A Theoretical Study of Carbon-Based Functionalized Materials
by
Reed Nieman, B.S.
A Dissertation
In
Chemistry
Submitted to the Graduate Faculty
of Texas Tech University in
Partial Fulfillment of
the Requirements for
the Degree of
DOCTOR OF PHILOSOPHY
Approved
Clemens Krempner, Ph.D.
Chair of Committee
Hans Lischka, Ph.D.
Adelia J. A. Aquino, Ph.D.
Mark Sheridan
Dean of the Graduate School
May, 2019
Copyright 2019, Reed Nieman
Texas Tech University, Reed Nieman, May 2019
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ACKNOWLEDGMENTS
I would first like to thank my advisor, Dr. Hans Lischka, whose passion for science
and dedication to his work are truly inspiring. Working with him on my various projects
has been a tremendous learning experience these past several years. One of his foremost
characteristics that I have made an attempt to emulate is his tireless attention to detail, no
matter how seemingly small, as all contributing parts are significant to a finished product.
Finally, I have always been able to rely on his patience and encouragement to see through
any difficult moment in a project. I would also like to thank Dr. Adelia Aquino who first
saw potential in me when I did my undergraduate research project with her. She provided
my first lessons in performing computational chemistry calculations and I still fondly
remember sitting at her side hurriedly taking notes and trying to keep up as she quickly
went through commands. Her constant kindness and support over the years is very much
appreciated. I am very grateful too for Dr. Clemens Krempner for not only serving as the
chairperson of my committee, but for going above and beyond in, among other things,
helping me to secure several of my fellowships.
I also thank Dr. Anita Das, and Nadeesha Jayani Silva for being good group
members and great friends. I am thankful as well to Megan Gonzalez and Dr. Veronica
Lyons for their friendship and support, without which graduate school would not have been
nearly as colorful let alone bearable. I am also very thankful for my family and their
unending encouragement even through our lowest moments.
I acknowledge and thank our collaborators, specifically Dr. Francisco Machado
with whom I worked closely while he was visiting. Thanks is also given to the groups of
Dr. Toma Susi at the University of Vienna and Dr. Sergei Tretiak at Los Alamos National
Laboratory for their excellent work and contributions.
Finally, I gratefully acknowledge the following for helping to fund my education:
the National Science Foundation, Texas Tech University for the Provost Scholarship and
Doctoral Dissertation Completion Fellowship, the AT&T Foundation for the AT&T
President’s Fellowship, and the Community Foundation of West Texas for the Lubbock
Achievement Rewards for College Scientists Chapter (ARCS) Memorial Scholarship.
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TABLE OF CONTENTS
ACKNOWLEDGMENTS ii
ABSTRACT vi
LIST OF TABLES viii
LIST OF FIGURES xiii
LIST OF ABBREVIATIONS xxi
I. BACKGROUND 1
1.1 Graphene 2
1.2 Organic Photovoltaic Bulk Heterojunctions and Electronic Excited
States 4
1.3 Radical Character in Polycyclic Aromatic Hydrocarbons 6
1.4 Bibliography 10
II. THEORETICAL METHODS 16
2.1 Density Functional Theory 18
2.2 Second-Order Møller-Plesset Perturbation Theory 22
2.3 Second-Order Algebraic Diagrammatic Construction 25
2.4 Basis Set Superposition Error and the Counterpoise Correction
Method 27
2.5 Environmental Simulation Approaches 29
2.6 Multiconfigurational Self-Consistent Field 31
2.7 Multireference Configuration Interaction 32
2.8 Multireference Averaged Quadratic Coupled-Cluster 34
2.9 Bibliography 36
III. SINGLE AND DOUBLE CARBON VACANCIES IN PYRENE AS
FIRST MODELS FOR GRAPHENE DEFECTS: A SURVEY OF THE
CHEMICAL REACTIVITY TOWARD HYDROGEN
38
3.1 Abstract 39
3.2 Introduction 40
3.3 Computational Details 44
3.4 Results and Discussion 45
3.4.1 Potential Energy Scan 45
3.4.2 Geometry Optimizations 49
3.4.3 Energetic Stabilities 52
3.5 Conclusions 59
3.5 Acknowledgments 61
3.7 Bibliography 61
IV. STRUCTURE AND ELECTRONIC STATES OF A GRAPHENE
DOUBLE VACANCY WITH EMBEDDED SI DOPANT 63
4.1 Abstract 64
4.2 Introduction 65
4.3 Computational Details 66
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4.4 Results and Discussion 68
4.4.1 The Pyrene-2C+Si Defect Structure 68
4.4.2 The Circumpyrene-2C+Si Defect Structure 71
4.4.3 The Graphene-2C+Si Defect Structure 73
4.4.4 Natural Bond Orbital Analysis 73
4.4.5 Molecular Electrostatic Potential 79
4.4.6 Electron Energy Loss Spectroscopy 80
4.5 Conclusions 82
4.5 Acknowledgments 83
4.7 Bibliography 83
V. THE CRUCIAL ROLE OF A SPACER MATERIAL ON THE
EFFICIENCY OF CHARGE TRANSFER PROCESSES IN ORGANIC
DONOR-ACCEPTOR JUNCTION SOLAR CELLS
87
5.1 Abstract 88
5.2 Introduction 89
5.3 Methods 92
5.3.1 Experimental 92
5.3.1.1 Bilayer Solar Cells Fabrication 92
5.3.1.2 Device Characterization 93
5.3.2 Computational Details 93
5.4 Results and Discussion 95
5.4.1 Experimental Observations 95
5.4.2 Analysis of Energy Levels 98
5.4.3 Characterization of Electronic States 102
5.5 Conclusions 104
5.6 Acknowledgments 106
5.7 Bibliography 106
VI. BENCHMARK AB INITIO CALCULATIONS OF POLY(P-
PHENYLENEVINYLENE) DIMERS: STRUCTURE AND EXCITON
ANALYSIS
111
6.1 Abstract 112
6.2 Introduction 112
6.3 Computational Details 116
6.4 Results and Discussion 119
6.4.1 Structural Description 119
6.4.2 Excited States 124
6.5 Conclusions 133
6.6 Bibliography 134
VII. THE BIRADICALOID CHARACTER IN RELATION TO
SINGLET/TRIPLET SPLITTINGS FOR CIS-/TRANS-
DIINDENOACENES AND TRANS-
INDENOINDENODI(ACENOTHIOPHENES) BASED ON EXTENDED
MULTIREFERENCE CALCULATIONS
139
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7.1 Abstract 140
7.2 Introduction 141
7.3 Computational Details 143
7.4 Results and Discussion 146
7.4.1 Trans-diindenoacenes 146
7.4.2 Cis-diindenoacenes 152
7.4.3 Trans-indenoindenodi(acenothiophenes) 156
7.5 Conclusions 160
7.6 Bibliography 161
VIII. CONCLUDING REMARKS 169
APPENDICES
A. ADDITIONAL TABLES AND FIGURES NOT INCLUDED IN
CHAPTER III 174
B. ADDITIONAL TABLES AND FIGURES NOT INCLUDED IN
CHAPTER IV 181
C. ADDITIONAL TABLES AND FIGURES NOT INCLUDED IN
CHAPTER V 189
D. ADDITIONAL TABLES AND FIGURES NOT INCLUDED IN
CHAPTER VI 198
E. ADDITIONAL TABLES AND FIGURES NOT INCLUDED IN
CHAPTER VII 216
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ABSTRACT
This dissertation will examine the electronic properties of several materials used in
state-of-the art technology and ongoing scientific investigation by means of high-level ab
initio quantum chemical calculations. In this way, the application of such calculations to
graphene reactivity and defects, organic photovoltaic processes, and excited states and
biradical character of polycyclic aromatic hydrocarbons (PAH) is presented.
Graphene reactivity and defect characterization is currently the source of intense
scrutiny. The reactivity of pristine graphene and single and double carbon vacancies toward
a hydrogen radical was assessed using complete active space self-consistent field theory
(CASSCF) and multireference configuration interaction singles and doubles (MR-CISD)
calculations using a pyrene model. While all carbon centers formed stable, novel C-H
bonds, the single carbon defect was found to be most stabilizing thanks to the dangling
bond at the six-membered ring. Bonding at this carbon center shows interaction with
several closely-spaced electronic excited states. A rigid scan was also used to show the
dynamic landscape of the potential energy surfaces of these structures.
Double carbon vacancy pyrene and circumpyrene were used to investigate the
structure and electronic states of an intrinsic silicon impurity in graphene by density
functional theory (DFT) methods. The ground state was found to be low spin, non-planar
and D2 symmetric with a nearby low spin, planar state with D2h symmetry. Characterization
with natural bond orbital (NBO) analyses showed that the hybridization in the non-planar
structure is sp3 and sp2d in the planar structure. Charge transfer from the silicon dopant to
the surrounding carbon atoms is demonstrated via analysis with natural charges and
molecular electrostatic potential (MEP) plots.
Experiment shows drastically enhanced power conversion efficiency (PCE) in a
bulk heterojunction (BHJ) device consisting of poly(3-hexylthiophene-2, 5-diyl) (P3HT),
terthiophene (T3), and fullerene (C60). This BHJ scheme was investigated using DFT and
several environmental models. The state-specific (SS) environment was found to best
reproduce the experimental findings showing that initial photoexcitation of a local, bright
exciton at P3HT was able to decay through several charge transfer (CT) bands of
P3HT→C60 and T3→C60 character explaining the increased PCE. The spectra calculated
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for the isolated system and linear response (LR) environment were very similar to each
other and showed the initial photoexcitation at P3HT decayed instead into bands of local
C60 excitonic states quenching the CT states.
Crucial benchmark calculations of the structure and electronic excited states
missing from the current literature were performed for oligomers of poly(p-
phenylenevinylene) (PPV) dimers using high-level scaled opposite-spin (SOS) second-
order Møller-Plesset (SOS-MP2) and second-order algebraic diagrammatic construction
(SOS-ADC(2)) calculations. Comparisons were also made to DFT methods. The dimer
structures were studied in two conformations: sandwich-stacked and displaced; the latter
exhibited greater stacking interactions. Good agreement was found between the cc-pVQZ
and aug-cc-pVQZ basis sets and the cc-pVTZ basis. Comparisons of the interaction
energies were made to the complete basis set limit. Basis set superposition error was also
accounted for. The excited state spectra presented a varied and dynamic picture that was
analyzed using transition density matrices and natural transition orbitals (NTOs).
The assessment of the biradical character of several trans-diindenoacenes, cis-
diindenoacenes, and trans-indenoindenodi(acenothiophenes) was accomplished utilizing
the multireference averaged quadratic coupled-cluster (MR-AQCC) method. These
systems have been previously identified as highly reactive in experiments, often subject to
unplanned side-reactions. The biradical character was characterized using the singlet-
triplet splitting energy (∆ES-T), effective number of unpaired electrons (NU), and unpaired
electron density. For all structures, low spin ground states were found. In addition,
increasing the number of core benzene rings decreased the ∆ES-T, increased the NU, and
increased the unpaired electron density. Good agreement to experimental values for the
∆ES-T is also demonstrated.
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LIST OF TABLES
3.1 Vertical energy differences (eV) of the lowest two A’ and A” states relative
to the C15 ground state computed for the C1, C8 and C15 pyrene-1C+Hr
structures using the SA4-CASSCF(5,5), MR-CISD and MR-CISD+Q
approaches and the 6-31G* basis set. 54
3.2 Dissociation energies Ediss (eV) for the C15 global minimum pyrene-
1C+Hr structure and estimated energy barriers Ebarr(est.) (eV) for the
approach of Hr on the ground state surface toward the different carbon
atoms of pyrene-1C using different theoretical methods and the 6-31G*
basis set. 57
3.3 Dissociation energies Ediss (eV) for the double vacancy pyrene-2C+Hr and
pristine pyrene + Hr cases using different methods and the 6-31G* basis se 58
4.1 Open-shell character, structural symmetry, spin multiplicity, bond distances
(Å), and relative energies (eV) of selected optimized structures of pyrene-
2C+Si. 69
4.2 Structural symmetry, open-shell character, spin multiplicity, bond distances
(Å), and relative energies (eV) of selected optimized structures of
circumyrene-2C+Si. 72
4.3 NBO bonding character analysis of pyrene-2C+Si using B3LYP/6-
311G(2d,1p). The discussed carbon atom is one of the four bonded to
silicon. 74
4.4 NBO bonding character analysis of circumpyrene-2C+Si. The discussed
carbon atom is the one of the four bonded to silicon. 76
5.1 Stabilization energy of the lowest P3HT→C60 CT state of P3HT-T3-C60 and
P3HT-C60 system calculated in the isolated system and the SS environmen 102
6.1 Interchain distances (Å) and ΔE (kcal/mol) in the stacked PPV2 dimer
computed at SOS-MP2 and SOS-MP2/BSSE levels with several basis sets
using C2h symmetry. 120
6.2 Interchain distances (Å) and ΔE (kcal/mol) in the stacked PPV3 dimer
computed at SOS-MP2 and SOS-MP2/BSSE levels with several basis sets
using C2h symmetry. 121
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6.3 Interchain distances (Å) and ΔE (kcal/mol) in the displaced PPV2 dimer
computed at SOS-MP2 level using C1 symmetry. 122
6.4 Interchain distances (Å) and ΔE (kcal/mol) in the displaced PPV3 dimer
computed at SOS-MP2 level using C1 symmetry 122
6.5 Stacked PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVQZ and
SOS-MP2/cc-pVQZ optimized geometry (ΔE (eV) – vertical excitation
energy, f – oscillator strength, CT – interchain charge transfer) 125
6.6 Displaced PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ
and SOS-MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation
energy, f – oscillator strength, CT – interchain charge transfer) 127
6.7 Displaced PPV3 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ
and SOS-MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation
energy, f – oscillator strength, CT – interchain charge transfer) 131
6.8 Interchain distances (Å) in the stacked PPV dimers optimized for π-π* S1
(1 1Bg) excited state using the cc-pVTZ basis and C2h symmetry 132
7.1 Singlet/triplet splitting energy (∆ES-T, E(13Bu) - E(11Ag)) calculated using
the MR-AQCC/6-31G* method for trans-diindenoacenes, 1-4. 149
7.2 Singlet/triplet splitting energy (∆ES-T, E(13B1) - E(11A1)) calculated using
the MR-AQCC/6-31G* method for cis-diindenoacenes, 5-8. 153
7.3 Singlet/triplet splitting energy (∆ES-T, E(13Bu) - E(11Ag)) calculated using
the MR-AQCC/6-31G*(6-311G*)S method for trans-
indenoindenodi(acenothiophene) (9-12). 157
A.1 Relative energy differences ΔErel (eV) of the local energy minima in the
electronic ground state computed for the different structures of pyrene-
1C+Hr using different theoretical methods and the 6-31G* approach. 174
A.2 Relative energy differences ΔErel (eV) of the local energy minima in the
electronic ground state computed for the different structures of pyrene-
2C+Hr using different theoretical methods and the 6-31G* approach. 174
A.3 Relative energy differences ΔErel (eV) of the local energy minima in the
electronic ground state computed for the different structures of pyrene+Hr
using different theoretical methods and the 6-31G* basis set. 175
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B.1 NBO bonding character analysis of pyrene-2C+Si using the CAM-
B3LYP/6-311G(2d,1p) method. The carbon atom discussed is the carbon
bonded to silicon. 181
C.1 Excitation energies Ω of the isolated standard C60-T3-P3HT system using
the CAM-B3LYP/6-31G approach 189
C.2 Excitation energies Ω of the isolated standard C60-T3-P3HT system using
the CAM-B3LYP/6-31G* approach 190
C.3 Lowest excitation energies Ω of the isolated components P3HT, T3 and C60
using the CAM-B3LYP/6-31G* approach. 191
C.4 Excitation energies Ω of the standard C60-T3-P3HT system the using the
CAM-B3LYP/6-31G* approach in the LR environment 192
C.5 Excitation energies Ω of the standard C60-T3-P3HT system the using the
CAM-B3LYP/6-31G approach in the LR environment 193
C.6 Excitation energies Ω of the standard C60-T3-P3HT system using the CAM-
B3LYP/6-31G* approach in the SS environment 193
C.7 Excitation energies Ω of the standard C60-T3-P3HT system using the CAM-
B3LYP/6-31G approach in the SS environment 194
C.8 Excitation energies Ω of the isolated standard C60-T3-P3HT system using
the CAM-B3LYP/6-31G* approach (only core orbitals were frozen) 194
C.9 Excitation energies Ω of the isolated extended (C60)3-(T3)3-P3HT system
using the CAM-B3LYP/6-31G* approach 195
C.10 Excitation energies Ω of the isolated extended (C60)3-(T3)3-P3HT system
using the CAM-B3LYP/6-31G approach 196
C.11 Excitation energies Ω of the extended C60-T3-P3HT system using the
CAM-B3LYP/6-31G approach in the SS environment 197
D.1 Interchain distances (Å) and ΔE (kcal/mol) in the stacked PPV2 dimer
computed at MP2 and MP2/BSSE levels with several basis sets using C2h
symmetry. 198
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D.2 Interchain distances (Å) and ΔE (kcal/mol) in the stacked PPV4 dimer
computed at SOS-MP2 and SOS-MP2/BSSE levels with several basis sets
using C2h symmetry. 199
D.3 Interchain distances (Å) and ΔE (kcal/mol) in the diplaced PPV4 dimer
computed at SOS-MP2 level with several basis sets using C2h symmetry. 200
D.4 Single chain PPV2 vertical excitations using SOS-ADC(2) and SOS-MP2
optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator
strength) 200
D.5 Stacked PPV2 dimer vertical excitations using SOS-ADC(2) and SOS-MP2
optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator
strength, CT – interchain charge transfer) 201
D.6 Stacked PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and
SOS-MP2/cc-pVTZ optimized geometry showing the largest contributing
hole/electron NTO pairs (Isovalue = ±0.03 Bohr3) 202
D.7 Displaced PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ
and SOS-MP2/cc-pVTZ optimized geometry showing the largest
contributing hole/electron NTO pairs (Isovalue = ±0.03 Bohr3) 205
D.8 Stacked PPV3 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and
SOS-MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation
energy, f – oscillator strength, CT – interchain charge transfer) 208
D.9 Stacked PPV4 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and
SOS-MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation
energy, f – oscillator strength, CT – interchain charge transfer) 208
D.10 Displaced PPV3 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ
and SOS-MP2/cc-pVTZ optimized geometry showing the largest
contributing hole/electron NTO pairs (Isovalue = ±0.03 Bohr3) 209
D.11 Stacked PPV3 dimer vertical excitations using B3LYP-D3/cc-pVTZ and the
B3LYP-D3/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation
energy, f – oscillator strength, CT – interchain charge transfer) 212
D.12 Stacked PPV3 dimer vertical excitations using CAM-B3LYP-D3/cc-pVTZ
and the CAM-B3LYP-D3/cc-pVTZ optimized geometry (ΔE (eV) –
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vertical excitation energy, f – oscillator strength, CT – interchain charge
transfer) 213
E.1 Active orbitals for each MR-AQCC reference space 216
E.2 NU value (e) of trans-diindenoacene (1-4) singlet 1 1Ag state and vertically
excited 1 3Bu state using the MR-AQCC(CAS(4,4))/6-31G* method. 217
E.3 NU value (e) of cis-diindenoacene (5-8) singlet 1 1A1 state and vertically
excited 1 3B1 state using the MR-AQCC(CAS(4,4))/6-31G* method. 218
E.4 NU value (e) of trans-indenoindenodi(acenothiophene) (9-12) singlet 1 1Ag
state and vertically excited 1 3Bu state using the MR-AQCC(CAS(4,4))/6-
31G*(6-311G*)S method. 219
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LIST OF FIGURES
1.1 Structures: a) n-acene, b) triangulene, c) 1,2:9,10-dibenzoheptazethrene, d)
(ma,nz) periacene (graphene nanoribbon model), e) indeno[1,2-b]fluorene 7
3.1 PES scans in terms of the location of Hr for the four lowest states of pyrene-
1C+Hr using the SA4-CASSCF(5,5)/6-31G* approach. The atom Hr is
located in a plane parallel and 1 Å above the plane defined by the frozen
carbon atoms. Lowest energies: a -574.222889 Hartree, b -574.198174
Hartree, c -574.149530 Hartree, d -574.117820 Hartree. The energy (color
scale) is given in eV, relative to the ground state lowest minimum. 46
3.2 PES scan in terms of the location of Hr for the lowest state of pyrene-2C+
Hr using the SA4-CASSCF(5,5)/6-31G* approach. The atom Hr is located
in a plane parallel and 1 Å above the plane defined by the frozen carbon
atoms. Lowest energy = -536.385046 Hartree. The energy (color scale) is
given in eV, relative to the ground state lowest minimum. 47
3.3 PES scan in terms of the location of Hr for the 22A state of pyrene-2C+ Hr
using the SA4-CASSCF(5,5)/6-31G* approach. The atom Hr is located in a
plane parallel and 1 Å above the plane defined by the frozen carbon atoms.
Lowest energy = -536.347629 Hartree. The energy (color scale) is given in
eV, relative to the ground state lowest minimum at -536.385046 Hartree. 48
3.4 PES scan of the lowest state of pyrene+Hr using the CASSCF(5,5)/6-31G*
approach. The atom Hr is located in a plane parallel and 1 Å above the plane
defined by the frozen carbon atoms. Lowest energy = -612.270253 Hartree.
The energy (color scale) is given in eV, relative to the ground state lowest
minimum. 48
3.5 Plots of the different minimum structures (labeled according to the bonding
carbon atom) on the ground state energy surface of pyrene-1C+Hr using the
SA4-CASSCF(5,5)/6-31G* approach. Bond distances are given in Å. 50
3.6 Plots of the different minimum structures (labeled according to the bonding
carbon atom) of the ground state energy surface of pyrene-2C+Hr using the
SA4-CASSCF(5,5)/6-31G* approach. Bond distances are given in Å. 51
3.7 Plots of the different minimum structures (labeled according to the bonding
carbon atom) on the ground state energy surface of pyrene+Hr using the
CASSCF(5,5)/6-31G* approach. Bond distances are given in Å. 53
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3.8 Location and relative stability (eV) of the minima on the ground state
energy surface of pyrene-1C+Hr using the MR-CISD+Q/6-31G* method.
Blue: 2A ground state, red: lowest vertical excitation. Note: the rigid pyrene-
1C structure shown does not reflect the actual structure of the optimized
minima. 54
3.9 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is
bonded to C15 using the MR-CISD/6-31G* approach. Minimum energy =
-574.277274 Hartree 56
3.10 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is
bonded to C1 using the MR-CISD/6-31G* approach. Minimum energy = -
574.277274 Hartree 56
3.11 Location and relative stability (eV) of the minima on the ground state
energy surface of pyrene-2C+Hr defect using the MR-CISD+Q/6-31G*
method. Blue: 2A ground state, red: lowest vertical excitation. Note: the
rigid pyrene-2C structure shown does not reflect the actual structure of the
optimized minima. 58
3.12 Location and relative stability (eV) of the minima on the ground state
energy surface of pyrene+Hr using the MR-CISD+Q/6-31G* method. Note:
the rigid pyrene structure shown does not reflect the actual structure of the
optimized minima. 59
4.1 Optimized pyrene-2C+Si structure P1 (D2 symmetry) using the B3LYP/6-
311G(2d,1p) method with frozen carbon atoms marked by gray circles 68
4.2 Optimized circumpyrene-2C+Si structure CP1 (D2 symmetry) using the
B3LYP/6-311G(2d,1p) method with frozen carbon atoms marked by gray
circles. 68
4.3 NBO plots for pyrene-2C+Si using the B3LYP/6-311G(2d,1p) approach for
structures P1 and P2. Isovalue = ±0.02 e/Bohr3 77
4.4 Natural charges (e) of carbon and silicon in circumpyrene-2C+Si using the
B3LYP/6-311G(2d,1p) method. 78
4.5 Molecular electrostatic potential for circumpyrene-2C+Si mapped onto the
electron density isosurface with an isovalue of 0.0004 e/Bohr3 (B3LYP/6-
311G(2d,1p)). 80
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4.6 Comparison of simulated and experimental EELS spectra of the Si-C4
defect in graphene. Despite the out-of-plane distorted structure being the
ground state, the originally proposed flat spectrum provides an overall
better match to the experiment, apart from the small peak at ~96 eV that is
not predicted for the flat structure. The inset shows a colored medium angle
annular dark field STEM image of the Si-C4 defect (the Si atom shows
brighter due to its greater ability to scatter the imaging electrons). 81
5.1 (a) Standard model of the molecular arrangement in the trimer. (b) Front
and (c) side views of the extended model. The picture shows molecular
notations, geometrical arrangement and selected nonbonding distances (Å).
Atoms kept fixed during the geometry optimization are circled in black. 92
5.2 (a) Scheme of bilayer device structure used in this study, the interface layer
with vertical alignment was produced by Langmuir-Blodgett (LB) method
illustrated on right. (b) photocurrent measured at short circuit condition as
function of illumination wavelength for bilayer device without spacer and
with 1-bilayer of O3 between P3HT and C60. (c) GIWAXS map for 1 bilayer
of O3 deposited on P3HT by LB method. (d) peak photocurrent at 550 nm
illumination as a function of number of O3 bilayer at interface. 97
5.3 Excited state energies Ω of the standard P3HT-T3-C60 trimer using the
CAM-B3LYP method and (a) the 6-31G* and (b) the 6-31G basis sets both
as isolated systems and in the LR and SS environments (for numerical data
see Table C.1, Table C.2, and Table C.4-Table C.8). For the isolated system
using the 6-31G* basis set (farthest left of (a)), the fifteen lowest-energy
locally excited C60 states have been computed also. Excited states of (c) the
extended P3HT-T3-C60 trimer using the 6-31G* and (d) 6-31G basis sets
both as isolated systems and in the SS environment (Table C.9-Table C.11).
100
5.4 Natural Transition Orbitals (NTO), particle/hole populations and MEP of
(a, b, c) the lowest-energy P3HT→(C60)3 CT state, (d, e, f) the lowest-
energy T3→(C60)3 CT state, (j, k, l) the lowest-energy P3HT bright
excitonic state of the extended thiophene trimer calculated using the CAM-
B3LYP/6-31G method in the SS environment, and (g, h, i) the lowest-
energy C60 dark excitonic state of the isolated standard thiophene trimer
calculated using the CAM-B3LYP/6-31G* method. Here we use NTO
isovalue ±0.015 e/Bohr3 and MEP isovalue ±0.0003 e/Bohr3. 103
6.1 Top/side views of the stacked PPV oligomers and labeling scheme. 117
6.2 Omega matrices for vertical excitations of stacked PPV2 dimer using the
SOS-ADC(2)/cc-pVQZ method with the SOS-MP2/cc-pVQZ optimized
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geometry. Each vinylene and phenylene group is a separate fragment:
fragments 1-3 belong to one chain, fragments 4-6 to the other. The y-axis
represents the electron and the x-axis represents the hole. Omega matrix
plots calculated using the TheoDORE program package. 126
6.3 Omega matrices for vertical excitations of stacked PPV3 dimer using the
SOS-ADC(2)/cc-pVTZ method with the SOS-MP2/cc-pVTZ optimized
geometry. Each vinylene and phenylene group is a separate fragment:
fragments 1-5 belong to one chain, fragments 6-10 to the other. The y-axis
represents the electron and the x-axis represents the hole. Omega matrix
plots calculated using the TheoDORE program package. 128
6.4 Calculated line spectra (right y-axis) and density of states (left y-axis) for
the stacked PPV oligomers optimized using the SOS-ADC(2), B3LYP-D3,
and CAM-B3LYP-D3 methods. Stacked a) PPV2 (Table 6.5), b-d) PPV3
(Table D.8, S11-S12), e) PPV4 (Table D.9). 130
7.1 Quinoid Kekule and biradical resonance forms of trans-diindenoanthracene,
cis-diindenoanthracene, and trans-indenoindenodi(anthracenothiophene). 142
7.2 Comparison of the singlet/triplet energy (∆ES-T, E(13Bu) - E(11Ag)) and
number of unpaired electrons (NU and NU(H-L), Table E.2) of the 11Ag state
with the number of interior six-membered rings of trans-diindenoacenes (1-
4) using the MR-AQCC(CAS(4,4))/6-31G* method. 148
7.3 Correlation of the singlet/triplet energy (∆ES-T) with a) the total number of
unpaired electrons (NU) and b) the number of unpaired electrons in the
HONO-LUNO pair (NU(H-L)) of the 1 1Ag state (Table E.2) of trans-
diindenoacenes (1-4) using the MR-AQCC(CAS(4,4))/6-31G* method. 150
7.4 Total unpaired electron density (sum over all NOs, Table E.2), in red, of the
1 1Ag state of trans-diindenobenzene (1-4) using the MR-
AQCC(CAS(4,4))/6-31G* method. (isovalue=0.004 e/Bohr3). HOMA
index is shown in the center of each ring. 151
7.5 Comparison of the singlet/triplet energy (∆ES-T, E(13B1) - E(11A1)) and
number of unpaired electrons (NU and NU(H-L), Table E.3) of the 11A1 state
with the number of interior six-membered rings of cis-diindenoacenes (5-8)
using the MR-AQCC(CAS(4,4))/6-31G* method. 153
7.6 Correlation of the singlet/triplet energy (∆ES-T) with a) the total number of
unpaired electrons (NU) and b) the number of unpaired electrons in the
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HONO-LUNO pair (NU(H-L)) of the 1 1A1 state (Table E.3) of cis-
diindenoacenes (5-8) using the MR-AQCC(CAS(4,4))/6-31G* method. 154
7.7 Total unpaired electron density (sum over all NOs, Table E.3), in red, of the
1 1A1 state of cis-diindenobenzene (5-8) using the MR-AQCC(CAS(4,4))/6-
31G* method. (isovalue=0.004 e/Bohr3). HOMA index is shown in the
center of each ring. 155
7.8 Comparison of the singlet/triplet energy (∆ES-T, E(13Bu) - E(11Ag)) and
number of unpaired electrons (NU and NU(H-L), Table E.4) of the 11Ag state
with the number of interior six-membered rings of trans-
indenoindenodi(acenothiophene) (9-12) using the MR-
AQCC(CAS(4,4))/6-31G*(6-311G*)S method. 157
7.9 Correlation of the singlet/triplet energy (∆ES-T) with a) the total number of
unpaired electrons (NU) and b) the number of unpaired electrons in the
HONO-LUNO pair (NU(H-L)) of the 1 1Ag state (Table E.4) of trans-
indenoindenodi(acenothiophene) (9-12) using the MR-
AQCC(CAS(4,4))/6-31G*(6-311G*)S method. 159
7.10 Total unpaired electron density (sum over all NOs, Table E.4), in red, of the
1 1Ag state of trans-indenoindenodi(acenothiophene) (9-12) using the MR-
AQCC(CAS(4,4))/6-31G*/6-311G* method. (isovalue=0.004 e/Bohr3).
HOMA index is shown in the center of each ring. 160
A.1 Plots of the active orbitals of the optimized ground states of pyrene+Hr
using the CASSCF(5,5)/6-31G* approach. 175
A.2 Plots of the active orbitals of the optimized ground states of pyrene-1C+Hr
using the SA4-CASSCF(5,5)/6-31G* approach. 176
A.3 Plots of the active orbitals of the optimized ground states of pyrene-2C+Hr
using the SA4-CASSCF(5,5)/6-31G* approach. 176
A.4 Potential energy scans in terms of the location of Hr for the 3 2A and 4 2A
states of pyrene-2C+Hr using the SA4-CASSCF(5,5)/6-31G* approach.
The atom Hr is located in a plane parallel and 1 Å above the plane defined
by the frozen carbon atoms. Lowest energies: a -574.149530 Hartree, b -
574.117820 Hartree. The energy (color scale) is given in eV, relative to the
ground state lowest minimum at -536.385046 Hartree. 177
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A.5 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is
bonded to C2 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum
energy = -574.277274 Hartree 177
A.6 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is
bonded to C3 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum
energy = -574.277274 Hartree 178
A.7 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is
bonded to C4 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum
energy = -574.277274 Hartree 178
A.8 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is
bonded to C5 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum
energy = -574.277274 Hartree 179
A.9 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is
bonded to C6 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum
energy = -574.277274 Hartree 179
A.10 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is
bonded to C7 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum
energy = -574.277274 Hartree 180
A.11 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is
bonded to C8 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum
energy = -574.277274 Hartree 180
B.1 NBO plots for circumpyrene-2C+Si using the B3LYP/6-311G(2d,1p)
approach (CP2 D2h). Isovalue = ±0.02 e/Bohr3 182
B.2 Natural charges of carbon and silicon in units of e of pyrene-2C+Si using
the B3LYP/6-311G(2d,1p) method. 183
B.3 Natural charges of carbon and silicon in units of e of pyrene-2C+Si using
the CAM-B3LYP/6-311G(2d,1p) method. 184
B.4 Molecular electrostatic potential plot for pyrene-2C+Si mapped onto the
electron density isosurface with an isovalue of 0.0004 e/Bohr3 using the
B3LYP/6-311G(2d,1p) method. 185
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B.5 Molecular electrostatic potential plot for pyrene-2C+Si mapped onto the
electron density isosurface with an isovalue of 0.0004 e/Bohr3 using the
CAM-B3LYP/6-311G(2d,1p) method. 187
B.6 Unprocessed EEL spectrum of the Si-C4 site. 187
B.7 First two unoccupied states immediately above the Fermi level spanning
about 1 eV for the flat and corrugated Si/graphene structures from the
periodic DFT calculations. Electron density isovalue is 0.277 e/Bohr3. 188
D.1 Omega matrices for vertical excitations of displaced PPV2 dimer using the
SOS-ADC(2)/cc-pVTZ method with the SOS-MP2/cc-pVTZ optimized
geometry. Each vinylene and phenylene group is a separate fragment:
fragments 1-3 belong to one chain, fragments 4-6 to the other. The y-axis
represents the electron and the x-axis represents the hole. Omega matrix
plots calculated using the TheoDORE program package. 214
D.2 Omega matrices for vertical excitations of stacked PPV4 dimer using the
SOS-ADC(2)/cc-pVTZ method with the SOS-MP2/cc-pVTZ optimized
geometry. Each vinylene and phenylene group is a separate fragment:
fragments 1-7 belong to one chain, fragments 8-14 to the other. The y-axis
represents the electron and the x-axis represents the hole. Omega matrix
plots calculated using the TheoDORE program package. 214
D.3 Omega matrices for vertical excitations of displaced PPV3 dimer using the
SOS-ADC(2)/cc-pVTZ method with the SOS-MP2/cc-pVTZ optimized
geometry. Each vinylene and phenylene group is a separate fragment:
fragments 1-5 belong to one chain, fragments 6-10 to the other. The y-axis
represents the electron and the x-axis represents the hole. Omega matrix
plots calculated using the TheoDORE program package. 215
E.1 Quinoid Kekule and biradical resonance forms of trans-diindenoacenes (1-
4), cis-diindenoacenes (5-8), and trans-indenoindenodi(acenothiophenes)
(9-12). 220
E.2 Total unpaired electron density (sum over all NOs, Table E.2), in red, of the
vertically excited 1 3Bu state of trans-diindenobenzene (1-4) using the MR-
AQCC(CAS(4,4))/6-31G* method. (isovalue=0.004 e/Bohr3). HOMA
index is shown in the center of each ring. 221
E.3 Total unpaired electron density (sum over all NOs, Table E.3), in red, of the
vertically excited 1 3B1 state of cis-diindenobenzene (5-8) using the MR-
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AQCC(CAS(4,4))/6-31G* method. (isovalue=0.004 e/Bohr3). HOMA
index is shown in the center of each ring. 221
E.4 Total unpaired electron density (sum over all NOs, Table E.4), in red, of the
vertically excited 1 3Bu state of trans-indenoindenodi(acenothiophene) (9-
12) using the MR-AQCC(CAS(4,4))/6-31G*(6-311G*)S method.
