chemistry is covered in detail, from the selection of a

2

computational model (Chapter 2) and basis sets (Chapter 3); the development of systematic

basis sets for density functional theory (Chapters 4 and 5) and ab initio methods (Chapter 6); to

the application and benchmarking of these basis sets (Chapters 7‐10).

First, a basic working knowledge of the mathematics behind computational chemistry,

specifically ab initio methods and density functional methods, is presented in some detail in

Chapter 2. From there, Chapter 3 discusses a special class of basis sets, called the correlation

consistent basis sets, which are the focal point of this dissertation as every subsequent chapter

employs them.

In Chapters 4 and 5, work is described that has been done to gain a greater

understanding of the basis set requirements in density functional theory calculations. By

examining the effects of systematically truncating and recontracting the existing correlation

consistent basis sets, a set of compact, fine‐tuned basis sets is presented that provides accurate

results relative to the original correlation consistent basis sets at a reduced computational cost.

Further, the recontracted basis sets, augmented with diffuse functions and a correction for

basis set superposition error, restore the smooth, monotonic behavior inherent to the

correlation consistent basis sets in Kohn‐Sham calculations. Chapter 6 discusses the

development and benchmarks of new correlation consistent basis sets for the s‐block atoms

lithium, beryllium, sodium, and magnesium. As discussed in the chapter, these atoms present a

unique challenge in the development of accurate basis sets due to their electronic structure.

Chapter 7 describes the implementation of the resolution of the identity approximation

into the correlation consistent composite approach. Specifically, this chapter presents the

implementation and discusses the need for specialized correlation consistent basis sets

3

optimized for the resolution of the identity. The chapter is concluded by demonstrating the

incredible efficiency of resolution of the identity methods, which are the key to accessing larger

and larger electronic systems on current computational hardware.

The last three chapters of this dissertation are benchmark/predictive studies employing

the correlation consistent basis sets in FOOF‐like molecules, germanium arsenides, and silicon

hydrides/halides. The calculated structures, spectroscopy, and thermochemistry of many

molecules is presented and discussed along side experimental data, where available. In the

work on FOOF‐like molecules, the electronic structure is specifically focused upon due to the

unique geometry of these molecules stemming from their electronic structure. All of the

chemical properties presented for the germanium arsenides are purely predictive since these

molecules have never been reported in the literature. Finally, the silicon molecule benchmarks

critically examine how the correlation consistent composite approach performs compared with

coupled cluster theory and present high‐level ab initio predictions of geometries, spectroscopic

properties, and thermochemistry of several transient silicon hydrides and halides.

4

CHAPTER 2

COMPUTATIONAL QUANTUM CHEMISTRY

2.1 The Schrödinger Equation

The central goal of computational chemistry is to model chemical systems by computing

their properties using the Schrödinger equation.1‐4 The Schrödinger equation is an eigen‐

equation in which an energy operator, the Hamiltonian, operates on a wavefunction describing

the electronic and nuclear structure of a given system to produce the system’s total energy as

an eigenvalue. There is no formal proof of the Schrödinger equation, but the fact that highly

accurate solutions to this equation agree with experimental observations leads to its continued

use. Since chemical properties are usually more dependent on the electronic, rather than the

nuclear structure of a chemical system, the time‐independent form of the electronic

Schrödinger equation within the Born‐Oppenheimer (BO) approximation5 is often employed:

Ψ Ψ (2.1)

In the equation above, is the Hamiltonian, Ψ is the electronic wavefunction, and is the

total electronic energy of the wavefunction. The Hamiltonian is, in atomic units:6,7

12 | |

12

1

(2.2)

where is the number of electrons interacting with nuclei. The first term of the Hamiltonian

is the kinetic energy of the electrons, where is the Laplacian; the second term is the nuclear‐

electron attraction operator, where is the charge on nucleus and | | is the distance

5

between electron and nucleus ; and the third term is the electron‐electron repulsion, where

is the distance between two electrons. The Hamiltonian is more compactly written as

, where is the kinetic term, is the nuclear‐electron term, and is

the electron‐electron term. Once the nuclear‐electron potential (plus any external potential)

has been specified, the Hamiltonian is uniquely defined for a given electronic system.

The time‐independent wavefunction Ψ is a mathematical function of the electron

positions within a field of fixed nuclei. All information regarding the static discrete energy states

of the electrons is contained within this wavefunction and is extracted using the energy

operators of the Hamiltonian.6,7 In general, the wavefunction is complex‐valued and has no

direct physical interpretation, but, the squared modulus of the wavefunction, |Ψ | , is the

electron density (discussed later), which is observable by experiment using x‐ray photoelectron

spectroscopy. Considering this, the wavefunction itself may be interpreted from a statistical

point of view as a position function of the electrons. Despite the statistical interpretation, the

wavefunction may not be used to find the exact position of the electrons – a consequence of

the Heisenberg uncertainty principle.8

The wavefunction must have various mathematical properties so that it remains

physically viable (i.e. spans a Hilbert space): 1) it must be single‐valued at all points in space, 2)

it must be continuous at all points in space, 3) it must be zero‐valued at the boundaries (usually

taken to be infinity), 4) its derivatives at the boundaries must also be zero‐valued, and 5) the

wavefunction must be square integrable.6 Further, the electronic wavefunction must be anti‐

symmetric (change sign) with respect to the interchange of two electron coordinates.6,7 That is

to say, the wavefunction must obey the Pauli exclusion principle,9,10

6

Ψ , , … , , , … , Ψ , , … , , , … , (2.3)

One method of writing an N‐electron wavefunction with the necessary anti‐symmetry, |Ψ , is

the Slater determinant:7,11

|Ψ1√ !

1√ !

1!

(2.4)

where the columns represent electron orbitals and the rows represent electrons. The factor

1 √ !⁄ is a normalization constant, 1 is the parity of the th term, and is the

permutation operator, which interchanges the orbital indices. The term in brackets is a product

wavefunction called the Hartree product.12‐14 From linear algebra, it is known that the

properties of a determinant are such that the interchange of two rows (electrons in this case)

leads to a change of sign in the value of the determinant.15 Thus, the necessary anti‐symmetry

is taken care of naturally using a determinant as the wavefunction. The form of the individual

electron orbitals may be any mathematical function that obeys the five conditions previously

discussed, and are usually taken from Gaussian‐type basis sets (discussed later).

A subtle point worth making here is that the non‐relativistic Hamiltonian does not

intrinsically account for the spin of the electrons. This phenomenon is accounted for a

posteriori by using generalized electron coordinates of the form , where contains

the spatial components and the spin component. The generalized coordinates used in (2.4)

lead to spin orbitals, which may be separated into their spatial and spin components as:

|

|

(2.5)

The spin functions | and | are orthonormal, | . When the Hamiltonian (more

7

specifically, the operator) is applied to the determinant wavefunction, the anti‐symmetry,

along with the orthogonal nature of the spin orbitals produces an exchange interaction. This

interaction (also called exchange correlation or same spin correlation) only occurs between

electrons of the same spin, has no classical analog, and lowers the classical Coulomb repulsion

energy a wavefunction without anti‐symmetry (i.e. a Hartree product). Spin anti‐symmetry is

one of the underlying phenomenon responsible for Hund’s rules.16,17

The Hamiltonian operator contains two 2‐particle interaction terms: the term and

the term. The nuclear positions are fixed in the BO approximation, reducing to a 1‐

electron problem. However, remains a 2‐electron term. It is well‐known that there is no

closed form solution to the interacting 3‐particle (or higher) problem, and thus no closed form

solution to the electronic Schrödinger equation for more than one electron. This is the central

problem of computational quantum chemistry.6,7

To overcome the intractable problem of finding wavefunctions that satisfy the many‐

electron Schrödinger equation, approximate wavefunctions may be constructed using

variational calculus. Once constructed, the energy of an approximate wavefunction may be

computed as the expectation value of the Hamiltonian,

Ψ | |Ψ d d Ψ ,… , Ψ ,… , (2.6)

where the wavefunction is assumed to be normalized, Ψ |Ψ 1. To obtain the best

approximation to the exact solution of the Schrödinger equation, an approximate wavefunction

is constructed with variational parameters that may be adjusted until the expectation value of

the energy is minimized. Once minimized, the wavefunction is the best solution (in a variational

sense) to the Schrödinger equation. The question then arises as to how the variational

8

parameters are chosen and how the initial wavefunction is constructed. The answer to this

question leads to the many approximate methods of constructing many‐electron wavefunctions

and basis set theory. There are two categories of computational methods that will be focused

on in this dissertation: ab initio methods and density functional theory. Basis set theory is

discussed later in this chapter.

2.2 Ab initio Methodology

Ab initio, or “from the beginning,” methods are so‐called because only fundamental

physical laws and constants (e.g. Coulomb’s law, the elementary charge, the masses of

subatomic particles, Planck’s constant, the speed of light, etc.) are employed in them, and thus,

no bias from experimental data.6,18,19 Ab initio methods first construct an approximate

wavefunction as a reference, then use that reference in more rigorous methods to approximate

the exact solution to the Schrödinger equation.

The first ab initio approximation to the solution of the Schrödinger equation is the

Hartree‐Fock approximation,12‐14,20,21 which recasts the many‐electron Hamiltonian as a sum of

one‐electron operators and an average electron‐electron repulsion potential. This procedure

typically accounts for greater than 99% of the exact energy obtained with the Schrödinger

equation. However, as a result of this approximation, the Hartree‐Fock method recovers all but

the opposite spin correlation energy (hereafter referred to as just correlation energy), which is

physically interpreted as the instantaneous changes in the motions of electrons due to

repulsions by other electrons. The correlation energy is defined as the difference between the

exact ( ) and Hartree‐Fock ( HF) energies:7,22

9

C HF (2.7)

Although a small fraction of the total energy, the correlation energy is responsible for the

quantitative and, in many cases, qualitative description of chemical properties. Computing the

correlation energy is computationally demanding, and in order to obtain all of the correlation

energy for an electronic system, every electron must be correlated or allowed to interact with

every other electron simultaneously in the wavefunction. In practice, this corresponds to a full

configuration interaction (FCI)7,23‐25 treatment of the system, which is only possible to employ,

using current computing technology, on systems of a few electrons (i.e. ten or less). The task of

approximating a FCI treatment is the so‐called N‐electron problem,18 and correlated ab initio

methods are distinguished from one another in how they recover the correlation energy from a

HF reference wavefunction.7,19,22

Correlated ab initio methods become more computationally demanding as the number

of electrons increases, and so, there are two approximations that are often employed to reduce

the computational demand: 1) the frozen core approximation and 2) truncation of the

correlation space. The frozen core approximation assumes that chemical properties are

dominated by valence electron effects, and, thus, only valence electrons need to be correlated.

The remaining core electrons are left frozen, that is, uncorrelated. Truncation of the correlation

space involves only correlating a certain number of electrons simultaneously. For example, it is

well known that the largest contributor to correlation energy is the electron pair energy,

followed by the single electron correlation energy. Thus, some correlated ab initio methods are

able to recover a significant amount of the total correlation energy by only correlating up to

two electrons simultaneously.19

10

2.2.1 The Hartree‐Fock Approximation

Since the Schrödinger equation may not be solved exactly as an eigenvalue equation for

a many‐electron system, the Hartree‐Fock (HF) method is invoked as a first approximation to

the exact many‐electron wavefunction. Starting from a trial one‐electron orbitals, the HF

method seeks to find an optimal set of one‐electron orbitals that minimize the energy

expectation value of the determinant.7 In molecular HF calculations, each one‐electron orbital

is approximated as a linear combination of fixed one‐electron functions . This is the

so‐called linear combination of atomic orbitals (LCAO) theory,7,18 in which the molecular orbitals

(MOs) forming are taken to be a weighted sum of fixed atomic orbitals (AOs):

(2.8)

The coefficients may be varied so that the energy expectation value reaches a minimum,

leading to the variational nature of the HF method. In this way, the HF wavefunction is known

to be the best possible wavefunction constructed from the set of AOs when the total energy

has been minimized. The energy produced by the HF procedure will always be an upper bound

to the exact ground state energy.

The actual HF approximation replaces the exact electron‐electron repulsion operator

with an average effective potential (called the HF potential) by fixing an electron’s coordinates

and computing its interaction with the other 1 electrons. To arrive at this potential,

consider the electronic Hamiltonian recast in terms of a one‐electron core‐Hamiltonian

operator and the term, taking :

11

12 | |

12

(2.9)

According to Slater’s rules,7,26 the electron‐electron repulsion operator connects all pairs of

electrons in a Slater determinant such that:

12 Ψ Ψ

12

(2.10)

where the following shorthand has been introduced:

1 (2.11)

The permutation operator interchanges the orbital indices of electrons and , a

consequence of the anti‐symmetry of the Slater determinant. The first integral that results is

the classical Coulomb repulsion integral, denoted , while the second integral is the afore‐

mentioned exchange interaction between two electrons, denoted , and only arises when

electrons and have the same spin. Now, we define two operators that correspond to the

values of and :

d

d

(2.12)

Note that these new operators are one‐electron operators, and each represents a piece of the

average electron‐electron interaction experienced by electron in the field of electron . By

summing over all the occupied MOs, we obtain the HF potential for a single electron:

12

HF (2.13)

Next, a Fock operator for a single electron21 is defined and a new one‐electron Hamiltonian is

written in terms of these Fock operators:

HF

(2.14)

The operator is often referred to as the HF, or zeroth‐order,22 Hamiltonian. Thus, the HF

approximation is to turn the complicated, many‐electron Schrödinger equation into a one‐

electron problem. As a consequence of replacing the exact electron‐electron repulsion with the

average HF potential, the correlation of the electrons is lost.

The HF approximation leads to a new set of Schrödinger‐like equations called the HF

equations. The Fock operator operates on a given orbital to produce an eigenvalue – the

orbital energy within the averaged‐out electron‐electron repulsion:

| |

(2.15)

The HF orbitals are constrained to be orthogonal, and the orbital energy is

| | (2.16)

According to Koopmans’ theorem,27 this orbital energy may be interpreted as the ionization

potential of the corresponding orbital. However, Koopmans’ theorem neglects both the

correlated motions of electrons and orbital relaxation upon ionization.7

To solve the HF equations, (2.8) is inserted into (2.15):7,22,28

13

d d

(2.17)

Note that the integral on the right‐hand side is not the Kronecker delta since the AOs are

allowed to overlap. If we define the Fock integral on the left‐hand side as an element of a Fock

matrix, ; the overlap integral on the right‐hand side as an element of an overlap matrix, ; and

write the coefficients as to form the coefficient matrix, ; then (2.17) is written in

matrix notation as:

(2.18)

Solving (2.18) is equivalent to solving each of the HF equations simultaneously. We are

interested in obtaining a matrix such that (2.18) holds. This leads to an iterative procedure for

solving (2.18) called the self‐consistent field (SCF). The column vectors of , called the canonical

HF orbitals, are the MOs of the optimal wavefunction. These orbitals are not unique and may

be mixed among themselves to form other solutions to the HF equations with the same total

energy. The Slater determinant formed from the set of HF orbitals is the HF wavefunction,

denoted |Ψ . Applying the full Hamiltonian to the HF wavefunction gives the HF energy:

HF | |12

(2.19)

There are three types of wavefunctions that the SCF procedure may produce, depending

on the electronic structure of the system of interest: 1) a restricted closed‐shell, 2) a restricted

open‐shell, or 3) an unrestricted wavefunction (cf. Figure 2.1).7,18,19 A closed‐shell wavefunction

14

is one in which each electron is paired up with another electron of the opposite spin, whereas

an open‐shell wavefunction contains one or more unpaired electrons. In the restricted

wavefunction, each pair of electrons occupies a single spatial orbital, while in the unrestricted

case, the α electrons occupy different spatial orbitals than the corresponding β electrons. The

restricted open‐shell case uses restricted orbitals for the electron pairs and unrestricted orbitals

for the unpaired electrons. In the unrestricted case, the β orbitals are higher in energy relative

to the α orbitals due to spin polarization of the α electrons. The advantage of using an

unrestricted wavefunction is a lower energy, and, hence, better wavefunction. However, the

consequence of the electron spins not being perfectly paired leads to a phenomenon called spin

contamination, such that the unrestricted wavefunction is not a pure spin state.7,22

The proper selection of a restricted versus unrestricted HF wavefunction is crucial to

exploiting the size consistency of the HF method.7,22 A method is size consistent if it correctly

describes the dissociation of a bond, e.g.

Figure 2.1 Representations of restricted closed‐shell (I), restricted open‐shell (II), andunrestricted (III) wavefunctions.

3

2

1

4

2

1

3

4α 4

β

1α 1

β

2α 2

β

3α 3

β

4α

4β

|ΨI |ΨII |ΨIII

15

H | HH 2 H (2.20)

For example, if a restricted closed‐shell wavefunction is used to dissociate H2, the dissociation

energy will be too high (see Figure 2.2), but an unrestricted wavefunction will give the

qualitatively correct dissociation behavior into two hydrogen atoms. The restricted

wavefunction does not correctly dissociation the H2 bond because the wavefunction always

forces the two electrons in the system to occupy the same spatial orbital, despite the fact that,

at long bond lengths, the occupied orbitals are localized on each hydrogen atom and are

spatially very different. Further, there are ionic terms corresponding to H‐ + H+ in the restricted

wavefunction that do not disappear as the bond elongates, which lead to the inflated

dissociation energy of the SCF curve in Figure 2.2. The unrestricted wavefunction will dissociate

the H2 molecule correctly, but the wavefunction at the dissociation limit will be spin

contaminated by the triplet configuration. The general consequence of size consistency at the

HF level is that a restricted wavefunction should be used when dissociation results in fragments

that can be properly described by restricted wavefunctions, while an unrestricted (or restricted

open‐shell) wavefunction should be used when the dissociation fragments are open‐shells.

A closely related concept that has been alluded to, but not explicitly discussed is the use

of more than one determinant to describe the reference electronic state. The HF formulism

introduced thus far has only used a single determinant in the total wavefunction. There are

many systems in which the electronic state may not be qualitatively described by a single

determinant (e.g. atoms and molecules with orbital degeneracies, open‐shell singlets, etc.) and

a multi‐determinant method is needed. In these special cases, multi‐configuration self‐

consistent field (MCSCF) is employed to obtain the HF wavefunction. The MCSCF wavefunction

16

is constructed as a linear combination of HF determinants that each describe a different pure

electronic state,

|Ψ |Ψ (2.21)

where both the coefficients and determinants |Ψ are simultaneously optimized subject to

the constraints Ψ |Ψ and Ψ |Ψ 1. The MCSCF energy expectation value is

minimized using the HF operator. By introducing more determinants in the reference

wavefunction, so‐called static correlation is introduced, leading to a lower energy solution that

the single reference wavefunction. Static correlation is the interaction between two or more

configurations that arises when the ground‐state wavefunction is written as in (2.21).

2.2.2 Configuration Interaction

As already discussed, the HF orbitals may be mixed among themselves without affecting

Figure 2.2 A comparison of the dissociation of H2 using a single determinant (SCF) and atwo‐determinant (MCSCF) wavefunction.

‐0.200

‐0.150

‐0.100

‐0.050

0.000

0.050

0.100

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

E(H2) ‐2E(H) / hartree

R(HH) / angstrom

SCF

MCSCF

17

a change in the total energy.7,22 Thus, it can be said that the occupied HF orbitals form an

orbital basis for the single determinant ground state wavefunction. However, the entire

collection of HF orbitals, occupied and virtual (unoccupied), form an orbital basis for the fully‐

interacting ground state wavefunction. Another way to look at it is to consider the entire set of

HF orbitals, arranged from lowest to highest orbital energy. By the aufbau principle, the HF

ground state will have the electrons filling up the lowest energy orbitals. But, this is only one

possible configuration of the electrons. In fact, there are possible configurations (where

is the number of one‐electron basis functions and is the number of electrons). These other

configurations (Slater determinants) form a basis for the exact N‐electron wavefunction.

Determining how these configurations interact with each other leads to the method of

configuration interaction (CI).7,19,22‐25

The fully‐interacting wavefunction may be written as a linear combination of excited

Slater determinants formed from the HF orbitals:

|Φ |Ψ |Ψ14 Ψ

136 Ψ

(2.22)

Here, |Ψ represents the HF reference wavefunction, while |Ψ , Ψ , and Ψ represent

connected singly‐, doubly‐, and triply‐excited determinants formed from the HF reference,

respectively (the term connected will be discussed in subsection 2.2.3). A few examples of these

types of excited determinants are shown in Figure 2.3. The set … is the occupied ( )

orbitals and … are the virtual ( ) orbitals. The script notation “|Ψ ” denotes the th

occupied orbital is excited to the th virtual orbital, etc., and the factors 1/4 and 1/36 ensure

that no excited configuration enters the overall wavefunction more than once. Finally, the sets

18

, , and are called the singles, doubles, and triples amplitudes, respectively.

The energy of the CI wavefunction is, recalling the definition in (2.7):

|Φ |Φ HF C |Φ (2.23)

The correlation energy C is obtained by left‐projecting (2.23) onto the HF wavefunction:

C Ψ |Φ Ψ | HF |Φ

C Ψ |Φ 1 HF Ψ | |Ψ14 Ψ Ψ

(2.24)

Assuming intermediate normalization, Ψ |Φ 1, which implies that 1, the CI

correlation energy C is written:

Figure 2.3 Examples of singly‐, doubly‐, and triply‐excited configurations of the HFreference wavefunction.

|Ψ0

3

2

1

4

5

6

7 Occup

ied

Virtual

|Ψ Ψ Ψ

19

C Ψ | |Ψ14 Ψ Ψ

136 Ψ Ψ

(2.25)

As a consequence of Brillouin’s theorem,7,22 which states that the HF wavefunction does not mix

with its singly‐excited determinants, the first summation in (2.25) equals zero. Also, according

to Slater’s rules,26 the Hamiltonian does not connect determinants that vary by more than two

permutations of the electrons. Thus, the triply‐excited, and higher, determinants drop out of

the expansion in (2.25) leaving:

C14 Ψ Ψ

(2.26)

Now, all that must be obtained to find the correlation energy of the fully‐interacting

wavefunction is the set of doubles amplitudes. Equation (2.26) demonstrates why two‐electron

correlation is the predominate contributor to the total correlation energy.

In practice, obtaining the doubles amplitudes is non‐trivial since they are coupled to not

only the other doubles amplitudes, but also to the reference, singles, triples, and quadruples

amplitudes. This means that in order to compute the correlation energy, all of the singly‐,

doubly‐, triply‐, and quadruply‐excited determinants must be formed from the HF reference.

Even more complication arises since the reference, singles, triples, etc. amplitudes are also

coupled to other excited determinants. As an example, if (2.24) is left‐projected by an arbitrary

singly‐excited determinant Ψ , the result would be:

C Ψ HF Ψ14

Ψ Ψ136

Ψ Ψ (2.27)

Note that, even though (2.27) is an expression for the amplitude , the presence of the other

20

singles amplitudes in this equation precludes that a method of obtaining each of the sets of

amplitudes simultaneously is needed. This leads to the construction of the FCI matrix, which, in

general, is very large and requires computing expectation values over all possible combinations

of configurations.7,19,22 For this reason, the FCI method is computationally demanding and can

only be performed on systems with a few electrons.

Instead of constructing the entire FCI matrix, the frozen core and truncated

wavefunction approximations discussed earlier may be employed. If the correlated

wavefunction of (2.22) is truncated after, say, the doubly‐excited determinants, then the size of

the CI matrix is reduced to a much more tractable form, called CI singles and doubles (CISD).

There are two major problems in truncated CI theory. One of these problems is the lack of size

consistency, despite the correct choice of a restricted or unrestricted HF reference. Truncated

CI wavefunctions do not have multiplicative separability, and, thus, not all excitations necessary

to properly describe bond dissociation are included when two fragments are pulled apart. The

other problem with truncated CI theory is the lack of size extensivity.29 A correlated method is

size extensive if the correlation energy recovered scales properly with the number of electrons.

Further, size extensivity ensures that the same amount of correlation energy is recovered

everywhere on the potential energy surface of an electronic system. The lack of size extensivity

in truncated CI methods stems from the fact that disconnected excitations (discussed in the

following section) do not enter the correlated wavefunction.

2.2.3 Coupled Cluster Theory

Instead of constructing all or some of the possible excited determinants from the HF

21

reference wavefunction, it is possible to recover correlation energy by writing the correlated

wavefunction in terms of cluster functions.30 A cluster function is a term added to the HF

wavefunction that correlates electrons. For example, consider a reference wavefunction

describing a system containing three electrons,

|Ψ1√3!

(2.28)

A cluster function that correlates electrons in orbitals and may be constructed as follows:

,12

(2.29)

Here, … and have the same meaning as in subsection 2.2.2, and is a doubles cluster

amplitude. If this cluster function is added to the orbital product , resembling a

sort of CI expansion, and replaced in the reference wavefunction as follows, a correlated

wavefunction results:

|Φ ,

|Φ |Ψ12

|

(2.30)

The correlated wavefunction in (2.30) includes only pair correlation stemming from electrons

occupying the orbitals and . If cluster functions correlating all possible orbital

combinations (not just pairs) were included in the reference wavefunction, then the FCI

wavefunction would result.30 This process is called the coupled cluster (CC) method.31‐36 It can

be shown that the CC wavefunction may be written in an exponential ansatz:

22

|ΦCC |Ψ1!

|Ψ (2.31)

Note that the Taylor series expansion of has been included for later discussion. The cluster

operator is composed of single ( ), double ( ), etc., up through N‐

tuple ( ) cluster functions, each defined by how they operate on the reference wavefunction:

|Ψ |Ψ

|Ψ14 Ψ

|Ψ1! …

… Ψ ……

……

(2.32)

If the cluster operator were applied directly to the reference wavefunction, the FCI

wavefunction (2.22) would result. However, the principle difference between CI and CC

theories is the exponential ansatz of (2.31), which gives multiplicative separability to the CC

wavefunction.29,37

Truncated CC methods are defined by restricting the cluster operator . For example,

the singles and doubles coupled cluster (CCSD) wavefunction is determined by setting

and inserting into the exponential ansatz (2.31):

|ΦCCSD1!

|Ψ

|ΦCCSD 112

16

|Ψ .

(2.33)

Consider the system composed of three electrons in (2.28); the highest possible excitation is a

23

triple excitation, thus, only cluster operators and products of cluster operators that correspond

to, at most, a triple excitation will be included in the CCSD wavefunction.30,38 In this case, (2.33)

then reduces to

|ΦCCSD 112 2

16

|Ψ

|ΦCCSD |ΦCISD12 2

16

|Ψ .

(2.34)

It is clear from this equation that the CCSD wavefunction includes more configurations than the

CISD wavefunction, and hence, includes more correlation than the CISD wavefunction. In fact,

not only do single and double excitations enter the CCSD wavefunction, but triple excitations

enter through the disconnected excitation terms and . Overall, the exponential ansatz

allows for disconnected excitations to enter truncated CC wavefunctions, which are the reason

that CC methods are both size consistent and size extensive.30

Despite the recovery of more correlation in the wavefunction compared with truncated

CI methods, the most popular implementations of CC are not variational.30 Although, it is

possible evaluate the truncated CC energy in a manner similar to that of truncated CI. Many

conventional CC methods exploit simplifications that arise from using a similarity transformed

Hamiltonian, . This transformed Hamiltonian comes from the bra‐representation of

(2.31), whereby the inverse exponential arises as the adjoint of . Due to Slater’s rules,26

naturally truncates at quadruple excitations making it more computationally efficient

than computing the full CC matrix (analogous to the FCI matrix) introduced in subsection 2.2.2.

As a consequence of employing the transformed Hamiltonian, both the variational and

hermitian nature of the CC equations are lost, but the resultant energy eigenvalues of

24

are, typically, very close to those of , which is motivation for the continued use of this

implementation of CC theory.30

2.2.4 Many‐Body Perturbation Theory

The correlated methods discussed up to this point have used different configurations

formed from the HF orbitals as a basis for constructing correlated wavefunctions. However,

another method of obtaining correlation energy is many‐body perturbation theory

(MBPT).7,19,22,34 Perturbation methods add corrections (perturbations) to the Hamiltonian and

wavefunction, which, in turn, give rise to a perturbed total energy:

|Φ |Ψ |Ψ |Ψ

(2.35)

Here, |Ψ and are the th‐order wavefunction and energy, respectively, of the th

eigenstate of the given system. Inserting these perturbed quantities into the Schrödinger

equation and collecting in orders of yields the th‐order perturbed Schrödinger equation:

|Ψ |Ψ |Ψ (2.36)

Next, MBPT, like the CI and CC methods, assumes that the eigenstates of the zeroth‐order

wavefunction form a basis for the th‐order wavefunction such that

|Ψ |Ψ (2.37)

If the eigenstates of the zeroth‐order wavefunction are orthonormal, Ψ Ψ , then

25

to find the th‐order energy, left‐project (2.36) by the zeroth‐order eigenstate to obtain

Ψ Ψ (2.38)

Now, it appears that the th‐order energy may be calculated from the 1 ‐order

wavefunction, but Wigner has shown that the th‐order wavefunction may be used to obtain up

to the energy up to order 2 1 .22,39 To obtain a given order wavefunction, we need to

calculate the set of coefficients , which are obtained by inserting (2.37) into (2.36), left‐

projecting by an arbitrary eigenstate of the zeroth‐order wavefunction, and using algebra to

clean up the resulting equation. Because this dissertation will only discuss MBPT results up

through second‐order, the derivation of the necessary quantities for obtaining the second‐

order energy is now presented.

According to Wigner’s theorem, the first‐order wavefunction is all that is necessary for

obtaining the second‐order energy. Thus, inserting (2.37) into (2.36), where 1, and left‐

projecting by Ψ yields

Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ (2.39)

With some algebra and recalling the orthogonality of the eigenstates of the zeroth‐order

wavefunction, the coefficient for the th component of the first‐order wavefunction is

Ψ Ψ

(2.40)

Inserting (2.40) into (2.37), then into (2.38) gives the second‐order energy as

Ψ Ψ Ψ Ψ

(2.41)

Now, consider the form of the perturbation operator . If the operator is the HF

26

operator (2.14) then the perturbation is defined using (2.9), (2.14), and (2.35):

12

(2.42)

The zeroth‐order wavefunction is taken to be the HF wavefunction, and the different

eigenstates are the excited configurations of the HF wavefunction. This form of MBPT is known

as Møller‐Plesset perturbation theory (MPn, where n denotes the order of perturbation).7,19,22,40

Using (2.36), the zeroth‐order ground state energy is just the sum of the HF orbital energies,

Ψ Ψ | | (2.43)

The first‐order correction to the energy is found using (2.10) and (2.38):

Ψ Ψ12

(2.44)

Note that the total energy to first order is the HF energy (2.19), MPHF. This

very important result demonstrates that the HF wavefunction is stable to first order, which is

proof that mixing the HF orbitals among themselves will not lower the total energy. As a result,

eigenstates outside the HF ground state must be considered to compute the second‐order

energy. Further, note that the first‐order energy correction effectively removes the double‐

counting of the electron‐electron interactions inherent to the zeroth‐order energy. In the

energy expression of (2.41), the eigenstates that correspond to |Ψ must be doubly‐excited

configurations of the HF wavefunction since 1) the HF wavefunction is stable to first‐order and

2) the singles do not mix with the HF ground state by Brillouin’s theorem. Thus, using script

notation defined earlier, the second‐order energy is

27

14

Ψ Ψ

(2.45)

Here, and are the orbital energies of the and occupied orbitals connected by to the

virtual orbitals and , whose energies are and defined by (2.15). Again it is shown that

the predominate contributor to the correlation energy is the double excitations (specifically the

pair correlation) since it is the leading correction to the HF energy.

Perturbative methods, in general, are not variational, and, thus, the energies obtained

are not upper bounds to the exact energy.22 However, MP2 recovers a significant fraction of the

total correlation energy at a lower computational cost than both CISD and CCSD. Further, MPn

is both size consistent and size extensive, making it a more desirable method than CISD. The

size consistency of MPn, like CI and CC methods, hinges on the proper selection of the

underlying HF wavefunction. If the zeroth‐order wavefunction is sufficient for describing the

electronic state, then the MPn energies will be size consistent. In homolytic bond cleavage or

dissociation of a molecule into open‐shell fragments, MPn, as discussed so far, will lose size

consistency since the underlying reference wavefunction is not size consistent.

2.3 Density Functional Theory

Instead of deriving electronic properties from a wavefunction, it is possible to derive

electronic properties from an electron density. The concept of using an electron density in place

of the wavefunction to solve the Schrödinger equation was explored by Thomas, Fermi, and

Dirac from 1927‐1930 using a simplified quantum model called the homogeneous electron

gas.41‐43 It was not until 1964, however, that the formal proof of principle was introduced by

28

Hohenberg and Kohn,44 the landmark theorems of whom have become the basis for a modern

quantum model called density functional theory (DFT).45‐48 There are several fundamental

differences between ab initio methodology and DFT. For one, the wavefunction cannot simply

be replaced by the electron density in the Schrödinger equation and be expected to produce

meaningful information. This is because the Hamiltonian is specified using the positions of the

electrons, the information of which is not explicitly available in the electron density. Thus, the

whole of DFT is dependent upon designing density functionals, which are operators analogous

to and that act on densities instead of wavefunctions to extract the energy of the

electronic system. Before discussing density functionals in detail, it is pertinent to define the

concept of an electron density and discuss its properties. In the sections that follow, the

fundamental theorems comprising atomic and molecular DFT are discussed.

As already discussed, the wavefunction is a function describing the spatial and spin

coordinates of the electrons in an electronic system. The distribution function or density of

electrons is generally defined by

; Ψ ,… , Ψ ,… , (2.46)

where the primed and unprimed electron coordinates represent two independent sets.45,49 If

the primed and unprimed sets are equal, then is just the squared modulus of the

wavefunction, and the integration of (2.46) over all coordinates is unity (assuming

normalization). Equation (2.46) is defined as the reduced density function (RDF) of order N. The

two independent sets are interpreted as coordinates for an element of the reduced density

matrix (RDM) of order N.50 Other important RDFs are the first‐ and second‐order RDFs, defined

respectively as

29

; d d Ψ , … , Ψ , … ,

;1

2 d d Ψ , , … , Ψ , , … ,

(2.47)

When and , both RDFs of (2.47) represent diagonal elements of their

respective RDMs (corresponding to physical observables) such that

; d d |Ψ |

;1

2 d d |Ψ | , ,/

(2.48)

where is a spin orbital, and , is a spin geminal, or two‐electron function. Both

the set of spin orbitals and spin geminals form orthonormal sets. The product of ; and

a coordinate element is statistically equivalent to the probability of finding any electron within

the coordinate element. If all possible electron coordinates are integrated over, then the result

for is the number of electrons, while the result for is the number of unique electron pairs:

d ; d

d d ; d d , ,/

12

(2.49)

Using algebra and the results of (2.49), a relationship between and may be established:

;21 d ;

(2.50)

Equation (2.50) is an important property that stems from a more general observation that first‐

order RDFs may be used to construct higher order RDMs.49,50

30

The diagonal first‐ and second‐order RDFs can be further simplified to their respective

spatial densities, denoted , by integrating out the spin component:45,48

d ; d d d |Ψ |

, d d ;1

2 d d d d |Ψ |

(2.51)

This assumes that the electron spin has been included a posteriori as in (2.5). A similar process

may be used to obtain the spin densities and , by integrating out the spatial

components. If the orbitals are defined by the LCAO theory (2.8), then may be

written in terms of AOs as

(2.52)

where is an element of the charge density matrix.7 The electron label is spurious since the

anti‐symmetric nature of the wavefunction does not allow individual electrons to be

distinguished from one another, and will be dispensed with in the following sections. Equation

(2.52) is the fundamental quantity that will be focused on in the Hohenberg‐Kohn and Kohn‐

Sham theorems below.

2.3.1 The Hohenberg‐Kohn Theorems

Much of the following derivations follow directly from the original paper of Hohenberg

and Kohn.51 Recall that the Hamiltonian (2.2) for a given electronic system is uniquely defined

for a given nuclear‐electron potential , and thus, so are the wavefunction Ψ and energy .

The extension that uniquely defines the density and that , in turn, uniquely

31

defines is the first Hohenberg‐Kohn (HK) theorem. To prove this, first consider two

systems with different potentials: and . Now, assume that the potentials and

produce the same density. Since the Hamiltonians and

are uniquely defined, it follows that Ψ Ψ. The respective energies are

Ψ| |Ψ Ψ Ψ Ψ Ψ

Ψ Ψ Ψ Ψ Ψ Ψ

(2.53)

The integrals on the right‐hand side of the above equations may be further simplified to:

Ψ Ψ d

Ψ Ψ d

(2.54)

where the spatial density is used instead of the total density since the potential only depends

on the spatial coordinates. Inserting (2.54) into (2.53) and adding the resulting equations

together gives a contradiction, , disproving that two different potentials can lead

to the same density. It is said that is a unique functional of , where a functional is a

mathematical function that maps one function to another, and is written as . The

implication of the first HK theorem is that given any density, the exact nuclear‐electron

potential can be deduced.

The second HK theorem demonstrates a variational approach to minimizing the total

energy with respect to the electron density. Written as a density functional, the energy is

Ψ| |Ψ Ψ| |Ψ Ψ| |Ψ

d

(2.55)

where Ψ| |Ψ is a universal energy functional (universal in the sense that it

32

may be applied to any number of electrons in any potential) describing the kinetic and Coulomb

energies. The total density is used in the expression since the term contains the

exchange interaction. If the number of particles in the wavefunction remains constant (i.e.

equation (2.51) is enforced), then Ψ may be varied to minimize the energy. If a trial

wavefunction Ψ is based on a different potential than the ground state wavefunction Ψ, then

d d (2.56)

in which is the density of Ψ and is the density of Ψ. This equation demonstrates that

the energy of an electron density may be minimized by varying the density arbitrarily with

respect to a given potential.

For convenience, the classical Coulomb repulsion is separated from the universal

functional since it may be directly evaluated with one‐electron densities:

12 d d | |

(2.57)

Here, is another universal functional of the density that contains information on the

kinetic, exchange, and correlation energies. The universal energy formula of Hohenberg and

Kohn is written as follows, without any recourse as to the form of .

d 12 | |

(2.58)

Given a trial density and the exact form of , the HK theorems demonstrate that the

exact, non‐relativistic energy of an electronic system may be obtained by variationally

minimizing the energy with respect to the density. In practice, a density must have certain

properties, namely it must be N‐representable and V‐representable.45,46 For an electron density

to be N‐representable, there must be some anti‐symmetric wavefunction that corresponds to

33

it; while to be V‐representable, the density must be N‐representable and correspond to a

unique potential (via the first HK theorem). Parr and Yang give a proof that one‐electron

densities are automatically N‐representable if they can be decomposed as in (2.48) with

occupation numbers between 0 and 1 (that is, the density obeys the Pauli principle).45 By

extension, two‐electron densities are also N‐representable by (2.50) if the one‐electron density

is N‐representable. The conditions for a density to be V‐representable are not known, and, as a

result, the variational nature of the second HK theorem is not strictly enforceable.45

2.3.2 The Kohn‐Sham Method

Following the seminal paper of Hohenberg and Kohn, Kohn and Sham formulated the HK

theorems into a self‐consistent procedure – entirely analogous to the HF‐SCF procedure – with

respect to one‐electron densities.52 The first step in the Kohn‐Sham (KS) procedure is to recast

the universal functional in terms of directly calculable quantities; writing it as a sum of

the remaining energy terms,

KS XC (2.59)

where KS is the KS kinetic energy functional and XC is an exchange‐correlation (XC)

functional. The kinetic energy functional KS is chosen to be that of the non‐interacting electron

model, similar to the HF approximation:

KS12 d

12 d

(2.60)

Because a non‐interacting model is assumed, the kinetic energy of KS is deficient such that,

KS Δ (2.61)

34

in which is the (unknown) exact kinetic energy functional and the residual Δ 0. The use of

KS removes sole dependency of the model on the density and introduces orbitals called the KS

orbitals. It is worth noting that (2.60) is not the only form that the kinetic energy functional may

take. As Kohn and Sham and others have discussed, the use of orbitals in this manner helps to

properly describe the shell structures of atoms.

Using equations (2.59), (2.60), and (2.58), a KS equation can be written using the density

functionals of each energy contribution as

12

12 d | |

XC (2.62)

where is the KS orbital energy and the fourth term in the parentheses is the exchange‐

correlation potential for a single electron. Solving the KS equations becomes an iterative

procedure since the KS energy is a functional of the density:

KS KS 12 | | XC

(2.63)

For the KS energy functional to be exact, the form of the XC functional must include not only

exchange and correlation effects, but also the kinetic energy residual:45,46

XC X C Δ (2.64)

The energy expression of (2.64) is the attention of most modern DFT research. The exchange

term X has a form that is known exactly (from the HF approximation), however, the form of

the correlation functional C is not known. In practice, the exact HF exchange is only used in a

class of density functionals called hybrid functionals, and even then, generally no more than

50% of the exact exchange is implemented. Another class of density functionals called pure

functionals do not use HF exchange at all. When approximate exchange functionals are used in

35

place of the exact HF exchange, self‐interaction occurs, and must be corrected.45,46 Self‐

interaction is the spurious interaction of an electron with itself that occurs when the Coulomb

and exchange energies do not exactly cancel as they would, cf. (2.13), if the exact HF exchange

were employed.

In using any class of density functionals to evaluate the KS equations, the KS orbitals do

not retain any physical meaning, only a mathematical convenience for expressing the electron

density.45,48 As a result, Koopmans’ theorem does not strictly apply. Further, the variational

nature of the second HK theorem is lost since the exact energy functional is not known,

nor are the conditions for V‐representability. Despite these facts, the KS procedure does

provide a much faster method of including (approximate) correlation effects in atoms,

molecules, and extended systems without the computational cost associated with correlated ab

initio methods.

2.4 Basis Set Theory

Broadly speaking, a basis set is a collection of fixed elementary functions, called basis

functions, the linear combination of which will describe any function in a given function space.15

A basis set is called complete if it spans all space for any function, and orthogonal if the basis

functions are linearly independent. In computational quantum chemistry, a basis set is a set of

fixed elementary functions that are used to construct the electronic wavefunction.6,7,18,19,22 As

already discussed, the basis set must have the five characteristics of a physically viable

wavefunction.

It is known that the exact radial solution to the Schrödinger equation for the hydrogen

36

atom takes the form of a combination of Laguerre and Legendre polynomials in spherical

coordinates.6 In quantum chemistry, these functions are called Slater‐type orbitals (STOs):6

Ψ , , , (2.65)

Here, is an integer called the principle quantum number and is the exponential parameter

that may assume any positive value and determines the radial extent of the wavefunction. The

angular dependence, and hence, the orbital and magnetic quantum numbers (the integers and

, respectively) of the exact solution enters through the spherical harmonic . The values of

determine the shape of the orbital ( 0 is an s‐type function, 1 is a p‐type function, 2

is a d‐type function, etc.), while the value of takes values as – , and determines the

spatial orientation of the orbital. The number of radial nodes in a wavefunction is 1 and

the number of angular nodes is . Since the many‐electron Schrödinger equation has no closed

form solution, it is assumed that a wavefunction describing two or more electrons may be

written as a linear combination of these one‐electron, or hydrogen‐like STO solutions since the

Laguerre and Legendre polynomials form a complete set. This assumption is entirely analogous

to the expansion of an electric potential in terms of Legendre polynomials in classical

electrodynamics.53,54 A many‐electron wavefunction expanded in a basis set of STOs is quite

accurate, having the correct short‐range and long‐range behavior about the nucleus.22

Variational methods employing STO basis sets produce ground state energies that quickly

converge as the number of STOs increases. However, the 2‐electron integrals that arise due to

the use of STOs in solving the many‐electron Schrödinger equation are non‐trivial in molecular

computations and must be solved using numerical techniques.55 While many efficient numerical

techniques exist for computing integrals, the computational demand dramatically increases

37

with the number of electrons in the system. As a result, computations involving STOs are

generally reserved for atoms and small molecules (i.e. diatomics).

For convenience, a Gaussian‐type orbital (GTO) is typically used in place of an STO.56‐58

This idea was introduced by Boys,56,58 who postulated that the radial part of an STO could be

represented, to a tolerable degree of accuracy, as a linear combination of GTOs,

| | (2.66)

where is a contraction coefficient and is the GTO exponential parameter. In fact, only an

infinite sum of GTOs will exactly describe an STO. The summation above is called a contracted

basis function or simply contraction, while the individual exponential functions comprising the

sum are called primitive basis functions or just primitives.

A wavefunction expanded in a basis set of GTOs exhibits some fundamental problems

when compared with a wavefunction expanded in a basis set of STOs. First, the short‐range

behavior of a primitive GTO varies quite markedly from that of an STO and results in an

incorrect description of the wavefunction at the nuclear cusp.22 Second, the long‐range

behavior of a primitive GTO is such that it decays more rapidly than the corresponding STO.

Both of these behavioral differences, demonstrated in Figure 2.4, may be circumvented (within

tolerable accuracy) by using a large number of GTOs in the basis set. For example, the cusp may

better be described by using more high‐exponent or tight functions, while the long range

behavior is better described by more low‐exponent or diffuse functions in (2.66).

The tradeoff between using a larger basis set composed of GTOs, compared with a

smaller STO basis set, is the ease of computing 2‐electron integrals. Whether employed in

atomic or molecular computations, a GTO basis set produces 2‐electron integrals that may be

38

reduced to error functions through either a Fourier or Laplace transform.55 The bottleneck of

either of these transformed integrals is less computationally demanding than the numerical

techniques required to compute 2‐electron integrals using STOs. There are other mathematical

tricks, such as projective or decomposition techniques, that may be employed to lower the

computational bottleneck of the 2‐electron integrals. One type of projective technique called

the resolution of the identity is discussed in Chapter 7.

The basis sets discussed in this dissertation are atom‐centered, as opposed to a

common origin or between nuclei (bond‐centered), and are GTO‐based. The atom‐centered

GTOs are optimized to describe the AOs of a free atom. When employed in molecular

calculations, the LCAO theory is invoked. The question arises as to how many basis functions,

and what values, should be chosen to form the basis set. The answer is dependent upon the

system to be described and the degree of accuracy sought for the many‐electron wavefunction.

For example, a free carbon atom may be described by three basis functions: two s‐type

functions ( 0) for the 1s and 2s orbitals and a single p‐type function ( 1) for the 2p

Figure 2.4 Plots comparing the behavior of a single STO to a single GTO (left) and that of asingle STO with a linear combination (LC) of three GTOs (right). The units are arbitrary.

0.000

0.200

0.400

0.600

0.800

1.000

‐6.0 ‐4.0 ‐2.0 0.0 2.0 4.0 6.0

Value

of Fun

ction

Distance from Nucleus

STOGTO

0.000

0.200

0.400

0.600

0.800

1.000

‐6.0 ‐4.0 ‐2.0 0.0 2.0 4.0 6.0

Value

of Fun

ction

Distance from Nucleus

STO

LC‐GTO

39

orbital. This is an example of a minimal basis set, where there are just enough functions to

cover the occupied AOs of each atom. Each occupied orbital is described by a contracted

function rather than a single GTO due to the deficiencies of GTOs described earlier. The

problem with employing a minimal basis set is two‐fold: 1) the basis set is not flexible enough

to describe the many possible chemical environments an atom might be in and 2) employing a

minimal basis set in correlated ab initio methods does not recover much correlation energy.7

The electronic structure of a free atom is much different than the same atom in a

molecule, and the electronic structure from atom to atom varies markedly as well. Thus, it is

imperative to have a basis set that will be flexible enough, that is, have basis functions

describing the proper orbital space, from one chemical environment to another. Consider a π

bond between two carbon atoms, formed by the overlap of their 2p orbitals. If d‐type functions

are added to the carbon basis sets, then some of the d orbitals will mix with the p orbitals by

symmetry and cause the overlap forming the π bond to increase; a sort of p‐d hybridization

occurs, lowering the bond energy.18 These additional d‐type basis functions are called

polarization functions because they polarize (distort) the 2p orbitals. It is typical of basis sets to

include at least some polarization functions since these functions aid in the qualitative and

quantitative description of chemical bonds. Polarization functions also become important in

systematically recovering correlation energy (discussed in Chapter 3). An investigation reported

by Schwartz on two‐electron systems demonstrated that the MP2 correlation energy is

proportional to the highest orbital quantum number in the basis set.59 This observation lends

credence to the use of polarization functions with ‐values well beyond the minimal basis set.

40

2.5 Model Chemistries

The discussion thus far has covered different types of computational methods that

construct approximate many‐electron wavefunctions (or densities) and basis sets that can be

used to expand such wavefunctions. A dichotomy arises as to which computational method

should be selected for a particular chemical system and which basis set will provide the most

accurate expansion of the wavefunction. Ideally, one would like to pair the FCI method with a

basis set that spans all possible AO space. This would give the exact solution to the non‐

relativistic Schrödinger equation within the BO approximation. However, a basis set that spans

all possible AO space requires an infinite linear combination of GTOs, making the approach

intractable. Instead, a compromise must be made as to how many and what type of basis

functions need to be employed to properly describe the electronic structure. Further, a

compromise as to which computational method is appropriate and applicable to the system of

interest must be made. Hehre et al. coined the term model chemistry, or level of theory to

describe the coupling of a computational method and a given basis set.18 This concept is

graphically represented in Figure 2.5.

Since doubles correlation is the dominant fraction of the total correlation energy,

compared with other excitation levels, ab initio methods such as CISD, CCSD, and MP2 are

generally sufficient and tractable for systems up to 25 non‐hydrogen atoms and a modest‐sized

basis set (the inclusion of singles in addition to doubles does not appreciably increase the

computational cost). In some electronic systems, triple and quadruple excitations make

appreciable contributions to the total correlation energy. To recover triples and some

quadruples, one may use the CCSDT method (note that CISDT is not usually employed since it is

41

not size consistent). However, the computational expense of such methods dramatically

increases with the number of electrons in the system. To avoid using CCSDT, the perturbative

correction to triple excitations, CCSD(T), is used, which recovers a significant portion of the

triples correlation energy without the computational overhead of a full CCSDT calculation.

As already discussed, a basis set beyond the minimal set of AOs is usually employed in

ab initio methods. This is generally true of DFT methods, as discussed in Chapter 4. The

composition of a basis set varies from system to system and chemical property to chemical

property. For example, a polarized basis set optimized with HF will not have basis functions

optimal for recovering correlation energy if employed in a correlated ab initio model chemistry.

As a result, it is desirable to have a basis set with basis functions optimized for both HF and

correlated ab initio methods. The correlation consistent basis sets are of this type and are

discussed in detail in Chapter 3. Further, the computational methods discussed so far have

Figure 2.5 A conceptual view of basis set size versus level of sophistication in correlated abinitio methods. The number of determinants with respect to excitation level (n), number ofelectrons (N), and basis set size (K) is given by the formula on the lower axis.

Exact

Full CI

Complete Basis Set Limit

No Correlation

Applicable Model Chemistries

HF Limit

2

∞

0

Basis Set S

ize

Excited Determinants

42

been non‐relativistic, and if relativistic effects are to be included, a basis set optimized with a

non‐relativistic method will not recover all possible relativistic effects. It is pertinent that the

basis set employed in relativistic computations have core AOs optimized for the relativistic

Hamiltonian employed since the major fraction of relativistic effects come from these orbitals.

Relativistic basis sets are discussed in detail in Chapters 3 and 6.

Finally, there is a special class of model chemistries called composite methods in which

individually energy components of the total energy of a system are computed by different

levels of theory and specialized basis sets in an effort to achieve an approximation to a higher

(or intractable) level of theory.18,19 Often time, for example, only the valence correlation energy

is computed, but the correlation between core and valence electrons (called core‐valence

correlation) is important for accurately computing thermochemical properties. Further, most

computational methods are non‐relativistic, but atoms from the second row of the main group,

and heavier, have significant relativistic effects. To include these two energy components, a

composite method may compute the core‐valence energy and relativistic corrections at a low

level of theory like MP2 using specialized basis sets. These individual energy components are

then added to a previously computed reference energy that was computed at a higher level of

theory. There are several composite methods in popular use including the Gaussian‐n (Gn),60‐62

complete basis set (CBS‐n),63‐69 high accuracy extrapolated ab initio thermochemistry (HEAT),70

Weizmann‐n (Wn),71,72 and correlation consistent Composite Approach (ccCA)73‐80 methods.

Chapter 7 discusses the ccCA formulism in detail, including how the method recovers core‐

valence and relativistic effects.

43

CHAPTER 3

CORRELATION CONSISTENT BASIS SETS

3.1 Introduction

Having established the basics of basis set theory in computational quantum chemistry in

the previous chapter, a specific class of basis sets called the correlation consistent polarized

valence basis sets (cc‐pVnZ, where n = D, T, Q, etc.)81‐109 are discussed at length in this chapter.

The correlation consistent basis sets were introduced by Dunning as a means to systematically

recover correlation energy in correlated ab initio computations. The novel approach of

designing basis sets to help recover correlation energy was first studied by Ahlrichs et al.110‐112

and by Almlöf and Taylor.113,114 Dunning’s approach was similar to that of Almlöf and Taylor in

which 1) high angular momentum functions (d, f, g, etc.) are employed beyond the minimal

basis set and 2) basis functions are grouped according to their contribution to the total

correlation energy. Further, the optimized atomic orbitals (AOs) in the basis set are taken to be

the natural orbitals from correlated ab initio wavefunctions. The natural orbitals (NOs) are

obtained by diagonalizing the correlated density matrix,7,22,50 and have the intrinsic property of

forming a basis in which the correlated wavefunction converges most rapidly (as opposed to,

say, the molecular orbital basis). The primary difference between the basis sets developed by

Almlöf and Taylor and those of Dunning is the smaller sets of functions used by Dunning. In fact,

Dunning’s basis sets recover more than 99% of the correlation energy obtained by the basis

sets of Almlöf and Taylor, yet contain less basis functions.

44

The correlation consistent basis sets for the main group atoms were constructed using

Gaussian‐type orbital (GTO) basis sets developed by van Duijneveldt.115 The AOs represented in

each basis set were taken from non‐relativistic multi‐configuration self‐consistent field (MCSCF)

calculations, yielding a set of AOs that adequately describe the valence orbital degeneracy

inherent to most atoms. Each AO is represented by a general contraction of GTOs,114,116,117 in

which all symmetry‐equivalent basis functions are grouped in the contraction. From this point

on, this minimal set of AOs will be called the Hartree‐Fock (HF) basis set.

Within each family of correlation consistent basis sets, there is a hierarchy called the

ζ‐level, where the quality and size of the basis set increases with the ζ‐level. More than that, the

ζ‐level tells how many valence basis functions are used per atom. For example, a double‐ζ basis

set contains two functions per valence orbital; a triple‐ζ basis set contains three functions; etc.

The underlying principle of the ζ‐level is that more functions in the valence space give more

flexibility to the basis set. The core orbitals are always described by a single contracted GTO at

each ζ‐level. Because only the valence orbitals are described by multiple functions, the

correlation consistent basis sets are often referred to as split‐valence basis sets. Table 3.1 lists

the composition of the correlation consistent basis sets for main group atoms (including H and

He) according to each ζ‐level. Note that the primitive, or uncontracted, basis functions are

denoted by parentheses, while the contracted functions are denoted by brackets. The notation

“3s 2p 1d” indicates that three s‐type functions, two p‐type functions, and one d‐type function

comprise the basis set, and are not to be confused with the principle and orbital quantum

numbers. The following sections describe the process by which the correlation consistent basis

sets were developed by Dunning and coworkers, including basis sets for valence‐only

45

correlation, long range correlation, core‐valence correlation, and the recovery of scalar

relativistic effects.

3.2 Valence and Tight d Basis Sets

As already discussed, the HF basis set is insufficient for recovering correlation energy in

correlated ab initio computations. Further, high angular momentum (polarization) functions are

necessary for the qualitative and quantitative description of chemical bonds and necessary for

expedient recovery of all possible correlation energy by a given ab initio method. Polarization

Table 3.1 The composition of the correlation consistent basis sets for the first three rows of the main group.

Row / Atoms ζ‐level Primitives Polarization Contracted Basis Set

H, He D ( 4s ) ( 1p ) [ 2s 1p ]

T ( 5s ), ( 6s )a ( 2p 1d ) [ 3s 2p 1d ]

Q ( 6s ), ( 7s )a ( 3p 2d 1f ) [ 4s 3p 2d 1f ]

5 ( 8s ) ( 4p 3d 2f 1g ) [ 5s 4p 3d 2f 1g ]

1st / B – Ne D ( 9s 4p ) ( 1d ) [ 3s 2p 1d ]

T ( 10s 5p ) ( 2d 1f ) [ 4s 3p 2d 1f ]

Q ( 12s 6p ) ( 3d 2f 1g ) [ 5s 4p 3d 2f 1g ]

5 ( 14s 8p ) ( 4d 3f 2g 1h ) [ 6s 5p 4d 3f 2g 1h ]

2nd / Al – Ar D ( 12s 8p ) ( 1d ) [ 4s 3p 1d ]

T ( 15s 9p ) ( 2d 1f ) [ 5s 4p 2d 1f ]

Q ( 16s 11p ) ( 3d 2f 1g ) [ 6s 5p 3d 2f 1g ]

5 ( 20s 12p ) ( 4d 3f 2g 1h ) [ 7s 6p 4d 3f 2g 1h ]

3rd / Ga‐Kr D ( 14s 11p 5d ) ( 1d ) [ 5s 4p 2d ]

T ( 20s 13p 8d ) ( 2d 1f ) [ 6s 5p 3d 1f ]

Q ( 21s 16p 11d ) ( 3d 2f 1g ) [ 7s 6p 4d 2f 1g ]

5 ( 26s 17p 12d ) ( 4d 3f 2g 1h ) [ 8s 7p 5d 3f 2g 1h ]

a. Indicates the primitive set for He.

46

functions optimized using HF will aid in the non‐correlated description of chemical bonds,

however, HF polarization functions are far from optimal as virtual orbitals in correlated

methods. Therefore, the polarization functions of the correlation consistent basis sets were

optimized as an even‐tempered expansion using singles and doubles configuration interaction

(CISD). An even‐tempered expansion of basis functions is a method of determining the

exponential parameters using a geometric series:116,118

, 1,2, … (3.1)

The parameters and are unique for each group of basis functions with the same value

and are varied so as to minimize the correlation energy. The advantages of using (3.1) instead

of explicit optimization of each polarization function are 1) there are only two parameters to

optimize for the entire group of functions and 2) the polarization functions are evenly

distributed in log‐space, providing even coverage of the orbital space.22

The high angular functions were optimized using (3.1) in groups according to their

values, and were added to the HF basis set of each atom, as suggested by Schwartz,59 until the

correlation energy did not change appreciably. By adding functions this way, the correlation

space for each particular value is saturated. Dunning noted that certain functions within each

group contributed similar amounts of correlation energy. Specifically, it was noted that the

first d function contributed most to the correlation energy; while the second d and first f

functions contributed similar amounts to the correlation energy; the third d, second f, and first

g functions contributed similar amounts to the correlation energy; etc. From this, Dunning

suggested that high angular functions should be added to the HF basis sets in a correlation

consistent manner, that is, in shells as ( 1d ), ( 2d 1f ), ( 3d 2f 1g ), etc. In this way, the

47

correlation energy is systematically recovered with each additional shell of high angular

functions. Further tests were performed by explicitly optimizing the high angular functions

within each group, but no appreciable change was noted in the correlation energy, so the

even‐tempered scheme was kept. Even‐tempered optimizations of s and p functions in addition

to the HF basis sets were also performed with respect to the correlation energy. Dunning found

that uncontracting the outermost s and p functions had the same effect of lowering the

correlation energy as explicit optimization. The split valence nature of the correlation

consistent basis sets stems from the number of s and p functions that are uncontracted: a

single s and p function are uncontracted at the double‐ζ level, two s and p functions are

uncontracted at the triple‐ζ level, etc. (cf. Table 3.1)

After being introduced, studies by Bauschlicher and Partridge119 and Martin et al.120‐122

employing the cc‐pVnZ basis sets showed that including an additional tight d function

significantly improved the computed atomization energies of molecules containing second row

atoms. While the most dramatic improvements were shown for energetics of sulfur species,

Bauschlicher and Partridge also showed that the atomization energies of SiH3 and SiH4 were

affected by 1.1 and 1.6 kcal/mol, respectively, when a tight d function was employed in the

triple‐ζ basis set. The role of an additional tight d function in second row atoms was

commented on by Woon and Dunning in their original paper, however, the need for such a

function did not become evident until the properties of a wider variety of molecular species

than were in the original benchmarking papers were studied.123 As a result, Dunning, Peterson,

and Wilson modified the sets, developing the tight d correlation consistent basis sets,

cc‐pV(n+d)Z,124 and in numerous studies, have demonstrated the importance of the augmented

48

tight d basis sets in coupled cluster and density functional theory studies.101,125‐128

The correlation consistent build‐up of high angular functions with increasing ζ‐level

leads to smooth, monotonic behavior of total energies. As the basis set becomes saturated at

high ζ‐levels, more and more of the AO space is covered by the basis functions, resulting in a

nearly complete basis set. Once the basis set reaches completeness, any remaining error

between a computed chemical property and a reliable experimental measurement is due solely

to the computational method. As the ζ‐level increases, the total energy computed with the

cc‐pVnZ basis sets asymptotically approaches the complete basis set (CBS) limit.93 The exact

mathematical form of the monotonic behavior is not known, but resembles an exponential‐type

decay. To this end, several empirical extrapolation techniques have been developed to

approximate the CBS limit.85,95,129‐132 Two popular extrapolation schemes are below:

CBS (3.2)

CBS (3.3)

Equation (3.2) is a standard exponential fit due to Feller,132 while (3.3) is a mixed

Gaussian/exponential fit due to Woon, Dunning, and Peterson.85,95 In each equation, and

are fitting parameters, is the chemical property (i.e. energy) computed at the th ζ‐level

( 2 for double‐ζ, 3 for triple‐ζ, etc.), and CBS is the estimated CBS limit. Both equations

are analytic for three points, but are numerical for any more. Taking to be the lowest ζ‐level in

a series of three points, the analytic forms of the parameters in (3.2) are

49

CBS 2

2

ln

(3.4)

while those in (3.3) are

CBS

(3.5)

Another extrapolation formula in common use is that of Halkier et al.,133 a two‐point scheme

based on the convergence of correlation energies in atoms:

CBS11

(3.6)

in which takes on the same meaning as in (3.2) and (3.3). The virtue of using (3.6) over (3.2)

or (3.3) is that it only requires two points to estimate the CBS limit. However, it is not possible

to estimate higher ζ‐levels with (3.6) as it is with (3.2) and (3.3). Further, using one of the

exponential‐type extrapolation formulas will sometimes provide a better approximation to the

true CBS limit since more points are used in the fit.