(isovalue=0.004 e/Bohr3). HOMA index is shown in the center of each ring.
222
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LIST OF ABBREVIATIONS
ADC(2) Second-Order Algebraic Diagrammatic Construction
AUX Auxiliary Orbitals
BHJ Bulk Hetero-Junction
BSSE Basis Set Superposition Error
C60 Fullerene
CASSCF Complete Active Space Self-Consistent Field
CI Configuration Interaction
CP Counterpoise Correction
CSF Configuration State Function
CT Charge Transfer
DFT Density Functional Theory
HF Hartree-Fock
HOMA Harmonic Oscillator Measure of Aromaticity
K-S Kohn-Sham
LR Linear Response
MCSCF Multiconfigurational Self-Consistent Field
MEP Molecular Electrostatic Potential
MP2 Second-Order Møller-Plesset
MR-AQCC Multireference-Averaged Quadratic Coupled-Cluster
MR-CISD Multireference-Configuration Interaction Singles and Doubles
NBO Natural Bond Orbital
NTO Natural Transition Orbital
NU Number of Unpaired Electrons
OPV Organic Photovoltaic
P3HT Poly(3-hexylthiophene-2, 5-diyl)
PAH Polycyclic Aromatic Hydrocarbon
PCE Power Conversion Efficiency
PES Potential Energy Surface
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PP Polarization Propagator
PPV Poly(p-phenylenevinylene)
SA State-Averaged
SOS Scaled Opposite-Spin
SS State-Specific
T3 Terthiophene
TD-DFT Time-Dependent Density Functional Theory
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CHAPTER I
BACKGROUND
This chapter will present a general overview of each research project of this
dissertation. Research into graphene, an allotrope of carbon in which the atoms are
arranged in a hexagonal network, has grown into a major area of investigation since its
discovery and characterization as a semi-metal with promise of semi-conductor capability.
Experimental and theoretical investigations of graphene have concentrated on, among a
wide variety other topics, band gap engineering by way of defects. Though many classes
of graphene defects have been characterized, the first two projects in this dissertation will
discuss two defects in detail: i) the ground state and three lowest excited states involved in
the chemisorption of a hydrogen radical to pristine graphene and graphene with a one and
two carbon atom vacancy utilizing pyrene as a graphene model system, and ii) the structure
and electronic states of silicon doped in two-carbon vacancy graphene using pyrene and
circumpyrene as graphene model systems. In the former case, CASSCF and MR-CISD
calculations were utilized to show that the bonding of hydrogen to graphene, specifically
at the dangling bond (a radical site) found in the single-carbon vacancy defect, involves
several closely spaced electronic excited states, and in ii) the non-planar ground state and
nearby planar excited state both exhibit hybridization at the silicon dopant that would
interrupt graphene’s sp2 hybridized network creating a band gap.
Another area of intense research is organic photovoltaics (OPV), which show great
promise in providing an alternative to traditional inorganic photovoltaic devices as they
consist of more abundant and potentially cheaper materials. Unfortunately, they do not yet
operate at the same PCE in converting photons into electricity, necessitating both
experimental and theoretical attention in order to both optimize the operational device
makeup and understand the contributing physical processes. The third research project in
this dissertation presents theoretical calculations of the electronic excited states an OPV
BHJ device that was found to have dramatically increased PCE upon addition of a
terthiophene spacer material between the electron donor/acceptor layers. The utilization of
the SS solvation environment was instrumental in the correct ordering of excited states.
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The fourth research project presents high-level benchmark calculations of the structure and
electronic excited states of several PPV oligomer dimers using the SOS-MP2 and SOS-
ADC(2) methods. PPV is an important material for the elucidation of excitation energy
transfer (EET) mechanisms in organic polyconjugated systems.
Finally, PAHs are a class of materials consisting of fused aromatic rings primarily
found to have close-shell, low spin ground states. Some cases have been found, however,
to have unpaired electron spins in their low spin ground state. These molecules have proven
to be highly reactive compared to close-shell PAHs and are subject to unforeseen side
reactions and with better characterization of their radical properties, their reactivity can be
better controlled in the future. In the final research project of this dissertation, MR-AQCC
calculations were utilized to determine the singlet-triplet splitting energy, effective number
of unpaired electrons, and unpaired electron density in order to characterize a series of
singlet biradical indenofluorenes and thiophene-fused indenofluorenes.
1.1 Graphene
The initial discovery of graphene by Geim and Novoselov1 using the aptly-named
“Scotch-Tape” method revolutionized material science by characterizing an allotrope of
carbon composed of a one-atom thick layer of hexagonally-arranged carbon atoms forming
a diffuse, delocalized sp2 hybridized network. Graphene was found to be a semi-metal with
a plethora of mechanical, thermal, and electronic properties making it very useful in next-
generation technologies like spin-tronics devices, single molecule sensors, photovoltaic
devices and more. Graphene has two main difficulties that prevent its wide-spread adoption
to industrial applications. The first is that there is no effective method for obtaining the
large amounts of pristine, one-atom thick graphene sheets that would be necessary, and the
second is that, as a semi-metal, graphene has a zero-energy band gap.
One of the primary reasons single-layer graphene has proven difficult to obtain, is
that once graphene layers are separated from graphite, the individual layers will bond
together and reform graphite. One of the first methods used to separate graphene layers
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was using solvents with similar surface energies as graphene to exfoliate graphene from
graphite.2-3 Sonication of graphite dispersed in liquid mediums has also been shown to
transform flakes of graphite into single- or multi-layer graphene.4-5 The use of dispersion
stabilizing agents with high adsorption energy (higher than that of graphene) promotes
exfoliation by adsorbing to the separate layers and preventing reaggregation.6-7
The problem of a zero-band gap material like graphene presents an interesting
avenue towards exploiting its many electronic properties as band gap engineering allows
for the semi-metal to be transformed into a semi-conductor via chemical
functionalization.8-9 Hydrogenation, the chemisorption of hydrogen onto graphene
surfaces, has shown to be an effective method towards opening and tuning its band gap.10-
11 The chemisorption of a species such as hydrogen opens the band gap of graphene by
disrupting the local sp2 hybridized bond network in the diffuse π system creating a sp3
hybridized bond with the incoming chemisorbate. This new sp3 hybridized bond disrupts
the once-continuous flow of conducting electrons opening the band gap, and, as an
additional benefit, increasing the reactivity of nearby carbon centers toward atomic
hydrogen chemisorption. Regions of the graphene surface made of sp3 hybridized carbons
centers are referred to as graphane islands and have been confirmed using scanning
tunneling microscopy (STM) and angle-resolved transmission electron microscopy
(TEM).12 Graphane island arrangement and their proximity to one another contribute to the
overall electronic character of the graphene sheet and have been shown to open band
gaps.12-13
Defects in the carbon network of graphene sheets such as vacancies (removed
carbon atoms in the sheet) and dopant species (carbon atoms replaced with heteroatoms)
have also displayed the capability to induce a band gap compared to pristine, unaltered
graphene.8-9 The removal of carbon atoms from the hexagonal structure either leaves
dangling bonds, radical centers that are much more reactive, or results in bond restructuring
using Jahn-Teller rearrangements. Such defects have demonstrated several properties such
as magnetism,14-15 and increased reactivity by way of greater binding energy was found for
vacancy defects in graphene toward species such as metals, hydrogen, fluorine, aryl groups
and thiol groups.16-18 Using pyrene as a model for a larger graphene sheet, it was shown
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using multireferernce (MR) calculations that in single (SV)19 and double (DV)20 vacancy
defects there are several closely-spaced singlet and triplet electronic states. These closely
spaced states were later shown to be instrumental in bond formation with atomic hydrogen
at these sites.21
Graphene doping with heteroatoms is another interesting method to tuning and
regulating the electronic properties of graphene sheets. Dopants such as nitrogen, boron,
and oxygen have been substituted into the carbon network using a variety of techniques
(e.g. annealing, arc discharge, plasma exposure, chemical vapor deposition (CVD), etc.)
and have shown to change the band character.22-24 Silicon is another common graphene
dopant species that is especially interesting as it is isovalent to carbon and can be found as
intrinsic impurities when graphene is grown using CVD25-26 and by thermal decomposition
of silicon carbide.27 Two identified forms of silicon impurity have been identified and
studied: the nonplanar threefold- coordinated and planar fourfold-coordinated cases, Si-C3
and Si-C4 respectively. Experimental techniques such as scanning tunneling electron
microscopy (STEM) and annular dark field (ADF) imaging together with DFT calculations
confirmed that the hybridization at Si-C3 is sp3 and at Si-C4 is sp2d.28 The d-band
hybridization of the Si-C4 case is thought to be responsible for the disruption of electron
density at the site of the defect.29 Further theoretical work suggested that the planar Si-C4
structure is an equilibrium between a planar and more stable non-planar structure.30
1.2 Organic Photovoltaic Bulk Heterojunctions and Electronic Excited
States
OPV BHJs consist of interfaces between electron donor layers and electron
acceptor layers both of which are composed of π-conjugated organic polymers. BHJ
devices are fundamental to the understanding in electron mobility processes (dissociation
of excitons generating CT states) necessary to generate electricity and to improving their
light conversion efficiency.31-32 Bulk heterojunction devices convert light into electricity
in a four-step process: i) light is absorbed by bands of bright π→π* transitions of the
conjugated organic polymer generating excitonic states, ii) diffusion and migration of the
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exciton through the bulk polymer to the interface and eventual segregation, iii) excitons
dissociate at the heterojunction interfaces creating CT states, leading to iv) charge
separation collected at the contacts.33 Heterogeneous distributions of donor and acceptor
materials in real BHJ devices causes the electricity generating processes to be more
complicated than what has been described here due to numerous structural defects and
conformations, large numbers of internal degrees of freedom for the various polymer
chains, and complicated manifolds of electronic states.34-35 In addition, the rate of exciton
diffusion at the interface competes with the rate of charge recombination, and controlling
these kinetic factors is key to the utilization of BHJ devices.
BHJ devices have been previously studied in a wide variety of compositions. A
PCE of 8.2% was obtained with a BHJ setup consisting of the electron donor, poly[4,8-
bis(5-(2-ethylhexyl)thiophen-2-yl)benzo[1,2-b; 4,5-b']dithiophene-2,6-diyl-alt-(4-(2-
ethylhexyl)-3-fluorothieno[3,4-b]thiophene-)-2-carboxylate-2-6-diyl)] (PTB7-Th), and
electron acceptor, [6,6]-phenyl-C70-butyric acid methyl ester (CN-PC70BM).36 Switching
the common electron acceptor, fullerene (C60) for one based on a fluroene-core design saw
an increase in the PCE to 10.7%.37 A BHJ PV device with porphyrin modifications showed
photoresponse beyond 800 nm together with a high open-circuit voltage and PCE,38-39
while hybrid organic/inorganic BHJ PV devices utilizing both porphyrin parts and
perovskite parts achieved a PCE of 19%.40 In another BHJ OPV device consisting of
electron donor, P3HT, and electron acceptor, fullerene, a dramatic increase in photocurrent,
open-circuit voltage, and overall PCE (2-5 times greater) was observed when utilizing a
spacer layer consisting of either lithium fluoride, terthiophene-derivative (for which the
greatest increase in PCE was found), or one of two metal-organic complexes involving
iridium.41
PPV and is a paradigmatic electron donor material for the understanding of EET
processes in organic π-conjugated polymers at the molecular level. Such understanding
will aid in the fine-tuning of their attractive absorption and emission properties useful in,
among other things, electroluminescent and OPV devices.42-44 It has been concluded using
experimental evidence that coherent PPV dynamics are a result of structural effects
coupling to vibrational modes.43, 45 The evolution of excited states resulting in EET
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dynamics is a product of structural relaxation and these processes are coupled to electronic
states requiring non-adiabatic considerations as state-crossing occurs with structural
evolution.33
BHJ devices have been investigated utilizing PPV as the electron donor. A study
of the compositional dependence of performance of a OPV BJH composed of (poly(2-
methoxy-5-(3’,7’-dimethyloctyloxy)-p-phenylene vinyl-ene):methanofullerene [6,6]-
phenyl C61-butyric acid methyl ester) (OC1C10-PPV:PCBM) showed that the hole mobility
in the PPV phase increases with increasing composition of the PCBM phase until the point
of hole saturation, allowing for the optimal design tuning.46 Another device showed that
polymer/polymer BHJ devices where electron-withdrawing cyano-group substituted PPV
could act as the electron acceptor with another PPV derivative, poly[2-methoxy-5-(2-
ethylhexyloxy)-1,4-phenylenevinylene] (MEH-PPV), acting as the electron donor.47
Quantum chemical calculations on electron donor/acceptor complexes are
important in describing the electronic processes in BHJ devices. The electronic excited
state spectrum of a system composed of poly(thieno[3,4-b]-thiophene benzodithiophene)
(PTB1) and [6,6]-phenyl-C61-butyric acid methyl ester (PCBM) was calculated showing
how internal conversion processes can convert the bright, light-absorbing π→π* transition
to CT states which were stabilized by inter-chain charge delocalization.48 Local, low
energy, dark excitonic transitions on fullerene units were shown to be problematic as they
can possibly quench CT states once formed, emphasizing the correct ordering and
characterizing of electronic excited states.49 The CT states were investigated in a system of
n-acene/tetracyanoethylene (TCNE) showing the necessity of the SOS approach with the
second-order algebraic diagrammatic construction (ADC(2)) to the accurate description of
the electronic excited states of donor/acceptor complexes, the strongly stabilizing effects
of solvent interactions on the lowest calculated CT states, and the change of decay
mechanisms from radiative to nonradiative going from benzene to anthracene with acene-
type electron donors.50
1.3 Radical Character in Polycyclic Aromatic Hydrocarbons
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PAHs are composed of fused aromatic rings, whose ground state in most cases is
expected to be closed-shell and low spin. Several types of PAH have been found to exist
however where the ground state is low spin, but open-shell in character suggesting the
presence of radicals.51 When found in nature, PAHs are often environmental pollutants52
and are good indicators of chemical reactivity of black carbon interfaces in soil.53 They can
even be found in surprising abundance in the interstellar medium54 and are hypothesized
to be the origin of unidentified mid-infrared range bands in the of the spectrum.55-56 Many
classes of radical PAHs have been studied such as zethrenes,57 anthenes,58 phenalenyl,59
besphenalenyls,60 triangulene,61 quinodimethanes,62 indofluorenes,63-65 high-order-
acenes,66-67 and graphene nanoribbons.68-70 Several of these classes will be discussed
below.
Figure 1.1 Structures: a) n-acene, b) triangulene, c) 1,2:9,10-dibenzoheptazethrene, d)
(ma,nz) periacene (graphene nanoribbon model), e) indeno[1,2-b]fluorene
Acenes (Figure 1.1a) are linear arrangements of n number of fused benzene rings
(referred to n-acenes) that are studied not only for their own properties, but their value as
models for larger PAH systems. Acenes are interesting cases for the investigation of radical
character as their number of unpaired electrons in their low spin or singlet ground state has
been found to increase with increasing chain length.71 Fluorine substituted pentacene (five
rings or 5-acene), 2,3,9,10-tetrafluoropentacene, was found to have a larger energy gap
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between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular
orbital (LUMO) when compared to unsubstituted pentacene and persistent radical cationic
and dicationic states when exposed to oxidizing solvents.72 Synthesizing acenes with
extended chain lengths larger than pentacene is difficult due to developing radical
character, but it is possible by functionalizing with additional groups or heteroatom
doping.73 Theoretical studies found that doping the outer more aromatic rings of octacene
increased the radical character by breaking the π-electron bonding, while doping in the
center acted to quench radical character and decrease the effective number of unpaired
electrons.74
Triangulene (Figure 1.1b) is a non-Kekulé extended phenalenyl derivative with an
two traditional radical centers. Its radical nature makes it difficult to synthesize as
polymerization side reactions occur. The ground state of triangulene was determined to be
a high spin triplet so the introduction of bulky groups such as tert-butyl is necessary to
protect reactive carbon sites.61 Tert-butyl substituted triangulene was also thought to have
ferromagnetic potential due to its triplet ground state.75 This magnetic behavior has been
exploited in interesting ways such as with spin-photovoltaic devices where pure spin
current can be generated without charge current upon irradiation with light.76 Theoretical
studies using high-level ab initio calculations support the high spin ground state, showing
degenerate first excited states, relatively small triplet-singlet splitting (0.8 eV), and
unpaired electron density that is delocalized over the entire molecule but is greatest at the
edges.77 The σ- and π-dimerization schemes of trioxo-triangulene monomers have also
been investigated using unrestricted density functional theory (UDFT).78
Zethrenes, so named for their z-shaped configurations of fused rings (Figure 1.1c),
are molecules possessing a singlet biradical ground state that is stabilized by enhanced
aromaticity as described by Clar’s aromatic sextet rule.79 Functionalized zethrene
molecules have shown singlet-fission processes by forming a bound, spin-entangled pair
of triplet states that can thermally dissociate and is enhanced with greater diradical
character.80 The large, laterally extended zethrene, superoctazethrene, was found to be
relatively stable, have significant radical character, have a smaller single-triplet gap than
previous octazethrenes (showing that its first triplet excited state is more thermally
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accessible), and have aromatic character covering the whole surface.81 High-level
multireference calculations found singlet-triplet splitting energies for heptazethrene in
good agreement with experiment and decreased with increasing number of Clar’s sextet
rings.77 Doping zethrenes with heteroatoms has been found to efficiently tune the electronic
properties of PAHs.82 This was shown utilizing multireference calculations in nitrogen and
boron doped heptazene where the effective number of unpaired electrons and singlet-triplet
splitting energy provided an excellent measure of radical character.83
Graphene nanoribbons (Figure 1.1d) are created by cutting graphene sheets into
slices possessing distinct open edges that have been shown to influence the overall
electronic properties of the nanoribbon with respect to bulk graphene. The edges come in
two varieties: zigzag edges that have a localized non-bonding π-state, and armchair edges
that do not. Because of the difference in character between these two edges, edge effects
and reactivity are intensely studied. Quarteranthene was synthesized and shown to have
magnetic susceptibility above 50 K, a small singlet triplet splitting of 30 meV, and a large
biradical index, y, of 0.91 (the singlet ground state possessed 91% biradical character).84
Multireference calculations revealed a multiradical ground state with a near-zero singlet-
triplet splitting.85 Additionally the unpaired electron density is concentrated at the zigzag
edges. The effects of ionization were also investigated in periacene cases showing
decreased unpaired electron density at the edges resulting from removing an electron.86
Indenofluroenes (Figure 1.1e) consist of fused indene and fluorene moieties and
have unique electronic characteristics that are targeted for intense investigation and
exploitation. Thanks to the two five-membered rings incorporated into the structures,
indenofluorenes have demonstrated reversible electron donation useful for OPV devices
and organic field effect transistors (OPET).87 Ambipolar charge transport properties were
found for diindenobenzene derivatives in singlet-crystal and thin-film OFETs.88 Fluorene-
based dispiro[fluorene-indenofluorene] was synthesized and functioned as a hole transport
material in perovskite solar cells with PCE of about 18.5 %.89 Magnetic susceptibility
studies of thiophene-fused structures found the high-spin structure of
indenoindenodi(naphthalenothiophene) exists only above 450 K while maintaining
moderate biradical character in the low-spin ground state and a relatively small singlet-
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triplet splitting.90 In this same study, indenofluorenes were found to have increasing singlet
biradical character and smaller singlet-triplet splitting with increasing core acene length.
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Zakharov, L. N.; Lonergan, M. C.; Nuckolls, C.; Haley, M. M., 6,12-
Diarylindeno[1,2-b]fluorenes: Syntheses, Photophysics, and Ambipolar OFETs.
Journal of the American Chemical Society 2012, 134 (25), 10349-10352.
(89) Yu, W.; Zhang, J.; Wang, X.; Liu, X.; Tu, D.; Zhang, J.; Guo, X.; Li, C., A Dispiro-
Type Fluorene-Indenofluorene-Centered Hole Transporting Material for Efficient
Planar Perovskite Solar Cells. Solar RRL 2018, 2 (7), 1800048.
(90) Dressler, J. J.; Teraoka, M.; Espejo, G. L.; Kishi, R.; Takamuku, S.; Gómez-García,
C. J.; Zakharov, L. N.; Nakano, M.; Casado, J.; Haley, M. M., Thiophene and its
sulfur inhibit indenoindenodibenzothiophene diradicals from low-energy lying
thermal triplets. Nature Chemistry 2018, 10 (11), 1134-1140.
Texas Tech University, Reed Nieman, May 2019
17
CHAPTER II
THEORETICAL METHODS
The Schrödinger equation (shown in Equation 2.1 in its one-dimension, time-
independent form) allows the total energy of a system, E, to be calculated from the system
of mass, m, with wave function, 𝜓(𝑥). The wave function is important as it describes the
state of the system. The first term is the kinetic energy, and the second term, 𝑉(𝑥), is the
potential energy function. As the Schrödinger equation can only be solved exactly for very
specific cases, several approximations have been made to make it accessible to a larger
range of systems. The Born-Oppenheimer (BO) approximation for instance states that since
the mass of electrons is much less than that of atomic nuclei, their velocities can be
decoupled. The consequences of this is that the nuclei are stationary from the perspective
of electrons, so the wave function depends only on electrons moving in the field of the
stationary nuclei (that is, it depends only parametrically on the nuclear coordinates). The
Hartree-Fock (HF) treatment for a many-electron system, itself utilizing the BO
approximation, and upon which several more advanced method (called post-HF methods)
are built, calculates the total energy of a system of electrons in only the average potential
created by the other particles.
−ħ2
2𝑚
𝑑2𝜓(𝑥)
𝑑𝑥2+ 𝑉(𝑥)𝜓(𝑥) = 𝐸𝜓(𝑥) (2.1)
The choice of which method is key when one wants to calculate specific molecular
properties. One priority is simply computational rigor, i.e. how much time will the
calculation require? The HF method for instance scales exponentially with the number of
basis functions, N, as N4, second-order Møller Plesset perturbation theory (MP2) is more
accurate but scales as N5, and multiconfigurational and multireference are more complex
as they depend not only on the number of basis functions, but also the number of reference
configurations considered (scaling linearly).
Besides the computational cost, one must consider the properties one wishes to
calculate. For example, HF theory provides nice molecular orbital shapes, but as it is only
Texas Tech University, Reed Nieman, May 2019
18
a single reference method, electron correlation energy is lost and it will not reliably
describe correct dissociation behavior. To account for this, multiconfigurational and
multireference methods should be used. DFT is quite efficient with respect to
computational cost as it foregoes reliance on the wave function for the total electron
density, but the correct choice of DFT functional is non-trivial and not easily determined.
This remainder of chapter will provide a brief discussion and derivation of the
various quantum chemical methods implemented in the chapters that follow. Unless
otherwise specified, the derivations and discussion presented in this chapter are adapted
from Levine’s Quantum Chemisty.1
2.1 Density Functional Theory
Hohenberg and Kohn provided the foundation for DFT by showing that the total
energy of a non-degenerate ground state, 𝐸0, is determined by the ground state electron
probability density, 𝜌0(𝑟) in a given external field, υ(r), where r represents the electron
coordinates.2 In addition, as the electron probability density can be used to find the ground-
state total energy, the ground state wave function, ψ0, can be determined which would allow
one to find excited state wave functions and energies as well.
The general expression for the ground state total energy is given in Equation 2.2.
The term 𝑇[𝜌0] is the kinetic energy, 𝑉𝑁𝑒[𝜌0] is the nuclear-electron attraction energy, and
𝑉𝑒𝑒[𝜌0] is the electron-electron repulsion energy. Of these, only 𝑉𝑁𝑒[𝜌0] is known
(Equation 2.3), while 𝑇[𝜌0] and 𝑉𝑒𝑒[𝜌0] are not explicitly known.
𝐸0 = 𝐸𝜐[𝜌0] = 𝑇[𝜌0] + 𝑉𝑁𝑒[𝜌0] + 𝑉𝑒𝑒[𝜌0] (2.2)
𝑉𝑁𝑒[𝜌0] = ⟨𝜓0| ∑ υ(r𝑖)𝑛𝑖=1 |𝜓0⟩ = ∫ 𝜌0υ(r)dr (2.3)
With this definition of 𝑉𝑁𝑒[𝜌0], Equation 2.2 can be simplified into Equation 2.2,
and by gathering the kinetic energy and electron-electron repulsion energy into a new term,
𝐹[𝜌0], showing a functional dependent on the electron probability density, Equation 2.5.
Texas Tech University, Reed Nieman, May 2019
19
This new functional is independent of the external potential, but cannot be used to calculate
the total ground state energy from 𝜌0 since 𝐹[𝜌0] is not explicitly defined.
𝐸0 = 𝐸𝜐[𝜌0] = ∫ 𝜌0υ(r)dr + 𝑇[𝜌0] + 𝑉𝑒𝑒[𝜌0] (2.4)
𝐸0 = 𝐸𝜐[𝜌0] = ∫ 𝜌0υ(r)dr + 𝐹[𝜌0] (2.5)
The Hohenberg-Kohn theorem that was just defined shows how the density, 𝜌0, can
be used in lieu of the wave function to calculate the ground state properties of a molecular
system, but since 𝐹[𝜌0] and the method of obtaining 𝜌0 without a wave function are
unknown, it does not explain how this is accomplished. Kohn and Sham provided a method
by which 𝜌0 could be determined and used to calculate 𝐸0.
The Kohn-Sham (K-S) approach3 to DFT begins with a reference system composed
of non-interacting electrons all experiencing the same external potential energy function,
υref(r). The function υref(r) gives the same ground state electron probability density,
𝜌𝑟𝑒𝑓(𝑟), for the reference as for the exact ground state electron probability density, 𝜌0(𝑟).
Finding 𝜌𝑟𝑒𝑓(𝑟) in this way allows for the determination of the external potential for the
reference system.
The K-S Hamiltonian, ��𝑟𝑒𝑓, for this non-interacting reference system is a sum of
one-electron Hamiltonians, ℎ𝑖, according to Equation 2.6, and the K-S reference system is
related back to the real, fully interacting system by Equation 2.7. The factor λ describes the
amount of electron-electron interaction and varies between λ=0 for the non-interacting
system and λ=1 for the real system (fully interacting). The external potential is such that it
makes the ground state electron probability density with the Hamiltonian, ��𝜆, equal to the
ground state electron probability density of the real system. Eigenfunctions of the one-
electron Hamiltonian are the spatial part of the K-S spin orbitals and the eigenvalues are
the K-S orbital energies as shown by Equation 2.8.
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��𝑟𝑒𝑓 = ∑ [−1
2∇𝑖
2 + 𝜐𝑟𝑒𝑓(𝑟𝑖)] ≡ ∑ ℎ𝑖
𝑛
𝑖=1
𝑛
𝑖 = 1
(2.6)
��𝜆 ≡ �� + ∑ 𝜐𝜆(𝑟𝑖)
𝑖
+ 𝜆��𝑒𝑒 (2.7)
ℎ𝑖 𝜓𝑖𝐾𝑆 = 𝜀𝑖
𝐾𝑆 𝜓𝑖𝐾𝑆 (2.8)
With these, another equation can be derived from Equation 2.5 to find the total
energy of the ground state, Equation 2.9. In Equation 2.9, Δ��[𝜌] is the difference between
the ground state electronic kinetic energy and that of the non-interacting system, and
Δ��𝑒𝑒[𝜌] is the difference between the average ground state electron-electron repulsion
energy and the electrostatic repulsion for a system of electrons that has been smeared into
a continuous distribution of charge with a density, ρ. The sum of Δ��[𝜌] and Δ��𝑒𝑒[𝜌], each
of which is unknown, gives the exchange-correlation energy functional, 𝐸𝑋𝐶[𝜌], rewriting
Equation 2.9 as Equation 2.10.
𝐸𝜐[𝜌0] = ∫ 𝜌0υ(r)dr + ��𝑟𝑒𝑓[𝜌0] +1
2∬
𝜌(𝑟1)𝜌(𝑟2)
𝑟12𝑑𝑟1𝑑𝑟2 + Δ��[𝜌]
+ Δ��𝑒𝑒[𝜌]
(2.9)
𝐸𝜐[𝜌0] = ∫ 𝜌0υ(r)dr + ��𝑟𝑒𝑓[𝜌0] +1
2∬
𝜌(𝑟1)𝜌(𝑟2)
𝑟12𝑑𝑟1𝑑𝑟2 + 𝐸𝑋𝐶[𝜌] (2.10)
The first three terms in Equation 2.10 are found from the ground state electron
density and provide the largest contribution. The last term, the exchange-correlation
energy, contributes less, but is more difficult to obtain accurately. All four terms in
Equation 2.10 require the ground state electron density for the reference system of non-
interacting electrons such that it equals the electron density of the real system. The electron
density of the n-electron system can then be found using a wave function expressed as a
Slater determinant as in Equation 2.11, where the K-S spatial orbitals are restricted to
remain orthonormal.
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21
𝜌 = 𝜌𝑟𝑒𝑓 = ∑|𝜓𝑖𝐾𝑆|
2𝑛
𝑖=1
(2.11)
With this definition of the electron density, Slater-Condon rules4-5 allow one to
rewrite nuclear-electron (nuclear coordinates given by α) attraction energy in Equation 2.10
as Equation 2.12. One can see that finding K-S orbitals that minimize the total ground state
energy will follow Equation 2.13 (following Equation 2.8), where 𝜐𝑋𝐶(1) is the exchange-
correlation potential and is the functional derivative of the exchange-correlation energy
with respect to the ground state electron density. Finally, Equation 2.13 can be written
succinctly as Equation 2.14.
𝐸𝜐[𝜌0] = − ∑ 𝑍𝛼 ∫𝜌(𝑟1)
𝑟1𝛼𝑑𝑟1
𝛼
−1
2∑⟨𝜓𝑖
𝐾𝑆(1)|∇12|𝜓𝑖
𝐾𝑆(1)⟩
𝑛
𝑖=1
+1
2∬
𝜌(𝑟1)𝜌(𝑟2)
𝑟12𝑑𝑟1𝑑𝑟2 + 𝐸𝑋𝐶[𝜌]
(2.12)
[−1
2∇1
2 − ∑𝑍𝛼
𝑟1𝛼+ ∫
𝜌(𝑟1)
𝑟1𝛼𝑑𝑟1 + 𝜐𝑋𝐶(1)
𝛼
] 𝜓𝑖𝐾𝑆(1) = 𝜀𝑖
𝐾𝑆(1)𝜓𝑖𝐾𝑆(1) (2.13)
ℎ𝑖 𝜓𝑖𝐾𝑆(1) = 𝜀𝑖
𝐾𝑆(1)𝜓𝑖𝐾𝑆(1) (2.14)
The most difficult part in evaluating DFT energies is the finding the correct
functional form of the exchange-correlation energy, 𝐸𝑋𝐶, as it makes the most significant
contribution to the total energy. One common procedure for overcoming this problem
called the Local Density Approximation (LDA) is to assume a uniform electron gas and
split 𝐸𝑋𝐶 into exchange and correlation energies, 𝐸𝑋 and 𝐸𝐶 respectively, which are then
summed together. A specific example of a LDA functional is from Vosko, Wilk, and Nuair
(VWN).6 Gradient-corrected functionals build on the LDA formalism and take into account
non-uniform gases utilizing gradients. This approach improve accuracy by having the
exchange-correlation energy depend on the electron density as well as the first derivative
of the electron density. Common examples of gradient-corrected functionals are Perdew,
Burke and Ernzerhof (PBE)7-8, and Becke, Lee, Yang, Parr (BLYP).9-11 Hybrid density
functionals were developed to combine Hartree-Fock exchange energy with the exchange-
Texas Tech University, Reed Nieman, May 2019
22
correlation energy of gradient-corrected functionals. An example of a hybrid functional is
the popular Becke 3-parameter Lee, Yang, Parr (B3LYP).12 Lastly, range-separated
functionals were developed where the Coulomb potential is treated separately at short and
long range in the form of a hybrid functional. Examples of such range-separated functional
are the Coulomb attenuated method B3LYP (CAM-B3LYP)13 and ωB97XD14.
Non-covalent interactions are especially important in weakly bonded complexes,
aggregates, and condensed phases made up of separable units, but these dispersion/Van der
Waals interactions require special consideration outside of computationally laborious ab
initio methods. One approach is DFT-D (D stands for dispersion),15 where the dispersion
energy correction, 𝐸𝑑𝑖𝑠𝑝, to the DFT energy, 𝐸𝐷𝐹𝑇, is applied via Equations 2.15 and 2.16.
In Equation 2.16, 𝑁𝑎𝑡 is the number of atoms, 𝐶6𝑡𝑓
is the dispersion coefficient for the atom
pairs f and t, 𝑆6 is a global scaling factor which is dependent on the density functional,
𝑅𝑡𝑓is the distance between atoms t and f, and 𝑓𝑑𝑎𝑚𝑝 is a dampening function applied to
correct for small R and electron double-counting.
𝐸𝐷𝐹𝑇−𝐷 = 𝐸𝐾𝑆−𝐷𝐹𝑇 + 𝐸𝑑𝑖𝑠𝑝 (2.15)
𝐸𝑑𝑖𝑠𝑝 = −𝑆6 ∑ ∑𝐶6
𝑡𝑓
𝑅𝑡𝑓6
𝑁𝑎𝑡
𝑓=𝑡+1
𝑁𝑎𝑡−1
𝑡=1
𝑓𝑑𝑎𝑚𝑝(𝑅𝑡𝑓) (2.16)
2.2 Second-Order Møller-Plesset Perturbation Theory
This section will cover perturbation theory (PT) as described by Møller and Plesset
(MP)16 for second-order corrections (MP2)17-21 for closed-shell, ground state systems and
utilizing spin orbitals, 𝑢𝑖. They detailed a method for the treatment of molecules in which
the HF wave function is the unperturbed wave function. The Hartree-Fock equation for an
electron, m, in a many-electron system is given by Equation 2.17 where the Fock operator
expression is Equation 2.18.
𝑓(𝑚)𝑢𝑖(𝑚) = 𝜀𝑖𝑢𝑖(𝑚) (2.17)
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23
𝑓(𝑚) = −1
2∇𝑚
2 − ∑𝑍𝛼
𝑟𝑚𝛼𝛼
+ ∑[𝑗��(𝑚) − ��𝑗(𝑚)]
𝑛
𝑗=1
(2.18)
The MP unperturbed Hamiltonian is a sum of one-electron Fock operators,
Equation 2.19, with the HF wave function, Φ0, also referred to in this context as the zeroth-
order wave function, is shown in Equation 2.20 and is a Slater determinant composed of
any of the n possible different spin orbitals. Each term in its expansion is a permutation of
the electrons in the spin orbitals, and is an eigenfunction of ��0, the MP zeroth-order
Hamiltonian. The total wave function, Φ0, is like-wise an eigenvalue of ��0 with
eigenvalues given by Equation 2.21.
��0 ≡ ∑ 𝑓(𝑚)
𝑛
𝑚=1
(2.19)
|𝑢1𝑢2𝑢3 … 𝑢𝑛| = Φ0 (2.20)
��0Φ0 = ( ∑ 𝜀𝑚
𝑛
𝑚=1
) Φ0 = 𝐸(0)Φ0 (2.21)
The Hamiltonian describing the perturbation, 𝐻′ is the difference between the real
molecular Hamiltonian, ��, and the zeroth-order Hamiltonian, ��0, as in Equation 2.22. In
this way, 𝐻′ is the difference between the real interelectronic repulsion and the HF
interelectronic repulsion. The first-order correction to the ground state energy is given in
Equation 2.23, which is also the variational integral for the HF wave function, Φ0, giving
the HF energy, E𝐻𝐹, Equation 2.24.
𝐻′ = �� − ��0 = ∑ ∑1
𝑟𝑙𝑚− ∑ ∑[𝑗��(𝑚) − ��𝑗(𝑚)]
𝑛
𝑗=1
𝑛
𝑚=1𝑚>𝑙𝑙
(2.22)
𝐸0(1)
= ⟨Φ0|𝐻′|Φ0⟩ (2.23)
𝐸0(0)
+ 𝐸0(1)
= E𝐻𝐹 (2.24)
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24
With Equation 2.24 in mind, corrections up to the second-order are shown to be
necessary in order to improve upon the HF energy, 𝐸0(2)
, as in Equation 2.25, where the
unperturbed functions, 𝜓𝑠(0)
, are all possible Slater determinants formed from n possible
different spin orbitals. One must also apply excitations in the ground state electron
configuration such that 𝜓𝑠(0)
differs from Φ0 by the number of unoccupied orbitals, also
referred to virtual orbitals, (represented by indices a, b, c, d, …) replacing occupied orbitals
(represented by indices i, j, k, l, …). An excitation level of one is shown as Φ𝑖𝑎, where
occupied spin orbital, 𝑢𝑖 is replaced by unoccupied spin orbital, 𝑢𝑎, an excitation level of
two is shown as Φ𝑖𝑗𝑎𝑏, where occupied spin orbitals, 𝑢𝑖 and 𝑢𝑗 , are replaced by unoccupied
spin orbitals, 𝑢𝑎 and 𝑢𝑏, and so on for excitation levels of three and higher. Single
excitations can be neglected as the integral, ⟨𝜓𝑖𝑎|𝐻′|Φ0⟩, is zero for all occupied orbitals,
i, and all unoccupied orbitals, a. Similarly, triple excitations and higher,
⟨𝜓𝑖𝑗𝑘𝑎𝑏𝑐|𝐻′|Φ0⟩, 𝑒𝑡𝑐 …, are zero following Slater-Condon rules4-5. Therefore, only an
excitation level of two is needed to be evaluated, ⟨𝜓𝑖𝑗𝑎𝑏|𝐻′|Φ0⟩. This same discussion is the
reason why only double excitations are present in the first order correction, 𝜓𝑠(1)
, to the
unperturbed wave function, 𝜓𝑠(0)
.
𝐸0(2)
= ∑⟨𝜓𝑠
(0)|𝐻′|Φ0⟩
2
𝐸0(0)
− 𝐸𝑠(0)
𝑠≠0
(2.25)
The doubly excited wave functions, Φ𝑖𝑗𝑎𝑏, are eigenfunctions of Equation 2.19
whose eigenvalues differ from those of the HF wave function, Φ0, by replacing occupied
orbital energies, 𝜀𝑖 and 𝜀𝑗, by unoccupied orbital energies, 𝜀𝑎 and 𝜀𝑏, as in Equation 2.26,
(which is the denominator of Equation 2.25) when 𝜓𝑠(0)
is equal to Φ𝑖𝑗𝑎𝑏. With the
expression for 𝐻′ in Equation 2.22 and Slater-Condon rules, the evaluation of integrals
with Φ𝑖𝑗𝑎𝑏 proceeds as in Equation 2.27 with the integral in Equation 2.28, where n is the
number of electrons.
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25
𝐸0(0)
− 𝐸𝑠(0)
= 𝜀𝑖 + 𝜀𝑗 + 𝜀𝑎 + 𝜀𝑏 (2.26)
𝐸0(2)
= ∑ ∑ ∑ ∑|⟨𝑎𝑏|𝑟−1|𝑖𝑗⟩ − ⟨𝑎𝑏|𝑟−1|𝑗𝑖⟩|2
𝜀𝑖 + 𝜀𝑗 + 𝜀𝑎 + 𝜀𝑏
𝑛−1
𝑗=1
𝑛
𝑖=𝑗+1
∞
𝑎=𝑛+1
∞
𝑏=𝑎+1
(2.27)
⟨𝑎𝑏|𝑟−1|𝑖𝑗⟩ = ∫ ∫ 𝑢𝑎∗ (1)𝑢𝑏
∗ (2)𝑟12−1𝑢𝑖(1)𝑢𝑗(2)𝑑𝜏 (2.28)
Integrals involving spin orbitals (including the actual spin) are evaluated with
respect to the electron-repulsion integrals, and sums over all a, b, i, and j in Equation 2.27
allow for all possible doubly excited terms of 𝜓𝑠(0)
. With this, the molecular MP2 energy,
𝐸𝑀𝑃2, is finally given in Equation 2.29. Consequently, MP2 energies are size-extensive,
but since they are not variational, they can produce energies below the true energy.
𝐸(0) + 𝐸(1) + 𝐸(2) = 𝐸𝐻𝐹 + 𝐸(2) = 𝐸𝑀𝑃2 (2.29)
2.3 Second-Order Algebraic Diagrammatic Construction
The algebraic diagrammatic construction (ADC)22 for the polarization propagator
(PP) allows for the calculation of electronic excited state information for a molecular
system and the following derivation is adapted from Ref 23. The derivation of the ADC
approach starts with many-body Green’s function theory that is used to solve
inhomogeneous differential equations.24 The electronic Hamiltonian for a many-bodied
system does not have a uniquely defined Green’s function, however, but propagators can
be created from the foundations of a Green’s function in order to solve the problem.25
Several propagators can be constructed such as the one-electron propagator which
gives the probability of an electron travelling a given length, l, in a given time, t; the two-
electron propagator gives the same property but for two correlated electrons. The
expectation values for the calculated properties can be found by knowing these two
propagators.26 The one-electron propagator can also be used for other one-electron
properties such as ionization energies and electron affinities.27-29 The PP describes the time-
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26
evolution of polarization in a many-body system by acting on a time-dependent (TD)
ground state wave function and propagates the TD change in density.25
The electronic Hamiltonian needs the wave function of the electronic excited states
and the PP contains the implicit information of the electronic excited states for the
molecular system. The spectral representation of the PP as a matric function, also called
the Lehmann representation, is given in Equation 2.30, where 𝜓0 is the ground state wave
function, 𝐸0𝑁 is the grounds state energy, 𝑐𝑞
† is the creation operator for the one-electron
state, q, and 𝑐𝑝 is the annihilation operator for the one-electron state, p (both the creation
and annihilation operators are associated with HF orbitals). In this form, the PP gives the
eigenstates of the molecular system to diagonalize the Hamiltonian matrix. The summation
in the PP is done for all electronic excited states, 𝜓𝑛, with total energy, 𝐸𝑛𝑁. This will give
the exciation spectrum since the poles of the PP (when 𝜔 = 𝐸𝑛𝑁 − 𝐸0
𝑁) are at the vertical
excitation energies and the residues are the transition probabilities. The spectral
representation of the PP can therefore be written in a compact, diagonalized form as in
Equation 2.31, where 𝛀 is the matrix of vertical excitation energies, 𝜔𝑛, and 𝒙 is the matrix
of transition or spectroscopic amplitudes.25
𝚷𝑝𝑞,𝑟𝑠(𝜔) = ∑⟨𝜓0|𝑐𝑞
†𝑐𝑝|𝜓𝑛⟩⟨𝜓𝑛|𝑐𝑟†𝑐𝑠|𝜓0⟩
𝜔 + 𝐸0𝑁 − 𝐸𝑛
𝑁𝑛≠0
+ ∑⟨𝜓0|𝑐𝑟
†𝑐𝑠|𝜓𝑛⟩⟨𝜓𝑛|𝑐𝑞†𝑐𝑝|𝜓0⟩
−𝜔 + 𝐸0𝑁 − 𝐸𝑛
𝑁𝑛≠0
(2.30)
𝚷(𝜔) = 𝒙†(𝜔 − 𝛀)−1𝒙 (2.31)
A non-diagonal representation of the PP is needed for the Feynmann-Goldstone
diagrammatic perturbation scheme, and is shown in Equation 2.32, where M and f are
respectively the non-diagonal matrix representation of the effective Hamiltonian and the
matrix of effective transition moments.22 The effective Hamiltonian and transition
amplitudes are expanded to the PT order in the fluctuation potential, and, following that,
the correlation energy in the MP Hamiltonian partitioning is shown in Equations 2.33 and
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2.34. Using the PT up to order n expands 𝚷(𝜔) to give the expression of 𝑀𝜇𝜈(𝑛)
and 𝑓𝜇(𝑛)
(μ
and ν give the excitation level).
𝚷(𝜔) = 𝐟†(𝜔 − 𝐌)−1𝐟 (2.32)
𝑴 = 𝑴(0) + 𝑴(1) + 𝑴(2) + ⋯ (2.33)
𝐟 = 𝐟(0) + 𝐟(1) + 𝐟(2) + ⋯ (2.34)
The ADC approximation up to order n, ADC(n), has all the terms necessary for a
PT application of 𝚷(𝜔) to that same order. The ADC scheme is thusly a reformulation of
the diagrammatic perturbation series of the PP. Since the excitation energies are the poles
of PP, they can be found by diagonalizing the M matrix to the specified order using PT.
The Hamiltonian eigenvalue problem gives the vertical excitation energies, 𝜔𝑛, whose
eigenvectors, y, give the spectroscopic amplitudes using Equation 2.36.
𝒙 = 𝒚†𝐟 (2.35)
In summary, the ADC method allows for the use of PT to reformulate the
approximations of the PP, which formally contains the information on the exact excited
states of a given molecular system, so that it only has approximate information on the
excited states. The diagrammatic method together with the PT series up to a specified order
gives the exact algebraic formulas for the computation of the excited states, with higher
orders being more accurate.
2.4 Basis Set Superposition Error and the Counterpoise Correction Method
Basis set superposition error (BSSE) manifests as artificially shorter intermolecular
distances and greater (more stabilizing) interaction energy in weakly bonded dimers or
clusters; the error is more prominent for smaller basis set sizes (fewer basis functions) and
is vanishing at the complete basis set limit.30-31 The following is adapted from Ref. 32. In
dimer systems composed of two monomers, A and B, the dimer is stabilized artificially as
monomer A, as a result of proximity, begins to take advantage of the basis functions
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localized to the atoms on monomer B (and vice versa). The increase in basis set size as a
result of the extra basis functions is not necessarily the source of the error, but rather it is
the unequal treatment found for each monomer: smaller intermolecular distances are able
to access more basis function from the other monomer than larger intermolecular distances
where the overlap integral becomes too small.33
The counterpoise correction (CP) method by Boys and Bernardi34 is a way to
removing BSSE from interacting systems. The usual method for calculating the interaction
energy, Δ𝐸𝑖𝑛𝑡(𝐴𝐵), of a dimer system, AB, composed of monomer A interacting with
monomer B without correcting for BSSE is shown in Equation 2.36, where the
parenthetical symbol is the system of interest, the superscripts are the basis set, and the
subscripts are the geometries. In this way, 𝐸𝐴𝐵𝐴𝐵(𝐴𝐵), is the energy of the dimer with the
dimer geometry in the basis of the dimer. The same procedure is carried out for the energies
of both monomers A and B (respectively the second and third terms in Equation 2.36).
Δ𝐸𝑖𝑛𝑡(𝐴𝐵) = 𝐸𝐴𝐵𝐴𝐵(𝐴𝐵) − 𝐸𝐴
𝐴(𝐴) − 𝐸𝐵𝐵(𝐵) (2.36)
The CP method corrects for BSSE by finding the energy by which monomer A (or
B) is stabilized artificially by the accessible basis functions of monomer B (or A).
Equations 2.37 and 2.38 represent the differences between monomer A in the basis of the
dimer and monomer A in the basis of monomer A (same for monomer B). The CP energy
is then negative as the energy of a monomer in the basis of the dimer must be lower than
that of the monomer in its own basis. Inserting Equations 2.37 and 2.38 into Equation 2.36
gives the expression for the CP correction energy, Equation 2.39.
𝐸𝐵𝑆𝑆𝐸(𝐴) = 𝐸𝐴𝐴𝐵(𝐴) − 𝐸𝐴
𝐴(𝐴) (2.37)
𝐸𝐵𝑆𝑆𝐸(𝐵) = 𝐸𝐵𝐴𝐵(𝐵) − 𝐸𝐵
𝐵(𝐵) (2.38)
Δ𝐸𝑖𝑛𝑡𝐶𝑃 (𝐴𝐵) = 𝐸𝐴𝐵
𝐴𝐵(𝐴𝐵) − 𝐸𝐴𝐴𝐵(𝐴) − 𝐸𝐵
𝐴𝐵(𝐵) (2.39)
In practice, evaluating monomer A in the basis of the dimer requires that the basis
functions of monomer B are placed on the atomic centers of monomer B while
simultaneously neglecting both the atomic charges and electrons found on monomer B. In
this case, the atoms and basis functions of monomer B are thusly referred to as ghost atoms
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and ghost functions, respectively. The procedure is performed also for monomer B in the
basis of the dimer with the ghost atoms and ghost functions of monomer A.
2.5 Environmental Simulation Approaches
Since most chemistry occurs in solution, not the gas phase with isolated molecules,
solvent effects are critical. For example, for a given solute in solution, the solute can induce
a dipole moment in the surrounding solvent molecules and the bulk of the solvent is then
polarized in the region of each solute molecule generating an electric field, or reaction
field, at every solute molecule. The reaction field distorts the solute’s electronic wave
function from the gas phase giving an induced dipole moment in addition to the solute gas
phase dipole moment.
There are two methods for simulating solvent or environmental effects. Explicit
solvation is where the solvent molecules are directly modelled and a potential is applied
such as the Lennard-Jones 6-12 or TIP4P approaches. More common however is implicit
solvation utilizing a continuum model. In continuum models, the molecular structure of the
solvent is ignored and the solvent environment is constructed as a continuous dielectric
(something that cannot conduct electricity) surrounding a cavity centered on the solute
molecule. The continuous dielectric is described using the dielectric constant of the chosen
solvent.
In quantum mechanical calculations, a potential term, ��𝑖𝑛𝑡 , is included in the
electronic Hamiltonian, ��(0) , of the isolated molecular system to account for the
interactions of the solute and dielectric continuum of the solvent environment. For a given
calculation using the continuum solvation method, the electronic wave function and
electronic probability density of the solute change iteratively going from vacuum to
solution until self-consistency between the solute’s charge distribution and the solvent
reaction field. This method is referred to the self-consistent reaction field (SCRF) method.
The polarizable continuum model (PCM) is an application of the SCRF approach
where a sphere is centered on each nucleus of the solute with radius equal to 1.2 times the
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van der Waals radius of the atom. A smoothing effect is generally applied to take care of
sharp crevices created by regions of the overlapping spheres. Apparent surface charges
(ASC) are distributed on the surface to create an electric potential equal to the electric
potential, 𝜙𝜎, created by the polarized dielectric constant. The ASC are approximated by
using point charges on the cavity surface and the cavity is divided into many small regions,
k, each with an apparent charge, 𝑄𝑘 (Equation 2.40), where 𝑟𝑘 is the point 𝑄𝑘 is located,
𝜙𝜎(𝑟) is the electric potential (Equation 2.41) due to the polarization of the dielectric, 𝐴𝑘
is the area of the region, ∇𝜙𝑖𝑛(𝑟𝑘) is the gradient of the electric potential in the cavity, and
𝑛𝑘 is a unit vector perpendicular to the cavity surface at 𝑟𝑘 pointing out of the cavity. The
electric potential in the cavity, 𝜙𝑖𝑛, is the sum of the charge distribution of the solute, 𝜙𝜇,𝑖𝑛,
and the polarized dielectric, 𝜙𝜎,𝑖𝑛: 𝜙𝑖𝑛 = 𝜙𝜇,𝑖𝑛 + 𝜙𝜎,𝑖𝑛. The electric potential, 𝜙𝑖𝑛, and 𝑄𝑘
are not known and are found iteratively until convergence of the charges, which are then
used to find the interacting potential term, ��𝑖𝑛𝑡, as in Equation 2.43. The interacting
potential, ��𝑖𝑛𝑡, is then added to the electronic Hamiltonian, ��(0), and the entire process is
repeated iteratively until convergence.
𝑄𝑘 = [𝜀𝑟 − 1
𝑟𝜋𝜀𝑟] 𝐴𝑘∇𝜙𝑖𝑛(𝑟𝑘) ∙ 𝑛𝑘 (2.40)
𝜙𝜎(𝑟) = ∑𝑄𝑘
|𝑟 − 𝑟𝑘|𝑘
(2.41)
��𝑖𝑛𝑡 = − ∑ 𝜙𝜎(𝑟𝑖)
𝑖
+ ∑ 𝑍𝛼𝜙𝜎(𝑟𝛼)
𝛼
(2.42)
A similar treatment to the PCM method is the conductor-like screening model
(COSMO).35 In this approach realistic solute-solvent shapes are still constructed and
surface charges are still used on the solute cavity. The difference comes when the charges
are calculated initially with a condition appropriate for the solvent that is not a dielectric,
but a conductor.
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2.6 Multiconfigurational Self-Consistent Field
Multiconfigurational self-consistent field (MCSCF)36 is a method that corrects the
dissociation behavior of molecules and incorrect description of systems at increasing
intermolecular distances in comparison to the HF method. The HF method does not account
for static electron correlation effects as only a single electron configuration is considered
in a HF wave function and MCSCF is an improvement over HF since it utilizes several
electron configurations in the MCSCF wave function. The MCSCF wave function,
𝜓𝑀𝐶𝑆𝐶𝐹 , is a linear combination of different electron configurations represented by
configuration state functions (CSF), Φ𝑛, as in Equation 2.43, where n is the total number
of CSFs and 𝑏𝑛 are the expansion coefficients that are minimized in the variational integral.
The MCSCF method, in addition to optimizing the expansion coefficients, also optimizes
the form of the molecular orbitals (MOs) by varying the expansion coefficients, c, that
connect the MOs, 𝜙, to the basis functions, 𝜒, shown in Equation 2.44 for the ground state
of He in the minimal basis set. Uncovered terms inside of bars represent α spin orbitals and
the covered terms are β spin orbitals. The coefficients b and c are varied to minimize the
variational integral in an iterative process similar to a HF SCF procedure and while keeping
𝜙 orthonormal and 𝜓𝑀𝐶𝑆𝐶𝐹 normalized. Because both the MOs and the CSF expansion
coefficients are optimized in the process, only relatively few CSFs are necessary for
normally acceptable results.
|𝜓𝑀𝐶𝑆𝐶𝐹⟩ = ∑ 𝑏𝑛|Φ𝑛⟩
𝑛
(2.43)
|𝜓𝑀𝐶𝑆𝐶𝐹⟩ = 𝑏1| 𝜙1 𝜙1 | + 𝑏2| 𝜙2 𝜙2
|
= 𝑏1|(𝑐11𝜒1 + 𝑐21𝜒1)(𝑐11𝜒1 + 𝑐21𝜒2 )|
+ 𝑏2|(𝑐12𝜒1 + 𝑐22𝜒2)(𝑐12𝜒1 + 𝑐22𝜒2 )|
(2.44)
The CASSCF method is the most common approach to MCSCF calculations.37 The
orbitals are still expanded as a linear combination of CSFs, though in CASSCF, the CSFs
are divided into active and inactive orbitals. The inactive orbitals are kept either doubly
occupied (if in the occupied space) or unoccupied (if in the virtual space) in all CSFs. The
remaining electrons and orbitals constitute the active space. The CASSCF wave function
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is then a linear combination of all CSFs, Φ𝑛, formed by distributing all the active electrons
in the active orbitals in all possible permutations while maintaining the desired state to be
calculated. An active space consisting of x number of electrons in y number of orbitals has
the notation, CAS(x,y). A MCSCF procedure is performed optimizing the coefficients, b
and c.
The selection of the active space for a CASSCF calculation is a non-trivial process
and of great consequence. One reliable method is found in Pulay’s unrestricted natural
orbital-complete active space (UNOCAS) approach.38 Here an unrestricted HF calculation
is performed in the chosen basis and the natural orbitals are analyzed. All natural orbitals
(NOs) whose occupation is between 1.98 e and 0.02 e are considered fractionally occupied
and will constitute the largest extent of the active space.
2.7 Multireference Configuration Interaction
Configuration interaction (CI) is a natural improvement of the single configuration
HF method and corrects for dissociation behavior and increasing intermolecular distances
The full CI wave function is a linear combination of CSFs, where the CSFs are first the
reference wave function (the HF wave function, 𝜓0) and subsequent CSFs represent
excitations of an electron in an occupied orbital (indices i, j, k, etc…) into a virtual orbital
(indices a, b, c, etc…). For full CI, the wave function, shown in Equation 2.45, contains all
possible excitations (single, double, triple excitations, etc…) of the reference wave
function while respecting the symmetry of the state of interest. The variational principle is
then used to minimize the CI energy with respect to the CSF expansion coefficients, 𝑏𝑛.
Full CI, by taking into account all excitations of the reference, produces a very large
number of CSFs and is computationally infeasible for all but the smallest systems (number
of atoms) and basis sets (number of basis functions). In light of this, truncations can be
made on the CI wave functions restricting the excitation level most commonly to either
single and/or double, CIS and CISD respectively, though truncated CI methods, including
MCSCF, are not size-extensive.
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33
|𝜓𝐶𝐼⟩ = 𝑏0|𝜓0⟩ + ∑ 𝑏𝑖𝑎|Φ𝑖
𝑎⟩
𝑖,𝑎
+ ∑ 𝑏𝑖𝑗𝑎𝑏|Φ𝑖𝑗
𝑎𝑏⟩
𝑖>𝑗,𝑎>𝑏
+ ∑ 𝑏𝑖𝑗𝑘𝑎𝑏𝑐|Φ𝑖𝑗𝑘
𝑎𝑏𝑐⟩
𝑖>𝑗>𝑘,𝑎>𝑏>𝑐
+ ⋯
(2.45)
Multireference configuration interaction (MR-CI)36 is built from the traditional CI
and MCSCF methods. A MCSCF calculation is first performed to optimize the MOs and
CSFs of the MCSCF wave function. Reference CSFs are created for the MR-CI wave
function by exciting electrons from occupied orbitals into virtual ones. In practice, the
excitation level for MR-CI calculations is truncated to only singles and doubles (MR-
CISD). The variational principle is again used to minimize the energy by varying the CSF
expansion coefficients. When considering computational cost, the savings are great when
compared to full CI (in that the calculations are at all feasible at modest system sizes), but
the cost still scales linearly with the number of CSFs.
Since MR-CISD is a truncated CI method, it is not size-extensive. This is not a
large issue as the amount of correlation energy lost is not great, but it becomes increasingly
important with increasing system size. The Davidson correction36, 39-40 (designated “+Q”)
is an a posteriori treatment that recovers some of the correlation energy lost because the
wave function neglects higher order terms. The expression for the energy correction, 𝐸𝑄,
is shown in Equation 2.46 where 𝑏02 is the sum of the square of the coefficients of the
reference CSFs in the MR-CISD expansion. It can be seen that this is only a correction to
the energy and does not improve any other properties calculated at the MR-CISD level, but
the improvement to the energy gives favorable agreement with full CI and comes at
substantially reduced computational cost.
𝐸𝑄 =(1 − 𝑏0
2)(𝐸𝑀𝑅−𝐶𝐼𝑆𝐷 − 𝐸𝑀𝐶𝑆𝐶𝐹)
𝑏02 (2.46)
MR-CI calculations are ideal for calculations of systems where there is a high
amount of resonance and aromaticity or in the case of radical systems. The multireference
character of the wave function provided by multiple configurations provides the flexibility
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34
to characterize and predict reactivity in these species so long as the computational cost, and
therefore the choice of reference space and basis set, is closely considered.
2.8 Multireference Averaged Quadratic Coupled-Cluster
The MCSCF and MR-CI methods account for static and dynamic electron
correlation, but the problem with size-extensivity in larger molecular systems remains even
with the Davidson correction to MR-CI. In an effort to remedy this, several other
multireference methods were developed such as the multireference linearized coupled-
cluster method (MR-LCCM), coupled electron pair approximation (MR-CEPA), and
averaged couple pair functional (MR-ACPF) that were shown to be approximately size-
extensive alternatives to MR-CI.41 The MR-AQCC method was developed to approach
nearly size-extensive results while simultaneously utilizing smaller reference spaces. The
following derivation follows what is presented in Ref.41
For a given reference space consisting of 𝑛𝑟 reference functions, Φ𝑟, the solution
to the Schrödinger equation is the zeroth-order reference, Ψ0 = ∑ 𝑐𝑟Φ𝑟𝑟 , and the correlated
wave function is found to be Equation 2.47. In this expression, 𝑇𝑟 is the cluster operator,
exp(𝑇𝑟) and 𝑃𝑟 = |Φ𝑟⟩⟨Φ𝑟| are found relative to each Φ𝑟 to define the cluster operator,
and Ψ𝑃 is the part of the wave function made up of the remaining orthogonal combinations
of reference functions in the set of all Φ𝑟 in the space of P. The singly (Ψ𝑖𝑎(𝑟)) and doubly
(Ψ𝑖𝑗𝑎𝑏) excited portion of the wave function is presented in Equation 2.48. With this, the
full correction, Ψ𝑐, to the zeroth-order wave function is Ψ𝑐 = Ψ𝑃 = Ψ𝑄.
Ψ = ∑ exp(𝑇𝑟) 𝑃𝑟Ψ0 + Ψ𝑃
𝑟
(2.47)
Ψ𝑄 = ∑ (∑ 𝑐𝑖𝑎Ψ𝑖
𝑎(𝑟)
𝑖𝑎
+ ∑ 𝑐𝑖𝑗𝑎𝑏Ψ𝑖𝑗
𝑎𝑏(𝑟)
𝑖𝑗𝑎𝑏
)
𝑟
(2.48)
The functional form of the expectation value for the energy of the single reference
case from the cluster operator, T, and the reference wave function, Φ0, is shown in Equation
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35
2.49. In this equation, 𝐸0 is the reference energy and the c indicates that only connected
terms are included. Exclusion principle violating (EPV) terms (terms do not come from
pure interactions, but from groupings of interactions) are included as well, so the
expression is considered infinite. Equation 2.49 is for true coupled-cluster which is
computationally expensive even without considering multireference, so approximations
are made to neglect the virtual EPV terms in the disconnected diagrams of the cluster
expansion. This is accomplished by writing the cluster expansion in its disconnected form
and substituting the MR-CISD correlation energy, ∆E𝑀𝑅−𝐶𝐼𝑆𝐷, into the coupled-cluster
energy functional, Equation 2.50. Doing this illustrates that the MR-AQCC method,
despite its name, is not a true coupled-cluster method, but is in actuality an advanced form
of MR-CISD.
𝐹(Δ𝐸) = ⟨𝑒𝑇Φ0|�� − 𝐸0|𝑒𝑇Φ0⟩𝑐 (2.49)
𝐹(Δ𝐸) = ∆E𝑀𝑅−𝐶𝐼𝑆𝐷 + ∑ 𝑐𝑖𝑗𝑎𝑏2
𝑖𝑗𝑎𝑏
⟨Ψ𝑖𝑗𝑎𝑏|Ψ𝑖𝑗
𝑎𝑏⟩ ∑⟨Ψ0|�� − 𝐸0|Ψ𝑘𝑙𝑐𝑑⟩
𝑘𝑙𝑐𝑑
c𝑘𝑙𝑐𝑑 (2.50)
By also neglecting the virtual EPV terms and considering the rest in only an average
manner, the second term of Equation 2.50 becomes that shown in Equation 2.51, where 𝑛𝑒
is the number of electrons. Finally, the functional form of the MR-AQCC correlation
energy can be found in Equation 2.52 by combining Equations 2.50 and 2.51 and by writing
∆E𝑀𝑅−𝐶𝐼𝑆𝐷 in a more explicit fashion.
⟨Ψ𝑐|Ψ𝑐⟩(𝑛𝑒 − 2)(𝑛𝑒 − 3)
𝑛𝑒(𝑛𝑒 − 1)∆E𝑀𝑅−𝐶𝐼𝑆𝐷 (2.51)
𝐹𝑀𝑅−𝐴𝑄𝐶𝐶(∆E) =⟨Ψ0 + Ψ𝑐|�� − 𝐸0|Ψ0 + Ψ𝑐⟩
1 + [1 −(𝑛𝑒 − 2)(𝑛𝑒 − 3)
𝑛𝑒(𝑛𝑒 − 1)] ⟨Ψ𝑐|Ψ𝑐⟩
(2.52)
This approach is still only nearly size-extensive since only the quadratic unlinked
EPV terms are included. The primary advantage of the MR-AQCC method is that it allows
for high-level calculations that do not have the computational cost of full MR-CI. In
addition, the Davidson correction to the MR-CI method only adds size-extensivity to the
total energy, whereas the size-extensivity correction with the MR-AQCC method is applied
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36
a priori, so meaningful electronic properties of the investigated system can be found such
as orbital occupations and spin-couplings among others.
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(16) Møller, C.; Plesset, M. S., Note on an Approximation Treatment for Many-Electron
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(17) Frisch, M. J.; Head-Gordon, M.; Pople, J. A., A direct MP2 gradient method.
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(19) Head-Gordon, M.; Pople, J. A.; Frisch, M. J., MP2 energy evaluation by direct
methods. Chemical Physics Letters 1988, 153 (6), 503-506.
(20) Sæbø, S.; Almlöf, J., Avoiding the integral storage bottleneck in LCAO
calculations of electron correlation. Chemical Physics Letters 1989, 154 (1), 83-89.
(21) Head-Gordon, M.; Head-Gordon, T., Analytic MP2 frequencies without fifth-order
storage. Theory and application to bifurcated hydrogen bonds in the water hexamer.
Chemical Physics Letters 1994, 220 (1), 122-128.
(22) Schirmer, J., Beyond the random-phase approximation: A new approximation
scheme for the polarization propagator. Physical Review A 1982, 26 (5), 2395-2416.
(23) Dreuw, A.; Wormit, M., The algebraic diagrammatic construction scheme for the
polarization propagator for the calculation of excited states. Wiley Interdisciplinary
Reviews: Computational Molecular Science 2015, 5 (1), 82-95.
(24) Bayin, S. S., Mathematical Methods in Science and Engineering. 2006.
(25) Fetter, A. L.; Walecka, J. D., Quantum theory of many-particle systems. McGraw-
Hill: New York, 1971.
(26) Linderberg, J.; Öhrn, Y., Propagators in quantum chemistry. Academic Press: New
York, 1973.
(27) Danovich, D., Green's function methods for calculating ionization potentials,
electron affinities, and excitation energies. Wiley Interdisciplinary Reviews:
Computational Molecular Science 2011, 1 (3), 377-387.
(28) Von Niessen, W.; Schirmer, J.; Cederbaum, L. S., Computational methods for the
one-particle green's function. Computer Physics Reports 1984, 1 (2), 57-125.
(29) Cederbaum, L. S., Green's Functions and Propagators for Chemistry. John Wiley
& Sons: Chichester, 1998.
(30) Jansen, H. B.; Ros, P., Non-empirical molecular orbital calculations on the
protonation of carbon monoxide. Chemical Physics Letters 1969, 3 (3), 140-143.
(31) Liu, B.; McLean, A. D., Accurate calculation of the attractive interaction of two
ground state helium atoms. The Journal of Chemical Physics 1973, 59 (8), 4557-
4558.
(32) Sherrill, C. D., Lecture Notes on Counterpoise Correction and Basis Set
Superposition Error (http://vergil.chemistry.gatech.edu/notes/cp.pdf). 2010.
(33) van Duijneveldt, F. B.; van Duijneveldt-van de Rijdt, J. G. C. M.; van Lenthe, J.
H., State of the Art in Counterpoise Theory. Chemical Reviews 1994, 94 (7), 1873-
1885.
(34) Boys, S. F.; Bernardi, F., The calculation of small molecular interactions by the
differences of separate total energies. Some procedures with reduced errors.
Molecular Physics 1970, 19 (4), 553-566.
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(35) Klamt, A.; Jonas, V.; Bürger, T.; Lohrenz, J. C. W., Refinement and
Parametrization of COSMO-RS. The Journal of Physical Chemistry A 1998, 102
(26), 5074-5085.
(36) Szalay, P. G.; Müller, T.; Gidofalvi, G.; Lischka, H.; Shepard, R.,
Multiconfiguration Self-Consistent Field and Multireference Configuration
Interaction Methods and Applications. Chemical Reviews 2012, 112 (1), 108-181.
(37) Roos, B. O., The Complete Active Space Self-Consistent Field Method and its
Applications in Electronic Structure Calculations. Advances in Chemical Physics
1987, 69, 399.
(38) Pulay, P.; Hamilton, T. P., UHF natural orbitals for defining and starting MC‐SCF
calculations. The Journal of Chemical Physics 1988, 88 (8), 4926-4933.
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nitrogen molecule. International Journal of Quantum Chemistry 1974, 8 (1), 61-
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(40) Luken, W. L., Unlinked cluster corrections for configuration interaction
calculations. Chemical Physics Letters 1978, 58 (3), 421-424.