50

3.3 Augmented Basis Sets

In order to quantitatively describe chemistry stemming from long‐range phenomena

such as van der Waals interactions, Rydberg states, multipole moments, polarizabilities,

hydrogen bonding, and the binding of excess electrons, diffuse functions must be employed in

the basis set. Dunning and coworkers have developed augmented correlation consistent basis

sets (aug‐cc‐pVnZ) describing such chemical phenomena.86,88,91,93,95,99,100

The diffuse functions added at each ζ‐level are comprised of a single function of each

value present. For example, to the double‐ζ basis set is added a set of ( 1s 1p 1d ) functions, to

the triple‐ζ basis set is added a set of ( 1s 1p 1d 1f ) functions, etc. The diffuse s and p functions

were optimized on top of the HF basis set in MCSCF calculations of the atomic anions, while the

high angular functions were optimized with CISD on top of the polarization functions already

present.

3.4 Core‐Valence Basis Sets

Valence‐only, or valence‐valence (VV) correlation is generally accepted to be adequate

for accurately computing most chemical properties since the qualitative and quantative

description of many chemical properties relies on changes in the valence electronic structure.

As already discussed, the cc‐pVnZ and aug‐cc‐pVnZ basis sets recover this correlation

systematically with increasing ζ‐level. Also well‐known is that in order to achieve ±1.0 kcal/mol

accuracy, or better, in thermodynamic properties, core‐valence (CV) correlation must also be

included.93,134 The cc‐pVnZ and aug‐cc‐pVnZ basis sets are not optimal for the recovery of CV

correlation and were reoptimized specifically for the recovery of CV correlation by Woon,

51

Peterson, and Dunning.81,98

There are two types of CV correlation consistent basis sets, the original CV basis sets

(cc‐pCVnZ)98 and the weighted core‐valence (wCV) correlation consistent basis sets

(cc‐pwCVnZ).81 The principle difference between the CV and wCV basis sets is that the former

are designed to recover more core‐core (CC) correlation energy, while the wCV basis sets are

intentionally biased towards CV (almost no CC) correlation energy. The optimization of the

core‐correlation (tight) functions proceeded by adding H/He‐like basis sets for correlating the

1s electrons to the first row, main group atoms. Woon and Dunning found that adding groups

of tight functions as ( 1s 1p ) to the double‐ζ basis set, ( 2s 2p 1d ) to the triple‐ζ basis set, etc.

resulted in smooth, monotonic behavior of the CC, CV, and VV energies, as opposed to the

erratic behavior of these energies when the valence‐only cc‐pVnZ basis sets were used. The

following formula was used to optimize the core‐correlating functions:

Δ CISD CC CV VV CISD CC CISD CV CISD VV (3.7)

The weight, , in the above equation is taken to be 1.00 in the cc‐pCVnZ basis sets, and to be

0.01 in the cc‐pwCVnZ basis sets from an in‐depth study of CC and CV effects of first and second

row molecules by Peterson and Dunning.81 It has been noted by Peterson81,93 that both the

cc‐pCVnZ and cc‐pwCVnZ basis sets converge on similar CBS limits, although the cc‐pwCVnZ

basis sets tend to produce lower energies in valence‐only calculations because of the more

diffuse nature of the weighted core‐correlation functions.

3.5 Scalar Relativistic Basis Sets

As discussed in section 2.4, basis sets optimized with non‐relativistic methods are not

52

optimal for use in relativistic calculations. The correlation consistent basis sets discussed so far

were all developed using non‐relativistic formulisms of HF and CISD. One of the most common

relativistic models is the spin‐free Douglas‐Kroll (DK) Hamiltonian,135‐137 which approximates the

full, four‐component Dirac equation138,139 by a transformation of the electron orbitals. The

resulting orbitals are then used in a subsequent HF or correlated ab initio calculation to include

scalar relativistic effects such as the mass‐velocity and Darwin corrections, the two largest

contributors to the relativistic energy.137,140,141 These corrections are termed scalar relativistic

since they do not include inner products containing spin components (i.e. spin‐orbit coupling,

spin‐spin coupling, etc.) The largest effect of scalar relativity occurs in the core orbitals, which

contract inward toward the nucleus due to their increased kinetic energy. Because the core

orbitals of the cc‐pVnZ basis sets are represented by a single contracted GTO, there is no

flexibility in the core to change under the DK transformation, resulting in a loss of accuracy

within the relativistic model. Thus, the core contractions must be reoptimized. Instead of

recontracting only the core orbitals, all of the contractions are typically reoptimized to account

for variations in the valence orbitals due to fluctuations in the core.

The first to report on correlation consistent basis sets optimized for scalar relativistic

computations, denoted cc‐pVnZ‐DK, were de Jong et al.142 Their scheme for recontracting the

cc‐pVnZ basis sets is as follows: 1) remove the polarization functions (d, f, g, and h) from the

cc‐pVnZ basis sets, and uncontract the s and p basis set; 2) run a HF calculation using the

uncontracted sp basis set and the DK Hamiltonian; and 3) the optimized s and p contraction

coefficients are then determined by a population analysis. A similar scheme is used in Chapter

6, in which DK‐optimized basis sets for the s‐block atoms are reported.

53

CHAPTER 4

SYSTEMATIC TRUNCATION OF THE CORRELATION CONSISTENT BASIS SETS IN DENSITY

FUNCTIONAL THEORY CALCULATIONS†

4.1 Introduction

The plethora of density functionals available46 has brought about a tremendous

challenge in applying density functional theory (DFT)44,45,52 to the study of chemical systems,

namely, the choice of a density functional that is suitable for the problem of interest.

Unfortunately, no simple means currently exists to aid in density functional selection, barring

whether or not a density functional was specifically developed for a particular problem, and

reliance upon benchmark studies, when available, is essential. However, it is of great interest to

have a simple, efficient means to gauge the intrinsic error of a density functional. To accomplish

this, a first step is to develop a greater understanding of the behavior of density functionals

with respect to the basis set.

As shown in Chapter 3, the correlation consistent basis sets exhibit smooth, monotonic

convergence in energetic properties with increasing basis set size, and that the complete basis

set (CBS) limit may be numerically approximated through the use of various extrapolation

schemes. Unfortunately, it is not clear whether or not such an approach is possible for DFT

calculations since the correlation consistent basis sets were developed specifically for ab initio

† This entire chapter is adapted from B.P. Prascher, B.R. Wilson, and A.K. Wilson, “Behavior of Density Functionals with Respect to Basis Set. VI. Truncation of the Correlation Consistent Basis Sets.” J. Chem. Phys. 2007, 127, 124110, with permission from the American Institute of Physics.

54

methods.

Energetic and structural properties are generally known to converge more quickly with

respect to increasing basis set size for DFT than for ab initio methods.125,126,143‐156 In fact, it is

commonly assumed that small changes in energetic properties beyond the triple‐ζ basis set

level typically occur, but recent work has demonstrated that significant changes in energetic

properties sometime do occur beyond the triple‐ζ level.148,149 Further, recent work has shown

non‐monotonic behavior of energetic properties can occur with respect to increasing basis set

size in DFT calculations.125,126,152‐158 This observation has been made using a number of

molecules and density functionals, with no obvious trends appearing as to when the non‐

monotonic behavior does or does not occur. When monotonic behavior does occur, then it has

been shown that various extrapolation formulas can be used to estimate the CBS limit, or in the

case of density functional theory, the Kohn‐Sham (KS) limit.144,145,154‐156 However, with non‐

monotonic behavior, such an approach is not reliable. To date, no single solution to the non‐

monotonic problem has evolved; rather, a number of possible solutions have emerged.

Provided a reasonable grid size is utilized for the numerical calculation of the DFT energy, grid

selection does not seem to impact the basis set convergence.156 Other factors that influence

convergence include basis set superposition error152 and basis set contraction,156 which, when

combined, have been shown to lead to monotonic convergence (discussed in the next

chapter).148 It is important to note that systematic convergence to the KS limit should not

necessarily be expected from the correlation consistent basis sets as they were explicitly

developed for ab initio methods. However, Jensen has developed a series of polarization

consistent basis sets (pc‐n, where n = 1‐4) explicitly for DFT,143‐145 and non‐monotonic behavior

55

using various density functionals is still observed for these basis sets.155

Ab initio methods rely directly on the inclusion of many excited determinants,

constructed from high angular momentum functions, while density functionals do not. High

angular momentum functions in DFT computations have the same effect as in Hartree‐Fock

(HF), that is, to polarize the electron density. As a result, DFT computations are less basis set

dependent than ab initio computations, and DFT computations, with systematic increases in

basis set size, tend to converge quickly. Rapid convergence towards the KS limit is desirable, but

the high angular momentum functions included in many basis sets may not have a significant

effect on a computed property. Thus, these functions create unnecessary overhead in terms of

computational expense and, as shown in this chapter, may be removed without the

introduction of significant error.

Earlier work has utilized truncated or auxiliary basis sets as a means to reduce

computational cost in DFT calculations. For example, the resolution of the identity Coulomb

approximation (RI‐J) has been studied by Skylaris et al.159 and by Head‐Gordon et al.150 In the

RI‐J approximation, an auxiliary basis set is used to compute the Coulomb integrals, which can

reduce the computational cost without introducing errors of more than a few millihartrees in

the total energy.159

Head‐Gordon et al. have also investigated a dual basis approach, first proposed by Van

Alsenoy and Vahtras et al.160,161 in which a truncated form of a basis set, such as cc‐pVTZ

without f functions or the inner d functions, is used to construct an initial approximation to the

full cc‐pVTZ basis density. Subsequent parameterization is included to help account for the

difference in densities between the full cc‐pVTZ basis set and the truncated basis set.151 Earlier

56

work by Jensen examined truncation of core basis functions from double‐ and triple‐ζ polarized

basis sets (DZP and TZP, respectively) as well as the pc‐n basis sets.147 The conclusions from his

investigation indicate that basis sets truncated of certain core functions lead to results similar

to those obtained when the full basis sets are employed. However, truncation of core functions

is not suitable for the accurate determination of KS limits due to the incomplete coverage of

the exponent space.

The present investigation examines the truncation of valence basis functions from the

correlation consistent basis sets without parameterization, and does not suffer from reduced

coverage of the necessary exponent space since neither the s or p functions are removed from

the HF basis set. By analyzing the change in computed properties as basis functions are

systematically truncated, the necessity of high angular momentum functions becomes clear, as

was demonstrated in a similar study by Mintz et al. in which truncated correlation consistent

basis sets were studied in ab initio methods.162,163 Understanding how high angular momentum

functions impact not only computed properties, but also basis set convergence, will aid in

designing basis sets for DFT computations.

4.2 Computational Methodology

Two popular density functionals have been included in this study, including pure and

hybrid density functionals. The pure density functional is the gradient‐corrected Becke (B)164

exchange functional, coupled with the Lee‐Yang‐Parr (LYP)165 gradient‐corrected correlation

functional, BLYP. For the hybrid DFT computations, the Becke three‐parameter scheme (B3)166

is employed with the LYP correlation functional, utilizing the third parameterization of the

57

Vosko, Wilk, and Nusair local correlation functional (VWN‐3).167

The correlation consistent basis sets, cc‐pVnZ,94‐101 through quintuple‐ζ quality have

been employed. Systematic truncation of these basis sets follows that of the scheme published

by Mintz et al. for the hydrogen atom,162,163 but is extended to include non‐hydrogen atoms.

The truncation scheme involves removing the highest ‐value basis function first, then the most

diffuse function from the 1 shell, then the second‐most diffuse function from the 1

shell, etc. The truncation continues through the polarization functions, but does not include the

contracted s and p functions. The notation used is as follows: the full triple‐ζ basis is denoted as

usual: cc‐pVTZ; the first truncation of this basis is denoted cc‐pVTZ(–1f); the second truncation

is denoted cc‐pVTZ(–1f 1d); and the third truncation is denoted cc‐pVTZ(–1f 2d) which reflects

the removal of one f function and the two most diffuse d functions (for cc‐pVTZ, this means

removal of all of the d functions). It is implied that the first truncation of a specific ‐shell is the

most diffuse (i.e. lowest exponent).

In atomic computations it is common practice to perform a symmetry‐averaging of

degenerate electronic states since this leads to a more physically‐sound wavefunction. If the

resulting wavefunction is spherically symmetric ( 0), then basis functions of angular

momentum not present in the filled, state‐averaged orbitals can be separated from the total

wavefunction without consequence to the total energy. This phenomenon has been

investigated in great detail by Bauschlicher and Taylor.168 This chapter strictly employs single

reference wavefunctions in all computations. Atomic symmetry breaking of the type used here

has been discussed in rigorous detail elsewhere, and is often necessary to correctly compute

electronic properties.46,169 As atom‐centered basis sets are typically used in molecular

58

computations, where atoms are not spherically‐symmetric, broken symmetry in atomic

computations is an appropriate means to effectively gauge basis set truncation effects.

Vertical ionization potentials and electron affinities have been determined using

experimental geometries for the neutral molecules taken from the National Institute of

Standards and Technology (NIST) Webbook.170‐172 The geometry was fixed so that the impact of

removing high angular momentum functions could be gauged without a geometry optimization

affecting the energy. Optimized geometries and their atomization energies are reported after

gauging the impact of truncation. The use of diffuse functions in the basis set has been

restricted to only the s and p functions here, which are taken from the augmented

(aug‐cc‐pVnZ) basis sets, and have been employed in computations of electron affinities.173

Atoms of the first and second rows, their dimers, and the molecules CH4, SiH4, NH3, PH3, H2O,

and H2S have been investigated. All DFT computations of this study have been performed using

the Gaussian software suite of programs.174 The default pruned grid (75,302)175 has been used

in all computations, and total energies in all computations are converged to 10‐8 Eh, or better.

4.3 Results and Discussion

4.3.1 Atoms

Total energies for the first row atoms carbon and oxygen are given in Table 4.1. Ideally,

the energy difference between a truncated basis and the full basis set should not be more than

1.0 mEh (approximately 1.0 kcal/mol) since this is the typical accuracy sought for computed

energetics when compared with experiment. As shown in the table, truncation effects in BLYP

59

computations are similar to those in B3LYP computations. The largest difference between BLYP

and B3LYP as a result of truncation occurs for carbon and is 0.110 mEh at the cc‐pVTZ(‐1f 2d)

level. Comparing carbon to oxygen, the effect of truncation on the total energy of oxygen is

almost twice that of carbon. For example, at the cc‐pV5Z(‐1h 2g 3f 4d) level for carbon, the

change in the BLYP energy with respect to the full basis set is 1.190 mEh, while for the oxygen

atom the change is 2.266 mEh. The greatest difference between the truncated and full basis

sets occurs for the fully truncated quintuple‐ζ basis set, cc‐pV5Z(‐1h 2g 3f 4d), for all atoms

investigated. Generally, the fully truncated cc‐pV5Z value for the total energy is comparable to

the full cc‐pVQZ basis set value for atoms with non‐spherical wavefunctions.

Computed ionization potentials and electron affinities of carbon and oxygen are shown

in Table 4.2 and Table 4.3, respectively. It is immediately obvious that electron affinities are not

as affected by truncation as ionization potentials (note that the electron affinities have been

computed using the s and p diffuse functions from the aug‐cc‐pVnZ basis sets). In the oxygen

atom, significant changes (i.e. more than 0.04 eV, or 1.0 kcal/mol) in the computed ionization

potential do not occur until f functions are completely removed. The carbon atom ionization

potentials show little change relative to the full basis, even with complete truncation of all high

angular momentum functions. Electron affinities, on the other hand, do not change until d

functions are truncated, and even then, the differences between the full and truncated basis

sets are less than 0.04 eV. There is little dependence on the density functional used since

truncation effects appear at the same level for B3LYP and BLYP. For example, the ionization

potential of oxygen begins to be affected by truncation at the cc‐pVTZ(‐1f), cc‐pVQZ(‐1g 1f), and

cc‐pV5Z(‐1h 2g 1f) levels in both BLYP and B3LYP. A similar trend results for electron affinities.

60

Figure 4.1 is a plot of the full basis set versus truncated basis set ionization potential of oxygen

determined with BLYP. Because basis functions of 3 do not significantly affect the

computed ionization potentials of atoms, a plot of each basis set containing only s, p, d, and f

functions is included for comparison in Figure 4.1. The 3 values coincide well with the full

basis set values. The inset in the figure demonstrates the effect of different truncation levels at

the cc‐pVQZ level on computed ionization potentials. Based on these observations for first row

atomic properties, a basis set for use in DFT computations need only contain up through f

functions to achieve full cc‐pVnZ basis set results for total energies, ionization potentials, and

electron affinities within 1.0 kcal/mol for atoms.

4.3.2 Homonuclear Diatomics

The BLYP and B3LYP total energies for the first row diatomics C2 and O2 are shown in

Table 4.4. In general, truncation affects the diatomic total energies by almost an order of

magnitude more than in atoms. This is expected, since the high angular momentum functions

are necessary for a better description of bonding and anti‐bonding orbitals, specifically d and f

functions which polarize p functions in σ‐ and π‐type bonds. This is readily seen for each

diatomic investigated, where truncation of f functions from the cc‐pV5Z basis set typically

causes the difference in total energy relative to the full basis to increase on the order of mEh,

and the truncation of d functions increases on the order of tens of mEh. Table 4.5 lists

computed ionization potentials, electron affinities, and optimized bond lengths for the first row

dimers with BLYP. The ionization potentials remain unaffected by truncation until d functions

have been truncated, but the electron affinities are affected with truncation of the f functions.

61

The general trend in truncation of d functions affecting ionization potentials and f functions

affecting electron affinities remains consistent. Note how truncation affects ionization

potentials and electron affinities in diatomics as compared with atoms. In atoms, ionization

potentials computed with a truncated basis are lower than the full basis set results, but are

higher for diatomics (cf. Figure 4.1 and Figure 4.2). The electron affinities of atoms computed

with a truncated basis are typically higher than the full basis result, and in diatomics they are

lower than the full basis set result, with the exception being when d functions are truncated.

Truncation leads to a systematic change in the computed property for atoms, but this trend is

not observed for diatomics. Figure 4.2 demonstrates the impact of truncation upon the

Figure 4.1 BLYP ionization potentials of the oxygen atom computed with truncated basissets (hash marks) and plotted against the full basis set values. The inset shows more detail atthe quadruple‐ζ level. Points denoted ‘l > 3’ are the truncated basis sets with only s, p, d, and fbasis functions.

62

computed ionization potential of the oxygen molecule.

It is clear that at least some of the f functions must remain to maintain atomization

energies that are within 1.0 kcal/mol of the full basis set values. An example of how f functions

impact atomization energies is shown by the truncation at the double‐ and triple‐ζ levels for the

oxygen molecule in Table 4.5. The removal of the only f function from cc‐pVTZ lowers the

computed atomization energy by 1.4 kcal/mol and the removal of the most diffuse d function

increases this deviation almost an order of magnitude. At the quadruple‐ζ level, removing one f

function significantly increases the deviation from the full basis – more so than the removal of

the higher g function. Truncation through the most diffuse f function of the quintuple‐ζ basis

Figure 4.2 BLYP ionization potentials of the O2 molecule (3Σg) computed with truncatedbasis sets (hash marks) and plotted against the full basis set values. Points denoted ‘l > 3’ arethe truncated basis sets with only s, p, d, and f basis functions.

63

does not significantly alter the computed atomization energy, but removing the second f

function does. It is clear that the f shell must be retained in the homonuclear diatomics to

minimize the deviation of the truncated basis atomization energy from that of the full basis.

Optimized bond lengths follow a trend similar to that of molecular electron affinities in

that they depend heavily on the d functions in the basis set. Target accuracy for truncated basis

set bond lengths is 0.01 Å relative to the bond length determined using the full basis set. The

removal of the d function from the double‐ζ basis set in a BLYP computation of the oxygen

molecule affect the optimized bond length by 0.060 Å. At the triple‐ζ level, truncation of the

most diffuse d function significantly affects the bond length and removal of the entire d shell

causes the bond length to deviate by 0.064 Å from the full basis set result. The most diffuse d

function of the quadruple‐ and quintuple‐ζ basis sets does not significantly affect the bond

length upon truncation, but removal of any of the other d functions from these bases

significantly impacts the computed value. These deviations in truncated basis set bond lengths

for BLYP are also typical in B3LYP computations.

4.3.3 CH4, SiH4, NH3, PH3, H2O, and H2S

Table 4.6, Table 4.7, and Table 4.8 provide B3LYP optimized geometries for CH4, NH3,

H2O, SiH4, PH3, and H2S. The truncation notation used in these tables has two terms. The first

represents the truncation of the non‐hydrogen basis set, and the second represents the

truncation of the hydrogen basis set. For example, at the triple‐ζ level, ‘‐1f 2d ; 1d 2p’ denotes

truncation of one f and two d functions from the non‐hydrogen basis and one d and two p

functions from the hydrogen atom basis set. Both truncations are performed simultaneously.

64

Included in the tables for CH4 and SiH4 are atomization energies. Table 4.6 shows that the

atomization energies in heteronuclear molecules are sensitive to the basis set truncation,

varying more than 1.0 kcal/mol when truncation of d functions from the non‐hydrogen basis

(including p functions from the hydrogen basis) occurs. Compared with the atomization

energies for homonuclear diatomics, the atomization energies for larger heteronuclear

molecules is less sensitive to basis set truncation overall. In fact, the atomization energy of the

oxygen molecule suffers almost a 30.0 kcal/mol deviation from the full basis set result when

truncation of all of the high angular momentum functions from each basis set is done. For

methane, full truncation of the basis sets only results in a 10.0 kcal/mol deviation from the full

basis set atomization energy. The atomization energy for silane is comparable to that of oxygen

in that the largest angular momentum functions required are f functions in the non‐hydrogen

basis set to maintain energies within 1.0 kcal/mol of those of the full basis set. Therefore, f

functions remain necessary in the non‐hydrogen basis set for computed atomization energies.

In molecules composed of first row atoms, the computed bond lengths do not vary by

more than 0.01 Å when the d and p shells are truncated from the non‐hydrogen and hydrogen

basis sets, respectively. In contrast, molecules which include a second row atom do depend on

the d and p shells from the non‐hydrogen and hydrogen basis sets, respectively, to avoid

deviating by more than 0.01 Å from the full basis set bond length. This is not surprising

considering that the second row atoms are known to form bonds with more 3p character than

3s, and the 3p orbitals are polarized by d functions present in the basis set. Further, angles of

second row systems are not as dependent on the d and p functions of the non‐hydrogen and

hydrogen basis sets, respectively. This is the direct result of more p character in the bonding

65

molecular orbitals for these compounds (cf. Table 4.7 and where the H‐P‐H angle is 93° and the

H‐N‐H angle is 106°; the H‐S‐H angle is 92° and the H‐O‐H angle is 104°). As a result of the

different bonding characteristics of first and second row systems, truncation of the first row

non‐hydrogen/hydrogen basis sets of their f/d functions the second row non‐

hydrogen/hydrogen basis sets of their d/p functions affects bond angles more than 0.1°. Basis

set truncation is therefore only effective in keeping the accuracy of the full basis set in bond

lengths and angles (within 0.01 Å and 0.1°) when the d and p shells are present in non‐hydrogen

and hydrogen basis sets, respectively, for second row systems; and when f and d functions are

present, respectively, in first row systems.

4.3.4 Computational Time Savings

In atomic computations, it is observed that truncating most of the high angular

momentum functions from the quintuple‐ζ basis could produce total energies that are

comparable with the full quadruple‐ζ basis energies at a fraction of the computational cost. In

heteronuclear molecules, if the basis set is no longer truncated when the computed angles

begin to vary by more than 0.1° from the full basis set result, then the computational cost is

significantly reduced without the introduction of significant error compared with the full basis.

The benefits are two‐fold: 1) computational time can be reduced significantly and 2) both the

bond length and the angle will be within 0.01 Å and 0.1°, respectively, from the full basis set

results. Bond lengths computed with truncated basis sets remain steadily within 0.01 Å of the

full basis. Table 4.9 lists the percent central processing unit (CPU) time saved in single‐point

computations for the heteronuclear molecules, and Figure 4.3 illustrates how much time is

66

saved with respect to energetic properties at various truncation levels of the cc‐pV5Z basis set.

The percent CPU time saved over the full basis computation escalates quickly as high angular

momentum functions are removed, reaching more than 50% when the highest angular

momentum functions are removed from the cc‐pVQZ and cc‐pV5Z basis sets. CPU time savings

are greater than 80% at the cc‐pV5Z level when the two highest angular momentum shells are

removed from both the non‐hydrogen and the hydrogen basis. Since truncation impacts

molecular ionization potentials, electron affinities, and atomization energies when truncating

the f functions, by only removing the higher angular momentum functions, the percent time

saved in computing these properties can be as large as 90% without deviating more than 1.0

kcal/mol (0.04 eV) from the full basis set results. Further, in bond lengths (without considering

angles) where the d shell is important for the non‐hydrogen basis, truncating the higher angular

momentum shells saves over 95% of the computational time without deviating from the full

basis set bond length by more than 0.01 Å.

4.4 Conclusions

Truncation of the correlation consistent basis sets in DFT is an excellent way to save

computational time and resources when computing various energetic properties without the

introduction of significant error. In the atoms and molecules studied, truncation of high angular

functions such as g and h functions affects ionization potentials and election affinities by less

than 0.01 eV, and affects atomization energies by less than 1.0 kcal/mol. Typical CPU time

savings upon truncation of only g and h functions from non‐hydrogen and hydrogen basis sets

are on the order of 60‐70%.

67

Unlike ab initio studies of cc‐pVnZ basis set truncation,162,163 DFT allows for the

truncation of not only the hydrogen basis but also the non‐hydrogen atom basis set.

Simultaneous truncation of both the hydrogen f and g functions and non‐hydrogen atom g and

h functions from the cc‐pV5Z basis set impacts optimized geometries by less than 0.01 Å in

bond lengths and 0.1° in angles as compared with the full basis geometry. In all molecules

investigated here, the removal of f functions from the non‐hydrogen basis with simultaneous

removal of d functions from the hydrogen basis does not affect bond lengths by more than

0.01 Å and affects angles by no more than 0.47°. The geometries of molecules composed of first

row atoms are not affected as greatly as those composed of second row atoms.

With truncation of all but the s, p, d, and f functions from the correlation consistent

basis sets, full basis set values for total energies, atomization energies, ionization potentials,

electron affinities (using s and p diffuse functions), and geometries are accessible (within

1.0 kcal/mol in energetics and 0.01 Å/0.1° in bond lengths/angles) with a large reduction in CPU

computational overhead on the order of 70‐90%. The trends of truncation versus property hold

for both functionals investigated here. When the correlation consistent basis sets of double‐

through quintuple‐ζ quality are employed in DFT computations, much computational overhead

can be alleviated by truncating the basis sets of all but the s, p, d, and f functions without the

introduction of significant error in energetics or geometries compared with the full basis sets.

68

Figure 4.3 Comparisons of CPU time savings (closed circles), ionization potentials(triangles), electron affinities (upside‐down triangles), atomization energies (open squares), andtotal energies (closed squares) between the full and truncated cc‐pV5Z basis set using B3LYP.Truncation levels: ‘0’ is the full basis, ‘1’ is cc‐pV5Z(‐1h ; 1g), ‘2’ is cc‐pV5Z(‐1h1g ; 1g1f), etc.

69

Table 4.1 Total energies (Eh) of the carbon and oxygen atoms computed with full and truncated basis sets; energy differences are listed in mEh.

BLYP B3LYP

Atom Basis Set Truncation Energy Δ Energy Δ

C (3P) cc‐pVDZ none ‐37.837836 ‐‐‐ ‐37.851974 ‐‐‐ –1d ‐37.836850 0.986 ‐37.850886 1.089 cc‐pVTZ none ‐37.845501 ‐‐‐ ‐37.858575 ‐‐‐ –1f ‐37.845492 0.009 ‐37.858571 0.004 –1f 1d ‐37.845093 0.408 ‐37.858129 0.445 –1f 2d ‐37.844387 1.114 ‐37.857351 1.224 cc‐pVQZ none ‐37.847806 ‐‐‐ ‐37.860592 ‐‐‐ –1g ‐37.847791 0.015 ‐37.860583 0.009 –1g 1f ‐37.847791 0.015 ‐37.860583 0.009 –1g 2f ‐37.847766 0.040 ‐37.860571 0.021 –1g 2f 1d ‐37.847640 0.167 ‐37.860422 0.170 –1g 2f 2d ‐37.847015 0.791 ‐37.859737 0.855 –1g 2f 3d ‐37.846634 1.172 ‐37.859324 1.268 cc‐pV5Z none ‐37.849077 ‐‐‐ ‐37.861508 ‐‐‐ –1h ‐37.849072 0.005 ‐37.861505 0.003 –1h 1g ‐37.849068 0.010 ‐37.861503 0.005 –1h 2g ‐37.849054 0.024 ‐37.861495 0.013 –1h 2g 1f ‐37.849053 0.024 ‐37.861495 0.013 –1h 2g 2f ‐37.849053 0.024 ‐37.861495 0.013 –1h 2g 3f ‐37.849021 0.056 ‐37.861478 0.030 –1h 2g 3f 1d ‐37.848969 0.108 ‐37.861417 0.091 –1h 2g 3f 2d ‐37.848557 0.520 ‐37.860955 0.553 –1h 2g 3f 3d ‐37.848042 1.036 ‐37.860392 1.116 –1h 2g 3f 4d ‐37.847887 1.190 ‐37.860228 1.280

O (3P) cc‐pVDZ none ‐75.054526 ‐‐‐ ‐75.068499 ‐‐‐ –1d ‐75.053435 1.091 ‐75.067322 1.177 cc‐pVTZ none ‐75.080286 ‐‐‐ ‐75.091864 ‐‐‐ –1f ‐75.079439 0.847 ‐75.091015 0.850 –1f 1d ‐75.078980 1.306 ‐75.090510 1.354 –1f 2d ‐75.078227 2.059 ‐75.089704 2.160 cc‐pVQZ none ‐75.087251 ‐‐‐ ‐75.098201 ‐‐‐ –1g ‐75.087239 0.012 ‐75.098194 0.006

(continued on next page)

70

Table 4.1 (continued)

BLYP B3LYP


O (3P) cc‐pVQZ –1g 1f ‐75.086825 0.426 ‐75.097774 0.427 –1g 2f ‐75.086243 1.008 ‐75.097200 1.001 –1g 2f 1d ‐75.086113 1.138 ‐75.097049 1.152 –1g 2f 2d ‐75.085423 1.828 ‐75.096298 1.902 –1g 2f 3d ‐75.085013 2.238 ‐75.095868 2.332 cc‐pV5Z none ‐75.090069 ‐‐‐ ‐75.100485 ‐‐‐ –1h ‐75.090066 0.002 ‐75.100484 0.001 –1h 1g ‐75.090063 0.005 ‐75.100482 0.003 –1h 2g ‐75.090051 0.018 ‐75.100476 0.009 –1h 2g 1f ‐75.089872 0.196 ‐75.100297 0.188 –1h 2g 2f ‐75.089385 0.684 ‐75.099799 0.685 –1h 2g 3f ‐75.089028 1.040 ‐75.099457 1.028 –1h 2g 3f 1d ‐75.088990 1.079 ‐75.099408 1.077 –1h 2g 3f 2d ‐75.088586 1.483 ‐75.098950 1.535 –1h 2g 3f 3d ‐75.087988 2.081 ‐75.098310 2.175 –1h 2g 3f 4d ‐75.087803 2.266 ‐75.098119 2.366

71

Table 4.2 Ionization potentials (eV) of the carbon and oxygen atoms computed with full and truncated basis sets; relative differences are listed below the full basis set values.

BLYP B3LYP

Basis Set Truncation C (3P) O (3P) C (3P) O (3P)

cc‐pVDZ none 11.36 13.84 11.52 13.31 –1d 0.01 ‐0.03 0.01 ‐0.03

cc‐pVTZ none 11.38 14.07 11.53 14.09 –1f ‐‐‐ ‐0.02 ‐‐‐ ‐0.02 –1f 1d ‐‐‐ ‐0.04 ‐‐‐ ‐0.04 –1f 2d 0.01 ‐0.06 0.01 ‐0.06

cc‐pVQZ none 11.39 14.12 11.54 14.12 –1g ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1g 1f ‐‐‐ ‐0.01 ‐‐‐ ‐0.01 –1g 2f ‐‐‐ ‐0.03 ‐‐‐ ‐0.03 –1g 2f 1d ‐‐‐ ‐0.03 ‐‐‐ ‐0.03 –1g 2f 2d ‐‐‐ ‐0.05 ‐‐‐ ‐0.05 –1g 2f 3d 0.01 ‐0.06 0.01 ‐0.06

cc‐pV5Z none 11.40 14.15 11.54 14.14 –1h ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 1g ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 2g ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 2g 1f ‐‐‐ ‐0.01 ‐‐‐ ‐0.01 –1h 2g 2f ‐‐‐ ‐0.02 ‐‐‐ ‐0.02 –1h 2g 3f ‐‐‐ ‐0.03 ‐‐‐ ‐0.03 –1h 2g 3f 1d ‐‐‐ ‐0.03 ‐‐‐ ‐0.03 –1h 2g 3f 2d ‐‐‐ ‐0.04 ‐‐‐ ‐0.04 –1h 2g 3f 3d 0.01 ‐0.06 0.01 ‐0.06 –1h 2g 3f 4d 0.01 ‐0.06 0.01 ‐0.06

72

Table 4.3 Electron affinities (eV) of the carbon and oxygen atoms computed with full, truncated, and augmented truncated basis sets (aug); relative differences are listed below the full basis set values.

BLYP B3LYP

Atom Basis Set Truncation EA EA(aug) EA EA(aug)

C (3P) cc‐pVDZ none 0.12 1.34 0.33 1.37 –1d 0.03 0.02 0.03 0.03 cc‐pVTZ none 0.67 1.34 0.82 1.37 –1f ‐‐‐ 0.01 ‐‐‐ ‐‐‐ –1f 1d 0.01 0.01 0.01 0.01 –1f 2d 0.03 0.03 0.03 0.03 cc‐pVQZ none 0.91 1.35 1.03 1.37 –1g ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1g 1f ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1g 2f ‐‐‐ 0.01 ‐‐‐ ‐‐‐ –1g 2f 1d 0.01 0.01 0.01 0.01 –1g 2f 2d 0.02 0.02 0.02 0.02 –1g 2f 3d 0.03 0.03 0.03 0.03 cc‐pV5Z none 1.14 1.37 1.22 1.38 –1h ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 1g ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 2g ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 2g 1f ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 2g 2f ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 2g 3f ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 2g 3f 1d ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 2g 3f 2d 0.01 0.01 0.02 0.01 –1h 2g 3f 3d 0.03 0.03 0.03 0.03 –1h 2g 3f 4d 0.03 0.03 0.03 0.03

O (3P) cc‐pVDZ none ‐0.58 1.80 1.85 1.67 –1d ‐‐‐ 0.01 ‐‐‐ 0.01 cc‐pVTZ none 0.57 1.81 0.62 1.68 –1f ‐‐‐ 0.01 ‐‐‐ ‐‐‐ –1f 1d ‐‐‐ 0.01 ‐‐‐ ‐‐‐ –1f 2d ‐‐‐ 0.01 0.01 0.01 cc‐pVQZ none 1.04 1.82 1.04 1.68


73


BLYP B3LYP

Atom Basis Set Truncation EA EA(aug) EA EA(aug)

O (3P) cc‐pVQZ –1g ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1g 1f ‐‐‐ 0.01 ‐‐‐ ‐‐‐ –1g 2f ‐‐‐ 0.01 ‐‐‐ 0.01 –1g 2f 1d ‐‐‐ 0.01 ‐‐‐ 0.01 –1g 2f 2d 0.01 0.01 0.01 0.01 –1g 2f 3d 0.01 0.01 0.01 0.01 cc‐pV5Z none 1.47 1.84 1.41 1.69 –1h ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 1g ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 2g ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 2g 1f ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 2g 2f ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 2g 3f 0.01 ‐‐‐ 0.01 ‐‐‐ –1h 2g 3f 1d 0.01 ‐‐‐ 0.01 ‐‐‐ –1h 2g 3f 2d 0.01 0.01 0.01 ‐‐‐ –1h 2g 3f 3d 0.01 0.01 0.01 0.01 –1h 2g 3f 4d 0.01 0.01 0.01 0.01

74

Table 4.4 Total energies (Eh) of the C2 and O2 molecules computed with full and truncated basis sets; energy differences are listed in mEh.

BLYP B3LYP


C2 (1Σg) cc‐pVDZ none ‐75.893728 ‐‐‐ ‐75.888710 ‐‐‐ –1d ‐75.873740 19.988 ‐75.873958 14.752 cc‐pVTZ none ‐75.916662 ‐‐‐ ‐75.907631 ‐‐‐ –1f ‐75.914735 1.926 ‐75.905205 2.426 –1f 1d ‐75.910817 5.845 ‐75.903585 4.046 –1f 2d ‐75.888437 28.224 ‐75.885075 22.556 cc‐pVQZ none ‐75.922052 ‐‐‐ ‐75.912333 ‐‐‐ –1g ‐75.921801 0.251 ‐75.912097 0.236 –1g 1f ‐75.921430 0.622 ‐75.911820 0.513 –1g 2f ‐75.919783 2.269 ‐75.909817 2.516 –1g 2f 1d ‐75.919094 2.958 ‐75.909747 2.586 –1g 2f 2d ‐75.907697 14.355 ‐75.900794 11.539 –1g 2f 3d ‐75.895122 26.930 ‐75.890878 21.455 cc‐pV5Z none ‐75.924393 ‐‐‐ ‐75.9140147 ‐‐‐ –1h ‐75.924348 0.046 ‐75.913965 0.049 –1h 1g ‐75.924276 0.117 ‐75.913917 0.098 –1h 2g ‐75.924110 0.284 ‐75.913736 0.278 –1h 2g 1f ‐75.924009 0.384 ‐75.913725 0.290 –1h 2g 2f ‐75.923431 0.963 ‐75.912952 1.063 –1h 2g 3f ‐75.922179 2.214 ‐75.911538 2.477 –1h 2g 3f 1d ‐75.921971 2.422 ‐75.911528 2.486 –1h 2g 3f 2d ‐75.917177 7.216 ‐75.908848 5.166 –1h 2g 3f 3d ‐75.902165 22.229 ‐75.896025 17.990 –1h 2g 3f 4d ‐75.897683 26.711 ‐75.892618 21.397

O2 (3Σg) cc‐pVDZ none ‐150.328047 ‐‐‐ ‐150.334047 ‐‐‐ –1d ‐150.280817 47.230 ‐150.282835 51.212 cc‐pVTZ none ‐150.377164 ‐‐‐ ‐150.380941 ‐‐‐ –1f ‐150.373216 3.948 ‐150.376785 4.156 –1f 1d ‐150.352941 24.223 ‐150.354860 26.081 –1f 2d ‐150.328045 49.119 ‐150.328372 52.569 cc‐pVQZ none ‐150.391260 ‐‐‐ ‐150.394129 ‐‐‐ –1g ‐150.390835 0.425 ‐150.393699 0.431


75


BLYP B3LYP


O2 (3Σg) cc‐pVQZ –1g 1f ‐150.389332 1.929 ‐150.392094 2.036 –1g 2f ‐150.386938 4.322 ‐150.389574 4.555 –1g 2f 1d ‐150.382798 8.463 ‐150.384994 9.135 –1g 2f 2d ‐150.351403 39.858 ‐150.351451 42.678 –1g 2f 3d ‐150.340489 50.771 ‐150.339871 54.258 cc‐pV5Z none ‐150.396251 ‐‐‐ ‐150.398385 ‐‐‐ –1h ‐150.396194 0.056 ‐150.398331 0.053 –1h 1g ‐150.396074 0.177 ‐150.398203 0.181 –1h 2g ‐150.395935 0.316 ‐150.398058 0.327 –1h 2g 1f ‐150.395751 0.499 ‐150.397879 0.506 –1h 2g 2f ‐150.393444 2.807 ‐150.395394 2.991 –1h 2g 3f ‐150.392519 3.732 ‐150.394432 3.953 –1h 2g 3f 1d ‐150.391698 4.553 ‐150.393515 4.870 –1h 2g 3f 2d ‐150.373542 22.709 ‐150.373975 24.409 –1h 2g 3f 3d ‐150.351483 44.768 ‐150.350460 47.925 –1h 2g 3f 4d ‐150.347558 48.693 ‐150.346289 52.095

76

Table 4.5 BLYP vertical ionization potentials (eV), electron affinities (eV), atomization energies (kcal/mol), and optimized bond lengths (Å) of the C2 and O2 molecules computed with full and truncated basis sets; electron affinities include diffuse functions and relative energy differences are listed below the full basis set values.

Molecule Basis Set Truncation IP EA(aug) D0 ReC2 (

1Σg) cc‐pVDZ none 12.57 3.92 134.2 1.271 –1d ‐0.02 ‐0.02 ‐11.3 0.005 cc‐pVTZ none 12.66 3.95 139.0 1.256 –1f 0.01 0.01 ‐1.2 ‐0.001 –1f 1d ‐‐‐ ‐‐‐ ‐3.2 ‐0.002 –1f 2d ‐0.04 ‐0.02 ‐16.2 0.014 cc‐pVQZ none 12.67 3.96 139.5 1.255 –1g ‐‐‐ ‐‐‐ ‐0.1 ‐‐‐ –1g 1f 0.05 ‐‐‐ ‐0.4 ‐‐‐ –1g 2f 0.02 0.01 ‐1.4 0.001 –1g 2f 1d 0.31 ‐‐‐ ‐1.6 0.001 –1g 2f 2d ‐‐‐ ‐‐‐ ‐8.0 ‐‐‐ –1g 2f 3d ‐0.04 ‐0.02 ‐15.4 0.012 cc‐pV5Z none 12.68 3.96 139.3 1.255 –1h ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 1g ‐‐‐ ‐‐‐ ‐0.1 ‐‐‐ –1h 2g ‐‐‐ ‐‐‐ ‐0.2 ‐‐‐ –1h 2g 1f ‐‐‐ ‐‐‐ ‐0.2 ‐‐‐ –1h 2g 2f 0.01 0.01 ‐0.5 ‐0.001 –1h 2g 3f 0.01 0.01 ‐1.3 0.001 –1h 2g 3f 1d ‐‐‐ 0.01 ‐1.3 0.001 –1h 2g 3f 2d ‐0.01 ‐‐‐ ‐3.9 ‐0.001 –1h 2g 3f 3d ‐0.02 ‐0.01 ‐12.6 0.007 –1h 2g 3f 4d ‐0.04 ‐0.02 ‐15.2 0.013

O2 (3Σg) cc‐pVDZ none 12.35 ‐0.09 135.3 1.235 –1d 0.29 0.14 ‐27.9 0.060 cc‐pVTZ none 12.52 ‐0.10 133.8 1.232 –1f ‐0.01 ‐‐‐ ‐1.4 0.006 –1f 1d 0.17 0.09 ‐13.4 0.021 –1f 2d 0.28 0.15 ‐28.0 0.064 cc‐pVQZ none 12.57 ‐0.09 133.9 1.230 –1g ‐‐‐ ‐‐‐ ‐0.2 ‐‐‐


77


Molecule Basis Set Truncation IP EA(aug) D0 ReO2 (

3Σg) cc‐pVQZ –1g 1f ‐‐‐ ‐‐‐ ‐0.7 ‐‐‐ –1g 2f ‐‐‐ ‐‐‐ ‐1.4 0.004 –1g 2f 1d 0.05 0.03 ‐3.8 0.004 –1g 2f 2d 0.18 0.12 ‐22.5 0.050 –1g 2f 3d 0.23 0.14 ‐28.8 0.070 cc‐pV5Z none 12.59 ‐0.07 133.5 1.229 –1h ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 1g ‐‐‐ ‐‐‐ ‐0.1 ‐‐‐ –1h 2g ‐‐‐ ‐‐‐ ‐0.2 ‐‐‐ –1h 2g 1f ‐‐‐ ‐‐‐ ‐0.1 ‐‐‐ –1h 2g 2f ‐‐‐ ‐‐‐ ‐0.9 0.002 –1h 2g 3f ‐‐‐ ‐‐‐ ‐1.0 0.004 –1h 2g 3f 1d 0.01 0.01 ‐1.5 0.004 –1h 2g 3f 2d 0.10 0.08 ‐12.2 0.017 –1h 2g 3f 3d 0.20 0.13 ‐25.2 0.056 –1h 2g 3f 4d 0.22 0.14 ‐27.5 0.063

78

Table 4.6 B3LYP atomization energies (kcal/mol) and optimized bond lengths (Å) for CH4 and SiH4 (both

1A1); relative differences are listed below the full basis set values.

CH4 SiH4

Basis Set Truncation ΣD0 Re ΣD0 Recc‐pVDZ none 385.9 1.100 296.9 1.496 –1d ; 1p ‐8.6 0.002 ‐14.6 0.002cc‐pVTZ none 393.1 1.088 302.6 1.483 –1f ; 1d ‐0.5 ‐‐‐ ‐1.0 0.002 –1f 1d ; 1d 1p ‐0.8 ‐0.001 ‐1.9 ‐0.001 –1f 2d ; 1d 2p ‐9.8 0.001 ‐21.4 0.016cc‐pVQZ none 393.4 1.088 304.1 1.480 –1g ; 1f ‐0.1 ‐‐‐ ‐0.2 ‐‐‐ –1g 1f ; 1f 1d ‐0.2 ‐‐‐ ‐0.4 ‐‐‐ –1g 2f ; 1f 2d ‐0.7 ‐‐‐ ‐1.0 ‐‐‐ –1g 2f 1d ; 1f 2d 1p ‐0.8 ‐‐‐ ‐1.3 ‐‐‐ –1g 2f 2d ; 1f 2d 2p ‐3.1 ‐0.002 ‐7.0 ‐0.008 –1g 2f 3d ; 1f 2d 3p ‐9.7 0.002 ‐22.6 0.012cc‐pV5Z none 393.3 1.088 304.6 1.479 –1h ; 1g ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 1g ; 1g 1f ‐0.1 ‐‐‐ ‐‐‐ ‐‐‐ –1h 2g ; 1g 2f ‐0.1 ‐‐‐ ‐0.1 ‐‐‐ –1h 2g 1f ; 1g 2f 1d ‐0.1 ‐‐‐ ‐0.1 ‐‐‐ –1h 2g 2f ; 1g 2f 2d ‐0.3 ‐‐‐ ‐0.3 ‐‐‐ –1h 2g 3f ; 1g 2f 3d ‐0.6 ‐‐‐ ‐0.8 0.001 –1h 2g 3f 1d ; 1g 2f 3d 1p ‐0.7 ‐‐‐ ‐0.8 0.001 –1h 2g 3f 2d ; 1g 2f 3d 2p ‐1.4 ‐0.001 ‐6.7 ‐0.007 –1h 2g 3f 3d ; 1g 2f 3d 3p ‐6.9 ‐0.001 ‐19.8 0.006 –1h 2g 3f 4d ; 1g 2f 3d 4p ‐9.6 0.003 ‐22.9 0.013

79

Table 4.7 B3LYP optimized geometries (Å and degrees) for NH3 and PH3 (both 1A1); relative

differences are listed below the full basis set values.

NH3 PH3

Basis Set Truncation Re AHNH Re AHPH

cc‐pVDZ none 1.025 104.33 1.435 93.09 –1d ; 1p ‐0.005 9.01 0.012 1.49cc‐pVTZ none 1.014 106.64 1.421 93.40 –1f ; 1d 0.001 ‐0.39 ‐‐‐ ‐0.09 –1f 1d ; 1d 1p ‐0.005 2.93 ‐0.002 0.36 –1f 2d ; 1d 2p ‐0.006 8.08 0.027 0.99cc‐pVQZ none 1.013 106.75 1.418 93.45 –1g ; 1f ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1g 1f ; 1f 1d ‐‐‐ ‐0.05 ‐‐‐ ‐0.02 –1g 2f ; 1f 2d ‐‐‐ ‐0.30 ‐‐‐ ‐0.05 –1g 2f 1d ; 1f 2d 1p ‐0.002 1.25 0.001 0.15 –1g 2f 2d ; 1f 2d 2p ‐0.010 6.20 ‐0.005 0.97 –1g 2f 3d ; 1f 2d 3p ‐0.005 8.22 0.027 1.07cc‐pV5Z none 1.012 107.19 1.418 93.45 –1h ; 1g ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 1g ; 1g 1f ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 2g ; 1g 2f ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 2g 1f ; 1g 2f 1d ‐‐‐ ‐0.02 ‐‐‐ ‐0.01 –1h 2g 2f ; 1g 2f 2d ‐‐‐ ‐0.40 ‐‐‐ ‐0.03 –1h 2g 3f ; 1g 2f 3d 0.001 ‐0.47 0.001 ‐0.03 –1h 2g 3f 1d ; 1g 2f 3d 1p ‐‐‐ 0.36 0.001 0.07 –1h 2g 3f 2d ; 1g 2f 3d 2p ‐0.006 3.88 ‐0.005 1.12 –1h 2g 3f 3d ; 1g 2f 3d 3p ‐0.009 7.66 0.018 1.32 –1h 2g 3f 4d ; 1g 2f 3d 4p ‐0.005 8.09 0.026 1.27

80

Table 4.8 B3LYP optimized geometries (Å and degrees) for H2O and H2S (both 1A1); relative

differences are listed below the full basis set values.

H2O H2S

Basis Set Truncation Re AHOH Re AHSH

cc‐pVDZ none 0.969 102.69 1.358 92.25 –1d ; 1p 0.021 4.28 0.022 1.98cc‐pVTZ none 0.961 104.53 1.345 92.53 –1f ; 1d 0.001 ‐0.07 ‐‐‐ ‐0.06 –1f 1d ; 1d 1p ‐0.001 2.67 ‐0.002 0.58 –1f 2d ; 1d 2p 0.013 5.22 0.038 1.45cc‐pVQZ none 0.960 104.98 1.343 92.54 –1g ; 1f ‐‐‐ ‐0.01 ‐‐‐ ‐‐‐ –1g 1f ; 1f 1d ‐‐‐ 0.03 ‐‐‐ 0.02 –1g 2f ; 1f 2d ‐‐‐ ‐0.08 ‐‐‐ ‐0.01 –1g 2f 1d ; 1f 2d 1p ‐0.001 0.39 ‐‐‐ 0.16 –1g 2f 2d ; 1f 2d 2p ‐0.001 4.46 ‐0.001 1.12 –1g 2f 3d ; 1f 2d 3p 0.016 4.79 0.038 1.41cc‐pV5Z none 0.961 105.19 1.341 92.51 –1h ; 1g ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 1g ; 1g 1f ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h 2g ; 1g 2f ‐‐‐ ‐0.01 ‐‐‐ ‐0.01 –1h 2g 1f ; 1g 2f 1d ‐‐‐ ‐‐‐ ‐‐‐ ‐0.02 –1h 2g 2f ; 1g 2f 2d ‐‐‐ 0.02 ‐‐‐ ‐0.03 –1h 2g 3f ; 1g 2f 3d ‐‐‐ ‐0.06 0.001 ‐0.06 –1h 2g 3f 1d ; 1g 2f 3d 1p ‐‐‐ ‐0.01 0.001 ‐0.09 –1h 2g 3f 2d ; 1g 2f 3d 2p ‐‐‐ 2.86 ‐0.001 1.20 –1h 2g 3f 3d ; 1g 2f 3d 3p 0.007 4.94 0.029 1.60 –1h 2g 3f 4d ; 1g 2f 3d 4p 0.014 4.91 0.037 1.55

81

Table 4.9 Percent CPU time and average time saved computing B3LYP single‐point energies with truncated basis sets, relative to the full basis sets.

Basis Set Truncation CH4 NH3 H2O SiH4 PH3 H2S Average

cc‐pVDZ none ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1d ; 1p 7 12 4 9 9 6 8cc‐pVTZ none ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1f ; 1d 28 21 13 29 23 15 22 –1f 1d ; 1d 1p 34 11 15 35 30 19 24 –1f 2d ; 1d 2p 40 32 20 42 36 25 33cc‐pVQZ none ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1g ; 1f 55 59 74 49 69 78 64 –1g 1f ; 1f 1d 68 73 80 71 78 83 76 –1g 2f ; 1f 2d 79 80 84 80 84 86 82 –1g 2f 1d ; 1f 2d 1p 83 82 85 83 86 87 84 –1g 2f 2d ; 1f 2d 2p 84 83 85 86 88 88 86 –1g 2f 3d ; 1f 2d 3p 86 85 86 87 89 89 87cc‐pV5Z none ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ –1h ; 1g 59 60 60 54 56 56 58 –1h 1g ; 1g 1f 77 78 37 69 73 73 68 –1h 2g ; 1g 2f 89 89 90 82 85 85 87 –1h 2g 1f ; 1g 2f 1d 92 93 96 87 88 96 92 –1h 2g 2f ; 1g 2f 2d 95 96 97 90 95 97 95 –1h 2g 3f ; 1g 2f 3d 96 97 98 95 97 97 97 –1h 2g 3f 1d ; 1g 2f 3d 1p 97 98 98 96 97 98 97 –1h 2g 3f 2d ; 1g 2f 3d 2p 98 98 98 97 98 98 98 –1h 2g 3f 3d ; 1g 2f 3d 3p 98 98 98 97 98 98 98 –1h 2g 3f 4d ; 1g 2f 3d 4p 98 98 98 98 98 98 98

82

CHAPTER 5

SYSTEMATIC RECONTRACTION OF THE CORRELATION CONSISTENT BASIS SETS FOR DENSITY

FUNCTIONAL THEORY CALCULATIONS†

5.1 Introduction

With correlated ab initio approaches, such as coupled cluster theory with single, double,

and perturbative triple excitations, CCSD(T),35,38 a high level of accuracy in properties such as

energetics can be achieved (e.g., atomization energies within 1.0 kcal/mol from reliable

experiment) when paired with a large basis set. However, CCSD(T) becomes so costly in terms

of computational time, memory, and disk space that this method is generally restricted to

molecules composed of no more than 10‐15 non‐hydrogen atoms with a modest‐sized basis

set. A practical alternative is density functional theory (DFT),44,45,52 which includes correlation

energy without the additional computational overhead of correlated ab initio methods.

One of the challenges in using DFT is determining the best and most appropriate

functional for a problem of interest. There are hundreds of density functionals available, each

designed for different tasks or with respect to a test set of atoms and molecules. Because of the

increasing number of density functionals available and the popularity of DFT, a quick and

efficient means to determine the intrinsic error of a density functional is needed.

The use of systematically constructed basis sets has proven to be a valuable tool in

† This entire chapter is adapted from B.P Prascher and A.K. Wilson, “The Behaviour of Density Functionals with Respect to Basis Set. V. Recontraction of Correlation Consistent Basis Sets.” Mol. Phys. 2007, 105(19‐22), 2899, with permission from Taylor & Francis.

83

determining the successes and failures of ab initio methods.122,133,176 For example, the

monotonic convergence towards the complete basis set (CBS) limit exhibited by a number of

molecular properties computed with the correlation consistent basis sets in Hartree‐Fock (HF),

DFT, and electron correlation methods, as the basis set size increases, allows for the systematic

elucidation of the intrinsic error of ab initio methods. Unlike numerical methods for

determining the HF CBS limit, which tend to be computationally expensive for systems larger

than two atoms, accurate approximations of the CBS limit by extrapolating a series of

properties computed with the correlation consistent basis sets can be done at the HF level for

molecules of many atoms.

Recently, a set of polarization consistent basis sets (pc‐n, where n = 1‐4) specifically

optimized for DFT were developed.143‐145,147 Wang and Wilson showed that using the pc‐n basis

sets in DFT calculations do not necessarily result in an improvement in the accuracy of

properties computed using the correlation consistent basis sets.155,156 For example, Wang and

Wilson showed that the performance of the correlation consistent basis sets is similar to that of

the pc‐n basis sets within 0.10 kcal/mol in computed energetics for a test set of 17 first row,

main group molecules.155 Yet, the correlation consistent basis sets have fewer basis functions

than the comparable polarization consistent basis set beyond the double‐ζ level.

The impact of employing segmented contraction versus general contraction schemes in

DFT has been investigated by Jensen.147 A segmented contraction does not include all the basis

functions of a given symmetry type in a basis set, whereas a general contraction does.19 Jensen

concluded that core functions are typically not important for either valence orbital descriptions

or for energetic properties computed with DFT that are largely impacted by the valence space

84

such as ionization potentials and atomization energies. This means that if a segmented

contraction scheme is used, high exponent basis functions need not be included in the

contracted valence functions; or, anaglogously, if a general contraction scheme is used, the

core functions do not need large contraction coefficients in the valence basis functions. In a

subsequent paper, Jensen discussed how the contraction of a basis set (independent of the

type of contraction scheme used) in DFT computations tends to impact computed properties

more than the choice of primitive functions.146 The overall conclusion of his papers was that so‐

called core‐pruned basis sets should be sufficient for valence energetic properties in DFT

computations, and that optimization of the contraction coefficients rather than the primitive

basis functions leads to a greater impact on the energetic properties. As a result of their

construction, the contracted valence basis functions in the correlation consistent basis sets

have small weights on the core functions, making them a type of core‐pruned basis sets.

Another important effect to examine when studying atomization energies is basis set

superposition error (BSSE).152,177‐179 In the linear combination of atomic orbitals (LCAO)

approximation, the interaction between two subunits is described by the overlap of their

respective basis sets. In the overlap region, there is a better description of the orbital space

than within the individual subunits. This leads to interaction energies that are too high,

compared with a more even description of the orbital space. To account for this effect, the ex

post facto Boys‐Bernardi correction is typically employed:178

AB AB A B (5.1)

where AB represents the dissociation energy of the system composed of subunits A and B

(or atomization energy of a molecule). The terms A and B are total energies of the

85

respective subunits computed in the presence of the basis sets of both subunits. Applying (5.1)

is a way of accounting for the over‐described regions of space by the overlapping basis sets; in

effect, smoothing out the basis set description so that the entire molecule is evenly described

by the basis set. As the basis set increases in size (i.e. becomes more complete) the BSSE

decreases. For example, the BSSE for the CO2 atomization energy in BLYP computations was

7.11 kcal/mol at the cc‐pVDZ level, 2.25 kcal/mol at the cc‐pVTZ level, and 1.36 kcal/mol at the

quadruple‐ζ level.152 It has also been reported that correcting for BSSE in computations with the

cc‐pVnZ basis sets can lead to monotonic behavior in computed atomization energies.156

Based on the observations of Jensen regarding basis set contractions in DFT, the success

of recontracting the correlation consistent basis sets for the recovery of scalar relativistic

effects (discussed in Chapter 3), and in continuing with the theme of Chapter 4 in determining

the basis sets requirements for DFT, this chapter reports on reoptimization of the contractions

of the correlation consistent basis sets specifically for DFT computations. The effect on the

convergent behavior of various energetic properties including atomization energies, vertical

ionization potentials, and vertical electron affinities is examined in addition to its impact on

optimized geometries. Further, the effects of incorporating diffuse functions in the basis sets

and the removal of BSSE are examined.


As in Chapter 4, the Becke 1988 gradient‐corrected exchange functional (B)164 has been

coupled with the Lee‐Yang‐Parr correlation functional (LYP)165 in pure DFT computations, and

the Becke three‐parameter exchange scheme (B3)166 has been coupled with the LYP functional

86

in hybrid DFT computations. In the latter case, the third parameterization of the Vosko‐Wilk‐

Nusair local correlation functional (VWN‐3)167 is used.

The correlation consistent basis sets, cc‐pVnZ, were employed from double‐ to

quintuple‐ζ quality (n = D, T, Q, and 5). Recontraction of the s and p functions in these basis sets

was performed in a similar manner to that described in Chapter 3: 1) all high angular

momentum functions were removed from the basis set; 2) the s and p functions were

uncontracted; 3) computations of atomic energies with BLYP and B3LYP were performed; and

4) the atomic orbital (AO) coefficients from Kohn‐Sham (KS) orbital analyses were then taken as

the new contraction coefficients. After replacing the high angular momentum functions (d, f,

etc.) from the original cc‐pVnZ basis sets, the new basis sets are denoted cc‐pVnZ[rc], where rc

stands for recontracted. Diffuse functions have also been added to these recontracted basis

sets to form augmented basis sets; the diffuse functions are unaltered from the original

aug‐cc‐pVnZ basis sets.

Recontraction of the correlation consistent basis sets was performed with the Molpro

software package180 using a variable grid size to achieve better than mEh convergence in the

total atomic energy. The Gaussian 03 software suite174 was then employed with the default grid

(75,302) to perform the BLYP and B3LYP computations with the recontracted basis sets. All

computations were performed using the spin‐unrestricted formalism, and the impact of BSSE

has been examined. Two extrapolation schemes including the Halkier et al. two‐point (3.6)133

and Feller three‐point exponential approach (3.2)132 were employed in estimating KS limits.

87


For oxygen and fluorine, there are two sets of contraction coefficients possible for the

2p manifold: the singly‐occupied p eigenstate or the doubly‐occupied p eigenstate. For

example, the oxygen p manifold has the electronic configuration , which leads to two

possibilities for contraction coefficients since and are degenerate. The properties

computed in this investigation used contracted coefficients from the doubly‐occupied

eigenstates since they produce lower atomic energies.