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method: a size-extensive modification of multi-reference CI. Chemical Physics
Letters 1993, 214 (5), 481-488.
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39
CHAPTER III
SINGLE AND DOUBLE CARBON VACANCIES IN PYRENE AS
FIRST MODELS FOR GRAPHENE DEFECTS: A SURVEY OF THE
CHEMICAL REACTIVITY TOWARD HYDROGEN
Authors: Reed Nieman,a Anita Das,a Adelia J. A. Aquino,a,b Rodrigo G. Amorim,c,d
Francisco B. C. Machado,d and Hans Lischkaa,b,e
Affiliations: aDepartment of Chemistry and Biochemistry, Texas Tech University,
Lubbock, TX 79409-1061, USA, bSchool of Pharmaceutical Sciences and Technology,
Tianjin University, Tianjin 300072, PR China, cDepartamento de Física, Universidade
Federal Fluminense, Volta Redonda, 27213-145, Rio de Janeiro, Brazil, dDepartamento de
Química, Instituto Tecnológico de Aeronáutica, São José dos Campos 12228-900, São
Paulo, Brazil, eInstitute for Theoretical Chemistry, University of Vienna, Währingerstrasse
17, A-1090 Vienna, Austria
Published: January 2017 in Chemical Physics, DOI:10.1016/j.chemphys.2016.08.007
3.1 Abstract
Graphene is regarded as one of the most promising materials for nanoelectronics
applications. Defects play an important role in modulating its electronic properties and also
enhance its chemical reactivity. In this work the reactivity of single vacancies (SV) and
double vacancies (DV) in reaction with a hydrogen atom Hr is studied. Because of the
complicated open shell electronic structures of these defects due to dangling bonds,
multireference configuration interaction (MR-CI) methods are being used in combination
with a previously developed defect model based on pyrene. Comparison of the stability of
products derived from C-Hr bond formation with different carbon atoms of the different
polyaromatic hydrocarbons is made. In the single vacancy case the most stable structure is
the one where the incoming hydrogen is bound to the carbon atom carrying the dangling
bond. However, stable C-Hr bonded structures are also observed in the five-membered ring
of the single vacancy. In the double vacancy, most stable bonding of the reactant Hr atom
is found in the five-membered rings. In total, C-Hr bonds, corresponding to local energy
minimum structures, are formed with all carbon atoms in the different defect systems and
the pyrene itself. Reaction profiles for the four lowest electronic states show in the case of
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a single vacancy a complex picture of curve crossings and avoided crossings which will
give rise to a complex nonadiabatic reaction dynamics involving several electronic states.
3.2 Introduction
Graphene has many extremely appealing electrical, physical, and thermal
properties which have made it an exciting material that is the subject of intensive research
ever since it was first isolated.1 Graphene is a particular allotrope of carbon in which a two-
dimensional hexagonal lattice is composed of benzene rings. Though it has been suggested
that graphene’s ability to conduct electricity could allow it to replace silicon in next-
generation organic semi-conductors, two problems remain, however, which detract from
this potential. The first issue is that there are no effective methods for obtaining amounts
of pristine graphene sheets of one atom thickness necessary for industrial applications. The
second problem is that graphene’s electronic properties come with a down side, though it
conducts electricity well, it is a semi-metal with a zero-energy band gap.
The chemical functionalization of graphene sheets for the purpose of tuning its band
gap has shown great promise.2-3 The hydrogen atom is a natural starting point when
beginning the investigation of the ability of graphene sheets to form sp3 bonds between an
incoming atom or molecule and the bonding carbon atom. Hydrogen chemisorption on a
graphene surface has been studied in detail by Casolo et al.4 using density functional theory
(DFT) and periodic boundary conditions. In fact, hydrogenation has been shown to be an
efficient tool for band gap tuning of graphene.5-8 Formation of the sp3 bond disrupts the
existing sp2 bond network by breaking the local π system. This has the effect of not only
increasing the reactivity of the carbon atoms neighboring the new C-H bond, but it also
creates a potential barrier to the conducting electrons opening a band gap, which is tunable
depending on the concentration of the chemisorbed species and properties of the sp3
network that has been formed. By reacting graphene sheets with atomic hydrogen,
hydrogenated regions of graphene are formed which are referred to as graphane islands.
The presence and extent of hydrogenation has been previously confirmed through the use
of scanning tunneling microscopy (STM) and angle-resolved transmission electron
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microscopy (TEM).6 These images also show details pertaining to the arrangement of
graphane islands with respect to each other. PAHs and their reaction with hydrogen are not
only of interest as molecular models for graphene but also as possible catalysts for H2
formation in interstellar space.9 Hydrogenation studies of pyrene have been performed10 to
investigate energetic stabilities of hydrogen attached to different carbon atoms and the
existence of energy barriers.
The arrangement of bonded hydrogen atoms in graphane islands and the proximity
of islands to one another are also contributing factors to the overall electronic character of
the hydrogenated graphene sheet due the dependence of the created band gap on such. A
small group of low hydrogen-concentration graphane islands have shown clearly the
beginning of the formation of a band gap.6 On a larger scale, graphane islands created in
strips parallel to each other, called graphene roads, have been shown to create environments
similar to graphene nanoribbons.11 Hydrogen atoms adsorbed on graphene have also been
reported to induce magnetic moments.12 The spin-polarized state is shown to be essentially
located on the carbon sublattice opposite to the one belonging to the adsorbed hydrogen
atom. This finding goes very well along with previous success in experimentally
engineering band gaps and has been shown by the Haddon group using substituted benzene
radicals, nitrophenylene, interacting with graphene sheets.3 As a consequence of this
functionalization, residual spins are available to couple and allow the possibility of
ferromagnetism at room temperatures.
Vacancies defects, absences of carbon atoms in the hexagonal structure, have
shown to be an interesting area with regards to the investigation of the chemical reactivity
of graphene. When one or multiple carbons are removed, the created dangling bonds
represent areas of greater chemical reactivity. The presence of these dangling bonds then
implies high polyradical character with many closely spaced locally excited electronic
states which are difficult to describe computationally. This situation then strongly advises
the use of flexible multireference theory13 in which quasi-degenerate orbitals are treated at
equal footing and the correct construction of wavefunctions of well-defined symmetry and
spin properties is possible.
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The electronic structures of both SV and DV pyrene (Scheme 3.1) which has been
used as a prototype for a graphene defect structure, have recently been described using
mulireference configuration interaction MR-CI14-15 and DFT.16 The single vacancy
investigation of pyrene using MR-CI calculations14 showed several closely spaced
electronic states of triplet and singlet multiplicity (Scheme 3.1, Pyrene-1C). The electronic
ground state of the double vacancy defect15 (Scheme 3.1, pyrene-2C) possesses 50% closed
shell character which indicates a significantly different character as compared to the SV
case. The pristine pyrene is depicted in Scheme 3.1, top, and as is, of course, to be expected
the chemically most stable structure of the three candidates discussed.
The purpose of this work is to explore the chemical reactivity of pyrene and the two
defect structures by studying their interaction with hydrogen radical (Hr). The role of Hr is
two-fold. It is used in the spirit of a test particle by moving it in a grid search across the
different pyrene structures as the arrows in Scheme 3.1 indicate. In this way a stability
surface with respect to C-Hr bond formation is constructed. In the second step
optimizations of C-Hr structures are performed and C-Hr dissociation curves are
constructed to the lowest dissociation products. The previous MR-CI calculations on the
SV and DV pyrene defect structures had shown the existence of several low-lying excited
states. We wanted to reproduce this asymptotic limit at complete C-H dissociation and,
thus, computed in all respective cases potential energy scans and C-H dissociation curves
for the four lowest electronic states. In addition to the determination of local energy minima
for different C-H bonding situations, these data should provide an overview of the most
important aspects for the approach and bonding of a hydrogen atom to a pristine or defect
graphene sheet surface.
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Scheme 3.1 Atomic numbering scheme for pristine pyrene, pyrene-1C (single vacancy)
and pyrene-2C (double vacancy), respectively. Hr is the hydrogen atom moving across the
different pyrene structures as shown by arrows.
Pyrene
Pyrene-1C
Pyrene-2C
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3.3 Computational Details
The geometries of each pyrene structure (Scheme 1) were optimized using complete active
space (CAS) self-consistent field (CASSCF) theory.17-18 Based on the experience with
previous pyrene defect calculations14-15 a CAS(5,5) was used comprised of five electrons
in five orbitals where, in Cs symmetry, three orbitals belonged to the A’ irreducible
representation and two orbitals belonged to the A” irreducible representation. The
CAS(4,4) is chosen for the description of the isolated pyrene defect and one active orbital
and electron is added to account for the hydrogen atom. The active orbitals used in the CAS
for pyrene+Hr were 31a’, 32a’, 33a’, 22a”, 23a” (Figure A.1 of Appendix A). For pyrene-
1C+Hr active orbitals were 29a’, 30a’, 31a’, 21a”, 22a” (Figure A.2) and for pyrene-2C+Hr
active orbitals were 27a’, 28a’, 29a’, 20a”, 21a” (Figure A.3). The 6-31G* basis set19 was
used. The state-averaging included four states (SA4). In case of Cs symmetry the two
lowest 2A’ and two lowest 2A” states were chosen. Single point calculations were carried
out using SA4-CASSCF(5,5)/6-31G* and MR-CI with all singles and doubles (MR-CISD)
again using the 6-31G* basis set. As reference space, the same CAS(5,5) as used in the
CASSCF(5,5) calculations was selected. Single and double excitations were constructed
from the doubly-occupied and active orbitals applying the interactive space restriction.20
The 1s orbitals of all carbon atoms were kept frozen. Size-extensivity corrections were
included by means of the extended Davidson correction13, 21-22 denoted as +Q (MR-
CISD+Q). 2
0C is the sum of square coefficients of the reference configurations in the MR-
CISD expansion and EREF denotes the reference energy.
12
12
0
2
0
C
EECE REFCI
Q (3.1)
The potential energy surface (PES) scan was carried out using the SA4-
CASSCF(5,5)/6-31G* approach for pyrene-1C+Hr and pyrene-2C+Hr and the
CASSCF(5,5)/6-31G* approach for pyrene+Hr. For that purpose, the previously relaxed
pyrene defect (pyrene-1C and -2C) structures computed earlier14-15 and an optimized
pyrene structure using the DFT and the hybrid Becke, 3-parameter, Lee-Yang-Parr
(B3LYP) functional23 together with the 6-31G**19 basis set were used to move Hr around
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in a grid with an increment of 0.3 Å and at a distance of 1.0 Å from the plane structures.
Single-point calculations were performed for each point on the grid.
Using the local minima in the PES´s scans as starting structures, geometry
optimizations were carried out. To simulate the constraints present in a larger graphene
system, the coordinates of the carbons connected to hydrogen atoms at the periphery of the
pyrene and pyrene defect systems were always frozen in all geometry optimizations since
they would be connected to neighboring carbon atoms in a graphene sheet. All other atoms
were allowed to relax. Geometry optimizations using SA4-CASSCF(5,5)/6-31G* for the
defect cases plus Hr and CASSCF(5,5) calculations for the pristine pyrene+Hr were
performed by placing Hr at an initial distance of 1.1 Å over each unique carbon in each of
the pyrene system.
The CASSCF, MR-CISD and MR-CISD+Q calculations were carried out using
COLUMBUS;24-28 the geometry optimizations used the program system DL-FIND29
interfaced to COLUMBUS. The DFT calculations were performed with the Turbomole
suite of programs.30
3.4 Results and Discussion
3.4.1 Potential Energy Scans
The two-dimensional PES scans of the single vacancy pyrene-1C+Hr for the ground
state and the first three excited states are shown in Figure 3.1. For that purpose, the Hr atom
has been moved systematically in a grid 1.0 Å above the fixed, planar structure of pyrene-
1C. These scans detail the ability of the test particle, Hr, to bond in a particular region of
pyrene-1C. For the ground state, deep local minima can be seen at the carbons in the five-
membered ring, C1, C2, C3, C13 and C14 (see Scheme 1 for numbering of atoms). The
lowest minimum at C15 persists through all four states; for the excited states the local
minimum at the five-membered ring is considerably higher in energy. Shallower minima
are present for C5, C7, C9, and C11. Generally, the minima for carbons in the six-
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membered ring, aside from C15, give minima much shallower than for the remaining
carbon atoms.
Figure 3.1 PES scans in terms of the location of Hr for the four lowest states of pyrene-
1C+Hr using the SA4-CASSCF(5,5)/6-31G* approach. The atom Hr is located in a plane
1 2Aa
2 2Ab
3 2Ac
4 2Ad
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parallel and 1 Å above the plane defined by the frozen carbon atoms. Lowest energies: a -
574.222889 Hartree, b -574.198174 Hartree, c -574.149530 Hartree, d -574.117820 Hartree.
The energy (color scale) is given in eV, relative to the ground state lowest minimum.
3-D pictures of the PES scan of the double vacancy defect pyrene-2C+Hr for the
electronic ground- and first excited states are displayed in Figure 3.2 and Figure 3.3,
respectively. The plots for the second and third excited state are shown in Figure A.4. For
the ground state deep potential wells are seen at C1, C2, C14 and C7, C8, C9, these being
the carbons at the outer sections of the five-membered rings, with the strongest minima
present at C1 and C8. Additional shallower minima are seen at each of the remaining
carbons. The ground and first excitation are seen to be on average energetically closer as
compared to the single vacancy defect just discussed above. On the other hand, the scans
for the second and third vertical excitations (Figure A.4) are closely spaced and quite
higher, by ~ 2-2.5 eV, than the lowest minimum.
Figure 3.2 PES scan in terms of the location of Hr for the lowest state of pyrene-2C+ Hr
using the SA4-CASSCF(5,5)/6-31G* approach. The atom Hr is located in a plane parallel
and 1 Å above the plane defined by the frozen carbon atoms. Lowest energy = -536.385046
Hartree. The energy (color scale) is given in eV, relative to the ground state lowest
minimum.
The ground state PES for pyrene+Hr is shown in Figure 3.4. Pronounced minima
can be seen at C4, C5, C11, and C12. While for each of the remaining carbon atoms in the
pyrene system local minima can be found, C2, C7, C9, and C14 show to be especially
stabilized among them.
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Figure 3.3 PES scan in terms of the location of Hr for the 22A state of pyrene-2C+ Hr using
the SA4-CASSCF(5,5)/6-31G* approach. The atom Hr is located in a plane parallel and 1
Å above the plane defined by the frozen carbon atoms. Lowest energy = -536.347629
Hartree. The energy (color scale) is given in eV, relative to the ground state lowest
minimum at -536.385046 Hartree.
Figure 3.4 PES scan of the lowest state of pyrene+Hr using the CASSCF(5,5)/6-31G*
approach. The atom Hr is located in a plane parallel and 1 Å above the plane defined by
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the frozen carbon atoms. Lowest energy = -612.270253 Hartree. The energy (color scale)
is given in eV, relative to the ground state lowest minimum.
3.4.2 Geometry Optimizations
The PES scans discussed in the previous section give a good global overview of the
chemical reactivity of the different pyrene defects (including the pristine case) concerning
the strength of different C-Hr bonds as well as to the location of the centers of most stable
bond formation. In the following sections more detailed information on the properties of
individual structures will be discussed.
In the search for local minima structures containing C-Hr bonds, geometry
optimizations were performed as described in the Computational Details by placing Hr on
top of each carbon atom and performing a geometry optimization in each case only freezing
selected carbon atoms as discussed above to simulate geometrical restrictions due to
neighboring carbon atoms in a graphene sheet. Each carbon tested was found to give rise
to a local minimum for all structures investigated.
Starting with the single vacancy pyrene-1C+Hr case, the nine ground state
structures found are described in Figure 3.5. The global minimum is given by bonding Hr
to C15 with a C-Hr bond distance of 1.060 Å. This is the smallest C-H bond distance
determined for the nine structures and is much smaller than the C-Hr bond distance of the
other minima whose bond distance is around 1.095 Å. When Hr is bonded to C1, C2, C4,
and C5, planarity of the ring system is mostly maintained. However, the existing hydrogen
is moved out-of-plane away from the incoming Hr due to the formation of a tetrahedral sp3
hybrid at this carbon atom. This behavior repeated in all analogous cases. When Hr is
bonded to C3 and C6, the carbon atoms reacted by rising out of plane. When Hr was bonded
to C7 and C8, the bonding carbon could not move since it was frozen, but C6, C10, and
C15 moved out of plane. Finally, when Hr bonded to C15 in the global minimum structure,
again C6, C10, and C15 moved out of plane.
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Figure 3.5 Plots of the different minimum structures (labeled according to the bonding
carbon atom) on the ground state energy surface of pyrene-1C+Hr using the SA4-
CASSCF(5,5)/6-31G* approach. Bond distances are given in Å.
In the double vacancy pyrene-2C+Hr case, the four ground state structures are
characterized in Figure 3.6. The global minimum was found when Hr is bonded to C2 in
the five-membered ring with a C-Hr distance of 1.093 Å and gave the smallest C-Hr
distance of all the minimum structures. The largest C-Hr bond distance of 1.121 Å to C1 is
C1 C2
C3 C4
C5 C6
C7 C8
C15
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notable here as it is the largest one of all three pyrene structures. For C3, as before, since
it was not frozen and only bonded to other carbons, the carbon atom rose out of the plane
of the frozen carbon atoms when rehybridizing from sp2 to sp3.
Figure 3.6 Plots of the different minimum structures (labeled according to the bonding
carbon atom) of the ground state energy surface of pyrene-2C+Hr using the SA4-
CASSCF(5,5)/6-31G* approach. Bond distances are given in Å.
Finally, for pyrene+Hr, the characterization of the five ground states is presented in
Figure 3.7. The global minimum was found when Hr was bonded to C4 with a C-Hr bond
distance of 1.097 Å. This C-Hr distance was the smallest one of the four minima found,
though it can be noted that the C-Hr bond distances for this structure do not vary greatly as
the largest C-Hr bond distance is only slightly larger with 1.103 Å given for C1-Hr. In cases
where the bonding carbon was already bonded to hydrogen (carbons 1, 2, 11), Hr formed a
C1
C2
C3
C4
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C-H bond by pushing the other hydrogen away, as reported also in previous cases. When
Hr bonded to a hydrogen-free carbon (carbons 3 and 16), the bonding carbon responded by
rising out of the plane of the frozen carbon atoms by about 0.5 Å. A similar observation
was made by Rasmussen et al.10 by means of DFT calculations where binding of a hydrogen
atom on the interior positions (C3 and C16) of the pyrene led to a significant puckering of
these carbon atoms (~0.4 Å) out of the molecular plane. This behavior on hydrogenation
of graphene is generally feasible in the free graphene sheet4 but is also true for the pyrene
defect plus Hr cases as explained above
3.4.3 Energetic Stabilities
Table 3.1 shows the vertical excitation energies for pyrene-1C+Hr the lowest two
A’ and A” states for structures having Cs symmetry, i.e. Hr bonded to C15 (global
minimum), C1 and C8 using the SA4-CASSCF, MR-CISD and MR-CISD+Q approaches.
The 12A’ state is the ground state. At MR-CISD+Q level, in the C15 structure the first
excited state is only ~0.8 eV above the ground state, followed by states about 2.4 eV and
3.6 eV above the ground state. The ground state C1 structure is about 1.4 eV higher at MR-
CISD+Q level than the respective one for C15; the C8 structure is located significantly
higher (4.2 eV) above the C15 ground state. The order of the states computed at MR-CISD
and MR-CISD+Q levels is reproduced well by the SA4-CASSCF(5,5) calculations even
though there are non-negligible numerical differences up to 0.8 eV.
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Figure 3.7 Plots of the different minimum structures (labeled according to the bonding
carbon atom) on the ground state energy surface of pyrene+Hr using the CASSCF(5,5)/6-
31G* approach. Bond distances are given in Å.
C1
C2
C3
C4
C16
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Table 3.1 Vertical energy differences (eV) of the lowest two A’ and A” states relative to
the C15 ground state computed for the C1, C8 and C15 pyrene-1C+Hr structures using the
SA4-CASSCF(5,5), MR-CISD and MR-CISD+Q approaches and the 6-31G* basis set.
CASSCF MR-CISD MR-CISD+Q
State ΔErel ΔErel ΔErel
C15
1 2A’ 0.000a,d 0.000b,d 0.000c,d
1 2A” 1.154 1.023 0.834
2 2A’ 2.571 2.504 2.375
2 2A” 4.100 3.904 3.594
C1
1 2A’ 0.636d 0.960d 1.393d
2 2A’ 3.775 4.073 4.444
1 2A” 4.828 4.973 5.074
2 2A” 5.854 6.069 6.186
C8
1 2A’ 3.542d 3.930d 4.246d
2 2A’ 3.900 4.214 4.456
1 2A” 4.216 4.604 4.947
2 2A” 4.643 4.994 5.187
a Total energy= -574.277274 Hartree, b Total energy= -575.610208 Hartree, cTotal
Energy= -576.244081 Hartree, d The SA4-CASSCF(5,5) optimized geometry was used.
Figure 3.8 Location and relative stability (eV) of the minima on the ground state energy
surface of pyrene-1C+Hr using the MR-CISD+Q/6-31G* method. Blue: 2A ground state,
red: lowest vertical excitation. Note: the rigid pyrene-1C structure shown does not reflect
the actual structure of the optimized minima.
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In Figure 3.8 (see also Table A.1) are the relative energies of each carbon minimum
in pyrene-1C+Hr given with respect to the global minimum structure at C15 computed at
MR-CISD+Q level. The first vertical excitations are given in the figure as well. C15 marks
the global minimum followed by C2, C3, and C1, respectively, in the five-membered ring.
Except for the structure C15, all first excited states are located in energy higher by ~3.5 eV
or more. This shows that in case of the reaction of Hr with a single vacancy graphene defect
at thermalized kinetic energies nonadiabatic interactions between different electronic states
will be relevant mostly for the C15 approach, which, is however, energetically the most
relevant one.
The energy profiles for the reaction of Hr with the pyrene-1C defect are shown in
Figure 3.9 for the case of bond formation with C15. The reaction curves have been created
by fixing the C15-Hr distance and optimizing the remaining part of the pyrene defect
structure for the ground state at SA4-CASSCF(5,5)/6-31G* level. Selected carbon atoms
as discussed above were frozen to simulate also in this case geometrical restrictions due to
neighboring carbon atoms in a graphene sheet. Vertical excitations have been used to
compute the higher electronic states. These calculations have been performed in C1
symmetry. Nevertheless, the Cs symmetry analysis of Table 3.1 for the ground state
minimum, as indicated in Figure 3.9, can be used for the analysis of states. For the 12A’
ground state the figure shows a smooth dissociation without change of the character of the
wave function. The second state is 12A” at the energy minimum. A crossing between this
state and the 22A’ state is observed at a C15-Hr distance of about 2 Å. For large distances
the two 2A’states become practically degenerate. The 22A’ and 22A” states show a spike at
1.5 Å indicating the crossing with higher states of different character. These states are
repulsive and are responsible for the correct asymptotic behavior of the dissociation. The
energy minima of the first three states are located below the lowest dissociation limit so
that on reaction with an incoming hydrogen atom several electronic states could be
involved in the relaxation process to the ground state.
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Figure 3.9 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is
bonded to C15 using the MR-CISD/6-31G* approach. Minimum energy = -574.277274
Hartree
Figure 3.10 shows the dissociation/bonding process of Hr from/to C1. Evident is
the lack of a potential barrier in the ground state curve, though potential energy barriers
and avoided crossings exist in the excited state curves. However, differently to the case of
the C15-Hr bond dissociation, the excited states are not expected to play a major role for
the relaxation process to the energy minimum on formation of the C1-Hr bond due to their
high energies around the ground state minimum.
Figure 3.10 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is
bonded to C1 using the MR-CISD/6-31G* approach. Minimum energy = -574.277274
Hartree
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The Hr dissociation curves computed at SA4-CASSCF(5,5) level for bonding to C2
to C8, respectively, are collected in the SI. For C2 and C3, Figure A.5 and Figure A.6, there
is no potential barrier for the incoming Hr ground state. In the remaining cases (Figure A.7
– Figure A.11), however, there are potential barriers present for the bond formation to C4,
C5, C6, C7 and C8. Estimated barrier heights with respect to the dissociation limit are
collected in Table 3.2. They show a variation between 0.4 eV to 1.4 eV indicating that the
approach from that side of pyrene-1C is not so favorable.
The dissociation energy of the C15 global minimum pyrene-1C + Hr structure into
the lowest state of pyrene-1C + Hr is given in Table 3.2. At MR-CISD+Q level a value of
4.61 eV is found. Comparison with the MR-CISD (4.13 eV) and SA4-CASSCF(5,5) (3.66
eV) values shows the importance of including size-extensivity and dynamic electron
correlation contributions to the dissociation energy. Dissociation of Hr for the other
structures can be obtained from Figure 3.8 by subtracting the relative stabilities form the
just-mentioned dissociation limit of the C15 structure. The strongest C-Hr bonds following
the one to C15 can be found for C-H bonding to carbons in the five-membered ring. Some
of the other structures, though being local minima, are close in energy to the C-Hr bond
dissociation limit.
Table 3.2 Dissociation energies Ediss (eV) for the C15 global minimum pyrene-1C+Hr
structure and estimated energy barriers Ebarr(est.) (eV) for the approach of Hr on the
ground state surface toward the different carbon atoms of pyrene-1C using different
theoretical methods and the 6-31G* basis set.
Ediss
SA4-CASSCF(5,5) MR-CISD MR-CISD+Q
3.660 4.128 4.605
Ebarr(est.)a (SA4-CASSCF(5,5))
C1, C2, C3, C15: no barrier
C4 C5 C6
0.914 0.426 1.403
C7 C8
1.138 1.051
The relative stabilities in the electronic ground state and the lowest vertical
excitations of pyrene-2C+Hr are presented in Figure 3.11 for the MR-CISD+Q/6-31G*
approach. SA4-CASSCF(5,5) and MR-CISD results are collected in Table A.2. Numerical
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agreement between the different methods is within 0.1 – 0. 2 eV. The global minimum was
found to be for Hr bonded to C2 in the five-membered ring. The minima are ordered from
most stable to least stable as C2<C1<C3<C4. The variation in the relative energies is much
smaller than observed for the pyrene-1C+Hr case (Figure 3.9).
Figure 3.11 Location and relative stability (eV) of the minima on the ground state energy
surface of pyrene-2C+Hr defect using the MR-CISD+Q/6-31G* method. Blue: 2A ground
state, red: lowest vertical excitation. Note: the rigid pyrene-2C structure shown does not
reflect the actual structure of the optimized minima.
The dissociation energy for the process pyrene-2C+Hr is given in Table 3.3 for the
global minimum structure with Hr bound to C2 for the CASSCF, MR-CISD and MR-
CISD+Q methods. Compared to the pyrene-1C+Hr case (Table 3.2), the C-Hr bond strength
is significantly reduced. However, it should be noted that all local ground state minima
(Figure 3.11) are below the dissociation limit of 2.269 eV.
Table 3.3 Dissociation energies Ediss (eV) for the double vacancy pyrene-2C+Hr and
pristine pyrene + Hr cases using different methods and the 6-31G* basis set.
SA4-CASSCF(5,5) MR-CISD MR-CISD+Q
pyrene-2C+Hr
Ediss(C2) 2.224 2.257 2.269
pyrene+Hra
Ediss(C2) 0.833 0.902 0.943
For the reaction of pyrene with Hr, pyrene+Hr, only the electronic ground state has
been investigated. Relative stabilities of the different C-Hr bonded local minima are shown
in Figure 3.12. CASSCF(5,5), MR-CISD and MR-CISD+Q results are collected in Table
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A.3. A similar good agreement between different methods is obtained as for the pyrene-
2C+Hr case discussed above in Table A.2. The minima are ordered energetically as
C2<C4<C3<C1<C16. Though the structure where Hr bound to C2 is most stable than C4,
the difference is negligibly small, which is in good agreement with literature.10 The
stabilization energy of the various minima was found to be quite close to each other, with
all five minima being found within 0.7 eV of the global minimum. With the dissociation
energy of 0.943 eV computed at MR-CISD+Q/6-31G* level (Table 3.3) all of the minima
are located below the dissociation limit, even though several cases are very close to it. A
similar value has been found by Casolo et al.4 in their hydrogenation studies of graphene
performed at DFT level.
Figure 3.12 Location and relative stability (eV) of the minima on the ground state energy
surface of pyrene+Hr using the MR-CISD+Q/6-31G* method. Note: the rigid pyrene
structure shown does not reflect the actual structure of the optimized minima.
3.5 Conclusions
The chemical reactivity of a single and double vacancy in graphene and of a pristine
surface toward C-H bond formation with a reactive hydrogen atom Hr has been investigated
using a pyrene model with peripheral carbon atoms connected to H atoms frozen.
Multireference methods (MR-CISD and MR-CISD+Q) have been used to describe (i) the
dissociation processes properly and (ii) the manifold of quasi-degenerate states of the
isolated defect.14-15 In a first step energy grids have been computed by moving an atom Hr
across the rigid surface of the pyrene defects structures in order to obtain an overview of
the stabilization energies in the different regions in space. The results show for the single
vacancy a preference for bonding to C15 which possessing a dangling bond. The relative
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stabilization in this position is also observed in the lowest excited states. However, the
carbon atoms in the five-membered ring show also strongly attractive positions in the
ground state. The double vacancy displays similar stability preferences in the five
membered rings; also the pristine pyrene model indicates attractive interactions primarily
around the peripheral carbon atoms.
Geometry optimizations performed for the ground state at CASSCF level followed
by single-point MR-CISD and MR-CISD+Q calculations refine this picture. Importantly,
separate local energy minima for C-Hr bond formation have been determined for each of
the carbon atoms in all defect structures and in pyrene itself. In the single vacancy case,
C15 remains the position with greatest stability followed by C2 towards the top of the five-
membered ring. In case of the double vacancy position C2 in the five-membered ring gives
rise to the strongest C-Hr bond and in pyrene itself these are positions C2 and C4.
C-Hr dissociation curves have been computed for the four lowest electronic states
for the single vacancy model starting from all minima found. Barrierless ground state
processes have been found for C15 and the C-Hr bonds involving the five-membered ring.
An incoming hydrogen atom though approaching the pyrene-C defect toward the other
carbon atoms would encounter barriers of about 0.5 – 1 eV, depending on the incoming
approach angle. Curve crossings and avoided crossings are found especially for Hr
approaches to the most stable carbon positions which lead to the expectation of a complex
nonadiabatic reaction dynamics. In summary, a remarkably complex picture of the bonding
options of a hydrogen atom reacting with a graphene vacancy structures is observed both
in terms of different bonding positions within the defect but also in terms of the reaction
dynamics involving several excited states. It is expected that these properties found for the
pyrene model are fairly representative for a localized graphene defect. Preliminary density
functional studies confirm this view. More detailed investigations are in preparation.
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3.6 Acknowledgments
This material is based upon work supported by the National Science Foundation under
Project No. CHE-1213263 and by the Austrian Science Fund (SFB F41, ViCoM). We are
grateful for computer time at the Vienna Scientific Cluster (VSC), project 70376. FBCM
wishes to thank Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) under
Projects Process Nos. 2014/24155-6 and 2015/500118-9, to Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior (CAPES) under Project CAPES/ITA
005/2014 and to Conselho Nacional de Desenvolvimento Científico and Tecnológico
(CNPq) for the research fellowship under Process No. 304914/2013-4. We also want to
thank the FAPESP/Texas Tech University SPRINT program (project no. 2015/50018-9)
for travel support.
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F.; Hammer, B.; Pedersen, T. G.; Hofmann, P.; Hornekaer, L., Bandgap opening in
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Multiconfiguration Self-Consistent Field and Multireference Configuration
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Electronic States Generated by a Single Carbon Defect in a Graphene Sheet:
Multireference Calculations Using a Pyrene Defect Model. ChemPhysChem 2014,
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(16) Haldar, S.; Amorim, R. G.; Sanyal, B.; Scheicher, R. H.; Rocha, A. R., Energetic
stability, STM fingerprints and electronic transport properties of defects in
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combined use of natural orbitals and the brillouin-levy-berthier theorem.
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(CASSCF) using a density matrix formulated super-CI approach. Chemical Physics
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(19) Hehre, W. J.; Ditchfield, R.; Pople, J. A., Self—Consistent Molecular Orbital
Methods. XII. Further Extensions of Gaussian—Type Basis Sets for Use in
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1972, 56 (5), 2257-2261.
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(20) Bunge, A., Electronic Wavefunctions for Atoms. III. Partition of Degenerate
Spaces and Ground State of C. The Journal of Chemical Physics 1970, 53 (1), 20-
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P. t. G.; Seth, M.; Kedziora, G. S.; Yabushita, S.; Zhang, Z., High-level
multireference methods in the quantum-chemistry program system COLUMBUS:
Analytic MR-CISD and MR-AQCC gradients and MR-AQCC-LRT for excited
states, GUGA spin–orbit CI and parallel CI density. Physical Chemistry Chemical
Physics 2001, 3, 664-673.
(25) Lischka, H.; Müller, T.; Szalay, P. G.; Shavitt, I.; Pitzer, R. M.; Shepard, R.,
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parallel multireference configuration interaction program: The parallel
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B. Brown, R. Ahlrichs, H. J. Böhm, A. Chang, D. C. Comeau, R. Gdanitz, H.
Dachsel, C. Ehrhardt, M. Ernzerhof, P. Höchtl, S. Irle, G. Kedziora, T. Kovar, V.
Parasuk, M. J. M. Pepper, P. Scharf, H. Schiffer, M. Schindler, M. Schüler, M.
Seth, E. A. Stahlberg, J.-G. Zhao, S. Yabushita, Z. Zhang, M. Barbatti, S. Matsika,
M. Schuurmann, D. R. Yarkony, S. R. Brozell, E. V. Beck, and J.-P. Blaudeau, M.
Ruckenbauer, B. Sellner, F. Plasser, and J. J. Szymczak COLUMBUS 7.0.
(29) Kästner, J.; Carr, J. M.; Keal, T. W.; Thiel, W.; Wander, A.; Sherwood, P., DL-
FIND: an open-source geometry optimizer for atomistic simulations. The journal
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calculations on workstation computers: The program system turbomole. Chemical
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CHAPTER VI
STRUCTURE AND ELECTRONIC STATES OF A GRAPHENE
DOUBLE VACANCY WITH EMBEDDED SI DOPANT
Authors: Reed Nieman,a Adelia J. A. Aquino,a,b Tevor P. Hardcastle,c,d Jani Kotakoski,e
Toma Susi,e and Hans Lischkaa,b
Affiliations: aDepartment of Chemistry and Biochemistry, Texas Tech University,
Lubbock, TX 79409-1061, USA, bSchool of Pharmaceutical Sciences and Technology,
Tianjin University, Tianjin 300072, PR China, cSuperSTEM Laboratory, STFC Daresbury
Campus, Daresbury WA4 4AD, United Kingdom, dFaculty of Engineering, School of
Chemical and Process Engineering, University of Leeds, 211 Clarendon Rd., Leeds LS2
9JT, United Kingdom, eFaculty of Physics, University of Vienna, Boltzmanngasse 5, A-
1090 Vienna, Austria
Published: October 2017 in The Journal of Chemical Physics, DOI:10.1063/1.4999779
4.1 Abstract
Silicon represents a common intrinsic impurity in graphene, commonly bonding to
either three or four carbon neighbors respectively in a single or double carbon vacancy. We
investigate the effect of the latter defect (Si-C4) on the structural and electronic properties
of graphene using density functional theory (DFT). Calculations based both on molecular
models and with periodic boundary conditions have been performed. The two-carbon
vacancy was constructed from pyrene (pyrene-2C) which was then expanded to
circumpyrene-2C. The structural characterization of these cases revealed that the ground
state is slightly non-planar, with the bonding carbons displaced from the plane by up to
±0.2 Å. This non-planar structure was confirmed by embedding the defect into a 10×8
supercell of graphene, resulting in 0.22 eV lower energy than the previously considered
planar structure. Natural bond orbital (NBO) analysis showed sp3 hybridization at the
silicon atom for the non-planar structure and sp2d hybridization for the planar structure.
Atomically resolved electron energy loss spectroscopy (EELS) and corresponding
spectrum simulations provide a mixed picture: a flat structure provides a slightly better
overall spectrum match, but a small observed pre-peak is only present in the corrugated
simulation. Considering the small energy barrier between the two equivalent corrugated
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conformations, both structures could plausibly exist as a superposition over the
experimental timescale of seconds.
4.2 Introduction
The modification of the physical properties of graphene sheets and nanoribbons, in
particular the introduction of a band gap via chemical adsorption, carbon vacancies, and
the incorporation of dopant species has gained a great deal of attention.1-2 Silicon is an
interesting element as a dopant as it is isovalent to carbon. It is also a common intrinsic
impurity in graphene sheets grown by chemical vapor deposition3-4 (CVD) and in graphene
epitaxially grown by the thermal decomposition of silicon carbide.5-6 Further, dopant
impurities have been shown to have significant effects on the transport properties of
graphene.7-9
Silicon dopants in graphene exist predominantly in two previously identified forms:
a non-planar, threefold-coordinated silicon atom in a single carbon vacancy, referred to as
Si-C3, and a fourfold-coordinated silicon atom in a double carbon vacancy, called Si-C4,
which is also thought to be planar. An experimental work from 2012 by Zhou and
coauthors10 showed via simultaneous scanning transmission electron microscopy (STEM)
annular dark field imaging (ADF) and electron energy loss spectroscopy (EELS) and,
additionally, by DFT calculations that the two forms of single silicon doped graphene differ
energetically, and that the silicon atom in Si-C3 has sp3 orbital hybridization and that the
silicon atom in Si-C4 is sp2d hybridized. Similarly, EELS experiments combined with DFT
spectrum simulations confirmed the puckering of the Si in the Si-C3 structure and supported
the planarity of the Si-C4 structure.11 However, a less satisfactory agreement of the
simulated spectrum with the experimental one was noted in the latter case. The results
suggested that d-band hybridization of the Si was responsible for electronic density
disruption found at these sites. Experiments and dynamical simulations performed on these
defects have demonstrated12 how the Si-C4 structure is formed when an adjacent carbon
atom to the silicon is removed. However, it was concluded that the Si-C3 structure is more
stable and can be readily reconstructed by a diffusing carbon atom.
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Past investigations of the electronic structure of graphene defects have shown the
usefulness of the molecule pyrene as a compact representation of the essential features of
single (SV) and double vacancy (DV) defects. Using this model, chemical bonding and the
manifold of electronic states were investigated by multireference configuration interaction
(MR-CI) calculations.13-16 In case of the SV defect,13 the MR-CI calculations showed four
electronic states (two singlet and two triplet) within a narrow margin of 0.1 eV, whereas
for the DV structure,14 a gap of ~1 eV to the lowest excited state was found. For the DV
defect, comparison with density functional theory using the hybrid density functional
Becke three-parameter Lee, Yang, and Parr (B3-LYP) functional17 showed good agreement
with the MR-CI results. Whereas for the covalently bonded Si-C3 structure no further low-
lying states are to be expected, the situation is different for the Si-C4 defect since in this
case an open-shell Si atom is inserted into a defect structure containing low-lying electronic
states.
Based on these experiences, a pyrene model will be adopted also in this work to
investigate the geometric arrangement and orbital hybridization around the silicon atom in
a double carbon vacancy (Figure 4.1). In a second step, a larger circumpyrene structure
(Figure 4.2) will be used to study the effect of increasing number of surrounding benzene
rings. Two types of functionals will be employed, the afore-mentioned B3LYP and, for
comparison, the long-range corrected Coulomb-attenuating B3LYP method (CAM-
B3LYP).18 Special emphasis will be devoted to verifying the electronic stability of the
closed-shell wavefunction with respect to triplet instability19-20 and of the optimized
structures with respect to structural relaxation. Finally, periodic DFT simulations of
graphene with the Perdew-Burke-Ernzerhof (PBE) functional21-22 will be compared to the
molecular calculations and an EELS spectrum simulated based on the computed Si-C4
structure compared to an atomic resolution STEM/EELS measurement.
4.3 Computational Details
Pyrene doped with a single silicon atom was investigated by first optimizing the
structure of pristine pyrene using the density functional B3LYP17 with the 6-31G*23 basis
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67
set. The two interior carbon atoms were then removed and replaced with one silicon atom,
referred to in this investigation as pyrene-2C+Si. This unrelaxed structure was subjected to
a variety of closed-shell [restricted DFT (RDFT)] and open-shell [unrestricted DFT
(UDFT)] geometry optimizations using the hybrid density functionals B3LYP and CAM-
B3LYP in combination with the 6-311G(2d,1p)24-28 basis set. To extend to a larger system,
pyrene-2C+Si was surrounded by benzene rings (circumpyrene-2C) and this structure was
optimized using the B3LYP/6-31G** approach. Finally, the geometry optimization of
cirumpyrene-2C+Si was performed using the B3LYP functional and the 6-311G(2d,1d)
basis.
To simulate the geometric restrictions present in a large graphene sheet, the
Cartesian coordinates of all carbon atoms on the outer rims of pyrene-2C+Si (Figure 4.1)
and cirumpyrene-2C+Si (Figure 4.2) that are bonded to hydrogen atoms were frozen during
the course of geometry optimization to their values in the pristine structures. The
coordinates of both pyrene-2C+Si and circumpyrene-2C+Si were oriented along the y-axis
(the long axis) in the xy-plane.
The orbital occupations of the open-shell structures were determined using the
natural orbital population analysis (NPA)29 calculated from the total density matrix, while
the bonding character at silicon for each structure was investigated using the natural bond
orbital analysis.30-34
The discovered corrugated ground state structure of the Si-C4 defect was then
introduced into a periodic 10×8 supercell of graphene, and the structure relaxed to confirm
that this corrugation is reproduced at this level of theory. Finally, the Si L2,3 EELS response
of the defect was simulated11, 35 by evaluating the perturbation matrix elements of
transitions from the Si 2p core states to the unoccupied states calculated up to 3042 bands,
with no explicit core hole.36 The resulting densities of state were broadened using the
OptaDOS package37 with a 0.4 eV Gaussian instrumental broadening and semi-empirical
0.015 eV Lorentzian lifetime broadening.
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The Gaussian 0938 and TURBOMOLE39 program packages were used for the
molecular calculations, and the GPAW40 and CASTEP41 packages for the graphene
simulations.
Figure 4.1 Optimized pyrene-2C+Si structure P1 (D2 symmetry) using the B3LYP/6-
311G(2d,1p) method with frozen carbon atoms marked by gray circles
Figure 4.2 Optimized circumpyrene-2C+Si structure CP1 (D2 symmetry) using the
B3LYP/6-311G(2d,1p) method with frozen carbon atoms marked by gray circles.
4.4 Results and Discussion
4.4.1 The Pyrene-2C+Si Defect Structure
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The geometry of the unrelaxed, planar pyrene-2C+Si was first optimized using the
B3LYP/6-311G(2d,1p) method at closed-shell level resulting in the D2h structure denoted
P2. There were two imaginary frequencies present in the Hessian matrix. A displacement
of the geometry of structure P2 along either of these imaginary frequencies led to a
distortion of the planarity, lowering the point group to D2 and producing the minimum
energy structure P1, which was 0.736 eV lower in energy than structure P2 (Table 4.1).
The carbon atoms bonded to the silicon atom (atoms 3, 6, 10, and 13) were displaced from
planarity by about ±0.2 Å while the silicon atom remained co-planar with the frozen carbon
atoms on the periphery of pyrene-2C+Si. A similar tetrahedral geometry was found for
doping Al and Ga into DV graphene.42 No triplet instability was found for either of the D2h
and D2 structures. Optimizing the planar pyrene-2C+Si structure at UDFT/B3LYP/6-
311G(2d,1p) level for the high spin case resulted in structure P3, which was planar and
belonging to the D2h point group. No imaginary frequencies were found in the Hessian
matrix of structure P3; it is located 0.922 eV above the minimum energy structure P1.
Table 4.1 Open-shell character, structural symmetry, spin multiplicity, bond distances (Å),
and relative energies (eV) of selected optimized structures of pyrene-2C+Si.
Structure
Closed/
Open
Shell
Triplet
instab. #imaga
High/
Low
Spin Sib-C
C3-
C13 C3-C6 ΔE
B3LYP/6-311G(2d,1p)
P1 (D2)c Closed no 0 Low 1.940 2.652 2.860 0.000
P2 (D2h) Closed no 2 Low 1.932 2.634 2.826 0.736
P3 (D2h) Open - 0 High 1.933 2.610 2.852 0.922
CAM-B3LYP/6-311G(2d,1p)
P4 (D2)c Open no 0 Low 1.933d 2.645d 2.847d 0.000
P5 (D2)c Weakly
open
no 0 Low 1.936 2.648 2.852 0.040
P6 (D2)c Closed yes 0 Low 1.938 2.648 2.857 0.056
P7 (D2h) Open no 1 Low 1.923 2.616 2.820 0.602
P8 (D2h) Closed yes 2 Low 1.931 2.631 2.828 0.734
P9 (D2)e Open - 0 High 1.900 2.599 2.777 0.770
P10 (D2h) Open - 1 High 1.900 2.598 2.772 0.826
aNumber of imaginary frequencies. bSilicon always located in-plane. cThe out-of-plane
distance of carbon atoms that were not frozen in D2 structures is ±0.2 Å dAveraged value
of the four bonding carbons. eThe out-of-plane distance of carbon atoms that were not
frozen is ±0.07 Å
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Structures P1-P3 were investigated also using the CAM-B3LYP/6-311G(2d,1p)
method (Table 4.1). For the planar geometry (D2h symmetry) two low spin structures were
found. The first one was the closed-shell structure P8 that, however, turned out to be triplet
instable. Re-optimization at UDFT/CAM-B3LYP level under D2h restrictions led to the
open-shell structure P7 that was 0.13 eV more stable than the corresponding closed-shell
structure. Structure P8 also had two imaginary vibrational frequencies, both of which were
out-of-plane, leading to D2 symmetry. Displacement along these modes and optimization
at closed-shell level gave the non-planar structure P6 of D2 symmetry. Similarly, following
the out-of-plane imaginary frequency found in the Hessian of structure P7 led to a new
structure P5 (D2 symmetry) that was 0.016 eV lower in energy than structure P6. It had
only a small open-shell character as can be seen from a natural orbital occupation of 0.089
and 1.911 e respectively for the lowest unoccupied natural orbital (LUNO) and highest
occupied natural orbital (HONO). This is much smaller than the NO occupation of the
open-shell low spin structure P7 whose LUNO-HONO occupations were 0.302 and 1.698
e. Searching along the imaginary frequencies found for the previous structures and
considering the triplet instability present in the wave functions led to structure P4 (D2
symmetry), the new low spin, open-shell minimum-energy CAM-B3LYP structure which
was 0.04 eV more stable than structure P5. The HONO-LUNO occupation for structure P4
was 1.899 e and 0.101 e respectively. Comparing the occupations from CAM-B3LYP to
B3LYP shows that the range-corrected functional CAM-B3LYP presents a more varied
picture than found with the B3LYP functional. In particular, open shell structures with non-
negligible deviations of NO occupations from the closed shell reference values of two and
zero, respectively, were found. However, the energy differences between these different
states are quite small, only a few hundredths of an eV.
Structure P10 of Table 4.1 was obtained using the unrestricted high spin approach
in the geometry optimization. The Hessian contained one imaginary frequency, an out-of-
plane bending mode leading to D2 symmetry. Following this imaginary frequency led to
structure P9 that was about 0.06 eV more stable than structure P10. The bonding carbons
of this new high spin structure were slightly out-of-plane by 0.07 Å.
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Selected bond distances of the silicon-2C+Si complex are given in Table 4.1 for the
B3-LYP and CAM-B3LYP results. Looking first at the distances from the silicon atom to
its bonding carbon atoms computed at B3LYP/6-311G(2d,1p) level, the Si-C bond
distances are found to vary by less than 0.01 Å in the different structures. The D2 structure
has the longest Si-C bond distance as would be expected from moving the bonding carbon
atoms out-of-plane from the D2h to D2 symmetry. The internuclear distance between
adjacent, non-bonded carbon atoms C3 and C13 along the direction perpendicular to the
long axis is 2.65 Å for structure P1, which is the largest among the three structures. This is
again a result of the out-of-plane character of structure P1. The distance C3-C6 along the
long axis of pyrene is ~0.2 Å longer than the perpendicular distance, reflecting the
restrictions imposed by freezing the carbon atoms which would be connected to the
surrounding graphene sheet (Figure 4.1).
Next, we characterize the structures computed with the CAM-B3LYP/6-
311G(2d,1p) method. The largest C3-C13 bond distance (Table 4.1) is 2.65 Å for the low
spin structures P5 and P6 (D2), while the smallest, 2.60 Å, is found for the high spin, open-
shell structure P10. The dependence of the Si-C distances on the different structures and
spin states is not very pronounced, though the high spin planar structures, structures P9 and
P10, were found to have the smallest Si-C and C3-C13 distances; the same situation that
was seen using B3LYP/6-311G(2d,1p).
Comparing the two above methods for pyrene-2C+Si, the most noticeable effect is
that there is greater open-shell character for the low spin structures when using the CAM-
B3LYP functional. The Si-C bond distance and C3-C13 intranuclear distances are also
slightly smaller when using the CAM-B3LYP functional. But overall, agreement between
the two methods is quite good.
4.4.2 The Circumpyrene-2C+Si Defect Structure
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Pyrene-2C+Si was then surrounded by benzene rings to create the larger
circumpyrene-2C+Si structure (Figure 4.2) as a better model for embedding the defect into
a graphene sheet. The edge carbon atoms linked to hydrogen atoms were again frozen.
Description of the geometry of circumpyrene-2C+Si is presented in Table 4.2 for the
B3LYP/6-311G(2d,1p) method. The geometry of the planar, unrelaxed circumpyrene-
2C+Si was optimized using a closed-shell approach that led to the D2h structure CP2, which
had one out-of-plane imaginary frequency. Following this mode, the non-planar D2
symmetric structure CP1 was obtained that contained no imaginary frequencies.
Optimizing the geometry of the planar, unrelaxed circumpyrene-2C+Si using an open-
shell, high spin approach resulted in a geometrically stable (no imaginary frequencies)
planar structure CP3 with D2h symmetry. The wave functions of circumpyrene-2C+Si
structures CP1 and CP2 were triplet stable.
Table 4.2 Structural symmetry, open-shell character, spin multiplicity, bond distances (Å),
and relative energies (eV) of selected optimized structures of circumyrene-2C+Si.
Structure
Closed/
Open
Shell
Triplet
instab. #imaga
High/
Low
Spin Sib-C
C29-
C39
C29-
C32 ΔE
B3LYP/6-311G(2d,1p)
CP1 (D2)c Closed no 0 Low 1.909 2.690 2.741 0.000
CP2 (D2h) Closed no 1 Low 1.902 2.668 2.713 0.118
CP3 (D2h) Open - 0 High 1.912 2.662 2.745 1.254 aNumber of imaginary frequencies. bSilicon always stays in-plane cThe out-of-plane
distance of non-frozen carbon atoms is ±0.2 Å in structures that are D2 symmetric
The Si-C bond distances for the three investigated structures are very similar (Table
4.2). The C29-C39 distance is largest for the non-planar, closed-shell structure P1 and
smallest for the planar structures P2 and P3, but the differences are less than 0.03 Å.
The difference in energy between the non-planar and planar low spin structures
decreases markedly when the 2C+Si defect is embedded in a larger hexagonal sheet
(compare ΔE values in Table 4.1 and Table 4.2). The energy difference decreases by ~0.6
eV from 0.74 eV in pyrene-2C+Si to 0.12 eV in circumpyrene-2C+Si. Also noteworthy,
when comparing pyrene-2C+Si and circumpyrene-2C+Si is that the Si-C bond distances in
circumpyrene-2C+Si are shorter by about 0.03 Å on average for the low spin cases. The
differences in the internuclear distance between adjacent, non-bonded carbon atoms C29
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and C39 (perpendicular to the long axis) and C29-C32 (parallel) is much less pronounced
than in the pyrene case, which is a consequence of the more flexible embedding into the
carbon network where none of the four C atoms surrounding Si is bonded to another atom
which is frozen.
4.4.3 The Graphene-2C+Si Defect Structure
To confirm the obtained geometry and to validate that periodic DFT simulations
reproduce the observed ground state structure, the Si-C4 defect with an alternating ±0.2 Å
corrugation of the four C atoms was placed in a 10×8 supercell of graphene and the
structure and cell relaxed using the PBE functional. The structural optimization preserved
the out-of-plane corrugation, and resulted in a total energy 0.22 eV lower than for the flat
structure. Thus, standard DFT is able to find the correct, non-planar structure when
initializing away from the flat geometry. Further, a nudged elastic band calculation43 shows
that the two equivalent conformations (alternating which C atoms are up and which are
down) are separated only by this energy barrier, leading to a Boltzmann factor of 1.7×10-4
at room temperature and thus a rapid oscillation between the two equivalent structures for
any reasonable vibration frequency.
Bader analysis44 of the charge density shows that the Si-C bonds are rather
polarized, with the Si donating 2.61 electrons shared by the four neighboring carbons. This
is even larger than the bond polarization in 2D-SiC, where a Si donates 1.2 electrons
(although each C has three Si neighbors, bringing the total to 1.2 per C instead of 0.65
here).45
4.4.4 Natural Bond Orbital Analysis
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NBO analysis was used to further characterize the bonding of pyrene-2C+Si and
circumpyrene-2C+Si and to describe in detail the charge transfer from silicon to carbon in
the various structures.
The NBO analysis of pyrene-2C+Si is presented in Table 4.3. The linear
combination factor of the Si is less than half of that of the bonding C, a sign of the greater
electronegativity of carbon compared to silicon. The non-planar, low spin, closed-shell
structure P1 (D2 symmetry) shows sp3 orbital hybridization at the silicon and sp3 orbital
hybridization at the bonding carbon, consistent with the slight tetrahedral arrangement
about the silicon center. The planar closed-shell structure P2 shows sp2d hybridization at
the silicon center, consistent with previous experimental and computational
interpretations,10 and approximately sp3 hybridization at the bonding carbons. The high
spin structure P3, which also has D2h symmetry, shows also sp2d hybridization at the
silicon. Compared to B3LYP (structures P1-P3), bonding from CAM-B3LYP (Table B.1
of Appendix B) is essentially the same.
Table 4.3 NBO bonding character analysis of pyrene-2C+Si using B3LYP/6-311G(2d,1p).
The discussed carbon atom is one of the four bonded to silicon.
Structure Bonding Character
B3LYP/6-311G(2d,1p)
P1 (D2)
closed sh. 0.54(sp2.98d0.02)Si+0.84(sp3.24d0.01)C
P2 (D2h)
closed sh. 0.46(sp2.00d1.00)Si+0.89(sp2.97d0.01)C
P3 (D2h)
open sh.,
high spin
α: 0.46(sp2.00d1.00)Si+0.89(sp2.81d0.01)C
β: 0.55(sp2.00d1.00)Si+0.83(sp3.61d0.01)C
The NBO analysis of circumpyrene-2C+Si is presented in Table 4.4. The bonding
character of structure CP1 is essentially sp3 for both the silicon and the bonding carbon as
in structure P1 of pyrene-2C+Si, although the p orbital character on the carbon atom of this
structure is reduced by about 0.60 e. Structure CP2 presents an interesting case, as no
bonding NBOs were found. Instead, a series of four unoccupied valence lone pairs (LP*),
one of s orbital character and three of p character, were located on the silicon and a
corresponding set of occupied valence lone pairs (LP), one for each bonding carbon, was
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found with sp3.58d0.01 character. It is noted that the Si-C bonding character of the high spin
structure CP3 changes markedly between the alpha shell and beta shell. In the alpha shell,
the silicon and carbon atoms both exhibit sp1 orbital hybridization with no d orbital
contribution on the silicon, but in the beta shell, the silicon atom shows sp2d hybridization
and the carbon atoms show sp3 hybridization as in the other planar cases.
The shapes of the NBOs describing the bond between Si and C is depicted in Figure
4.3 for structures P1 (D2 symmetry) and P2 (D2h symmetry) using the B3LYP/6-
311G(2d,1p) method. They appear very similar in spite of the different hybridization on Si
(sp3 vs. sp2d, Table 4.3). The plots of the NBOs for all other structures, even those for
circumpyrene-2C+Si, are almost identical. The exception is structure CP2 of
circumpyrene-2C+Si in which no Si-C bonds were found during the NBO analysis. The
NBOs shown in Figure B.1 for this structure consist of the valence lone pairs found on
each carbon atom (Table 4.4) possessing sp3.58 orbital hybridization with a lobe of electron
density found facing the silicon center. The NBOs of silicon are low-occupancy valence
lone pairs, representing one s and three p NBOs. Even though this classification of bonding
by means of lone pairs according to the NBO analysis looks quite different, the
combination of these lone pairs is not expected to look substantially different from the
localized bonds shown in the other structures, especially considering the strong polarity of
the silicon-carbon bonds as discussed below.
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Table 4.4 NBO bonding character analysis of circumpyrene-2C+Si. The discussed carbon
atom is the one of the four bonded to silicon.
Structure Bonding Character
B3LYP/6-311G(2d,1p)
CP1 (D2)
closed sh. 0.51(sp2.96d0.04)Si+0.86(sp2.62d0.01)C
CP2 (D2h)
closed sh. Four LP*: one [1.0(s)Si] + three [1.00(p)Si]
a
four[1.00(sp3.58d0.01)C]b
CP3 (D2h)
open sh.,
high spin
α: 0.68(sp1.00d0.00)Si+0.74(sp1.00d0.00)C
β: 0.55(sp2.00d1.00)Si+0.84(sp3.41 d0.01)C
aFour unfilled valence lone pairs (LP*) on silicon (occupation less than 1 e), bOne occupied
valence lone pair on each bonding carbon
Natural charges are instructive for illustrating the polarity of the silicon-carbon
bonds and the charge transfer between silicon and the surrounding graphitic lattice. These
are shown in Figure 4.4 for circumpyrene-2C+Si (Structures CP2 and CP3) using the
B3LYP/6-311G(2d,1p) method. The silicon center is strongly positive in all cases and more
positive in circumpyrene-2C+Si for the low spin cases (1.7–1.8 e, structures CP1 and CP2)
than the high spin case (1.4 e, structure CP3). It thus seems that the Bader analysis
presented above overestimates the charge transfer by over 40 %, despite being qualitatively
correct. The bonding carbon in each case shows a larger negative charge compared to the
remaining carbon atoms of circumpyrene. The charge on the bonding carbon in the low
spin structures CP1 and CP2 is more negative than that of the high spin structure CP3.
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Figure 4.3 NBO plots for pyrene-2C+Si using the B3LYP/6-311G(2d,1p) approach for
structures P1 and P2. Isovalue = ±0.02 e/Bohr3
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Figure 4.4 Natural charges (e) of carbon and silicon in circumpyrene-2C+Si using the
B3LYP/6-311G(2d,1p) method.
The natural charges of the remaining carbon atoms in circumpyrene-2C vary
depending on whether they are bonded to other carbon atoms or to hydrogen. For the low
spin structures CP1 and CP2, the natural charges on C atoms on the periphery of
circumpyrene bonded to other C atoms is about one-third as negative as the natural charges
of C atoms bonded to H atoms, while the natural charges of carbon atoms in the interior of
circumpyrene and not bonded to Si are much smaller. Taken together, the magnitude of the
Si-graphene charge transfer can be seen to be largest for the planar, closed-shell, low spin
structure CP2, and smallest for the planar, high spin structure CP3.
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The natural charges for pyrene-2C+Si using B3LYP/6-311G(2d,1p) are plotted in
Figure B.2. The low spin structures P1 and P2 have respective positive charges on the
silicon atom of 1.596 and 1.953 e, a larger difference than in the low spin structures CP1
and CP2 of circumpyrene-2C+Si where the silicon atom has positive charges of 1.726 and
1.821 e. Correspondingly, the difference in the negative charges of the bonding carbon
atoms is also larger between these two structures than in circumpyrene. For the other
carbons in pyrene, the magnitude of their charges varies less than in circumpyrene. For
pyrene-2C+Si using the CAM-B3LYP/6-311G(2d,1p) method, the natural charges are
plotted in Figure B.3. The same trend is seen for this method as in our other cases, namely
that the closed-shell, low spin, planar structure P8 has the largest positive charge on silicon
and the largest negative charge on the bonding carbon compared to the other structures.
4.4.5 Molecular Electrostatic Potential
The molecular electrostatic potential (MEP) was computed for each structure and
is presented in Figure 4.5 for circumpyrene-2C+Si using the B3LYP/6-311G(2d,1p)
method. The plots illustrate the natural charge transfer discussed previously: for structures
CP1-CP3, the positive potential at the silicon corresponds to the positive charge buildup as
discussed above. This positive potential at the silicon is more positive for structures CP1
and CP2 than for CP3 following the trend of the natural charges. There is also a negative
potential diffusely distributed on the carbon atoms in circumpyrene.
The MEP plots for pyrene-2C+Si using the B3LYP/6-311G(2d,1p) method is
shown in Figure B.4. Similar results are obtained when using the CAM-B3LYP functional
(shown in Figure B.5). Finally, the low spin planar structures P2, P7, and P8 have a much
more positive potential than either the non-planar structures or the high spin planar
structures, represented by the deeper blue coloring in the figures.
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Figure 4.5 Molecular electrostatic potential for circumpyrene-2C+Si mapped onto the
electron density isosurface with an isovalue of 0.0004 e/Bohr3 (B3LYP/6-311G(2d,1p)).
4.4.6 Electron Energy Loss Spectroscopy
The Si-C4 defect in graphene was originally identified through a combination of
atomic resolution STEM and atomically resolved EELS. In that study11 it was concluded
that a Si-C3 defect is non-planar due to a significantly better match between the measured
and simulated EELS spectrum; however, the simulated spectrum of a flat Si-C4 defect
matched the experiment less well. Because of this discrepancy and the findings described
above, we also simulated the EELS spectrum of the corrugated Si-C4 graphene defect.
Figure 4.6 displays the simulated spectra of both the flat and the corrugated defect
(normalized to the π* peak at 100 eV), overlaid on a new background-subtracted
experimental signal that we have recorded with a higher dispersion than the original
spectrum11 (average of 20 spot spectra, with the Si-C4 structure verified after acquisition
by imaging; see Figure B.6 for the unprocessed spectrum).
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Figure 4.6 Comparison of simulated and experimental EELS spectra of the Si-C4 defect in
graphene. Despite the out-of-plane distorted structure being the ground state, the originally
proposed flat spectrum provides an overall better match to the experiment, apart from the
small peak at ~96 eV that is not predicted for the flat structure. The inset shows a colored
medium angle annular dark field STEM image of the Si-C4 defect (the Si atom shows
brighter due to its greater ability to scatter the imaging electrons).
The new spectrum closely resembles the originally reported one apart from a small
additional peak around 96 eV. Comparing the recorded signal to the simulated spectra, the
overall relative intensities of the π* and σ* contributions match the flat structure better, but
both simulations overestimate the π* intensity at 100 eV. Intriguingly, despite the better
overall match with the flat structure, the small pre-peak present in the new signal is only
present in the spectrum simulated for the lowest energy corrugated structure, providing
spectroscopic evidence for its existence. The lowest unoccupied states responsible for this
peak are π* in character and form a „cross-like“ pattern across the Si site, as shown in Figure
B.7, markedly different from the lowest unoccupied states of the flat structure. However,
considering the small energy barrier between the equivalent corrugated conformations
discussed above, even at room temperature the spectrum may be integrated over a
superposition of corrugated and nearly flat morphologies.
100 110 120 130
0.0
0.5
1.0
1.5
´1
04
co
un
ts
Energy loss (eV)
Signal
Flat
Corrugated
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4.5 Conclusions
Density functional theory calculations (both restricted and unrestricted) were
performed on pyrene-2C+Si and circumpyrene-2C+Si producing a variety of structures that
were characterized using natural orbital occupations, natural bond orbital analyses, and
molecular electrostatic potential plots. A non-planar low spin structure of D2 symmetry
was found to be the minimum for both cases, followed energetically by a low spin planar
structure, 0.6 eV higher in energy for pyrene-2C+Si, but with a much smaller (only ~0.1
eV) energy difference for circumpyrene-2C+Si. A periodic graphene model with the non-
planar defect had a 0.22 eV lower energy than a flat one. In the molecular calculations, the
out-of-plane structure was shown to be a minimum by means of harmonic frequency
calculations. The structures of the high spin state are much higher in energy than the
minimum: about 0.85 eV higher for pyrene-2C+Si and about 1.3 eV higher for
circumpyrene-2C+Si.
For pyrene-2C+Si, B3LYP and CAM-B3LYP were compared showing that there
was not much difference either energetically, in the NBO characterization, or in the MEP
plots. One feature of interest is that CAM-B3LYP shows greater open-shell character in
the low spin case than was seen for B3LYP. This open-shell character demonstrates the
variety of electronic structures which can be found in seemingly ordinary closed-shell
cases.
NBO analyses showed the expected bonding configurations for the two structural
symmetries present. The bonding of the silicon atom in the planar structure was found to
have sp2d orbital character while in the non-planar structures, the bonding orbitals are sp3
hybridized. The natural charges illustrated the charge transfer that occurs, demonstrating
the buildup of positive charge at the silicon and the diffusion of negative charge into the
pyrene-2C/cirumpyrene-2C structure, which was further evident in the MEP plots.
Despite carefully establishing the non-planar ground state nature of the Si-C4 defect
in graphene, electron energy loss spectra still seem to match the flat structure better, but
imperfectly especially concerning a small peak at ~96 eV. Considering the large amounts
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of kinetic energy that the energetic electrons used in transmission electron microscopy can
impart, and the available thermal energy in the system, perhaps the defect is not able to
stay in its ground state, but rather exists in a superposition of slightly different
configurations that result in an average spectrum resembling the flat state.
4.6 Acknowledgments
We are grateful to Michael Zehetbauer for continuous support and interest in this
work and also thank Quentin Ramasse for many useful discussions. T.S. and J.K.
acknowledge the Austrian Science Fund (FWF) for funding via projects P 28322-N36 and
I 3181-N36. H.L. and T.S. are grateful to the Vienna Scientific Cluster (Austria) for
computational resources (H.L.: project no. P70376). H.L. also acknowledges computer
time at the computer cluster of the School for Pharmaceutical Science and Technology of
the Tianjin University, China, T.P.H. acknowledges the Engineering and Physical Sciences
Research Council (EPSRC) Doctoral Prize Fellowship that funded this research, and the
ARC1 and ARC2 high-performance computing facilities at the University of Leeds.
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Mangler, C.; Drummond-Brydson, R.; Scott, A. J.; Meyer, J. C.; Kotakoski, J.
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CHAPTER V
THE CRUCIAL ROLE OF A SPACER MATERIAL ON THE
EFFICIENCY OF CHARGE TRANSFER PROCESSES IN ORGANIC
DONOR-ACCEPTOR JUNCTION SOLAR CELLS
Authors: Reed Nieman,a Hsinhan Tsai,b Wanyi Nie,b Adelia J. A. Aquino,a,c Aditya D.
Mohite,b Sergei Tretiak,b Hao Li,d Hans Lischkaa,c
Affiliations: aDepartment of Chemistry and Biochemistry, Texas Tech University
Lubbock, TX 79409-1061, USA, bLos Alamos National Laboratory, Los Alamos, New
Mexico 87545, USA, cSchool of Pharmaceutical Sciences and Technology, Tianjin
University, Tianjin, 300072 P.R.China, dDepartment of Chemistry, University of Houston,
Houston, Texas 77204. USA
Published: December 2017 in Nanoscale, DOI:10.1039/C7NR07125F
5.1 Abstract
Organic photovoltaic donor-acceptor junction devices composed of π-conjugated polymer
electron donors (D) and fullerene electron acceptors (A) show greatly increased
performance when a spacer material is inserted between the two layers.1 For instance,
experimental results reveal significant improvement of photocurrent when a terthiophene
oligomer derivative is inserted in between π-conjugated poly(3-hexylthiophene-2, 5-diyl
(P3HT) donor and C60 acceptor. These results indicate favorable charge separation
dynamics, which is addressed by our present joint theoretical/experimental study
establishing the beneficial alignment of electronic levels due to the specific morphology of
the material. Namely, based on the experimental data we have constructed extended
structural interface models containing C60 fullerenes and P3HT separated by aligned
oligomer chains. Our time-dependent density functional theory (TD-DFT) calculations
based on a long-range corrected functional, allowed us to address the energetics of essential
electronic states and analyze them in terms of charge transfer (CT) character. Specifically,
the simulations reveal the electronic spectra composed of a ladder of excited states evolving
excitation toward spatial charge separation: an initial excitonic excitation at P3HT
decomposes into charges by sequentially relaxing through bands of C60-centric,
oligomer→C60 and P3HT→C60 CT states. Our modeling exposes a critical role of dielectric
environment effects and electronic couplings in the self-assembled spacer oligomer layer
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on the energetics of critical CT states leading to a reduced back-electron transfer,
preventing recombination losses, and thus rationalizes physical processes underpinning
experimental observations.