Several energetic properties were computed and examined using the recontracted

correlation consistent basis sets including vertical ionization potentials, electron affinities

(vertical attachment energies). The effect of adding diffuse functions to the recontracted basis

sets has been examined with respect to electron affinities and atomization energies.

Recontracted basis sets that include the diffuse functions of the original aug‐cc‐pVnZ basis

sets95,99,100 were used.

Molecules from the test set of Wang et al.152,154‐156 have been included in this study,

specifically H2, N2, F2, HF, CO, CO2, O3, H2O, and CH3OH. In addition, several molecules have

been added to augment the set, including O2, Ne2, CH, NH, OH, CH2 (singlet and triplet states),

NH3, CH4, (H2O)2, and (HF)2.

5.3.1 Atoms

Table 5.1 displays the changes in total energies of hydrogen and the first row atoms B‐

Ne using the recontracted basis sets compared with the original basis sets. The change in

energy is greatest at the double‐ and triple‐ζ levels and tends to increase as the atom size

88

increases. The energy changes are smaller for B3LYP than for BLYP, with the largest changes

occurring at the double‐ζ level for both methods. For example, the total energy for the carbon

atom is 3.160 mEh lower with the recontracted basis sets as compared with the total energy

arising from original basis set using BLYP, but is only 2.015 mEh lower using B3LYP. The

differences in energy grow to 7.285 and 4.626 mEh for BLYP and B3LYP, respectively, for the

neon atom.

Table 5.1 also helps to gauge the importance of specific density functional optimizations

of the contraction. The B3LYP recontraction of the basis sets, denoted cc‐pVnZ[B3LYP], was

utilized to compute BLYP energies, and the BLYP recontraction, denoted cc‐pVnZ[BLYP], was

used to calculate B3LYP energies. Overall, when the same functional is used for both the

recontraction and the calculation of the total energy, lower energies are obtained. The

exceptions occur for the B3LYP/cc‐pVDZ[BLYP] calculations for oxygen and fluorine, where the

total energies are lower than those obtained using the B3LYP recontraction by ‐0.030 and

‐0.178 mEh, respectively. The largest differences in total energies for the B3LYP and BLYP

recontraction schemes are observed for the BLYP calculations, with differences of 0.475 and

0.707 mEh, for oxygen and fluorine, respectively.

The ionization potentials and electron affinities of the first row atoms B‐Ne and H using

the double‐ and triple‐ζ basis sets are given in Table 5.2. Overall, comparing the cc‐pVnZ basis

sets to the recontracted sets shows very little difference in ionization potentials for each atom.

The largest differences occur at the double‐ζ level. For oxygen and fluorine, the impact of

recontraction increases the ionization potential by 0.06 and 0.07 eV, respectively, using BLYP.

These increases are above the threshold of chemical accuracy (0.04 eV). Using B3LYP, the

89

oxygen and fluorine ionization potentials are increased by 0.04 and 0.05 eV, respectively, by

recontraction. The magnitude of changes between the original basis sets and the recontracted

sets is smaller using B3LYP than using BLYP.

Electron affinities listed in Table 5.2 are computed using the diffuse functions from the

respective aug‐cc‐pVnZ basis sets. Electron affinities are typically not affected by recontraction

as much as ionization potentials, and do not display differences as compared with the original

basis sets beyond 0.02 eV (approximately 0.5 mEh). In general, atomic electron affinities are

affected by recontraction only at the double‐ζ level. Note that Ne, which does not bind an

electron, has been included for comparison with the other first row atoms. Both BLYP and

B3LYP correctly predict a negative value for the Ne electron affinity.

5.3.2 Molecules

Table 5.3 lists total energies, vertical ionization potentials, and electron affinities

(vertical attachment energies) for H2, B2, C2, N2, O2, F2, and Ne2 computed with fixed geometries

from experiment: rHH = 0.74144 Å, rBB = 1.590 Å, rCC = 1.2425 Å, rNN = 1.09768 Å, and

rOO = 1.20752 Å, rFF = 1.41193 Å, and rNeNe = 3.150 Å.170 In general, the largest change between

the original and recontracted basis sets occurs at the double‐ζ level. The changes in the BLYP

total energies due to recontraction are, for example 3.704, 5.088, 7.719, 10.268, 6.412, 9.457,

and 14.569 mEh, respectively, for H2, B2, C2, N2, O2, F2, and Ne2. Compared with atomic

computations, the ionization potentials for these molecules tend to be less sensitive to

recontraction of the basis set, and vary at most between the original and recontracted basis

sets in Table 5.3 by 0.15 eV with BLYP and 0.11 eV with B3LYP (each occurring in the Ne2

90

complex). Electron affinities included in Table 5.3 are also affected the most at the double‐ζ

level, with the exception of H2 at the quintuple‐ζ level. The electron affinities of N2 and Ne2 are

included for the sake of comparison despite the fact that these systems do not stably bind an

electron. However, both BLYP and B3LYP are qualitatively correct in predicting a negative value

for the electron affinity of these two systems.

The double‐ and triple‐ζ optimized geometries of the molecule test set used are shown

in Table 5.4. The largest deviation in bond lengths and angles occurs at the double‐ζ level for

each molecule of Table 5.4, but are typically less than 0.01 Å and 0.1°, respectively. Any

observed changes at the quadruple‐ and quintuple‐ζ levels are not considered significant and

are not included in Table 5.4. The bond lengths reported for (H2O)2 and (HF)2 are the

intermolecular hydrogen bonds. The difference between the original and recontracted double‐ζ

bond lengths are 0.024 Å and 0.016 Å for BLYP and B3LYP, respectively. The largest differences

between the original and recontracted basis set bond lengths and angles in the molecules of

Table 5.4 are consistently at the double‐ζ level for both BLYP and B3LYP. Bond lengths tend to

vary less than 0.01 Å at the double‐ζ level, except in a few cases: Ne2, (H2O)2, and (HF)2, which

are all weakly‐bound systems. Angles tend to vary only on the order of a few tenths of a degree

between the two families of basis sets.

To further illustrate how geometries are affected little by recontraction, Figure 5.1

shows a comparison between the O2 Σ potential energy curve computed with the original

and recontracted cc‐pVDZ basis set. Note that the minima reside in the same location (a

consequence of the geometry not being significantly affected by recontraction, even at the

double‐ζ level), but energies vary by almost 0.01 Eh. The energy difference between the two

91

curves increases as the inter‐nuclear distance increases.

Atomization energies (ΣD0) are tabulated in Table 5.5 for BLYP and B3LYP. In general,

atomization energies are affected most at the double‐ζ level. For example, the atomization

energy of O2 is impacted by 1.55 kcal/mol using B3LYP and 2.02 kcal/mol using BLYP at the

double‐ζ level. For O3, the impact of recontraction on the computed atomization energy is 2.81

kcal/mol using B3LYP and 3.59 kcal/mol using BLYP at the double‐ζ level. Other molecules like

NH3, CH4, and CH3OH have atomization energies that are affected by more than 2.0 kcal/mol

using both BLYP and B3LYP at the double‐ζ level, with CH4 affected most at 3.91 kcal/mol using

BLYP.

As shown in Table 5.5 and Table 5.6, not all atomization energies using the cc‐pVnZ basis

sets converge monotonically as the basis set size is increased, a point which was investigated by

Wang et al.125,154,156 Employing the recontracted cc‐pVnZ basis sets does not remedy non‐

Figure 5.1 The BLYP potential energy curve of O2 (

3Σg) computed with the original andrecontracted correlation consistent basis sets.

92

monotonic behavior for all of the molecules. However, for some species, as shown in Figure 5.2

for CO2 and O3, the recontracted basis sets can change the behavior observed for the

atomization energies as the basis set size is increased. For example, the atomization energies of

CO2 at the triple‐ and quadruple‐ζ levels using the original cc‐pVnZ basis sets change by almost

0.5 kcal/mol, whereas the energies resulting from use of the equivalent recontracted basis sets

only change by almost 0.05 kcal/mol. For O3 atomization energies computed with the cc‐pVnZ

basis sets, the behavior is oscillatory. Using the cc‐pVnZ[rc] basis sets, the oscillatory behavior is

not observed, and the behavior resembles that of CO2. Often, as evidenced by Figure 5.2, a

series of properties calculated at the double‐ through quadruple‐ζ levels exhibits monotonic

behavior respect to increasing basis set size.

5.3.3 Kohn‐Sham Limits

Table 5.6 lists extrapolated atomization energies for each molecule in Table 5.5 using

Figure 5.2 A comparison of the BLYP atomization energies computed with the original andrecontracted correlation consistent basis sets.

93

the two‐ and three‐point approaches described earlier. In the two‐point scheme, the first

extrapolation is performed using double‐ and triple‐ζ points, (DT), while the second

extrapolation uses the triple‐ and quadruple‐ζ points, (TQ). In the three‐point scheme, the

first extrapolation uses double‐ through quadruple‐ζ points, (DTQ), and the second uses

triple‐ through quintuple‐ζ points, (TQ5).

In general, KS limits should not be calculated for each extrapolation approach for all of

the molecules in Table 5.6 because of the non‐monotonic convergence of the atomization

energies with respect to increasing basis set size. Out of the 20 molecules represented, the

atomization energies of H2, N2, O2, F2, CO, CO2, O3, CH4, and (H2O)2 cannot be extrapolated

using all four extrapolation techniques employed. Typically, if non‐monotonic behavior is

observed in BLYP, then it is also observed in B3LYP. The obvious exceptions to this trend in

Table 5.6 are N2, O2, F2, and O3. The N2 molecule is unique in that its B3LYP atomization

energies can be extrapolated with all four extrapolation approaches, but those of BLYP cannot

be extrapolated with all four schemes. The F2 molecule is even more peculiar since only the

BLYP atomization energies employing the original cc‐pVnZ basis sets can be extrapolated.

Recontraction affects the computed F2 atomization energy ies such that the extrapolations of

the BLYP values are not possible beyond (DT). Further, all B3LYP computations on the F2

atomization energy cannot be extrapolated due to non‐monotonic behavior beyond the (DT)

scheme.

The largest difference between the original and recontracted KS limit extrapolations

occurs for the (DT) scheme for most molecules studied. For O3, CH4, and NH3, the differences

are 1.99, 1.57 and 1.26 kcal/mol for BLYP and 1.54, 1.16, and 1.02 kcal/mol for B3LYP,

94

respectively. This is expected since the largest impact on energetics due to recontraction occurs

at the double‐ζ level. The largest molecule of this study, CH3OH, varies in (DT) extrapolations

of its computed atomization energy by 0.91 kcal/mol and 0.71 kcal/mol at the BLYP and B3LYP

levels, respectively. In BLYP computations on Ne2, only the double‐ζ level atomization energy is

affected by recontraction (cf. Table 5.5). Therefore, the Ne2 molecule experiences the smallest

difference between the cc‐pVnZ and cc‐pVnZ[rc] basis set extrapolated energies, with

differences ranging from 0.01 to 0.33 kcal/mol in magnitude.

5.3.4 Basis Set Superposition Error

As discussed, BSSE can be corrected using the Boys‐Bernardi counterpoise correction.

Table 5.7 and Figure 5.3 demonstrate how counterpoise corrections can be important in

restoring monotonic behavior expected with the correlation consistent basis sets using BLYP.

Recall from Table 5.6 that BLYP atomization energies of N2, F2, and O3 computed with cc‐pVnZ

or cc‐pVnZ[rc] were not extrapolated due to non‐monotonic behavior except with the (DT)

scheme. Correcting each ζ‐level for BSSE improves the behavior so that it is monotonic for N2

with both the original and recontracted basis sets (cf. Table 5.7). For F2, employing the

counterpoise correction does not alleviate the non‐monotonic behavior in the recontracted

basis set computations; it also does not affect any convergent behavior of the original basis set

computations. The O3 computed atomization energies, when corrected for BSSE, become

monotonic only in the original basis set computations. Unfortunately, including a counterpoise

correction in CO computations with BLYP does not result in monotonic behavior. Non‐

monotonic behavior, even after a counterpoise correction has been applied, is not surprising in

95

light of the work of Wang et al.,152,156 who also found that some first row molecules did not

display monotonic behavior after accounting for BSSE (e.g. CO).

Accounting for BSSE using the counterpoise correction lowers the atomization energy at

the double‐ and triple‐ζ levels for each of the five systems in Figure 5.3. For F2, the difference

between the BLYP/cc‐pVDZ[rc] energy and the cc‐pVDZ[rc] energy including a counterpoise

correction is almost 4.5 kcal/mol, while in CO and O3 the difference is almost 2.5 kcal/mol. At

the triple‐ζ level, F2 has an energy difference of almost 2.0 kcal/mol when counterpoise‐

corrected; and at the quadruple‐ζ level, the difference is almost 1.0 kcal/mol. The N2 molecule

displays the smallest difference between the cc‐pVnZ[rc] atomization energies and their

respective counterpoise‐corrected values.

5.3.5 Diffuse Functions

The impact of including diffuse functions in BLYP and B3LYP computations employing

the cc‐pVnZ basis sets has been investigated by Wang.156 Eight molecules that do not display

monotonic behavior with either the original or recontracted cc‐pVnZ basis sets have been

reexamined after the addition of diffuse functions to the recontracted basis sets. The BLYP‐ and

B3LYP‐computed atomization energies of H2, N2, O2, F2, CO, CO2, O3, and CH4 using the

aug‐cc‐pVnZ[rc] basis sets with and without a counterpoise correction for BSSE are included in

Table 5.8 and are compared with aug‐cc‐pVnZ results.

It is evident that the inclusion of diffuse functions improves the convergence of

atomization energies as compared with the original cc‐pVnZ basis sets (cf. Table 5.6), which is in

accord with the conclusions of Wang.156 However, including diffuse functions for each of the

96

eight molecules of Table 5.8 does not completely alleviate non‐monotonic behavior as the basis

set size increases. Specifically, O2, F2, and O3 with BLYP and B3LYP do not display monotonic

behavior, nor does CH4 with BLYP when the aug‐cc‐pVnZ basis sets are utilized. For each of

these systems with the exception of F2, the convergence problem in both BLYP and B3LYP is

eliminated by using the aug‐cc‐pVnZ[rc] basis sets. Further employing a counterpoise correction

to the F2 molecule alleviates non‐monotonic behavior. It is now clear that simply employing

diffuse functions does not correct non‐monotonic behavior for any of the first row systems

studied. Rather, recontraction, in addition, to diffuse functions and a correction for BSSE is

needed to achieve monotonic behavior with the correlation consistent basis sets.

Figure 5.4 shows the atomization energies of H2, N2, F2, CO, CO2, and O3 computed with

the augmented and recontracted basis sets (a counterpoise correction is added to the latter)

compared with the pc‐n basis sets. The pc‐n data is from Wang,156 which does not include O2 or

CH4. Examining the plots, it is evident that the three different basis set families converge on

similar KS limits. The molecule F2 stands out in that it is the system in which the pc‐n basis sets

display erratic behavior, while the correlation consistent basis sets result in monotonic

convergence with respect to increasing basis set size.

5.4 Conclusions

Fine‐tuning the cc‐pVnZ basis sets for DFT through optimization of the contraction

coefficients for BLYP and B3LYP has lead to a revised family of cc‐pVnZ basis sets, which are

referred to as cc‐pVnZ[rc] basis sets. For atomic energies, the most significant impact of

recontraction occurs at the double‐ζ level and, overall, is on the order of several mEh for each

97

first row atom and increases as the atom size increases. Atomic ionization potentials and

electron affinities are also affected most at the double‐ζ level with recontraction of the basis

sets. Several small first row molecules have been used in the benchmarking of these

recontracted basis sets and show that the double‐ and triple‐ζ level ionization potentials and

atomization energies are most affected (with and without a correction for BSSE). Overall,

optimized bond lengths are not affected by recontraction by more than 0.01 Å at any ζ‐level.

When non‐monotonic behavior is observed in computed atomization energies,

recontraction alone does not provide a means to restoring monotonic behavior when

employing the cc‐pVnZ basis sets. Previous studies have shown that both the contraction of the

cc‐pVnZ basis sets and BSSE effects can affect the convergence of computed properties with

increasing basis set size. When a counterpoise correction is applied to the original cc‐pVnZ basis

sets, for example, it has been shown to give monotonic behavior in some molecules where

computational atomization energies converge non‐monotonically with increasing basis set size.

Applying the Boys‐Bernardi counterpoise correction to the cc‐pVnZ[rc] basis sets can result in

monotonic behavior for some molecules that do not converge monotonically when the

standard correlation consistent basis sets are used.

In comparing the counterpoise corrected aug‐cc‐pVnZ[rc] basis sets to the pc‐n basis

sets, the recontracted sets exhibit monotonic behavior for some molecular properties where

the pc‐n sets do not (i.e. the atomization energy of F2). In general, the recontracted basis sets

perform as well as the pc‐n sets for the molecules investigated, despite the fact that the pc‐n

primitive basis functions are explicitly optimized for DFT, while only the contraction coefficients

of the cc‐pVnZ[rc] set have been optimized for DFT.

98

Employing the Boys‐Bernardi counterpoise correction with these recontracted basis sets

can correct non‐monotonic behavior in energetic properties at the double‐, triple‐, and

quadruple‐ζ levels. When a counterpoise correction on the cc‐pVnZ[rc] basis sets does not

alleviate non‐monotonic behavior, the addition of diffuse functions acts to correct non‐

convergent behavior in most systems of this study. However, it is necessary to also include the

Boys‐Bernardi scheme with the aug‐cc‐pVnZ[rc] basis sets in the case of F2 to achieve

monotonic behavior with BLYP and B3LYP atomization energies. Thus, all three corrections:

recontracting the basis sets, adding diffuse functions, and including the counterpoise correction

can alleviate non‐monotonic behavior in the molecules of this study. Further, it is

recommended that the cc‐pVnZ[rc] basis sets be used in BLYP and B3LYP computations due to

their smaller size as compared with the pc‐n basis sets.

99

Figure 5.3 A comparison of BLYP atomization energies computed with the original and recontracted correlation consistent basis sets, and the recontracted basis sets with basis set superposition error (BSSE) removed.

100

Figure 5.4 A comparison of BLYP atomization energies computed with the pc‐n, aug‐cc‐pVnZ, and aug‐cc‐pVnZ[rc] basis sets with basis set superposition error (BSSE) removed.

101

Table 5.1 A comparison of the BLYP and B3LYP total energies (Eh) of the first row atoms (including hydrogen) computed with the original and recontracted cc‐pVnZ basis sets; relative differences to the original basis sets are shown in mEh under [rc].

BLYP B3LYP

Atom ζ cc‐pVnZ [rc] [B3LYP]b cc‐pVnZ [rc] [BLYP]b

Ha D ‐0.496403 ‐0.405 0.026 ‐0.501258 ‐0.228 0.027(1S) T ‐0.497555 ‐0.002 ‐‐‐ ‐0.502156 ‐0.001 ‐‐‐ Q ‐0.497781 ‐‐‐ ‐‐‐ ‐0.502346 ‐‐‐ ‐‐‐ 5 ‐0.497889 ‐0.001 ‐‐‐ ‐0.502428 ‐‐‐ ‐‐‐B D ‐24.647768 ‐2.438 0.104 ‐24.660873 ‐1.552 0.123(2P) T ‐24.651259 ‐1.063 0.062 ‐24.663759 ‐0.627 0.062 Q ‐24.652475 ‐0.950 0.042 ‐24.664792 ‐0.590 0.043 5 ‐24.653437 ‐0.215 0.009 ‐24.665439 ‐0.130 0.008Ca D ‐37.837837 ‐3.160 0.139 ‐37.851975 ‐2.015 0.151(3P) T ‐37.845501 ‐1.060 0.060 ‐37.858575 ‐0.627 0.062 Q ‐37.847806 ‐0.946 0.040 ‐37.860592 ‐0.594 0.041 5 ‐37.849077 ‐0.241 0.010 ‐37.861508 ‐0.148 0.010Na D ‐54.572571 ‐4.203 0.190 ‐54.589136 ‐2.686 0.189(4S) T ‐54.586935 ‐1.080 0.061 ‐54.601781 ‐0.642 0.062 Q ‐54.590896 ‐0.953 0.039 ‐54.605328 ‐0.604 0.040 5 ‐54.592689 ‐0.267 0.011 ‐54.606704 ‐0.165 0.011Oa D ‐75.054526 ‐4.838 0.475 ‐75.068499 ‐3.034 ‐0.030(3P) T ‐75.080286 ‐1.097 0.071 ‐75.091864 ‐0.653 0.052 Q ‐75.087251 ‐0.954 0.043 ‐75.098201 ‐0.607 0.034 5 ‐75.090069 ‐0.306 0.012 ‐75.100485 ‐0.191 0.013Fa D ‐99.713359 ‐5.718 0.707 ‐99.726602 ‐3.536 ‐0.178(2P) T ‐99.752932 ‐1.145 0.082 ‐99.762867 ‐0.685 0.044 Q ‐99.763470 ‐0.981 0.046 ‐99.772527 ‐0.625 0.032 5 ‐99.767416 ‐0.324 0.013 ‐99.775818 ‐0.203 0.014

Ne D ‐128.894856 ‐7.285 0.322 ‐128.909463 ‐4.626 0.324(1S) T ‐128.951679 ‐1.226 0.065 ‐128.961856 ‐0.748 0.065 Q ‐128.966632 ‐0.998 0.041 ‐128.975665 ‐0.636 0.041 5 ‐128.972023 ‐0.346 0.015 ‐128.980262 ‐0.217 0.015

a. The original cc‐pVnZ energies come from Wang, X., The Performance of Density Functionals with Respect to the Correlation Consistent Basis Sets. Ph.D. thesis, University of North Texas (2004).

b. The recontracted set employed was optimized for the bracketed density functional, and the differences are relative to the [rc] values.

102

Table 5.2 A comparison of the BLYP and B3LYP ionization potentials and electron affinities (eV) of the first row atoms (including hydrogen) computed with the original and recontracted cc‐pVnZ basis sets; relative differences to the original basis sets are shown under [rc].

BLYP B3LYP

Atom ζ IP IP[rc] EAa EA[rc]a IP IP[rc] EAa EA[rc]a

H D 13.51 0.01 0.82 0.01 13.64 0.01 0.89 0.01 T 13.54 ‐‐‐ 0.84 ‐‐‐ 13.66 ‐‐‐ 0.91 ‐‐‐

B D 8.64 ‐0.03 0.42 ‐‐‐ 8.76 ‐0.03 0.45 ‐‐‐ T 8.62 ‐‐‐ 0.43 ‐‐‐ 8.74 ‐‐‐ 0.46 ‐‐‐

C D 11.36 ‐0.01 1.34 ‐‐‐ 11.52 ‐0.01 1.37 ‐‐‐ T 11.38 ‐‐‐ 1.37 ‐0.02 11.53 ‐‐‐ 1.37 ‐‐‐

N D 14.37 0.02 0.29 0.02 14.57 0.01 0.17 0.02 T 14.46 ‐‐‐ 0.34 ‐‐‐ 14.64 ‐‐‐ 0.21 ‐‐‐

O D 13.84 0.06 1.80 0.02 13.92 0.04 1.68 0.01 T 14.07 ‐‐‐ 1.81 ‐‐‐ 14.09 ‐‐‐ 1.68 ‐‐‐

F D 17.30 0.07 3.69 0.02 17.44 0.05 3.56 0.02 T 17.60 ‐‐‐ 3.67 ‐‐‐ 17.66 ‐‐‐ 3.53 ‐‐‐

Ne D 21.13 0.13 ‐6.73 ‐‐‐ 21.32 0.09 ‐6.72 ‐‐‐ T 21.53 ‐‐‐ ‐5.67 ‐‐‐ 21.62 ‐‐‐ ‐5.65 ‐‐‐

a. Diffuse functions from the aug‐cc‐pVnZ basis sets have been included in these calculations.

103

Table 5.3 A comparison of the BLYP and B3LYP total energies (Eh), ionization potentials, and electron affinities (eV) of the first row dimers (including hydrogen) computed with the original and recontracted cc‐pVnZ basis sets; relative differences to the original basis sets are shown in mEh (total energies) and eV under [rc].

Method Molecule ζ Energy [rc] IP IP[rc] EAa EA[rc]a

BLYP H2 D ‐1.161740 ‐3.704 16.33 0.03 ‐1.26 ‐0.01 T ‐1.169568 ‐0.051 16.38 ‐‐‐ ‐1.10 ‐‐‐ Q ‐1.170136 ‐‐‐ 16.39 ‐‐‐ ‐0.97 ‐‐‐ 5 ‐1.170293 0.003 16.39 ‐‐‐ ‐0.59 ‐0.26 B2 D ‐49.400390 ‐5.088 9.67 ‐0.01 1.76 ‐‐‐ T ‐49.409266 ‐2.010 9.65 ‐‐‐ 1.77 ‐‐‐ Q ‐49.411852 ‐1.763 9.66 ‐‐‐ 1.77 ‐‐‐ 5 ‐49.413586 ‐0.427 9.66 ‐‐‐ 1.77 ‐‐‐ C2 D ‐75.901894 ‐7.719 12.80 0.03 3.56 ‐0.01 T ‐75.923863 ‐2.051 12.86 ‐‐‐ 3.49 ‐‐‐ Q ‐75.929160 ‐1.795 12.87 ‐‐‐ 3.48 ‐‐‐ 5 ‐75.931444 ‐0.499 12.87 ‐‐‐ 3.47 ‐‐‐ N2 D ‐109.517908 ‐10.268 15.11 0.07 ‐1.93 ‐0.04 T ‐109.555976 ‐2.276 15.31 ‐‐‐ ‐2.57 ‐0.01 Q ‐109.565159 ‐1.869 15.34 ‐‐‐ ‐2.71 ‐0.01 5 ‐109.568630 ‐0.547 15.35 ‐‐‐ ‐3.00 ‐0.01 O2 D ‐150.328040 ‐6.412 12.27 0.07 0.36 ‐0.01 T ‐150.377180 ‐2.477 12.43 ‐‐‐ 0.01 ‐‐‐ Q ‐150.391251 ‐1.915 12.48 ‐‐‐ ‐0.04 ‐‐‐ 5 ‐150.396247 ‐0.629 12.50 ‐‐‐ ‐0.05 ‐‐‐ F2 D ‐199.508997 ‐9.457 14.99 0.09 1.38 ‐0.05 T ‐199.586196 ‐2.487 15.23 ‐‐‐ 0.81 ‐‐‐ Q ‐199.606002 ‐1.923 15.29 ‐‐‐ 0.72 ‐‐‐ 5 ‐199.612895 ‐0.626 15.32 ‐‐‐ 0.72 ‐‐‐ Ne2 D ‐257.789685 ‐14.569 17.67 0.15 ‐4.02 ‐0.17 T ‐257.903443 ‐2.453 18.28 ‐‐‐ ‐4.14 0.01 Q ‐257.933299 ‐1.994 18.44 ‐‐‐ ‐3.63 ‐‐‐ 5 ‐257.943896 ‐0.694 18.52 ‐‐‐ ‐2.63 ‐‐‐

B3LYP H2 D ‐1.173323 ‐2.596 16.65 ‐‐‐ ‐1.58 0.02 T ‐1.179997 ‐0.040 16.67 ‐‐‐ ‐1.38 ‐‐‐ Q ‐1.180536 ‐‐‐ 16.67 ‐‐‐ ‐1.25 ‐‐‐ 5 ‐1.180689 0.002 16.67 ‐‐‐ ‐0.87 ‐0.26


104


Method Molecule ζ Energy [rc] IP IP[rc] EAa EA[rc]a

B3LYP B2 D ‐49.415388 ‐3.190 9.81 ‐0.01 1.45 ‐‐‐ T ‐49.423119 ‐1.174 9.79 ‐‐‐ 1.45 ‐‐‐ Q ‐49.425421 ‐1.098 9.79 ‐‐‐ 1.45 ‐‐‐ 5 ‐49.426633 ‐0.261 9.79 ‐‐‐ 1.45 ‐‐‐ C2 D ‐75.888709 ‐5.275 12.49 0.03 4.40 ‐0.01 T ‐75.907630 ‐1.238 12.51 ‐‐‐ 4.37 ‐‐‐ Q ‐75.912336 ‐1.125 12.51 ‐‐‐ 4.36 ‐‐‐ 5 ‐75.914014 ‐0.312 12.51 ‐‐‐ 4.36 ‐‐‐ N2 D ‐109.533243 ‐6.899 15.64 0.06 ‐1.66 ‐0.02 T ‐109.568318 ‐1.403 15.79 ‐‐‐ ‐1.75 ‐‐‐ Q ‐109.576828 ‐1.195 15.81 ‐‐‐ ‐1.43 ‐‐‐ 5 ‐109.579654 ‐0.342 15.82 ‐‐‐ ‐1.16 ‐‐‐ O2 D ‐150.334040 ‐3.601 12.88 0.05 0.47 ‐‐‐ T ‐150.380947 ‐1.506 12.97 ‐‐‐ 0.15 ‐‐‐ Q ‐150.394121 ‐1.218 13.00 ‐‐‐ 0.10 ‐‐‐ 5 ‐150.398385 ‐0.391 13.01 ‐‐‐ 0.08 ‐‐‐ F2 D ‐199.513377 ‐5.715 15.72 0.06 1.46 ‐0.02 T ‐199.586223 ‐1.513 15.87 ‐‐‐ 0.94 ‐‐‐ Q ‐199.604711 ‐1.212 15.89 ‐‐‐ 0.85 ‐‐‐ 5 ‐199.610627 ‐0.386 15.91 ‐‐‐ 0.83 ‐‐‐ Ne2 D ‐257.818924 ‐9.252 18.80 0.11 ‐4.36 ‐0.12 T ‐257.923824 ‐1.496 19.29 ‐‐‐ ‐4.28 ‐‐‐ Q ‐257.951405 ‐1.270 19.41 ‐‐‐ ‐3.86 ‐‐‐ 5 ‐257.960467 ‐0.435 19.46 ‐‐‐ ‐3.33 ‐‐‐

a. Diffuse functions from the aug‐cc‐pVnZ basis sets have been included in these calculations.

105

Table 5.4 The BLYP and B3LYP optimized geometries (Å and degrees) of several first row molecules computed with the original and recontracted basis sets; relative differences to the original basis set are shown under [rc].

BLYP B3LYP

Molecule ζ cc‐pVnZ [rc] cc‐pVnZ [rc]

H2a Re D 0.767 ‐0.008 0.762 ‐0.006 T 0.747 ‐‐‐ 0.743 ‐‐‐

N2a Re D 1.117 ‐0.003 1.104 ‐0.003 T 1.103 ‐‐‐ 1.091 ‐‐‐ O2 Re D 1.235 0.001 1.209 ‐‐‐ T 1.232 ‐‐‐ 1.206 ‐‐‐

F2a Re D 1.444 0.009 1.410 0.006 T 1.433 ‐‐‐ 1.398 ‐‐‐

Ne2 Re D 2.534 0.011 2.532 0.008 T 2.822 ‐0.002 2.802 ‐0.001

CH Re D 1.154 ‐0.005 1.142 ‐0.004 T 1.134 ‐‐‐ 1.124 ‐‐‐

NH Re D 1.068 ‐0.004 1.055 ‐0.003 T 1.052 ‐‐‐ 1.041 ‐‐‐

OH Re D 0.996 ‐0.003 0.984 ‐0.002 T 0.985 ‐‐‐ 0.975 ‐‐‐

HFa Re D 0.938 ‐0.003 0.927 ‐0.002 T 0.933 ‐‐‐ 0.923 ‐‐‐

CH2 (1A1) Re D 1.139 ‐0.006 1.127 ‐0.004 T 1.120 ‐‐‐ 1.110 ‐‐‐ A D 99.19 0.55 100.27 0.36 T 100.83 ‐0.01 101.62 ‐0.01

CH2 (3B1) Re D 1.098 ‐0.004 1.091 ‐0.003 T 1.083 ‐‐‐ 1.077 ‐‐‐ A D 134.87 ‐0.19 134.40 ‐0.09 T 135.38 0.03 134.91 0.02

COa Re D 1.147 ‐0.003 1.135 ‐0.002 T 1.138 ‐‐‐ 1.126 ‐‐‐

CO2a Re D 1.182 ‐0.002 1.167 ‐0.002

T 1.174 ‐‐‐ 1.160 ‐‐‐


106


BLYP B3LYP

Molecule ζ cc‐pVnZ [rc] cc‐pVnZ [rc]

O3a Re D 1.295 0.003 1.260 0.001 T 1.292 ‐‐‐ 1.256 ‐‐‐ A D 117.90 ‐0.03 117.95 0.01 T 118.00 ‐‐‐ 118.14 ‐‐‐

H2Oa Re D 0.980 ‐0.003 0.969 ‐0.002

T 0.972 ‐‐‐ 0.961 ‐‐‐ A D 101.77 0.47 102.74 0.32 T 103.75 0.05 104.50 0.06

NH3 Re D 1.035 ‐0.004 1.025 ‐0.003 T 1.023 ‐‐‐ 1.014 ‐‐‐ AHNH D 103.36 0.66 104.74 ‐0.37 T 105.79 ‐0.01 106.46 ‐‐‐

CH4 Re D 1.107 ‐0.004 1.100 ‐0.003 T 1.094 ‐‐‐ 1.088 ‐‐‐

(H2O)2 Re(O…H) D 2.149 ‐0.016 2.148 0.010

Ci T 2.254 0.001 2.245 0.001 (HF)2 Re(F

…H) D 1.960 0.024 1.996 0.016 C2h T 2.138 ‐0.002 2.154 ‐0.002

CH3OHa Re(OH) D 0.979 ‐0.003 0.968 ‐0.002

Cs T 0.971 ‐‐‐ 0.961 ‐‐‐ A(COH) D 106.79 0.12 107.49 0.07 T 107.77 0.02 108.51 ‐0.03


107

Table 5.5 The BLYP and B3LYP atomization energies (kcal/mol) of several first row molecules computed with the original and recontracted basis sets; relative differences to the original basis set are shown under [rc].

BLYP B3LYP

Molecule ζ ΣD0 [rc] ΣD0 [rc]

H2a D 100.17 1.57 101.11 1.19 T 103.27 0.03 103.93 0.02 Q 103.35 ‐0.01 104.03 ‐‐‐ 5 103.31 ‐0.01 104.02 ‐‐‐

N2a D 231.22 0.97 219.32 0.91 T 236.49 0.08 225.45 0.08 Q 237.27 ‐0.03 226.38 ‐0.01 5 237.19 0.01 226.43 ‐‐‐ O2 D 135.90 ‐2.02 121.32 ‐1.55 T 134.27 0.17 121.43 0.12 Q 134.28 ‐‐‐ 121.75 ‐‐‐ 5 133.86 ‐‐‐ 121.57 ‐‐‐

F2a D 50.67 ‐1.05 36.32 ‐0.90 T 49.21 0.12 36.54 0.10 Q 48.41 ‐0.03 36.04 ‐0.03 5 47.78 ‐0.02 35.63 ‐0.02

Ne2 D 0.18 ‐0.03 0.21 ‐0.02 T 0.02 ‐‐‐ 0.05 0.01 Q ‐0.09 ‐‐‐ ‐0.05 ‐0.01 5 ‐0.14 0.01 ‐0.14 0.01

CH D 78.69 0.75 78.45 0.61 T 81.32 ‐‐‐ 81.01 0.01 Q 81.69 ‐0.05 81.38 ‐0.02 5 81.77 ‐0.01 81.48 ‐0.01

NH D 80.27 0.93 78.77 0.78 T 84.10 0.03 82.50 0.02 Q 84.82 ‐0.03 83.16 ‐0.02 5 85.09 ‐0.01 83.41 ‐0.01

OH D 97.92 1.03 96.58 0.83 T 103.02 0.04 101.46 0.03 Q 104.18 ‐0.01 102.49 ‐‐‐


108


BLYP B3LYP

Molecule Ζ ΣD0 [rc] ΣD0 [rc]

OH 5 104.66 ‐0.01 102.90 ‐0.01 HFa D 126.08 0.89 124.58 0.71 T 133.07 0.06 131.30 0.03 Q 134.74 ‐0.01 132.78 ‐0.01 5 135.43 ‐0.01 133.37 ‐0.02

CH2 (1A1) D 164.23 1.81 164.93 1.43 T 169.31 0.02 169.77 0.03 Q 169.94 ‐0.08 170.40 ‐0.04 5 170.06 ‐0.01 170.56 ‐0.01

CH2 (3B1) D 174.81 1.68 176.85 1.29 T 179.36 0.01 181.15 0.03 Q 179.75 ‐0.08 181.55 ‐0.05 5 179.75 ‐0.01 181.60 ‐0.01

COa D 256.35 0.89 248.42 0.81 T 259.25 0.11 252.12 0.08 Q 259.76 ‐0.06 252.84 ‐0.03 5 259.26 0.01 252.56 ‐‐‐

CO2a D 390.46 0.68 375.18 0.70

T 394.00 0.23 380.63 0.16 Q 394.46 ‐0.17 381.49 ‐0.10 5 393.52 ‐0.01 380.95 ‐‐‐

O3a D 168.32 ‐3.59 133.76 ‐2.81 T 166.96 0.33 135.68 0.24 Q 167.28 ‐‐‐ 136.58 0.02 5 166.79 0.03 136.45 0.03

H2Oa D 207.84 1.98 206.14 1.60

T 216.81 0.07 214.85 0.05 Q 218.91 ‐0.01 216.78 ‐‐‐ 5 219.81 0.01 217.57 0.01

NH3 D 268.97 3.26 268.01 2.66 T 278.74 0.08 277.50 0.08 Q 280.45 ‐0.07 279.12 ‐0.04 5 281.14 ‐‐‐ 279.78 ‐0.01


109


BLYP B3LYP

Molecule Ζ ΣD0 [rc] ΣD0 [rc]

CH4 D 382.05 3.91 385.76 2.98 T 389.98 0.05 393.10 0.07 Q 390.19 ‐0.13 393.41 ‐0.08 5 389.96 ‐0.02 393.34 ‐0.01

(H2O)2 D ‐18.39 ‐0.31 ‐20.23 ‐0.15 T ‐22.08 0.04 ‐23.22 0.04 Q ‐22.69 ‐0.04 ‐24.30 ‐‐‐ 5 ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐

(HF)2 D ‐2.45 ‐0.44 ‐4.11 ‐0.25 T ‐7.13 0.02 ‐7.91 0.01 Q ‐8.63 ‐0.03 ‐9.12 ‐0.02 5 ‐9.59 ‐‐‐ ‐9.84 ‐‐‐

CH3OHa D 469.32 2.54 468.45 2.02

T 478.88 0.17 478.25 0.14 Q 480.34 ‐0.15 479.76 ‐0.10 5 480.40 0.01 480.00 ‐‐‐


110

Table 5.6 Two‐ and three‐point extrapolated BLYP and B3LYP Kohn‐Sham limits of atomization energies (kcal/mol) computed with the original and recontracted basis sets. Non‐monotonic behavior is denoted by ‘‐‐‐’.

BLYP B3LYP

Molecule Extrapolation ΣD0 [rc] ΣD0 [rc]

H2 P∞(DT) 104.62 103.95 105.11a 104.65 P∞(TQ) 103.38 103.36 104.11a 104.06 P∞(DTQ) 103.35 103.34 104.03 104.03 P∞(TQ5) ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ N2 P∞(DT) 238.71 238.41 228.04a 227.76 P∞(TQ) 237.59 237.52 227.05a 226.72 P∞(DTQ) 237.40 237.36 226.54a 226.52 P∞(TQ5) ‐‐‐ ‐‐‐ 226.43 226.44 O2 P∞(DT) 133.58 134.68 121.48 122.30 P∞(TQ) 134.28 134.21 121.89 121.84 P∞(DTQ) ‐‐‐ ‐‐‐ ‐‐‐ 121.78 P∞(TQ5) ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ F2 P∞(DT) 48.61 49.21 36.64a 37.15 P∞(TQ) 48.07 47.98 35.82 35.74 P∞(DTQ) 47.37 ‐‐‐ ‐‐‐ ‐‐‐ P∞(TQ5) 45.42 ‐‐‐ ‐‐‐ ‐‐‐

Ne2 P∞(DT) ‐0.05 ‐0.03 ‐0.01 0.00 P∞(TQ) ‐0.13 ‐0.13 ‐0.10 ‐0.10 P∞(DTQ) ‐0.28 ‐0.61 ‐0.32 ‐0.55 P∞(TQ5) ‐0.18 ‐0.17 ‐0.49 ‐0.28

CH P∞(DT) 82.43 82.11 82.09 81.85 P∞(TQ) 81.85 81.78 81.54 81.50 P∞(DTQ) 81.75 81.71 81.44 81.42 P∞(TQ5) 81.79 81.82 81.51 81.54

NH P∞(DT) 85.72 85.36 84.07 83.77 P∞(TQ) 85.13 85.07 83.44 83.41 P∞(DTQ) 84.99 84.99 83.31 83.31 P∞(TQ5) 85.25 85.30 83.55 83.58

OH P∞(DT) 105.17 104.79 103.51 103.2 P∞(TQ) 104.67 104.64 102.93 102.91 P∞(DTQ) 104.53 104.58 102.77 102.81


111


BLYP B3LYP


OH P∞(TQ5) 104.99 105.02 103.15 103.17 HF P∞(DT) 136.07 135.71 134.13a 133.87 P∞(TQ) 135.45 135.40 133.86a 133.37 P∞(DTQ) 135.25 135.29 133.20a 133.22 P∞(TQ5) 135.91 135.94 133.72 133.73

CH2 (1A1) P∞(DT) 171.45 170.71 171.81 171.25 P∞(TQ) 170.21 170.09 170.67 170.59 P∞(DTQ) 170.03 169.97 170.50 170.46 P∞(TQ5) 170.09 170.15 170.61 170.66

CH2 (3B1) P∞(DT) 181.27 180.59 182.96 182.46

P∞(TQ) 179.92 179.79 181.71 181.63 P∞(DTQ) 179.79 179.70 181.59 181.54 P∞(TQ5) 179.75 179.77 181.60 181.63

CO P∞(DT) 260.49 260.25 253.68a 253.44 P∞(TQ) 259.98 259.84 253.37a 253.07 P∞(DTQ) 259.87 259.76 253.02 252.97 P∞(TQ5) ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐

CO2 P∞(DT) 395.52 395.53 382.93a 382.86 P∞(TQ) 394.65 394.32 382.11a 381.64 P∞(DTQ) 394.53 394.29 381.64 381.47 P∞(TQ5) ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ O3 P∞(DT) 166.38 168.37 136.48a 138.01 P∞(TQ) 167.46 167.28 137.24a 136.88 P∞(DTQ) ‐‐‐ ‐‐‐ 137.53 136.70 P∞(TQ5) ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐

H2O P∞(DT) 220.65 219.85 218.52a 217.92 P∞(TQ) 219.82 219.75 218.19a 217.57 P∞(DTQ) 219.57 219.71 217.33a 217.44 P∞(TQ5) 220.50 220.58 218.13 218.17

NH3 P∞(DT) 282.86 281.60 281.50 280.48 P∞(TQ) 281.17 281.04 279.79 279.71 P∞(DTQ) 280.81 280.86 279.44 279.49 P∞(TQ5) 281.62 281.85 280.24 280.38


112


BLYP B3LYP


CH4 P∞(DT) 393.32 391.75 396.20 395.04 P∞(TQ) 390.28 390.07 393.54 393.40 P∞(DTQ) 390.20 390.06 393.43 393.34 P∞(TQ5) ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐

(H2O)2 P∞(DT) ‐9.10 ‐8.88 ‐24.47 ‐24.36 P∞(TQ) ‐9.26 ‐9.31 ‐24.76 ‐24.77 P∞(DTQ) ‐9.33 ‐9.56 ‐24.92 ‐25.04 P∞(TQ5) ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐

(HF)2 P∞(DT) ‐9.10 ‐8.88 ‐9.51 ‐9.39 P∞(TQ) ‐9.26 ‐9.31 ‐9.63 ‐9.66 P∞(DTQ) ‐9.33 ‐9.56 ‐9.68 ‐9.81 P∞(TQ5) ‐11.34 ‐11.00 ‐10.88 ‐10.72

CH3OH P∞(DT) 483.01 482.08 482.48a 481.73 P∞(TQ) 480.98 480.67 480.96a 480.20 P∞(DTQ) 480.62 480.40 480.13a 479.90 P∞(TQ5) 480.43 480.46 480.06 480.12


113

Table 5.7 The BLYP and B3LYP atomization energies (kcal/mol) and their extrapolated Kohn‐Sham limits of four molecules corrected for basis set superposition error (BSSE).

cc‐pVnZ cc‐pVnZ[rc]

Molecule ζ / P∞ ΣD0 − BSSE ΣD0 − BSSE

N2a D 229.69 230.65 T 236.11 235.97 Q 236.94 236.98 5 237.10 237.12 P∞(DT) 238.95 238.21 P∞(TQ) 237.27 237.72 P∞(DTQ) 236.75 237.22 P∞(TQ5) 237.28 237.15F2 D 45.07 44.98 T 47.53 47.63 Q 47.64 47.73 5 47.69 47.69 P∞(DT) 48.57 48.75 P∞(TQ) 47.71 47.80 P∞(DTQ) 47.64 47.73 P∞(TQ5) 47.74 ‐‐‐

COa D 253.69 254.74 T 258.44 258.06 Q 259.22 259.49 5 259.14 259.37 P∞(DT) 260.44 259.46 P∞(TQ) 259.79 260.53 P∞(DTQ) 259.37 260.57 P∞(TQ5) ‐‐‐ ‐‐‐O3 D 163.57 162.37 T 165.45 166.96 Q 166.03 167.27 5 166.31 167.07 P∞(DT) 166.24 168.89 P∞(TQ) 166.46 167.50 P∞(DTQ) 166.29 167.29 P∞(TQ5) 166.56 ‐‐‐


114

Table 5.8 A comparison of BLYP and B3LYP atomization energies (kcal/mol) and their extrapolated Kohn‐Sham values using the aug‐cc‐pVnZ, aug‐cc‐pVnZ[rc], and aug‐cc‐pVnZ[rc] basis sets corrected for basis set superposition error (BSSE).

BLYP B3LYP

Molecule ζ / P∞ ΣD0 [rc] [rc]‐BSSE ΣD0 [rc] [rc]‐BSSE

H2a D 99.86 101.44 101.43 100.90 102.08 102.08 T 103.08 103.10 103.09 103.82 103.83 103.83 Q 103.25 103.25 103.24 103.98 103.98 103.97 5 103.29 103.28 103.28 104.01 104.01 104.01 P∞(DT) 104.44 103.79 103.79 105.05 104.57 104.57 P∞(TQ) 103.37 103.35 103.35 104.10 104.08 104.08 P∞(DTQ) 103.26 103.26 103.26 103.99 103.99 103.99 P∞(TQ5) 103.30 103.29 103.29 104.02 104.02 104.02

N2a D 230.53 231.68 231.28 219.36 220.41 219.95 T 236.25 236.21 236.09 225.42 225.42 225.28 Q 237.33 237.26 237.22 226.52 226.47 226.43 5 237.38 237.37 237.37 226.60 226.59 226.59 P∞(DT) 238.66 238.12 238.12 227.97 227.53 227.53 P∞(TQ) 238.12 238.02 238.04 227.32 227.24 227.27 P∞(DTQ) 237.58 237.57 237.57 226.76 226.75 226.75 P∞(TQ5) 237.38 237.39 237.39 226.61 226.61 226.61O2 D 133.13 131.88 131.08 120.29 119.34 118.45 T 133.12 133.23 133.05 120.82 120.91 120.70 Q 133.93 133.90 133.85 121.66 121.65 121.58 5 133.97 133.96 133.95 121.72 121.71 121.70 P∞(DT) 133.11 133.80 133.88 121.05 121.56 121.65 P∞(TQ) 134.52 134.39 134.44 122.27 122.19 122.23 P∞(DTQ) ‐‐‐ 134.58 134.40 ‐‐‐ 122.31 122.15 P∞(TQ5) ‐‐‐ 133.97 133.97 ‐‐‐ 121.72 121.72

F2a D 46.70 46.14 45.55 33.94 33.43 32.79 T 47.68 47.76 47.60 35.59 35.66 35.47 Q 47.77 47.72 47.67 35.68 35.63 35.59 5 47.77 47.73 47.72 35.68 35.66 35.66 P∞(DT) 48.09 48.45 48.46 36.28 36.60 36.60 P∞(TQ) 47.84 47.69 47.72 35.75 35.61 35.67 P∞(DTQ) 47.78 ‐‐‐ 47.67 35.69 ‐‐‐ 35.59


115


BLYP B3LYP

Molecule ζ / P∞ ΣD0 [rc] [rc]‐BSSE ΣD0 [rc] [rc]‐BSSE

F2a P∞(TQ5) ‐‐‐ ‐‐‐ 47.86 ‐‐‐ ‐‐‐ 35.79

COa D 254.29 254.73 254.22 247.20 247.65 247.09 T 258.66 258.06 257.91 251.36 251.37 251.20 Q 259.16 259.09 259.04 252.47 252.43 252.37 5 259.19 259.18 259.17 252.53 252.52 252.51 P∞(DT) 260.50 259.47 259.46 253.11 252.93 252.93 P∞(TQ) 259.52 259.84 259.87 253.28 253.20 253.23 P∞(DTQ) 259.22 259.54 259.54 252.87 252.85 252.85 P∞(TQ5) 259.19 259.19 259.19 252.53 252.53 252.53

CO2a D 386.01 386.41 385.57 372.82 373.28 372.30

T 392.31 391.75 391.43 379.21 379.25 378.85 Q 393.38 393.17 393.10 380.85 380.72 380.59 5 393.32 393.28 393.27 380.86 380.83 380.82 P∞(DT) 394.96 393.99 393.90 381.90 381.76 381.61 P∞(TQ) 394.16 394.21 394.32 382.05 381.79 381.87 P∞(DTQ) 393.60 393.69 393.77 381.42 381.20 381.23 P∞(TQ5) ‐‐‐ 393.29 393.28 380.86 380.84 380.86

O3a D 165.88 163.72 162.76 134.41 132.79 131.66 T 165.82 166.01 165.76 135.39 135.54 135.24 Q 166.98 166.93 166.82 136.67 136.65 136.57 5 166.98 166.99 166.95 136.70 136.71 136.71 P∞(DT) 165.79 166.97 167.02 135.80 136.69 136.76 P∞(TQ) 167.83 167.60 167.60 137.60 137.46 137.54 P∞(DTQ) ‐‐‐ 167.55 167.41 ‐‐‐ 137.40 137.35 P∞(TQ5) ‐‐‐ 166.99 166.97 ‐‐‐ 136.72 136.73

CH4 D 382.41 384.75 384.22 386.06 387.97 387.40 T 389.85 389.23 389.11 392.67 392.68 392.56 Q 389.91 389.72 389.67 393.26 393.14 393.09 5 389.91 389.86 389.85 393.32 393.29 393.28 P∞(DT) 392.99 391.12 391.17 395.45 394.66 394.73 P∞(TQ) 392.99 390.07 390.07 393.69 393.48 393.49 P∞(DTQ) 389.95 389.78 389.74 393.32 393.19 393.16 P∞(TQ5) ‐‐‐ 389.93 389.94 393.32 393.36 393.37


116

CHAPTER 6

THE DEVELOPMENT OF S‐BLOCK CORRELATION CONSISTENT BASIS SETS†

6.1 Introduction

In correlated ab initio methodology, achieving the exact solution to the non‐relativistic

Schrödinger equation requires two criteria: 1) a complete N‐electron treatment and 2) a

complete (infinite) basis set.7,18 The former can be achieved through a full configuration

interaction (FCI) treatment, which is rarely applicable to any but the smallest of electronic

systems (i.e. less than ten electrons). Even in small systems where FCI can be applied, the use of

modest‐sized basis sets is a limiting factor in achieving the exact solution of the Schrödinger

equation since FCI scales factorially in basis set size. Further, the use of an infinite basis set is

computationally infeasible. In a practical sense, the N‐electron treatment and basis set must be

chosen considering both the desired level of accuracy and the computational feasibility.

As already discussed in Chapter 2, the use of an infinite basis set is desirable but

infeasible, and the best method of approximating the exact solution to the Schrödinger

equation would be to select as large a basis set as possible. However, this quickly limits both

the system size and the N‐electron treatment that can be employed. The correlation consistent,

polarized valance basis sets (cc‐pVnZ)81‐109 were specifically designed so that ab initio methods

recover correlation energy more quickly than other basis sets (e.g. basis sets developed with

Hartree‐Fock). Further, the correlation consistent basis sets offer a way to circumvent the

† Work reported in this chapter was performed in collaboration with Kirk A. Peterson, David E. Woon, and Thom H. Dunning, Jr.

117

infeasibility of infinite basis sets by systematically approaching the complete basis set (CBS)

limit as the ζ‐level, or cardinal number, increases.

Correlation consistent basis sets have been developed for much of the main group,

including all‐electron basis sets for the first three rows and pseudopotential basis sets for the

fourth and fifth rows.81‐109 Further, correlation consistent basis sets have been developed for

the 3d and 4d transition metals, while development of basis sets for the heavier transition

metals continues. There are several families of correlation consistent basis sets beyond the

standard valence sets, including the tight d basis sets for inner valence correlation in second

row, main group atoms [cc‐pV(n+d)Z];101 the augmented basis sets for long range interactions

[aug‐cc‐pVnZ];95,99,100 the core‐valence basis sets for sub‐valence correlation energy [cc‐pCVnZ

and cc‐pwCVnZ];81,98,103 and the Douglas‐Kroll (DK) basis sets for accurate recovery of scalar

relativistic energy [cc‐pVnZ‐DK].142 Finally, there has been recent development of correlation

consistent basis sets for resolution of the identity (RI) and explicitly correlated methods.104,181

Correlation consistent basis sets for Li, Be, Na, and Mg93 have been available for some

time via the extensible Basis Set Exchange.182,183 However, only unpublished basis sets through

quadruple‐ζ quality have ever been available. Further, no tight d basis sets are available through

the Basis Set Exchange for the Na and Mg atoms. The goal of this chapter is to present the

correlation consistent basis sets for Li, Be, Na, and Mg, while at the same time revisiting the

construction of their polarization functions (d, f, g, and h) using the robust Broyden‐Fletcher‐

Goldfarb‐Shanno (BFGS)184 numerical optimization technique. The preexisting Hartree‐Fock (HF)

sp basis sets are left intact, and sp sets for the quintuple‐ζ level are presented. The new basis

sets formed from the BFGS‐optimized polarization functions have already found a niche in

118

benchmarking the correlation consistent Composite Approach (ccCA)73‐77 and studying the ab

initio potential energy surfaces of BeOH and MgOH.185,186 The first half of this chapter discusses

the details of building the different families of correlation consistent basis sets for s‐block

atoms, while the second half focuses on various benchmark calculations of atomic and

molecular properties utilizing these new basis sets. In particular, double‐ through quintuple‐ζ

basis sets are presented for standard valence, tight d valence (for Na and Mg only), diffuse,

core‐valence, and DK scalar relativistic computations.


The construction of correlation consistent basis sets follows that of previous work. First,

a set of HF atomic orbitals are developed, then shells of correlation consistent polarization

functions are developed. The development of the HF basis sets for Li, Be, Na, and Mg differs

from that of the main group atoms in that there are no filled valence p orbitals. However, the

inclusion of optimized p orbitals in the valence space is paramount to accurately describing the

bonding environments of these atoms in molecules. The valence p functions were optimized

using the excited states of the atoms ( for Li, Na and for Be, Mg) in multi‐configuration

self‐consistent field (MCSCF) calculations. The HF basis set was then contracted using the

general contraction scheme (see Chapter 3). The distinguishing characteristic of the Li, Be, Na,

and Mg basis sets, compared with those of the main group atoms, is the contracted valence p

function is unoccupied in the ground state. The number and type of primitive and contracted

basis functions used for Li, Be, Na, and Mg are listed in Table 6.1.

119

The polarization functions (d, f, g, and h) are optimized for the atoms using singles and

doubles configuration interaction (CISD). A challenge in developing polarization functions with

CISD for Li and Na is the one electron valence space, since no correlation energy is recovered in

valence CISD with only one electron. To circumvent the one electron valence, the Li and Na

atom polarization functions were optimized in Li2 and Na2 CISD calculations, analogous to the

way the hydrogen basis sets were originally developed.94 The polarization functions of Be and

Mg were optimized correlating only the two valence electrons. The polarization functions of

each angular quantum number were optimized using the even‐tempered expansion scheme

(3.1). The polarization functions are grouped together in shells, where each function

contributes a similar amount of correlation energy (cf. Figure 6.1).

Diffuse functions have been developed to augment the new correlation consistent basis

sets for Li, Be, Na, and Mg. Following the prescription established in developing basis sets for

main group atoms, a single diffuse function has been added for each angular momentum group

in the valence basis set. For example, a set of ( 1s 1p 1d ) diffuse functions is added to the

Table 6.1 The composition of the correlation consistent basis sets for s‐block atoms.

Atom cc‐pVDZ cc‐pVTZ cc‐pVQZ cc‐pV5Z

Li (9s 4p 1d) [3s 2p 1d]

(11s 5p 2d 1f) [4s 3p 2d 1f]

(12s 6p 3d 2f 1g) [5s 4p 3d 2f 1g]

(14s 8p 4d 3f 2g 1h) [6s 5p 4d 3f 2g 1h]

Be (9s 4p 1d) [3s 2p 1d]

(11s 5p 2d 1f) [4s 3p 2d 1f]

(12s 6p 3d 2f 1g) [5s 4p 3d 2f 1g]

(14s 8p 4d 3f 2g 1h) [6s 5p 4d 3f 2g 1h]

Na (12s 8p 1d) [4s 3p 1d]

(16s 10p 2d 1f) [5s 4p 2d 1f]

(19s 12p 3d 2f 1g) [6s 5p 3d 2f 1g]

(20s 14p 4d 3f 2g 1h) [7s 6p 4d 3f 2g 1h]

Mg (12s 8p 1d) [4s 3p 1d]

(15s 10p 2d 1f) [5s 4p 2d 1f]

(16s 12p 3d 2f 1g) [6s 5p 3d 2f 1g]

(20s 14p 4d 3f 2g 1h) [7s 6p 4d 3f 2g 1h]

120

double‐ζ basis, a set of ( 1s 1p 1d 1f ) functions is added to the triple‐ζ basis, and so on. The

diffuse functions are optimized in atomic anion calculations.

All calculations were performed using the Molpro software package.180 Energy gradients

were converged to 10‐4 Eh or better in all geometry optimizations, leading to a precision of at

least 10‐7 Eh in total energies. The reported vibrational frequencies and zero‐point energies

were computed using the harmonic oscillator approximation. Thermochemical values were

computed using enthalpies of formation from the National Institute of Standards and

Technology (NIST) database.170‐172

6.3 Basis Set Construction

6.3.1 Tight d Functions for Inner Valence Correlation

To construct the cc‐pV(n+d)Z basis sets, a single tight (high exponent) d function was

added to the double‐, triple‐, quadruple‐, and quintuple‐ζ cc‐pVnZ basis sets. All of the d

functions at each basis set level were then reoptimized in CISD calculations (using the Na2

molecule, so as to avoid the one electron valence as previously discussed, and the Mg atom).

The exponents of the cc‐pVnZ and cc‐pV(n+d)Z basis sets are plotted in Figure 6.3 for

comparison. It can be seen from the figure that there is little change in the original valence

functions upon optimization with a tight d function, and that the additional d function moves to

describe more of the inner valence space with increasing basis set size.

121

6.3.2 Core‐Valence Functions for Sub‐Valence Correlation

The importance of valence correlation energy cannot be over emphasized in the

accurate calculation of atomic and molecular properties. Moreover, the importance of core‐

valence (CV) correlation energy (the energy of interaction between valence and sub‐valence

electrons) is also to be emphasized, especially when accuracy below 1.0 kcal/mol in atomic and

molecular properties (i.e thermochemistry) is required. For that reason, CV basis sets for Li, Be,

Na, and Mg that monotonically approach the CBS limit are introduced.

The CV basis sets are optimized in an analogous manner as those for main group atoms

using atomic CISD calculations using (3.7). For the Li and Be atoms, all of the electrons are

correlated, but the 1s orbitals of Na and Mg are constrained to be doubly‐occupied (frozen) in

each correlated calculation. Following the core‐valence prescription previously discussed in

Chapter 3, shells of tight functions are added to the cc‐pVnZ basis sets of Li and Be as follows:

Figure 6.1 The incremental CISD correlation energy lowering of the Mg atom due to eachpolarization function.

0.01

0.1

1

10

100

0 1 2 3 4 5

|ΔE c

orr| (m

illihartree)

Number of Functions

Mg

3d4f5g6h

122

( 1s 1p 1d ) to cc‐pVDZ; ( 2s 2p 2d 1f ) to cc‐pVTZ; ( 3s 3p 3d 2f 1g ) to cc‐pVQZ; and ( 4s 4p 4d 3f

2g 1h ) to cc‐pV5Z. Similarly, shells of tight functions are added to the double‐ through

quadruple‐ζ basis sets of Na and Mg. To avoid near linear dependence in the s functions at the

quintuple‐ζ level for Na and Mg, an additional four s‐type functions are uncontracted from the

HF basis set and are taken to be the tight s functions for CV correlation. That is, the ( 20s 14p )

primitive set is recontracted to ( 10s 6p ). The usual ( 4p 4d 3f 2g 1h ) tight functions are then

added. As a consequence, the cc‐pCV5Z and cc‐pwCV5Z basis sets do not contain a contracted

s‐type function for the 3s orbital of Na and Mg in the HF basis set.

Plots comparing the exponents of the cc‐pVnZ and cc‐pCVnZ basis sets are found in

Figure 6.2, while plots comparing the amount of valence correlation energy recovered by the

Figure 6.2 Comparisons of the Na atom valence (cc‐pVnZ) exponents with the core‐valence(cc‐pCVnZ) and weighted core‐valence (cc‐pwCVnZ) exponents; the exponents are grouped byangular momentum.

0.001 0.010 0.100 1.000 10.000 100.000

Primitive Exponents

Valence

Core‐Valence

s

h

g

d

p

f

0.001 0.010 0.100 1.000 10.000 100.000

Primitive Exponents

Valence

Weighted Core‐Valence

s

h

g

d

p

f

123

cc‐pVnZ, cc‐pCVnZ, and cc‐pwCVnZ basis sets are found in Figure 6.4. Figure 6.5 demonstrates

the necessity of using basis sets designed for CV correlation. It is shown in atomic calculations

that if the valence‐only basis sets are used, a loss of more than 20 mEh (12.5 kcal/mol) in the CV

energy of Li and Be results, while greater than 100 mEh (62.5 kcal/mol) is lost in Na and greater

than 150 mEh (94.0 kcal/mol) is lost in Mg calculations.

6.3.3 Recontracted Basis Sets for Scalar Relativistic Computations

The necessity of systematically altering a basis set to accommodate a relativistic

Hamiltonian has been thoroughly discussed by de Jong et al. with regard to the DK

Hamiltonian.142 The problem in using basis sets contracted with non‐relativistic methods is the

improper description of the radial contraction that occurs in core s‐ and p‐type orbitals.

Figure 6.3 The spacing of the d exponents in the cc‐pVnZ basis sets compared with those inthe cc‐pV(n+d)Z basis sets.

0.01 0.10 1.00 10.00

Basis S

et

Primitive Exponent

cc‐pVnZ

cc‐pV(n+d)ZDZ

5Z

QZ

TZ

124

Specifically, atomic contractions optimized with non‐relativistic HF produce core s and p orbitals

that are too diffuse. The use of these diffuse core orbitals results in scalar relativistic energies

that are too high. To that end, a family of DK‐contracted basis sets for Li, Be, Na, and Mg are

introduced. To optimize the s‐ and p‐type contractions for use with the DK Hamiltonian, the

steps outlined in section 5 of Chapter 3 have been employed.

While the differences between the amount of scalar relativistic energy recovered with

non‐relativistic and scalar relativistic basis sets is small for Li and Be (less than 10‐4 Eh; less than

0.1 kcal/mol), the difference compounded over several atoms in a molecule, say, will become

significant (i.e. greater than 1.0 kcal/mol). Further, as Figure 6.6 shows, the scalar relativistic

energy recovered by the original contractions of the correlation consistent basis sets actually

decreases as the basis set increases in size for Li and Be. The DK‐contracted basis sets increase

the amount of scalar relativistic energy recovered with increasing basis set size. In the Na and

Mg atoms, the use of the DK contractions is necessary to achieve kcal/mol accuracy since the

Figure 6.4 CCSD(T) valence correlation energies recovered by the cc‐pVnZ (V/V), cc‐pCVnZ(V/CV), and cc‐pwCVnZ (V/wCV) basis sets.

44.000

44.500

45.000

45.500

46.000

46.500

D T Q 5

|ΔE c

orr| (m

illihartrees)

Basis Set (cc‐pVnZ)

V/V

V/CV

V/wCV

BeBe

32.800

33.000

33.200

33.400

33.600

33.800

34.000

34.200

D T Q 5

|ΔE c

orr| (m

illihartrees)


V/V

V/CV

V/wCV

Mg

125

difference between the original and recontracted scalar relativistic energies are on the order of

14 kcal/mol for Na and 22 kcal/mol for Mg.

6.4 Benchmark Computations

6.4.1 Ionization Potentials and Electron Affinities

To demonstrate the ability of the correlation consistent basis sets to correctly describe

the frontier orbital properties of the alkali and alkaline earth metal atoms, ionization potentials

and electron affinities have been computed and are listed in Table 6.2. The first and second

ionization potentials and electron affinities are computed in the typical manner. Due to the one

electron valence of Li and Na, the ionization potentials are the same as the HF computed

ionization potentials, and it is seen that there is very little difference between successive

ζ‐levels. Further, comparing the Li and Na aug‐cc‐pV5Z ionization potentials with experiment

shows errors of 0.050 eV and ‐0.188 eV, respectively. It is only when CV correlation is included

do the ionization potentials of these two atoms compare well with experiment. At the cc‐pCVDZ

level, for example, the difference between including CV correlation and not is enough to bring

the Li ionization potential within 0.04 eV of experiment. Using the cc‐pwCVDZ basis set instead

of the cc‐pCVDZ basis set makes a larger impact on the computed ionization potentials of Li, Be,

Na, and Mg. For example, employing the cc‐pCVDZ basis set in computing the Li ionization

potential shows an error relative to experiment of 0.036 eV, but the cc‐pwCVDZ basis set shows

and error of 0.027 eV. This observation is typical for the rest of the atomic ionization potentials

of Table 6.2: the cc‐pwCVnZ basis sets give results closer to the experimental value when

126

compared with the cc‐pCVnZ basis sets. Further, it has been remarked previously that both

basis sets will converge on the same CBS limit for properties of interest, and it is shown to be

true for the Li, Be, Na, and Mg atoms.

The electron affinities of Be and Mg are not reported since it is generally accepted that a

bound electron should not be stable on these atoms. The use of diffuse functions in the

computation of anionic properties is vital to achieving results that compare with experiment.

Figure 6.5 CCSD(T) core‐valence plus valence correlation energies recovered by the cc‐pVnZ(CV/V), cc‐pCVnZ (CV/CV), and cc‐pwCVnZ (CV/wCV); the 1s orbitals of Na and Mg were keptfrozen in each calculation.

0.000

5.000

10.000

15.000

20.000

25.000

30.000

35.000

40.000

45.000

50.000

D T Q 5

|ΔE c

orr| (m

illihartrees)


CV/V

CV/CV

CV/wCV

Li

40.000

50.000

60.000

70.000

80.000

90.000

100.000

D T Q 5

|ΔE c

orr| (m

illihartrees)


CV/V

CV/CV

CV/wCV

Be

0.000

50.000

100.000

150.000

200.000

250.000

300.000

350.000

D T Q 5

|ΔE c

orr| (m

illihartrees)


CV/V

CV/CV

CV/wCV

Na

0.000

50.000

100.000

150.000

200.000

250.000

300.000

350.000

400.000

D T Q 5

|ΔE c

orr| (m

illihartrees)


CV/V

CV/CV

CV/wCV

Mg

127

This claim is supported by the fact that, at the cc‐pVDZ level, the Li and Na electron affinities

are in error by ‐0.204 eV and ‐0.142 eV, respectively. When the diffuse functions are included

(aug‐cc‐pVDZ), the errors in the Li and Na electron affinities drop to ‐0.026 eV and ‐0.022 eV,

respectively. Further, the inclusion of core‐valence correlation does not have a significant

impact on the electron affinities of Li or Na at the triple‐ζ level or higher. In fact, the electron

affinities of the cc‐pV5Z, cc‐pCV5Z, and cc‐pwCV5Z basis sets are quite similar.

Figure 6.6 Comparisons of the scalar relativistic corrections to the atomic Hartree‐Fockenergy using the original contractions (diamonds) and the Douglas‐Kroll contractions (squares)with increasing basis set size.

0.650

0.655

0.660

0.665

0.670

0.675

0.680

D T Q 5

|ΔE r

el| (m

illihartree

s)


Li2.400

2.420

2.440

2.460

2.480

2.500

2.520

2.540

D T Q 5

|ΔE r

el| (m

illihartree

s)


Be

180.000

185.000

190.000

195.000

200.000

205.000

210.000

215.000

D T Q 5

|ΔE r

el| (m

illihartree

s)


Na260.000

270.000

280.000

290.000

300.000

310.000

320.000

D T Q 5

|ΔE r

el| (m

illihartree

s)


Mg

128

6.4.2 Optimized Geometries and Vibrational Frequencies

The optimized geometries of several di‐ and triatomics and their corresponding

harmonic vibrational frequencies computed with the cc‐pV(T+d)Z basis set are listed in Table

6.3 and Table 6.4, respectively. The target accuracy of computed structures relative to

experiment is 0.01 Å. The impact of core‐valence correlation is readily seen in the optimized

geometries of the diatomics. For example, the bond length of Li2 (the smallest system of Table

6.3) at the cc‐pV5Z level deviates from experiment by 0.027 Å, while the corresponding core‐

valence basis sets only deviate by 0.002 Å. A similar trend is observed in the bond lengths of LiF,

Na2, NaF, and MgF. Interestingly, it is the standard valence basis sets that more accurately

predict the bond length of MgO as opposed to the core‐valence basis sets. The difference

between experiment and theory at the cc‐pV5Z level is 0.005 Å, but is 0.011 Å at the cc‐pCV5Z

level (0.012 Å at the cc‐pwCV5Z level). Examining the bond lengths of the cc‐pwCVnZ basis sets

more closely reveals that the bond length has a very low error at the double‐ζ level, but

diverges from the experimental value as the basis set increases in size.

Another basis set effect that is observed is the general improvement in the convergence

of the bond lengths with the cc‐pV(n+d)Z basis sets relative to the cc‐pVnZ basis sets. The

largest tight d effects observed are in the bond lengths of MgO and MgF. Using the cc‐pVDZ

basis set, the bond lengths of MgO and MgF are 1.774 Å and 1.773 Å, respectively, which

decrease to 1.753 Å and 1.756 Å, respectively, when the cc‐pV(D+d)Z basis set is used. At the

higher ζ‐levels, the changes in the bond length due to the tight d function are less than 0.01 Å.

A similar effect of the tight d function is seen in the aug‐cc‐pVDZ/aug‐cc‐pV(D+d)Z bond lengths

of MgO and MgF, and in the cc‐pVDZ/cc‐pV(D+d)Z bond lengths of MgF2.

129

6.4.3 Thermochemistry

The computed enthalpies of formation for a set of twelve s‐block molecules are found in

Table 6.5. The desired accuracy of computed enthalpies of formation with respect to

experiment is ±1.0 kcal/mol. It is readily seen that the Li2 enthalpies of formation are within 1.0

kcal/mol of experiment at the cc‐pVTZ and aug‐cc‐pVTZ levels and higher, while the Na2

enthalpies are within 1.0 kcal/mol of experiment at all ζ‐levels with each family of basis sets.

Using the cc‐pVnZ and aug‐cc‐pVnZ basis sets, only the NaF molecule reaches 1.0 kcal/mol of

experiment at the quintuple‐ζ level; the other molecules are still outside the desired accuracy

range. The overall effect of adding the diffuse functions is to accelerate the convergence of the

enthalpies of formation. For example, in LiF, NaF, MgF, BeF2, and MgF2, the cc‐pVDZ and aug‐cc‐

pVDZ enthalpies of formation differ by more than 10 kcal/mol, with the diffuse functions aiding

convergence towards the experimental value.

Adding core‐valence correlation does have a noticeable impact on the enthalpies of

formation in the larger basis sets (i.e. quadruple‐ and quintuple‐ζ levels). For example, at the

cc‐pCV5Z and cc‐pwCV5Z levels, the BeO enthalpy of formation differs from the cc‐pV5Z

enthalpy by 1.65 kcal/mol and 1.83 kcal/mol, respectively. Still other noticeable differences are

between the BeF2 and MgF2 enthalpies of formation at the cc‐pCV5Z and cc‐pwCV5Z levels with

respect to the cc‐pV5Z level. The BeF2 enthalpy of formation computed with the cc‐pCV5Z and

cc‐pwCV5Z basis sets, for example, is different from the cc‐pV5Z enthalpy by 1.92 kcal/mol and

2.12 kcal/mol, respectively.

Using the largest basis sets to compute the enthalpies of formation of Li2, LiF, BeO, and

BeF, gives results that more closely agree with experiment than the other basis set families.

130

However, the enthalpies of formation of NaF and MgF2 computed with the cc‐pCVnZ and

cc‐pwCVnZ basis sets are slightly better (relative to experiment) than those computed with the

aug‐cc‐pCVnZ and aug‐cc‐pwCVnZ basis sets.

6.5 Conclusions

The unpublished basis sets for Li, Be, Na, and Mg have been reexamined using robust

numerical optimization techniques, and new official correlation consistent basis sets for these

four atoms are presented. Specifically, new cc‐pVnZ and aug‐cc‐pVnZ basis sets of double‐,

triple‐, quadruple‐, and quintuple‐ζ quality have been developed and benchmarked. In addition,

a set of tight d basis sets, cc‐pV(n+d)Z, have been developed for Na and Mg in correlated

valence calculations.

For subvalence correlation, new core‐valence and weighted core‐valence (cc‐pCVnZ and

cc‐pwCVnZ) basis sets have been developed. Benchmark calculations of atomic CV correlation

energies demonstrate the necessity of including core‐valence functions when subvalence

correlation is needed (i.e. for high accuracy thermochemistry). Benchmarks of molecular

properties also demonstrate the necessity of CV correlation, especially in Li and Na, whose one

electron valence is not a large enough valence space for correlated molecular calculations when

0.01 Å and 1.0 kcal/mol accuracy is desired.

Finally, the HF sp basis sets have been recontracted for scalar relativistic DK calculations,

resulting in cc‐pVnZ‐DK basis sets that consistently recover more scalar relativistic energy than

the original HF contractions at each basis set level.

131

Table 6.2 CCSD(T) ionization potentials and electron affinities of Li, Be, Na, and Mg; the tight d correlation consistent basis sets for Na and Mg have been used.

Li Be Na Mg

Basis IP EA IP(1) IP(2) IP EA IP(1) IP(2)

cc‐pVDZ 5.342 0.414 9.290 18.086 4.951 0.406 7.521 14.711cc‐pVTZ 5.342 0.480 9.285 18.124 4.952 0.437 7.527 14.721cc‐pVQZ 5.342 0.569 9.296 18.125 4.951 0.523 7.531 14.722cc‐pV5Z 5.342 0.580 9.298 18.125 4.951 0.522 7.533 14.722

aug‐cc‐pVDZ 5.342 0.592 9.287 18.091 4.952 0.526 7.522 14.711aug‐cc‐pVTZ 5.342 0.615 9.286 18.124 4.952 0.544 7.527 14.721aug‐cc‐pVQZ 5.342 0.617 9.296 18.125 4.951 0.545 7.531 14.722aug‐cc‐pV5Z 5.342 0.617 9.299 18.125 4.951 0.545 7.533 14.722cc‐pCVDZ 5.356 0.474 9.273 18.145 5.000 0.444 7.542 14.793cc‐pCVTZ 5.379 0.481 9.303 18.192 5.077 0.442 7.598 14.934cc‐pCVQZ 5.388 0.570 9.316 18.205 5.114 0.527 7.619 14.982cc‐pCV5Z 5.390 0.581 9.319 18.207 5.127 0.526 7.630 15.002cc‐pwCVDZ 5.365 0.476 9.278 18.154 5.092 0.443 7.567 14.869cc‐pwCVTZ 5.386 0.482 9.309 18.199 5.113 0.443 7.613 14.975cc‐pwCVQZ 5.390 0.570 9.318 18.207 5.127 0.527 7.627 15.001cc‐pwCV5Z 5.391 0.581 9.320 18.208 5.131 0.526 7.633 15.008

aug‐cc‐pwCVDZ 5.365 0.610 9.277 18.157 5.094 0.539 7.571 14.872aug‐cc‐pwCVTZ 5.386 0.615 9.310 18.199 5.114 0.543 7.613 14.976aug‐cc‐pwCVQZ 5.390 0.617 9.318 18.207 5.127 0.547 7.628 15.001aug‐cc‐pwCV5Z 5.391 0.617 9.320 18.208 5.131 0.548 7.633 15.008Experimenta 5.392 0.618 9.323 18.211 5.139 0.548 7.646 15.035

a. Martin, W. C.; Wiese, W. L. “Atomic Spectroscopy.” in Atomic, Molecular, & Optical Physics Handbook; Drake, G. W. F., Ed.; American Institute of Physics: Woodbury, NY, 1996; pp 135.

132

Table 6.3 CCSD(T) optimized geometries (Re, Å) of some s‐block molecules computed with various families of correlation consistent basis sets.

Molecule ζ VnZ V(n+d)Z CVnZ wCVnZ aVnZ aV(n+d)Z aCVnZ awCVnZ Expt.

Li2 D 2.731 ‐‐‐ 2.699 2.701 2.727 ‐‐‐ 2.700 2.702 T 2.701 ‐‐‐ 2.680 2.677 2.701 ‐‐‐ 2.680 2.677 Q 2.699 ‐‐‐ 2.676 2.674 2.699 ‐‐‐ 2.676 2.674 5 2.699 ‐‐‐ 2.674 2.674 2.699 ‐‐‐ 2.674 2.674 2.6729a

LiF D 1.592 ‐‐‐ 1.579 1.556 1.606 ‐‐‐ 1.596 1.587 T 1.588 ‐‐‐ 1.572 1.557 1.591 ‐‐‐ 1.574 1.568 Q 1.578 ‐‐‐ 1.564 1.561 1.582 ‐‐‐ 1.567 1.565 5 1.580 ‐‐‐ 1.565 1.563 1.581 ‐‐‐ 1.565 1.564 1.56386a

BeO D 1.369 ‐‐‐ 1.365 1.357 1.370 ‐‐‐ 1.366 1.361 T 1.344 ‐‐‐ 1.338 1.333 1.346 ‐‐‐ 1.341 1.336 Q 1.338 ‐‐‐ 1.331 1.330 1.339 ‐‐‐ 1.333 1.331 5 1.337 ‐‐‐ 1.331 1.330 1.337 ‐‐‐ 1.331 1.330 1.3309a

BeF D 1.413 ‐‐‐ 1.409 1.399 1.414 ‐‐‐ 1.407 1.397 T 1.371 ‐‐‐ 1.365 1.359 1.374 ‐‐‐ 1.369 1.365 Q 1.367 ‐‐‐ 1.361 1.359 1.369 ‐‐‐ 1.363 1.362 5 1.367 ‐‐‐ 1.361 1.360 1.368 ‐‐‐ 1.362 1.361 1.3610a

Na2 D 3.207 3.205 3.163 3.113 3.204 3.203 3.148 3.095 T 3.178 3.180 3.095 3.088 3.178 3.179 3.083 3.081 Q 3.178 3.179 3.087 3.080 3.178 3.179 3.087 3.080 5 3.179 3.179 3.083 3.079 3.179 3.179 3.073 3.081 3.0788a

NaF D 1.934 1.928 1.910 1.899 1.986 1.982 1.950 1.938 T 1.980 1.971 1.928 1.919 1.996 1.990 1.939 1.934 Q 1.989 1.982 1.928 1.923 1.994 1.990 1.931 1.928 5 1.989 1.987 1.928 1.926 1.991 1.989 1.928 1.927 1.92594a

MgO D 1.774 1.753 1.768 1.753 1.786 1.770 1.774 1.764 T 1.759 1.752 1.743 1.736 1.766 1.760 1.748 1.743 Q 1.756 1.753 1.739 1.737 1.759 1.756 1.741 1.740 5 1.754 1.753 1.738 1.737 1.755 1.754 1.739 1.738 1.749a

MgF D 1.773 1.756 1.771 1.762 1.785 1.775 1.776 1.771 T 1.768 1.760 1.754 1.746 1.777 1.770 1.761 1.755 Q 1.766 1.762 1.751 1.748 1.770 1.766 1.753 1.752 5 1.765 1.764 1.751 1.750 1.766 1.765 1.751 1.751 1.7500a


133


Molecule ζ VnZ V(n+d)Z CVnZ wCVnZ aVnZ aV(n+d)Z aCVnZ awCVnZ Expt.

BeH2 D 1.339 ‐‐‐ 1.336 1.333 1.340 ‐‐‐ 1.336 1.336 (D∞h) T 1.334 ‐‐‐ 1.330 1.327 1.333 ‐‐‐ 1.330 1.328

Q 1.331 ‐‐‐ 1.326 1.326 1.331 ‐‐‐ 1.327 1.326 5 1.331 ‐‐‐ 1.326 1.326 1.331 ‐‐‐ 1.326 1.326 ‐‐‐

BeF2 D 1.417 ‐‐‐ 1.412 1.406 1.417 ‐‐‐ 1.407 1.402 (D∞h) T 1.382 ‐‐‐ 1.377 1.372 1.384 ‐‐‐ 1.380 1.377

Q 1.378 ‐‐‐ 1.373 1.372 1.380 ‐‐‐ 1.375 1.374 5 1.379 ‐‐‐ 1.373 1.373 1.379 ‐‐‐ 1.374 1.373 ‐‐‐

MgH2 D 1.710 1.707 1.707 1.703 1.717 1.713 1.709 1.706 (D∞h) T 1.712 1.710 1.700 1.698 1.713 1.712 1.701 1.698

Q 1.711 1.710 1.698 1.697 1.711 1.710 1.698 1.697 5 1.710 1.710 1.696 1.696 1.710 1.710 1.697 1.696 ‐‐‐

MgF2 D 1.760 1.744 1.756 1.748 1.769 1.760 1.761 1.756 (D∞h) T 1.756 1.749 1.743 1.736 1.763 1.757 1.747 1.742

Q 1.754 1.751 1.739 1.737 1.757 1.754 1.741 1.739 5 1.753 1.752 1.738 1.737 1.754 1.753 1.739 1.738 ‐‐‐a. Constants of Diatomic Molecules; Number 69 ed.; Huber, K. P.; Hertzberg, G., Eds.; National Institute of

Standards and Technology: Gaithersburg, MD, 2005.

134

Table 6.4 CCSD(T) harmonic vibrational frequencies (Å) of some s‐block molecules computed with various families of correlation consistent basis sets.

Molecule VnZ V(n+d)Z CVnZ wCVnZ aVnZ aV(n+d)Z aCVnZ awCVnZ Expt.

Li2 346.5 ‐‐‐ 353.0 355.4 346.7 ‐‐‐ 353.4 354.2 351.43a

LiF 904.1 ‐‐‐ 905.6 937.5 887.4 ‐‐‐ 894.1 908.6 910.34a

BeO 1458.4 ‐‐‐ 1468.0 1477.6 1451.7 ‐‐‐ 1458.7 1466.1 1487.32a

BeF 1260.7 ‐‐‐ 1270.2 1280.0 1242.4 ‐‐‐ 1248.1 1256.6 1247.36a

Na2 151.5 152.4 159.4 161.9 151.5 152.5 159.7 161.1 159.124a

NaF 548.7 554.9 543.6 554.9 532.3 536.4 524.9 530.8 536a

MgO 782.1 796.5 805.3 817.5 770.9 781.7 794.2 802.6 785.0a

MgF 722.9 723.2 726.0 731.5 698.5 702.1 702.0 710.2 711.69a

BeH2 710.3 ‐‐‐ 717.9 719.7 707.9 ‐‐‐ 715.8 718.6 708.5b

(D∞h) 2030.0 ‐‐‐ 2039.7 2047.4 2025.4 ‐‐‐ 2032.4 2041.6 ‐‐‐ 2238.7 ‐‐‐ 2249.1 2255.3 2232.3 ‐‐‐ 2243.1 2250.1 2172.2b

BeF2 335.4 ‐‐‐ 344.2 345.9 336.8 ‐‐‐ 342.7 344.2 342.61c

(D∞h) 727.1 ‐‐‐ 731.4 737.0 719.7 ‐‐‐ 722.8 726.4 769.09c

1584.3 ‐‐‐ 1591.4 1599.1 1566.2 ‐‐‐ 1568.9 1573.7 1555.05c

MgH2 439.2 434.8 457.0 446.2 439.0 434.0 447.8 442.1 450.4d

(D∞h) 1605.5 1603.4 1624.5 1620.5 1595.6 1594.7 1615.6 1615.2 ‐‐‐ 1632.3 1629.0 1650.0 1644.2 1618.6 1618.3 1637.9 1630.9 1576.8d

MgF2 156.6 154.7 160.0 156.9 149.2 145.3 155.3 153.7 254e

(D∞h) 571.3 571.0 575.5 579.1 556.7 558.9 561.4 567.0 ‐‐‐ 891.8 890.5 896.4 900.8 864.8 867.5 872.6 879.9 862e

a. Constants of Diatomic Molecules; Number 69 ed.; Huber, K. P.; Hertzberg, G., Eds.; National Institute of Standards and Technology: Gaithersburg, MD, 2005.

b. Wang, X.; Andrews, L. Inorg. Chem. 2005, 44, 610. c. Yu, S.; Shayesteh, A.; Bernath, P. F.; Koput, J. J. Chem. Phys. 2005, 123, 134303. d. Wang, X.; Andrews, L. J.Phys. Chem. A 2004, 108, 11511. e. Snelson, A. J.Phys. Chem. 1966, 70, 3208.

135

Table 6.5 CCSD(T) standard state enthalpies of formation at 298.15 K (kcal/mol) for some s‐block molecules computed with various families of correlation consistent basis sets; tight d functions have been used for the Na and Mg atoms.

Molecule ζ VnZ CVnZ wCVnZ aVnZ aCVnZ awCVnZ Expt.a

Li2 D 53.40 52.55 52.51 53.22 52.46 52.42

T 52.11 51.90 51.80 52.05 51.82 51.75

Q 51.84 51.66 51.63 51.83 51.64 51.62

5 51.80 51.61 51.59 51.79 51.60 51.58 51.60

LiF D ‐64.78 ‐65.79 ‐67.85 ‐72.64 ‐73.91 ‐74.52

T ‐74.17 ‐75.23 ‐76.79 ‐76.80 ‐77.86 ‐78.64

Q ‐78.38 ‐79.39 ‐79.79 ‐79.31 ‐80.32 ‐80.53

5 ‐79.39 ‐80.48 ‐80.68 ‐79.78 ‐80.89 ‐80.99 ‐81.45

BeO D 56.39 55.16 53.87 52.37 51.01 50.00

T 41.63 40.00 38.62 39.40 38.25 37.14

Q 36.75 35.11 34.74 35.63 34.07 33.78

5 35.04 33.39 33.21 34.63 33.00 32.85 32.60

BeF D ‐20.27 ‐20.88 ‐21.62 ‐25.37 ‐26.52 ‐27.38

T ‐32.90 ‐34.22 ‐35.49 ‐34.22 ‐35.06 ‐35.99

Q ‐36.36 ‐37.48 ‐37.73 ‐36.95 ‐38.01 ‐38.17

5 ‐37.24 ‐38.40 ‐38.52 ‐37.46 ‐38.60 ‐38.70 ‐40.60

Na2 D 34.76 33.98 33.83 34.68 33.68 33.48

T 33.95 33.27 33.40 33.94 33.11 33.28

Q 33.86 33.50 33.47 33.86 33.45 33.42

5 33.81 33.27 33.45 33.81 33.11 33.41 33.96

NaF D ‐53.86 ‐53.65 ‐54.38 ‐64.77 ‐65.18 ‐65.33

T ‐63.61 ‐63.54 ‐64.01 ‐67.47 ‐67.86 ‐67.88

Q ‐67.45 ‐67.76 ‐68.00 ‐69.09 ‐69.42 ‐69.52

5 ‐69.07 ‐69.34 ‐69.39 ‐69.65 ‐70.04 ‐69.98 ‐69.42

MgO D 52.23 53.69 53.11 43.50 43.87 43.52

T 40.80 40.95 40.39 37.29 36.95 36.65

Q 36.17 36.29 36.19 34.86 34.68 34.68

5 34.69 34.78 34.73 34.15 34.08 34.07 36.18

MgF D ‐37.45 ‐35.88 ‐35.67 ‐47.55 ‐47.23 ‐46.86

T ‐47.62 ‐46.64 ‐47.01 ‐50.70 ‐50.13 ‐50.32

Q ‐51.79 ‐50.76 ‐50.77 ‐52.87 ‐52.14 ‐52.04


136


Molecule ζ VnZ CVnZ wCVnZ aVnZ aCVnZ awCVnZ Expt.a

MgF 5 ‐52.94 ‐51.96 ‐51.97 ‐53.38 ‐52.51 ‐52.49 ‐56.60

BeH2 D 57.11 56.67 56.36 55.11 54.31 54.32

(D∞h) T 50.67 49.84 49.24 49.99 49.39 48.85

Q 48.69 47.85 47.76 48.55 47.72 47.64

5 48.30 47.44 47.39 48.24 47.38 47.34 30.00

BeF2 D ‐152.60 ‐153.85 ‐154.82 ‐162.55 ‐165.29 ‐166.16

(D∞h) T ‐176.30 ‐178.60 ‐180.77 ‐178.99 ‐180.26 ‐181.79

Q ‐182.76 ‐184.57 ‐184.98 ‐183.94 ‐185.64 ‐185.90

5 ‐184.41 ‐186.33 ‐186.53 ‐184.84 ‐186.72 ‐186.87 ‐190.25

MgH2 D 51.63 52.37 52.95 48.16 47.95 48.83

(D∞h) T 44.56 45.05 45.38 43.76 43.88 44.14

Q 42.99 43.97 44.15 42.79 43.65 43.86

5 42.62 43.76 43.81 42.51 43.57 43.60 ‐‐‐

MgF2 D ‐145.67 ‐142.77 ‐142.49 ‐164.91 ‐164.41 ‐163.75

(D∞h) T ‐165.38 ‐164.27 ‐164.98 ‐170.66 ‐170.34 ‐170.66

Q ‐173.16 ‐171.77 ‐171.84 ‐174.97 ‐174.08 ‐173.99

5 ‐175.09 ‐173.91 ‐173.97 ‐175.87 ‐174.89 ‐174.93 ‐173.70a. Chase, M. W. J. Phys. Chem. Ref. Data 1998, 9, 1.

137

CHAPTER 7

THE RESOLUTION OF THE IDENTITY APPROXIMATION APPLIED TO THE CORRELATION

CONSISTENT COMPOSITE APPROACH†

7.1 Introduction

In correlated ab initio methods, the number of four‐index two‐electron Coulomb

integrals (2.11) increases dramatically as the number of electrons increases.7,19,22,55 The

computation and storage of these integrals is the bottleneck in applying correlated methods,

and so it is highly desirable to have efficient ways of computing and storing them. Typically,

integral screening techniques187 are employed to avoid computing very small (i.e 10‐14 Eh)

integrals, which helps reduce the number of four‐index integrals stored. To deal efficiently with

the other integrals, a novel approach is to employ a projective technique.

The resolution of the identity (RI) is a well‐known projective technique from vector

algebra.188 Consider a ket vector | expanded in a complete, orthonormal basis :

| | (7.1)

By projecting | onto an arbitrary basis vector in , the component of | along that vector is

resolved:

| | (7.2)

Insertion of (7.2) into (7.1) yields the following result:

† Work reported in this chapter was performed in collaboration with Jeremy D. Lai, and has been submitted for publication in the Journal of Chemical Physics.

138

| | | | | | (7.3)

where the summation in parentheses apparently takes the form of an algebraic ‘1’:

| | (7.4)

The object in (7.4) has several names in the literature including identity operator, completeness

relation, and unit dyadic, the latter of which is used throughout this chapter. The summation

terms are projection operators instead of scalar products, since the ket is written before the

bra. When (7.4) operates on a ket vector, the result is the same ket represented in whatever

basis set comprises the unit dyadic, provided that basis is complete. This is why this

mathematical construct is called the resolution of the identity, because it resolves a vector in

one basis to another. For example, a unit dyadic in the basis may be constructed as

| | (7.5)

then, since | | , the vector | may be resolved in the new basis using (7.1):

| | | | | |

| | |

(7.6)

The components of | in are now defined in terms of its components in and the overlap

between the two basis sets | .

More generally, if the domain of the bra vector in (7.4) is different from the domain in

the ket, then the resulting operator is the Dirac delta function:7

139

| | (7.7)

Instead of resolving vectors from one basis to another, the Dirac delta function resolves a

function in one domain into another as follows:

d (7.8)

As a sidebar, the definition of the Dirac delta function in (7.7) is entirely analogous to the

definition of the first‐order reduced density function in (2.47). The Dirac delta function is

related to the following Fourier integral, where is the momentum vector:

d , (7.9)

which is why RI methods are sometimes referred to as pseudospectral methods in the

literature.189 Finally, it is readily seen from (7.9) that the inverse Fourier transform of the Dirac

delta function, and, by extension the unit dyadic, is unity.

Boys and Shavitt first introduced the idea of reducing the computational cost of four‐

index integrals using the RI technique,190 while Löwdin discussed the use of RI in terms of

describing quantum phenomena.191 Since then, the RI approximation has been employed in

density functional theory (DFT),161,192 Hartree‐Fock self‐consistent field (HF‐SCF),160 second‐

order Møller‐Plesset perturbation theory (MP2),160,193 and coupled cluster theory.194 For a

review of the RI approximation in computational chemistry, see Kendall and Früchtl.190

Following the derivations of Vahtras et al. with regards to various RI approximations,160

any four‐index integral can be approximated as a linear combination of three‐ or two‐index

integrals by insertion of the unit dyadic. Vahtras et al. showed that describing charge

distributions with the RI approximation leads to three different formulas for approximating

140

four‐index Coulomb integrals:160

| | 1 | (7.10)

| | 1 1 (7.11)

| | 1 (7.12)

in which the basis sets denoted by , , , are called auxiliary basis sets. The bracketed terms

on the right‐hand side are three‐center overlap integrals, the terms are two‐center overlap

integrals (in matrix form) between two auxiliary basis sets, and the terms are two‐center,

two‐electron Coulomb integrals (also in matrix form). Equations (7.10) and (7.11) are called the

S and SVS approximations, respectively, in which the individual charge distributions are

projected onto an auxiliary basis set. Note that the auxiliary basis set for the bra vector is not

necessarily the same used for the ket vector in the S and SVS approximations, which is why the

additional overlap integrals enter the equations. Equation (7.12) is called the V approximation,

in which the Coulomb operator is projected onto an auxiliary basis set. The V approximation

more closely approximates the exact four‐index integral in practice, and, consequently, is the

most common RI approximation in use for approximating the four‐index integrals in SCF, DFT,

and MP2 calculations.

In more recent implementations of RI methods, a single auxiliary basis is used instead of

multiple auxiliary basis sets as indicated by the equations above. When a single auxiliary basis

set is used in the V approximation, the number of four‐index integrals is reduced from to

approximately , where is the number of orbital basis functions and is the number

of auxiliary basis functions. So long as does not exceed , there will be an effective

141

reduction in the number, and hence, the computational scaling of four‐index integrals. This is

the primary motivation behind employing the RI approximation in correlated ab initio methods,

where the number of four‐index integrals over excited determinants becomes very large. In DFT

methods, the RI approximation takes on a slightly different role, namely, that of projecting the

exchange‐correlation functional, which may not have a trivial integral form, onto an auxiliary

basis set in which integrals over the exchange‐correlation functional become more tractable. In

practice, a different auxiliary basis set is used for each computational method, each having

been optimized specifically for that method’s ansatz. For example, a density‐fit or Coulomb‐fit

auxiliary basis set might be used for all or some of the HF‐SCF integrals, while a different

auxiliary basis set is used for correlated integrals that arise in an MP2 calculation employing

those HF orbitals.

An additional means to reduce the computational expense of ab initio methods is via

composite methods (discussed in Chapter 2). These approaches approximate a high‐level

correlated ab initio/large basis set calculation with a series of lower‐level/small basis set

calculations. Examples of composite methods include the Gaussian‐n methods (Gn),60‐62

Weizmann‐n methods (Wn),71,72 the high‐accuracy extrapolated ab initio thermochemistry

(HEAT) method,70 complete basis set (CBS‐n) methods,63‐69 and the recently developed

correlation consistent Composite Approach (ccCA).73‐80 In general, each composite method

listed varies in the target accuracy with respect to reliable experimental data and in

formulation. The ccCA method makes use of the well‐known monotonic behavior of the

correlation consistent basis sets, cc‐pVnZ,94‐101 with respect to increasing basis set size for

energetic properties (e.g., thermochemical properties, atomization energies), and considers

142

additional effects such as high‐order correlation effects, core‐valence effects, and scalar

relativistic effects. Further, ccCA has been applied to over 700 main group molecules, resulting

in an overall mean absolute deviation (MAD) of approximately 1 kcal/mol from reliable

experiment.73‐80 The use of composite methodology may be able to approximate high‐level

computational methods with a lower computational expense, however, further modifications

to the handling of four‐index integrals can be introduced to further reduce the computational

scaling.

In this Chapter, the RI approximation is applied to the ccCA method and the resulting

formulism (RI‐ccCA) is discussed. The accuracy of properties computed with RI‐ccCA is

compared with those computed with the original formulation of ccCA. Further, relative timings

and storage requirements are presented for RI‐ccCA and are compared with the original ccCA

formulation.


The set of molecules examined is a subset of the G2/97 test set61 containing 102 closed

shell systems, including both first and second row, main group atoms. All calculations were

performed with the Molpro 2006.1 software package180 without employing symmetry and using

the default semi‐direct algorithm. Energy gradients and hessians were converged to at least

0.1 mEh in the geometry optimizations and frequency calculations. Following the formulation

for the ccCA method,73‐80 the geometry optimization of each molecule is performed using

Becke’s 3‐parameter exchange scheme coupled with the Lee‐Yang‐Parr correlation functional,

B3LYP,165,166 and the cc‐pVTZ basis set. Harmonic frequencies are then computed at the same

143

level of theory and scaled by 0.9854 to account for anharmonicity.73,74 In order to compare

energetic properties without bias from a change in the geometry, we have used the same

B3LYP/cc‐pVTZ geometries and scaled harmonic frequencies in each of the different ccCA

formulations discussed in the next section.

From the B3LYP/cc‐pVTZ geometry, the ccCA reference energy (Eref), higher‐order

correlation (ECC), core‐valence correlation (ECV), and scalar relativistic (ESR) corrections are

computed as follows:

MP2/aug‐cc‐pV∞Z

∆ CCSD T /cc‐pVTZ MP2/cc‐pVTZ

∆ MP2 full /aug‐cc‐pCVTZ MP2 fc /aug‐cc‐pVTZ

∆ DK‐MP2/cc‐pVTZ‐DK MP2/cc‐pVTZ

∆ ∆ ∆

(7.13)

The reference energy, MP2/aug‐cc‐pV∞Z (where aug‐cc‐pV∞Z denotes the CBS limit), is

extrapolated using the mixed Gaussian/exponential formula (3.3) from double‐, triple‐, and

quadruple‐ζ single‐point energies. In the core‐valence step, MP2(full) correlates the standard

atomic valence plus the (n‐1)s and (n‐1)p orbitals for main group atoms, while the MP2(fc)

denotes using a frozen core of (n‐1)s and (n‐1)p electrons. The scalar relativistic correction

employs the second‐order, spin‐free Douglas‐Kroll (DK) Hamiltonian135,136 and the DK‐

recontracted correlation consistent basis sets.142 Spin‐orbit (SO) effects (not included in the DK

Hamiltonian) have been included a posteriori for the atoms.74

Where thermochemical properties have been computed, atomic enthalpies of

formation and thermal corrections have been taken from CODATA,195 except for the Si atom,

144

which has been taken from Karton and Martin.196


7.3.1 RI‐ccCA Implementation

An RI‐based ccCA approach has been constructed by substituting the conventional SCF

and MP2 steps in ccCA with their RI analogs. The general formulation is listed below, and is

hereafter denoted RI‐ccCA:

RI‐MP2/aug‐cc‐pV∞Z‐RI

Δ CCSD T /cc‐pVTZ‐RI RI‐MP2/cc‐pVTZ‐RI

Δ RI‐MP2 full /aug‐cc‐pCVTZ‐RI RI‐MP2/aug‐cc‐pVTZ‐RI

Δ DK‐RI‐MP2/cc‐pVTZ‐DK‐RI RI‐MP2/cc‐pVTZ‐RI

‐ Δ Δ Δ

(7.14)

Each of the conventional MP2 steps of (7.13) have been replaced with RI‐MP2. Also, the

underlying SCF reference calculations have been replaced with RI‐SCF. For the CCSD(T) step,

two different routes have been examined: one in which conventional CCSD(T) is used and

another in which the density‐fit (DF, which is entirely analgous to the RI approximation) local

CCSD(T) method, denoted DF‐LCCSD(T), of Schütz and Werner197‐199 is used. In the latter part of

this section, the difference between employing the conventional CCSD(T) method in RI‐ccCA

versus the DF‐LCCSD(T) method in RI‐ccCA (denoted RI‐ccCA+L for clarity) is analyzed and

discussed.

145

7.3.2 Auxiliary Basis Sets for RI‐ccCA

To determine the optimal auxiliary basis sets to be used in the RI‐SCF and RI‐MP2

calculations, a comparison has been made between RI‐ccCA, which utilizes conventional

CCSD(T), and ccCA employing several different auxiliary basis sets. Utilimately, the auxiliary

basis sets should be chosen so that the convergent nature of the correlation consistent basis

sets remains intact, and so that the RI‐ccCA energies are as close to the ccCA energies as

possible.

In the RI‐SCF steps, the use of completely unconctracted cc‐pVnZ basis sets one ζ‐level

higher than the orbital basis set, denoted cc‐pV(n+1)Z(unc), and the basis sets of Weigend,

denoted ‘JKFIT,’200 have been studied. (It is common practice when using RI methods to assume

that an uncontracted basis set that is larger than the orbital basis set is an adequate auxiliary

basis set; i.e. if a cc‐pVTZ orbital basis set is used, then the uncontracted cc‐pVQZ basis set is

assumed to be adequate as an auxiliary basis set.) The JKFIT basis sets are studied here because

they have been reported to reproduce conventional SCF atomic energies within 120 μEh at any

ζ‐level through quadruple‐ζ.200 Further, the use of the cc‐pV(n+1)Z(unc) and the cc‐pVnZ‐RI

basis sets of Weigend et al.181 have been studied as potential auxiliary basis sets in the RI‐MP2

steps of RI‐ccCA. The latter auxiliary sets were optimized specifically for use with the correlation

consistent basis sets, which is why they were selected, and have been reported to reproduce

conventional MP2 correlation energies withing 101 μEh per atom in molecules containing first

and second row atoms.181

The results of the test calculations described above for H2P+, Cl‐, CH3

‐, NH2‐, OH‐, SiH3

‐,

PH2‐, SH‐, CN‐, NH4

+, H3O+, C2H3

+, SiH5+, PH4

+, H3S+, and H2Cl

+ are listed in Table 7.1 according to

146

the auxiliary basis sets used. The relative energy differences of RI‐ccCA, as compared with ccCA,

are shown in the table, as well as the relative computational cost of RI‐ccCA in terms of CPU

time and disk space. Overall, a computational savings is observed in each RI implementation of

Table 7.1, except for a few isolated cases (i.e. Cl‐, OH‐, and SH‐). A mean absolute deviation

(MAD) of 213 mEh is observed in the test set when the cc‐pV(n+1)Z(unc) auxiliary set is used.

Employing the JKFIT auxiliary basis sets in the RI‐SCF reference calculation and using the

cc‐pV(n+1)Z(unc) auxiliary sets in the RI‐MP2 step lowers the MAD of the energies dramatically

(compared with the previous approach) to 0.296 mEh. Thus, the JKFIT sets have demonstrated

much better accuracy than the cc‐pV(n+1)Z(unc) set as the auxiliary basis set in RI‐SCF

calculations. Comparing the different auxiliary basis sets used in the RI‐MP2 step shows the cc‐

pVnZ‐RI basis sets of Weigend et al. to have the lower MAD in total energies (0.270 mEh),

relative to ccCA. In light of these test calculations, it is recommended that the RI‐ccCA

implementation use the JKFIT auxiliary basis sets in the RI‐SCF reference calculations and the

cc‐pVnZ‐RI basis sets in the RI‐MP2 and RI‐LCCSD(T) steps of (7.14). This recommended

formulation is employed from here on in this investigation.

7.3.3 Energetic Properties

Total energies computed with ccCA, RI‐ccCA, and RI‐ccCA+L listed in Table 7.2. The

largest difference in total energies between ccCA and RI‐ccCA is 2.214 mEh which occurs for

SiCl4, while the smallest difference between the two methods is 0.007 mEh, which occurs for H2.

The average difference in total energies between ccCA and RI‐ccCA for the entire test set is

0.433 mEh. The total energy differences between RI‐ccCA+L and ccCA are almost an order of

147

magnitude larger than those of RI‐ccCA and ccCA. The average total energy difference between

RI‐ccCA+L and ccCA is 4.193 mEh, with the worst difference being 14.231 mEh for ClF3.

Examining each additive component (included in the supplemental material of the submitted

manuscript)201 of the RI‐ccCA reveals that the largest differences relative to the ccCA method

occur in the core‐valence and scalar relativistic steps. There is no total energy difference larger

than 1.0 kcal/mol between the original ccCA coupled cluster step and the RI‐ccCA coupled

cluster step in any of the molecules studied.

Enthalpies of formation computed at 298 K are listed in Table 7.3. Overall, good

agreement is observed between ccCA and RI‐ccCA. The largest deviation of the RI‐ccCA

enthalpies of formation from those of ccCA, not surprisingly, comes from SiCl4 and is

1.39 kcal/mol. The MAD of RI‐ccCA enthalpies of formation from ccCA for the entire test set is

0.27 kcal/mol. The minimum and maximum deviations from experiment are similar between

RI‐ccCA and ccCA, and the MAD is 1.00 and 1.03 kcal/mol, respectively, for each method. The

average difference between ccCA and RI‐ccCA+L enthalpies of formation is an order of

magnitude larger than those of RI‐ccCA and ccCA at 2.63 kcal/mol, with the largest deviation

being 8.93 kcal/mol from ClF3. Compared with experiment, RI‐ccCA+L has a MAD of

2.30 kcal/mol, and deviates the worst by 9.20 kcal/mo in the enthalpy of formation of SiF4.

7.3.4 Computational Cost

The CPU time and disk space savings of RI‐ccCA and RI‐ccCA+L, relative to ccCA, are

provided in Table 7.4. The relative CPU times range from 10% more computational time (worst

case: FH) to 94% less time (best case: C2H6), with an average CPU time savings of 72% compared

148

with ccCA. Employing RI‐ccCA+L affords a slightly better average CPU time savings of 76% and

97% savings in the best case, but increases the computational cost to 129% in the worst case

(H2). Comparing RI‐ccCA to RI‐ccCA+L, we observe an average CPU time savings of 28% using

RI‐ccCA+L, and in the best case, an 86% CPU time reduction is observed using RI‐ccCA+L.

Overall, for the test set we have employed, the RI‐ccCA+L method affords lower CPU times.

Figure 7.1 CPU times of RI‐ccCA (top) and RI‐ccCA+L (bottom) versus ccCA; each data pointrepresents a molecule in the test set.

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

1.0E+05

1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05

RI‐ccCA CPU

Tim

e (s)

ccCA CPU Time (s)

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

1.0E+05

1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05

RI‐ccCA+L CPU

Tim

e (s)

ccCA CPU Time (s)

149

Further, dramatic reductions in the disk space requirements are observed, ranging from 84% to

99% for RI‐ccCA, compared with ccCA, with an average disk space savings of 97%; and ranging

from 77% to almost 100% for RI‐ccCA+L, with an average savings of 96%. Comparing RI‐ccCA to

RI‐ccCA+L, an average 17% less disk space is required in the latter. In the largest molecule

examined, benzene, the disk space saved using RI‐ccCA and RI‐ccCA+L is 97% and 98%,

respectively. Further, for the series CH3Cl, CH3CH2Cl, and CH3CH2CH2Cl, the CPU time saved

employing RI‐ccCA over ccCA is 87%, 89%, and 93%, respectively, while using RI‐ccCA on each

molecule affords a 98% disk space savings over ccCA. Using RI‐ccCA+L on the same series

produces CPU time savings of 84%, 90%, and 96%%, respectively, with disk space savings of

99% over ccCA.

The CPU time and disk space savings of RI‐ccCA and RI‐ccCA+L, relative to ccCA, are

reflected in Figure 7.1 and Figure 7.2, respectively. Figure 7.1 clearly demonstrates that the use

of the RI approximation does not always result in a computational time savings, as is seen in the

left‐most three points in the plots (cf. the Cl‐ atom and OH‐ and SH‐ diatomics of Table 7.1).

Employing the RI‐ccCA+L method exacerbates this phenomenon, as is seen in the bottom plot

of Figure 7.1 in which the left‐most three points are above the line indicating a CPU time

increase relative to ccCA. However, as Figure 7.1 shows, there is clearly more of a

computational time savings using RI‐ccCA+L over RI‐ccCA since the calculations that require the

most computational time are lower on the vertical axis of the bottom plot than in the top plot.

As Figure 7.2 demonstrates, there is always a clear disk space savings when RI‐ccCA or RI‐

ccCA+L is used, compared with ccCA.

150

In general, it is observed that there is a correlation between molecular size and amount

of computational resources needed for RI‐ccCA and RI‐ccCA+L, compared with ccCA, which is

demonstrated in Figure 7.3. The values shown are averages over each group of molecules

containing a certain number of atoms. There is a clear increase in both CPU and disk space

savings as molecular size increases. The ‘cross‐over’ point at which the RI‐ccCA+L method

Figure 7.2 Disk space usage of RI‐ccCA (top) and RI‐ccCA+L (bottom) versus ccCA; each datapoint represents a molecule in the test set.

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

1.0E+05

1.0E+06

1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06

RI‐ccCA Disk Usage

(MB)

ccCA Disk Usage (MB)

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

1.0E+05

1.0E+06

1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06

RI‐ccCA

+L Disk Usage

(MB)

ccCA Disk Usage (MB)

151

becomes more efficient in CPU time is observed to be between three and four atoms in a

molecule, while the point is between four and five atoms in terms of disk space.

Practical experience has shown that the computational bottlenecks in the original ccCA

formulation are the MP2/aug‐cc‐pVQZ step in the calculation of the reference energy and the

MP2(full)/aug‐cc‐pCVTZ core‐valence step. Although the CCSD(T) step formally scales as ,

versus the scaling of MP2 ( is the number of basis functions), the drastic difference in

the size of the basis sets used between the CCSD(T)/cc‐pVTZ and the MP2/aug‐cc‐pVQZ steps

makes the MP2 step more computationally demanding. In the core‐valence step, the

correlation of the (n‐1)s and (n‐1)p orbitals precludes this step as more computationally

demanding than the CCSD(T) step. An analysis of each step in RI‐ccCA with the corresponding

Figure 7.3 Correlation between molecular size and average CPU time and disk space savedusing RI‐ccCA and RI‐ccCA+L, relative to ccCA.

50%

55%

60%

65%

70%

75%

80%

85%

90%

95%

100%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Savings Relative to ccCA

Number of Atoms

CPU (RI‐ccCA)

CPU (RI‐ccCA+L)

Disk (RI‐ccCA)

Disk (RI‐ccCA+L)

152

step in the original ccCA implementation shows that every molecule of this study has a

significant CPU time reduction in the MP2/aug‐cc‐pVQZ step – the average CPU reduction being

91%. Further, 95 of the 102 molecules of this study have a significant CPU time reduction in the

MP2(full)/aug‐cc‐pCVTZ step as well, with an average CPU time reduction of 77%. Thus, both

bottlenecks of the original ccCA implementation are greatly reduced when the RI

approximation is invoked.

7.4 Conclusions

A new formulation of the ccCA method employing the RI approximation in the MP2

steps has been developed and benchmarked for total energies, atomization energies, and

enthalpies of formation for a test set containing 102 molecules containing first and second row,

main group atoms. This new formulation of ccCA, denoted RI‐ccCA, reproduces total energies

within 0.433 mEh, on average, compared to the original implementation of ccCA. Further, the

average error in enthalpies of formation introduced by the RI approximation in ccCA is

0.27 kcal/mol. There are dramatic CPU time and disk space reductions afforded by using

RI‐ccCA over ccCA, both of which correlate with molecular size. The average CPU time reduction

by RI‐ccCA over ccCA for our test set is 72%, while the average disk space reduction is 97%. If

the conventional CCSD(T) step of ccCA is replaced with DF‐LCCSD(T), then the computational

cost of RI‐ccCA may be further reduced when larger molecules are to be investigated. It has

been shown here that employing RI‐ccCA+L in molecules composed of four, or more, atoms

affords a greater computational savings in terms of CPU time and disk space than with RI‐ccCA:

the average CPU time reduction for RI‐ccCA+L being 76% with a disk space reduction of 96%,

153

relative to ccCA. However, employing RI‐ccCA+L results in an average deviation from the ccCA

total energies of 4.193 mEh and 2.63 kcal/mol in enthalpies of formation.

In summary, the RI‐ccCA method is recommended as a means to significantly reducing

the computational cost of ccCA calculations, thus accessing much larger molecules in correlated

ab initio computations. The original ccCA method is limited severely by bottlenecks in the

MP2/aug‐cc‐pVQZ reference energy step and MP2(full)/aug‐cc‐pCVTZ core‐valence energy step;

RI‐ccCA has demonstrated that these bottlenecks can successfully be overcome without the

introduction of significant error. When even larger molecules (i.e. 14, or more, atoms) are to be

studied, RI‐ccCA+L affords greater computational cost reduction, but at a significant (i.e.

>1.0 kcal/mol) cost in accuracy relative to ccCA.

154

Table 7.1 Differences between ccCA and RI‐ccCA total energies (mEh), relative CPU times, and relative disk space using different combinations of RI‐SCF and RI‐MP2 auxiliary basis sets.a

RI‐SCF: cc‐pV(n+1)Z (unc) JKFIT JKFIT RI‐MP2: cc‐pV(n+1)Z (unc) cc‐pV(n+1)Z (unc) cc‐pVnZ‐RI

Molecule ΔE ΔCPUb ΔDisk ΔE ΔCPU ΔDisk ΔE ΔCPU ΔDisk

H2P+ 350.516 ‐41% ‐97% 0.099 ‐48% ‐97% 0.497 ‐50% ‐97%Cl‐ 474.637 140% ‐66% ‐0.875 132% ‐67% 0.068 137% ‐67%CH3

‐ 0.254 ‐65% ‐98% 0.026 ‐69% ‐98% ‐0.228 ‐69% ‐98%H2N

‐ 0.264 ‐35% ‐97% 0.106 ‐40% ‐97% ‐0.198 ‐46% ‐97%HO‐ 0.261 57% ‐89% 0.152 33% ‐89% ‐0.286 35% ‐89%H3Si

‐ 327.021 ‐64% ‐98% 0.208 ‐72% ‐98% 0.445 ‐71% ‐98%H2P

‐ 356.559 ‐42% ‐97% ‐0.197 ‐48% ‐97% 0.254 ‐48% ‐97%HS‐ 404.713 22% ‐89% ‐0.622 2% ‐89% ‐0.040 4% ‐89%CN‐ 0.315 ‐20% ‐93% 0.485 ‐30% ‐93% 0.113 ‐29% ‐93%H4N

+ 0.147 ‐82% ‐99% 0.278 ‐85% ‐99% 0.072 ‐84% ‐99%H3O

+ 0.075 ‐69% ‐98% 0.328 ‐74% ‐98% 0.072 ‐73% ‐98%C2H3

+ 0.284 ‐78% ‐98% 0.437 ‐82% ‐98% 0.126 ‐82% ‐98%H5Si

+ 319.564 ‐85% ‐98% 0.269 ‐87% ‐98% 0.495 ‐87% ‐98%H4P

+ 341.686 ‐79% ‐98% 0.158 ‐82% ‐98% 0.570 ‐83% ‐98%H3S

+ 389.406 ‐68% ‐98% ‐0.066 ‐74% ‐98% 0.439 ‐73% ‐98%ClH2

+ 446.055 ‐42% ‐97% ‐0.421 ‐53% ‐97% 0.422 ‐50% ‐97%

MAD 213.235 ‐‐‐ ‐‐‐ 0.296 ‐‐‐ ‐‐‐ 0.270 ‐‐‐ ‐‐‐ Average 213.235 ‐34% ‐94% 0.023 ‐42% ‐94% 0.176 ‐42% ‐94%Worst 474.637 140% ‐66% 0.875 132% ‐67% 0.570 137% ‐67%Best 0.075 ‐85% ‐99% 0.026 ‐87% ‐99% 0.040 ‐87% ‐99%a. The designation unc denotes an uncontracted basis set, while n+1 indicates the use of the next highest ζ‐

level basis set as an auxiliary basis set; diffuse or core‐valence functions where employed as appropriate. b. A negative CPU time indicates a savings, while a positive CPU time indicates additional computational cost

relative to the ccCA formulation.

155

Table 7.2 Total energies (Eh) computed with ccCA, RI‐ccCA, and RI‐ccCA+L; differences (mEh) are relative to the ccCA energies.

Molecule ccCA RI‐ccCA Δ RI‐ccCA+L Δ

H2 ‐1.175407 ‐1.175400 0.007 ‐1.175435 ‐0.029FH ‐100.542079 ‐100.542018 0.062 ‐100.541326 0.753CO ‐113.382281 ‐113.382141 0.140 ‐113.381165 1.116N2 ‐109.589789 ‐109.589645 0.144 ‐109.588652 1.136ClH ‐462.098616 ‐462.098244 0.372 ‐462.097963 0.653F2 ‐199.691783 ‐199.691647 0.136 ‐199.690894 0.888CS ‐437.190630 ‐437.190179 0.451 ‐437.189259 1.371SiO ‐365.257919 ‐365.257305 0.614 ‐365.256470 1.449FCl ‐561.342347 ‐561.341851 0.496 ‐561.338518 3.830P2 ‐684.080694 ‐684.079763 0.931 ‐684.078040 2.654Cl2 ‐922.948692 ‐922.947899 0.793 ‐922.946226 2.466CH2 (

1A1) ‐39.143711 ‐39.143654 0.058 ‐39.143045 0.666H2O ‐76.486631 ‐76.486570 0.061 ‐76.484975 1.655HCN ‐93.465715 ‐93.465588 0.128 ‐93.464445 1.271SiH2 (

1A1) ‐291.083557 ‐291.083118 0.440 ‐291.082906 0.651H2S ‐400.355464 ‐400.355073 0.391 ‐400.354201 1.263CO2 ‐188.702734 ‐188.702538 0.196 ‐188.699995 2.739N2O ‐184.770279 ‐184.770111 0.168 ‐184.768027 2.252O3 ‐225.572994 ‐225.572791 0.203 ‐225.568743 4.252HOCl ‐537.306204 ‐537.305728 0.477 ‐537.301985 4.220F2O ‐274.892314 ‐274.892085 0.229 ‐274.889664 2.650OCS (1Σ+) ‐512.564911 ‐512.564404 0.508 ‐512.561960 2.951SO2 ‐549.702915 ‐549.701722 1.193 ‐549.698728 4.186ClNO ‐591.458141 ‐591.457653 0.488 ‐591.452826 5.315CS2 ‐836.424684 ‐836.423843 0.841 ‐836.421280 3.405NH3 ‐56.588920 ‐56.588850 0.070 ‐56.586946 1.974C2H2 ‐77.353816 ‐77.353694 0.122 ‐77.352388 1.428CH2O ‐114.565162 ‐114.565025 0.137 ‐114.563153 2.009PH3 ‐343.833344 ‐343.832821 0.522 ‐343.831837 1.507H2O2 ‐151.657387 ‐151.657257 0.130 ‐151.655386 2.001C2N2 (cyanogen) ‐185.731564 ‐185.731322 0.241 ‐185.731833 ‐0.270CF2O ‐313.267733 ‐313.267447 0.286 ‐313.262409 5.324BF3 ‐324.849794 ‐324.849496 0.298 ‐324.838528 11.266


156



NF3 ‐354.385557 ‐354.385217 0.340 ‐354.379991 5.566AlF3 ‐542.788964 ‐542.787986 0.978 ‐542.779940 9.025PF3 ‐641.973120 ‐641.972078 1.042 ‐641.963711 9.408ClF3 ‐761.076734 ‐761.075790 0.944 ‐761.062503 14.231BCl3 ‐1409.452639 ‐1409.451398 1.242 ‐1409.444572 8.068AlCl3 ‐1627.434618 ‐1627.433047 1.571 ‐1627.427542 7.076CH4 ‐40.524638 ‐40.524577 0.061 ‐40.522940 1.698SiH4 ‐292.354689 ‐292.354178 0.511 ‐292.353490 1.199H2CCO ‐152.671361 ‐152.671160 0.200 ‐152.667884 3.477HCOOH ‐189.881673 ‐189.881471 0.202 ‐189.878317 3.356CH3Cl ‐501.412376 ‐501.411928 0.448 ‐501.408947 3.429CH2F2 ‐239.182375 ‐239.182172 0.202 ‐239.177620 4.755CHF3 ‐338.531489 ‐338.531202 0.287 ‐338.523792 7.698CH2Cl2 ‐962.300361 ‐962.299511 0.851 ‐962.295417 4.945CF4 ‐437.876533 ‐437.876153 0.380 ‐437.865977 10.556SiF4 ‐690.014081 ‐690.012771 1.309 ‐690.000711 13.370CHCl3 ‐1423.185544 ‐1423.184278 1.266 ‐1423.179789 5.755CCl4 ‐1884.065161 ‐1884.063465 1.695 ‐1884.059607 5.554SiCl4 ‐2136.160888 ‐2136.158674 2.214 ‐2136.151404 9.484C2H4 ‐78.607303 ‐78.607174 0.130 ‐78.604732 2.572CH3OH ‐115.787113 ‐115.786990 0.122 ‐115.784263 2.850H4N2 ‐111.925364 ‐111.925217 0.148 ‐111.922832 2.533CH3CN ‐132.802880 ‐132.802697 0.183 ‐132.801448 1.432CH2CHF ‐177.937846 ‐177.937642 0.203 ‐177.933273 4.573CH3SH ‐439.673890 ‐439.673417 0.474 ‐439.669506 4.385C2H2O2 (glyoxal) ‐227.947083 ‐227.946801 0.282 ‐227.943490 3.593CH2CHCl ‐539.502427 ‐539.501902 0.525 ‐539.498001 4.426CF3CN ‐430.787698 ‐430.787283 0.415 ‐430.777846 9.852C2F4 ‐475.907903 ‐475.907454 0.449 ‐475.896113 11.791C2Cl4 ‐1922.168969 ‐1922.167177 1.792 ‐1922.160485 8.484CH3NH2 ‐95.895784 ‐95.895655 0.130 ‐95.893119 2.665C3H4 (propyne) ‐116.685968 ‐116.685792 0.176 ‐116.684600 1.368C3H4 (allene) ‐116.683773 ‐116.683570 0.202 ‐116.680163 3.610C3H4 (cyclopropene) ‐116.648595 ‐116.648407 0.188 ‐116.644572 4.023


157



CH3CHO ‐153.902179 ‐153.901977 0.201 ‐153.899086 3.093C2H4O (oxirane) ‐153.860359 ‐153.860181 0.178 ‐153.857261 3.098CH2CHCN ‐170.889667 ‐170.889421 0.247 ‐170.888004 1.664C2H4S (thiirane) ‐477.769073 ‐477.768548 0.524 ‐477.763997 5.076CH3NO2 ‐245.153126 ‐245.152849 0.278 ‐245.150450 2.676CH3ONO ‐245.148955 ‐245.148682 0.273 ‐245.144513 4.443CH3COF ‐253.263187 ‐253.262916 0.271 ‐253.257901 5.286CH3COCl ‐614.814794 ‐614.814212 0.582 ‐614.808962 5.832C2H6 ‐79.845000 ‐79.844877 0.123 ‐79.842448 2.553C2H5N (aziridine) ‐133.969353 ‐133.969166 0.187 ‐133.966802 2.551CH3SiH3 ‐331.693833 ‐331.693237 0.596 ‐331.689192 4.641CH3COOH ‐229.216864 ‐229.216597 0.267 ‐229.212217 4.646Si2H6 ‐583.527283 ‐583.526269 1.014 ‐583.523184 4.100CH3CH2Cl ‐540.739582 ‐540.739070 0.513 ‐540.735142 4.440C3H6 (propene) ‐117.935232 ‐117.935038 0.194 ‐117.932089 3.143C3H6 (cyclopropane) ‐117.923244 ‐117.923064 0.180 ‐117.920378 2.866CH3CH2SH ‐478.998514 ‐478.997979 0.535 ‐478.993309 5.205CH3SCH3 ‐478.996589 ‐478.996026 0.562 ‐478.989100 7.488C4H4O (furan) ‐230.115063 ‐230.114735 0.328 ‐230.110649 4.414C4H4S (thiophene) ‐554.015825 ‐554.015100 0.725 ‐554.008405 7.421C4H6 (trans‐butadiene) ‐156.030557 ‐156.030298 0.258 ‐156.027003 3.554C4H6 (methylene cyclopropane) ‐155.998952 ‐155.998703 0.249 ‐155.995909 3.043C4H6 (bicyclo[1.1.0]butane) ‐155.987935 ‐155.987705 0.230 ‐155.986121 1.814C4H6 (cyclobutene) ‐156.012210 ‐156.011959 0.252 ‐156.008233 3.978CH3COCH3 ‐193.237065 ‐193.236802 0.263 ‐193.232670 4.395C4H5N (pyrrole) ‐210.238145 ‐210.237822 0.323 ‐210.233647 4.498CH3CH2CH3 ‐119.168795 ‐119.168616 0.179 ‐119.165132 3.663CH3CH2CH2Cl ‐580.063634 ‐580.063067 0.567 ‐580.057960 5.673C4H8 (cyclobutane) ‐157.249681 ‐157.249437 0.244 ‐157.246822 2.859C4H8 (isobutene) ‐157.264516 ‐157.264266 0.250 ‐157.260594 3.922CH3CHOHCH3 ‐194.445233 ‐194.444984 0.249 ‐194.438735 6.499C6H6 (benzene) ‐232.305096 ‐232.304724 0.372 ‐232.298473 6.622N(CH3)3 ‐174.526943 ‐174.526693 0.250 ‐174.521397 5.546CH3OCH2CH3 ‐194.424232 ‐194.423989 0.243 ‐194.418414 5.818


158



C5H8 (spiropentane) ‐195.318795 ‐195.318505 0.291 ‐195.314928 3.867C4H10 (trans‐butane) ‐158.492755 ‐158.492504 0.251 ‐158.487871 4.884C4H10 (isobutane) ‐158.495109 ‐158.494867 0.242 ‐158.490073 5.036

Average 0.433 4.193Best 0.007 ‐0.270

Worst 2.214 14.231

159

Table 7.3 Enthalpies of formation (ΔHf°) at 298 K (kcal/mol) computed with ccCA, RI‐ccCA, and RI‐ccCA+L.