5.2 Introduction
The interfaces of organic photovoltaic bulk heterojunction (BHJ) donor
(D)/acceptor (A) devices between π-conjugated electron donor polymers and electron
acceptors, such as fullerenes, are of fundamental importance for understanding organic
photovoltaic processes and for improving their efficiency.2-4 These interfaces control the
dissociation of excitons and generation of CT states. The conversion of light into electricity
in a BHJ photovoltaic device is accomplished by a four-step process: (i) light absorption
by bright π→π* transition of the organic conjugated polymer and generation of excitons,
(ii) diffusion of the excitons through the bulk polymer and segregation to the interface, (iii)
dissociation of the excitons at afore-mentioned heterojunctions and creation of CT states,
and (iv) charge separation and collection at contacts.2, 5 The actual processes occurring in
the BHJ material are significantly more complex because of the heterogeneous distribution
of donor and acceptor in the BHJ material, a broad variety of structural defects and
conformations, the large number of internal degrees of freedom of the polymer chains, and
the complicated manifold of electronic states.6-8 Therefore, finessing the mechanism of
ultrafast and loss-less free charge generation in BHJ devices via intercede design is still a
very active research area.9-13
While traditional organic photovoltaic (OPV) BHJ devices involve only a pair of
materials, i.e., electron donor and acceptor complexes, recent experimental work1 has
employed a new strategy of using a third material acting as a spacer between the donor and
acceptor regions,14-15 which mitigates the electron-hole recombination rate (i.e. the
backward charge transfer processes) but without affecting the efficiency of charge
separation (i.e., forward electron transfer). In particular, these studies in a simple bi-layer
device have shown a large increase in both the photocurrent (up to 800%) and open circuit
voltage (VOC) when a spacer material such as a terthiophene-derivative (O3) is added
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between the donor, a P3HT, and a fullerene (C60) acceptor. Subsequently, in the BHJ device
a significant increase in the power conversion efficiency (PCE) from 4% to greater than
7% was observed.1 These experimental findings followed the hypothesis of a “fast energy
transfer of the exciton from the donor across O3 to the acceptor and the back cascading of
the hole to the donor” due to a favorable alignment of the corresponding energy levels.
Such arrangement ensures that the spacer material is also physically separating the donor
and acceptor interface thus working to mitigate the CT recombination rate.
Quantum-chemical modeling of these materials can provide important atomistic
insights into the nature of underlining processes and ultimately help to formulate design
strategies toward optimizing the photophysical dynamics. The first important step is the
determination of the energy level alignment occurring at the complex donor-acceptor
interfaces. For such large systems density functional theory (DFT) and time-dependent
(TD)-DFT are the most widely used techniques providing access to the ground and excited
state properties, respectively, being a reasonable compromise between accuracy and
numerical cost. However, in spite of the overwhelming success of DFT in ground-state
computation, the correct description of charge-separated excited states is still problematic
for many developed exchange-correlation functionals due to the significant delocalized
nature of the density, especially for intermolecular CT states at BHJ interfaces.16-19 A
variety of strategies have been developed to correctly account for delocalized excitations.
Long-range corrected functionals20-21 in combination with optimization of the parameters
determining the separation range22-25 have been used successfully to correct several TD-
DFT artifacts. In previous work,26 we have shown that long-range corrected functionals of
CAM-B3LYP27-28 and ωB97XD29 gave very good agreement for excitonic and CT states
for thiolated oligothiophene dimers in comparison to the second-order algebraic
diagrammatic construction (ADC(2)) approach. Thus, long-range corrected functionals
appear to be well applicable for the calculation of electronic spectra of π-conjugated
semiconductor polymers including capacity of describing both excitonic and CT states.30
They are also computationally efficient so that the capability of treating large molecular
aggregate systems can be well managed.
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Regarding the discussed bilayer interfaces,31 previous calculations have shown that
the bright, light absorbing π→π* transition can be located above CT states and can convert
to the CT state via internal conversion processes. However, it has been concluded from
TD-DFT calculations32 that dark fullerene states could be located even lower in energy and
would have the possibility to quench the CT states after they have been formed. Other
calculations based on the already mentioned ADC(2) method have shown a slightly
different picture indicating CT states as the lowest singlet states.31, 33
The energetic ordering of states and the stabilization of CT states obviously
represents a crucial issue affecting the reduction of unwanted leaking processes. This joint
experimental and computational study wants to demonstrate the specific factors affecting
especially the stability of the CT states. For that purpose, the experimental P3HT/O3/C60
system discussed above is being used together with our previously benchmarked TD-DFT
methodology.26 Even though the calculations are specific for this system, as we will show,
the conclusions are quite general and can certainly be transferred to many other polar
interface systems. Two structural models were employed in order to represent the donor-
spacer-acceptor interface: (i) the donor P3HT polymer is connected via one spacer T3 chain
to the acceptor C60 (Figure 5.1a, denoted as standard model) and (ii) an extended model
(Figure 5.1b,c) where P3HT is connected via three T3 spacer molecules to three C60
electron acceptors. A spacer material, thiolated terthiophene, T3, was used which can be
considered as a good approximation to O3 used in the experiments. This specific mutual
arrangement of molecules underscores our experimental fabrication technique utilizing the
Langmuir-Blodgett (LB) method (Figure 5.2a) ensuring orthogonal orientation of the
oligomer with respect to the polymer P3HT layer, as demonstrated by our X-ray
measurements. Furthermore, the extended model aims to account for π-stacking effects in
the oligomer, which were shown to be important for conventional, simple donor-acceptor
interfaces.34-39
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Figure 5.1 (a) Standard model of the molecular arrangement in the trimer. (b) Front and (c)
side views of the extended model. The picture shows molecular notations, geometrical
arrangement and selected nonbonding distances (Å). Atoms kept fixed during the geometry
optimization are circled in black.
The article is organized as follows: Section 5.3 describes experimental and
computational methodology, Section 5.4 presents our results, and finally Section 5.5
summarizes our findings and conclusions.
5.3 Methods
5.3.1 Experimental
5.3.1.1 Bilayer Solar Cells Fabrication
Solar cell fabrication started from the cleaning of indium tin oxide (ITO)
transparent substrate with sonication bath in water, acetone and isopropyl alcohol for 15
min respectively. The hole transporting polymer poly(3,4-ethylenedioxythiophene)
polystyrene sulfonate (PEDOT:PSS) solution was then coated on the cleaned ITO
substrates by spin coating method. After drying, the coated substrates were transferred to
an argon filled glovebox for P3HT coating. The in-stock P3HT solution (2 mg/ml in
chlorobenzene) was spin coated on the substrates at 2000 rpm for 45 Sec, forming a
uniform, 40 nm thin film without further annealing. The coated substrates were then taken
outside the glovebox for interfacial layer deposition by LB technique between air-water
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interfaces on a spin coated P3HT film. After LB deposition, the devices were completed
by thermally evaporating C60 (30nm) for acceptor layer and electrode deposition through
shadow mask.
5.3.1.2 Device Characterization
The completed solar cells were mounted in cryostat with BNC connections from
the electrodes to the instruments. The photocurrent was collected in AC mode by
illuminating the device with monochromatic light while measuring the short circuit current
using lock-in amplifier.
5.3.2 Computational Details
The computational methods are based on DFT using the Perdew–Burke–Ernzerhof
(PBE)40 functional with the SV(P)41 basis for geometry optimizations, and the long-range
separated density functional, CAM-B3LYP, for the calculation of excited states using two
basis sets, 6-31G42 and 6-31G*.43-44 Solvent effects were taken into account using the
conductor-like polarizable continuum model (C-PCM)45. Environmental effects on the
electronically excited states were investigated on the basis of the linear response (LR)46-47
and state-specific (SS)48 approaches. Dichloromethane (dielectric constant, ε=8.93,
refractive index, n=1.42) was chosen to act as a modestly polar environment. Geometry
optimizations were performed by freezing selected atoms as shown in Figure 5.1 in order
to maintain the desired structure of the aggregate as deduced from experiment.
The two types of trimer structures are shown in Figure 5.1a (standard trimer) and
Figure 5.1b,c (extended trimer). All structures were optimized as isolated complexes using
DFT together with the PBE40 functional, the SV(P)41 basis, and the Multipole-Accelerated
Resolution of the Identity for the Coulomb energy (MARI-J).49 Empirical dispersion
corrections were added by means of the D350 approach. Excited states were calculated
using the long-range separated density functional, CAM-B3LYP, and two basis sets, 6-
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31G42 and 6-31G*43-44. D3 dispersion corrections were used for the CAM-B3LYP TD-DFT
calculations as well.
Full geometry optimizations of the standard and extended trimer structures would
lead to distorted geometries which do not resemble the intended structural arrangements of
a donor-spacer-C60 interface since geometrical restrictions of the surrounding environment
are missing. To achieve the desired model arrangement, geometry optimization was
performed in steps and by freezing a few selected atoms (see Figure 5.1) to restrict the
structural manifold. For the standard trimer, the C60 and T3 monomers were combined and
optimized freezing selected atoms. To this C60-T3 structure, P3HT was added as shown in
Figure 5.1a. The whole complex (C60-T3-P3HT) was then optimized. The frozen atoms
were located at the end of the chains and at the opposite side of C60, respectively, so that
the subunits had sufficient flexibility to come in contact allowing the intersystem distances
to be optimized. For the extended trimer, an analogous restricted geometry optimization
technique was used as shown in Figure 5.1b, c.
Since the main objective of this investigation was to determine the relative stability
of the CT states with excitations from the π-conjugated polymers to the C60s and the local
excitations in the polymers, excitations within the fullerenes were suppressed in most of
the calculations by freezing in the TD-DFT calculations the occupied orbitals located on
the C60 units. Otherwise, about a dozen of local excitations per C60 unit would have
impeded the already expensive calculations and would not have allowed the focus on the
actual states of interest. In practice, all the core orbitals along with twenty percent of the
highest occupied orbitals, which were localized at the C60 and (C60)3 units, were kept
frozen. Comparison between full and frozen orbital calculations have been performed as
well, showing good agreement in energetic ordering of the remaining states in the two
calculations.
Solvent effects were taken into account using the conductor-like polarizable
continuum model (C-PCM).45 Environmental effects on the electronically excited states
were investigated on the basis of the LR46-47 and SS48 approaches. In the LR approach,
Hartree and exchange-correlation (XC) potentials are linearly expanded with respect to the
variation of the time-dependent density of the ground state. In the SS method, the solvation
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field adapts to changes in the solute wave function for electronic excitations where the
energy correction from fast electronic relaxation of the solvent is state-dependent. SS is
considered more accurate for describing the excited electronic states51-53 with charge
distributions significantly different from the ground state such as with polar solvents or CT
states in the solute. The SS method is computationally significantly more demanding
though and requires that each state be calculated independently. As such, only five states
were calculated using the CAM-B3LYP/6-31G* method for the extended trimer which
contained, however, representatives of the most important excitation types. For the
extended trimer using the CAM-B3LYP/6-31G method, twenty states were calculated, and
for the standard trimer using both basis sets, ten states were considered. Dichloromethane
(dielectric constant, ε=8.93, refractive index, n=1.42) was chosen to act as a modestly polar
environment.
Geometry optimizations were carried out using the TURBOMOLE54 program suite,
while all TD-DFT calculations were performed with the Gaussian G09 program package.55
The character of the electronic states has been analyzed by means of natural transition
orbitals (NTOs)56-58 and molecular electrostatic potential plots (MEP). NTOs represent the
weighted contribution to each excited state transition from a hole to a particle (electron)
derived from the one-electron transition density matrix. NTOs and particle/hole
populations were performed with the TheoDORE program package.59-61
5.4 Results and Discussion
5.4.1 Experimental Observations
We fabricate the bilayer device as illustrated in Figure 5.2a comprised with 20 nm
P3HT as donor and 40 nm of fullerene (C60) as acceptor for photocurrent measurements as
summarized in Figure 5.2. To insert oriented O3 between the (D/A) interface, we employed
the O3 with Langmuir-Blodgett (LB) method after spin coating P3HT on hole transporting
material, followed by C60 deposition by thermal evaporation (Figure 5.2a). The obtained
bilayer device was taken for photocurrent spectrum measurement in Figure 5.2b. The
photocurrent follows the absorption spectrum of P3HT (band gap 1.9 eV), while the
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magnitude of the photocurrent increase by 1.8 times from the device without spacer layer
(black) to the device with 1-bilayer of O3 inserted (red). The optical band gap remains the
same in both cases indicating the oligomer absorption does not contribute to the
photocurrent increase. To characterize the molecular orientation of the O3 molecules at the
interface, we took the LB deposited O3 molecules on P3HT thin-film for gracing incidence
wide angle X-ray scattering (GIWAXS) measurement. The result in Figure 5.2c clearly
shows a diffraction spot along the qz axis. The spot is located near qz = 0.322 Å-1
corresponding to a d-spacing of 19.4 Å which matches with the molecule length. In
addition, the Bragg spots are parallel to the qy axis (substrate) indicating that the O3
molecules are aligning in the out-of-plane62 orientation by LB method (Figure 5.2a) and
pointing from P3HT towards C60. The distance between the donor-acceptor is thus
determined by the length of molecule and number of bilayers deposited. This is consistent
with previous measurements of three self-assembled oligomers.1, 63 We further performed
device photocurrent measurement as a function of the number of O3 bilayers as shown in
Figure 5.2d, we found the maximum gain in photocurrent can be obtained when inserting
two bilayers of O3 molecules between D/A interface. This is suggesting that such
separation between donor and acceptor materials is optimal to suppress the electron-hole
recombination through CT state at the interface after a suitable thickness of spacer layer
insertion. The subsequent decrease of the photocurrent is associated with the decrease of
efficiency for direct CT process.
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Figure 5.2 (a) Scheme of bilayer device structure used in this study, the interface layer with
vertical alignment was produced by Langmuir-Blodgett (LB) method illustrated on right.
(b) photocurrent measured at short circuit condition as function of illumination wavelength
for bilayer device without spacer and with 1-bilayer of O3 between P3HT and C60. (c)
GIWAXS map for 1 bilayer of O3 deposited on P3HT by LB method. (d) peak photocurrent
at 550 nm illumination as a function of number of O3 bilayer at interface.
At open circuit condition, the photo-voltage value is determined by the carrier
generation and recombination, following the equation:
𝑉𝑂𝐶 = 𝐸𝑔 −𝑛𝐾𝑏𝑇
𝑞𝑙𝑛(
𝐽𝑅𝑒𝑐
𝐽𝑆𝐶) (5.1)
where Eg is the effective band gap, Kb is the Boltzmann constant, T is the temperature, q is
the elementary charge, JRec is the recombination current and JSC is the final short circuit
photocurrent density.64 Considering a dielectric constant of 3~4 in a typical organic system,
the Coulomb radius is estimated to be 16 nm,65 and the energetic disorder further reduces
this value to 4 nm, probed experimentally.11 This means that if the electron-hole pair
bonded at the charge transfer states can be separated to a distance greater than the Coulomb
radius, significant amount of carrier recombination can be suppressed, that reduces JREC
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(10
-9 A
)
700650600550500450400Wavelength (nm)
No spacer
1-bilayer O3
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and increases JSC, which ultimately reduces the voltage loss through non-radiative
recombination. In fact, our experimental data in Figure 5.2 show an increase in JSC
(photocurrent at short circuit condition) which is an indication of reduced recombination
that will lead to increase in VOC.
The purpose of the following electronic structure calculations is to show that such
a large electron-hole separation is energetically feasible using the detailed interface models
shown in Figure 5.1. At this point it should be mentioned that in the single spacer level
model used because of computational economy, the electron-hole pair will have a
separation of only ~3 nm.
5.4.2 Analysis of Energy Levels
The geometrical arrangement of the three components of the standard model is
depicted in Figure 5.1. In this model only a single T3 layer is used because of computational
efficiency. Nevertheless, as the results below will show, the major stabilization effects for
charge separated states can be seen already in this case. The spacer T3 chain faces the C60
unit via the thiol group with interatomic distances of 2.6 – 2.8 Å and P3HT via the alkane
chain at distances of 2.7 – 2.85 Å. The distance between the center of C60 and P3HT is
~28.6 Å. In the extended model (Figure 5.1b, c) three C60 units are arranged horizontally
with distances of around 3.3 Å. The three T3 chains are stacked (side view, Figure 5.1c).
Typical intermolecular distances are displayed in Figure 5.1.
The calculations on the standard trimer have been used to establish the general
structure of the electronic spectrum and to test the validity of the smaller 6-31G basis set
for use with the significantly larger extended trimer. The scheme of the lowest excited
singlet states of the isolated complex is given in Figure 5.3a, b using the 6-31G* and 6-
31G basis sets, respectively. The full list of the 20 excited singlet state energies Ω
calculated can be found in Table C.1 (6-31G basis) and Table C.2 (6-31G* basis) of
Appendix C. For the 6-31G* basis (Figure 5.3a, isolated system), the local C60 states have
been calculated as well. This figure shows that in case of the calculation for the isolated
system they form the lowest band of excitonic states. If this would be the true energy
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ordering, these states would act as a sink for conversion of CT states, which would lead to
serious losses for the efficiency of the photovoltaic process. The C60 states are followed by
a local π-π* state on P3HT which has the largest oscillator strength, f, of all states
calculated. Next appears another band of C60 states followed by two closely spaced triples
of CT states from P3HTC60 and T3C60 at 3.3 and 3.4 eV, respectively. There are two
more locally excited states, one on T3 and the other one on P3HT. Triples of CT states
alternating between excitations from P3HT and T3 follow. This sequence is interrupted
only by one local state on P3HT. Inclusion of the environment via the LR method Figure
5.3a, b and Table C.4 and Table C.5) does not change much in the relative ordering of
states compared to the isolated system. However, the more sophisticated, but also
numerically more expensive SS solvent calculations change the situation dramatically
(Figure 5.3a, Table C.6). With respect to the P3HT excitonic bright state, the SS approach
leads to a quite dramatic stabilization of nine CT states to be located below the P3HT state.
The energetic spacing between the P3HT→C60 CT states and the T3→C60 CT states is
rather large being about 0.6 eV. There is also a large energy gap of about 0.75 eV between
the P3HT state and the following T3→C60 CT state. Thus, from this model one can already
see the crucial environmental effect which significantly stabilizes the CT state relative to
the others. On the other side, the environmental effect on the energetic location of the
locally excited P3HT and C60 states is small. As a result, the CT states are the lowest ones
and the leaking of the CT states into locally excited C60 states is avoided.
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Figure 5.3 Excited state energies Ω of the standard P3HT-T3-C60 trimer using the CAM-
B3LYP method and (a) the 6-31G* and (b) the 6-31G basis sets both as isolated systems
and in the LR and SS environments (for numerical data see Table C.1, Table C.2, and Table
C.4-Table C.8). For the isolated system using the 6-31G* basis set (farthest left of (a)), the
fifteen lowest-energy locally excited C60 states have been computed also. Excited states of
(c) the extended P3HT-T3-C60 trimer using the 6-31G* and (d) 6-31G basis sets both as
isolated systems and in the SS environment (Table C.9-Table C.11).
The strongly different behavior of the two solvation models is to be expected since
in the LR method dynamic solvent polarization is computed from the transition density
which is, however, very small for the present case of charge transfer states. Consequently,
the solvent effects are correspondingly small and not adequately represented by this
approach in the case of charge transfer states. On the other hand, the SS approach has been
developed having charge-transfer transitions in mind by representing the dynamic solvent
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polarization by means of the difference of the electronic densities of the initial and final
states. For a more detailed discussion of this point see e.g. Ref. 66. Therefore, it is
concluded that the SS scheme allows for a better characterization of environmental effects
on CT states and the LR approach will not be considered further here.
Comparison between the 6-31G* and 6-31G basis sets shows that the smaller 6-
31G basis represents the energy spectrum of the standard complex very well and also that
omission of the C60 states by freezing respective orbitals does not affect the remaining
spectrum. These findings are important for the calculations on the extended trimer where
the SS solvent calculations are time consuming. Therefore, they have been performed only
with the 6-31G basis (see below).
The extended trimer was investigated next as an extension of our standard trimer
structure providing a significantly enhanced interface model. (Figure 5.3c, d, Table C.9-
Table C.11). For the isolated system, the lowest bright state is located on P3HT at ~2.65
eV, very similar to the excitation energy in the standard model. The difference to the
standard model is that now several (T3)3→(C60)3 CT states are located below even for the
isolated complex. A rich variety of states is also found above the P3HT state (Figure 5.3c).
Including the SS environment using the 6-31G basis, Figure 5.3d, results in
significant changes with respect to the CT states, the most important one being that the
P3HT CT states are lowest in energy. Below the P3HT bright excitonic state, we computed
nineteen CT states: fifteen (T3)3→(C60)3 CT states and four P3HT→(C60)3 CT states. The
(T3)3→(C60)3 CT states occupy a CT band of 0.44 eV, while the P3HT→(C60)3 CT states
are found in a much narrower range of 0.02 eV. As has already been found for the standard
model, the mentioned CT states are located significantly below the C60 states (which are
around 2.5 eV) and that, therefore, the former states cannot leak into C60 states as soon as
they are formed. In contrast to the results presented here, in TD-DFT thiophene/fullerene
complexes32 locally excited C60 states comprised the lowest-energy band of states. These
differences are probably due to the use of the LR solvation model used in this reference. It
is important to point out that the presence of any -conjugated chromophores adds an
effective dielectric medium and facilitates further stabilization of CT states even though
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the wavefunction is not directly delocalized into these molecules. To illustrate this trait, we
removed the spacer and calculated only the C60/P3HT system. Table 5.1 records the
calculated energies of the lowest P3HT→C60 CT states. In the P3HT-T3-C60 system, the
solvent stabilization is 1.62 eV, but in the P3HT-C60 system only 0.42 eV. This comparison
shows the strong sensitivity of the electrostatic interaction in the charge separated systems
which are better stabilized by the environment in the case where the charge separation is
more pronounced due to the spacer in between.
Table 5.1 Stabilization energy of the lowest P3HT→C60 CT state of P3HT-T3-C60 and
P3HT-C60 system calculated in the isolated system and the SS environment.
P3HT-T3-C60 P3HT-C60
Isolated system 2.91 2.20
SS environment 1.29 1.78
ΔE (eV) 1.62 0.42
5.4.3 Characterization of Electronic States
The properties of the different types of electronic states are characterized in Figure
5.4 by means of two descriptors. The first one is represented by the most important NTO’s
of typical electronic transitions of the systems calculated in the SS environment (Figure
5.4). All transitions are well represented with one hole/particle pair as the corresponding
weights are above 0.9 e. As discussed above, the lowest excited singlet state band has
P3HT→(C60)3 CT character. This can be nicely seen from the NTO picture of Figure 5.4a,
b, which shows, respectively, the hole density in P3HT and the particle density in the
central C60 unit. These two densities are well separated with essentially no overlap between
the two systems. As second descriptor, the MEP plot displayed in Figure 5.4c shows that
the P3HT unit has a strongly positive potential which extends significantly into the (T3)3
moiety. The electrostatic potential of the middle C60 molecule of the (C60)3 unit is strongly
negative in agreement with the location of the particle NTO in Figure 5.4b. The other two
C60 molecules are slightly negative as well. Overall, the MEP plots give a stronger
delocalized picture of the CT process in comparison to the NTOs, since the localized CT
density affects much larger regions via Coulomb interactions.
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Figure 5.4 Natural Transition Orbitals (NTO), particle/hole populations and MEP of (a, b,
c) the lowest-energy P3HT→(C60)3 CT state, (d, e, f) the lowest-energy T3→(C60)3 CT
state, (j, k, l) the lowest-energy P3HT bright excitonic state of the extended thiophene
trimer calculated using the CAM-B3LYP/6-31G method in the SS environment, and (g, h,
i) the lowest-energy C60 dark excitonic state of the isolated standard thiophene trimer
calculated using the CAM-B3LYP/6-31G* method. Here we use NTO isovalue ±0.015
e/Bohr3 and MEP isovalue ±0.0003 e/Bohr3.
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The next band consists of (T3)3(C60)3 CT states. Here the lowest-energy state of
this type (Figure 5.4d,e) shows a transition from an occupied orbital diffusely distributed
across the (T3)3 unit to an orbital located on a single C60 molecule in the (C60)3 unit. The
MEP plot given in Figure 5.4f depicts similar character to the P3HT→(C60)3 delocalized
CT character, however as expected the positive region is spread over the T3 unit and P3HT
is essentially neutral.
The C60 exciton band is represented in Figure 5.4g,h by the lowest-energy C60 state
calculated for the isolated standard trimer. This band is optically dark consisting of an
excitation localized to the C60 molecule, while the MEP plot (Figure 5.4i) shows very little
polarity, which is typical for neutral excitonic states. For the P3HT bright excitonic state,
the primary contribution to the electronic excitation is localized on the P3HT unit (Figure
5.4j,k). The MEP plot of this state displayed in Figure 5.4l shows only little polarity.
5.5 Conclusions
Efficient operation of organic solar cells based on the BHJ architecture is ensured
by tuning sophisticated organic-organic interfaces to align excitonic and charge transfer
excited states to provide robust photo-generated exciton dissociation, while minimizing the
recombination of the dissociated electron and hole across the donor-acceptor interface. The
present study focuses on a detailed molecular model of a polar interface in a bulk
heterojunction, which is suggested by our experiment. Here a self-assembled spacer unit
(oligomer T3) is included in between a P3HT donor and C60 acceptor materials. Following
a non-trivial geometry optimization routine, extensive TD-DFT calculations based on long-
range corrected functional mode and including environmental effects have been performed
to characterize the series of lowest excited singlet states in terms of charge transfer and
excitonic character. The calculations show unequivocally the strong stabilization of the CT
states by environmental effects. The amount of stabilization increases with the distance of
the charges (P3HT→C60 vs. T3→C60 CT) due to improved environmental stabilization. By
comparison with a system of an ion pair without spacer layer (P3HT→C60), a stabilization
of more than 1 eV is computed for the single layer model. Inclusion of more layers are
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expected to increase the stabilization somewhat further but are currently too expensive in
the framework of an explicit molecular model and state-specific TDDFT. Both spacer
models show that with consideration of environmental interaction the lowest excited states
are formed by bands of CT states, the lowest ones being of P3HTC60 character followed
by T3C60 transitions. The bright excitonic P3HT state is located 0.5 eV above the first
CT below and 1.15 eV above the lowest CT state. Local C60 excitations were found below
the P3HT state, but significantly (~0.9 eV) above the lowest CT state.
Such calculated and analyzed electronic structure fully support our hypothesis
which emerged from experimental study1: photo-generated exciton in the P3HT component
at the BHJ interface either goes to C60 via energy transfer and subsequently dissociates into
a T3C60 CT state, or directly dissociates into a P3HTC60 state. This state is followed
by a transition to the next P3HTC60 state with lower energy, which has even stronger CT
character. Our calculation of the stability of the CT states clearly indicate via the use of
Equation 5.1 that the spacer facilitates separation of the electron and hole, thus diminishing
their Coulomb binding energy and reducing the efficiency of recombination. Moreover,
our simulations demonstrate how the interface ordering and structure affects the energetics
of relevant states via electronic coupling in self-assembled oligomer layer, which further
translates to macroscopic device performance. Such physical processes are general and
underpin manipulation in any donor-acceptor based electronic devices, such as organic
light emitting diodes, photodetectors, sensors, and comprise the design principles for
tailoring the interface properties of organic electronic devices. All our simulations of
electronically excited states are based on the equilibrium molecular geometries. To capture
the effects of vibrational broadening and conformational disorder, it is possible to use
multiple snapshots of structures obtained from classical molecular dynamics simulations.
However, modeling excited states in such a case is currently a numerically intractable task
for the present high level of theory. Likewise, direct modeling of excited state relaxation
with, for instance, surface hopping approaches,6762 are numerically costly and require
reduced Hamiltonian models. This work goes far beyond the scope of this paper and will
be addressed in future investigations.
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5.6 Acknowledgments
The work at Los Alamos National Laboratory (LANL) was supported by the LANL LDRD
program (W.N., H.T., A.D.M., S.T.). This work was done in part at Center for Nonlinear
Studies (CNLS) and the Center for Integrated Nanotechnologies CINT, project no.
2017AC0027), a U.S. Department of Energy and Office of Basic Energy Sciences user
facility, at LANL. This research used resources provided by the LANL Institutional
Computing Program. We also thank for computer time at the Linux cluster arran of the
School for Pharmaceutical Science and Technology of the University of Tianjin, China.
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Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A.;
Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.;
Zheng, G.; Sonnenberg, J. L.; Hada, M.; Ehara, M.; Toyota, K.; Fukuda, R.;
Hasegawa, J.; Ishida, M.; Nakajima, T.; Honda, Y.; Kitao, O.; Nakai, H.; Vreven,
T.; Montgomery Jr., J. A.; Peralta, J. E.; Ogliaro, F.; Bearpark, M. J.; Heyd, J.;
Brothers, E. N.; Kudin, K. N.; Staroverov, V. N.; Kobayashi, R.; Normand, J.;
Raghavachari, K.; Rendell, A. P.; Burant, J. C.; Iyengar, S. S.; Tomasi, J.; Cossi,
M.; Rega, N.; Millam, N. J.; Klene, M.; Knox, J. E.; 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.; Martin, R. L.; Morokuma, K.;
Zakrzewski, V. G.; Voth, G. A.; Salvador, P.; Dannenberg, J. J.; Dapprich, S.;
Daniels, A. D.; Farkas, Ö.; Foresman, J. B.; Ortiz, J. V.; Cioslowski, J.; Fox, D. J.
Gaussian 09, Gaussian, Inc.: Wallingford, CT, USA, 2009.
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(62) Smith, K. A.; Lin, Y. H.; Mok, J. W.; Yager, K. G.; Strzalka, J.; Nie, W. Y.; Mohite,
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H. L., Synthesis, electrochemistry, STM investigation of oligothiophene self-
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CHAPTER VI
BENCHMARK AB INITIO CALCULATIONS OF POLY(P-
PHENYLENEVINYLENE) DIMERS: STRUCTURE AND EXCITON
ANALYSIS
Authors: Reed Nieman,a Adelia J. A. Aquino,a,b and Hans Lischkaa,b
Affiliations: aDepartment of Chemistry and Biochemistry, Texas Tech University,
Lubbock, TX 79409-1061, USA, bSchool of Pharmaceutical Sciences and Technology,
Tianjin University, Tianjin 300072, PR China
6.1 Abstract
Crucial benchmark calculations of the structure and electronic excited states
missing from the current literature were performed for oligomers of poly(p-
phenylenevinylene) (PPV) dimers using high-level scaled opposite-spin (SOS) second-
order Møller-Plesset (SOS-MP2) and second-order algebraic diagrammatic construction
(SOS-ADC(2)) calculations. Comparisons were also made to DFT methods. The dimer
structures were studied in two conformations: sandwich-stacked and displaced; the latter
exhibited greater stacking interactions. Good agreement was found between the cc-pVQZ
and aug-cc-pVQZ basis sets and the cc-pVTZ basis. Comparisons of the interaction
energies were made to the complete basis set limit. Basis set superposition error was also
accounted for. The excited state spectra presented a varied and dynamic picture that was
analyzed using transition density matrices and natural transition orbitals (NTOs).
6.2 Introduction
Poly(p-phenylenevinylene) (PPV) is paradigmatic for the study of the mechanisms
affecting the photophysical excitation energy transfer (EET) behavior of organic π-
conjugated polymers on the molecular level that allows the fine-tuning of their absorption
and emission properties which make them so attractive as materials for use in
electroluminescent and photovoltaic devices.1-6 Experiments conclude that features of
coherent PPV dynamics can be attributed to structural effects from coupling to a vibrational
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mode,5, 7 and structural relaxation is connected to the evolution of excited states necessary
for EET dynamics.
Accurate electronic structure calculations are a prerequisite for the characterization
of these properties, but are hindered by the necessity of large molecular systems and
reliable descriptions of excited states. The Pariser-Parr-Pople (PPP) π electron
Hamiltonian combined with a mechanical force field described the effect of the σ electrons
has been used previously to model surface-hopping dynamics of PPV,8 while later the
Pariser-Parr-Pople-Peierls (PPPP) Hamiltonian was used together with the Configuration
Interaction Singles (CIS) method to compute and classify PPV excited states.9 Calculations
of PPV chains with the collective electronic oscillator (CEO) method together with the
Austin Model 1 (AM1) approach showed exciton self-trapping over six PV units and
demonstrated close agreement with experimental spectra.10 The vibronic structure of the
lowest optical transition of PPV was found using the CIS method with an empirical
description of the electron-phonon coupling.11 A time-dependent density functional theory
(TDDFT) method was used to calculate the singlet-triplet splitting of PPV among other π-
conjugated systems.12 The exchange-correlation (XC) functional dependence of the
localization of the electronic excitation was investigated for polymeric PPV chains.13
Charged polaron localization was found to be sensitive to the polarization of the
environment whereas neutral states are not affected to the same degree.14 A diabatic
Hamiltonian has been developed for use studying wavepacket dynamics in the torsional
modes of PPV vinylene single bonds.15 The kinetic Monte-Carlo method was used to model
fluorescent depolarization to study the interactions between linked chromophoric units in
poly-[2-methoxy-5-((2-ethylhexyl)oxy)-phenylenevinylene (MEH-PPV).16
Though more computationally expensive than semi-empirical or TDDFT
calculations, ab initio methods have also been used to model PPV and PPV-like systems,
though they focus on smaller molecular sizes. The electronic spectrum of stilbene (PPV2)
has been investigated using complete active space perturbation theory to second-order,17
and the electronic spectrum of PPV oligomers up to four repeat PV units (PPV4) has been
studied with symmetry-adapted cluster-configuration interaction (SAC-CI).18 Multi-
configurational time-dependent Hartree (MCTDH) simulations found that dynamic
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electron-hole transients involve rapid expansion and contraction of the exciton coherence
size in PPV-type systems.19
There is a need to calculate larger oligomer lengths and the approximate coupled-
cluster method to second-order (CC2) has shown to be adequate for the calculations of
methylene-bridged oligofluorenes and oligo-p-phenylenes.20-21 The resolution of the
identity method (RI) together with CC2 has allowed for efficient, less expensive
calculations of excited states as well as having analytical gradients available.22 The second-
order algebraic diagrammatic construction (ADC(2)) is related to the CC2 method and
provides similar results, though it has an advantage over the CC2 method in that excited
states are obtained as eigenvalues of a Hermitian matrix instead of a non-Hermitian Jacobi
matrix as with the coupled-cluster response in the CC2 method.23 Calculations based on
the ADC(2) method investigated the exciton and charge transfer (CT) states of nucleobase
dimers.24 Benchmark calculations using the equation of motion excitation energy coupled-
cluster (EOMEE-CC) method on DNA nucleobases showed that CC2 was able to
reproduce π-π* excitations well compared to EOMEE-CC with singles, doubles, and non-
iterative triples (CCSD(T)).25 CC2 has shown deficiencies however in describing n-π*
states, though these are not types of excitations normally relevant to the UV spectrum of
PPV.25
Excited state properties of PPV oligomers has been calculated in the past by the RI-
ADC(2) method when investigating vertical excitation energies, torsional potentials, and
defect localization in the S1 state in varying sized oligomers.26 The ADC(2) method has
also been used to calculate the absorbance and fluorescence band spectra of PPV oligomers
of increasing chain length,27 while high-level ADC(3) benchmark calculations on PPV
oligomers have shown that the ADC(2) method is adequate for excited state energies,
oscillator strengths, and wave function behavior.28 The scaled opposite spin (SOS)
approach applied to ADC(2) (SOS-ADC(2)) described well the pi-pi* electronic excited
states of several electron donor-acceptor complexes with excitation energies that compared
favorably to experimental results.29
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Most calculations of PPV oligomers focus only on single chain systems, with few
calculations of dimer systems. Semi-empirical calculations were carried out using the CEO
method together with the AM1 approach to show the dependence on dimer conformation
and intermolecular distance of the formation of interchain electronic excitations in PPV
dimers.30 The AM1 method was used in another investigation finding that the luminescence
capabilities of cofacial PPV dimers is strongly quenched by transition from ground state
to the first excited state.31 The Intermediate Neglect of the Differential Overlap (INDO)
Hamiltonian together with the Single Configuration Interaction (SCI) approach was used
to show that the most stable photogenerated species in interacting PPV chains are
intrachain excitons.32 Mixed quantum/classical calculations of non-adiabatic excitation
processes of long-chain, stacked PPV oligomers showed that the photoexcitation process
leads to thermally-driven polaron hopping between adjacent chains.33 A study on linearly
coupled PPV chains showed the effects of intrinsic conjugation defects on the mobility of
excitons using a tight-binding method with the Su–Schrieffer–Heeger (SSH) model for
which an interchain coupling term is included in the Hamiltonian.34 TDDFT comparisons
of the method to experimental two-photon absorption measurements of PPV derivatives
showed that the hybrid exchange-correlation density functional, Becke three-parameter
Lee, Yang, and Parr (B3LYP) compared well to experimental observations.35 Experimental
results on interchain interactions suggest that interchain electronic phenomena such as
excimers may be responsible for the promotion of charge transport in PPV chains.36-38
The goal of this work is to provide crucial reference data lacking in the literature
by characterizing the geometry and excited states of PPV oligomer dimer systems with
high-level ab initio calculations and wave function analysis tools. To that end, PPV
oligomers chains of from two (stilbene) to four repeating phenylenevinylene (PV) units
(referred to here as PPV2, PPV3, and PPV4) were arranged in either in stacked-sandwich or
displaced dimer conformations. The basis set dependence of the geometric characteristics
was investigated using the SOS approach39-40 applied to second-order Møller-Plesset
Perturbation theory (MP2)41 with several correlation-consistent basis sets with emphasis
on extrapolation of the stabilization energy to the complete basis set (CBS) limit. Excited
states and S1 excited state geometries were studied using the SOS-ADC(2) method.42-43
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The SOS-ADC(2) method was also compared to DFT utilizing B3LYP and the long-range
corrected Coulomb Attenuating Method with B3LYP (CAM-B3LYP).