Molecule Expt. ccCA Δexpt RI‐ccCA ΔccCA Δexpt RI‐ccCA+L ΔccCA Δexpt

H2 0.00 ‐1.09 ‐1.09 ‐1.08 0.00 ‐1.08 ‐1.11 ‐0.02 ‐1.11FH ‐65.10 ‐66.68 ‐1.58 ‐66.64 0.04 ‐1.54 ‐66.21 0.47 ‐1.11CO ‐26.40 ‐27.18 ‐0.78 ‐27.10 0.09 ‐0.70 ‐26.48 0.70 ‐0.08N2 0.00 ‐0.71 ‐0.71 ‐0.62 0.09 ‐0.62 0.01 0.71 0.01ClH ‐22.10 ‐22.84 ‐0.74 ‐22.60 0.23 ‐0.50 ‐22.43 0.41 ‐0.33F2 0.00 0.08 0.08 0.16 0.09 0.16 0.64 0.56 0.64CS 66.90 66.81 ‐0.09 67.09 0.28 0.19 67.67 0.86 0.77SiO ‐24.60 ‐23.59 1.01 ‐23.21 0.39 1.39 ‐22.68 0.91 1.92FCl ‐13.20 ‐13.82 ‐0.62 ‐13.51 0.31 ‐0.31 ‐11.41 2.40 1.79P2 34.30 34.91 0.61 35.50 0.58 1.20 36.58 1.67 2.28Cl2 0.00 0.11 0.11 0.60 0.50 0.60 1.65 1.55 1.65CH2 (

1A1) 102.50 101.65 ‐0.85 101.69 0.04 ‐0.81 102.07 0.42 ‐0.43H2O ‐57.80 ‐60.61 ‐2.81 ‐60.57 0.04 ‐2.77 ‐59.57 1.04 ‐1.77HCN 31.50 30.87 ‐0.63 30.95 0.08 ‐0.55 31.67 0.80 0.17SiH2 (

1A1) 65.20 63.22 ‐1.98 63.50 0.28 ‐1.70 63.63 0.41 ‐1.57H2S ‐4.90 ‐6.14 ‐1.24 ‐5.89 0.25 ‐0.99 ‐5.34 0.79 ‐0.44CO2 ‐94.10 ‐94.43 ‐0.33 ‐94.30 0.12 ‐0.20 ‐92.71 1.72 1.39N2O 19.60 19.28 ‐0.32 19.38 0.11 ‐0.22 20.69 1.41 1.09O3 34.10 35.49 1.39 35.62 0.13 1.52 38.16 2.67 4.06HOCl ‐17.80 ‐19.18 ‐1.38 ‐18.88 0.30 ‐1.08 ‐16.53 2.65 1.27F2O 5.90 6.19 0.29 6.33 0.14 0.43 7.85 1.66 1.95OCS (1Σ+) ‐33.10 ‐34.33 ‐1.23 ‐34.01 0.32 ‐0.91 ‐32.48 1.85 0.62SO2 ‐70.90 ‐71.11 ‐0.21 ‐70.36 0.75 0.54 ‐68.48 2.63 2.42


160



ClNO 12.40 13.23 0.83 13.54 0.31 1.14 16.57 3.34 4.17CS2 28.00 27.42 ‐0.58 27.95 0.53 ‐0.05 29.55 2.14 1.55NH3 ‐11.00 ‐12.44 ‐1.44 ‐12.40 0.04 ‐1.40 ‐11.20 1.24 ‐0.20C2H2 54.20 54.78 0.58 54.86 0.08 0.66 55.68 0.90 1.48CH2O ‐26.00 ‐27.07 ‐1.07 ‐26.99 0.09 ‐0.99 ‐25.81 1.26 0.19PH3 1.30 0.77 ‐0.53 1.09 0.33 ‐0.21 1.71 0.95 0.41H2O2 ‐32.50 ‐33.72 ‐1.22 ‐33.64 0.08 ‐1.14 ‐32.46 1.26 0.04C2N2 (cyanogen) 73.30 74.98 1.68 75.13 0.15 1.83 74.81 ‐0.17 1.51CF2O ‐149.10 ‐145.47 3.63 ‐145.29 0.18 3.81 ‐142.13 3.34 6.97BF3 ‐271.40 ‐270.52 0.88 ‐270.33 0.19 1.07 ‐263.45 7.07 7.95NF3 ‐31.60 ‐31.87 ‐0.27 ‐31.66 0.21 ‐0.06 ‐28.38 3.49 3.22AlF3 ‐289.00 ‐290.15 ‐1.15 ‐289.54 0.61 ‐0.54 ‐284.49 5.66 4.51PF3 ‐229.10 ‐228.41 0.69 ‐227.76 0.65 1.34 ‐222.51 5.90 6.59ClF3 ‐38.00 ‐39.08 ‐1.08 ‐38.49 0.59 ‐0.49 ‐30.15 8.93 7.85BCl3 ‐96.30 ‐94.73 1.57 ‐93.95 0.78 2.35 ‐89.67 5.06 6.63AlCl3 ‐139.70 ‐141.26 ‐1.56 ‐140.27 0.99 ‐0.57 ‐136.82 4.44 2.88CH4 ‐17.90 ‐18.76 ‐0.86 ‐18.73 0.04 ‐0.83 ‐17.70 1.07 0.20SiH4 8.20 6.58 ‐1.62 6.90 0.32 ‐1.30 7.33 0.75 ‐0.87H2CCO ‐11.40 ‐11.66 ‐0.26 ‐11.54 0.13 ‐0.14 ‐9.48 2.18 1.92HCOOH ‐90.50 ‐91.52 ‐1.02 ‐91.39 0.13 ‐0.89 ‐89.41 2.11 1.09CH3Cl ‐19.60 ‐20.40 ‐0.80 ‐20.12 0.28 ‐0.52 ‐18.25 2.15 1.35CH2F2 ‐107.70 ‐108.78 ‐1.08 ‐108.65 0.13 ‐0.95 ‐105.80 2.98 1.90CHF3 ‐166.60 ‐167.33 ‐0.73 ‐167.15 0.18 ‐0.55 ‐162.49 4.83 4.11


161



CH2Cl2 ‐22.80 ‐22.79 0.01 ‐22.25 0.53 0.55 ‐19.68 3.10 3.12CF4 ‐223.00 ‐223.70 ‐0.70 ‐223.46 0.24 ‐0.46 ‐217.08 6.62 5.92SiF4 ‐386.00 ‐385.19 0.81 ‐384.37 0.82 1.63 ‐376.80 8.39 9.20CHCl3 ‐24.70 ‐23.77 0.93 ‐22.97 0.79 1.73 ‐20.16 3.61 4.54CCl4 ‐22.90 ‐22.00 0.90 ‐20.94 1.06 1.96 ‐18.52 3.48 4.38SiCl4 ‐158.40 ‐157.13 1.27 ‐155.74 1.39 2.66 ‐151.18 5.95 7.22C2H4 12.50 11.92 ‐0.58 12.00 0.08 ‐0.50 13.54 1.61 1.04CH3OH ‐48.20 ‐49.58 ‐1.38 ‐49.50 0.08 ‐1.30 ‐47.79 1.79 0.41H4N2 22.80 21.29 ‐1.51 21.38 0.09 ‐1.42 22.88 1.59 0.08CH3CN 17.70 17.60 ‐0.10 17.71 0.11 0.01 18.50 0.90 0.80CH2CHF ‐33.20 ‐34.66 ‐1.46 ‐34.53 0.13 ‐1.33 ‐31.79 2.87 1.41CH3SH ‐5.50 ‐6.60 ‐1.10 ‐6.30 0.30 ‐0.80 ‐3.85 2.75 1.65C2H2O2 (glyoxal) ‐50.70 ‐51.99 ‐1.29 ‐51.81 0.18 ‐1.11 ‐49.73 2.25 0.97CH2CHCl 8.90 5.01 ‐3.89 5.34 0.33 ‐3.56 7.79 2.78 ‐1.11CF3CN ‐118.40 ‐118.54 ‐0.14 ‐118.28 0.26 0.12 ‐112.36 6.18 6.04C2F4 ‐157.40 ‐161.16 ‐3.76 ‐160.88 0.28 ‐3.48 ‐153.76 7.40 3.64C2Cl4 ‐5.80 ‐4.94 0.86 ‐3.82 1.12 1.98 0.38 5.32 6.18CH3NH2 ‐5.50 ‐6.60 ‐1.10 ‐6.51 0.08 ‐1.01 ‐4.92 1.67 0.58C3H4 (propyne) 44.20 44.50 0.30 44.61 0.11 0.41 45.36 0.86 1.16C3H4 (allene) 45.50 45.43 ‐0.07 45.55 0.13 0.05 47.69 2.27 2.19C3H4 (cyclopropene) 66.20 67.81 1.61 67.93 0.12 1.73 70.33 2.52 4.13CH3CHO ‐39.70 ‐40.34 ‐0.64 ‐40.21 0.13 ‐0.51 ‐38.40 1.94 1.30C2H4O (oxirane) ‐12.60 ‐13.34 ‐0.74 ‐13.22 0.11 ‐0.62 ‐11.39 1.94 1.21


162



CH2CHCN 43.20 45.44 2.24 45.59 0.15 2.39 46.48 1.04 3.28C2H4S (thiirane) 19.60 17.65 ‐1.95 17.98 0.33 ‐1.62 20.83 3.19 1.23CH3NO2 ‐17.80 ‐18.93 ‐1.13 ‐18.76 0.17 ‐0.96 ‐17.25 1.68 0.55CH3ONO ‐15.90 ‐16.61 ‐0.71 ‐16.44 0.17 ‐0.54 ‐13.82 2.79 2.08CH3COF ‐105.70 ‐106.22 ‐0.52 ‐106.05 0.17 ‐0.35 ‐102.91 3.32 2.79CH3COCl ‐58.00 ‐58.55 ‐0.55 ‐58.18 0.37 ‐0.18 ‐54.89 3.66 3.11C2H6 ‐20.60 ‐21.18 ‐0.58 ‐21.10 0.08 ‐0.50 ‐19.57 1.60 1.03C2H5N (aziridine) 30.20 29.59 ‐0.61 29.71 0.12 ‐0.49 31.19 1.60 0.99CH3SiH3 ‐7.00 ‐7.34 ‐0.34 ‐6.96 0.37 0.04 ‐4.42 2.91 2.58CH3COOH ‐103.40 ‐104.51 ‐1.11 ‐104.34 0.17 ‐0.94 ‐101.59 2.92 1.81Si2H6 19.00 16.58 ‐2.42 17.22 0.64 ‐1.78 19.15 2.57 0.15CH3CH2Cl ‐26.80 ‐27.48 ‐0.68 ‐27.15 0.32 ‐0.35 ‐24.69 2.79 2.11C3H6 (propene) 4.80 4.20 ‐0.60 4.32 0.12 ‐0.48 6.18 1.97 1.38C3H6 (cyclopropane) 12.70 12.38 ‐0.32 12.49 0.11 ‐0.21 14.18 1.80 1.48CH3CH2SH ‐11.10 ‐12.04 ‐0.94 ‐11.70 0.34 ‐0.60 ‐8.77 3.27 2.33CH3SCH3 ‐9.00 ‐10.23 ‐1.23 ‐9.87 0.35 ‐0.87 ‐5.53 4.70 3.47C4H4O (furan) ‐8.30 ‐7.90 0.40 ‐7.70 0.21 0.60 ‐5.13 2.77 3.17C4H4S (thiophene) 27.50 27.56 0.06 28.01 0.46 0.51 32.21 4.66 4.71C4H6 (trans‐butadiene) 26.30 26.72 0.42 26.88 0.16 0.58 28.95 2.23 2.65C4H6 (methylene cyclopropane) 47.90 46.29 ‐1.61 46.45 0.16 ‐1.45 48.20 1.91 0.30C4H6 (bicyclo[1.1.0]butane) 51.90 53.63 1.73 53.78 0.14 1.88 54.77 1.14 2.87C4H6 (cyclobutene) 37.40 38.53 1.13 38.69 0.16 1.29 41.03 2.50 3.63CH3COCH3 ‐51.90 ‐53.63 ‐1.73 ‐53.46 0.16 ‐1.56 ‐50.87 2.76 1.03


163



C4H5N (pyrrole) 25.90 26.06 0.16 26.26 0.20 0.36 28.88 2.82 2.98CH3CH2CH3 ‐25.00 ‐26.19 ‐1.19 ‐26.07 0.11 ‐1.07 ‐23.89 2.30 1.11CH3CH2CH2Cl ‐31.50 ‐33.22 ‐1.72 ‐32.87 0.36 ‐1.37 ‐29.66 3.56 1.84C4H8 (cyclobutane) 6.80 5.98 ‐0.82 6.13 0.15 ‐0.67 7.77 1.79 0.97C4H8 (isobutene) ‐4.00 ‐5.15 ‐1.15 ‐4.99 0.16 ‐0.99 ‐2.68 2.46 1.32CH3CHOHCH3 ‐65.20 ‐65.71 ‐0.51 ‐65.56 0.16 ‐0.36 ‐61.64 4.08 3.56C6H6 (benzene) 19.70 20.95 1.25 21.19 0.23 1.49 25.11 4.16 5.41N(CH3)3 ‐5.70 ‐7.18 ‐1.48 ‐7.02 0.16 ‐1.32 ‐3.70 3.48 2.00CH3OCH2CH3 ‐51.70 ‐53.97 ‐2.27 ‐53.82 0.15 ‐2.12 ‐50.32 3.65 1.38C5H8 (spiropentane) 44.30 44.20 ‐0.10 44.38 0.18 0.08 46.62 2.43 2.32C4H10 (trans‐butane) ‐30.00 ‐31.88 ‐1.88 ‐31.72 0.16 ‐1.72 ‐28.82 3.06 1.18C4H10 (isobutane) ‐32.10 ‐33.07 ‐0.97 ‐32.91 0.15 ‐0.81 ‐29.91 3.16 2.19

Average ‐0.50 0.27 ‐0.23 2.63 2.13Minimum ‐3.89 0.00 ‐3.56 ‐0.17 ‐1.77Maximum 3.63 1.39 3.81 8.93 9.20

MAD 1.03 0.27 1.00 2.63 2.30

164

Table 7.4 CPU times and disk space usage of RI‐ccCA and RI‐ccCA+L relative to ccCA.

CPU Disk

Molecule RI‐ccCA RI‐ccCA+L RI‐ccCA RI‐ccCA+L

H2 ‐32% 129% ‐87% ‐91% FH 10% 58% ‐84% ‐84% CO ‐45% ‐24% ‐90% ‐83% N2 ‐42% ‐19% ‐90% ‐83% ClH ‐2% 36% ‐85% ‐79% F2 ‐40% ‐17% ‐89% ‐80% CS ‐46% ‐27% ‐91% ‐81% SiO ‐46% ‐25% ‐91% ‐78% FCl ‐47% ‐31% ‐89% ‐79% P2 ‐57% ‐42% ‐90% ‐79% Cl2 ‐56% ‐44% ‐88% ‐77% CH2 (

1A1) ‐66% ‐55% ‐96% ‐96% H2O ‐66% ‐55% ‐96% ‐96% HCN ‐57% ‐44% ‐94% ‐89% SiH2 (

1A1) ‐61% ‐50% ‐97% ‐95% H2S ‐66% ‐56% ‐96% ‐95% CO2 ‐70% ‐59% ‐93% ‐87% N2O ‐60% ‐65% ‐96% ‐93% O3 ‐62% ‐69% ‐97% ‐96% HOCl ‐58% ‐64% ‐97% ‐97% F2O ‐59% ‐76% ‐97% ‐97% OCS (1Σ+) ‐62% ‐68% ‐96% ‐93% SO2 ‐71% ‐77% ‐97% ‐96% ClNO ‐60% ‐71% ‐97% ‐96% CS2 ‐64% ‐70% ‐96% ‐92% NH3 ‐86% ‐83% ‐98% ‐99% C2H2 ‐75% ‐70% ‐96% ‐93% CH2O ‐80% ‐77% ‐97% ‐96% PH3 ‐83% ‐79% ‐98% ‐98% H2O2 ‐84% ‐83% ‐98% ‐98% C2N2 (cyanogen) ‐66% ‐79% ‐97% ‐96% CF2O ‐65% ‐85% ‐97% ‐97% BF3 ‐64% ‐87% ‐97% ‐97%


165


CPU Disk


NF3 ‐61% ‐87% ‐98% ‐98% AlF3 ‐64% ‐86% ‐97% ‐96% PF3 ‐66% ‐87% ‐97% ‐98% ClF3 ‐56% ‐87% ‐97% ‐97% BCl3 ‐68% ‐88% ‐96% ‐97% AlCl3 ‐66% ‐86% ‐96% ‐96% CH4 ‐88% ‐86% ‐99% ‐99% SiH4 ‐86% ‐83% ‐98% ‐98% H2CCO ‐74% ‐82% ‐98% ‐98% HCOOH ‐74% ‐83% ‐98% ‐98% CH3Cl ‐77% ‐84% ‐98% ‐99% CH2F2 ‐72% ‐85% ‐98% ‐99% CHF3 ‐67% ‐89% ‐98% ‐99% CH2Cl2 ‐72% ‐85% ‐98% ‐99% CF4 ‐63% ‐92% ‐98% ‐98% SiF4 ‐63% ‐91% ‐97% ‐98% CHCl3 ‐69% ‐91% ‐97% ‐98% CCl4 ‐66% ‐92% ‐97% ‐98% SiCl4 ‐69% ‐93% ‐96% ‐98% C2H4 ‐89% ‐89% ‐98% ‐98% CH3OH ‐91% ‐91% ‐98% ‐99% H4N2 ‐91% ‐92% ‐98% ‐99% CH3CN ‐79% ‐87% ‐98% ‐99% CH2CHF ‐74% ‐81% ‐98% ‐99% CH3SH ‐80% ‐87% ‐98% ‐99% C2H2O2 (glyoxal) ‐72% ‐88% ‐98% ‐99% CH2CHCl ‐76% ‐83% ‐98% ‐99% CF3CN ‐66% ‐95% ‐98% ‐99% C2F4 ‐63% ‐95% ‐97% ‐98% C2Cl4 ‐68% ‐93% ‐96% ‐97% CH3NH2 ‐85% ‐89% ‐99% ‐99% C3H4 (propyne) ‐83% ‐91% ‐99% ‐99% C3H4 (allene) ‐84% ‐92% ‐98% ‐99%


166


CPU Disk


C3H4 (cyclopropene) ‐84% ‐91% ‐98% ‐99% CH3CHO ‐80% ‐89% ‐98% ‐99% C2H4O (oxirane) ‐82% ‐91% ‐98% ‐99% CH2CHCN ‐78% ‐91% ‐98% ‐99% C2H4S (thiirane) ‐80% ‐89% ‐98% ‐99% CH3NO2 ‐72% ‐90% ‐98% ‐99% CH3ONO ‐74% ‐91% ‐98% ‐99% CH3COF ‐73% ‐88% ‐98% ‐99% CH3COCl ‐75% ‐89% ‐98% ‐99% C2H6 ‐94% ‐96% ‐99% ‐99% C2H5N (aziridine) ‐83% ‐92% ‐99% ‐99% CH3SiH3 ‐84% ‐90% ‐99% ‐99% CH3COOH ‐75% ‐90% ‐98% ‐99% Si2H6 ‐87% ‐92% ‐98% ‐99% CH3CH2Cl ‐81% ‐90% ‐98% ‐99% C3H6 (propene) ‐85% ‐94% ‐99% ‐100% C3H6 (cyclopropane) ‐85% ‐94% ‐99% ‐100% CH3CH2SH ‐80% ‐90% ‐98% ‐100% CH3SCH3 ‐83% ‐92% ‐98% ‐100% C4H4O (furan) ‐73% ‐83% ‐98% ‐98% C4H4S (thiophene) ‐75% ‐86% ‐98% ‐99% C4H6 (trans‐butadiene) ‐85% ‐92% ‐97% ‐99% C4H6 (methylene cyclopropane) ‐88% ‐93% ‐98% ‐99% C4H6 (bicyclo[1.1.0]butane) ‐87% ‐92% ‐98% ‐99% C4H6 (cyclobutene) ‐86% ‐89% ‐98% ‐99% CH3COCH3 ‐79% ‐92% ‐99% ‐100% C4H5N (pyrrole) ‐75% ‐87% ‐98% ‐99% CH3CH2CH3 ‐89% ‐93% ‐98% ‐99% CH3CH2CH2Cl ‐91% ‐96% ‐98% ‐99% C4H8 (cyclobutane) ‐88% ‐94% ‐98% ‐99% C4H8 (isobutene) ‐90% ‐96% ‐98% ‐99% CH3CHOHCH3 ‐90% ‐96% ‐98% ‐99% C6H6 (benzene) ‐88% ‐94% ‐97% ‐98%


167


CPU Disk


C5H8 (spiropentane) ‐88% ‐96% ‐98% ‐99% C4H10 (trans‐butane) ‐89% ‐97% ‐98% ‐99% C4H10 (isobutane) ‐90% ‐97% ‐98% ‐99%

Average ‐72% ‐76% ‐97% ‐96% Best ‐94% ‐97% ‐99% ‐100%

Worst 10% 129% ‐84% ‐77%

168

CHAPTER 8

A SYSTEMATIC INVESTIGATION OF DIHALOGEN‐μ‐DICHALCOGENIDES†

8.1 Introduction

Mixed halogen‐chalcogen species, in particular the fluorine‐ and oxygen‐containing class

of compounds, have drawn much attention in the literature over the past seven decades both

for their applications and unique bonding. Molecules containing a fluorine‐oxygen bond, such

as FOOF, have many applications, such as a near‐room‐temperature fluorinating agent for

various actinides and xenon.202‐205 Sometimes referred to as oxyfluorides, these compounds

have also been reported to be of use in various rocket fuels.206 They are also speculated to

participate in various reactions of atmospheric importance, such as the depletion of ozone by a

photolysis mechanism that produces FO radicals.207,208 The sulfur analog of FOOF, FSSF, has also

been implicated in the depletion of ozone.209 Other halogenated persulfides, ClSSCl, BrSSBr,

ClSeSeCl, and BrSeSeBr, are used extensively as plasma etchants in manufacturing various

microelectronic devices.210‐213 Disulfide and diselenide bonds are particularly important in

biological systems, specifically cystine‐rich proteins in the case of disulfides. Molecules with

disulfide bridges are studied to gain insight into the relationship between the strength of the

S‐S bond with respect to varying the terminal substituents. An understanding of this

relationship can provide insight into the structure‐activity relationship of proteins, especially in

† This entire chapter is adapted from B.P. Prascher and A.K. Wilson, “A Computational Study of Dihalogen‐μ‐dichalcogenides: XAAX (X = F, Cl, Br; A = S, Se),” J. Mo. Struct. THEOCHEM, 2007, 814(1‐3), 1, with permission from Elsevier.

169

proteins where a disulfide bond is involved in the chemistry of the active site or if the disulfide

bond stabilizes the tertiary structure of the protein.214‐218

While the applications of FOOF and a few of its sulfur analogs are known in the

literature, there has been disagreement between computational and experimental geometries.

In the earliest microwave spectroscopy experiment reported by Jackson219 on FOOF, a short O‐

O bond length resembling that of O2 and a long F‐O bond relative to FO were observed.

Subsequent reports in the literature fail to reproduce Jackson’s geometry, instead reporting a

peroxide‐like (longer) O‐O bond and shorter F‐O bonds. A careful electron diffraction study

undertaken by Hedberg and coworkers220 verified the structure Jackson had observed, and

noted that a better understanding of the structure of FOOF is crucial to elucidating the

mechanism by which its fluorinating abilities arise.

The structure of FOOF has F‐O bond lengths of 1.575 Å,219,220 longer than that of FO at

1.354 Å.221 FOOF also has an O‐O bond length of 1.216 Å,219,220 resembling the oxygen molecule,

1.208 Å,222,223 and not a typical peroxide (1.44 – 1.46 Å).224 From microwave spectroscopy, it is

known that FSSF also has a short A‐A bond and long A‐X bonds.225‐228 Another structural

anomaly of XSSX species is the increase in the S‐S bond length when the halogen size is

increased,229 a trend which is observed in XOOX species also.222,230

The structural anomalies of FOOF have been addressed in much detail elsewhere207 as

the result of overlap between an oxygen lone pair orbital and the adjacent unoccupied F‐O σ*

orbital – a phenomenon called anomeric delocalization (see Figure 8.1). This phenomenon is

not unique to the FOOF molecule, but also exists in molecules such as α‐fluoroethers231 and

other molecules where highly electronegative atoms are in close proximity to one another.

170

From an experimental standpoint, anomeric effects can account for the shortened O‐O bond

and lengthened F‐O bond, while, computationally, anomeric effects can present significant

challenges to predicting structures accurately, even for some of the most sophisticated

theoretical approaches.232

Computational methods such as multi‐reference configuration interaction (MRCI) and

coupled cluster (CC) with basis sets of varying sizes have been employed to study the FOOF

molecule.232 Additionally, composite approaches such as Gaussian‐1 and Gaussian‐2 (G1 and

G2, respectively) have been used to study FOOF.233 Both density functional theory (DFT) and CC

methods reproduce the experimental geometry with the least amount of error compared with

the other methods listed above.232 Absolute errors of 0.006 – 0.06 Å in bond lengths and 0.2 –

1.2° in angles occur with various pure and hybrid density functionals, while errors of 0.008 –

0.048 Å in bond lengths and 0.1 – 1.3° in angles occur with coupled cluster computations

employing various basis sets.

Small basis sets (i.e. the double‐ζ level), coupled with electron correlation methods,

have been shown to fortuitously yield accurate structures of FOOF compared with

experiment.232,234,235 Structures with deviations from experiment less than 0.01 Å compared

Figure 8.1 The orbital representation of anomeric delocalization in XAAX systems: the lonepair on a bridging A atom delocalizes into the adjacent σ*(AX) orbital.

171

have been computed in the past by adding non‐optimized polarization235 and diffuse

functions236 to a double‐ζ basis set. An improvement in the accuracy of the O‐O bond length is

typically observed with a loss of accuracy in the O‐F bond length using this approach.

Computational studies with smaller, more rigid basis sets, such as that of Scuseria237 where the

polarization functions of the oxygen and fluorine basis sets were constrained to have equal

exponents, tend to give results closer to experiment. This work, and others,232,235 do not

condone the use of such nonstandard approaches to predictive quantum chemistry since the

method cannot be readily applied to study other molecular systems. Instead, one would like to

apply a standard and systematic approach to studying similar systems.

In this chapter, the heavier analogs of the FOOF molecule with the structure XAAX (X =

F, Cl, Br and A = S, Se) are investigated. When this investigation was published, no systematic ab

initio or DFT study on the structure, frequencies, and anomeric effects had been performed on

the sulfur and selenium XAAX analogs of FOOF, nor had CC computations on the relative

stabilities of the XAAX conformation compared with other A2X2 isomers been reported.


Both hybrid DFT and CC methods have been utilized for this work. DFT is particularly

attractive because of its formal scaling (where is the number of basis functions). The

DFT method employed is Becke’s three‐parameter exchange scheme (B3)166 coupled with the

Lee‐Yang‐Parr (LYP) correlation functional.165 The third parameterization of Vosko, Wilk, and

Nusair correlation functional (VWN‐3) has been used. The coupled cluster method including

singles, doubles, and non‐iterative, perturbative triples is also employed, CCSD(T).31,38

172

A series of natural orbital (NO) analyses using singles and doubles configuration

interaction (CISD) has been performed. The CISD NOs provide insight into the amount of multi‐

reference character of the system and the most important orbitals involved in computing the

dynamic correlation energy. The NOs reported here also demonstrate how well correlated

methods, specifically truncated ab initio methods, describe XAAX systems with respect to

anomeric delocalization.

The correlation consistent, polarized valence basis sets, cc‐pVnZ (where n = D, T, and

Q)94‐101 have been employed in this study. In computations involving second row atoms, the

tight d correlation consistent basis sets, cc‐pV(n+d)Z,101 were employed. Additionally, the

augmented correlation consistent basis sets, aug‐cc‐pVnZ and aug‐cc‐pV(n+d)Z,95,99,100 were

used for describing long‐range effects, specifically lone pair interactions.

This investigation is concerned with the chemistry of disulfide and diselenide

compounds, following the results of experimental work regarding the stability of the

conformations of X2A2 compounds.215,216,225,238‐241 Structures have been fully optimized at each

level of theory and basis set, utilizing symmetry where present. The B3LYP calculations were

performed using the Gaussian 03 software suite,174 while the Molpro software package180 was

used for CCSD(T) and CISD computations. Harmonic vibrational frequencies were computed

analytically with B3LYP and numerically with CCSD(T) at the aug‐cc‐pVTZ level. Computed

anharmonic frequencies are included for comparison.242

173


8.3.1 Stability of X2A2 Compounds

Following the reported stabilities of several conformations of FOOF by Kraka et al.,232

computations have been performed at the aug‐cc‐pVTZ level to compare the stabilities of the

linear, cis, trans, and X2AA conformations of the heavier analogs with their gauche forms. The

computed relative stabilities (in kcal/mol) are tabulated in Table 8.1 for both B3LYP and

CCSD(T). It is found that all the conformations are significantly less stable than the gauche form.

For F2S2, the B3LYP value shows FSSF to be more stable than F2SS, but CCSD(T) does not. This

apparent contradiction to what is known experimentally regarding FSSF being more stable than

F2SS215,216,225 has been investigated further through a series of computations using the

augmented double‐ through quadruple‐ζ basis sets with both methods. As the basis set size

increases, B3LYP predicts that FSSF is the more stable compound. The CCSD(T) computations

with increasing basis set size show F2SS as the more stable conformation . However, when a

thermal correction (for 298 K) is applied to the CCSD(T) electronic energy, FSSF is shown to be

more stable than the F2SS conformation by 0.04 kcal/mol.

Other isomers of the XAAX‐type molecules have been investigated previously,

specifically the XAXA and AXXA isomers of iodine oxides.243,244 We have briefly examined these

two isomeric forms here using second‐order Møller‐Plesset perturbation theory (MP2) and the

aug‐cc‐pVTZ basis set. MP2 has been used since CCSD(T) computations without symmetry can

become time intensive, and MP2 will recover a significant percent of the CCSD(T) correlation

energy. Further, test computations on the cis, trans, and linear forms have shown MP2 relative

174

stabilities to be comparable to CCSD(T) stability energies. For most XAAX compounds of this

study, the corresponding XAXA compound is found to dissociate into two XA fragments as

doublet electronic states. Only in the case of FSFS does the molecule dissociate into FSF ( ) +

S ( ). Similarly, the AXXA compounds display no bound minima as tetraatomic molecules, but

dissociate into two XA fragments, each with a doublet electron configuration.

8.3.2 XAAX Structures

Optimized geometries are listed in Table 8.2, Table 8.3, and Table 8.4 for the fluorine,

chlorine, and bromine species, respectively. For the fluorine compounds, relative to experiment

both B3LYP and CCSD(T) result in the least error at the aug‐cc‐pVQZ level as compared with the

other basis sets – the errors being 0.013 Å and 0.011 Å, respectively. The CCSD(T) S‐F bond

length is in good agreement with experiment, with a difference of only 0.01 Å at the triple‐ζ

level. CCSD(T) only slightly outperforms B3LYP for the bond lengths and angles. B3LYP dihedral

angles tend to converge to the experimental value while CCSD(T) dihedrals converge away from

experiment. In FSeSeF, the optimized Se‐Se bond length converges away from the experimental

value for both methods and their smallest errors relative to experiment are 0.023 Å for B3LYP

and 0.014 Å for CCSD(T) at the cc‐pVDZ level. Just as in the case of FSSF, the Se‐F bonds with

CCSD(T) are within chemical accuracy by the triple‐ζ basis set. The dihedral in FSeSeF is closest

to the experimental value for both methods when the smallest basis set is employed. The

differences between the computed angles and the experimental angle range from 5.06° – 5.33°

for B3LYP and from 4.19° – 4.55° for CCSD(T), where an experimental value of 100° has been

reported by Haas.245 In either the case of B3LYP or CCSD(T), adding diffuse functions does not

175

significantly affect the bond lengths, but does affect the angles and dihedral angles. The

inclusion of the tight d function in FSSF is important, especially at the double‐ and triple‐ζ levels

where the bond length can be affected by up to 0.016 Å and 0.010 Å, respectively.

For the chlorine species, ClSSCl, the optimized S‐S and S‐Cl bond lengths in ClSSCl move

towards experiment as the basis set size is increased for both CCSD(T) and B3LYP. The results

are within 0.01 Å with CCSD(T) at the cc‐pV(Q+d)Z level. Angles are closer to experiment with

CCSD(T) than with B3LYP at the largest basis set level. Dihedrals show a similar trend as that of

the angles. The tight d basis set is again noted to improve upon the description of bond lengths

and angles as compared with the standard correlation consistent basis sets for both B3LYP and

CCSD(T), which suggests that core polarization can be important in DFT.126,246 For example, with

B3LYP, the error in the bond lengths is lowered by 0.019 Å – 0.036 Å at the double‐ζ level when

a tight d function is added to the valence and augmented basis sets. For ClSeSeCl, the B3LYP

computations using the quadruple‐ζ basis sets (original and tight d) converge to similar values

for the Se‐Cl bond length. It is observed that the addition of the tight d function will not

significantly affect bond lengths in DFT computations since this method is typically not very

basis set dependent. CCSD(T) is more basis set dependent, and we observe that the addition of

a tight d function also has little effect on the computed bond lengths of ClSeSeCl.

Optimized bond lengths for S‐S and S‐Br in BrSSBr are within 0.01 Å of experiment when

computed with the CCSD(T) method, however, the basis set level at which the errors are lowest

varies. Table 8.4 shows that for cc‐pVQZ and aug‐cc‐pVQZ, the CCSD(T) deviation from

experiment in the S‐S bond is 0.005 Å and 0.006 Å, respectively. The valence and augmented

tight d sets deviate by 0.003 Å and 0.005 Å, respectively, at the triple‐ζ level. Computed B3LYP

176

S‐S bond lengths and bond angles converge away from the experimental values when the basis

set size is increased. Additionally, the tight d function helps to accelerate this convergence

away from the experimental values. The opposite trend is observed in the S‐Br bond length

with B3LYP: the larger basis sets yield the more accurate geometries. In BrSeSeBr, as seen in

ClSeSeCl, there is no effect of adding diffuse functions since the valence and augmented sets

converge on the same values at the quadruple‐ζ level. The B3LYP computations show a

deviation of 0.023 Å in the Se‐Se bond and 0.143 Å in the Se‐Br bond relative to experiment at

the quadruple‐ζ level of both families of basis sets, while CCSD(T) shows deviations of

approximately 0.014 Å and 0.111 Å in the Se‐Se bond and Se‐Br bond, respectively. The addition

of diffuse functions does not significantly affect these results, which is not surprising since the

cc‐pVQZ basis set already contains functions that are rather diffuse.

8.3.3 Vibrational Frequency Analyses

Computed harmonic infrared (IR) vibrational frequencies at the aug‐cc‐pVTZ level for the

six molecules in this investigation are presented in Table 8.5. The assignments of computed

normal modes concur with experimental assignments. There is an artifact in the S‐Br

asymmetric stretch in which the computed frequencies deviate from the experimental

frequencies of Frenzel et al.241 by 39.6 cm‐1 and 67.3 cm‐1 for B3LYP and CCSD(T), respectively.

The magnitude of the deviation between theory and experiment in this particular case not

consistent with computed results of the other normal modes for this compound and the results

of two other experimental determinations of the spectrum of BrSSBr have been included for

comparison. An explanation of the discrepancy between later experiments and that of Frenzel

177

et al. is offered by Forneris et al.239 as due to liquid Br2 dissolved in the BrSSBr sample from

which the spectrum was taken. The dissolved Br2 impurity has the effect of dampening the

observed S‐Br asymmetric stretch frequency considerably.

To compare our computations with experimental IR spectra, computations of the

anharmonic corrections to the normal modes of FSSF have been performed at the

CCSD(T)/cc‐pVDZ level242 and documented in Table 8.6. The largest difference between the

harmonic and anharmonic frequencies in FSSF occurs in the S‐S stretching mode and has a

magnitude of 11.5 cm‐1. The correction for anharmonicity to the zero‐point energy is computed

to be 0.017 kcal/mol. From these anharmonic computations and the low deviation of the

computed harmonic frequencies from the experimental values, it is postulated that the

anharmonic corrections for the other dihalo‐μ‐dichalcogenides investigated here will be

negligible. As a result, the computed harmonic frequencies, as well as geometries, may be

compared to the experimental values without the introduction of significant error.

8.3.4 Anomeric Effects

The two lowest unoccupied NOs for FSSF are displayed in Figure 8.2. These orbitals

represent F‐S σ* orbitals with equal occupation numbers of 0.063. This indicates that the

heaviest contributors to the dynamic correlation energy are excited configurations involving

these σ* orbitals. This observation is in accord with the conclusion reported by Kraka et al. that

the unique geometry of FOOF is a result of anomeric delocalization.232 These F‐S σ* orbitals are

formally unoccupied in the ground state, but symmetry allows for overlap between them and

adjacent lone pairs. The result is a destabilization of the F‐S bond and a stabilization of the S‐S

178

bond, which accounts for the experimentally observed S2‐like bond in FSSF and the elongated

F‐S bonds. Further analysis of the truncated CI wavefunction indicates that the weighting

coefficient of the HF reference state is 0.925, which indicates some multi‐reference character in

the system. Previous analyses of the FOOF molecule by Kraka et al.232 and Feller et al.234

concluded that advanced ab initio methods that include excited configurations through

quadruple excitations would be important in the quantitative description of the FOOF structure

and the exact wavefunction. These highly‐excited configurations, they claimed, would be

important because the anomeric delocalization in FOOF involves two lone pairs simultaneously

delocalized. The wavefunction that would describe such delocalization is a quadruply‐excited

configuration. Considering the results obtained in this investigation do not include quadruply‐

excited configurations, there is evidence for anomeric effects in FSSF, but the impact on the

overall structure is diminished compared with FOOF. This is evidenced by the computed

geometries at the CCSD(T) level, where systematic improvements in the basis set lead to lower

Figure 8.2 The lowest unoccupied CISD natural orbitals of FSSF: both contour plots are F‐Sσ* orbitals (14A, left; 13B, right). The perspective is down the z‐axis (iso‐contour value: 0.095).

179

deviations from experiment.

Figure 8.3 shows plots of the lone pair orbitals involved in anomeric delocalization in

FSSF, ClSSCl, and BrSSBr. Analysis of the overlap between the lone pair orbitals and the

corresponding σ* orbitals demonstrates that as the halogen increases in size, the lone pair

orbitals do not have the radial extent to significantly overlap with the σ* orbitals and anomeric

delocalization is diminished. This is a plausible explanation for the change in the geometry of

XSSX to an S2‐type bond length and XS‐type halogen‐sulfur bond length when the halogen

increases in size.

Inspection of the computed and experimental geometries of FSeSeF indicates that

anomeric effects are markedly decreased compared with those of FSSF. The Se‐Se bond

measures 2.25 Å, compared with 2.166 Å in the selenium dimer.170 Additionally, the Se‐F bond

distance is 1.77 Å, while the SeF molecule has a bond length of 1.740 Å.170 Since the chalcogen

bond length is not shortened, nor is the chalcogen‐halogen bond lengthened as considerably as

in FOOF or FSSF, it is clear that anomeric effects do not play a significant role in determining the

Figure 8.3 The molecules FSSF (left), ClSSCl (center), and BrSSBr (right), and the lone pairorbitals responsible for anomeric delocalization. The prespective is down the z‐axis (iso‐contourvalues: 0.06, 0.03, and 0.03, respectively).

180

structure of the selenium compounds in this investigation.

The unique structure of these compounds is due to the presence of anomeric

delocalization, and this delocalization is directly related to the radial extent of the lone pair

orbitals. The other factor in determining the impact of anomeric effects in these species is

symmetry. In computations performed at the CCSD(T)/aug‐cc‐pV(T+d)Z level on trans‐FSSF

(C2h), the S‐S bond extends from 1.908 Å to 2.126 Å, and the S‐F bond recedes from 1.641 Å to

1.626 Å. The lone pairs cannot overlap with the F‐S σ* orbitals in this conformation, thus

leaving the structure with a typical disulfide bond length and a S‐F bond length resembling that

of diatomic SF. This is not new to the literature, as it has been reported by Samdal et al.229 for a

variety of dihalo‐μ‐dichalcogenides computed at the HF/6‐31G* level. The present investigation

reports for the first time correlated computations demonstrating that anomeric effects

diminish as the dihedral angle changes. Kraka et al.232 also performed similar computations with

B3LYP and reported that trans‐FOOF has a peroxide‐like structure with O‐F bond lengths similar

to that of molecular FO.

8.4 Conclusions

For the sulfur systems in this investigation, geometries accurate to 0.011 Å are possible

with CCSD(T) and 0.013 Å with B3LYP when the correlation consistent basis sets of quadruple‐ζ

quality are employed. The largest deviations, relative to experimental values occur for the

geometries of FSSF. The geometries of the selenium compounds optimized with CCSD(T)

deviate by no more than 0.111 Å in bond lengths, while B3LYP geometries deviate by as much

as 0.143 Å in bond lengths relative to experiment when computed with the cc‐pVQZ basis set.

181

Bond angles deviate up to 5.07° in CCSD(T)/cc‐pVQZ computations with respect to experiment,

however, overall CCSD(T) angles result in less deviation from experimental values as compared

with B3LYP.

The use of the cc‐pV(n+d)Z basis sets improves the computed geometries at the double‐

and triple‐ζ basis set levels for the CCSD(T) geometries, and, as expected, only slightly improves

B3LYP‐optimized geometries. The inclusion of diffuse functions in the basis has little impact on

the geometry of the species investigated here. For the computed harmonic vibrational

frequencies, the maximum deviation from experimental values is almost 67 cm‐1, which is

attributed to damping from Br2 impurities in one of the experimental studies referenced on

BrSSBr. Other compounds have computed frequencies that deviate no more than 25 cm‐1.

Computed anharmonic effects of the FSSF molecule revealed that their impact on bond lengths

and harmonic frequencies is negligible.

Anomeric effects have been investigated by others as the source of the unique

geometry of the FOOF molecule. The current investigation shows that the impact of anomeric

delocalization on FSSF is decreased as compared with the impact on FOOF, and that it has

almost no impact on the structure of FSeSeF. Upon increasing the halogen size to chlorine and

bromine, the radial extent of the lone pair orbitals is not sufficient for significant overlap with

the appropriate σ* orbitals, and thus anomeric effects on the geometric structure are almost

nonexistent. When FSSF is rotated into the trans conformation, anomeric delocalization cannot

occur due to broken symmetry. The natural consequence of this is an increase in S‐S bond

length and a decrease in the S‐F bond length.

182

Table 8.1 Stabilities (kcal/mol) of the different isomers of X2A2 molecules relative to the gauche isomer, XAAX, computed with B3LYP and CCSD(T).

System Method Linear cis trans X2AA

F2S2 B3LYP 103.1 63.5 29.3 1.9 CCSD(T) 107.4 65.5 27.3 ‐0.5a

Cl2S2 B3LYP 86.4 21.3 17.0 13.1 CCSD(T) 94.6 18.7 14.1 13.4

Br2S2 B3LYP 65.7 20.1 15.3 13.7 CCSD(T) ‐‐‐b 17.0 12.1 14.5

F2Se2 B3LYP 94.0 61.6 36.2 2.8 CCSD(T) 98.5 63.4 37.1 1.1

Cl2Se2 B3LYP 61.0 17.7 14.0 12.2 CCSD(T) ‐‐‐b 15.5 11.8 12.7

Br2Se2 B3LYP 57.7 16.4 12.6 13.0 CCSD(T) ‐‐‐b 13.9 10.1 14.0

a. Including a thermal correction to 298 K in F2SS and FSSF shows FSSF to be the more stable isomer. b. Optimization leads to dissociation: X2A2 2X + A2.

183

Table 8.2 B3LYP and CCSD(T) optimized geometries (Å and degrees) of the fluorine species using various correlation consistent basis sets.

B3LYP CCSD(T)

FSSF ζ VnZ V(n+d)Z aVnZ aV(n+d)Z VnZ V(n+d)Z aVnZ aV(n+d)Z Expt.

Re(AA) D 1.952 1.919 1.941 1.909 1.961 1.929 1.951 1.921

T 1.924 1.907 1.920 1.903 1.929 1.913 1.924 1.908

Q 1.913 1.902 1.912 1.901 1.909 1.900 1.908 1.899 1.888a

Re(AF) D 1.688 1.673 1.699 1.683 1.690 1.677 1.699 1.686

T 1.657 1.647 1.662 1.652 1.642 1.634 1.649 1.641

Q 1.654 1.648 1.655 1.649 1.636 1.631 1.639 1.633 1.635a

α D 107.40 108.58 107.35 108.54 106.04 107.13 106.44 107.53

T 107.95 108.67 107.96 108.68 107.26 107.95 107.17 107.86

Q 108.31 108.74 108.31 108.74 107.69 108.10 107.63 108.03 108.3a

δ D 89.28 89.21 88.15 88.06 89.04 88.93 87.95 87.87

T 88.22 88.17 87.80 87.78 87.89 87.86 87.48 87.46

Q 87.97 87.93 87.82 87.80 87.58 87.55 87.44 87.41 87.9a

FSeSeF

Re(AA) D 2.227 ‐‐‐ 2.218 ‐‐‐ 2.237 ‐‐‐ 2.228 ‐‐‐

T 2.213 ‐‐‐ 2.211 ‐‐‐ 2.219 ‐‐‐ 2.215 ‐‐‐

Q 2.211 ‐‐‐ 2.210 ‐‐‐ 2.207 ‐‐‐ 2.206 ‐‐‐ 2.25b

Re(AF) D 1.810 ‐‐‐ 1.819 ‐‐‐ 1.808 ‐‐‐ 1.811 ‐‐‐

T 1.790 ‐‐‐ 1.796 ‐‐‐ 1.770 ‐‐‐ 1.777 ‐‐‐

Q 1.790 ‐‐‐ 1.792 ‐‐‐ 1.767 ‐‐‐ 1.770 ‐‐‐ 1.77b

α D 105.33 ‐‐‐ 105.08 ‐‐‐ 104.19 ‐‐‐ 104.38 ‐‐‐

T 105.14 ‐‐‐ 105.06 ‐‐‐ 104.55 ‐‐‐ 104.37 ‐‐‐

Q 105.07 ‐‐‐ 105.06 ‐‐‐ 104.53 ‐‐‐ 104.46 ‐‐‐ 100b

δ D 90.14 ‐‐‐ 88.47 ‐‐‐ 90.03 ‐‐‐ 88.63 ‐‐‐

T 88.80 ‐‐‐ 88.17 ‐‐‐ 88.79 ‐‐‐ 88.33 ‐‐‐

Q 88.15 ‐‐‐ 88.18 ‐‐‐ 88.37 ‐‐‐ 88.26 ‐‐‐ 90b

a. Kuczkowski, R. L. J. Am. Chem. Soc. 1964, 86, 3617. b. Haas, A. J. Fluor. Chem. 1986, 32, 415.

184

Table 8.3 B3LYP and CCSD(T) optimized geometries (Å and degrees) of the chlorine species using various correlation consistent basis sets.

B3LYP CCSD(T)

ClSSCl ζ VnZ V(n+d)Z aVnZ aV(n+d)Z VnZ V(n+d)Z aVnZ aV(n+d)Z Expt.

Re(AA) D 1.987 1.961 1.983 1.956 2.011 1.985 2.005 1.980

T 1.964 1.950 1.963 1.949 1.983 1.970 1.980 1.968

Q 1.956 1.947 1.955 1.947 1.966 1.958 1.965 1.958 1.950a

Re(ACl) D 2.132 2.112 2.133 2.113 2.116 2.098 2.129 2.111

T 2.109 2.096 2.110 2.098 2.081 2.072 2.083 2.073

Q 2.102 2.094 2.101 2.094 2.067 2.061 2.067 2.061 2.055a

Α D 108.71 109.40 108.56 109.24 106.47 107.07 106.44 107.03

T 109.00 109.38 109.04 109.41 106.96 107.31 106.81 107.17

Q 109.16 109.42 109.20 109.42 107.08 107.31 107.05 107.27 107.66a

Δ D 87.33 87.15 87.39 87.29 86.27 86.17 86.17 86.09

T 87.14 87.10 87.16 87.10 85.76 85.73 85.47 85.45

Q 87.02 87.00 87.10 87.00 85.46 85.48 85.38 85.37 85.24a

ClSeSeCl

Re(AA) D 2.261 2.264 2.257 2.261 2.282 2.285 2.277 2.279

T 2.253 2.256 2.252 2.254 2.267 2.268 2.265 2.266

Q 2.252 2.253 2.252 2.253 2.256 2.257 2.257 2.257 2.28b

Re(ACl) D 2.246 2.238 2.248 2.240 2.235 2.227 2.246 2.238

T 2.227 2.222 2.229 2.224 2.204 2.200 2.206 2.202

Q 2.224 2.221 2.224 2.221 2.193 2.191 2.195 2.191 2.13b

α D 107.14 107.08 107.05 106.96 104.82 104.75 104.87 104.84

T 107.02 106.97 106.98 106.93 104.98 104.94 104.72 104.71

Q 106.98 106.96 106.97 106.95 104.83 104.81 104.80 104.73 106b

δ D 88.41 88.39 88.30 88.18 87.65 87.55 87.21 87.13

T 87.88 87.89 87.89 87.88 86.91 86.87 86.62 86.58

Q 87.78 87.83 87.88 87.74 86.62 86.59 86.33 86.49 83b

a. Marsden, C. J.; Brown, R. D.; Godfrey, P. D. J. Chem. Soc. Chem. Comm. 1979, 399. b. Forneris, R.; Hennies, C. E. J. Mol. Struct. 1970, 5, 449.

185

Table 8.4 B3LYP and CCSD(T) optimized geometries (Å and degrees) of the bromine species using various correlation consistent basis sets.

B3LYP CCSD(T)

BrSSBr ζ VnZ V(n+d)Z aVnZ aV(n+d)Z VnZ V(n+d)Z aVnZ aV(n+d)Z Expt.

Re(AA) D 1.992 1.961 1.988 1.957 2.017 1.987 2.014 1.984

T 1.969 1.952 1.968 1.952 1.991 1.977 1.990 1.975

Q 1.960 1.950 1.960 1.950 1.975 1.966 1.974 1.965 1.980a

Re(ABr) D 2.293 2.288 2.294 2.290 2.282 2.276 2.291 2.286

T 2.273 2.270 2.274 2.270 2.244 2.241 2.245 2.242

Q 2.269 2.267 2.269 2.267 2.231 2.229 2.231 2.230 2.24a

α D 109.35 110.01 109.18 109.90 106.83 107.44 106.69 107.33

T 109.56 109.98 109.64 109.98 107.16 107.50 107.04 107.38

Q 109.77 110.01 109.77 110.01 107.21 107.44 107.17 107.41 105a

δ D 87.29 87.37 87.33 87.61 85.59 85.68 85.53 85.65

T 86.99 87.06 86.98 87.06 85.06 85.11 84.62 84.70

Q 86.99 87.01 86.99 87.01 84.70 84.76 84.51 84.55 83.5a

BrSeSeBr

Re(AA) D 2.267 ‐‐‐ 2.263 ‐‐‐ 2.290 ‐‐‐ 2.286 ‐‐‐

T 2.259 ‐‐‐ 2.258 ‐‐‐ 2.276 ‐‐‐ 2.275 ‐‐‐

Q 2.257 ‐‐‐ 2.257 ‐‐‐ 2.266 ‐‐‐ 2.266 ‐‐‐ 2.28b

Re(ABr) D 2.401 ‐‐‐ 2.405 ‐‐‐ 2.393 ‐‐‐ 2.404 ‐‐‐

T 2.386 ‐‐‐ 2.387 ‐‐‐ 2.363 ‐‐‐ 2.364 ‐‐‐

Q 2.383 ‐‐‐ 2.383 ‐‐‐ 2.351 ‐‐‐ 2.351 ‐‐‐ 2.24b

α D 107.85 ‐‐‐ 107.76 ‐‐‐ 105.12 ‐‐‐ 105.03 ‐‐‐

T 107.72 ‐‐‐ 107.73 ‐‐‐ 105.21 ‐‐‐ 104.94 ‐‐‐

Q 107.69 ‐‐‐ 107.71 ‐‐‐ 104.92 ‐‐‐ 104.94 ‐‐‐ 106b

δ D 88.28 ‐‐‐ 88.15 ‐‐‐ 86.86 ‐‐‐ 86.50 ‐‐‐

T 87.88 ‐‐‐ 87.84 ‐‐‐ 86.10 ‐‐‐ 86.73 ‐‐‐

Q 87.80 ‐‐‐ 87.68 ‐‐‐ 85.58 ‐‐‐ 85.56 ‐‐‐ 83b

a. Hirota, E. Bull. Chem. Soc. Jpn. 1958, 31, 130. b. Forneris, R.; Hennies, C. E. J. Mol. Struct. 1970, 5, 449.

186

Table 8.5 B3LYP and CCSD(T) harmonic vibrational frequencies (cm‐1) computed with the aug‐cc‐pVTZ basis set.

Molecule Method Torsion A‐A‐X Bend

A‐A‐X Bend

A‐X Asymm. Stretch

A‐X Symm. Stretch

A‐A Stretch

FSSF B3LYP 178 282 310 596 662 697 CCSD(T) 180 283 315 600 691 724 Expt.a 182 301 319 614 680 717

ClSSCl B3LYP 96 201 231 422 435 538 CCSD(T) 89 199 236 458 466 531 Expt.b 102 205 238 434 446 540 Expt.c 102 206 240 436 449 540 Expt.d 105 208 242 (430) 448 540

BrSSBr B3LYP 59 164 193 343 351 533 CCSD(T) 55 163 197 371 373 521 Expt.b 68 170 199 304 357 534 Expt.c 66 172 200 351 365 529 Expt.d 67 170 202 356 357 534

FSeSeF B3LYP 125 170 189 320 569 593 CCSD(T) 143 181 201 328 607 628 Expt. ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐

ClSeSeCl B3LYP 75 124 141 290 352 364 CCSD(T) 72 125 144 297 374 382 Expt.c 87 130 146 288 367 367 Expt.d 91 133 150 289 349 357

BrSeSeBr B3LYP 47 98 113 255 258 290 CCSD(T) 43 97 116 273 275 292 Expt.c 50 107e 118e 265 265 292 Expt.d 63 105 119 265 265 290

a. Brown, R. D.; Pez, G. P. Spectroc. Acta 1970, 26A, 1375. b. Frenzel, C. A.; Blick, K. E. J. Chem. Phys. 1971, 55, 2715. c. Forneris, R.; Hennies, C. E. J. Mol. Struct. 1970, 5, 449. d. Frankiss, S. G. J. Mol. Struct. 1968, 2, 271. e. Hendra, P. J.; Park, P. J. D. J. Chem. Soc. A 1968, 908.

187

Table 8.6 A comparison of CCSD(T) harmonic and anharmonic vibrational frequencies (cm‐1) of FSSF computed with the cc‐pVDZ basis set.

Mode ωe ν0 Difference Torsion 173.0 170.9 ‐2.1

S‐S‐F Bend 260.7 258.0 ‐2.6 S‐S‐F Bend 293.7 292.4 ‐1.2

S‐F Asymm. Stretch 583.7 576.3 ‐7.4 S‐F Symm. Stretch 692.8 686.4 ‐6.4

S‐S Stretch 716.5 705.0 ‐11.5

188

CHAPTER 9

A SYSTEMATIC INVESTIGATION OF GERMANIUM ARSENIDES†

9.1 Introduction

Compounds containing heavy group 14 and 15 elements, especially compounds with

multiple bonds, have received considerable attention over the past three decades by both

experimentalists and theoreticians.247‐263 The surge in interest in these compounds came after

the synthesis of several non‐transient multiply‐bonded phosphorus and silicon systems in 1981,

and these species have been the subject of several reviews.250,252,260‐263 Before that time, it was

believed that multiple bonds between heavy main group elements would likely not exist due to

the so‐called double‐bond rule, which stated that elements such as silicon and phosphorus

would exhibit weak π bonds formed by their p orbitals.259,264 Subsequent attempts to synthesize

these types of compounds failed since the molecules would oligomerize in solution. The

oligomerization was found to be reduced by incorporating large, bulky substituents on the

heavy main group atoms.250,260 Since that time, several compounds that have multiply‐bonded

group 14 and 15 elements with low coordination numbers (i.e. P2, Si2, SiC, PC, etc.) have been

reported. However, compounds that are not found in earlier reviews are those with germanium

multiply‐bonded to an arsenic atom – so called arsagermenes and arsagermynes (germanium

doubly‐ and triply‐bonded to arsenic, respectively). While it is purported that these types of

systems should exist, and could be synthesized via similar mechanisms as analogous silicon

† This entire chapter is adapted from B.P. Prascher, A.M. Kavi, B. Mintz, S.M. Yockel, and A.K. Wilson, “Theoretical Investigation of the Germanium Arsenides,” Chem. Phys., 2008, 353(1‐3), 209, with permission from Elsevier.

189

phosphide systems, no such studies have been reported.

Germanium arsenides are important in various research fields. Although gallium

arsenides are routinely encountered in solid state research, germanium arsenide materials such

as GeAsS glasses and MGeAs2 (M = Zn, Cd) are of interest for their nonlinear optical properties

and electrical properties. Germanium arsenide materials have been studied both by

experimentalists and by theoreticians for their applications in solar cells and as ternary

semiconductors.247,248 Another example of an exotic germanium arsenide is the material

(Ge38As8)8+ ∙ 8 I‐, which has been characterized by x‐ray and neutron diffraction, and is a known

N‐type semiconductor.253

The only previous computational investigation of the germanium arsenide molecules

considered in the present study was performed by Lai et al.258 They utilized both correlated ab

initio methodology and density functional theory (DFT) with the 6‐311++G(d,p) and

6‐311++G(3df,3pd) basis sets (triple‐ζ quality basis sets with diffuse and polarization functions)

to describe the structures, bonding characteristics, and relative stabilities of L‐GeAs and GeAs‐L

molecules. They computed that the most stable L‐GeAs molecule of their test set should be

H2N‐GeAs, while the most stable GeAs‐L molecule should be GeAs‐AlH2. Further, Lai et al.

predicted with DFT that, for L = F, OH, and CH3, the GeAs‐L isomer should be more stable than

L‐GeAs, while their ab initio predictions showed the opposite (i.e. the L‐GeAs isomer is more

stable for these L groups). This apparent discrepancy between DFT and correlated ab initio

methodology has prompted this systematic re‐investigation of Lai’s molecules.

One of the differences between the present investigation and that of Lai et al.258 is the

use of the correlation consistent basis sets, cc‐pVnZ.94‐101 The systematic construction of the

190

correlation consistent basis sets leads to monotonic convergence towards the complete basis

set (CBS) limit of many energetic properties as the basis set size increases93 – a property not

inherent to the basis sets used by Lai et al.265

The present investigation reexamines the GeAs‐L and L‐GeAs molecules studied by Lai et

al.258 (where L = H, Li, Na, BeH, MgH, BH2, AlH2, CH3, SiH3, NH2, PH2, OH, SH, F, and Cl; and has

been extended to include Br) Reported are transition state energies, enthalpies of formation at

298 K, and enthalpies of isomerization at 298 K. Typically, the threshold for chemical accuracy

of main group molecules is considered to be 1.0 kcal/mol. To achieve this threshold of chemical

accuracy for third row, main group systems, it has been shown that scalar relativistic and spin‐

orbit effects must be included.62,266‐271 Since this is a predictive investigation into the properties

of third row, main group molecules yet to be characterized by experiment, relativistic effects

cannot be ignored.


Coupled cluster with single, double, and non‐iterative, perturbative triple excitations,

CCSD(T), is used.31,38 In addition, the hybrid three‐parameter exchange functional of Becke

(B3)166 coupled with the Lee‐Yang‐Parr (LYP)165 correlation functional, B3LYP, has been

employed. In all B3LYP computations, the third parameterization of the Vosko‐Wilk‐Nusair local

correlation functional (VWN‐3) is invoked.167

The correlation consistent basis sets94‐101 of double‐ through quadruple‐ζ quality have

been employed. For germanium arsenides that contain an alkali or alkaline earth metal, new

cc‐pVnZ basis sets discussed in Chapter 6 have been used; while for atoms in the second row of

191

the periodic table (Na, Mg, and Al‐Ar), the tight d correlation consistent basis sets,

cc‐pV(n+d)Z101 have been employed. To compute CBS limits, the exponential (3.2) and mixed

Gaussian/exponential (3.3) extrapolation schemes have been employed.85,95,132

The geometries of the L‐GeAs and GeAs‐L systems were fully optimized using both

B3LYP and CCSD(T) at each basis set level. Zero‐point energies were computed from cc‐pVTZ

harmonic vibrational frequencies using each method. Classical barriers were computed from

transition state structures with exactly one imaginary frequency along the isomerization

pathway. Enthalpies of formation and enthalpies of isomerization were determined using the

equations of Ockterski272 from atomic enthalpies of formation at 100 kPa and 298 K.195,273 All

B3LYP computations were performed using the Gaussian 03 software suite,174 while CCSD(T)

computations were performed using the Molpro software package.180

Scalar relativistic corrections for both the CCSD(T) and B3YLP energetics were

determined using both the Cowan‐Griffin (CG)274 and the second‐order, spin‐free Douglas‐Kroll

(DK) Hamiltonians.135‐137 The DK approach is variational, while the CG (perturbative) approach is

not.141,275 Relativistic corrections were computed using the B3LYP/cc‐pVTZ geometry at each

basis set level, since scalar relativistic effects are not very sensitive to changes in the

geometry.276 Spin‐orbit (SO) coupling of the L‐GeAs and GeAs‐L molecules is neglected because

they are closed shell molecules. First‐order SO corrections (see Table 9.1) are included a

posteriori in the atomic energies since the spin‐free implementations of the CG and DK

Hamiltonians do not include it intrinsically. These SO corrections were computed using multi‐

configuration self‐consistent field (MCSCF) wavefunctions and the Breit‐Pauli spin‐orbit

Hamiltonian180,277 implemented in the Molpro software package.

192


9.3.1 Optimized Structures

The bonding characteristics of L‐GeAs and GeAs‐L systems have been discussed in some

detail by Lai et al.258 The structures of L‐GeAs systems are linear, or very close to linear, and

exhibit a triple bond between the germanium and arsenic atoms not unlike a cyanide group.

However, the GeAs‐L systems exhibit a strongly bent structure as opposed to the linear

structure of an isocyanide molecule. Instead, there is a double bond between the germanium

and arsenic atoms, with a lone pair on the arsenic atom, resembling an extremely bent imine‐

type structure. The decrease in bond order in migrating the L group from the germanium to the

arsenic atom is reinforced by the increase in the Ge‐As bond length. For example (cf. Table 9.2

and Table 9.3), in the optimized structures of the L‐GeAs molecules of this study, the Ge‐As

bond length falls in the range of 2.134 – 2.187 Å for CCSD(T) and 2.106 – 2.149 Å for B3LYP.

Further evidence to support a triple bond between germanium and arsenic in the L‐GeAs

molecules is that the bond lengths resemble that of the arsenic dimer, which has a formal triple

bond and bond length of 2.103 Å.170 The Ge‐As bond length in the optimized structures of the

GeAs‐L molecules falls in the range of 2.222 – 2.389 Å for CCSD(T) and 2.205 – 2.370 Å for

B3LYP, resembling that of the GeAs diatomic at 2.283 Å.270 The Ge‐As‐L angle remains rather

small, even when L groups that are highly electropositive or highly electronegative are used,

making the bonding characteristics of germanium arsenides unique among main group

molecules.

In Table 9.2 and Table 9.3 the CCSD(T)‐ and B3LYP‐optimized geometries for the

193

different L groups of this study are listed. The results of Lai et al.258, which are included for

comparison in the tables, were computed using CCSD(T)/6‐311G++(3df,3pd)//

B3LYP/6‐311G++(d,p). Overall, for the L‐GeAs molecules optimized with CCSD(T), the L‐Ge and

Ge‐As bond lengths differ up to 0.057 Å and 0.049 Å, respectively, when compared with the

results of Lai et al. The same bond lengths differ up to 0.024 Å and 0.006 Å, respectively, when

optimized with B3LYP. The largest differences between the bond lengths of Lai et al. and this

study are in the CCSD(T) geometries of Li‐GeAs, Na‐GeAs, HBe‐GeAs, and HMg‐GeAs. The small

difference between the B3LYP results and those previously published reaffirms the fact that

B3LYP bond lengths for these species are not very basis set dependent. The average difference

between results reported here and those of Lai et al. are 0.017 Å and 0.007 Å for the L‐Ge bond

length and 0.029 Å and 0.004 Å for the Ge‐As bond length, for CCSD(T) and B3LYP, respectively.

In the GeAs‐L molecules, the variations between previously published bond lengths and

those of this study are more dramatic. The CCSD(T)‐optimized As‐L and Ge‐As bond lengths

differ the most from previous work258 up to 0.216 Å and 0.147 Å, respectively. However, the

B3LYP As‐L and Ge‐As bond lengths show less dramatic differences with respective variations

up to 0.108 Å and 0.162 Å. The largest differences between both the CCSD(T) and B3LYP bond

lengths and those of Lai et al.258 arise in the GeAs‐AlH2 molecule. Also, the increased variation

in bond lengths between the B3LYP results and those previously published indicate that B3LYP

is more sensitive to the choice of basis set for GeAs‐L systems, compared with the L‐GeAs

systems.

Bond angles are affected more than bond lengths in both the L‐GeAs and GeAs‐L

systems, however, the qualitative structure remains the same as previously reported – the

194

L‐GeAs systems are linear in X‐Ge‐As (within 5.0°), where X is the non‐hydrogen atom of the L

group, and GeAs‐L systems are severely bent in the Ge‐As‐X angle. In both the CCSD(T)‐ and

B3LYP‐optimized structures of the L‐GeAs molecules, the differences in the bond angles are in

the range 0.0 – 5.0°, compared with Lai et al.,258 but the differences in the bond angles of the

GeAs‐L molecules are in the range of 0.0 – 26.4°. The largest differences between the computed

bond angles and those of Lai et al. occur in the GeAs‐OH molecule with CCSD(T), and in the

GeAs‐AlH2 molecule with both CCSD(T) and B3LYP. Upon closer examination, the B3LYP/

6‐311++G(d,p) structures of the HO‐GeAs and GeAs‐OH isomers that Lai et al. reported were

reproduced, and it was found that explicit optimization with CCSD(T)/cc‐pVnZ, as performed in

this investigation, is responsible for the large variation in the bond angle.

Comparing the CCSD(T) to the B3LYP optimized geometries, the largest differences

between the two methods are: in the Ge‐F bond length (0.056 Å at the cc‐pVQZ level); in the

As‐OH bond length (0.186 Å at the cc‐pVTZ level); and in the Ge‐As bond length of the L‐GeAs

isomers, the largest differences between the two methods are 0.054 Å (L‐GeAs isomers) in

HBe‐GeAs at the cc‐pVTZ and cc‐pVQZ levels and 0.083 Å (GeAs‐L isomers) in GeAs‐Br at the

cc‐pVQZ level. Further, the X‐Ge‐As and Ge‐As‐X bond angles vary between the two methods up

to 3.7° and 38.1°, respectively. In general, CCSD(T) and B3LYP agree on the linearity of the

L‐GeAs molecules, however, the ranges of the bond lengths and angles of the GeAs‐L molecules

are significantly different between the two methods. The average differences in the geometries

of CCSD(T) compared with B3LYP are 0.014 Å in the Ge‐L bond, 0.031 Å in the Ge‐As bond of the

L‐GeAs isomers, 0.034 Å in the As‐L bond, 0.023 Å in the Ge‐As bond of the GeAs‐L isomers,

0.28° in the L‐Ge‐As angle, and 5.23° in the Ge‐As‐L angle.

195

9.3.2 Classical Barriers to Isomerization

The barriers to isomerization (migration) of the L group from the germanium terminus

to the arsenic terminus have been computed and are listed in Table 9.4. Each of the energies,

including the CBS limits, are computed at 0 K, with a zero‐point vibrational energy correction,

but without tunneling effects, which are only expected to be important for hydrogen migration.

Both the forward (f) and reverse (r) reactions are represented (cf. Figure 9.1). The minimum

structures are connected by a single transition state (the highest energy structure along the

lowest energy pathway connecting L‐GeAs and GeAs‐L), which is verified by exactly one

imaginary (negative) vibrational frequency along the isomerization pathway. These reported

double‐ through quadruple‐ζ classical barriers do not show monotonic behavior with respect to

increasing basis set size, yet a CBS value is still reported because the electronic energies (the

energy from which the barriers are computed) do converge monotonically.

There are some notable trends in the forward energy of activation (Eact) listed in Table

9.4. First, in the alkali metal and alkaline earth metal hydrides, which require the least energy to

migrate from the germanium atom to the arsenic atom, Li‐GeAs only requires 1.02 kcal/mol and

decreases to 0.43 kcal/mol for Na‐GeAs with the CCSD(T)/quadruple‐ζ level of theory, while Eact

Figure 9.1 A schematic of the forward (f) and reverse (r) reaction coordinates of L migrationfrom the germanium atom to the arsenic atom.

196

for HBe‐GeAs is 3.57 kcal/mol and decreases to 1.44 kcal/mol in HMg‐GeAs. A similar trend

holds for both computational methods at the quadruple‐ζ level, and in each of the other L

groups in the forward reaction: the larger the non‐hydrogen atom in the L substituent within

the periodic table group, the lower the Eact. For example, in the main group species, H3C‐GeAs

has an Eact = 24.20 kcal/mol versus H3Si‐GeAs with an Eact = 6.00 kcal/mol. The halogens, which

also follow the preceding trend, vary by less than 1.0 kcal/mol between one another in the

forward Eact. For example, the L = F, Cl, and Br values for Eact are 15.15, 14.96, and

14.68 kcal/mol, respectively for CCSD(T), and are 12.25, 11.93, and 11.23 kcal/mol, respectively

for B3LYP. Further, the forward value of Eact increases as the size of the non‐hydrogen atom

increases across the period, peaking at L = NH2 and PH2 in each period.

The same trend of increasing Eact values down a periodic group is also observed in the

reverse migration of the L = Li, Na, BeH, and MgH molecules, but is reversed for substituents of

the main group. For example, for GeAs‐BH2, the reverse Eact value is 15.63 kcal/mol and

increases to 18.38 kcal/mol when AlH2 is substituted for BH2 at the CCSD(T)/quadruple‐ζ level of

theory; the Eact is 21.48 kcal/mol for GeAs‐CH3 and increases to 22.61 kcal/mol for GeAs‐SiH3;

etc. The halogens do not have as similar Eact values in the reverse migration, where for L = F, Cl,

and Br the energies are 12.12, 18.39, and 18.34 kcal/mol at the CCSD(T)/quadruple‐ζ level.

Similar trends occur with B3LYP at the quadruple‐ζ basis set level, except that the reverse Eact

value is 19.32 kcal/mol for GeAs‐BH2 and decreases to 16.06 kcal/mol for AlH2.

The computed CCSD(T) and B3LYP Eact values differ significantly (>1.0 kcal/mol) from one

another for several of the molecules studied here. The variations in the activation energies

between CCSD(T) and B3LYP are tabulated in a supplemental table accompanying this

197

manuscript. Overall, the mean absolute difference is 2.42 kcal/mol for the forward Eact and

2.12 kcal/mol for the reverse Eact at the quadruple‐ζ basis set level between the two

computational methods. The absolute differences range from 0.07 kcal/mol in the forward Eact

for L = Li to as large as 9.31 kcal/mol in the reverse Eact for L = OH. The large variation in the

L = OH barrier is likely attributed to the large angle difference computed with our methodology

(see previous section for discussion). These variations between CCSD(T) and B3LYP indicate the

sensitivity of the barrier heights to the choice of method since the same basis sets are used

with each method.

9.3.3 Thermochemistry and Relative Stabilities

The enthalpies of formation reported here show how much energy is required to form

the molecule from the gas phase atoms. Table 9.5 and Table 9.6 include the CCSD(T)‐ and

B3LYP‐computed enthalpies of formation at 298 K for L‐GeAs and GeAs‐L, respectively, while

Table 9.7 lists the enthalpies of isomerization at 298 K. Included in the supplemental tables of

this article are the atomization energies of the CCSD(T) and B3LYP‐optimized structures of this

investigation.

The computed SO corrections in Table 9.1 agree with those referenced by Kedziora et

al.255 It is obvious from Table 9.5 and Table 9.6 that the impact of including scalar relativistic

and SO effects is significant (>1.0 kcal/mol). Consider the CG and the DK relativistic corrections

together for the entire test set. In all systems, except for L = Br, the relativistic contribution to

the enthalpies of formation of the L‐GeAs systems can be as large as 6.93 kcal/mol, while the

contribution is as large as 5.48 kcal/mol for the GeAs‐L systems. In the bromine systems, the

198

relativistic contributions are 9.20 kcal/mol for Br‐GeAs and 7.60 kcal/mol for GeAs‐Br, and are

the highest in the test set due to the presence of three third row, main group atoms. Overall,

the CG and DK corrections are similar for each molecule. The largest difference between the

two relativistic corrections is approximately 3 kcal/mol for the L = SiH3 molecules at the cc‐pVDZ

level.

Table 9.7 lists the enthalpies of isomerization at 298 K in the migration of the L group

from the germanium terminus to the arsenic terminus. The advantage of computing the

enthalpies of isomerization instead of simply computing the difference in classical barrier

heights as an indication of relative stability is that the former takes into account thermal effects

in the synthesis of the respective isomers. It is observed, from the non‐relativistic CCSD(T)

enthalpies in Table 9.7 (cf. Figure 9.2), that the GeAs‐L isomer is more stable than the L‐GeAs

isomer for each L group except L = NH2, OH, and F. The same is true of the non‐relativistic

B3LYP isomerization enthalpies for L = NH2 and F. Both L = CH3 and F have little appreciable

difference in the non‐relativistic B3LYP energy difference between the two isomers. For L = F,

non‐relativistic CCSD(T) with larger basis sets shows L‐GeAs to be the more stable isomer, but

including either the CG or DK corrections shows the GeAs‐L isomer to be more stable. The same

is true of the B3LYP non‐relativistic versus relativistic stabilities. The relative stabilities of the

isomers computed here using non‐relativistic B3LYP are in agreement with those of Lai et al.,

except in the case of L = F. Further, the CCSD(T) results show GeAs‐CH3 to be the more stable

isomer, and the GeAs‐OH isomer of this study is computed to be almost 10 kcal/mol more

stable than previously reported.

The choice of basis set can significantly affect the computed enthalpy of formation and

199

enthalpy of isomerization (cf. Table 9.5, Table 9.6, and Table 9.7). For example, increasing the

basis set from cc‐pVDZ to cc‐pVTZ lowers the enthalpy of formation of the L‐GeAs systems an

average of 20.00 kcal/mol, while increasing from cc‐pVTZ to cc‐pVQZ further lowers the

enthalpy of formation an average of 7.19 kcal/mol.

9.4 Conclusions

This investigation has reported energy‐minimized structures, classical barriers heights,

enthalpies of formation, and enthalpies of isomerization computed with both CCSD(T) and

B3LYP. In terms of optimized geometries, it is observed that both CCSD(T) and B3LYP agree

qualitatively for all of the molecules under investigation, but do not always agree quantitatively

(within 0.01 Å for bond lengths, and 1.0° in bond angles). The most notable differences

between the two computational methods stem from the GeAs‐L systems. Comparing structures

computed here with those of Lai et al., it is observed that the largest differences in the L‐GeAs

molecules occur when L = Li, Na, BeH, and MgH, while the largest differences in the GeAs‐L

molecules occur when L = AlH2.

In the computation of classical barrier heights, it is observed in the forward migration

reaction that the heavier the L substituent within a column of the periodic table, the lower the

energy of activation. Also, as the non‐hydrogen atom in the L substituent increases across the

periodic table, the forward activation energy increases, peaking at L = NH2 and PH2, then

decreasing. In the reverse migration reaction, the same trends hold for group 1 and 2

substituents, but the main group substituents follow an opposite trend of increasing the

activation energy upon reverse migration. The only exception to these periodic trends

200

encountered is in the B3LYP/quadruple‐ζ reverse energies of activation of L = BH2 and AlH2.

Predicted enthalpies of formation and isomerization agree within approximately

3 kcal/mol between CCSD(T) and B3LYP at the cc‐pVQZ level, and the present investigation

demonstrates that the inclusion of relativistic effects is vital to achieving kcal/mol chemical

accuracy. This is especially true for the newly reported enthalpies of formation and

isomerization of GeAsBr molecules, where the relativistic contributions are between 6.53 and

9.44 kcal/mol. The enthalpies of formation and isomerization should be useful as a reference

for experiment. We predict that the H2N‐GeAs and HO‐GeAs molecules should be the more

stable respective isomer at 298 K, while the other substituents of this investigation (except for

L = F) should be more stable in the GeAs‐L form. The F‐GeAs and GeAs‐F isomers have similar

extrapolated energies when relativistic CCSD(T) energies are considered. The large computed

barrier to isomerization (approximately 15 kcal/mol) indicates that both fluorine isomers should

coexist at 298 K, but isomerization should be slow.

201

Figure 9.2 A side‐by‐side comparison of non‐relativistic and Douglas‐Kroll (DK) relativistic CCSD(T) and B3LYP relative energies from mixed Gaussian/exponential extrapolations.

CCSD(T) DK‐CCSD(T) B3LYP DK‐B3LYP

202

Table 9.1 Non‐relativistic and spin‐orbit (SO) total energies (Eh) of some main group atoms computed at the MCSCF/cc‐pVTZ level of theory; SO corrections are given in kcal/mol.

Atom Term E0 E0 + ESO ESO B 2P1/2 ‐24.559329 ‐24.559373 ‐0.03 C 3P0 ‐37.704023 ‐37.704137 ‐0.07 O 3P2 ‐74.803078 ‐74.803435 ‐0.22 F 2P3/2 ‐99.399194 ‐99.399786 ‐0.37 Al 2P1/2 ‐241.893139 ‐241.893450 ‐0.19 Si 3P0 ‐288.862391 ‐288.862970 ‐0.36 S 3P2 ‐397.501584 ‐397.502416 ‐0.52 Cl 2P3/2 ‐459.478249 ‐459.479495 ‐0.78 Ge 3P0 ‐2075.366126 ‐2075.369752 ‐2.28 Br 2P3/2 ‐2572.438555 ‐2572.443599 ‐3.16

203

Table 9.2 A comparison of the CCSD(T) optimized geometries (Å and degrees) for the L‐GeAs and GeAs‐L isomers; previously reported geometries are included for comparison.

L‐GeAs GeAs‐L

L ζ Re(GeL) Re(GeAs) A(LGeAs) Re(AsL) Re(GeAs) A(GeAsL)

H D 1.540 2.152 180.0 1.606 2.276 59.3 T 1.537 2.140 180.0 1.605 2.264 59.0 Q 1.535 2.134 180.0 1.603 2.258 59.1 Lai et al. 1.529 2.111 180.0 1.593 2.251 61.8Li D 2.463 2.185 180.0 2.491 2.243 73.1 T 2.447 2.174 180.0 2.462 2.231 74.1 Q 2.449 2.168 180.0 2.458 2.225 74.5 Lai et al. 2.415 2.145 180.0 2.409 2.210 77.0

Na D 2.789 2.187 180.0 2.831 2.240 74.8 T 2.775 2.177 180.0 2.816 2.228 75.1 Q 2.781 2.171 180.0 2.814 2.222 75.6 Lai et al. 2.732 2.148 179.9 2.754 2.210 76.8

BeH D 2.181 2.172 180.0 2.142 2.264 78.1 T 2.159 2.177 180.0 2.134 2.253 78.8 Q 2.154 2.177 180.0 2.130 2.247 79.2 Lai et al. 2.156 2.128 180.0 2.109 2.239 82.6

MgH D 2.574 2.177 180.0 2.577 2.252 78.0 T 2.572 2.165 180.0 2.566 2.242 78.8 Q 2.575 2.159 180.0 2.563 2.236 79.4 Lai et al. 2.561 2.135 180.0 2.539 2.225 83.1

BH2 D 2.040 2.171 180.0 2.013 2.266 101.6 T 2.031 2.159 180.0 2.007 2.255 100.8 Q 2.028 2.153 180.0 2.003 2.248 101.5 Lai et al. 2.014 2.132 180.0 1.998 2.237 105.3

AlH2 D 2.476 2.172 180.0 2.437 2.257 91.0 T 2.465 2.160 180.0 2.429 2.246 90.5 Q 2.462 2.155 180.0 2.421 2.240 91.1 Lai et al. 2.460 2.132 179.9 2.311 2.387 69.4

CH3 D 1.981 2.153 180.0 2.063 2.296 78.1 T 1.969 2.142 180.0 2.052 2.281 77.1 Q 1.964 2.136 180.0 2.047 2.275 77.3 Lai et al. 1.966 2.117 180.0 2.071 2.260 77.5


204


L‐GeAs GeAs‐L


SiH3 D 2.399 2.161 180.0 2.382 2.287 75.2 T 2.390 2.149 180.0 2.372 2.289 74.4 Q 2.388 2.147 180.0 2.365 2.283 74.3 Lai et al. 2.394 2.121 179.9 2.350 2.298 72.9

NH2 D 1.861 2.156 175.0 1.956 2.322 85.6 T 1.834 2.147 174.7 1.870 2.306 112.7 Q 1.827 2.141 174.8 1.876 2.258 124.2 Lai et al. 1.832 2.124 175.1 1.887 2.304 108.9

PH2 D 2.350 2.160 173.9 2.301 2.366 63.2 T 2.332 2.149 173.6 2.282 2.356 63.0 Q 2.324 2.143 173.1 2.274 2.350 63.0 Lai et al. 2.339 2.123 172.1 2.293 2.360 62.9

OH D 1.799 2.157 180.0 1.978 2.357 57.7 T 1.781 2.149 180.0 1.829 2.344 96.0 Q 1.781 2.143 180.0 1.828 2.338 94.8 Lai et al. 1.779 2.126 175.0 2.044 2.301 58.0

SH D 2.226 2.159 180.0 2.394 2.324 66.4 T 2.212 2.149 180.0 2.349 2.324 64.3 Q 2.205 2.143 180.0 2.358 2.308 65.6 Lai et al. 2.223 2.125 175.6 2.379 2.310 65.1F D 1.779 2.159 180.0 1.989 2.330 58.0 T 1.780 2.153 180.0 1.948 2.313 59.9 Q 1.785 2.147 180.0 1.943 2.307 60.2 Lai et al. 1.753 2.128 179.9 2.027 2.305 58.0Cl D 2.158 2.155 180.0 2.376 2.335 65.7 T 2.136 2.149 180.0 2.353 2.320 64.7 Q 2.130 2.143 180.0 2.346 2.315 64.7 Lai et al. 2.150 2.126 180.0 2.400 2.310 65.2Br D 2.305 2.160 180.0 2.517 2.327 67.2 T 2.288 2.150 180.0 2.499 2.323 66.5 Q 2.282 2.144 180.0 2.483 2.389 68.0 Lai et al. ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐

205

Table 9.3 A comparison of the B3LYP optimized geometries (Å and degrees) for the L‐GeAs and GeAs‐L isomers; previously reported geometries are included for comparison.

L‐GeAs GeAs‐L


H D 1.537 2.116 180.0 1.603 2.254 61.3 T 1.529 2.107 180.0 1.596 2.246 60.7 Q 1.528 2.106 180.0 1.596 2.245 60.4 Lai et al. 1.529 2.111 180.0 1.593 2.251 61.8Li D 2.433 2.147 180.0 2.422 2.214 76.9 T 2.415 2.141 180.0 2.407 2.206 77.1 Q 2.413 2.140 180.0 2.403 2.205 77.2 Lai et al. 2.415 2.145 180.0 2.409 2.210 77.0

Na D 2.745 2.149 180.0 2.755 2.214 77.0 T 2.732 2.143 180.0 2.753 2.206 76.8 Q 2.729 2.142 180.0 2.751 2.205 76.7 Lai et al. 2.732 2.148 179.9 2.754 2.210 76.8

BeH D 2.158 2.130 180.0 2.111 2.242 82.3 T 2.157 2.123 180.0 2.107 2.235 82.5 Q 2.156 2.123 180.0 2.106 2.234 82.4 Lai et al. 2.156 2.128 180.0 2.109 2.239 82.6

MgH D 2.561 2.137 180.0 2.540 2.229 83.0 T 2.560 2.130 180.0 2.533 2.221 83.2 Q 2.560 2.130 180.0 2.533 2.220 83.2 Lai et al. 2.561 2.135 180.0 2.539 2.225 83.1

BH2 D 2.017 2.135 179.8 1.998 2.239 105.7 T 2.013 2.128 180.0 1.994 2.232 105.6 Q 2.012 2.127 180.0 1.993 2.231 105.7 Lai et al. 2.014 2.132 180.0 1.998 2.237 105.3

AlH2 D 2.466 2.134 180.0 2.419 2.234 95.6 T 2.460 2.127 179.7 2.415 2.226 95.6 Q 2.457 2.126 179.4 2.412 2.225 95.8 Lai et al. 2.460 2.132 179.9 2.311 2.387 69.4

CH3 D 1.971 2.120 180.0 2.072 2.262 75.8 T 1.963 2.113 180.0 2.068 2.255 76.0 Q 1.961 2.112 180.0 2.066 2.253 76.0 Lai et al. 1.966 2.117 180.0 2.071 2.260 77.5


206


L‐GeAs GeAs‐L


SiH3 D 2.391 2.124 180.0 2.340 2.309 72.3 T 2.387 2.118 180.0 2.333 2.301 72.2 Q 2.385 2.116 180.0 2.331 2.300 72.2 Lai et al. 2.394 2.121 179.9 2.350 2.298 72.9

NH2 D 1.843 2.126 177.8 1.899 2.301 110.1 T 1.822 2.120 178.5 1.877 2.294 111.4 Q 1.819 2.119 178.3 1.874 2.295 111.0 Lai et al. 1.832 2.124 175.1 1.887 2.304 108.9

PH2 D 2.341 2.127 173.4 2.283 2.370 62.6 T 2.330 2.120 172.7 2.270 2.363 62.4 Q 2.328 2.119 172.3 2.268 2.362 62.4 Lai et al. 2.339 2.123 172.1 2.293 2.360 62.9

OH D 1.788 2.129 179.8 2.034 2.301 57.9 T 1.767 2.123 180.0 2.015 2.290 57.9 Q 1.765 2.122 180.0 2.012 2.289 58.0 Lai et al. 1.779 2.126 175.0 2.044 2.301 58.0

SH D 2.222 2.129 180.0 2.366 2.316 65.0 T 2.211 2.122 180.0 2.350 2.309 64.7 Q 2.209 2.120 180.0 2.348 2.307 64.7 Lai et al. 2.223 2.125 175.6 2.379 2.310 65.1F D 1.752 2.130 180.0 2.012 2.304 56.6 T 1.731 2.125 180.0 1.988 2.294 57.5 Q 1.729 2.123 180.0 1.984 2.292 57.5 Lai et al. 1.753 2.128 179.9 2.027 2.305 58.0Cl D 2.159 2.130 180.0 2.388 2.314 65.1 T 2.141 2.123 180.0 2.370 2.305 64.8 Q 2.137 2.122 180.0 2.371 2.304 64.8 Lai et al. 2.150 2.126 180.0 2.400 2.310 65.2Br D 2.304 2.131 180.0 2.529 2.316 66.9 T 2.291 2.125 180.0 2.513 2.307 66.7 Q 2.289 2.123 180.0 2.511 2.306 66.7 Lai et al. ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐

207

Table 9.4 CCSD(T) and B3LYP classical reaction barriers (kcal/mol) for the forward (f) and reverse (r) reactions; previously reported reaction barriers are included for comparison.

L Method Eact cc‐pVDZ cc‐pVTZ cc‐pVQZ CBS(1)a CBS(2)a Lai et al.

H CCSD(T) (f) 12.80 12.67 12.60 12.59 12.57 12.20 (r) 22.17 22.63 22.64 22.63 22.64 22.50 B3LYP (f) 9.45 9.90 9.95 9.96 9.98 9.60 (r) 22.63 22.76 22.78 22.78 22.79 22.20Li CCSD(T) (f) 0.88 1.08 1.02 1.00 0.99 0.93 (r) 15.65 14.92 14.73 14.69 14.62 14.50 B3LYP (f) 0.75 1.06 1.09 1.09 1.11 1.05 (r) 16.18 15.42 15.17 15.15 15.03 15.30

Na CCSD(T) (f) 0.32 0.49 0.43 0.41 0.39 0.40 (r) 13.99 13.08 12.81 12.75 12.65 12.70 B3LYP (f) 0.30 0.56 0.60 0.60 0.62 0.51 (r) 14.55 13.42 13.28 13.27 13.20 13.40

BeH CCSD(T) (f) 3.42 3.79 3.57 3.50 3.43 3.26 (r) 20.62 21.06 21.04 21.03 21.03 20.80 B3LYP (f) 3.63 3.74 3.80 3.81 3.84 3.68 (r) 22.65 22.37 22.33 22.33 22.31 22.40

MgH CCSD(T) (f) 1.58 1.64 1.44 1.38 1.32 1.37 (r) 16.85 17.05 17.09 17.10 17.12 16.90 B3LYP (f) 1.47 1.44 1.52 1.53 1.57 1.47 (r) 18.11 17.61 17.52 17.52 17.48 17.50

BH2 CCSD(T) (f) 11.44 6.39 6.75 6.97 6.99 6.61 (r) 11.37 15.06 15.63 15.73 15.96 15.80 B3LYP (f) 9.19 9.17 9.20 9.21 9.23 7.08 (r) 19.69 19.36 19.32 19.32 19.30 17.40

AlH2 CCSD(T) (f) 3.19 4.70 4.70 4.68 4.70 2.03 (r) 17.58 18.29 18.38 18.40 18.44 22.90 B3LYP (f) ‐0.10 1.28 1.67 1.73 1.91 1.87 (r) 16.06 16.06 16.06 16.06 16.06 23.20

CH3 CCSD(T) (f) 23.96 23.78 24.20 24.34 24.44 21.00 (r) 21.27 21.08 21.48 21.62 21.72 19.70 B3LYP (f) 18.34 18.79 18.86 18.87 18.90 18.10 (r) 19.37 18.95 18.93 18.93 18.93 18.40


208


L Method Eact cc‐pVDZ cc‐pVTZ cc‐pVQZ CBS(1)a CBS(2)a Lai et al.

SiH3 CCSD(T) (f) 9.78 9.54 6.00 4.91 3.90 8.16 (r) 20.55 22.21 22.61 22.70 22.83 21.80 B3LYP (f) 7.68 7.95 8.05 8.07 8.11 7.35 (r) 22.35 22.54 22.57 22.58 22.59 21.40

NH2 CCSD(T) (f) 28.12 28.00 27.88 27.84 27.81 21.60 (r) 17.71 17.52 16.28 15.82 15.55 11.90 B3LYP (f) 21.17 22.21 22.18 22.17 22.17 17.80 (r) 14.74 15.06 15.00 14.99 14.96 11.00

PH2 CCSD(T) (f) 14.27 18.47 20.43 20.99 21.57 11.60 (r) 18.97 26.94 30.12 31.02 31.97 26.40 B3LYP (f) 11.53 12.55 12.55 12.54 12.54 9.19 (r) 26.88 28.61 28.63 28.62 28.63 23.90

OH CCSD(T) (f) 21.98 21.79 21.32 21.13 21.05 20.90 (r) 12.51 10.49 10.63 10.78 10.73 20.60 B3LYP (f) 16.99 17.50 17.28 17.23 17.15 16.20 (r) 23.02 20.42 19.94 19.87 19.67 20.00

SH CCSD(T) (f) 19.17 16.01 16.22 16.39 16.36 15.40 (r) 26.48 28.85 25.81 24.72 24.00 28.50 B3LYP (f) 14.91 14.79 15.94 16.08 16.63 11.10 (r) 30.98 30.68 31.70 31.82 32.31 26.40F CCSD(T) (f) 17.05 15.85 15.15 14.85 14.73 13.90 (r) 13.85 11.79 12.12 12.40 12.33 11.40 B3LYP (f) 12.76 12.63 12.25 12.17 12.03 10.30 (r) 14.42 11.65 11.29 11.24 11.09 11.50Cl CCSD(T) (f) 12.99 14.92 14.96 14.91 14.98 15.40 (r) 17.11 18.15 18.39 18.45 18.53 17.10 B3LYP (f) 10.70 11.65 11.93 11.97 12.09 10.40 (r) 17.81 18.28 18.12 18.09 18.02 16.50Br CCSD(T) (f) 15.14 15.08 14.68 14.56 14.45 ‐‐‐ (r) 18.44 19.85 18.34 17.83 17.43 ‐‐‐ B3LYP (f) 10.64 11.25 11.23 11.23 11.22 ‐‐‐ (r) 19.13 19.51 19.49 19.49 19.48 ‐‐‐

a. “CBS(1)” and “CBS(2)” denote the exponential and mixed Gaussian/exponential extrapolation schemes, respectively.

209

Table 9.5 Non‐relativistic (NR), Cowan‐Griffin (CG), and Douglas‐Kroll (DK) enthalpies of formation at 298 K (kcal/mol) of the L‐GeAs isomers computed with CCSD(T) and B3LYP; spin‐orbit effects have been included in the relativistic enthalpies.

CCSD(T) B3LYP

L ζ NR CG DK NR CG DK

H D 98.96 104.52 104.98 82.27 87.83 88.29 T 85.90 91.04 91.65 79.23 84.37 84.98 Q 80.81 86.35 86.53 78.92 84.46 84.64 CBS(1)a 77.57 83.84 83.34 78.89 ‐‐‐ 84.60 CBS(2)a 77.85 83.62 83.55 78.75 ‐‐‐ 84.45Li D 90.22 94.90 95.68 75.29 79.97 80.76 T 75.70 80.46 80.93 71.92 76.68 77.15 Q 70.10 75.08 75.32 71.50 76.48 76.72 CBS(1) 66.58 71.87 71.87 71.44 76.47 76.66 CBS(2) 66.83 71.94 72.05 71.27 76.37 76.48

Na D 85.38 90.00 90.41 71.54 76.17 76.58 T 70.24 75.07 75.24 68.37 73.20 73.37 Q 64.61 69.63 69.64 67.95 72.97 72.98 CBS(1) 61.28 66.50 66.36 67.89 72.95 72.92 CBS(2) 61.33 66.46 66.37 67.72 72.84 72.76

BeH D 125.82 130.71 131.51 100.74 105.63 106.43 T 109.81 114.66 115.04 97.03 101.88 102.25 Q 103.87 108.88 109.17 96.56 101.57 101.86 CBS(1) 100.37 105.63 105.93 96.49 101.54 101.82 CBS(2) 100.41 105.52 105.76 96.29 101.40 101.64

MgH D 110.57 115.55 116.37 94.47 99.46 100.28 T 93.55 98.66 99.10 91.09 96.20 96.63 Q 87.76 92.99 93.31 90.67 95.90 96.22 CBS(1) 84.77 90.13 90.39 90.61 95.87 96.16 CBS(2) 84.39 89.70 89.94 90.43 95.74 95.98

BH2 D 120.60 125.61 126.42 101.77 106.78 107.59 T 108.39 113.29 113.82 95.86 100.77 101.29 Q 101.38 106.57 106.80 95.35 100.54 100.77 CBS(1) 91.94 98.47 97.97 95.30 100.53 100.72 CBS(2) 97.27 102.63 102.68 95.07 100.43 100.49


210


CCSD(T) B3LYP


AlH2 D 122.04 127.45 125.80 100.21 105.63 103.98 T 100.30 105.80 105.37 94.91 100.41 99.98 Q 92.58 98.22 98.08 93.97 99.60 99.47 CBS(1) 88.33 94.12 94.04 93.77 99.45 99.40 CBS(2) 88.09 93.80 93.84 93.43 99.14 99.18

CH3 D 106.93 112.97 113.42 76.61 82.65 83.10 T 78.94 85.08 85.32 68.24 74.38 74.62 Q 70.14 76.48 76.52 67.62 73.97 74.00 CBS(1) 66.10 72.65 72.50 67.58 73.95 73.95 CBS(2) 65.03 71.50 71.41 67.29 73.76 73.67

SiH3 D 112.69 118.63 115.60 85.62 91.56 88.53 T 88.36 94.37 93.23 80.27 86.28 85.14 Q 82.62 88.83 88.31 79.61 85.81 85.29 CBS(1) 80.85 87.18 86.92 79.51 85.77 ‐‐‐ CBS(2) 79.32 85.63 85.47 79.24 85.55 ‐‐‐

NH2 D 117.13 123.32 123.71 81.53 87.72 88.11 T 85.61 91.87 92.09 70.94 77.19 77.42 Q 75.24 81.73 81.72 69.60 76.08 76.08 CBS(1) 70.15 76.90 76.65 69.40 75.95 75.89 CBS(2) 69.21 75.84 75.69 68.84 75.47 75.32

PH2 D 110.58 116.03 113.91 78.10 83.55 81.43 T 85.88 91.36 90.43 73.04 78.52 77.59 Q 76.62 82.42 81.84 72.66 78.46 77.88 CBS(1) 71.08 77.34 76.88 72.63 78.46 ‐‐‐ CBS(2) 71.23 77.22 76.84 72.46 78.45 ‐‐‐

OH D 61.35 67.39 68.28 35.16 41.20 42.08 T 34.14 40.21 40.78 25.96 32.03 32.60 Q 25.31 31.88 31.97 24.77 31.33 31.42 CBS(1) 21.07 28.20 27.81 24.59 31.27 31.25 CBS(2) 20.18 27.04 26.85 24.09 30.95 30.76

SH D 96.56 102.61 101.45 73.21 79.26 78.10 T 75.80 81.87 81.40 68.85 74.92 74.45 Q 67.66 74.13 73.69 68.47 74.94 74.50


211


CCSD(T) B3LYP


SH CBS(1) 62.40 69.52 68.87 68.43 ‐‐‐ ‐‐‐ CBS(2) 62.91 69.62 69.19 68.26 ‐‐‐ ‐‐‐F D 44.00 50.11 50.81 20.19 26.30 27.00 T 25.05 31.06 31.65 14.14 20.15 20.74 Q 17.85 24.47 24.50 13.22 19.84 19.86 CBS(1) 13.44 20.99 20.23 13.05 19.82 19.72 CBS(2) 13.66 20.64 20.33 12.70 19.67 19.36Cl D 75.74 81.94 81.93 54.96 61.16 61.15 T 56.48 62.65 62.77 50.85 57.01 57.14 Q 49.17 55.80 55.61 50.31 56.94 56.76 CBS(1) 44.69 52.02 51.34 50.23 56.94 56.72 CBS(2) 44.91 51.81 51.44 50.01 56.91 56.55Br D 74.70 83.05 84.14 57.17 65.52 66.60 T 59.02 68.03 68.29 54.12 63.13 63.39 Q 52.40 61.46 61.67 53.79 62.85 63.06 CBS(1) 47.54 56.36 56.93 53.75 62.82 63.03 CBS(2) 48.53 57.63 57.81 53.60 62.70 62.89


212

Table 9.6 Non‐relativistic (NR), Cowan‐Griffin (CG), and Douglas‐Kroll (DK) enthalpies of formation at 298 K (kcal/mol) of the GeAs‐L isomers computed with CCSD(T) and B3LYP; spin‐orbit effects have been included in the relativistic enthalpies.

CCSD(T) B3LYP


H D 89.38 93.17 93.80 68.97 72.76 73.39 T 75.79 79.77 79.97 66.44 70.42 70.62 Q 70.63 74.71 74.82 66.17 70.24 70.35 CBS(1) 67.49 71.65 71.76 66.13 70.23 70.32 CBS(2) 67.63 71.77 71.82 66.01 70.14 70.20Li D 76.09 80.17 81.07 59.57 63.65 64.55 T 62.50 66.98 67.22 57.27 61.76 61.99 Q 57.03 61.51 61.73 57.14 61.62 61.84 CBS(1) 53.33 57.63 58.13 57.13 61.61 61.83 CBS(2) 53.83 58.32 58.53 57.06 61.55 61.76

Na D 72.33 76.49 76.87 56.99 61.15 61.53 T 58.27 62.84 62.79 55.20 59.78 59.72 Q 52.85 57.43 57.40 54.97 59.55 59.52 CBS(1) 49.45 53.88 54.07 54.93 59.51 59.50 CBS(2) 49.69 54.28 54.27 54.83 59.42 59.41

BeH D 106.37 110.25 111.05 81.45 85.33 86.13 T 90.10 94.35 94.37 78.14 82.38 82.41 Q 83.96 88.07 88.32 77.76 81.87 82.12 CBS(1) 80.23 83.98 84.88 77.71 81.77 82.10 CBS(2) 80.38 84.41 84.80 77.55 81.58 81.97

MgH D 96.50 100.63 101.49 77.54 81.67 82.53 T 77.37 81.96 82.09 74.63 79.23 79.35 Q 71.05 75.53 75.82 74.37 78.85 79.14 CBS(1) 67.95 72.15 72.82 74.35 78.79 79.12 CBS(2) 67.39 71.79 72.18 74.24 78.64 79.02

BH2 D 121.88 125.99 126.86 91.28 95.40 96.26 T 100.92 105.39 105.60 85.66 90.13 90.33 Q 93.71 98.19 98.38 85.23 89.71 89.90 CBS(1) 89.91 94.32 94.67 85.20 89.68 89.87 CBS(2) 89.51 94.00 94.18 85.00 89.49 89.67


213


CCSD(T) B3LYP


AlH2 D 107.63 112.43 111.59 83.46 88.26 87.42 T 86.71 91.91 91.50 79.54 84.74 84.33 Q 78.90 84.10 83.88 79.00 84.20 83.98 CBS(1) 74.24 79.30 79.22 78.91 84.10 83.94 CBS(2) 74.35 79.55 79.44 78.69 83.89 83.79

CH3 D 109.98 113.90 114.77 75.37 79.29 80.16 T 78.22 82.64 82.76 68.02 72.43 72.55 Q 69.43 73.83 73.97 67.48 71.89 72.03 CBS(1) 66.06 70.37 70.65 67.44 71.84 71.99 CBS(2) 64.34 68.72 68.89 67.20 71.59 71.75

SiH3 D 102.74 107.28 103.96 70.60 75.15 71.83 T 76.67 81.84 80.14 65.35 70.52 68.82 Q 66.99 72.00 71.34 64.75 69.77 69.10 CBS(1) 61.27 65.81 66.20 64.67 69.62 ‐‐‐ CBS(2) 61.35 66.27 66.22 64.41 69.34 ‐‐‐

NH2 D 128.59 133.36 134.07 88.02 92.78 93.50 T 97.08 102.36 102.43 78.61 83.89 83.95 Q 89.87 95.17 95.22 77.31 82.60 82.65 CBS(1) 87.74 92.99 93.10 77.10 82.38 82.45 CBS(2) 85.72 91.02 91.07 76.57 81.87 81.92

PH2 D 106.24 110.67 107.83 62.31 66.74 63.90 T 77.80 82.89 81.11 56.53 61.63 59.84 Q 67.32 72.34 71.54 56.14 61.16 60.36 CBS(1) 61.21 65.88 66.20 56.11 61.11 ‐‐‐ CBS(2) 61.22 66.20 65.97 55.93 60.91 ‐‐‐

OH D 71.33 75.80 76.28 28.60 33.08 33.55 T 45.60 50.53 50.45 22.56 27.49 27.41 Q 36.16 40.97 41.03 21.62 26.43 26.49 CBS(1) 30.69 35.14 35.62 21.45 26.18 26.33 CBS(2) 30.66 35.40 35.54 21.09 25.82 25.97

SH D 89.89 94.27 92.81 56.66 61.04 59.59 T 63.48 68.51 67.33 52.48 57.51 56.32 Q 58.59 63.51 62.95 52.23 57.15 56.59


214


CCSD(T) B3LYP


SH CBS(1) 57.48 62.31 62.05 52.21 57.11 ‐‐‐ CBS(2) 55.80 60.65 60.46 52.09 56.95 ‐‐‐ F D 47.67 52.12 52.30 18.30 22.74 22.92 T 29.65 34.44 34.22 14.94 19.73 19.52 Q 21.41 26.00 26.02 14.00 18.60 18.62 CBS(1) 14.46 18.32 19.20 13.64 17.93 18.30 CBS(2) 16.59 21.07 21.23 13.46 17.95 18.10Cl D 72.63 77.02 76.87 47.62 52.01 51.86 T 54.04 58.98 58.45 43.99 48.93 48.40 Q 46.52 51.25 51.10 43.89 48.61 48.47 CBS(1) 41.43 45.45 46.22 43.89 48.58 ‐‐‐ CBS(2) 42.14 46.74 46.82 43.85 48.44 ‐‐‐ Br D 72.40 78.93 79.97 48.45 54.99 56.02 T 55.24 63.17 62.82 45.62 53.55 53.20 Q 49.74 57.06 57.33 45.29 52.61 52.88 CBS(1) 47.14 53.19 54.74 45.24 50.81 52.84 CBS(2) 46.54 53.49 54.14 45.10 52.05 52.70


215

Table 9.7 CCSD(T) and B3LYP enthalpies of isomerization (kcal/mol) from L‐GeAs to GeAs‐L at 298 K. Various basis set levels and CBS limits are represented as well as non‐relativistic (NR) and relativistic energies using the Cowan‐Griffin (CG) and Douglas‐Kroll (DK) approach.

CCSD(T) B3LYP


H D ‐9.58 ‐11.36 ‐11.18 ‐13.30 ‐15.07 ‐14.90 T ‐10.11 ‐11.28 ‐11.69 ‐12.79 ‐13.95 ‐14.36 Q ‐10.18 ‐11.64 ‐11.72 ‐12.76 ‐14.22 ‐14.30 CBS(1) ‐10.08 ‐12.19 ‐11.58 ‐12.76 ‐‐‐ ‐14.29 CBS(2) ‐10.21 ‐11.85 ‐11.73 ‐12.74 ‐‐‐ ‐14.26Li D ‐14.13 ‐14.73 ‐14.61 ‐15.72 ‐16.32 ‐16.21 T ‐13.20 ‐13.48 ‐13.72 ‐14.64 ‐14.92 ‐15.16 Q ‐13.07 ‐13.56 ‐13.59 ‐14.36 ‐14.86 ‐14.88 CBS(1) ‐13.24 ‐14.24 ‐13.73 ‐14.31 ‐14.85 ‐14.83 CBS(2) ‐13.00 ‐13.62 ‐13.52 ‐14.20 ‐14.82 ‐14.72

Na D ‐19.45 ‐20.46 ‐20.46 ‐19.29 ‐20.30 ‐20.30 T ‐19.71 ‐20.31 ‐20.66 ‐18.89 ‐19.50 ‐19.85 Q ‐19.91 ‐20.81 ‐20.85 ‐18.80 ‐19.69 ‐19.73 CBS(1) ‐20.14 ‐21.65 ‐21.05 ‐18.78 ‐19.77 ‐19.72 CBS(2) ‐20.04 ‐21.11 ‐20.96 ‐18.74 ‐19.81 ‐19.67

BeH D ‐13.05 ‐13.51 ‐13.54 ‐14.55 ‐15.02 ‐15.05 T ‐11.98 ‐12.23 ‐12.46 ‐13.17 ‐13.42 ‐13.65 Q ‐11.76 ‐12.20 ‐12.23 ‐12.98 ‐13.42 ‐13.45 CBS(1) ‐11.83 ‐12.62 ‐12.28 ‐12.96 ‐13.44 ‐13.42 CBS(2) ‐11.64 ‐12.18 ‐12.11 ‐12.88 ‐13.42 ‐13.35

MgH D ‐14.06 ‐14.92 ‐14.88 ‐16.93 ‐17.79 ‐17.75 T ‐16.19 ‐16.69 ‐17.01 ‐16.46 ‐16.97 ‐17.28 Q ‐16.70 ‐17.46 ‐17.49 ‐16.29 ‐17.05 ‐17.08 CBS(1) ‐16.82 ‐17.98 ‐17.57 ‐16.26 ‐17.09 ‐17.04 CBS(2) ‐17.00 ‐17.91 ‐17.77 ‐16.20 ‐17.10 ‐16.96

BH2 D 1.27 0.39 0.44 ‐10.49 ‐11.38 ‐11.33 T ‐7.46 ‐7.91 ‐8.22 ‐10.20 ‐10.64 ‐10.96 Q ‐7.67 ‐8.38 ‐8.42 ‐10.12 ‐10.83 ‐10.87 CBS(1) ‐2.03 ‐4.15 ‐3.30 ‐10.11 ‐10.86 ‐10.85 CBS(2) ‐7.76 ‐8.63 ‐8.50 ‐10.07 ‐10.94 ‐10.82


216


CCSD(T) B3LYP


AlH2 D ‐14.41 ‐15.02 ‐14.21 ‐16.75 ‐17.37 ‐16.56 T ‐13.60 ‐13.90 ‐13.87 ‐15.37 ‐15.67 ‐15.65 Q ‐13.69 ‐14.12 ‐14.20 ‐14.98 ‐15.41 ‐15.49 CBS(1) ‐14.09 ‐14.82 ‐14.81 ‐14.86 ‐15.35 ‐15.46 CBS(2) ‐13.75 ‐14.25 ‐14.40 ‐14.74 ‐15.25 ‐15.40

CH3 D 3.04 0.93 1.35 ‐1.24 ‐3.36 ‐2.94 T ‐0.72 ‐2.44 ‐2.57 ‐0.22 ‐1.94 ‐2.07 Q ‐0.72 ‐2.66 ‐2.55 ‐0.14 ‐2.08 ‐1.97 CBS(1) ‐0.05 ‐2.29 ‐1.85 ‐0.13 ‐2.11 ‐1.96 CBS(2) ‐0.70 ‐2.77 ‐2.52 ‐0.09 ‐2.17 ‐1.92

SiH3 D ‐9.96 ‐11.35 ‐11.64 ‐15.02 ‐16.42 ‐16.70 T ‐11.70 ‐12.53 ‐13.09 ‐14.93 ‐15.76 ‐16.32 Q ‐15.64 ‐16.82 ‐16.96 ‐14.86 ‐16.04 ‐16.19 CBS(1) ‐19.58 ‐21.38 ‐20.72 ‐14.84 ‐16.14 ‐‐‐ CBS(2) ‐17.97 ‐19.36 ‐19.25 ‐14.82 ‐16.22 ‐‐‐

NH2 D 11.46 10.04 10.36 6.49 5.07 5.38 T 11.47 10.49 10.33 7.67 6.70 6.54 Q 14.63 13.44 13.50 7.71 6.51 6.58 CBS(1) 17.59 16.09 16.44 7.70 6.43 6.56 CBS(2) 16.51 15.18 15.38 7.73 6.40 6.60

PH2 D ‐4.34 ‐5.36 ‐6.08 ‐15.79 ‐16.81 ‐17.53 T ‐8.08 ‐8.47 ‐9.32 ‐16.51 ‐16.89 ‐17.74 Q ‐9.31 ‐10.08 ‐10.30 ‐16.53 ‐17.30 ‐17.52 CBS(1) ‐9.87 ‐11.46 ‐10.69 ‐16.53 ‐17.35 ‐‐‐ CBS(2) ‐10.02 ‐11.02 ‐10.87 ‐16.54 ‐17.54 ‐‐‐

OH D 9.98 8.41 8.00 ‐6.56 ‐8.12 ‐8.53 T 11.46 10.33 9.67 ‐3.40 ‐4.54 ‐5.19 Q 10.85 9.09 9.06 ‐3.14 ‐4.90 ‐4.93 CBS(1) 9.62 6.95 7.80 ‐3.14 ‐5.09 ‐4.93 CBS(2) 10.48 8.35 8.69 ‐3.00 ‐5.13 ‐4.79

SH D ‐6.67 ‐8.35 ‐8.64 ‐16.54 ‐18.22 ‐18.51 T ‐12.32 ‐13.36 ‐14.07 ‐16.37 ‐17.42 ‐18.13 Q ‐9.06 ‐10.62 ‐10.73 ‐16.24 ‐17.79 ‐17.91


217


CCSD(T) B3LYP


CBS(1) ‐4.92 ‐7.21 ‐6.82 ‐16.22 ‐‐‐ ‐‐‐ CBS(2) ‐7.11 ‐8.97 ‐8.73 ‐16.16 ‐‐‐ ‐‐‐ F D 3.68 2.01 1.49 ‐1.89 ‐3.55 ‐4.08 T 4.60 3.38 2.57 0.80 ‐0.42 ‐1.23 Q 3.55 1.53 1.52 0.78 ‐1.23 ‐1.24 CBS(1) 1.02 ‐2.66 ‐1.03 0.59 ‐1.89 ‐1.42 CBS(2) 2.93 0.44 0.90 0.77 ‐1.73 ‐1.26Cl D ‐3.11 ‐4.92 ‐5.06 ‐7.34 ‐9.15 ‐9.29 T ‐2.45 ‐3.67 ‐4.32 ‐6.86 ‐8.09 ‐8.74 Q ‐2.65 ‐4.55 ‐4.51 ‐6.42 ‐8.33 ‐8.29 CBS(1) ‐3.26 ‐6.57 ‐5.12 ‐6.34 ‐8.36 ‐‐‐ CBS(2) ‐2.77 ‐5.07 ‐4.62 ‐6.16 ‐8.47 ‐‐‐ Br D ‐2.30 ‐4.11 ‐4.17 ‐8.72 ‐10.53 ‐10.59 T ‐3.78 ‐4.86 ‐5.47 ‐8.50 ‐9.58 ‐10.19 Q ‐2.66 ‐4.41 ‐4.34 ‐8.50 ‐10.25 ‐10.19 CBS(1) ‐0.41 ‐3.17 ‐2.19 ‐8.50 ‐12.00 ‐10.19 CBS(2) ‐1.99 ‐4.13 ‐3.67 ‐8.50 ‐10.65 ‐10.19


218

CHAPTER 10

A SYSTEMATIC INVESTIGATION OF SILICON HYDRIDES AND HALIDES†

10.1 Introduction

Silicon‐containing species are of vital importance in many industries, with the vast

majority of applications in the semiconductor industry. Specifically, crystalline silicon provides

the foundation for both micro‐ and macro‐electronic devices from computer processors to

light‐emitting devices and solar cells.278 Halogenated silanes, such as SiF4 and SiCl4, are found in

the etching of metal surfaces such as copper.279,280 Further, plasmas of silicon are used in

chemical vapor deposition (CVD) and chemical etching of crystalline silicon and silicon oxide

surfaces.281 In the CVD process, for example, electric discharge decomposes silanes into

transient species that react readily with the surface. Many small reactive compounds produced

in such plasmas have been characterized by experiment,282 yet other compounds believed to be

produced in the same plasmas have not, or cannot at present, be characterized due to their

transient nature and short lifetimes. This affords an excellent opportunity for theoretical

computations to model these transient intermediates and predict their properties.

Hydrides, fluorides, and chlorides of silicon have been the topic of both experimental

and theoretical studies. The primary focus of many of these studies has been upon the accurate

determination of thermodynamic properties, including atomization energies and enthalpies of

formation, enabling a better understanding of industrial processes. Numerous experimental

† Work reported in this chapter was performed in collaboration with Rebecca M. Lucente‐Schultz, and has been submitted for publication in Chemical Physics.

219

thermodynamic studies have been reviewed in a paper by Walsh.283 A review of the theoretical

treatments of silicon hydrides has been presented by Allen and Schaefer,282 while numerous

fluorine and chlorine silicon species were investigated by Ignacio and Schlegel284,285 and Ho et

al.286,287 Silicon hydrides, fluorides, and chlorides have been studied using methods that range

from Hartree‐Fock (HF)60 and density functional theory (DFT)288 to coupled cluster (CC)

theory,289‐292 in combination with basis sets ranging from 3‐21G (27 basis functions per silicon

atom) to aug‐cc‐pV6Z (227 basis functions per silicon atom).293 Further, Martin et al. and Begue

et al. have investigated silane and difluorosilane, respectively, using ab initio anharmonic force

fields.121,294 Despite the different theoretical approaches that have been used to study silicon

hydrides and halides, no extensive CC study has been reported utilizing the augmented tight d

correlation consistent basis sets for many (SiH4 is an exception)121,294 of the silicon‐containing

molecules of this study (SiH, SiF, SiCl, SiH2, SiF2, SiCl2, SiHF, SiHCl, SiH3, SiF3, SiCl3, SiH2F, SiH2Cl,

SiHF2, SiHCl2, SiF4, SiCl4, SiH3F, SiH3Cl, SiH2F2, SiH2Cl2, SiHF3, SiHCl3), nor has a comparison of

coupled cluster with single, double, and non‐iterative triple excitations, CCSD(T),31,38 and the

recently development correlation consistent Composite Approach (ccCA)73‐80 been reported.

The ccCA method has been shown to be successful in computing ionization potentials,

electron affinities, and enthalpies of formation for the Gaussian‐3 1999 (G3/99) test set with

small mean absolute deviations (MAD) relative to experiment.74,75 Compared with the

Gaussian‐n (Gn) methods, ccCA performs well without the need for an empirical high‐level

correction. Further, ccCA is more efficient than CCSD(T) since the bottleneck in ccCA is a

second‐order perturbation single point computation with a quadruple‐ζ basis set, which scales

, while CCSD(T) scales as , where is the number of basis functions. Despite the

220

fact that ccCA does contain a CCSD(T) step, it is coupled with a small basis set, which does not

make it more computationally demanding than the bottleneck perturbation step. The

demonstrated success and efficiency of ccCA has motivated employing it as a computational

method this investigation to further benchmark it against coupled cluster theory in the study of

silicon‐containing molecules (the bulk of which were not included in previous benchmark

papers with ccCA).73‐80


The standard and augmented correlation consistent, polarized valence basis sets

(cc‐pVnZ and aug‐cc‐pVnZ, where n = D, T, Q, and 5)94‐101 have been used, as well as the tight d

correlation consistent basis sets, cc‐pV(n+d)Z and aug‐cc‐pV(n+d)Z.101 The mixed

Gaussian/exponential extrapolation scheme (3.3) has been employed herein to approximate

the complete basis set (CBS) limit.85,93,95 In the following discussions, the term ‘CBS(DTQ)’

denotes an extrapolation using the double‐, triple‐, and quadruple‐ζ points, ‘CBS(TQ5)’ denotes

an extrapolation using the triple‐, quadruple‐, and quintuple‐ζ points, and ‘CBS(DTQ5)’ denotes

an extrapolation using each ζ point. All 4‐point extrapolations have been fit using a least‐

squares procedure with a residual less than 10‐5 hartree, while the 3‐point extrapolations utilize

the analytical equation (3.5) for the CBS limit.