6.3 Computational Details
The geometry of PPV dimers was investigated using PPV chains of from two to
four repeating PV units arranged in either stacked-sandwich, belonging to the C2h point
group, or displaced, belonging to the C1 point group, structural conformations (Figure 6.1).
Geometry optimizations for both structural conformations were carried out using the SOS-
MP2 method using the RI approximation.44-45 A variety of Dunning-type correlation
consistent basis sets were used in order to show the dependence of the basis on the
intermolecular stacking distances and the stabilization energies. For sandwich-stacked
PPV2 dimer, the geometry was studied with the largest number of basis sets, cc-pVXZ, and
aug-cc-pVXZ where X=D, T, and Q,46-47 as its relatively small size allowed for more
extensive investigations. Smaller basis sets were considered for sandwich-stacked PPV3
and PPV4 dimers, cc-pVDZ and cc-pVTZ. Though geometry optimization calculations for
the sandwich-stacked PPV3 and PPV4 dimers with the cc-pVQZ basis set would be too
computationally taxing, single-point calculations were made for both systems with the cc-
pVQZ basis set at the cc-pVTZ optimized geometry. The displaced dimers lack the
symmetry advantages of the sandwich-stacked dimers and so were treated with a fewer
basis sets, cc-pVXZ (X=D, T, and Q) and aug-cc-pVDZ for PPV2 dimer, cc-pVDZ and cc-
pVTZ for PPV3 and PPV4. For the sandwich-stacked PPV2 dimer, the effect of using the
SOS approach is compared to geometries optimized with the MP2 method. Sandwich-
stacked PPV3 dimer geometries were compared to density functional theory (DFT)
methods using the B3LYP48 and CAM-B3LYP49-50 methods together with the D351
dispersion correction approach and the cc-pVTZ basis set. Basis set superposition error
(BSSE) was included using the counterpoise (CP) correction method52 for MP2 and SOS-
MP2 calculations.
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Figure 6.1 Top/side views of the stacked PPV oligomers and labeling scheme.
Extrapolation to the CBS limit of the stabilization energies was carried out using
the two-point fit approach given by Halkier et al in Equation 6.1 for the TQ extrapolation,53
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𝐸𝑇𝑄∞ =
𝐸𝑄𝑄3 − 𝐸𝑇𝑇3
𝑄3 − 𝑇3 (6.1)
where 𝐸𝑇𝑄∞ is the CBS limit stabilization energy, 𝑄 and 𝑇 are the cardinal numbers four
(for the cc-pVQZ or aug-cc-pVQZ basis sets) and three (for the cc-pVTZ or aug-cc-pVTZ
basis sets), respectively, and 𝐸𝑄 and 𝐸𝑇 are the stabilization energies found for those
individual basis sets.
Vertical excitations were calculated using the SOS-ADC(2) method with the RI
approximation and the basis sets specified earlier for stacked PPV2, but only cc-pVTZ was
considered for the stacked PPV3 and PPV4 dimers. Two states per irreducible
representation were calculated for each case. The optimized geometry of the S1 excited
state was found for each stacked dimer using the SOS-ADC(2) method with the cc-pVTZ
basis set. For stacked PPV3 dimers, comparative TDDFT calculations were carried out
using the B3LYP-D3, CAM-B3LYP-D3 methods and the cc-pVTZ basis set. Due to the
computational efficiency afforded by TDDFT methods, thirty states were calculated with
each method.
All excited states were analyzed via natural transition orbitals (NTO)54 and the
NTO participation ratio (PRNTO), while charge transfer between fragments for a given
electronic transition was described using the q(CT) value55 computed from the transition
density, 𝐷0𝛼. The value, q(CT), is calculated first by using the transition density to define
the omega matrix, Ω𝐴𝐵𝛼 ,
Ω𝐴𝐵𝛼 =
1
2∑(𝐷0𝛼.[𝐴𝑂]𝑆[𝐴𝑂])
𝑎𝑏(𝑆[𝐴𝑂]𝐷0𝛼.[𝐴𝑂])
𝑎𝑏𝑎∈𝐴𝑏∈𝐵
(6.2)
such that α is the electronic state and AO are the atomic orbitals. The omega matrix, Ω𝐴𝐵𝛼 ,
describes the charge transfer contribution from fragment A to fragment B (A ≠ B), and
contributions from excitations occurring on the same fragment (A = B). The CT character
for a given system composed of multiple fragments is then given by q(CT):
𝑞(CT) =1
Ω𝛼∑ ∑ Ω𝐴𝐵
𝛼
𝐴≠𝐵𝐴
(6.3)
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where Ω𝛼 is the sum of the charge transfer values for all fragment pairs, A and B. When
q(CT) = 1 e, one electron has been transferred, and when q(CT) = 0 e, the transition is local
a excitation or Frenkel exciton state.
The value of PRNTO indicates the number of essential participating NTOs and,
therefore, how many configurations are important to the description of the excited state.
PRNTO is calculated as:
𝑃𝑅𝑁𝑇𝑂 =(∑ 𝜆𝑖𝑖 )2
∑ 𝜆𝑖2
𝑖
(4)
where 𝜆𝑖 is the weight for each hole-to-electron orbital transition.
All MP2, SOS-MP2, and SOS-ADC(2) calculations were performed using the
TURBOMOLE program package,56 and all DFT and TDDFT calculations were done with
the Gaussian G09 program package.57 Excited states were analyzed via natural transition
orbitals (NTO), q(CT), and omega matrices utilizing the wave function analysis program
TheoDORE.55, 58-59
6.4 Results and Discussion
6.4.1 Structural Description
The geometry of stacked PPV2 has been optimized utilizing the SOS-MP2 method
at several basis sets and described via intermolecular stacking distances in Table 6.1 with
labeling provided in Figure 6.1. In generally, these distances for the phenylene units were
larger than the vinylene unit. The aug-cc-pVDZ and aug-cc-pVTZ basis sets show
considerably smaller intermolecular distances compared to the cc-pVDZ and cc-pVTZ
basis sets, respectively, especially going toward the interior. Going from cc-pVDZ to cc-
pVTZ, the distances decrease only slightly and the structure found for cc-pVTZ agreed
well with the cc-pVQZ structure. The distances found for the cc-pVQZ and the aug-cc-
pVQZ basis sets are nearly identical. BSSE counterpoise corrections showed increased
intermolecular stacking distances of an average of about 0.37 Å for the cc-pVDZ basis set.
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This increase in distance is an average of about 0.14 Å for the cc-pVTZ basis, and is
relatively unchanged for the aug-cc-pVDZ and cc-pVQZ basis sets. In spite of the
relatively large changes in the distances, the differences in distances between the basis sets
with and without the diffuse functions and the BSSE corrections do converge, but only at
the QZ basis. Finally, when using the MP2 method without the SOS approach (Table D.1),
the stacking distances were smaller in general with each basis set utilized than those found
with the SOS-MP2 method, an effect that extended to the BSSE corrections as well.
Table 6.1 Interchain distances (Å) and ΔE (kcal/mol) in the stacked PPV2 dimer computed
at SOS-MP2 and SOS-MP2/BSSE levels with several basis sets using C2h symmetry.
1-1’ 2-2’ 3-3’ 4-4’ 5-5’ ΔE
SOS-MP2
cc-pVDZ 3.83 3.82 3.80 3.85 3.86 -3.99
aug-cc-pVDZ 3.67 3.61 3.61 3.66 3.72 -12.24
cc-pVTZ 3.80 3.79 3.77 3.82 3.83 -5.41
aug-cc-pVTZ 3.80 3.63 3.64 3.68 3.86 -8.26
cc-pVQZ 3.81 3.80 3.77 3.83 3.84 -5.07
aug-cc-pVQZ 3.81 3.80 3.77 3.83 3.84 -5.96
CBS limit extrapolation:
TQa -4.79
aug-TQb -4.27
SOS-MP2/BSSE
cc-pVDZ 4.21 4.18 4.17 4.21 4.23 -1.26
aug-cc-pVDZ 3.67 3.62 3.61 3.67 3.72 -2.58
cc-pVTZ 3.95 3.93 3.93 3.94 3.96 -3.40
cc-pVQZ 3.81 3.80 3.77 3.83 3.83 -4.04
CBS limit extrapolation:
TQa -4.51
Stacked PPV3 (Table 6.2) showed only a slight decrease in the intermolecular
stacking distances when going from cc-pVDZ to cc-pVTZ. BSSE counterpoise corrections
again calculate much larger distances between monomers. The average intermolecular
distances found using the cc-pVDZ and cc-pVTZ basis sets for stacked PPV3 are only
marginally smaller than those found for stacked PPV2 using the SOS-MP2 method.
Similarly, there were only small differences between the two stacked systems for the two
basis sets after BSSE counterpoise correction. Comparing the SOS-MP2 method to DFT
methods at the cc-pVTZ basis set for stacked PPV3, the stacking distances are marginally
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smaller when the B3LYP-D3 method was used, but are larger by about 0.05 Å with the
long range-corrected CAM-B3LYP-D3 method. The intermolecular stacking distances of
stacked PPV4 are shown in Table D.2 for the SOS-MP2 and SOS-MP2/BSSE methods. An
increase in basis set size from cc-pVDZ to cc-pVTZ showed a similar in intermolecular
distances as with stacked PPV3, and the average stacking distance decreased in general for
stacked PPV4 compared to stacked PPV2.
Table 6.2 Interchain distances (Å) and ΔE (kcal/mol) in the stacked PPV3 dimer computed
at SOS-MP2 and SOS-MP2/BSSE levels with several basis sets using C2h symmetry.
1-1’ 2-2’ 3-3’ 4-4’ 5-5’ 6-6’ 7-7’ 8-8’ ΔE
SOS-MP2
cc-pVDZ 3.82 3.81 3.77 3.77 3.77 3.80 3.82 3.83 -7.12
cc-pVTZ 3.81 3.81 3.76 3.76 3.76 3.79 3.82 3.83 -9.54
cc-pVQZa - - - - - - - - -8.97
CBS limit extrapolation:
TQb -8.55
SOS-MP2/BSSE
cc-pVDZ 4.16 4.15 4.07 4.08 4.07 4.14 4.18 4.18 -2.64
cc-pVTZ 3.94 3.93 3.88 3.87 3.87 3.91 3.95 3.96 -6.26
cc-pVQZa - - - - - - - - -7.37
CBS limit extrapolation:
TQb -8.18
B3LYP-D3
cc-pVTZ 3.77 3.83 3.75 3.75 3.73 3.79 3.83 3.77 -
10.62 CAM-B3LYP-D3
cc-pVTZ 3.89 3.84 3.81 3.80 3.80 3.85 3.85 3.89 -8.99 aSingle-point energy at SOS-MP2/cc-pVTZ optimized geometry, bExtrapolation made
using cc-pVTZ and cc-pVQC
The sandwich-stacked PPV dimers were compared to a displaced dimer geometry
configuration using the SOS-MP2 method, Table 6.3. Displaced PPV2 dimer displayed
smaller intermolecular distances than the sandwich stacked PPV2 dimer by an average of
about 0.3 Å, but showed similar trends. The intermolecular distances increased from cc-
pVDZ to cc-pVQZ, with increasing basis set size. Displaced PPV3 dimer, Table 6.4, using
the cc-pVTZ basis, had smaller intermolecular distances compared to the sandwich-stacked
PPV3 dimer and displaced PPV2 dimer using the same basis set. The intermolecular
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distances of displaced PPV4, Table D.3, were smaller compared to the sandwich-stacked
PPV4 dimer by about 0.3 Å, but larger than either displaced PPV2 or PPV3 dimers.
Table 6.3 Interchain distances (Å) and ΔE (kcal/mol) in the displaced PPV2 dimer
computed at SOS-MP2 level using C1 symmetry.
1-1’ 2-2’ 3-3’ 4-4’ ΔE
cc-pVDZ 3.52 3.44 3.44 3.52 -7.69
cc-pVTZ 3.54 3.44 3.45 3.54 -8.73
cc-pVQZ 3.54 3.45 3.45 3.54 -8.12
CBS limit extrapolation:
TQa -7.68 aExtrapolation made using cc-pVTZ and cc-pVQC
Table 6.4 Interchain distances (Å) and ΔE (kcal/mol) in the displaced PPV3 dimer
computed at SOS-MP2 level using C1 symmetry
1-1’ 2-2’ 3-3’ 4-4’ 5-5’ 6-6’ ΔE
cc-pVTZ 3.35 3.43 3.39 3.39 3.34 3.53 -15.10
cc-pVQZa - - - - - - -14.04
CBS limit extrapolation:
TQb -13.28 aSingle-point energy at SOS-MP2/cc-pVTZ optimized geometry, bExtrapolation made
using cc-pVTZ and cc-pVQC
The stability of these PPV dimer systems was described by calculating the
interaction energy with comparison to the extrapolated CBS limit. For the stacked PPV2
dimer (Table 6.1), the extrapolation of the CBS limit interaction energy is found to be less
stabilizing than that found with the cc-pVDZ basis. The interaction energy given for the
aug-cc-pVDZ is greatly overstabilized compared to the CBS limit energy, a larger
difference than is found for any other case. The cc-pVQZ interaction energy is found very
close to the CBS limit. The interaction energy at the CBS limit using the SOS-MP2/BSSE
method was less stabilizing than that found for the SOS-MP2 method, but was more
stabilizing than the interaction energies found for the individual basis sets using
counterpoise corrections; using the cc-pVDZ basis was much less stabilizing than the CBS
limit by about 3.3 kcal/mol, though the interaction energy at the cc-pVQZ basis was only
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slightly less stable. The interaction energy of the displaced PPV2 dimer is more stabilizing
compared to the sandwich stacked system, reflective of the smaller intermolecular
distances, and the energy at the CBS limit compares most closely to that found using the
cc-pVQZ basis. This stabilization of π-conjugated aromatic systems coming from
displacement has been rigorously shown in previous investigations of benzene and
naphthalene dimers.60-65
Without the SOS approach, the MP2 method, Table D.1, the interaction energy was
found to be more stabilizing for all basis sets by about 6 kcal/mol without BSSE. This
showed the necessity of the SOS approach to correct the over-estimation of the stabilization
energy of dimer complexes when using the MP2 method. A similar effect was
demonstrated for stacked complexes of benzene and naphthalene with tetracyanoethylene
(TCNE) (benzene/TCNE and naphthalene/TCNE, respectively)29 where the MP2 method
alone considerably overestimated the interaction energy compared to the SOS-MP2
approach. Calculations with the highly correlated CCSD(F12*)(T) method were also
performed for the benzene/TCNE system (which was found to agree quite well with
experimental findings). The SOS-ADC(2) and CCSD(F12*)(T) methods differed only
slightly from each other.
The interaction energies of the sandwich-stacked PPV3 dimer (Table 6.2) using the
SOS-MP2 method at the CBS limit was less stable than when using the cc-pVQZ and cc-
pVTZ basis sets but was more stable than for the cc-pVDZ basis. CP corrections for the
interaction energy at the CBS limit became slightly less stable compared to the extrapolated
value with the SOS-MP2 method. Using the CAM-B3LYP-D3/cc-pVTZ method compares
well to the SOS-MP2 interaction energy at the CBS limit, being only a little more stable,
though the B3LYP-D3/cc-pVTZ method showed an over-stabilization of about 2 kcal/mol.
The interaction energy at the CBS limit of the displaced PPV3 dimer (Table 6.4) was about
4.7 kcal/mol more stable than for the sandwich-stacked PPV3 dimer and is less stable
compared to the cc-pVQZ basis set.
The CBS limit of the interaction energy found for the stacked PPV4 dimer (Table
D.2) was more stable than the cc-pVQZ energy, and the displaced PPV4 dimer (Table D.3)
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is stabilized by about 6 kcal/mol at the CBS limit compared to the stacked dimer. The CBS
limit for the displaced dimer is less stable than the cc-pVQZ basis.
Several general trends are apparent across all investigated structures. With
increasing basis set size (not including augmented basis sets or BSSE corrections), the
interchain stacking distances increase slightly and interaction energies become more
stabilizing. Decreases in stacking distances and increasingly stable interaction energies are
found with larger chain lengths. The displaced geometries have much smaller stacking
distances compared to the sandwich stacked dimers, and are greatly stabilized as a result.
The BSSE counterpoise correction largely overestimates the interchain stacking distances
and greatly decreases the stabilization energies. The augmented basis sets in general
displayed much smaller interchain stacking distances and greatly overestimated interaction
energies. In light of this, augmented basis sets and counterpoise BSSE corrections are not
recommended for intermolecular geometries.
6.4.2 Excited States
The excited states of the stacked PPV dimers were investigated using the SOS-
ADC(2) method and compared to TDDFT for stacked PPV3. The vertical excitations for
stacked PPV2 using the SOS-ADC(2)/cc-pVQZ method are presented in Table 6.5. Two
states per irreducible representation were calculated. The S1 state (1 1Bg) is dark, possesses
partial interchain CT character, and is found at 4.07 eV. Two intensive transitions, or bright
states, were found, the S4 and S6 states (1 1Bu and 2 1Bu, respectively) where the latter has
greater oscillator strength. Both were found to be local exciton states with negligible
interchain charge transfer. In the spectrum of the monomer (Table D.4), the bright states,
S1 and S3, occur at approximately the same energy as the dimer, though the oscillator
strength of the S2 state is reduced by about 0.8.
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Table 6.5 Stacked PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVQZ and SOS-
MP2/cc-pVQZ optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator
strength, CT – interchain charge transfer)
State Excitation ΔE (eV) f CT (e)
S1 1 1Bg 4.07 0.00 0.17
S2 1 1Au 4.49 0.00 0.09
S3 2 1Bg 4.50 0.00 0.08
S4 1 1Bu 4.52 0.83 0.01
S5 1 1Ag 4.64 0.00 0.02
S6 2 1Bu 4.75 1.18 0.01
S7 2 1Au 5.59 0.00 0.21
S8 2 1Ag 5.96 0.00 0.02
Omega matrices revealed more detailed information about the nature of each
excitation and was examined for stacked PPV2 dimer using the SOS-MP2/cc-pVQZ
method (Figure 6.2). The small amount of off-diagonal character seen in the dark S1 and
S7 states correlates with calculated CT values (Table 6.5). Distinct nodes between the
individual chains (fragments 1-3 belong to one chain and fragments 4-6 belong to the other)
are visible for remaining six excitations. The bright 1 1Bu and 2 1Bu excitations, the S4 and
S6 states, respectively, are characterized by local excitations in the phenylene groups.
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Figure 6.2 Omega matrices for vertical excitations of stacked PPV2 dimer using the SOS-
ADC(2)/cc-pVQZ method with the SOS-MP2/cc-pVQZ optimized geometry. Each
vinylene and phenylene group is a separate fragment: fragments 1-3 belong to one chain,
fragments 4-6 to the other. The y-axis represents the electron and the x-axis represents the
hole. Omega matrix plots calculated using the TheoDORE program package.
As with the structural investigation, the basis set dependence of stacked PPV2 was
studied comparing the cc-pVQZ basis (Table 6.5) to the cc-pVDZ and cc-pVTZ basis sets
(Table D.5). The relative ordering of the states remains constant. The S1 state is stabilized
with increasing basis set size. For each basis, the oscillator strengths of the bright states (S4
and S6) were found to be sensitive to the basis set. Compared to the monomers, the first
bright state was found to be well preserved, though the oscillator strength of the second
bright state is greatly reduced as was seen with the cc-pVQZ case. An exception to this
trend was the cc-pVDZ basis where the second bright state is stabilized by about 0.07 eV
and the oscillator strengths are increased for the two bright states.
The NTOs for the vertical excitations were plotted for stacked PPV2 (Table D.6).
Each excitation had contributions from at least two primary transitions. For the S1 state,
the primary contributing hole NTO is located on the separated phenylene and vinylene
groups while the electron NTO is somewhat mixed. This same intrachain fragment
interaction is present in several other states and explains some of the off-diagonal character
seen in the omega matrices (Figure 6.2). The NTOs of the partial CT state, S7, displays,
S1 S2 S3 S4
S5 S6 S7 S8
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while not much interchain interaction, transfer of electron density between intrachain
fragments.
The spectrum of displaced PPV2 dimer was calculated (Table 6.6) to compare to
the stacked dimer configuration showing non-negligible destabilization of the first several
excitations and the introduction of interchain CT states. Displacement from the stacked
conformation raises the energy of the S1 state compared to the stacked, though it remains
a dark, local exciton. The first bright state is now the S2 state and was destabilized in the
displaced dimer by about 0.08 eV. The S2 and S3 states in the stacked dimers were both
destabilized to above the bright state. Two pronounced intermolecular CT states, S7 and
S8, were found for the displaced dimer at approximately 5.4 eV. In the associated omega
matrices (Figure D.1), the CT states can be seen to have primarily involved the phenylene
units. This can be seen in the NTOs (Table D.7) for the S7 state, though the NTOs for the
S8 state are more complex. The S4 and S5 states showed unique intrachain CT character
compared to the sandwich-stacked dimer involving only electron/hole density on one
phenylene group per chain. These two excitations both have NTO participation ratios
(PRNTO) of about four in which the intrachain exciton character was represented well.
Table 6.6 Displaced PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and
SOS-MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy, f –
oscillator strength, CT – interchain charge transfer)
State Excitation ΔE (eV) f CT (e)
S1 1 1A 4.40 0.00 0.03
S2 2 1A 4.58 0.53 0.05
S3 3 1A 4.64 0.01 0.05
S4 4 1A 4.64 0.00 0.09
S5 5 1A 4.66 0.00 0.09
S6 6 1A 4.77 1.43 0.04
S7 7 1A 5.39 0.00 0.95
S8 8 1A 5.40 0.08 0.94
The spectra of stacked PPV3 (Table D.8) and PPV4 (Table D.9) show that the S1
states occur at lower energies compared to stacked PPV2 (Table 6.5), and are also lower in
energy compared to the experimental values by 0.22 eV 0.27 eV, respectivley. The bright
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states, S2 for both PPV3 and PPV4, are likewise stabilized considerably going to longer
chain lengths. In the omega matrices (Figure 6.3) of the vertical excitations of stacked
PPV3, five of the eight calculated states are similar in character to the excitations found in
stacked PPV2: transitions beginning and ending on the capping phenylene groups. The
remaining three states, S1, S2, and S4, are transitions primarily involving the interior
phenlylene groups. The excitation with the largest interchain CT character (Table D.8,
theS1 state) is seen to mostly involve the interior phenylene groups on each chain. Other
partial interchain CT states, S5 and S6, appear to have very similar transitions: mainly
phenylene-phenelyene with some phenylene-vinylene involvement. The S1, S2, and S8
states are shown to have the most phenylene-vinylene CT character. The omega matrices
of stacked PPV4 (Figure D.2) were similar in character to those of stacked PPV3.
Figure 6.3 Omega matrices for vertical excitations of stacked PPV3 dimer using the SOS-
ADC(2)/cc-pVTZ method with the SOS-MP2/cc-pVTZ optimized geometry. Each
vinylene and phenylene group is a separate fragment: fragments 1-5 belong to one chain,
fragments 6-10 to the other. The y-axis represents the electron and the x-axis represents the
hole. Omega matrix plots calculated using the TheoDORE program package.
The density of states is shown in Figure 6.4a,b,e for stacked PPV2, stacked PPV3,
and PPV4 using the SOS-ADC(2) method. The energy range containing the highest density
of states was stabilized from the 4.5-5.0 eV range in stacked PPV2 (four states) to the 4.0-
4.5 eV range in stacked PPV3 and PPV4 (three and four states respectively) coinciding with
the stabilization of the bright states. While there are two states in the 5.5-6.0 eV range, this
too shifted to lower energies, as no calculated state was found higher in energy than 5.5 eV
for stacked PPV3 and 5.0 eV in stacked PPV4.
S1 S2 S3 S4
S5 S6 S7 S8
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Examining the spectrum of the displaced PPV3 dimer (Table 6.7), the same
destabilization compared to the sandwich stacked dimer is found that was seen for the PPV2
dimers. The S1 state was destabilized more than most of the calculated states. There was a
third state (S5) with significant oscillator strength appearing about 0.07 eV higher than the
second bright state that displayed partial interchain CT character (~0.5 e). The omega
matrices for this excitation (Figure D.3) showed off-diagonal elements corresponding to
transitions primarily from/to the interior phenylene groups of both chains, though also
involving the vinylene and capping phenylene groups. The NTOs (Table D.10) for the two
primary contributing transitions showed symmetric distribution of hole density shifting to
one side of both dimers. There were four other excitations in addition to the S5 state with
CT character greater than 0.1 e. Of these the S6 and S7 states had significant off-diagonal
elements in the omega matrices with transitions from a variety of fragments.
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Figure 6.4 Calculated line spectra (right y-axis) and density of states (left y-axis) for the
stacked PPV oligomers optimized using the SOS-ADC(2), B3LYP-D3, and CAM-B3LYP-
D3 methods. Stacked a) PPV2 (Table 6.5), b-d) PPV3 (Table D.8, S11-S12), e) PPV4 (Table
D.9).
a)
b)
c)
d)
e)
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Table 6.7 Displaced PPV3 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and
SOS-MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy, f –
oscillator strength, CT – interchain charge transfer)
State Excitation ΔE (eV) f CT (e)
S1 1 1A 3.77 0.00 0.04
S2 2 1A 4.02 3.47 0.06
S3 3 1A 4.34 0.00 0.17
S4 4 1A 4.38 0.32 0.07
S5 5 1A 4.55 0.11 0.52
S6 6 1A 4.58 0.00 0.39
S7 7 1A 4.59 0.02 0.30
S8 8 1A 4.61 0.00 0.13
TDDFT methods were compared to the SOS-ADC(2) using the cc-pVTZ basis for
stacked PPV3. The relative ordering of the first eight excited states is mostly preserved with
the B3LYP-D3 method (Table D.8). The S1 state is stabilized using both TDDFT methods
compared to the previous method, though much more so with B3LYP-D3. The first bright
state (Table D.8, S2) was found to be stabilized using the B3LYP-D3 (Table D.11, S3) and
CAM-B3LYP-D3 (Table D.12, S2) methods (about 0.7 and 0.3 eV respectively). The
oscillator strength of this transition is also reduced for B3LYP-D3 and CAM-B3LYP-D3.
Though only this one bright state was found when utilizing the B3LYP-D3 method, a third
bright state was found with the CAM-B3LYP-D3 method, located about 0.2 eV higher than
the second and is a partial CT state. The density of states (Figure 6.4c,d) using both TDDFT
methods is much larger than with SOS-ADC(2) approach. The majority of transitions occur
in the energy range of 4.0-4.5 eV (twelve states), similar to what was discussed before.
This peak in states is destabilized to the range of 5.0-5.5 eV (eight states) using the CAM-
B3LYP-D3 method.
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Tab
le 6
.8 I
nte
rchai
n d
ista
nce
s (Å
) in
the
stac
ked
PP
V d
imer
s o
pti
miz
ed f
or
π-π
* S
1 (
1
1B
g)
exci
ted s
tate
usi
ng t
he
cc-p
VT
Z b
asis
and C
2h s
ym
met
ry
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The B3LYP-D3 and CAM-B3LYP-D3 methods presented more varied CT
character compared to the SOS-ADC(2) method. Though only one bright state was found
using the B3LYP-D3 method in the thirty states calculated (S3), it is located energetically
higher than one prominent interchain CT state (S2) and one partial interchain CT state (S1).
The first prominent CT state using the CAM-B3LYP-D3 method (S3) was located about
0.3 eV higher in energy than the bright state and had oscillator strength greater than zero.
The geometry of the S1 state (1 1Bg) was optimized (Table 6.8) and compared to the
ground state. In all cases a decrease in intermolecular stacking distances was observed and
was greatest for stacked PPV2 and least for stacked PPV4. The largest decrease in stacking
distances occurred for the vinylene units (about 0.9 Å for stacked PPV2) and generally
becomes greater toward the interior of the stacked dimers. Comparing the SOS-ADC(2)
method to TDDFT in stacked PPV3, the CAM-B3LYP-D3 method deviated slightly less
on average than the B3LYP-D3 method which deviated quite strongly (over 0.1 Å) in the
interior phenylene and vinylene groups.
6.5 Conclusions
Ab initio calculations were performed on stacked PPV oligomers of from two to
four repeating PV units with the purpose of providing necessary benchmark calculations.
Geometry optimizations showed that the intermolecular stacking distances became
generally shorter with increasing basis set size. In stacked PPV2, the structure obtained
with the cc-pVTZ basis closely matched those found with the cc-pVQZ and aug-cc-pVQZ
basis sets and the CP method to correct reproduced the cc-pVQZ structure well, though the
cc-pVDZ and aug-cc-pVDZ basis sets were found to, respectively, overestimate and
underestimate intermolecular distances. Increasing system size increased the
intermolecular stacking distances going to PPV3 and PPV4. Comparisons with DFT
methods showed that the B3LYP-D3 method more closely reproduced the structure found
with the SOS-MP2 method. Displacement from the exact sandwich structure was found to
play a large role as the displaced PPV dimers had much smaller stacking distances than the
sandwich-stacked dimers.
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The calculated interaction energies were compared to the extrapolated CBS limit.
The interactions energy found for stacked PPV2 using the aug-cc-pVDZ basis set strongly
overestimated the stabilization, though the interaction energy found with the cc-pVQZ
basis was close to the CBS limit. CP corrections decreased the magnitude of the interaction
energy, though the calculated CBS limit for SOS-MP2/BSSE method was still relatively
close to stabilization energy found for the SOS-MP2 method. The CAM-B3LYP-D3
method gave stabilization energies slightly more stabilizing than the SOS-MP2 mothod,
but the B3LYP-D3 method differed more strongly.
Vertical excitation calculation using the SOS-ADC(2) method found two intensive
transitions in the eight calculated excited states for each case. The oscillator strength of the
first bright state increased with going from PPV2 to PPV3 and PPV4, but the oscillator
strength of the second decreased. Though interchain CT character is more or less constant
with increasing chain size, intrachain CT increases with a larger number of contributing
phenylene and vinylene fragments as analyzed using omega matrices and NTO transitions.
Comparative DFT methods displayed a much more diverse range of intermolecular CT
states than was found for the SOS-ADC(2) method.
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Nakatsuji, H.; Caricato, M.; Li, X.; Hratchian, H. P.; Izmaylov, A. F.; Bloino, J.;
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Dimer: High-Level Wave Function and Density Functional Theory Calculations.
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CHAPTER VII
THE BIRADICALOID CHARACTER IN RELATION TO
SINGLET/TRIPLET SPLITTINGS FOR CIS-/TRANS-
DIINDENOACENES AND TRANS-
INDENOINDENODI(ACENOTHIOPHENES) BASED ON
EXTENDED MULTIREFERENCE CALCULATIONS
Authors: Reed Nieman,a Nadeesha J. Silva,a Adelia J. A. Aquino,a,b and Hans Lischkaa,b
Affiliations: aDepartment of Chemistry and Biochemistry, Texas Tech University,
Lubbock, TX 79409-1061, USA, bSchool of Pharmaceutical Sciences and Technology,
Tianjin University, Tianjin 300072, PR China
7.1 Abstract
The assessment of the biradical character of select trans-diindenoacenes, cis-
diindenoacenes, and trans-indenoindenodi(acenothiophenes) was accomplished utilizing
the multireference averaged quadratic coupled-cluster (MR-AQCC) method. These
systems have been previously identified as highly reactive in experiments, often subject to
unplanned side-reactions. The biradical character of these species was characterized using
the singlet/triplet splitting energy (∆ES-T), effective number of unpaired electrons (NU), and
unpaired electron density. Additionally, the aromaticity has been described by the
harmonic oscillator measure of aromaticity (HOMA). For all structures, low spin singlet
ground states were found, in agreement with previous investigations. In addition,
increasing the number of core six-membered rings decreased the ∆ES-T, and increased the
NU and total unpaired electron density of the singlet state. The apical carbon of the five-
membered ring has the largest amount of unpaired electron density in each investigated
structure. The cis-diindenoacenes (5-8) displayed a greatly increased value of NU and
unpaired electron density corresponding to much smaller ∆ES-T values compared to the
other structures. The thiophene rings in the trans-indenoindenodi(acenothiophenes)
structures (9-12) were found to simultaneously limit the unpaired electron density in the
terminal six-membered rings and stabilize the density found in the core rings. Finally
excellent agreement to experimental values for the ∆ES-T has been demonstrated for
structures 5 and 10.