The CCSD(T) method31,38 has been employed along side the ccCA method.73‐80 Using a

fixed geometry optimized with the Becke three‐parameter exchange scheme (B3) and the Lee‐

Yang‐Parr (LYP) correlation functional, B3LYP, the ccCA method uses perturbation and CCSD(T)

theories in combination with the cc‐pVnZ basis sets to account for higher order correlation,

221

core‐valence (CV) correlation, and scalar relativistic corrections from second‐order Douglas‐

Kroll (DK) theory135‐137 as additive corrections (see Chapter 7 for a detailed discussion). These

corrections are then applied to an extrapolated, second‐order perturbation theory reference

energy. The ccCA method also includes atomic spin‐orbit (SO) coupling corrections. The target

accuracy of ccCA is the DK‐CCSD(T)/aug‐cc‐pCV(∞+d)Z level of theory for energetic properties.

In computations on open‐shell molecules, the restricted open‐shell CCSD(T) formulism was

used, while the unrestricted formalism was used in the ccCA computations.

All CCSD(T) computations were performed using the Molpro software package,180 while

ccCA computations utilized the Gaussian 03 software package.174 Gradients were converged to

at least 10‐4 Eh, while harmonic frequencies were computed at the cc‐pVTZ level.

Thermochemical properties were computed using atomic enthalpies of formation taken from

the JANAF or CODATA tables,195,273 except the enthalpy of formation of the silicon atom, which

was taken from Karton and Martin196 (see discussion in section 10.4). Atomic SO corrections (cf.

Table 9.1) were applied to the computed thermodynamic properties.

10.3 Optimized Structures

10.3.1 SiH, SiF, and SiCl

The CCSD(T)‐optimized structures of SiH, SiF, and SiCl are listed in Table 10.1. The JANAF

tables report a bond length for SiH (Re = 1.5201 Å) from theoretical (HF) studies,170,273 and many

subsequent experimental and theoretical investigations reference the JANAF value. However,

an extensive rovibrational investigation on the 28SiH radical isotopomer has been done by

222

Betrencourt et al.,295 and this value (Re = 1.51966 Å) has been included in Table 10.1. Similarly,

bond lengths from high resolution spectroscopic studies of SiF (Re = 1.60099 Å) and SiCl (Re =

2.05706 Å) have been used for comparison instead of the JANAF values.296,297 In comparison

with experiment, the SiH CCSD(T) bond lengths fall within 0.01 Å when using a triple‐ζ or higher

level basis set. The tight d function affects the double‐ζ bond length by shorting the bond from

1.540 Å to 1.533 Å with the standard basis sets and 1.543 Å to 1.535 Å in the augmented basis

sets, making the tight d basis sets closer to the experimental value of 1.51996 Å.

The tight d basis sets converge faster in the bond length than the standard basis sets (cf.

Figure 10.1). For example, the cc‐pV(T+d)Z basis set gives a bond length of SiF (1.611 Å) similar

to that arising from the cc‐pVQZ basis set (1.611 Å), while the cc‐pV(Q+d)Z basis set gives a

bond length (1.608 Å) similar to that of the cc‐pV5Z basis set (1.607 Å). Similar effects are

observed in the CCSD(T)‐computed bond lengths of SiCl. Overall, using the tight d basis sets

result in bond lengths at the (n+1) standard basis set level.

10.3.2 SiH2, SiF2, SiCl2, SiHF, and SiHCl

Table 10.2 lists the optimized geometries of the triatomic species. The desired 0.01 Å

accuracy is achieved at the quadruple‐ζ level with the standard and augmented basis sets for

SiH2, but is achieved at the triple‐ζ level with the tight d basis sets. Further, the SiF2 bond

lengths agree with experiment at the quadruple‐ζ level with all the basis set families employed.

Optimized bond angles of SiH2 agree with experiment within 0.5° at all basis set levels, while

that of SiF2 agrees within 0.5° at the triple‐ζ level and higher. Compared with the experimental

geometry reported in JANAF for SiCl2 (Re = 2.03 Å, A = 105° ± 3), the optimized geometries have

223

discrepancies of >0.04 Å and >3.5° at the quintuple‐ζ level. The underlying assumption in

JANAF, however, is that the bond lengths and bond angle of SiCl2 are the same as for Si‐Cl and

Cl‐Si‐Cl in SiH2Cl2. Fujitake and Hirota have reported the microwave spectrum of SiCl2,298 which

gives a structure (in Table 10.2 for comparison) much closer to the CCSD(T)‐optimized structure.

In SiHF and SiHCl, the Si‐H bond length is already within 0.01 Å at the triple‐ζ level, using

the standard and augmented basis sets, and is within this same accuracy at the double‐ζ level

using the tight d basis sets. Indeed, the Si‐F bond length converges toward the experimental

value with each family of basis set, however, 0.01 Å is not achieved until the quadruple‐ζ level.

The exception is for the cc‐pV(n+d)Z bond lengths, where the triple‐ζ level Si‐F bond length is

1.611 Å, compared with experiment (1.603 Å). The Si‐Cl bond of SiHCl, in constrast to the Si‐F

bond of SiHF does not reach 0.01 Å of experiment until the quintuple‐ζ level of the standard

and augmented basis sets and quadruple‐ζ level in the tight d basis sets.

10.3.3 SiH3, SiF3, SiCl3, SiH2F, SiH2Cl, SiHF2, and SiHCl2

Qualitative bond angles from electron paramagnetic resonance (EPR) studies of the SiH3

and SiF3 radicals have been reported,299,300 but no geometric parameters are well known for the

other tri‐substituted silicon radicals in this investigation. Microwave studies have reported

structures for SiH3 and SiF3,301,302 though they used a fixed F‐Si‐F bond angle in computing the

Si‐F bond length (reported in Table 10.3). One qualitative aspect that theory corroborates is the

pyramidal structure of the SiH3 radical, as compared with that of the isoelectronic CH3 radical,

which is planar.

Although very little structural information is available for these molecules from

224

experiment, trends in the computed structures from the mono‐, di‐, and tetra‐substituted

molecules lead to the following observations. One obvious trend is the decrease in the Si‐H

bond length in the procession from SiH to SiH4. For example, the bond length in SiH is 1.522 Å at

the largest basis set level and decreases to 1.517 Å in SiH2, then to 1.479 Å in SiH3 and SiH4.

Comparing the optimized structure of the SiH3 radical to that of gaseous silyl acetylene

(HCC‐SiH3) [Rz(SiH) = 1.4794 Å] determined by Brookman et al.,303 quantitative agreement for

the Si‐H bond length is observed. The Si‐F bonds show a decrease from 1.607 Å in SiF to 1.596 Å

in SiF2, then to 1.576 Å in SiF3, and finally to 1.563 Å in SiF4. Note the significant change in bond

length from SiF3 to SiF4, as compared with SiH3 to SiH4. A similar trend is observed in the

procession from SiCl2 to SiCl4 (the SiCl bond length is actually shorter than in SiCl2).

The Si‐H bond lengths are approximately 1.480 Å in the SiH2F, SiHF2, SiH2Cl, and SiHCl2

molecules. However, both the Si‐F and Si‐Cl bonds contract in going from SiH2X to SiHX2. This is

shown in Table 10.3: the Si‐F bond length is 1.600 Å at the largest basis set level of SiH2F and

contracts to 1.587 Å in SiHF2. Similarly, the Si‐Cl bond length is 2.054 Å in SiH2Cl and contracts

to 2.045 Å in SiHCl2. Finally, the H‐Si‐X and H‐Si‐H bond angles remain relatively constant in

each of the mixed hydride/halogen species, but the X‐Si‐X bond angles of SiHF2 and SiHCl2 are

markedly different. The F‐Si‐F angle, for example, is 107.15° at the largest basis set level and

expands to 109.99° for Cl‐Si‐Cl.

10.3.4 SiH4, SiF4, SiCl4, SiH3F, SiH3Cl, SiH2F2, SiH2Cl2, SiHF3, and SiHCl3

All of the optimized geometries of the tetra‐substituted silicon species examined are

listed in Table 10.4, and only basis sets through quadruple‐ζ have been used to optimize the

225

geometries. Considering the tetrahedral species, CCSD(T)‐optimized bond lengths of SiH4, SiF4,

and SiCl4 are within 0.005, 0.008, and 0.005 Å of experiment, respectively, at the quadruple‐ζ

level. The tight d effect is again pronounced by accelerating the convergence of the bond

lengths of each tetrahedral species.

The Si‐H bonds of SiHF3 and SiH3F agree with experiment at the quadruple‐ζ level for all

basis sets, including the tight d sets, while those of SiHCl3 and SiH3Cl agree with experiment by

the triple‐ζ level. The Si‐F bond length of SiHF3 is within 0.01 Å of experiment at the quadruple‐ζ

level using the tight d basis sets, but differs from experiment by 0.011 Å and 0.012 Å with the

standard and augmented basis sets, respectively, at the quadruple‐ζ level. In contrast, the Si‐F

bond length of SiH3F is well within 0.01 Å of experiment at triple‐ζ level for all basis set families

except the aug‐cc‐pVnZ sets. The Si‐Cl bond of SiHCl3, although approaching the experimental

value (Re = 2.020 Å) as the basis increases in size, differs from experiment by 0.009 Å and

0.010 Å at the cc‐pV(Q+d)Z and aug‐cc‐pV(Q+d)Z levels, respectively, but differs from

experiment by 0.013 Å and 0.014 Å at the cc‐pVQZ and aug‐cc‐pVQZ levels, respectively. A

similar observation is made for the Si‐Cl bond in SiH3Cl with differences, relative to experiment

(Re = 2.0506 Å), of 0.010 Å, 0.006 Å, 0.011 Å, and 0.007 Å at the cc‐pVQZ, cc‐pV(Q+d)Z, aug‐cc‐

pVQZ, and aug‐cc‐pV(Q+d)Z basis set levels, respectively. This shows that the inclusion of the

tight d function is vital to achieving 0.01 Å accuracy, relative to experiment. The bond angles of

SiHF3, SiH3F, SiHCl3, and SiH3Cl are within 0.5° of experiment at the double‐ζ level of each basis

set family.

226

10.4 Thermochemistry

The enthalpy of formation of the silicon atom reported by JANAF and CODATA has been

heavily criticized due to its large error bars (106.60 ± 1.91 kcal/mol).195,273 Ochterski et al.304

computed the enthalpy of formation of the silicon atom with lower error bars by using reliable

experimental enthalpies of formation of SiH4 and Si2H6 following the method of Grev and

Schaefer.290 Their recommended 0 K enthalpy of formation for silicon is 108.1 ± 0.5 kcal/mol.

Karton and Martin196 have recently reinvestigated the enthalpy of formation of the silicon atom

using the Weizmann‐4 (W4) composite approach. Karton and Martin’s reinvestigation of the

silicon enthalpy of formation did not rely on semi‐empirical corrections;305 their reported value

of the 0 K enthalpy of formation for silicon is 107.15 ± 0.20 kcal/mol, which is the value used in

this investigation.

Several atomization energies of Table 10.5 were calculated from the 0 K enthalpies of

formation taken from JANAF or CODATA. The experimental atomization energies (Σ ) for each

molecule have been calculated using the following equation:272,306

Σ Δ ,° Δ ,

° molecule (10.1)

In the formula, is the number of atoms of a particular type; the values for Δ ,° and

Δ ,° molecule are the experimental 0 K enthalpies of formation for the th atom and

molecule, respectively. Enthalpies that do not appear in JANAF or CODATA are taken from other

sources listed in Table 10.5.

There are six molecules for which neither an experimental atomization energy nor an

enthalpy of formation could be found: SiHF, SiHCl, SiH2F, SiH2Cl, SiHF2, and SiHCl2. The

227

enthalpies of formation of these molecules have been calculated from DK‐CCSD(T)/CBS(DTQ)

total energies (including atomic SO corrections) using the aug‐cc‐pCVnZ‐DK basis sets81,142 and

experimental infrared (IR) frequencies, where available, using statistical thermodynamics. The

thermal contributions of translational, rotational, and vibrational energy have been calculated

from the following equations from McQuarrie and Simon:306

(10.2)

Θ2

Θ /

1 / , where Θ (10.3)

It is assumed that there is no appreciable degeneracy in the electronic wavefunction, and the

thermal correction to the electronic energy is, therefore, negligible. The translational and

rotational thermal contributions, and , respectively, are taken as 3/2 each. The

vibrational frequencies of the molecules ( ) are taken directly from experimental infrared (IR)

frequencies. The first term in the summation of (10.3) is the zero‐point energy (ZPE) and the

second term is the thermal correction to the vibrational energy (Boltzmann correction); the

number 3 6 ( is the number of atoms) is the number of vibrational degrees of

freedom. The atomization energies computed in this manner are considerd in this study to be

the best estimate of the true value in the absence of a reliable experimental value.

10.4.2 Atomization Energies and Enthalpies of Formation

Computed atomization energies at 0 K are listed in Table 10.5 and enthalpies of

formation at 298.15 K are listed in Table 10.6. Experimental data are presented for comparison

with the atomization energy and the enthalpy of formation for each molecule. If the

228

atomization energy was not found to be directly available from experiment, it was calculated

from the experimental enthalpy of formation from equation (10.1). Conversely, if the enthalpy

of formation was not found to be available from experiment, then it was calculated from the

experimental atomization energy using equation (10.1).

The IR spectra for SiHF and SiHCl from JANAF were used to derive the respective

experimental atomization energies, but no spectral data for the SiH2F, SiH2Cl, SiHF2, and SiHCl2

molecules could be found, so harmonic frequencies from CCSD(T)/aug‐cc‐pV(T+d)Z

computations were employed. The atomization energies and enthalpies of formation for these

six molecules are listed in Table 10.5 and Table 10.6, respectively. It should be noted that

Ignacio and Schlegel have reported enthalpies of formation for SiHF (‐41 ± 5 kcal/mol), SiH2F

(‐49 ± 5 kcal/mol), and SiHF2 (‐144 ± 5 kcal/mol) using a linear interpolation scheme reported in

the JANAF tables.273,284 The DK‐CCSD(T)/CBS(DTQ) enthalpies reported here are within the error

bars of these three fluorine species.

The impact of employing atomization energies from sources other than the JANAF

tables can be quite significant. For example, in SiCl, when the JANAF‐derived atomization

energy (88.84 kcal/mol) is compared to the CCSD(T)/CBS(DTQ5) value, the difference is

12.02 kcal/mol for the aug‐cc‐pV(n+d)Z basis sets. Using a very recent value reported by

Hildenbrand et al.307 (98.68 kcal/mol), there is much better agreement between theory and

experiment, with the error reduced to 2.18 kcal/mol. Other similar cases of JANAF‐derived

atomization energies differing significantly from theory exist as well (e.g. SiF2, SiF3, and SiCl3),

and using the experimental data reported by various other sources (listed in Table 10.5) shows

much better agreement between theory and experiment. These large sources of error can be

229

attributed to the averaging of different experimental enthalpies of formation reported in the

JANAF tables.273

For each of the silicon fluorides represented, it is seen in Figure 10.2 that employing the

tight d function also accelerates the convergence of the enthalpy of formation, although

employing the diffuse functions more rapidly accelerates the convergence, especially at the

double‐ζ level, relative to the cc‐pVnZ results. More generally, the diffuse functions accelerate

convergence of the enthalpies of formation of all molecules of this investigation. However, the

cc‐pV(n+d)Z basis sets produce enthalpies of formation comparable to the aug‐cc‐pVnZ basis

sets in many of the silicon molecules at the computational cost of a single d function (five

angular functions) versus a shell of multiple diffuse functions.

The data in Table 10.5 and Table 10.6 demonstrates that large basis sets are needed to

converge the CCSD(T) energies to the kcal/mol accuracy regime, particularly for the larger

silicon molecules. The difference between the cc‐pVDZ and cc‐pVTZ energies is 64.97 kcal/mol

for SiF4, which decreases to 19.06 kcal/mol between cc‐pVTZ and cc‐pVQZ, and then to 6.85

kcal/mol between the cc‐pVQZ and cc‐pV5Z energies. Further, basis set effects are seen in SiF in

which the molecule is predicted to be enthalpically unstable at the double‐ζ level (cf. Figure

10.2), with the exception of the aug‐cc‐pV(n+d)Z basis sets, but instability is not observed for

basis sets of at least triple‐ζ quality.

Comparing the CBS limits of Table 10.5 and Table 10.6, it becomes evident that

extrapolating different data points can lead to significantly different limits. Differences greater

than 1.0 kcal/mol between the extrapolated CBS limits of Table 10.5 and experiment are found,

for example, with SiF2, SiCl2, SiCl3, SiH2F, SiHF2, SiHCl2, SiH4, SiF4, SiCl4, SiHF3, SiH2F2, SiH3F,

230

SiHCl3, and SiH2Cl2. Large differences between the extrapolated CBS limits, like for SiF4,

complicate the comparison with experiment, but examining the CBS(DTQ5) extrapolation can

be illuminating. For example, the difference between the SiF4 cc‐pV(n+d)Z CBS(DTQ) and

CBS(TQ5) atomization energies is 3.21 kcal/mol, while the CBS(DTQ5) is almost half‐way

between the CBS(DTQ) and CBS(TQ5) atomization energies. It is commonly assumed that a

CBS(TQ5) extrapolation is close to the actual CBS limit for the system since the difference

between the quintuple‐ζ point and the actual CBS limit is usually small. However, using the

double‐ζ point can provide an anchor point that changes the curvature of the CBS fitting

function, as is shown in Table 10.5 for the SiF4 atomization energy, causing significant

differences between the CBS limits. In CBS(DTQ5) extrapolations, like that of SiF4, the quintuple‐

ζ point is not as close to the basis set limit as predicted by the CBS(TQ5) extrapolation.

Comparing the SiF4 cc‐pV(n+d)Z CBS(DTQ5) limit with experiment shows a difference of 2.97

kcal/mol, while the CBS(DTQ) and CBS(TQ5) limits show differences of 4.51 and 1.30 kcal/mol,

respectively. Note that the CBS(TQ5) extrapolation, in this case, is closer to experiment than the

other extrapolations.

The MAD of the enthalpies of formation from experiment, excluding those of SiHF,

SiHCl, SiH2F, SiHF2, SiH2Cl, and SiHCl2 since there are no direct experimental enthalpies to

compare with, are, at the CCSD(T)/CBS(TQ5) level vs. the CCSD(T)/CBS(DTQ5) level: 2.25 vs. 2.07

kcal/mol (cc‐pVnZ); 1.98 vs. 2.17 kcal/mol (cc‐pV(n+d)Z); 2.35 vs. 2.24 kcal/mol (aug‐cc‐pVnZ);

and 2.13 vs. 2.34 kcal/mol (aug‐cc‐pV(n+d)Z). These MADs indicate that the CBS(TQ5)

extrapolation slightly outperforms the CBS(DTQ5) extrapolation in the tight d basis sets, but not

in the standard and augmented basis sets. The largest errors between CCSD(T)/CBS(DTQ5) and

231

experimental enthalpies of formation are in SiCl3 and SiCl4.

It should be noted that a direct comparison of ccCA results to CCSD(T) should not be

made since ccCA contains additive corrections for both CV and scalar relativistic effects. The

CCSD(T)/CBS(DTQ) extrapolations reported here were done as a reference point for comparison

with ccCA, since ccCA uses a CBS(DTQ) reference energy. The CCSD(T)/CBS(DTQ) enthalpy of

formation MAD is 2.03 kcal/mol [cc‐pVnZ], 2.53 kcal/mol [cc‐pV(n+d)Z], 2.25 kcal/mol [aug‐cc‐

pVnZ], and 2.68 kcal/mol [aug‐cc‐pV(n+d)Z]. The ccCA MAD from experiment is 2.62 kcal/mol.

Further, the average deviation from experiment for ccCA is 1.98 ± 2.99 kcal/mol, while the

CCSD(T)/CBS(DTQ) average deviation is ‐2.05 ± 2.37 kcal/mol using the aug‐cc‐pV(n+d)Z basis

sets.

In every molecule of this study, CCSD(T)/CBS(DTQ) extrapolations of the enthalpy of

formation are too low relative to experiment, and ccCA tends to be too high, but slightly closer

to experiment. This observation stems from the CCSD(T)/CBS(DTQ) atomization energies being

too high, while those of ccCA are too low. Assuming that all basis set error has been removed

by the extrapolated CCSD(T) energies, then the enthalpies of formation are too low due to the

lack of higher‐order correlation, CV correlation, and DK scalar relativistic effects. Computations

were performed explicitly including CV and DK scalar relativistic effects in the CCSD(T)

enthalpies of formation for a subset of silicon molecules. Overall, including CV and DK scalar

relativistic components in the CCSD(T) treatment does not appreciably improve the enthalpies

since these corrections tend to increase computed atomization energies.

Removing the CV and DK scalar relativistic corrections from ccCA approximates the

CCSD(T)/aug‐cc‐pV(∞+d)Z level of theory. When these two corrections are removed, the MAD

232

of the ccCA enthalpies of formation reduces to 1.95 kcal/mol, compared to the CCSD(T) MAD of

2.68 kcal/mol. This is because the additive corrections tend to overcorrect the ccCA atomization

energies and the remaining difference between ccCA and CCSD(T) is fortuitously due to the

additive correction approximation. Although the MAD from experiment is lowered when the CV

and DK scalar relativistic corrections are removed, they are necessary for accurately describing

the enthalpies of formation of SiH, SiF, SiH2, SiCl2, SiH4, and SiHF3. For example, removing these

corrections from the SiH4 enthalpy of formation raises the difference between ccCA and

experiment from 0.88 kcal/mol to 2.81 kcal/mol; and that of SiHF3 from 0.85 kcal/mol to 1.88

kcal/mol.

Further, comparison of CCSD(T)/aug‐cc‐pV(n+d)Z and ccCA to experiment shows that

neither method is within the experimental error bars for the enthalpies of formation of SiF,

SiH2, SiCl2, SiF3, SiCl3, SiH4, SiF4, SiCl4, and SiHCl3 (cf. Table 10.6, excluding the SiHF, SiHCl, SiH2F,

SiHF2, SiH2Cl, and SiHCl2 molecules due to the bias introduced from their experimental

enthalpies of formation being derived from CCSD(T) energies). For the SiH and SiCl molecules,

ccCA is within the experimental error bars, but the CCSD(T) extrapolations are not; for the SiF2

and SiH3 molecules, CCSD(T) is within the experimental errors bars, but ccCA is not. The latter

two enthalpies of formation computed by ccCA differ from experiment by 1.57 kcal/mol for SiF2

and differ by 1.88 kcal/mol for SiH3.

Comparing this investigation with other high accuracy studies, Martin et al. examined

the atomization energy and enthalpy of formation of SiH4 (experiment: Σ = 303.19 kcal/mol;

Δ ,° = 10.5 ± 0.5 kcal/mol) using CCSD(T) and an ab initio quartic force field. Their study of SiH4

was motivated by the purported problems with the experimental values.121 They included inner

233

valence, scalar relativistic, and SO corrections leading to 303.80 kcal/mol for the computed

atomization energy and 9.90 kcal/mol for the enthalpy of formation at 0 K. The aug‐cc‐pV(n+d)Z

CCSD(T)/CBS(DTQ) computations of this study include inner valence effects and atomic SO

corrections, giving an atomization energy of 305.68 kcal/mol and a 0 K enthalpy of formation of

8.01 kcal/mol (see supplemental material). The ccCA values for the atomization energy and 0 K

enthalpy of formation are, respectively, 304.05 and 9.62 kcal/mol. Thus, ccCA more closely

predicts the atomization energy and enthalpy of formation of SiH4 computed with an

anharmonic force field and the values from experiment than CCSD(T) at a comparable basis set

size.

10.4.3 Dissociation Reaction Enthalpies

Table 10.7, Table 10.8, and Table 10.9 repsectively show reaction enthalpies at 298 K for

the hydrogen‐, fluorine‐, and chlorine‐based silicon molecules of this study. In addition to the

JANAF enthalpies of formation of hydrogen, fluorine, and chlorine (recall that silicon comes

from Karton and Martin)196, the computed enthalpies of formation for HF [CCSD(T)/CBS(DTQ):

‐66.89 kcal/mol; CCSD(T)/CBS(TQ5): ‐66.35 kcal/mol; CCSD(T)/CBS(DTQ5): ‐65.82 kcal/mol;

ccCA: ‐65.99 kcal/mol] and HCl [CCSD(T)/CBS(DTQ): ‐24.66 kcal/mol; CCSD(T)/CBS(TQ5):

‐24.57 kcal/mol; CCSD(T)/CBS(DTQ5): ‐22.88 kcal/mol; ccCA: ‐22.14 kcal/mol] have been used

along with the JANAF values of ‐65.14 kcal/mol and ‐22.06 kcal/mol for HF and HCl,

respectively.

The enthalpies of Table 10.7 show CCSD(T) to perform slightly better than ccCA in

silicon‐hydrogen reactions. The average difference of the CCSD(T)/CBS(DTQ) reaction enthalpies

234

relative to experiment is 0.46 ± 1.31 kcal/mol, while that of ccCA is ‐0.67 ± 1.76 kcal/mol. In the

silicon‐fluorine and ‐chlorine reactions, the reactions involving SiHF, SiHCl, SiH2F, SiHF2, SiH2Cl,

and SiHCl2 enthalpies have been excluded as in the previous section so as to compare theory

directly with experimental enthalpies. In the fluorine reactions of Table 10.8, the

CCSD(T)/CBS(DTQ) and ccCA results compare with average differences relative to experiment of

0.37 ± 2.45 kcal/mol and ‐2.05 ± 2.31 kcal/mol, respectively. In Table 10.9, the

CCSD(T)/CBS(DTQ) and ccCA results compare with average differences relative to experiment of

0.21 ± 2.97 kcal/mol and ‐2.31 ± 3.61 kcal/mol, respectively. Since the SiCl3 enthalpy of

formation had the largest difference relative to experiment using both CCSD(T) and ccCA, the

recomputed MADs excluding reactions involving SiCl3 are also listed in Table 10.9 for

comparison. There is a drastic decrease, as expected, in the MADs as a result of excluding the

SiCl3 reactions, and comparing the average differences relative to experiment of

CCSD(T)/CBS(DTQ) with those of ccCA gives 0.23 ± 1.85 kcal/mol (MAD = 1.40 kcal/mol) and

‐2.18 ± 1.91 kcal/mol (MAD = 2.23 kcal/mol), respectively. Removing the SiCl3 outlier reveals

that ccCA, on average, more accurately predicts reaction enthalpies in the silicon‐chlorine

reactions of Table 9 than CCSD(T)/CBS(TQ5).

10.5 Conclusions

This investigation has presented CCSD(T)‐optimized geometries for 24 silicon hydrides

and halides, employing the cc‐pVnZ and aug‐cc‐pVnZ basis sets and their tight d analogs

cc‐pV(n+d)Z and aug‐cc‐pV(n+d)Z. Overall, agreement within 0.01 Å in bond lengths and within

0.5° in bond angles of experiment is achieved for the geometries reported as the basis set

235

reaches completeness. The effect of employing the tight d function is acceleration of the

convergence of the bond lengths (there is little impact on bond angles) to the complete basis

set limit. The CCSD(T)/aug‐cc‐pV(n+d)Z optimized geometries, especially for the tri‐substituted

radicals, where there is a general lack of directly observed experimental equilibrium structures,

are the best estimate of the true equilibrium structure compared with previous theoretical

studies.

Large basis set DK‐CCSD(T) computations with explicit inclusion of CV and DK scalar

relativistic effects have been performed to provide predictive values for the atomization

energies and enthalpies of formation for SiHF, SiHCl, SiH2F, SiHF2, SiH2Cl, and SiHCl2 in the

absence of direct experimental values. The recommended values for the enthalpies of

formation (Δ ,° ) from this investigation are: SiHF, ‐39.16 kcal/mol; SiHCl, 12.19 kcal/mol;

SiH2F, ‐45.37 kcal/mol; SiH2Cl, 5.06 kcal/mol; SiHF2, ‐143.40 kcal/mol; and SiHCl2,

‐35.11 kcal/mol with an uncertainty of ± 1.21 kcal/mol in each, based on comparisons between

DK‐CCSD(T) and experimental enthalpies of formation for a subset of the molecules

investigated.

The convergence of computed atomization energies and enthalpies of formation is also

accelerated by inclusion of the tight d function. Employing diffuse functions causes the

energetic properties to more rapidly converge, but at the computational cost of a shell of

diffuse functions versus one d function (five angular functions). Large basis sets are needed to

converge CCSD(T) energetic properties, and, in general, the CBS(TQ5) extrapolation gives

slightly better accuracy than CBS(DTQ5) compared with experiment for the cc‐pVnZ and

aug‐cc‐pVnZ basis sets. However, using the cc‐pV(n+d)Z and aug‐cc‐pV(n+d)Z basis sets, the

236

CBS(DTQ5) extrapolation performs slightly better than the CBS(TQ5) extrapolation. Comparing

ccCA to CCSD(T) shows that ccCA gives, on average, atomization energies and enthalpies of

formation that are closer to experiment than CCSD(T)/CBS(DTQ) and DK‐CCSD(T).

237

Figure 10.1 Plots comparing the CCSD(T) optimized geometries of SiH, SiF, and SiCl using four different families of correlation consistent basis sets.

1.515

1.520

1.525

1.530

1.535

1.540

1.545

D T Q 5

cc‐pVnZ

cc‐pV(n+d)Z

aug‐cc‐pVnZ

aug‐cc‐pV(n+d)Z

Experiment

Basis Set (n)

Bond

Length (an

gstrom

)

SiH

1.595

1.605

1.615

1.625

1.635

1.645

1.655

1.665

1.675

D T Q 5

cc‐pVnZ

cc‐pV(n+d)Z

aug‐cc‐pVnZ

aug‐cc‐pV(n+d)Z

Experiment

Basis Set (n)

Bond

Length (an

gstrom

)

SiF

2.050

2.060

2.070

2.080

2.090

2.100

2.110

2.120

2.130

2.140

D T Q 5

cc‐pVnZ

cc‐pV(n+d)Z

aug‐cc‐pVnZ

aug‐cc‐pV(n+d)Z

Experiment

Basis Set (n)

Bond

Length (an

gstrom

)

SiCl

238

Figure 10.2 Plots comparing the CCSD(T) enthalpies of formation at 298 K of the silicon fluorides using four families of correlation consistent basis sets.

‐20.00

‐15.00

‐10.00

‐5.00

0.00

5.00

10.00

15.00

D T Q 5

cc‐pVnZ

cc‐pV(n+d)Z

aug‐cc‐pVnZ

aug‐cc‐pV(n+d)Z

Basis Set (n)

Enthalpy

of Formation (298K, kcal/mol)

SiF‐160.00

‐150.00

‐140.00

‐130.00

‐120.00

‐110.00

‐100.00

D T Q 5

cc‐pVnZ

cc‐pV(n+d)Z

aug‐cc‐pVnZ

aug‐cc‐pV(n+d)Z

Basis Set (n)

Enthalpy


SiF2

‐250.00

‐240.00

‐230.00

‐220.00

‐210.00

‐200.00

‐190.00

‐180.00

‐170.00

‐160.00

D T Q 5

cc‐pVnZ

cc‐pV(n+d)Z

aug‐cc‐pVnZ

aug‐cc‐pV(n+d)Z

Basis Set (n)

Enthalpy


SiF3‐400.00

‐380.00

‐360.00

‐340.00

‐320.00

‐300.00

‐280.00

D T Q 5

cc‐pVnZ

cc‐pV(n+d)Z

aug‐cc‐pVnZ

aug‐cc‐pV(n+d)Z

Basis Set (n)

Enthalpy


SiF4

239

Table 10.1 CCSD(T) optimized geometries (Å) of SiH, SiF, and SiCl.

SiH (2Π) SiF (2Π) SiCl (2Π)

Basis Set Re Re Recc‐pVDZ 1.540 1.670 2.113cc‐pVTZ 1.528 1.620 2.084cc‐pVQZ 1.524 1.611 2.073cc‐pV5Z 1.522 1.607 2.066cc‐pV(D+d)Z 1.533 1.651 2.093cc‐pV(T+d)Z 1.525 1.611 2.076cc‐pV(Q+d)Z 1.522 1.608 2.068cc‐pV(5+d)Z 1.522 1.606 2.066aug‐cc‐pVDZ 1.543a 1.672a 2.131aug‐cc‐pVTZ 1.529a 1.625a 2.088aug‐cc‐pVQZ 1.526a 1.613a 2.074aug‐cc‐pV5Z 1.522a 1.610a 2.067aug‐cc‐pV(D+d)Z 1.535 1.654 2.112aug‐cc‐pV(T+d)Z 1.525 1.615 2.079aug‐cc‐pV(Q+d)Z 1.523 1.609 2.069aug‐cc‐pV(5+d)Z 1.522 1.607 2.066Experiment 1.51966b 1.60099c 2.05706d

a. Feller, D.; Dixon, D. A. J. Phys. Chem. A 1999, 103, 6413. b. Betrencourt, M.; Boudjaadar, D.; Chollet, P.; Guelachvili, G.; Morillon‐Chapey, M. J. Chem. Phys. 1986, 84,

4121. c. Tanaka, T.; Tamura, M.; Tanaka, K. J. Mol. Struct. 1997, 413‐414, 153. d. Calculated from the rotational constant of the 28Si35Cl isotopomer, see Mélen, F.; Dubois, I.; Bredohl, H. J.

Mol. Spectros. 1990, 139, 361.

240

Table 10.2 CCSD(T) optimized geometries (Å and degrees) of SiH2, SiF2, SiCl2, SiHF, and SiHCl.

SiH2 (1A1) SiF2 (

1A1) SiCl2 (1A1)

Basis Set Re A Re A Re A

cc‐pVDZ 1.533 92.31 1.656 100.09 2.114 101.67cc‐pVTZ 1.522 92.37 1.608 100.70 2.089 101.66cc‐pVQZ 1.518 92.34 1.600 100.69 2.078 101.41cc‐pV5Z 1.517 92.30 1.596 100.75 2.073 101.34cc‐pV(D+d)Z 1.526 92.33 1.637 100.68 2.095 101.99cc‐pV(T+d)Z 1.519 92.38 1.599 101.06 2.082 101.82cc‐pV(Q+d)Z 1.517 92.35 1.597 100.86 2.074 101.49cc‐pV(5+d)Z 1.516 92.30 1.596 100.77 2.072 101.34aug‐cc‐pVDZ 1.536a 92.1a 1.656a 99.3a 2.131 101.10aug‐cc‐pVTZ 1.523a 92.2a 1.612a 100.2a 2.092 101.23aug‐cc‐pVQZ 1.518a 92.3a 1.602a 100.5a 2.080 101.24aug‐cc‐pV5Z 1.517a 92.3a 1.599a 100.5a 2.074 101.27aug‐cc‐pV(D+d)Z 1.529 92.08 1.638 99.90 2.113 101.40aug‐cc‐pV(T+d)Z 1.519 92.25 1.603 100.56 2.085 101.41aug‐cc‐pV(Q+d)Z 1.517 92.29 1.598 100.70 2.075 101.33aug‐cc‐pV(5+d)Z 1.517 92.29 1.596 100.71 2.073 101.28Experiment 1.5141b 92.0b 1.591c 100.98c 2.0653c 101.32c

SiHF (1A´) SiHCl (1A´)

Re(SiH) Re(SiF) A Re(SiH) Re(SiCl) A

cc‐pVDZ 1.542 1.667 96.75 1.534 2.118 95.31cc‐pVTZ 1.533 1.620 96.94 1.524 2.092 95.39cc‐pVQZ 1.529 1.612 96.87 1.519 2.082 95.27cc‐pV5Z 1.527 1.609 96.86 1.518 2.077 95.26cc‐pV(D+d)Z 1.535 1.648 97.09 1.526 2.099 95.56cc‐pV(T+d)Z 1.530 1.611 97.14 1.520 2.085 95.50cc‐pV(Q+d)Z 1.527 1.609 96.95 1.518 2.078 95.34cc‐pV(5+d)Z 1.527 1.608 96.87 1.518 2.076 95.27aug‐cc‐pVDZ 1.545 1.672 95.97 1.536 2.138 94.60aug‐cc‐pVTZ 1.533 1.626 96.52 1.524 2.097 95.04aug‐cc‐pVQZ 1.529 1.614 96.70 1.520 2.084 95.14aug‐cc‐pV5Z 1.528 1.609 96.81 1.518 2.077 95.21


241


SiHF (1A´) SiHCl (1A´)

Re(SiH) Re(SiF) A Re(SiH) Re(SiCl) A

aug‐cc‐pV(D+d)Z 1.537 1.654 96.33 1.528 2.119 94.85aug‐cc‐pV(T+d)Z 1.530 1.616 96.72 1.521 2.089 95.16aug‐cc‐pV(Q+d)Z 1.528 1.611 96.80 1.518 2.079 95.21aug‐cc‐pV(5+d)Z 1.527 1.609 96.82 1.518 2.077 95.22Experiment 1.529d 1.603d 96.9d 1.515d 2.0700d 95.0d

a. Feller, D.; Dixon, D. A. J. Phys. Chem. A 1999, 103, 6413. b. Kalemos, A.; Dunning, T. H.; Mavridis, A. Mol. Phys. 2004, 102, 2597. c. Fujitake, M.; Hirota, E. Spectrochim. Acta A 1994, 50A, 1345. d. Hostutler, D. A.; Clouthier, D. J.; Judge, R. H. J. Chem. Phys. 2001, 114, 10728; Hostutler, D. A.; Ndiege, N.;

Clouthier, D. J.; Pauls, S. W. J. Chem. Phys. 2001, 115, 5485.

242

Table 10.3 CCSD(T) optimized geometries (Å and degrees) of SiH3, SiF3, SiCl3, SiH2F, SiH2Cl, SiHF2, and SiHCl2.

SiH3 (2A1) SiF3 (

2A1) SiCl3 (2A1)

Basis Re A(HSiH) Re A(FSiF) Re A(ClSiCl)

cc‐pVDZ 1.493 111.19 1.636 107.99 2.081 109.49 cc‐pVTZ 1.484 111.21 1.588 108.00 2.056 109.54 cc‐pVQZ 1.481 111.28 1.580 107.98 2.045 109.47 cc‐pV5Z 1.479 111.25 1.576 107.99 2.039 109.44 cc‐pV(D+d)Z 1.487 111.08 1.616 108.17 2.062 109.66 cc‐pV(T+d)Z 1.481 111.16 1.580 108.11 2.049 109.63 cc‐pV(Q+d)Z 1.479 111.24 1.577 108.03 2.041 109.52 cc‐pV(5+d)Z 1.479 111.25 1.575 108.00 2.039 109.45 aug‐cc‐pVDZ 1.495a 111.25 1.631 107.70 2.092 109.48 aug‐cc‐pVTZ 1.484a 111.28 1.590 107.96 2.058 109.41 aug‐cc‐pVQZ 1.482a 111.29 1.581 107.95 2.045 109.40 aug‐cc‐pV5Z 1.481a 111.26 1.576 107.98 2.040 109.41 aug‐cc‐pV(D+d)Z 1.489 111.14 1.613 107.85 2.074 109.65 aug‐cc‐pV(T+d)Z 1.481 111.24 1.582 108.06 2.050 109.50 aug‐cc‐pV(Q+d)Z 1.480 111.25 1.578 108.00 2.041 109.45 aug‐cc‐pV(5+d)Z 1.480 111.25 1.576 107.99 2.039 109.43 Experiment ‐‐‐ ‐‐‐ (1.565)b (109.94)b ‐‐‐ ‐‐‐

SiH2F (2A´) SiHF2 (

2A´)

Re(SiH) Re(SiF) A(HSiF) A(HSiH) Re(SiH) Re(SiF) A(HSiF) A(FSiF)

cc‐pVDZ 1.494 1.659 108.07 111.36 1.496 1.647 107.95 107.28cc‐pVTZ 1.486 1.612 108.17 111.31 1.488 1.599 108.04 107.24cc‐pVQZ 1.483 1.604 108.13 111.44 1.485 1.591 108.05 107.13cc‐pV5Z 1.481 1.601 108.12 111.41 1.483 1.587 108.04 107.16cc‐pV(D+d)Z 1.488 1.640 108.26 111.03 1.489 1.627 107.93 107.67cc‐pV(T+d)Z 1.483 1.603 108.27 111.16 1.484 1.590 108.04 107.46cc‐pV(Q+d)Z 1.481 1.601 108.17 111.37 1.483 1.588 108.04 107.23cc‐pV(5+d)Z 1.481 1.600 108.12 111.41 1.483 1.587 108.04 107.18aug‐cc‐pVDZ 1.496 1.661 107.52 112.05 1.499 1.646 107.92 106.46aug‐cc‐pVTZ 1.486 1.617 108.05 111.58 1.488 1.603 108.10 106.96aug‐cc‐pVQZ 1.483 1.606 108.06 111.54 1.485 1.593 108.05 107.05aug‐cc‐pV5Z 1.482 1.601 108.10 111.46 1.484 1.588 108.05 107.13


243


SiH2F (2A´) SiHF2 (

2A´)

Re(SiH) Re(SiF) A(HSiF) A(HSiH) Re(SiH) Re(SiF) A(HSiF) A(FSiF)

aug‐cc‐pV(D+d)Z 1.490 1.643 107.75 111.71 1.492 1.627 107.95 106.82aug‐cc‐pV(T+d)Z 1.483 1.607 108.16 111.43 1.485 1.594 108.09 107.19aug‐cc‐pV(Q+d)Z 1.482 1.602 108.10 111.45 1.484 1.589 108.04 107.16aug‐cc‐pV(5+d)Z 1.481 1.600 108.11 111.45 1.483 1.587 108.04 107.15Experiment ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐

SiH2Cl (2A´) SiHCl2 (

2A´)

Re(SiH) Re(SiCl) A(HSiCl) A(HSiH) Re(SiH) Re(SiCl) A(HSiCl) A(ClSiCl)

cc‐pVDZ 1.493 2.094 108.73 111.18 1.493 2.086 108.03 110.47cc‐pVTZ 1.484 2.069 108.75 111.21 1.485 2.061 108.18 110.32cc‐pVQZ 1.481 2.059 108.69 111.35 1.481 2.051 108.23 110.10cc‐pV5Z 1.480 2.054 108.69 111.34 1.480 2.045 108.24 110.05cc‐pV(D+d)Z 1.487 2.075 108.95 110.88 1.487 2.067 108.08 110.81cc‐pV(T+d)Z 1.481 2.062 108.84 111.09 1.481 2.054 108.20 110.48cc‐pV(Q+d)Z 1.479 2.055 108.75 111.28 1.479 2.047 108.25 110.18cc‐pV(5+d)Z 1.479 2.053 108.70 111.33 1.480 2.045 108.26 110.03aug‐cc‐pVDZ 1.495 2.110 108.21 111.67 1.496 2.099 107.96 110.07aug‐cc‐pVTZ 1.484 2.073 108.62 111.40 1.485 2.064 108.21 109.98aug‐cc‐pVQZ 1.481 2.060 108.63 111.43 1.481 2.051 108.24 109.94aug‐cc‐pV5Z 1.480 2.055 108.67 111.38 1.480 2.046 108.27 109.95aug‐cc‐pV(D+d)Z 1.489 2.091 108.44 111.38 1.489 2.081 108.03 110.39aug‐cc‐pV(T+d)Z 1.481 2.065 108.72 111.29 1.481 2.056 108.23 110.15aug‐cc‐pV(Q+d)Z 1.480 2.056 108.69 111.35 1.480 2.047 108.26 110.03aug‐cc‐pV(5+d)Z 1.480 2.054 108.68 111.36 1.480 2.045 108.28 109.99Experiment ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐

a. Feller, D.; Dixon, D. A. J. Phys. Chem. A 1999, 103, 6413. b. Tanimoto, M.; Saito, S. J. Chem. Phys. 1999, 111, 9242; Milligan, D. E.; Jacox, M. E.; Guillory, W. A. J. Chem.

Phys. 1968, 49, 5330.

244

Table 10.4 CCSD(T) optimized geometries (Å and degrees) of SiH4, SiF4, SiCl4, SiHF3, SiH3F, SiHCl3, SiH3Cl, SiH2F2, and SiH2Cl2.

SiH4 SiF4 SiCl4

Basis Set Re Re Re

cc‐pVDZ 1.491 1.613 2.057 cc‐pVTZ 1.483 1.570 2.036 cc‐pVQZ 1.480 1.563 2.025 cc‐pV(D+d)Z 1.485 1.593 2.040 cc‐pV(T+d)Z 1.480 1.562 2.029 cc‐pV(Q+d)Z 1.479 1.560 2.021 aug‐cc‐pVDZ 1.493a 1.607a 2.066 aug‐cc‐pVTZ 1.483a 1.571a 2.037 aug‐cc‐pVQZ 1.480a 1.564a 2.026 aug‐cc‐pV(D+d)Z 1.487 1.590 2.050 aug‐cc‐pV(T+d)Z 1.480 1.564 2.030 aug‐cc‐pV(Q+d)Z 1.479 1.560 2.022 Experiment 1.4741b 1.5524c 2.0173d

SiHF3 SiH3F

Re(SiH) Re(SiF) A(HSiF) Re(SiH) Re(SiF) A(HSiF)

cc‐pVDZ 1.463 1.626 110.71 1.485 1.654 108.26 cc‐pVTZ 1.457 1.581 110.66 1.478 1.608 108.40 cc‐pVQZ 1.456 1.573 110.70 1.476 1.601 108.34 cc‐pV(D+d)Z 1.456 1.605 110.49 1.479 1.634 108.47 cc‐pV(T+d)Z 1.454 1.572 110.55 1.475 1.599 108.50 cc‐pV(Q+d)Z 1.454 1.570 110.64 1.474 1.597 108.38 aug‐cc‐pVDZ 1.467 1.622 111.10 1.488 1.657 107.74 aug‐cc‐pVTZ 1.458 1.583 110.77 1.478 1.613 108.22 aug‐cc‐pVQZ 1.456 1.574 110.74 1.476 1.603 108.26 aug‐cc‐pV(D+d)Z 1.460 1.604 110.90 1.481 1.638 107.98 aug‐cc‐pV(T+d)Z 1.455 1.575 110.66 1.476 1.604 108.33 aug‐cc‐pV(Q+d)Z 1.455 1.571 110.69 1.475 1.599 108.31 Experiment 1.4468e 1.5624e 110.64e 1.4761f 1.5945f 108.269f


245


SiHCl3 SiH3Cl

Re(SiH) Re(SiCl) A(HSiCl) Re(SiH) Re(SiCl) A(HSiCl)

cc‐pVDZ 1.474 2.065 109.37 1.486 2.092 108.53 cc‐pVTZ 1.466 2.043 109.39 1.478 2.069 108.51 cc‐pVQZ 1.462 2.033 109.48 1.475 2.060 108.43 cc‐pV(D+d)Z 1.467 2.047 109.19 1.480 2.072 108.77 cc‐pV(T+d)Z 1.462 2.036 109.31 1.475 2.061 108.61 cc‐pV(Q+d)Z 1.461 2.029 109.44 1.474 2.056 108.49 aug‐cc‐pVDZ 1.475 2.076 109.53 1.487 2.108 107.93 aug‐cc‐pVTZ 1.466 2.045 109.56 1.478 2.072 108.34 aug‐cc‐pVQZ 1.463 2.034 109.56 1.475 2.061 108.37 aug‐cc‐pV(D+d)Z 1.469 2.059 109.35 1.481 2.090 108.18 aug‐cc‐pV(T+d)Z 1.462 2.038 109.48 1.475 2.065 108.43 aug‐cc‐pV(Q+d)Z 1.462 2.030 109.51 1.474 2.057 108.43 Experiment 1.464f 2.020f 109.5f 1.4749f 2.0506f 108.295f

SiH2F2 SiH2Cl2

Re(SiH) Re(SiF) A(HSiH) A(FSiF) Re(SiH) Re(SiCl) A(HSiH) A(ClSiCl)

cc‐pVDZ 1.476 1.639 114.98 107.90 1.480 2.077 112.81 110.28cc‐pVTZ 1.470 1.593 114.26 107.91 1.472 2.054 112.62 110.12cc‐pVQZ 1.468 1.586 114.31 107.79 1.469 2.045 112.72 109.91cc‐pV(D+d)Z 1.470 1.619 114.46 108.27 1.473 2.058 112.28 110.62cc‐pV(T+d)Z 1.467 1.585 114.04 108.09 1.469 2.047 112.41 110.27cc‐pV(Q+d)Z 1.467 1.583 114.22 107.88 1.468 2.041 112.59 109.98aug‐cc‐pVDZ 1.479 1.638 115.63 107.10 1.481 2.090 113.80 109.80aug‐cc‐pVTZ 1.470 1.597 114.49 107.60 1.472 2.057 112.95 109.75aug‐cc‐pVQZ 1.468 1.587 114.44 107.71 1.469 2.046 112.85 109.75aug‐cc‐pV(D+d)Z 1.470 1.624 114.93 107.92 1.475 2.072 113.25 110.13aug‐cc‐pV(T+d)Z 1.467 1.585 114.10 107.97 1.469 2.050 112.74 109.91aug‐cc‐pV(Q+d)Z 1.467 1.583 114.24 107.87 1.468 2.042 112.71 109.84Experiment 1.4652f 1.5760f 114.2f 107.7f 1.4671f 2.0316f 112.45f 109.67f

a. Feller, D.; Dixon, D. A. J. Phys. Chem. A 1999, 103, 6413. b. Allen, W. D.; Schaefer, H. F. Chem. Phys. 1986, 108, 243. c. Breidung, J.; Demaison, J.; Margulès, L.; Thiel, W. Chem. Phys. Lett. 1999, 313, 713. d. Chase, M. W. J. Phys. Chem. Ref. Data 1998, 9, 1. e. Hoy, A. R.; Bertram, M.; Mills, I. M. J. Mol. Spectrosc. 1973, 46, 429. f. Donald, K. J.; Böhm, M. C.; Lindner, H. J. J. Mol. Struct. (THEOCHEM) 2005, 713, 215.

246

Table 10.5 CCSD(T) and ccCA atomization energies (kcal/mol) and extrapolated energies; atomic spin‐orbit corrections have been applied to the CCSD(T) results.

Basis SiH SiF SiCl SiH2 SiF2 SiCl2 SiHF SiHCl SiH3 SiF3 SiCl3 SiH2F

cc‐pVDZ 63.08 115.42 84.18 132.75 248.05 173.51 184.33 151.41 196.63 331.07 225.46 236.41cc‐pVTZ 68.07 131.76 93.85 141.76 279.56 191.75 204.23 164.90 208.38 378.90 252.04 259.94cc‐pVQZ 69.63 137.26 97.83 144.60 289.74 199.37 210.77 170.14 212.41 393.76 263.29 267.76cc‐pV5Z 70.18 139.12 99.74 145.60 293.28 203.05 213.04 172.47 213.69 398.92 268.79 270.35CBS(DTQ) 70.57 140.56 100.22 146.31 295.85 203.94 214.69 173.28 214.83 402.67 270.03 272.45CBS(TQ5) 70.50 140.24 100.88 146.20 295.40 205.26 214.41 173.86 214.45 402.01 272.09 271.90CBS(DTQ5) 70.52 140.33 100.50 146.22 295.50 204.51 214.47 173.51 214.59 402.15 270.93 272.08cc‐pV(D+d)Z 63.96 117.89 85.95 134.53 253.55 177.13 187.83 154.10 199.10 340.39 231.82 240.98cc‐pV(T+d)Z 68.46 133.14 94.79 142.55 282.52 193.63 206.04 166.23 209.47 383.80 255.20 262.22cc‐pV(Q+d)Z 69.85 137.99 98.44 145.04 291.26 200.59 211.73 170.96 213.00 396.23 265.31 268.96cc‐pV(5+d)Z 70.23 139.33 99.85 145.69 293.64 203.26 213.27 172.62 213.78 399.48 269.09 270.61CBS(DTQ) 70.69 140.90 100.63 146.54 296.50 204.76 215.15 173.80 215.11 403.67 271.36 273.01CBS(TQ5) 70.45 140.13 100.69 146.08 295.07 204.86 214.19 173.60 214.26 401.43 271.37 271.59CBS(DTQ5) 70.55 140.45 100.62 146.28 295.67 204.73 214.60 173.65 214.64 402.39 271.25 272.22aug‐cc‐pVDZ 64.26 124.94 87.27 134.63 264.67 179.37 193.81 155.27 198.61 353.76 232.84 246.42aug‐cc‐pVTZ 68.66 134.48 94.99 142.65 283.84 193.88 206.98 166.41 209.41 384.43 254.91 262.89aug‐cc‐pVQZ 69.84 138.32 98.49 144.94 291.45 200.62 211.86 170.93 212.78 395.97 265.03 268.86aug‐cc‐pV5Z 70.29 139.65 100.04 145.77 294.10 203.61 213.56 172.83 213.87 400.01 269.56 270.89CBS(DTQ) 70.55 140.63 100.59 146.31 296.00 204.67 214.79 173.64 214.80 402.89 271.10 272.44CBS(TQ5) 70.55 140.45 100.97 146.27 295.69 205.39 214.59 173.97 214.52 402.43 272.27 272.10CBS(DTQ5) 70.53 140.49 100.74 146.26 295.76 204.96 214.63 173.75 214.62 402.52 271.57 272.20


247



aug‐cc‐pV(D+d)Z 65.12 127.35 88.93 136.37 270.01 182.74 197.19 157.82 201.02 362.97 238.82 250.84aug‐cc‐pV(T+d)Z 69.04 135.88 95.91 143.42 286.83 195.72 208.79 167.72 210.48 389.43 258.09 265.17aug‐cc‐pV(Q+d)Z 70.05 139.02 99.07 145.36 292.91 201.77 212.79 171.72 213.35 398.36 266.94 270.02aug‐cc‐pV(5+d)Z 70.32 139.79 100.17 145.84 294.39 203.84 213.74 172.98 213.94 400.46 269.93 271.08CBS(DTQ) 70.66 140.90 100.97 146.53 296.56 205.39 215.18 174.11 215.07 403.72 272.25 272.92CBS(TQ5) 70.48 140.25 100.83 146.12 295.27 205.08 214.31 173.74 214.30 401.72 271.71 271.72CBS(DTQ5) 70.56 140.54 100.86 146.30 295.85 205.17 214.70 173.88 214.65 402.61 271.89 272.26ccCA 70.20 139.82 99.53 146.21 294.74 202.93 214.20 172.73 213.06 398.04 266.17 269.93Experiment 69.18a 136.52b 98.68c 144.82d 296.29b 204.66a (215.44) (174.26) 214.35d 400.16b 277.02e (272.43)

SiH2Cl SiHF2 SiHCl2 SiH4 SiF4 SiCl4 SiHF3 SiH2F2 SiH3F SiHCl3 SiH2Cl2 SiH3Cl

cc‐pVDZ 204.71 283.42 215.06 281.57 472.96 320.54 425.25 373.95 324.33 310.92 300.28 289.86cc‐pVTZ 221.50 318.89 236.82 297.02 537.93 356.10 477.17 413.11 351.48 341.62 325.96 310.45cc‐pVQZ 228.02 330.28 245.75 302.24 556.99 371.28 492.97 425.54 360.42 354.39 336.27 318.26cc‐pV5Z 230.67 334.16 249.81 303.94 563.84 378.79 498.55 429.83 363.42 360.38 340.77 321.32CBS(DTQ) 231.93 337.10 251.10 305.37 568.40 380.37 502.44 432.98 365.77 362.05 342.46 322.94CBS(TQ5) 232.26 336.49 252.25 304.95 567.95 383.29 501.90 432.40 365.21 363.96 343.46 323.15CBS(DTQ5) 232.02 336.65 251.57 305.10 567.93 381.66 501.97 432.53 365.38 362.86 342.84 322.95cc‐pV(D+d)Z 208.41 290.28 220.07 285.13 487.11 330.34 436.49 382.43 330.25 319.02 306.71 294.77cc‐pV(T+d)Z 223.27 322.44 239.28 298.56 544.95 360.77 482.76 417.29 354.29 345.50 329.02 312.73cc‐pV(Q+d)Z 229.07 332.11 247.28 303.07 560.44 374.21 495.79 427.69 361.90 356.77 338.11 319.57cc‐pV(5+d)Z 230.84 334.57 250.05 304.07 564.58 379.20 499.14 430.26 363.70 360.72 341.04 321.52


248



CBS(DTQ) 232.55 337.91 252.07 305.77 569.73 382.26 503.59 433.91 366.46 363.53 343.55 323.67CBS(TQ5) 231.90 336.04 251.71 304.67 566.52 382.19 501.15 431.80 364.77 363.09 342.80 322.69CBS(DTQ5) 232.16 336.85 251.80 305.17 568.19 382.08 502.20 432.72 365.52 363.19 343.08 323.10aug‐cc‐pVDZ 208.86 300.18 221.00 283.81 500.10 329.00 447.12 389.92 334.03 318.33 306.40 294.31aug‐cc‐pVTZ 223.20 323.26 239.11 298.26 544.32 359.79 482.58 417.41 354.48 344.69 328.46 312.38aug‐cc‐pVQZ 228.85 331.97 247.03 302.66 559.60 373.51 495.10 427.14 361.48 356.15 337.58 319.12aug‐cc‐pV5Z 231.05 334.99 250.39 304.13 565.11 379.75 499.55 430.59 363.90 361.14 341.34 321.71CBS(DTQ) 232.23 337.19 251.77 305.30 568.76 381.73 502.60 432.97 365.67 363.02 343.04 323.17CBS(TQ5) 232.37 336.80 252.40 305.01 568.41 383.50 502.22 432.66 365.36 364.13 343.60 323.26CBS(DTQ5) 232.24 336.89 252.00 305.10 568.40 382.46 502.26 432.70 365.43 363.44 343.21 323.13aug‐cc‐pV(D+d)Z 212.38 306.87 225.73 287.27 513.98 338.19 458.08 358.83 339.72 325.94 312.46 298.96aug‐cc‐pV(T+d)Z 224.95 326.87 241.57 299.77 551.43 364.53 488.24 408.66 357.28 348.58 331.51 314.62aug‐cc‐pV(Q+d)Z 229.85 333.74 248.48 303.46 562.97 376.29 497.83 424.41 362.91 358.41 339.32 320.38aug‐cc‐pV(5+d)Z 231.23 335.31 250.66 304.24 565.73 380.26 500.06 428.92 364.14 361.57 341.66 321.92CBS(DTQ) 232.79 337.86 252.62 305.68 569.88 383.34 503.58 433.85 366.28 364.31 344.01 323.83CBS(TQ5) 232.05 336.25 251.97 304.71 567.39 382.64 501.40 431.62 364.88 363.46 343.06 322.85CBS(DTQ5) 232.36 336.98 252.22 305.15 568.49 382.87 502.37 432.54 365.52 363.77 343.45 323.27ccCA 229.45 333.76 247.95 304.05 563.61 375.92 498.69 430.07 363.65 358.64 339.66 320.86Experiment (232.17) (337.50) (249.69) 303.19a 565.70a 379.39a 499.49a 434.32a 368.28a 361.94a 342.41a 322.38a

a. Chase, M. W. J. Phys. Chem. Ref. Data 1998, 9, 1; CODATA Key Values for Thermodynamics; Cox, J. D.; Wagman, D. D.; Medvedev, V. A., Eds.; Hemisphere Publishing Corp.: New York, NY, 1989.

b. Fisher, E. R.; Kickel, B. L.; Armentrout, P. B. J. Phys. Chem. 1993, 97, 10204. c. Hildenbrand, D. L.; Lau, K. H.; Sanjurjo, A. J. Phys. Chem. 2003, 107, 5448. d. Berkowitz, J.; Greene, J. P.; Cho, H.; Ruscic, B. J. Chem. Phys. 1987, 86, 1235. e. Irikura, K. K.; Johnson III, R. D.; Hudgens, J. W. J. Phys. Chem. 1992, 96, 4306.

249

Table 10.6 CCSD(T) and ccCA enthalpies of formation at 298 K (kcal/mol) and extrapolated energies; atomic spin‐orbit corrections have been applied to the CCSD(T) results.


cc‐pVDZ 95.40 9.88 51.29 76.68 ‐104.86 ‐9.76 ‐8.08 35.04 63.50 ‐169.98 ‐33.41 ‐9.44cc‐pVTZ 90.42 ‐6.46 41.62 67.66 ‐136.34 ‐28.00 ‐27.97 21.55 51.75 ‐217.76 ‐59.98 ‐32.95cc‐pVQZ 88.85 ‐11.95 37.64 64.82 ‐146.50 ‐35.61 ‐34.50 16.31 47.72 ‐232.60 ‐71.22 ‐40.76cc‐pV5Z 88.31 ‐13.81 35.74 63.82 ‐150.03 ‐39.29 ‐36.77 13.98 46.45 ‐237.75 ‐76.72 ‐43.35CBS(DTQ) 87.91 ‐15.24 35.26 63.12 ‐152.60 ‐40.18 ‐38.41 13.17 45.31 ‐241.50 ‐77.96 ‐45.45CBS(TQ5) 87.98 ‐14.92 34.59 63.22 ‐152.15 ‐41.50 ‐38.13 12.58 45.69 ‐240.84 ‐80.02 ‐44.90CBS(DTQ5) 87.97 ‐15.01 34.97 63.21 ‐152.25 ‐40.75 ‐38.19 12.94 45.55 ‐240.98 ‐78.86 ‐45.08cc‐pV(D+d)Z 94.53 7.40 49.52 74.89 ‐110.36 ‐13.38 ‐11.59 32.34 61.04 ‐179.30 ‐39.77 ‐14.00cc‐pV(T+d)Z 90.03 ‐7.83 40.68 66.87 ‐139.29 ‐29.87 ‐29.78 20.21 50.66 ‐222.66 ‐63.14 ‐35.23cc‐pV(Q+d)Z 88.63 ‐12.68 37.03 64.38 ‐148.02 ‐36.82 ‐35.46 15.48 47.14 ‐235.07 ‐73.24 ‐41.97cc‐pV(5+d)Z 88.26 ‐14.01 35.62 63.73 ‐150.39 ‐39.50 ‐36.99 13.83 46.35 ‐238.31 ‐77.03 ‐43.61CBS(DTQ) 87.79 ‐15.58 34.84 62.89 ‐153.25 ‐40.99 ‐38.87 12.65 45.03 ‐242.50 ‐79.29 ‐46.01CBS(TQ5) 88.04 ‐14.81 34.78 63.35 ‐151.82 ‐41.10 ‐37.91 12.85 45.88 ‐240.26 ‐79.30 ‐44.59CBS(DTQ5) 87.93 ‐15.13 34.85 63.15 ‐152.42 ‐40.97 ‐38.32 12.80 45.50 ‐241.22 ‐79.18 ‐45.22aug‐cc‐pVDZ 94.22 0.35 48.20 74.79 ‐121.47 ‐15.62 ‐17.57 31.17 61.53 ‐192.67 ‐40.78 ‐19.44aug‐cc‐pVTZ 89.82 ‐9.18 40.48 66.77 ‐140.62 ‐30.12 ‐30.72 20.03 50.72 ‐223.29 ‐62.85 ‐35.91aug‐cc‐pVQZ 88.64 ‐13.01 36.98 64.49 ‐148.21 ‐36.86 ‐35.59 15.52 47.35 ‐234.81 ‐72.96 ‐41.86aug‐cc‐pV5Z 88.20 ‐14.33 35.43 63.65 ‐150.85 ‐39.84 ‐37.29 13.62 46.27 ‐238.84 ‐77.49 ‐43.89CBS(DTQ) 87.94 ‐15.31 34.88 63.12 ‐152.75 ‐40.90 ‐38.51 12.81 45.33 ‐241.72 ‐79.03 ‐45.44CBS(TQ5) 87.93 ‐15.13 34.50 63.16 ‐152.45 ‐41.63 ‐38.31 12.48 45.62 ‐241.26 ‐80.20 ‐45.10CBS(DTQ5) 87.95 ‐15.17 34.73 63.17 ‐152.51 ‐41.19 ‐38.36 12.70 45.52 ‐241.35 ‐79.50 ‐45.20


250



aug‐cc‐pV(D+d)Z 93.37 ‐2.06 46.54 73.05 ‐126.81 ‐18.98 ‐20.95 28.63 59.11 ‐201.88 ‐46.76 ‐23.87aug‐cc‐pV(T+d)Z 89.44 ‐10.58 39.56 66.01 ‐143.60 ‐31.97 ‐32.53 18.72 49.65 ‐228.29 ‐66.03 ‐38.18aug‐cc‐pV(Q+d)Z 88.43 ‐13.70 36.40 64.06 ‐149.67 ‐38.00 ‐36.52 14.73 46.79 ‐237.20 ‐74.88 ‐43.02aug‐cc‐pV(5+d)Z 88.16 ‐14.47 35.30 63.59 ‐151.14 ‐40.08 ‐37.47 13.47 46.19 ‐239.29 ‐77.86 ‐44.08CBS(DTQ) 87.83 ‐15.58 34.50 62.90 ‐153.31 ‐41.62 ‐38.91 12.34 45.07 ‐242.54 ‐80.18 ‐45.92CBS(TQ5) 88.00 ‐14.93 34.65 63.30 ‐152.03 ‐41.32 ‐38.04 12.71 45.84 ‐240.55 ‐79.64 ‐44.72CBS(DTQ5) 87.93 ‐15.22 34.61 63.13 ‐152.60 ‐41.41 ‐38.42 12.57 45.49 ‐241.44 ‐79.82 ‐45.26ccCA 88.88 ‐13.90 36.56 63.80 ‐150.83 ‐38.55 ‐37.33 14.32 47.67 ‐236.16 ‐73.45 ‐42.33Experiment 90.02a ‐11.20b 36.80c 64.61d ‐152.40b ‐40.30a (‐39.16) (12.19) 45.79d ‐238.40b ‐84.40e (‐45.37)


cc‐pVDZ 32.47 ‐89.44 ‐0.55 29.27 ‐293.82 ‐100.97 ‐213.34 ‐129.19 ‐46.61 ‐68.19 ‐35.07 ‐2.00cc‐pVTZ 15.68 ‐124.87 ‐22.30 13.82 ‐358.74 ‐136.52 ‐265.21 ‐168.32 ‐73.74 ‐98.88 ‐60.75 ‐22.58cc‐pVQZ 9.16 ‐136.24 ‐31.23 8.60 ‐377.76 ‐151.69 ‐280.99 ‐180.73 ‐82.67 ‐111.65 ‐71.06 ‐30.39cc‐pV5Z 6.51 ‐140.12 ‐35.30 6.90 ‐384.60 ‐159.20 ‐286.56 ‐185.01 ‐85.66 ‐117.63 ‐75.56 ‐33.44CBS(DTQ) 5.25 ‐143.06 ‐36.58 5.47 ‐389.16 ‐160.78 ‐290.45 ‐188.17 ‐88.02 ‐119.30 ‐77.24 ‐35.06CBS(TQ5) 4.93 ‐142.44 ‐37.73 5.89 ‐388.71 ‐163.70 ‐289.91 ‐187.58 ‐87.46 ‐121.21 ‐78.25 ‐35.27CBS(DTQ5) 5.17 ‐142.61 ‐37.05 5.74 ‐388.68 ‐162.06 ‐289.98 ‐187.72 ‐87.63 ‐120.11 ‐77.63 ‐35.08cc‐pV(D+d)Z 28.77 ‐96.29 ‐5.56 25.71 ‐307.98 ‐110.76 ‐224.58 ‐137.67 ‐52.53 ‐76.29 ‐41.51 ‐6.90cc‐pV(T+d)Z 13.91 ‐128.42 ‐24.77 12.28 ‐365.75 ‐141.19 ‐270.80 ‐172.50 ‐76.55 ‐102.76 ‐63.81 ‐24.86cc‐pV(Q+d)Z 8.11 ‐138.07 ‐32.76 7.77 ‐381.22 ‐154.62 ‐283.81 ‐182.88 ‐84.15 ‐114.03 ‐72.89 ‐31.69cc‐pV(5+d)Z 6.34 ‐140.53 ‐35.53 6.77 ‐385.34 ‐159.61 ‐287.15 ‐185.44 ‐85.95 ‐117.98 ‐75.83 ‐33.65


251



CBS(DTQ) 4.64 ‐143.86 ‐37.55 5.07 ‐390.49 ‐162.67 ‐291.60 ‐189.10 ‐88.71 ‐120.78 ‐78.34 ‐35.79CBS(TQ5) 5.29 ‐142.00 ‐37.19 6.17 ‐387.28 ‐162.60 ‐289.16 ‐186.98 ‐87.02 ‐120.34 ‐77.59 ‐34.82CBS(DTQ5) 5.03 ‐142.81 ‐37.28 5.68 ‐388.95 ‐162.48 ‐290.21 ‐187.91 ‐87.77 ‐120.44 ‐77.86 ‐35.23aug‐cc‐pVDZ 28.32 ‐106.19 ‐6.49 27.03 ‐320.97 ‐109.42 ‐235.21 ‐145.16 ‐56.31 ‐75.59 ‐41.20 ‐6.44aug‐cc‐pVTZ 13.98 ‐129.24 ‐24.60 12.58 ‐365.13 ‐140.21 ‐270.62 ‐172.62 ‐76.74 ‐101.95 ‐63.25 ‐24.51aug‐cc‐pVQZ 8.34 ‐137.94 ‐32.51 8.18 ‐380.38 ‐153.91 ‐283.12 ‐182.33 ‐83.73 ‐113.40 ‐72.36 ‐31.25aug‐cc‐pV5Z 6.13 ‐140.95 ‐35.87 6.71 ‐385.87 ‐160.16 ‐287.56 ‐185.78 ‐86.15 ‐118.39 ‐76.13 ‐33.83CBS(DTQ) 4.96 ‐143.15 ‐37.25 5.54 ‐389.52 ‐162.13 ‐290.61 ‐188.16 ‐87.92 ‐120.27 ‐77.83 ‐35.29CBS(TQ5) 4.81 ‐142.75 ‐37.88 5.83 ‐389.17 ‐163.90 ‐290.23 ‐187.84 ‐87.60 ‐121.38 ‐78.39 ‐35.38CBS(DTQ5) 4.95 ‐142.85 ‐37.48 5.74 ‐389.16 ‐162.86 ‐290.27 ‐187.88 ‐87.68 ‐120.70 ‐78.00 ‐35.26aug‐cc‐pV(D+d)Z 24.80 ‐112.89 ‐11.22 23.57 ‐334.85 ‐118.62 ‐246.17 ‐114.07 ‐62.00 ‐83.20 ‐47.25 ‐11.09aug‐cc‐pV(T+d)Z 12.23 ‐132.84 ‐27.06 11.07 ‐372.24 ‐144.95 ‐276.28 ‐163.86 ‐79.54 ‐105.84 ‐66.31 ‐26.75aug‐cc‐pV(Q+d)Z 7.34 ‐139.71 ‐33.96 7.38 ‐383.74 ‐156.70 ‐285.85 ‐179.60 ‐85.16 ‐115.67 ‐74.11 ‐32.50aug‐cc‐pV(5+d)Z 5.96 ‐141.27 ‐36.14 6.60 ‐386.49 ‐160.67 ‐288.07 ‐184.11 ‐86.39 ‐118.82 ‐76.45 ‐34.05CBS(DTQ) 4.40 ‐143.82 ‐38.10 5.17 ‐390.64 ‐163.75 ‐291.59 ‐189.04 ‐88.53 ‐121.56 ‐78.79 ‐35.96CBS(TQ5) 5.13 ‐142.21 ‐37.45 6.13 ‐388.15 ‐163.05 ‐289.41 ‐186.81 ‐87.13 ‐120.71 ‐77.85 ‐34.97CBS(DTQ5) 4.82 ‐142.93 ‐37.70 5.70 ‐389.25 ‐163.27 ‐290.38 ‐187.73 ‐87.76 ‐121.03 ‐78.24 ‐35.40ccCA 8.34 ‐139.09 ‐32.81 7.32 ‐383.82 ‐154.87 ‐286.15 ‐184.73 ‐85.37 ‐115.27 ‐73.83 ‐32.38Experiment (5.06) (‐143.40) (‐35.11) 8.20a ‐385.98a ‐158.40a ‐287.00a ‐189.00a ‐90.00a ‐118.60a ‐76.60a ‐33.90a

a. Chase, M. W. J. Phys. Chem. Ref. Data 1998, 9, 1; CODATA Key Values for Thermodynamics; Cox, J. D.; Wagman, D. D.; Medvedev, V. A., Eds.; Hemisphere Publishing Corp.: New York, NY, 1989.

b. Fisher, E. R.; Kickel, B. L.; Armentrout, P. B. J. Phys. Chem. 1993, 97, 10204. c. Hildenbrand, D. L.; Lau, K. H.; Sanjurjo, A. J. Phys. Chem. 2003, 107, 5448. d. Berkowitz, J.; Greene, J. P.; Cho, H.; Ruscic, B. J. Chem. Phys. 1987, 86, 1235. e. Irikura, K. K.; Johnson III, R. D.; Hudgens, J. W. J. Phys. Chem. 1992, 96, 4306.

252

Table 10.7 CCSD(T) and ccCA enthalpies of dissociation at 298.15 K (kcal/mol) of the silicon hydrides, computed with the aug‐cc‐pV(n+d)Z basis sets; the mean absolute deviation (MAD) is relative to the experimental values.

CCSD(T)

Reaction CBS(DTQ) CBS(TQ5) CBS(DTQ5) ccCA Expt.

SiH → Si + H 69.18 68.98 69.08 68.13 66.98

SiH2 → SiH + H 75.55 75.32 75.42 75.70 76.04

→ Si + H2 42.96 42.61 42.78 42.25 41.77

SiH3 → SiH2 + H 68.45 68.08 68.26 66.75 69.44

→ SiH + H2 42.24 41.72 41.97 40.88 44.23

→ Si + H2 + H 111.41 110.70 111.04 109.00 111.21

→ Si + (3/2)H2 60.53 59.86 60.19 58.22 60.59

SiH4 → SiH3 + H 90.53 90.32 90.42 90.97 88.21

→ SiH2 + H2 57.21 56.72 56.96 56.15 56.41

→ SiH + H2 + H 132.76 132.04 132.38 131.85 132.45

→ SiH + (3/2)H2 81.88 81.20 81.53 81.06 81.82

→ Si + 2H2 100.18 99.34 99.75 98.40 98.18

MAD 1.05 1.11 1.04 1.43 ‐‐‐

253

Table 10.8 CCSD(T) and ccCA enthalpies of dissociation at 298.15 K (kcal/mol) of the silicon fluorides, computed with the aug‐cc‐pV(n+d)Z basis sets; the mean absolute deviation (MAD) is relative to the experimental values.

CCSD(T)


SiF → Si + F 139.37 138.61 139.02 137.70 135.00

SiHF → SiH + F 144.15 143.35 143.77 143.62 144.47a

→ SiF + H 73.95 73.72 73.83 74.05 73.66a

→ Si + HF 78.40 77.94 78.99 77.72 79.59a

SiF2 → SiF + F 155.15 154.40 154.79 154.34 158.62

→ Si + F2 258.38 256.98 257.74 257.21 258.78 SiH2F → SiH2 + F 126.23 125.33 125.80 123.55 124.59a

→ SiHF + H 57.64 57.30 57.46 55.62 55.61a

→ SiH + HF 66.86 66.26 67.37 65.22 65.79a

→ SiF + H2 29.82 29.34 29.57 28.43 28.02a

→ Si + H2 + F 169.20 167.94 168.59 166.13 168.25a

SiHF2 → SiHF + F 122.33 121.48 121.92 119.17 120.58a

→ SiH + F2 230.34 228.80 229.63 227.97 230.22a

→ SiF + HF 61.35 60.81 61.89 59.19 59.94a

→ Si + H + F2 299.51 297.78 298.70 296.09 299.63a

SiF3 → SiF2 + F 106.65 105.83 106.26 102.75 103.42

→ SiF + F2 225.66 224.20 224.98 222.26 227.20

→ Si + (3/2)F2 346.96 344.79 345.97 342.55 344.78

SiH3F → SiH3 + F 151.02 150.28 150.67 150.46 153.21

→ SiH2 + HF 84.54 83.97 85.07 83.18 89.47

→ SiH2F + H 93.23 93.03 93.12 93.66 93.61a

→ SiHF + H2 49.10 48.65 48.87 48.04 47.98a

→ SiH + H2 + F 193.25 191.99 192.64 191.66 197.44

→ SiF + (3/2)H2 72.17 71.53 71.84 71.47 78.80

→ Si + HF + H2 127.50 126.59 127.85 125.76 131.24 SiH2F2 → SiH2F + F 160.53 159.35 159.88 159.81 160.72a

→ SiH2 + F2 250.63 248.65 249.62 248.53 253.61

→ SiHF2 + H 95.84 95.17 95.42 96.26 95.76a

→ SiF2 + H2 35.21 34.29 34.66 33.90 36.60

→ Si + 2HF 161.64 160.20 162.47 159.12 165.10

→ Si + H2 + F2 293.59 291.27 292.40 291.11 295.38


254


CCSD(T)


SiHF3 → SiHF2 + F 165.18 164.51 164.86 164.48 165.60a

→ SiHF + F2 251.37 249.97 250.72 248.82 251.34a

→ SiF2 + HF 71.39 70.92 71.96 69.34 69.46

→ SiF3 + H 99.66 99.49 99.56 100.61 99.22

→ SiH + (3/2)F2 379.41 377.09 378.31 375.03 377.02

→ Si + HF + F2 329.77 327.90 329.71 326.54 328.24

SiF4 → SiF3 + F 165.51 164.91 165.22 165.07 165.00

→ SiF2 + F2 236.02 234.71 235.41 233.00 233.58

→ SiF + (3/2)F2 373.10 371.09 372.17 369.92 374.78

→ Si + 2F2 494.40 491.69 493.16 490.20 492.36

MAD 1.93 1.84 1.83 2.32 ‐‐‐a. Enthalpies derived from the calculated enthalpies of formation in Table 10.6.

255

Table 10.9 CCSD(T) and ccCA enthalpies of dissociation at 298.15 K (kcal/mol) of the silicon chlorides, computed with the aug‐cc‐pV(n+d)Z basis sets; the mean absolute deviation (MAD) is relative to the experimental values.

CCSD(T)


SiCl → Si + Cl 99.37 99.14 99.27 97.32 97.07

SiHCl → SiH + Cl 102.98 102.71 102.85 102.05 104.78a

→ SiCl + H 72.79 72.56 72.66 72.86 72.50a

→ Si + HCl 69.38 69.00 70.93 69.92 71.56a SiCl2 → SiCl + Cl 103.62 103.38 103.51 102.60 104.59

→ Si + Cl2 145.10 144.38 144.79 144.93 146.68 SiH2Cl → SiH2 + Cl 85.99 85.59 85.80 82.96 85.83a

→ SiHCl + H 58.56 58.19 58.37 56.61 56.54a

→ SiH + HCl 58.76 58.22 60.23 58.40 58.69a

→ SiCl + H2 29.58 29.07 29.32 28.22 27.80a

→ Si + H2 + Cl 128.95 128.20 128.58 125.54 129.49a

SiHCl2 → SiHCl + Cl 77.93 77.57 77.76 74.62 77.70a

→ SiH + Cl2 123.03 122.14 122.63 121.69 127.50a

→ SiCl + HCl 47.94 47.44 49.43 47.22 47.57a

→ Si + H + Cl2 192.20 191.12 191.70 189.81 196.91a

SiCl3 → SiCl2 + Cl 66.05 65.74 65.90 62.39 71.59

→ SiCl + Cl2 111.78 110.98 111.42 110.00 121.20

→ Si + (3/2)Cl2 182.21 181.05 181.70 179.83 190.78

SiH3Cl → SiH3 + Cl 108.52 108.23 108.38 107.54 107.18

→ SiH2 + HCl 74.19 73.62 75.64 74.04 76.45

→ SiH2Cl + H 90.98 90.72 90.84 91.34 91.26a

→ SiHCl + H2 47.77 47.23 47.49 46.70 46.56a

→ SiH + H2 + Cl 150.75 149.94 150.35 148.75 151.42

→ SiCl + (3/2)H2 69.68 68.94 69.30 68.94 70.70

→ Si + HCl + H2 117.16 116.23 118.43 116.62 118.22 SiH2Cl2 → SiH2Cl + Cl 110.69 110.40 110.55 109.66 112.57a

→ SiH2 + Cl2 138.79 137.85 138.36 137.64 141.21

→ SiHCl2 + H 91.31 91.02 91.16 91.65 91.41a

→ SiCl2 + H2 36.65 36.08 36.36 35.28 36.30

→ Si + 2HCl 135.86 134.91 138.86 135.93 138.86

→ Si + H2 + Cl2 181.75 180.46 181.15 180.22 182.98


256


CCSD(T)


SiHCl3 → SiHCl2 + Cl 110.95 110.68 110.82 109.95 112.73a

→ SiHCl + Cl2 130.99 130.11 130.59 129.59 135.44a

→ SiCl2 + HCl 55.28 54.73 56.74 54.57 56.24

→ SiCl3 + H 92.00 91.68 91.83 92.44 84.82

→ SiH + (3/2)Cl2 209.38 208.48 208.95 204.14 208.62

→ Si + HCl + Cl2 200.38 199.11 201.53 199.51 202.92

SiCl4 → SiCl3 + Cl 111.06 110.82 110.95 108.92 101.49

→ SiCl2 + Cl2 119.23 118.42 118.86 116.32 118.10

→ SiCl + (3/2)Cl2 193.90 192.74 193.38 191.43 195.20

→ Si + 2Cl2 264.33 262.80 263.65 261.25 264.78

MAD 2.03 2.12 1.96 3.14 ‐‐‐MADb 1.40 1.40 1.30 2.23 ‐‐‐

a. Enthalpies derived from the calculated enthalpies of formation in Table 10.6. b. Calculated excluding reactions involving SiCl3 (see discussion in text).

257

CHAPTER 11

CONCLUDING REMARKS

This dissertation has covered several aspects of both predictive and comparative

computational quantum chemistry employing the correlation consistent basis sets. As shown in

this work, the systematic contruction of the correlation consistent basis sets may be employed

not only in ab initio calculations, but also in density functional theory (DFT) calculations to

evaluate the complete basis set (CBS) limit, and hence, the intrinsic accuracy of a given

computational method.

In DFT computations, the correlation consistent basis sets in their original form are not

optimal, efficient, nor entirely adequate for elucidating Kohn‐Sham (KS) limits. This dissertation

has discussed both the basis set requirements and the effects of recontracting (fine‐tuning) the

correlation consistent basis sets for DFT. Specifically, it has been shown that truncating the

correlation consistent basis sets of all but s, p, d, and f functions does not result in deviations

greater than a kcal/mol in computed atomic and molecular properties (relative to the full basis

set), making the correlation consistent basis sets more efficient in DFT computations. Further,

recontraction of the Hartree‐Fock (HF) minimal basis set for a specific density functional (e.g

BLYP and B3LYP) along with the inclusion of diffuse functions and a correction for basis set

superposition error (BSSE) has been shown to give smooth, monotonic convergence towards

the KS limit for main group atoms and molecules.

Regarding the development of correlation consistent basis sets for ab initio

258

methodologies, new basis sets for the s‐block atoms lithium, beryllium, sodium, and

magnesium have been reported and discussed. In particular, the standard valence, core‐

valence, augmented, and Douglas‐Kroll (DK) scalar relativistic correlation consistent basis sets

were benchmarked and discussed, and shown to give kcal/mol accuracy in many atomic and

molecular properties while exhibiting smooth, monotonic convergence towards the CBS limit.

Next, the resolution of the identity (RI) approximation was applied to the recently

developed correlation consistent Composite Approach (ccCA), a composite ab initio method

that exploits the convergent behavior of the correlation consistent basis sets. The newly

formulated RI‐ccCA method has been benchmarked against the original ccCA formulation, and

shown to give accuracy better 1.5 kcal/mol in molecular atomization energies. More novel than

the application of the RI approximation to ccCA is the dramatic difference in the computational

cost of RI‐ccCA compared with ccCA, namely, the reduction in computing time by almost an

order of magnitude and the reduction in disk space by almost two orders of magnitude.

Finally, three separate benchmark studies of heavy FOOF‐like molecules, germanium

arsenides, and silicon hydrides/halides utilizing the correlation consistent basis sets were

presented. Computed geometries, spectroscopic properties, and thermochemical properties

were discussed with respect to experimental observations. The convergent behavior of the

correlation consistent basis sets was exploited in the latter two chapters to compute CBS limits

of bond lengths, atomization energies, and enthalpies of formation, thus presenting highly

reliable predictions of experimental observations yet to be made.

259

REFERENCES

(1) Schrödinger, E. Ann. Phys. 1926, 385, 437. (2) Schrödinger, E. Ann. Phys. 1926, 384, 489. (3) Schrödinger, E. Ann. Phys. 1926, 384, 361. (4) Schrödinger, E. Phys. Rev. 1926, 28, 1049. (5) Born, M.; Oppenheimer, J. R. Ann. Phys. 1927, 389, 457. (6) Levine, I. N. Quantum Chemistry, 5 ed.; Prentice Hall, 2000. (7) Szabo, A.; Ostlund, N. S. Modern Quantum Chemistry: Introduction to Advanced

Electronic Structure Theory; McGraw‐Hill, Inc.: New York, NY, 1989. (8) Heisenberg, W. Z. Phys. 1927, 43, 172. (9) Pauli, W. Z. Phys. 1925, 31, 765. (10) Pauli, W. "Exclusion principle and quantum mechanics." (Nobel Prize Lecture) Geneva,

Switzerland, 1945. (11) Slater, J. C. Phys. Rev. 1929, 34, 1293. (12) Hartree, D. R. Proc. Camb. Phil. Soc. 1928, 24, 89. (13) Hartree, D. R. Proc. Camb. Phil. Soc. 1928, 24, 111. (14) Hartree, D. R. Proc. Camb. Phil. Soc. 1928, 24, 426. (15) Lay, D. C. Linear Algebra and its Applications, 2 ed.; Addison Wesley Longman, Inc.:

Reading, MA, 2000. (16) Hund, F. Z. Phys. 1925, 33, 345. (17) Hund, F. Z. Phys. 1925, 34, 296. (18) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory;

John Wiley & Sons, Inc.: New York, NY, 1986.

260

(19) Jensen, F. Introduction to Computational Chemistry; John Wiley & Sons Ltd.: Chichester, West Sussex, 1999.

(20) Hartree, D. R.; Hartree, W.; Swirles, B. Phil. Trans. Roy. Soc. (London) 1939, A238, 229. (21) Fock, V. Z. Phys. A 1930, 61, 126. (22) Helgaker, T.; Jørgensen, P.; Olsen, J. Molecular Electronic‐Structure Theory; John Wiley &

Sons Ltd.: Chichester, West Sussex, 2000. (23) Hylleraas, E. A. Z. Phys. 1928, 48, 469. (24) Hylleraas, E. A. Z. Phys. 1929, 54, 347. (25) Hylleraas, E. A. Z. Phys. 1930, 65, 209. (26) Slater, J. C. Phys. Rev. 1930, 36, 57. (27) Koopmans, T. A. Physica 1933, 1, 104. (28) Roothaan, C. C. J. Rev. Mod. Phys. 1951, 23, 69. (29) Nooijen, M.; Shamasundar, K. R.; Mukherjee, D. Mol. Phys. 2005, 103, 2277. (30) Crawford, T. D.; Schaefer III, H. F. "An Introduction to Coupled Cluster Theory for

Computational Chemists." In Reviews in Computational Chemistry; Boyd, D. B., Lipkowitz, K. B., Eds.; Wiley‐VCH: New York, NY, 2000; Vol. 14; pp 33.

(31) Bartlett, R. J. Journal of Physical Chemistry 1989, 93, 1697. (32) Paldus, J. "Coupled Cluster Theory." In Methods in Computational Molecular Physics;

Wilson, S., Diercksen, H. F., Eds.; Plenum Press: New York, NY, 1992. (33) Paldus, J.; Li, X. "A Critical Assessment of Coupled Cluster Method in Quantum

Chemistry." In Advances in Chemical Physics; Prigogine, I., Rice, S., Eds.; John Wiley and Sons, Inc., 1999; Vol. 110; pp 1.

(34) Bartlett, R. J. Ann. Rev. Phys. Chem. 1981, 32, 359. (35) Bartlett, R. J. "Coupled‐Cluster Theory: An Overview of Recent Developments." In

Modern Electronic Structure Theory, Part II; Yarkony, D. R., Ed.; World Scientific: Singapore, 1995.

261

(36) Bartlett, R. J.; Stanton, J. F. "Applications of Post‐Hartree‐Fock Methods: A Tutorial." In Reviews in Computational Chemistry; Lipkowitz, K. B., Boyd, D. B., Eds.; VCH Publishers, Inc.: New York, NY, 1994; Vol. 5.

(37) Lee, T. J.; Rendell, A. P.; Taylor, P. R. J. Phys. Chem. 1990, 94, 5463. (38) Raghavachari, K.; Trucks, G. W.; Pople, J. A.; Head‐Gordon, M. Chem. Phys. Lett. 1989,

157, 479. (39) Kvasnička, V.; Laurinc, V.; Biskupič, S. Mol. Phys. 1980, 39, 143 (40) Møller, C.; Plesset, M. S. Phys. Rev. 1934, 46, 618. (41) Dirac, P. A. M. Proc. Camb. Phil. Soc. 1930, 26, 376. (42) Fermi, E. Z. Phys. 1928, 48, 73. (43) Thomas, L. H. Proc. Camb. Phil. Soc. 1927, 23, 542. (44) Hohenberg, P.; Kohn, W. Phys. Rev. 1964, 136, B864. (45) Parr, R. G.; Yang, W. Density‐Functional Theory of Atoms and Molecules; Oxford

University Press, Inc.: New York, NY, 1989. (46) Koch, W.; Holthausen, M. C. A Chemist's Guide to Density Functional Theory, 2 ed.;

Wiley‐VCH Verlag GmbH: Weinheim, Germany, 2002. (47) A Primer in Density Functional Theory; Fiolhais, C.; Nogueira, F.; Marques, M., Eds.;

Springer‐Verlag: Berlin, 2003. (48) Density Functional Theory: A Bridge Between Chemistry and Physics; Geerlings, P.; De

Proft, F.; Langenaeker, W., Eds.; VUB University Press: Brussels, 1999. (49) Many‐Electron Densities and Reduced Density Matrices; Cioslowski, J., Ed.; Kluwer

Academic/Plenum Publishers: New York, NY, 2000. (50) Löwdin, P.‐O. Phys. Rev. 1955, 97, 1474. (51) Hohenberg, P.; Kohn, W. Physical Review 1964, 136, B864. (52) Kohn, W.; Sham, L. J. Phys. Rev. 1965, 140, A1133. (53) Jackson, J. D. Classical Electrodynamics, 3 ed.; John Wiley & Sons, Inc.: New York, NY,

1998.

262

(54) Griffiths, D. J. Introduction to Electrodynamics, 3 ed.; Prentice Hall: Upper Saddle River,

NJ, 1999. (55) Cook, D. B. Handbook of Computational Quantum Chemistry; Dover Publications, Inc.:

Mineola, NY, 2005. (56) Boys, S. F. Proc. Roy. Soc. (London) 1950, A200, 542. (57) Taketa, H.; Huzinaga, S.; O‐Ohata, K. J. Phys. Soc. Jpn. 1966, 21, 2313. (58) Boys, S. F. Proc. Roy. Soc. (London) 1950, A201, 125. (59) Schwartz, C. Phys. Rev. 1962, 126, 1015. (60) Curtiss, L. A.; Pople, J. A. Chem. Phys. Lett. 1988, 144, 38. (61) Curtiss, L. A.; Raghavachari, K.; Trucks, G. W.; Pople, J. A. J. Chem. Phys. 1991, 94, 7221. (62) Curtiss, L. A.; Redfern, P. C.; Rassolov, V. A.; Kedziora, G. S.; Pople, J. A. J. Chem. Phys.

2001, 114, 9287. (63) Montgomery Jr., J. A.; Frisch, M. J.; Ochterski, J. W.; Petersson, G. A. J. Chem. Phys. 1999,

110, 2822. (64) Montgomery Jr., J. A.; Frisch, M. J.; Ochterski, J. W.; Petersson, G. A. J. Chem. Phys. 2000,

112, 6532. (65) Montgomery Jr., J. A.; Ochterski, J. W.; Petersson, G. A. J. Chem. Phys. 1994, 101, 5900. (66) Ochterski, J. W.; Petersson, G. A.; Montgomery Jr., J. A. J. Chem. Phys. 1996, 104, 2598. (67) Petersson, G. A.; Al‐Laham, M. A. J. Chem. Phys. 1991, 94, 6081. (68) Petersson, G. A.; Bennett, A.; Tensfeldt, T. G.; Al‐Laham, M. A.; Shirley, W. A.; Mantzaris,

J. J. Chem. Phys. 1988, 89, 2193. (69) Petersson, G. A.; Tensfeldt, T. G.; Montgomery Jr., J. A. J. Chem. Phys. 1991, 94, 6091. (70) Harding, M. E.; Vazquez, J.; Ruscic, B.; Wilson, A. K.; Gauss, J.; Stanton, J. F. J. Chem.

Phys. 2008, 128, 114111. (71) Boese, A. D.; Oren, M.; Atasoylu, O.; Martin, J. M. L.; Kállay, M.; Gauss, J. J. Chem. Phys.

2004, 120, 4129.

263

(72) Martin, J. M. L.; De Oliveira, G. J. Chem. Phys. 1999, 111, 1843. (73) DeYonker, N. J.; Cundari, T. R.; Wilson, A. K. J. Chem. Phys. 2006, 124, 114104. (74) DeYonker, N. J.; Grimes, T.; Yockel, S. M.; Dinescu, A.; Mintz, B. J.; Cundari, T. R.; Wilson,

A. K. J. Chem. Phys. 2006, 125, 104111. (75) DeYonker, N. J.; Ho, D. S.; Wilson, A. K.; Cundari, T. R. J. Phys. Chem. A 2007, 111, 10776. (76) DeYonker, N. J.; Mintz, B. J.; Cundari, T. R.; Wilson, A. K. J. Chem. Theor. Comput. 2008,

4, 328. (77) DeYonker, N. J.; Peterson, K. A.; Steyl, G.; Wilson, A. K.; Cundari, T. R. J. Phys. Chem. A

2007, 111, 11269. (78) Grimes, T. V.; Cundari, T. R.; DeYonker, N. J.; Wilson, A. K. J. Chem. Phys. 2007, 127,

154117. (79) Williams, T. G.; Wilson, A. K. J. Sulf. Chem. 2008, 29, 353. (80) Ho, D. S.; DeYonker, N. J.; Wilson, A. K.; Cundari, T. R. J. Phys. Chem. A 2007, 110, 9767. (81) Peterson, K. A.; Dunning Jr., T. H. J. Chem. Phys. 2002, 117, 10548. (82) Woon, D. E.; Dunning Jr., T. H. J. Chem. Phys. 1993, 99, 1914. (83) Peterson, K. A.; Kendall, R. A.; Dunning Jr., T. H. J. Chem. Phys. 1993, 99, 1930. (84) Peterson, K. A.; Kendall, R. A.; Dunning Jr., T. H. J. Chem. Phys. 1993, 99, 9790. (85) Peterson, K. A.; Woon, D. E.; Dunning Jr., T. H. J. Chem. Phys. 1994, 100, 7410. (86) Woon, D. E.; Peterson, K. A.; Dunning Jr., T. H. J. Chem. Phys. 1998, 109, 2233. (87) Woon, D. E.; Dunning Jr., T. H. J. Chem. Phys. 1994, 101, 8877. (88) Peterson, K. A.; Dunning Jr., T. H. J. Chem. Phys. 1995, 102, 2032. (89) Peterson, K. A.; Dunning Jr., T. H. J. Chem. Phys. 1997, 106, 4119. (90) Wilson, A. K.; Dunning Jr., T. H. J. Chem. Phys. 1997, 106, 8718. (91) van Mourik, T.; Wilson, A. K.; Dunning Jr., T. H. Mol. Phys. 1999, 96, 529.

264

(92) Peterson, K. A.; Figgen, D.; Dolg, M.; Stoll, H. J. Chem. Phys. 2007, 126, 124101. (93) Peterson, K. A. Ann. Rep. Comp. Chem. 2007, 3, 195. (94) Dunning Jr., T. H. J. Chem. Phys. 1989, 90, 1007. (95) Woon, D. E.; Dunning Jr., T. H. J. Chem. Phys. 1992, 98, 1358. (96) Woon, D. E.; Dunning Jr., T. H. J. Chem. Phys. 1994, 100, 2975. (97) Wilson, A. K.; Woon, D. E.; Peterson, K. A.; Dunning Jr., T. H. J. Chem. Phys. 1999, 110,

7667. (98) Woon, D. E.; Dunning Jr., T. H. J. Chem. Phys. 1995, 103, 4572. (99) Wilson, A. K.; van Mourik, T.; Dunning Jr., T. H. J. Mol. Struct. (THEOCHEM) 1996, 388,

339. (100) van Mourik, T.; Dunning Jr., T. H. Int. J. Quant. Chem. 1999, 76, 205. (101) Dunning Jr., T. H.; Peterson, K. A.; Wilson, A. K. J. Chem. Phys. 2001, 114, 9244. (102) Prascher, B. P.; Wilson, A. K.; Dunning Jr., T. H.; Woon, D. E.; Peterson, K. A. (in

preparation). (103) Yockel, S. M. The evaluation, development, and application of the correlation consistent

basis sets. Ph.D. Thesis, University of North Texas, USA, 2006. (104) Peterson, K. A.; Adler, T. B.; Werner, H.‐J. J. Chem. Phys. 2008, 128, 084102. (105) Balabanov, N. B.; Peterson, K. A. J. Chem. Phys. 2005, 123, 064107. (106) Peterson, K. A.; Puzzarini, C. Theoret. Chem. Acc. 2005, 114, 283. (107) Peterson, K. A. J. Chem. Phys. 2003, 119, 11099. (108) Peterson, K. A.; Figgen, D.; Goll, E.; Stoll, H.; Dolg, M. J. Chem. Phys. 2003, 119, 11113. (109) Balabanov, N. B.; Peterson, K. A. J. Phys. Chem. A 2003, 107, 7465. (110) Jankowski, K.; Becherer, R.; Scharf, P.; Schiffer, H.; Ahlrichs, R. J. Chem. Phys. 1985, 82,

1413.

265

(111) Ahlrichs, R.; Scharf, P.; Jankowski, K. Chem. Phys. 1985, 98, 381. (112) Becherer, R.; Ahlrichs, R. Chem. Phys. 1985, 99, 389. (113) Almlöf, J.; Taylor, P. R. J. Chem. Phys. 1987, 86, 4070. (114) Almlöf, J.; Taylor, P. R. J. Chem. Phys. 1990, 92, 551. (115) van Duijneveldt, F. B. IBM Research Report 1971, RJ 945. (116) Raffenetti, R. C. J. Chem. Phys. 1973, 58, 4452. (117) Almlöf, J.; Taylor, P. R. J. Chem. Phys. 1987, 86, 4070. (118) Ruedenberg, K.; Raffenetti, R. C.; Bardo, R. D. “Energy, Structure, and Reactivity.”;

Proceedings from the 1972 Boulder Seminar Research Conference on Theoretical Chemistry, Boulder, Colorado.

(119) Bauschlicher, C. W.; Partridge, H. Chem. Phys. Lett. 1995, 240, 533. (120) Martin, J. M. L. J. Chem. Phys. 1998, 108, 2791. (121) Martin, J. M. L.; Baldridge, K. K.; Lee, T. J. Mol. Phys. 1999, 97, 945. (122) Martin, J. M. L.; Uzan, O. Chem. Phys. Lett. 1998, 282, 16. (123) Woon, D. E.; Dunning, T. H. J. Chem. Phys. 1992, 98, 1358. (124) Dunning, T. H.; Peterson, K. A.; Wilson, A. K. J. Chem. Phys. 2001, 114, 9244. (125) Wang, N. X.; Wilson, A. K. J. Phys. Chem. A 2003, 107, 6720. (126) Wang, N. X.; Wilson, A. K. J. Phys. Chem. A 2005, 109, 7187. (127) Wilson, A. K.; Dunning Jr., T. H. J. Chem. Phys. 2003, 119, 11712. (128) Wilson, A. K.; Dunning Jr., T. H. J. Phys. Chem. A 2004, 108, 3129. (129) Heckert, M.; Kállay, M.; Tew, D. P.; Klopper, W.; Gauss, J. J. Chem. Phys. 2006, 125,

044108. (130) Bakowies, D. J. Chem. Phys. 2007, 127, 084105. (131) Varandas, A. J. C. J. Chem. Phys. 2000, 113, 8880.

266

(132) Feller, D. J. Chem. Phys. 1992, 96, 6104. (133) Halkier, A.; Helgaker, T.; Jorgensen, P.; Klopper, W.; Koch, H.; Olsen, J.; Wilson, A. K.

Chem. Phys. Lett. 1998, 286, 243. (134) Taylor, P. R. Lecture Notes on Quantum Chemistry; Roos, B. O., Ed.; Springer‐Verlag:

Berlin, Germany, 1992; pp 406. (135) Douglas, M.; Kroll, N. M. Ann. Phys. 1974, 82, 89. (136) Hess, B. A. Phys. Rev. A 1986, 33, 3742. (137) Relativistic Effects in Heavy‐Element Chemistry and Physics; Hess, B. A., Ed.; John Wiley

and Sons, Ltd.: Chichester, West Sussex, England, 2003. (138) Dirac, P. A. M. Proc. Roy. Soc. (London) 1928, 117, 610. (139) Dirac, P. A. M. Proc. Roy. Soc. (London) 1930, 126, 360. (140) Balasubramanian, K. Relativistic Effects in Chemistry; John Wiley and Sons, Inc.: New

York, NY, 1997. (141) Dyall, K. G.; Fægri, K. Introduction to Relativistic Quantum Chemistry; Oxford University

Press: New York, NY, 2007. (142) de Jong, W. A.; Harrison, R. J.; Dixon, D. A. J. Chem. Phys. 2001, 114, 48. (143) Jensen, F. J. Chem. Phys. 2001, 115, 9113. (144) Jensen, F. J. Chem. Phys. 2002, 116, 7372. (145) Jensen, F. J. Chem. Phys. 2003, 118, 2459. (146) Jensen, F. Chem. Phys. Lett. 2005, 402, 510. (147) Jensen, F. J. Chem. Phys. 2005, 122, 74111. (148) Prascher, B. P.; Wilson, A. K. Mol. Phys. 2007, 105, 2899. (149) Prascher, B. P.; Wilson, B. R.; Wilson, A. K. J. Chem. Phys. 2007, 127, 124110. (150) Sodt, A.; Subotnik, J. E.; Head‐Gordon, M. J. Chem. Phys. 2006, 125, 194109.

267

(151) Steele, R. P.; Shao, Y.; DiStasio, R. A.; Head‐Gordon, M. J. Phys. Chem. A. 2006, 110, 13915.

(152) Wang, N. X.; Venkatesh, K.; Wilson, A. K. J. Phys. Chem. A. 2006, 110, 779. (153) Wang, N. X.; Wilson, A. K. (unpublished results). (154) Wang, N. X.; Wilson, A. K. J. Chem. Phys. 2004, 121, 7632. (155) Wang, N. X.; Wilson, A. K. Mol. Phys. 2005, 103, 345. (156) Wang, X. The performance of density functionals with respect to the correlation

consistent basis sets. Ph.D. Disstertation, University of North Texas, 2005. (157) Sekusak, S.; Frenking, G. J. Mol. Struct. (THEOCHEM) 2001, 541, 17. (158) Raymond, K. S.; Wheeler, R. A. J. Comput. Chem. 1999, 20, 207. (159) Skylaris, C.‐K.; Gagliardi, L.; Handy, N. C.; Ioannou, A. G.; Spencer, S.; Willetts, A. J. Mol.

Struct. (THEOCHEM) 2000, 501‐502, 229. (160) Vahtras, O.; Almlöf, J. E.; Feyereisen, M. W. Chem. Phys. Lett. 1993, 213, 514. (161) Van Alsenoy, C. J. Comp. Chem. 1988, 9, 620. (162) Mintz, B. J.; Lennox, K. P.; Wilson, A. K. J. Chem. Phys. 2004, 121, 5629. (163) Mintz, B. J.; Wilson, A. K. J. Chem. Phys. 2005, 122, 134106. (164) Becke, A. D. Phys. Rev. A. 1988, 38, 3098. (165) Lee, C.; Yang, W.; Parr, R. G. Phys. Rev. B 1987, 37, 785. (166) Becke, A. D. J. Chem. Phys. 1993, 98, 5648. (167) Vosko, S. H.; Wilk, L.; Nusair, M. Can. J. Phys. 1980, 58, 1200. (168) Bauschlicher, C. W.; Taylor, P. R. Theor. Chim. Acta 1988, 74, 63. (169) Dunlap, B. I. "Symmetry and degeneracy in Xa and density functional theory." In Ab

Initio Methods in Quantum Chemistry, Part II; Lawley, K. P., Ed.; John Wiley and Sons: Chicester, 1987.

268

(170) Constants of Diatomic Molecules; Number 69 ed.; Huber, K. P.; Hertzberg, G., Eds.; National Institute of Standards and Technology: Gaithersburg, MD, 2005.

(171) Lias, S. G. "Ionization Energy Evaluation." In NIST Chemistry WebBook, NIST Standard

Reference Database Number 69; Linstrom, P. J., Mallard, W. G., Eds.; National Institute of Standards and Technology: Gaithersburg, MD, 2005.

(172) Lias, S. G.; Bartmess, J. E.; Liebman, J. F.; Holmes, J. L.; Levin, R. D.; Mallard, W. G. "Ion

Energetics Data." In NIST Chemistry WebBook, NIST Standard Reference Database Number 69; Linstrom, P. J., Mallard, W. G., Eds.; National Institute of Standards and Technology: Gaithersburg, MD, 2005.

(173) Yamanaka, S.; Nakata, K.; Ukai, T.; Takada, T.; Yamaguchi, K. Int. J. Quant. Chem. 2006,

106, 3312. (174) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.;

Montgomery, J., J. A.; Vreven, T.; Kudin, K. N.; Burant, J. C.; Millam, J. M.; Iyengar, S. S.; Tomasi, J.; Barone, V.; Mennucci, B.; Cossi, M.; Scalmani, G.; Rega, N.; Petersson, G. A.; Nakatsuji, H.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.; Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Klene, M.; Li, X.; Knox, J. E.; Hratchian, H. P.; Cross, J. B.; Bakken, V.; Adamo, C.; Jaramillo, J.; Gomperts, R.; Stratmann, R. E.; Yazyev, O.; Austin, A. J.; Cammi, R.; Pomelli, C.; Ochterski, J. W.; Ayala, P. Y.; Morokuma, K.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Zakrzewski, V. G.; Dapprich, S.; Daniels, A. D.; Strain, M. C.; Farkas, O.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman, J. B.; Ortiz, J. V.; Cui, Q.; Baboul, A. G.; Clifford, S.; Cioslowski, J.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Martin, R. L.; Fox, D. J.; Keith, T.; Al‐Laham, M. A.; Peng, C. Y.; Nanayakkara, A.; Challacombe, M.; Gill, P. M. W.; Johnson, B.; Chen, W.; Wong, M. W.; Gonzalez, C.; Pople, J. A. Gaussian 03; version B.05, C.02, and D.01; Gaussian, Inc.: Wallingford, CT, 2004.

(175) Krack, M.; Köster, A. M. J. Chem. Phys. 1998, 108, 3226. (176) Martin, J. M. L. J. Chem. Phys. 1998, 108, 2791. (177) van Duijneveldt, F. B.; van Duijneveldt‐van de Rijdt, J. G. C. M.; van Lenthe, J. H. Chem.

Rev. 1994, 94, 1873. (178) Boys, S. F.; Bernardi, F. Mol. Phys. 1970, 19, 553. (179) van Mourik, T.; Wilson, A. K.; Peterson, K. A.; Woon, D. E.; Dunning Jr., T. H. Adv. Quant.

Chem. 1999, 31, 105. (180) Amos, R. D.; Bernhardsson, A.; Berning, A.; Celani, P.; Cooper, D. L.; Deegan, M. J. O.;

Dobbyn, A. J.; Eckert, F.; Hampel, C.; Hetzer, G.; Knowles, P. J.; Korona, T.; Lindh, R.;

269

Lloyd, A. W.; McNicholas, S. J.; Manby, F. R.; Meyer, W.; Mura, M. E.; Nicklass, A.; Palmieri, P.; Pitzer, R.; Rauhut, G.; Schütz, M.; Schumann, U.; Stoll, H.; Stone, A. J.; Tarroni, R.; Thorsteinsson, T.; Werner, H.‐J. MOLPRO, a package of ab initio programs designed by H.‐J. Werner and P. J. Knowles; version version 2002.6.

(181) Weigend, F.; Köhn, A.; Hättig, C. J. Chem. Phys. 2002, 116, 3175. (182) Feller, D. J. Comp. Chem. 1996, 17, 1571. (183) Schuchardt, K. L.; Didier, B. T.; Elsethagen, T.; Sun, L.; Gurumoorthi, V.; Chase, J.; Li, J.;

Windus, T. L. J. Chem. Inform. Mod. 2007, 47, 1045. (184) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes in

FORTRAN 77: The Art of Scientific Computing., 2 ed.; Cambridge University Press, 1992. (185) Koput, J.; Peterson, K. A. J. Phys. Chem. A 2003, 107, 3981. (186) Koput, J.; Carter, S.; Peterson, K. A.; Theodorakopoulos, G. J. Chem. Phys. 2002, 117,

1529. (187) Ruud, K.; Jonsson, D.; Norman, P.; Agren, H.; Saue, T.; Jensen, H. J. A.; Dahle, P.;

Helgaker, T. J. Chem. Phys. 1998, 108, 7973. (188) Schwartz, C. Ann. Phys. 1967, 42, 367. (189) Martinez, T. J.; Carter, E. A. "Pseudospectral Methods Applied to the Electron

Correlation Problem." In Modern Electronic Structure Theory; Yarkony, D. R., Ed.; World Scientific: River Edge, New Jersey, 1995; Vol. 2.

(190) Kendall, R. A.; Früchtl, H. A. Theor. Chem. Acc. 1997, 97, 158. (191) Löwdin, P.‐O. Rev. Mod. Phys. 1967, 39, 259. (192) Baerends, E. J.; Ellis, D. E.; Ros, R. Chem. Phys. 1973, 2, 41. (193) Feyereisen, M. W.; Fitzgerald, G.; Komornicki, A. Chem. Phys. Lett. 1993, 208, 359. (194) Rendell, A. P.; Lee, T. J. J. Chem. Phys. 1994, 101, 400. (195) CODATA Key Values for Thermodynamics; Cox, J. D.; Wagman, D. D.; Medvedev, V. A.,

Eds.; Hemisphere Publishing Corp.: New York, NY, 1989. (196) Karton, A.; Martin, J. M. L. J. Phys. Chem. A 2007, 111, 5936.

270

(197) Schütz, M. J. Chem. Phys. 2000, 113, 9986. (198) Schütz, M.; Werner, H.‐J. Chem. Phys. Lett. 2000, 318, 370. (199) Schütz, M.; Werner, H.‐J. J. Chem. Phys. 2001, 114, 661. (200) Weigend, F. Phys. Chem. Chem. Phys. 2006, 8, 1057. (201) Prascher, B. P.; Lai, J. D.; Wilson, A. K. J. Chem. Phys., (submitted). (202) Asprey, L. B.; Kinkead, S. A.; Eller, P. G. Nuc. Tech. 1986, 73, 69. (203) Erilov, P. E.; Titov, V. V.; Serik, V. F.; Sokolov, V. B. Atomic Energy 2002, 92, 57. (204) Morrow, S. I.; Young, A. R. Inorg. Chem. 1965, 4, 759. (205) Streng, A. G. Chem. Rev. 1963, 63, 607. (206) Dickinson, L. A.; Amster, A. B.; Capener, E. L. J. Spacecr. Rocket. 1968, 5, 1329. (207) Bridgeman, A. J.; Rothery, J. J. Chem. Soc. Dalt. Trans. 1999, 4077. (208) Francisco, J. S.; Maricq, M. M. Acc. Chem. Res. 1996, 29, 391. (209) Grabowski, J. J.; Zhang, L. J. Amer. Chem. Soc. 1989, 111, 1193. (210) Kadomura, S. (Sony Corp. Japan) Plasma etching in manufacture of semiconductor

device. In Sony Corp., Japanese Kokai Tokkyo Koho, Japan; JP‐08069995: Japan, 1996. (211) Nagayama, T. (Sony Corp. Japan) Plasma etching of organic antirelfective flim. In Sony

Corp., Japanese Kokai Tokkyo Koho, Japan; JP‐10312993: Japan, 1998. (212) Nagayama, T. (Sony Corp. Japan) Plasma etching of organic antireflective flim in

manufacture of semiconductor device. In Sony Corp., Japanese Kokai Tokkyo Koho, Japan; JP‐10312992: Japan, 1998.

(213) Yokoyama, S. (Sharp Corp., Japan) Manufacture of dielectric thin film device. In Sharp

Corp., Japanese Kokai Tokkyo Koho, Japan; JP‐11354505: Japan, 1999. (214) Bachrach, S. M.; Demoin, D. W.; Luk, M.; Miller Jr., J. V. J. Phys. Chem. A 2004, 108,

4040. (215) Cao, X.; Qian, X.; Qiao, C.; Wang, D. Chemical Physics Letters 1999, 299, 322.

271

(216) Cao, X.; Qiao, C.; Wang, D. Chem. Phys. Lett. 1998, 290, 405. (217) Kurney, G. W.; Turnbull, K. Chem. Rev. 1982, 82, 333. (218) Meyer, M. J. Mol. Struct. 1992, 273, 99. (219) Jackson, R. H. J. Chem. Soc. 1962, 4585. (220) Hedberg, L.; Hedberg, K.; Eller, P. G.; Ryan, R. R. Inorg. Chem. 1988, 27, 232. (221) Hammer, P. D.; Sinha, A.; Burkholder, J. B.; Howard, C. J. J. Mol. Spectrosc. 1988, 129,

199. (222) Freedman, P. A.; Jones, W. J. J. Chem. Soc. Farad. Trans. 2 1975, 71, 286. (223) Gouedard, G.; Lehmann, J. C. J. Phys. B 1976, 9, 2113. (224) Christen, D.; Mack, H.‐G.; Oberhammer, H. Tetrahedron 1988, 44, 7363. (225) Kuczkowski, R. L. Journal of the American Chemical Society 1963, 85, 3047. (226) Kuczkowski, R. L. J. Amer. Chem. Soc. 1964, 86, 3617. (227) Kuczkowski, R. L.; Wilson, E. B. J. Amer. Chem. Soc. 1963, 85, 2028. (228) Marsden, C. J.; Oberhammer, H.; Lösking, O.; Willner, H. J. Mol. Struct. 1989, 193, 233. (229) Samdal, S.; Mastryukov, V. S.; Boggs, J. E. J. Mol. Struct. (THEOCHEM) 1994, 309, 21. (230) Birk, M.; Friedl, R. R.; Cohen, E. A.; Pickett, H. M. J. Chem. Phys. 1989, 91, 6588. (231) Borden, W. T. Chem. Comm. 1998, 1919. (232) Kraka, E.; He, Y.; Cremer, D. J. Phys. Chem. 2001, 105, 3269. (233) Jursic, B. S. J. Mol. Struct. (THEOCHEM) 1999, 459, 23. (234) Feller, D.; Dixon, D. J. Phys. Chem. A 2003, 107, 9641. (235) Lucchese, R. R.; Schaefer III, H. F.; Rodwell, W. R.; Radom, L. J. Chem. Phys. 1978, 68,

2507. (236) Mack, H.‐G.; Oberhammer, H. Chem. Phys. Lett. 1988, 145, 121.

272

(237) Scuseria, G. E. J. Chem. Phys. 1991, 94, 442. (238) Colton, R. J.; Rabalais, J. W. J. Elec. Spectrosc. Rel. Phenom. 1974, 3, 345. (239) Forneris, R.; Hennies, C. E. J. Mol. Struct. 1970, 5, 449. (240) Frankiss, S. G. J. Mol. Struct. 1968, 2, 271. (241) Frenzel, C. A.; Blick, K. E. J. Chem. Phys. 1971, 55, 2715. (242) DeYonker, N., (personal communication). (243) Marshall, P.; Misra, A. J. Phys. Chem. A 1998, 102, 9056. (244) Papayannis, D. K.; Melissas, V. S.; Kosmas, A. M. Chem. Phys. Lett. 2001, 349, 299. (245) Haas, A. J. Flourin. Chem. 1986, 32, 415. (246) Denis, P. A. J. Chem. Theor. Comput. 2005, 1, 900. (247) Aitken, B. G.; Youngman, R. E.; Ponader, C. W. J. Noncrystal. Sol. 2001, 284, 34. (248) Asokamani, R.; Rita, R. Phys. Stat. Sol. 2001, 226, 375. (249) Bachelet, G. B.; Christensen, N. E. Phys. Rev. B 1985, 31, 879. (250) Barrau, J.; Escudié, J.; Satgé, J. Chem. Rev. 1990, 90, 283. (251) Basch, H. Inorg. Chim. Acta 1996, 252, 265. (252) Brook, A. G.; Baines, K. M. Adv. Organomet. Chem. 1986, 25, 1. (253) Chu, T. L.; Chu, S. S.; Ray, L. J. Appl. Phys. 1982, 53, 7102. (254) Curtiss, L. A.; Redfern, P. C.; Rassolov, V. A.; Kedziora, G. S.; Pople, J. A. J. Chem. Phys.

2001, 114, 9287. (255) Kedziora, G. S.; Pople, J. A.; Rassolov, V. A.; Ratner, M. A.; Redfern, P. C.; Curtiss, L. A. J.

Chem. Phys. 1999, 110, 7123. (256) Kedziora, G. S.; Pople, J. A.; Ratner, M. A.; Redfern, P. C.; Curtiss, L. A. J. Chem. Phys.

2001, 115, 718. (257) Koizumi, H.; Dávalos, J. Z.; Baer, T. Chem. Phys. 2006, 324, 385.

273

(258) Lai, C.‐H.; Su, M.‐D.; Chu, S.‐Y. Polyhedron 2002, 21, 579. (259) Pitzer, K. S. J. Am. Chem. Soc. 1948, 70, 2140. (260) Power, P. P. Chem. Rev. 1999, 99, 3463. (261) Raabe, G.; Michl, J. Chem. Rev. 1985, 85, 419. (262) West, R. Science 1984, 225, 1109. (263) Wiberg, J. J. Organomet. Chem. 1984, 273, 141. (264) Mulliken, R. S. J. Am. Chem. Soc. 1950, 72, 4493. (265) Del Bene, J. E.; Aue, D. H.; Shavitt, I. J. Am. Chem. Soc. 1992, 114, 1631. (266) Bachelet, G. B.; Christensen, N. E. Phys. Rev. B 1985, 31, 879. (267) Kedziora, G. S.; Pople, J. A.; Rassolov, V. A.; Ratner, M. A.; Redfern, P. C.; Curtiss, L. A. J.

Chem. Phys. 1999, 110, 7123. (268) Kedziora, G. S.; Pople, J. A.; Ratner, M. A.; Redfern, P. C.; Curtiss, L. A. J. Chem. Phys.

2001, 115, 718. (269) Koizumi, H.; Dávalos, J. Z.; Baer, T. Chem. Phys. 2006, 324, 385. (270) Kalcher, J. Phys. Chem. Chem. Phys. 2002, 4, 3311. (271) Yockel, S. M.; Mintz, B. J.; Wilson, A. K. J. Chem. Phys. 2004, 121, 60. (272) Ochterski, J. W. Thermochemistry in Gaussian; Gaussian, Inc.: Wallingford, CT., 2000. (273) Chase, M. W. J. Phys. Chem. Ref. Data 1998, 9, 1. (274) Cowan, R. D.; Griffin, D. C. J. Opt. Soc. Amer. 1976, 66, 1010. (275) Barysz, M.; Flocke, N.; Karwowski, J. Acta Physica Polonica A 2001, 99, 631. (276) de Jong, W. A.; Harrison, R. J.; Dixon, D. A. J. Chem. Phys. 2001, 114, 48. (277) Berning, A.; Schweizer, M.; Werner, H.‐J.; Knowles, P. J.; Palmieri, P. Mol. Phys. 2000, 98,

1823.

274

(278) Rohatgi, A. “Fundamental Research and Development for Improved Crystalline Silicon Solar Cells,” National Renewable Energy Laboratory, U.S. Department of Energy, 2007.

(279) Pelletier, J. J. Phys. D Appl. Phys. 1987, 20, 858. (280) Park, S. C.; Stansfield, R. A.; Clary, D. C. J. Phys. D Appl. Phys. 1987, 20, 880. (281) Jasinski, J. M.; Meyerson, B. S.; Scott, B. A. Ann. Rev. Phys. Chem. 1987, 38, 109. (282) Allen, W. D.; Schaefer, H. F. Chem. Phys. 1986, 108, 243. (283) Walsh, R. J. Chem. Soc., Faraday Trans. 1983, 79, 2233. (284) Ignacio, E. W.; Schlegel, H. B. J. Chem. Phys. 1990, 92, 5404. (285) Ignacio, E. W.; Schlegel, H. B. J. Phys. Chem. 1992, 96, 5830. (286) Ho, P.; Coltrin, M. E.; Binkley, J. S.; Melius, C. F. J. Phys. Chem. 1985, 89, 4647. (287) Ho, P.; Melius, C. F. J. Phys. Chem. 1990, 94, 5120. (288) Gutsev, G. L. J. Phys. Chem. 1994, 98, 1570. (289) Bauschlicher, C. W.; Partridge, H. Chem. Phys. Lett. 1997, 276, 47. (290) Grev, R. S.; Schaefer, H. F. J. Chem. Phys. 1992, 97, 8389. (291) Ricca, A.; Bauschlicher, C. W. J. Phys. Chem. A 1998, 102, 876. (292) Ricca, A.; Bauschlicher, C. W. Chem. Phys. Lett. 1998, 287, 239. (293) Feller, D.; Dixon, D. A. J. Phys. Chem. A 1999, 103, 6413. (294) Begue, D.; Carbonniere, P.; Barone, V.; Pouchan, C. Chem. Phys. Lett. 2005, 415, 25. (295) Betrencourt, M.; Boudjaadar, D.; Chollet, P.; Guelachvili, G.; Morillon‐Chapey, M. J.

Chem. Phys. 1986, 84, 4121. (296) Mélen, F.; Dubois, I.; Bredohl, H. J. Mol. Spectrosc. 1990, 139, 361. (297) Tanaka, T.; Tamura, M.; Tanaka, K. J. Mol. Struct. 1997, 413‐414, 153. (298) Fujitake, M.; Hirota, E. Spectrochim. Acta A 1994, 50A, 1345.

275

(299) Adrian, F. J.; Cochran, E. L.; Bowers, V. A. Advan. Chem. Ser. 1962, 36, 50. (300) Merritt, M. V.; Fessensen, R. W. J. Chem. Phys. 1972, 56, 2353. (301) Yamada, C.; Hirota, E. Phys. Rev. Lett. 1986, 56, 923. (302) Tanimoto, M.; Saito, S. J. Chem. Phys. 1999, 111, 9242. (303) Brookman, C. A.; Cradock, S.; Rankin, D. W. H.; Robertson, N.; Vefghi, P. J. Mol. Struct.

1990, 216, 191. (304) Ochterski, J. W.; Petersson, G. A.; Wiberg, K. B. J. Am. Chem. Soc. 1995, 117, 11299. (305) Karton, A.; Rabinovich, E.; Martin, J. M. L.; Ruscic, B. J. Chem. Phys. 2006, 125, 144108. (306) McQuarrie, D. A.; Simon, J. D. Molecular Thermodynamics; University Science Books:

Sausalito, CA, 1999. (307) Hildenbrand, D. L.; Lau, K. H.; Sanjurjo, A. J. Phys. Chem. 2003, 107, 5448.

Documents

Systematic Approaches to Predictive Computational