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7.2 Introduction
The study of the properties and reactivity of biradical polycyclic aromatic
hydrocarbons (PAHs) is important to a wide variety of scientific fields due to their unique
electronic nature. Since the discovery of open-shell PAHs systems,1 they have provided
invaluable insight into how the radicals influence the π-bonding network to generate
properties such as magnetism. Diradical PAH systems often display a range of properties
such as small energy gaps between the highest occupied molecular orbital (HOMO) and
the lowest unoccupied molecular orbital (LUMO),2 electron spin resonance and peak
broadening in proton NMR spectra,2 low-lying doubly-excited electron configurations
allowing low-energy electronic absorptions,3 and amphoteric redox character and
multicenter bonds.4 These expansive properties have led biradical PAHs to be prime
candidates for use in molecular electronic and spintronic devices,5 battery technology,6
nonlinear optical devices,7-9 and singlet fission materials.10-12 Several classes of radical and
biradical PAHs have been studied including zethrenes,13-17 anthenes,18-19 phenalenyl,5
bisphenalenyls,20-22 triangulene23-25 and extended triangulene,5 quinodimethanes,26-28
indofluorenes,29-35 high-order acenes,36-40 and graphene nanoribbons.40-44 The biradical
character in these cases is derived from open-shell and closed-shell resonance structures,45-
47 the interconversion between which cleaves the exocyclic π-bonds increasing the number
of Clar sextets and energetically driving the stability of the open-shell state.15, 48 Double
spin-polarization effects give biradical PAHs with singlet ground states low-lying triplet
excited states that are more easily populated.49
The unique characteristics of idenofluorenes (representative structures shown in
Figure 7.1) have been targeted for investigation and utilization. The five-member carbon
rings in indenofluorenes have shown to make the structures reversible electron donors.50
Diindenobenzene dervatives show ambipolar charge transport properties in single-crystal
and thin-film organic field effect transitors (OFETs).30, 51 Extensive work on trans-
diindenoanthracene found that it was chemically robust due to only modest biradical
character and moderate singlet/triplet splitting energy, ∆ES-T, of 4.2 kcal/mol,4 and this
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small energy gap showed that the triplet state was thermally accessible and could be
populated by an entropy-driven singlet-triplet fission process.52 Only the outer six-
membered rings of trans-diindenonaphthalene (fluoreno[3,2-b]fluorene) were shown to be
aromatic, while the core rings displayed anti-aromatic character.35 Experimental results for
cis-diindenobenzene (fluoreno[2,1-b]fluorene) show it has a singlet ground state, modest
∆ES-T of 4.2 kcal/mol, and a small HOMO-LUMO band gap (1.26 eV).31 The fusion of
thiophene rings into the indenofluorene frame yield interesting experimental results.
Indacenedithiophenes have smaller energy level gaps than pure indenofluorenes as they
have increased paratropicity in their indacene core.32 It has been shown that the inclusion
of fused benzothia groups in indenofluorenes leads to pro-aromatic properties that show
larger diradical character and longer lifetimes of singlet excited states.53
Indenoindenodibenzothiophene molecules were studied experimentally and theoretically
and were found to have relatively large ∆ES-T (8.0 kcal/mol) together with large diradial
character.54
Figure 7.1 Quinoid Kekule and biradical resonance forms of trans-diindenoanthracene, cis-
diindenoanthracene, and trans-indenoindenodi(anthracenothiophene).
Quantum chemical calculations are of great importance to the determination of
molecular properties. While the computationally efficient density functional theory (DFT)
is a natural first choice, the unrestricted DFT (UDFT) method suffers heavily from spin-
contamination effects especially for broken-symmetry singlet state biradical cases.37, 55 To
remedy this, PAH systems with singlet biradical cases have been investigated with a wide
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variety of methods such as spin-flip (SF) approaches,56-57 projected Hartree-Fock,58 active
space variational two-electron reduced density matrix (2RDM),59-60 density matrix
renormalization group (DMRG),61-62 and CCSD(T).63-64 The multireference averaged
quadratic coupled-cluster (MR-AQCC) method65 is able to measure reliably the
biradicaloid character of molecules as it includes quasi-degenerate configurations in the
reference wave function and dynamic electron correlation including size-extensivity
contributions by explicitly considering single and double excitations.65-66 Previous reports
have had success using the MR-AQCC method to describe n-acenes,40, 67-68 periacenes,
zethrenes,69-70 among other challenging biradical cases.71-72
To this end, the MR-AQCC method has been employed in this work to investigate
the biradical character of several extended indenofluorene and
indenoindenodi(acenothiophene) structures (Figure 7.1 and Figure E.1) by describing their
singlet ground state and first vertical and adiabatic triplet excited state. From these
calculations, relatively simple descriptors such as the singlet/triplet splitting energy, ∆ES-
T, effective number of unpaired electrons (NU), and density of unpaired electrons will be
used to characterize singlet biradical nature of twelve PAH molecules. The harmonic
oscillator measure of aromaticity (HOMA) index will also be calculated at the DFT level
in order to assess the aromaticity of each structure.
7.3 Computational Details
The biradicaloid character of the structures shown in Figure 7.1 and Figure E.1 of
the Supporting Information were investigated using the MR-AQCC65 method by relating
the ∆ES-T values to NU data. To begin, the low and high spin ground states of all structures
were optimized using the range-corrected ωB97XD73 density functional and the def2-
TZVP74-75 basis set. Systems 1-4 (trans-diindenoacenes) and 9-12 (trans-
indenoindenodi(acenothiophenes)) belong to the C2h point group while 5-8 (cis-
diindenoacenes) belong to the C2v point group. The correct low spin ground state electron
configuration was determined by way of wave function instability analysis76 of the Kohn-
Sham determinants.77 All structures, except 1, were found to have triplet instabilities
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present in their wave functions at restricted DFT (RDFT) level, which were followed to
reoptimize the geometry using an unrestricted DFT (UDFT) approach. The harmonic
vibrational frequencies for all structures were calculated and found to be non-imaginary in
all cases showing that each structure is a minimum in the potential curve.
Based on the optimized geometry using the ωB97XD/def2-TZVP method, the
aromaticity of each ring in each structure was determined using the harmonic oscillator
measure of aromaticity (HOMA):78-79
𝐻𝑂𝑀𝐴 = 1 −𝛼
𝑁∑(𝑅𝑜𝑝𝑡 − 𝑅𝑖)
2
𝑖
(6.1)
This method uses Kekulé benzene for the reference, and the index of aromaticity is
determined based on how much the bond distances, 𝑅𝑖, differ from the optimal C-C bond
length, 𝑅𝑜𝑝𝑡. A normalization constant, 𝛼, keeps the HOMA index of zero for Kekulé
benzene. The total number of bonds in a given ring is given by N. Here, 𝑅𝑜𝑝𝑡 for C-C bonds
is 1.388 Å, and for C-S bonds is 1.677 Å. The constant, 𝛼, for C-C bonds is 257.7 Å2, and
for C-S bonds is 94.09 Å2.
The starting orbitals for the MR-AQCC calculations were found using the complete active
space self-consistent field theory (CASSCF) method80 and the ωB97XD/def2-TZVP
optimized geometries. The 6-31G*81-83 basis set was used for all CASSCF and MR-AQCC
calculations. For 9-12, a split basis set was used, designated here as 6-31G*(6-311G*)S,
where 6-31G* was applied for carbon and hydrogen, and the triple-zeta basis set, 6-
311G*,84-85 was utilized for sulfur. The σ orbitals (both occupied and virtual) were frozen
in all CASSCF and MR-AQCC calculations at SCF level while the π orbitals were kept
active. For 1-4 and 9-12, orbitals belonging to the Ag and Bu irreducible representations
were σ in character, and orbitals belonging to the Au and Bg irreducible representations
were π in character. For 5-8, orbitals belonging to the A1 and B1 irreducible representations
were σ in character, and orbitals belonging to the A2 and B2 irreducible representations
were π in character. The molecular orbitals constituting the active space were determined
with the method of Pulay et al86 using the natural orbital (NO) occupation numbers from
the unrestricted Hartree-Fock (UHF) wave function. For all structures, the optimized
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orbitals from a two-state state-averaged (SA2) complete active space consisting of four
electrons in four orbitals (CAS(4,4)) where the state-averaging was performed over the
lowest singlet and triplet states. These orbitals were taken for the subsequent MR-AQCC
calculations with a CAS(4,4) reference space (MR-AQCC(CAS(4,4)). The states taken in
SA2 were the 1 1Ag and 1 3Bu states for 1-4 and 9-12, and 1 1A1 and 1 3B1 for 5-8. Expanded
reference spaces were also considered for 1, 2, and 9 to show the effect on the ∆ES-T by
utilizing in addition to the CAS a restricted active space (RAS) (doubly occupied orbitals
from which one electron may be excited to the CAS or auxiliary (AUX) orbitals) and AUX
space (unoccupied or virtual orbitals into which one electron may be excited). For 1 and 9,
CASSCF calculations were done with RAS(3)/CAS(4,4)/AUX(3) to optimize orbitals for
MR-AQCC(CAS(10,7)/AUX(3)) calculations, and for 9, CASSCF calculations were
performed with RAS(4)/CAS(4,4)/AUX(4) to optimize orbitals for MR-
AQCC(CAS(12,8)/AUX(4)) calculations. A further expanded reference space was
considered for 1, where CASSCF calculations were done with RAS(5)/CAS(4,4)/AUX(3)
to optimize orbitals for MR-AQCC(CAS(14,9)/AUX(3)) calculations. In addition, to show
the effect on the ∆ES-T when keeping the σ orbitals active, calculations were carried out for
1 using the MR-AQCC(CAS(4,4)) reference space whereby only the core orbitals were
frozen. For each reference space investigated, the active orbitals are given in Table E.1 of
the Supporting Information. In a few cases, intruder states (configurations not found in the
reference space and with ≥1% weight) were found to occur at the MR-AQCC level. This
was accounted for by utilizing more active orbitals to include the intruder state as a
reference configuration in the wave function.
The singlet/triplet splitting energy (∆ES-T) is calculated according to ∆ES-T = ET -
ES, such that positive values mean that the low spin or singlet state is lower in energy than
the high spin or triplet state. Three different cases of the ∆ES-T are considered: the vertical
splitting, ∆ES-T, vert., where the energy of the triplet state is found at the singlet state
geometry, the adiabatic splitting, ∆ES-T, adiab., where the energy of the triplet state is found
at the triplet state geometry, and the adiabatic splitting that includes the zero-point energy
correction (ZPE), ∆ES-T, adiab.+ZPE, for which the ZPE was calculated at the ωB97XD/def2-
TZVP level.
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The effective number of unpaired electrons (NU) and unpaired electron density are
calculated87-88 in order to assess the radical nature of the discussed structures by means of
the non-linear formula developed by Head-Gordon:89-90
𝑁𝑈 = ∑ 𝑛𝑖2
𝑚
𝑖=1
(2 − 𝑛𝑖)2 (6.2)
In this approach, orbitals with occupations close to one are emphasized over those orbitals
that are either nearly double-occupied or are unoccupied. m is the total number of NOs and
ni is the occupation of the ith NO. Two schemes for NU have been considered: as a sum
over all NO occupations (NU) to assess the total radical character, and as a sum only over
the HONO-LUNO pair (H-L, NU(H-L)) to account only for the biradical character. For each
triplet state calculated, the NU values and unpaired electron density are practically the same
for the vertically excited triplet state and the adiabatic triplet state, so only the values for
the vertical triplet states are reported here.
The geometry optimizations, stability analysis, and frequency calculations were
carried out using the Gaussian program package (Gaussian 09 Rev.E.01).91 The UHF and
natural orbitals calculations were done using the TURBOMOLE program.92 All CASSCF
and MR-AQCC calculations were performed using the COLUMBUS program suite.93-95
The unpaired electron population analysis was accomplished using the TheoDORE
program.96-97
7.4 Results and Discussion
7.4.1 Trans-diindenoacenes
The ∆ES-T values for 1-4 (trans-diindenoacenes) and the effective number of
unpaired electrons, respectively, have been plotted in Figure 7.2 against the number of
interior six-membered rings using the MR-AQCC(CAS(4,4))/6-31G* method. One finds
that with increasing number of rings, that the ∆ES-T decreases. This decrease is greatest
when going from 1 to 2 (the vertical ∆ES-T decreases by about 22.3 kcal/mol and the ∆ES-T
decrease by 10.6 and 9.0 kcal/mol for adiabatic and adiabatic+ZPE, respectively). The
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decrease in ∆ES-T going from 2-4 shows to be practically linear. The effect of using the
adiabatic and adiabatic+ZPE ∆ES-T is especially clear in 1, where the ∆ES-T is smaller than
the vertically excited case by about 14.1 and 16.2 kcal/mol, respectively. This decrease is
less pronounced in 2 (-2.5 and -2.9 kcal/mol, respectively), 3 (-0.8 and -0.9 kcal/mol,
respectively), and 4 (+0.05 and +0.07 kcal/mol, respectively). For 3 and 4, the ZPE
correction to the adiabatic ∆ES-T makes nearly no difference. For 4, the adiabatic and
adiabatic+ZPE ∆ES-T is increased from the vertical ∆ES-T, though by less than 0.1 kcal/mol.
In addition, 4 has the smallest ∆ES-T at only 6.5 kcal/mol, meaning that it has the most
energetically accessible triplet state of the investigated trans-diindenoacenes. Overall, there
is good agreement to recent theoretical ∆ES-T values calculated using the spin-flip non-
colinear time-dependent DFT (SF-NC-TDDFT) method for 1-3,54 though the MR-AQCC
values here are slightly larger in each case. The experimental value obtained for a
derivative of 3 (with substituted mesityl (2,4,6-trimethylphenyl) groups at the apical carbon
of the five-membered rings and (triisopropylsilyl)ethynyl groups at the center-most six-
membered ring) shows a ∆ES-T of 4.18 kcal/mol,4 smaller than the calculated
adiabatic+ZPE value of 10.5 kcal/mol reported here. The calculated ∆ES-T value for 3
reported in ref. 54 via the SF-NC-TDDFT method is about 7.1 kcal/mol, intermediate of
our MR-AQCC result and the experimental value, though the ∆ES-T values computed with
the complete active space configuration interaction (CASCI) method were approximately
twice as large as those presented here for 1-3 using the MR-AQCC(CAS(4,4)) reference
space.
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Figure 7.2 Comparison of the singlet/triplet energy (∆ES-T, E(13Bu) - E(11Ag)) and number
of unpaired electrons (NU and NU(H-L), Table E.2) of the 11Ag state with the number of
interior six-membered rings of trans-diindenoacenes (1-4) using the MR-
AQCC(CAS(4,4))/6-31G* method.
The standard MR-AQCC(CAS(4,4)) method is compared to expanded reference
spaces for 1 and 2 in Table 7.1. First looking at 1, the vertical ∆ES-T calculated using the
MR-AQCC(CAS(4,4)) method (frozen σ orbitals) compares well to those using the MR-
AQCC(CAS(10,7)/AUX(3)) and MR-AQCC(CAS(14,9)/AUX(3)) methods, decreasing
only by about 0.06 and 0.6 kcal/mol, respectively. Using the MR-
AQCC(CAS(10,7)/AUX(3)) method for 1, the adiabatic and adiabatic+ZPE ∆ES-T are
increased by about 0.9 and 0.7 kcal/mol, respectively. This is likely a result of intruder
states (configurations from outside the reference space) in the triplet state, the necessary
inclusion of which as reference configurations increased the total energy and thus the ∆ES-
T. The inclusion of sigma orbitals (excluding the core orbitals that remained frozen) in the
MR-AQCC(CAS(4,4)) method for 1 decreases the adiabatic and adiabatic+ZPE ∆ES-T by
about 2.0 and 2.1 kcal/mol, respectively, compared to the MR-AQCC(CAS(4,4)) method
where the σ orbitals remained frozen. The vertically excited ∆ES-T could not be obtained
with the MR-AQCC(CAS(4,4)) method and including the σ orbitals due to persistent
intruder states in the triplet state keeping the calculation from converging. When
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considering 2, the vertical ∆ES-T decreases with the MR-AQCC(CAS(12,8)/AUX(4))
method by about 1.9 kcal/mol compared to the MR-AQCC(CAS(4,4)) method, but the
adiabatic and adiabatic+ZPE ∆ES-T increase by about 0.6 kcal/mol.
Table 7.1 Singlet/triplet splitting energy (∆ES-T, E(13Bu) - E(11Ag)) calculated using the
MR-AQCC/6-31G* method for trans-diindenoacenes, 1-4.
System Reference Space ∆ES-T, vert.
(kcal/mol)
∆ES-T, adiab.
(kcal/mol)
∆ES-T, adiab.+ZPE
(kcal/mol)
1 MR-AQCC(CAS(4,4))a -b 23.7 21.6
MR-AQCC(CAS(4,4)) 39.8 25.7 23.6
MR-AQCC
(CAS(10,7)/AUX(3))
38.8 26.5 24.4
MR-AQCC
(CAS(14,9)/AUX(3))
38.2 23.0 20.9
2 MR-AQCC(CAS(4,4)) 17.5 15.1 14.7
MR-AQCC
(CAS(12,8)/AUX(4))
15.6 15.6 15.2
3 MR-AQCC(CAS(4,4)) 11.5 10.7 10.5
Expt.4 4.18
4 MR-AQCC(CAS(4,4)) 6.4 6.5 6.5 aonly core orbitals are frozen, bvertically excited triplet state had persistent intruder states.
The effective number of unpaired electrons of the 1 1Ag state increases linearly,
whether one considers the sum over all NOs or just the H-L pair, with the number of interior
rings going from 1-4. 1 has the smallest NU values of any of the trans-diindenoacenes (NU
is 0.55 e and NU(H-L) is 0.23 e), correlating with its large ∆ES-T, Figure 7.3 and Table E.2.
The total effective number of unpaired electrons for 2 is about 1.1 e, about 1.6 e for 3, and
2.2 e for 4. Using only the H-L pair reduces the total number of unpaired electrons by about
0.3 e for 1 to about 0.7 e for 4. The total number of unpaired electrons for 4 is 2.2 e, showing
a true singlet biradical system, which correlates to its small ∆ES-T, Figure 7.3. When
considering only the H-L pair for each vertically excited triplet state, Table E.2, the number
of unpaired electrons is almost constant at 2.0 e, consistent with triplet spin multiplicity
and singly occupied HONOs and LUNOs. The largest difference in NU values between the
singlet and vertical triplet state comes for 1 at about 1.8 e.
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a)
b)
Figure 7.3 Correlation of the singlet/triplet energy (∆ES-T) with a) the total number of
unpaired electrons (NU) and b) the number of unpaired electrons in the HONO-LUNO pair
(NU(H-L)) of the 1 1Ag state (Table E.2) of trans-diindenoacenes (1-4) using the MR-
AQCC(CAS(4,4))/6-31G* method.
The unpaired electron density of 1-4 for the 1 1Ag ground state is shown in Figure
7.4. The density is greatest on the carbon atom of the five-membered rings connected to
hydrogen, referred to as the apical carbon, correlating to the location of the radical center
shown in the valence bond (VB) structures shown in Figure 7.1 and Figure E.1, and the
density then alternates going toward the center of the molecules. The unpaired electron
density on carbon atoms not bonded to hydrogen is nearly negligible. The unpaired electron
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density of the vertically excited triplet state of 1-4 (Figure E.2) is greater than for the singlet
state. There is greater density found on the indene groups in the triplet state. The greatest
density is still found on the apical carbon atom of the five-membered ring in the triplet state
as in the singlet state, but the density decreases at this location with increasing system size.
Despite the decrease in unpaired electron density at the location of greatest density, the
slight increase of the NU (Table E.2) with increasing system size (about 0.1 e per additional
core ring) can be seen to come from the increase in number of centers of unpaired electron
density.
Figure 7.4 Total unpaired electron density (sum over all NOs, Table E.2), in red, of the 1 1Ag state of trans-diindenobenzene (1-4) using the MR-AQCC(CAS(4,4))/6-31G* method.
(isovalue=0.004 e/Bohr3). HOMA index is shown in the center of each ring.
The HOMA indices for the singlet ground state geometry are also shown in Figure
7.4 for each trans-diindenoacene structured investigated (1-4). Structure 1 is noteworthy as
the five-membered ring and interior six-membered ring are almost anti-aromatic in
character as the HOMA indices are nearly zero, consistent with the previous findings that
concluded the local anti-aromatic nature arises from the large odd-electron density
distribution.98 The six-membered ring of the indene is aromatic however. This is
substantially different from the triplet ground state geometry (Figure E.2) of 1 where the
five-membered ring has a HOMA index of 0.37 and the interior six-membered ring has an
index of 0.93. This great difference in HOMA indices between the singlet and triplet states,
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implying also a large change in the geometry, is the origin of the substantial difference in
the NU discussed above for 1. In the remaining structures, 2-4, the HOMA index of the
five-membered is about 0.3, with indices increasing going toward the center of each
molecule, reaching a maximum value of about 0.75 for 3 and 4. As shown, the aromaticity
of each ring (correlating to the HOMA index) increases as the unpaired electron density at
each carbon of that ring decreases. For the triplet ground states structures, the interior most
six-membered ring has a larger HOMA index than those found in the singlet ground state
structures, though they are nearly the same in 4.
7.4.2 Cis-diindenoacenes
The singlet/triplet splitting of the cis-diindenoacene structures, 5-8, Figure 7.5 and
Table 7.2, is much smaller compared to the trans-isomers, 1-4. The ∆ES-T decreases as the
number of rings in each system increases. The largest vertical ∆ES-T is found for 5 and is
only 3.9 kcal/mol while the vertical case for 8 is negative indicating that the 1 3B1 state is
more stable than the 1 1A1 state. For 7 though, the ∆ES-T, while negative, is close to zero so
the singlet and triplet states are almost isoelectronic. The adiabatic and adiabatic+ZPE ∆ES-
T for 5 and 6 are smaller than the vertical ∆ES-T by about 0.4 kcal/mol, but 7 and 8 become
positive when considering the adiabatic+ZPE ∆ES-T, increasing to 0.5 and 0.2 kcal/mol.
The ZPE correction does not contribute greatly to the adiabatic ∆ES-T, adding or subtracting
by only about 0.1 kcal/mol. The ∆ES-T calculated for 5, 3.6 kcal/mol, compares well with
the experimental value of 4.2 kcal/mol.31
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Figure 7.5 Comparison of the singlet/triplet energy (∆ES-T, E(13B1) - E(11A1)) and number
of unpaired electrons (NU and NU(H-L), Table E.3) of the 11A1 state with the number of
interior six-membered rings of cis-diindenoacenes (5-8) using the MR-
AQCC(CAS(4,4))/6-31G* method.
Table 7.2 Singlet/triplet splitting energy (∆ES-T, E(13B1) - E(11A1)) calculated using the
MR-AQCC/6-31G* method for cis-diindenoacenes, 5-8.
System Reference Space ∆ES-T, vert.
(kcal/mol)
∆ES-T, adiab.
(kcal/mol)
∆ES-T, adiab.+ZPE
(kcal/mol)
5 MR-AQCC(CAS(4,4)) 3.9 3.5 3.6
Expt.31 4.2
6 MR-AQCC(CAS(4,4)) 1.2 0.8 0.7
7 MR-AQCC(CAS(4,4)) -0.3 0.5 0.5
8 MR-AQCC(CAS(4,4)) -1.2 0.3 0.2
The decrease in ∆ES-T with increasing number of interior six-membered rings
correlates to an increase in the effective number of unpaired electrons of the 1 1A1 state
(Figure 7.5), and as was shown with the trans-isomers (Figure 7.3), the decrease in ∆ES-T
correlates to increasing effective unpaired electrons for the cis-diindenoacenes, shown in
Figure 7.6. The NU for 5 and 6 (Table E.3) are much larger than their counterparts, 1 and 2
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(Table E.2) by about 1.0 e, showing much more radical character, and the NU values for 7
and 8 are larger than those for 3 and 4 by about 0.6 e and 0.4 e, respectively. Taking only
the HONO-LUNO pair, the NU is least 1.0 e for 5 and increases to almost 2.0 e for 8. The
NU and NU(H-L) for the triplet state is very similar to those found in the trans-isomers.
Figure 7.6 Correlation of the singlet/triplet energy (∆ES-T) with a) the total number of
unpaired electrons (NU) and b) the number of unpaired electrons in the HONO-LUNO pair
(NU(H-L)) of the 1 1A1 state (Table E.3) of cis-diindenoacenes (5-8) using the MR-
AQCC(CAS(4,4))/6-31G* method.
The unpaired electron density of the 1 1A1 state for 5-8 is presented in Figure 7.7.
The The greatest amount of unpaired electron density is found located on the apical carbon
in the five-membered ring of the indene, like for 1-4 (Figure 7.4), correlating to the VB
a)
b)
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structures. The apical carbon atoms were also found by Shimizu et al. to have the largest
spin-density amplitudes in 1.31 For 7 and 8, nearly 0.5 e are present on these carbon atoms.
As was discussed for the NU values above, the unpaired electron density is greater for the
cis-diindenoacene structures than for trans-indenoacene. The same trend that was seen for
1-4 (Figure 7.4) is present for the singlet states of 5-8 (Figure 7.7) whereby the unpaired
density increases, especially at the apical carbon of the five-membered ring, with
increasing number of interior rings. There is greater unpaired electron density present on
the six-membered rings of the capping indene groups of the cis-isomers compared to
transisomers. The unpaired density of the vertically excited 1 3B1 states is similar to the
triplet states calculated for the trans-isomers (Figure E.3), where the value of the total NU
increases with increasing system size due to the larger number of centers of unpaired
electron density, but the density on the carbon in the five-membered ring decreases. While
the difference in density for 5 between the singlet and triplet states is rather large (with the
triplet density being larger), going from 6-7 the difference decreases, and for 8 the density
is only slightly larger in the triplet state than in the singlet state.
Figure 7.7 Total unpaired electron density (sum over all NOs, Table E.3), in red, of the 1 1A1 state of cis-diindenobenzene (5-8) using the MR-AQCC(CAS(4,4))/6-31G* method.
(isovalue=0.004 e/Bohr3). HOMA index is shown in the center of each ring.
HOMA indices were calculated for the singlet and triplet states of the cis-diindenoacenes
(5-8), Figure 7.7. The HOMA indices of the five-membered rings decrease from 5 (0.51)
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to 8 (0.31) while the indices of six-membered ring of the indene group remains practically
constant. The HOMA index also shows to increase from the five-membered ring toward
the center. The HOMA indices reported here for 5 and 6 are slightly larger for all rings
than those presented previously by Miyoshi and coworkers.34 For structures with odd
numbers of rings, 5 and 7, the HOMA index of the center-most ring decreases by 0.07,
while for even numbers of rings, 6 and 8, the HOMA indices for the center-most rings
increases slightly by 0.04. Comparing to the trans-isomers, the five-membered rings in the
cis-isomers have higher HOMA indices, while the six-membered rings of the indene groups
are decreases slightly. The interior rings are shown to be more aromatic in the cis-isomers,
as the HOMA is larger for 5-8 than 1-4. The HOMA indices of the 1 3B1 state (Figure E.3)
are smaller for all structures than the 1 1A1 state except for the six-membered ring of the
indene groups, which are only slightly larger.
7.4.3 Trans-indenoindenodi(acenothiophenes)
The ∆ES-T of the investigated trans-indenoindenodi(acenothiophenes) (9-12) is
compared to the effective NU and number of interior rings in Figure 7.8. The trend of
decreasing ∆ES-T with increasing ring number is found also for 9-12 as for the previous
structures. The adiabatic ∆ES-T value decreases going from 9 to 10 by about 16 kcal/mol
(primarily due to the large value found for 9), though the values for 11 and 12 are
practically the same. The large decrease in all considered ∆ES-T values is observed also
going from 1 to 2 due to the much larger ∆ES-T of 1. The ZPE correction to the adiabatic
∆ES-T is similarly small; while the adiabatic+ZPE ∆ES-T decreases going from 9-11 with
increasing number of rings, going from 11 to 12 is basically constant. As with 1 and 2, an
expanded reference space was calculated showing that the vertical ∆ES-T increased by about
0.3 kcal/mol, but the adiabatic and adiabatic+ZPE ∆ES-T decreased by about 2.3 kcal/mol
compared to the MR-AQCC(CAS(4,4)) reference space. There is excellent agreement with
previous SF-NC-TDDFT calculations for 9-1154 (the CASCI values for the ∆ES-T of these
structures54 are in general larger by about 10 kcal/mol for each structure compared to our
MR-AQCC values), and the experimental value (8.0 kcal/mol)54 was found to agree quite
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well and was only 0.4 kcal/mol smaller than the calculated value of the adiabatic+ZPE ∆ES-
T for 10, given in Table 7.3.
Figure 7.8 Comparison of the singlet/triplet energy (∆ES-T, E(13Bu) - E(11Ag)) and number
of unpaired electrons (NU and NU(H-L), Table E.4) of the 11Ag state with the number of
interior six-membered rings of trans-indenoindenodi(acenothiophene) (9-12) using the
MR-AQCC(CAS(4,4))/6-31G*(6-311G*)S method.
Table 7.3 Singlet/triplet splitting energy (∆ES-T, E(13Bu) - E(11Ag)) calculated using the
MR-AQCC/6-31G*(6-311G*)S method for trans-indenoindenodi(acenothiophene) (9-12).
System Reference Space ∆ES-T, vert.
(kcal/mol)
∆ES-T, adiab.
(kcal/mol)
∆ES-T, adiab.+ZPE
(kcal/mol)
9 MR-AQCC(CAS(4,4)) 24.5 19.7 19.2
MR-AQCC
(CAS(10,7)/AUX(3))
24.8 17.4 17.0
10 MR-AQCC(CAS(4,4)) 8.8 8.6 8.4
Expt.54 8.0
11 MR-AQCC(CAS(4,4)) 4.4 4.4 4.2
12 MR-AQCC(CAS(4,4)) 4.5 4.5 4.4
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As shown in Figure 7.9, the energy decreases with increasing NU in the 1 1Ag state.
The total NU values (Table E.4) for the trans-indenoindenodi(acenothiophene) (9-12) are
larger than those of the trans-diindenoacenes, 1-4 (Table E.2), by about 0.2 e to 0.5 e, and
0.1 e to 0.4 e when considering only the H-L pair. The total NU for the cis-diindenoacenes
(5-8, Table E.3) are larger than for 9-12 by 0.7 e comparing 5 and 9, but the difference
becomes smaller, about 0.1 e, with increasing molecular size for 8 and 12. The vertically
excited triplet state has larger values of NU than for the singlet state, though the difference
while large for 9 (by about 1.7 e) decreases with increasing system size and is only 0.45 e
for 12. As with 1-8, when taking only H-L, the NU values become essentially 2.0 e. The
total NU for the vertical triplet state is larger than the triplet states of 1-8 by about 0.1 e.
The unpaired electron density of the 1 1Ag state of trans-
indenoindenodi(acenothiophenes) (9-12) is presented in Figure 7.10. As for structures 1-8,
the greatest amount of unpaired density is located on the apical carbon atom of the five-
membered ring (again correlating to the VB structures as with 1-8) and increases with
increasing system size, 9 to 12. The unpaired electron density is much reduced in the
thiophene rings and terminal six-membered rings compared to that in the core six-
membered rings. This same feature is found for the 1 3Bu state, Figure E.4. The triplet state
features greater unpaired electron density compared to the singlet state, but the magnitude
of the density localized on each center decreases as the system size grows from 9 to 12. As
suggested by the NU values, Table E.4, the unpaired density of the singlet state is larger
than for the trans-diindenoacenes, though smaller than for the cis-diindenoacenes.
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Figure 7.9 Correlation of the singlet/triplet energy (∆ES-T) with a) the total number of
unpaired electrons (NU) and b) the number of unpaired electrons in the HONO-LUNO pair
(NU(H-L)) of the 1 1Ag state (Table E.4) of trans-indenoindenodi(acenothiophene) (9-12)
using the MR-AQCC(CAS(4,4))/6-31G*(6-311G*)S method.
a)
b)
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Figure 7.10 Total unpaired electron density (sum over all NOs, Table E.4), in red, of the 1 1Ag state of trans-indenoindenodi(acenothiophene) (9-12) using the MR-
AQCC(CAS(4,4))/6-31G*/6-311G* method. (isovalue=0.004 e/Bohr3). HOMA index is
shown in the center of each ring.
The HOMA indices for the 1 1Ag state structures is presented in Figure 7.10 as well.
The index of the thiophene group decreases by about 0.08 by increasing the system size
after 9. The capping six-membered rings have HOMA indices of approximately 1 for 9-12.
The HOMA index of the five-membered rings increases from 9 to 10, but then decreases
from 10 to 12. The innermost six-membered rings increase in HOMA index from 9 to 11,
but then decreases with 12. The HOMA indices of the singlet state structures indicate that
9-12 are more aromatic than 1-4, but less aromatic than 5-8. The triplet state structures’
HOMA indices are shown in Figure E.4, and are found to be more aromatic than the singlet
state structures. As was the case with 1 of the trans-diindenoacenes, the large difference in
NU of 9 between the singlet and triplet states originates in the large difference in HOMA
indices, indicating a large difference in geometry between the two states.
7.5 Conclusions
MR-AQCC calculations were carried out to investigate the biradicaloid character
of trans- diindenoacenes (1-4), cis-diindenoacenes (5-8), and trans-
indenoindenodi(acenothiophenes) (9-12) by calculating the ∆ES-T and NU and comparing
to the number of interior or core rings. For each structure, the ground state was found to be
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low spin, with the high spin state becoming increasing thermally accessible with increasing
number of core rings. The singlet ground state 1 and 9 have only slight radical or biradical
character judged by their small NU and NU(H-L) values, respectively, but these values
increase greatly when increasing the number of core rings by only one to obtain 2 and 10,
respectively. The cis-diindenoacene isomers are very different from the other eight
structures, as even the smallest considered structure, 5, has a small ∆ES-T (3.57 kcal/mol)
and large NU (1.41 e) and NU(H-L) (1.08 e). The greatest unpaired electron density for each
considered structure is found on the five-membered ring, and in the case of the thiophene-
fused structures, does not extend to the capping six-membered rings. The triplet structures
were stabilized with increasing core rings while the effective number of unpaired electrons
remained quite unchanged in magnitude, but became more diffuse and delocalized. In
summary, the MR-AQCC approach has been proven to be a single method able to
successfully assess the biradicaloid character of PAH molecules and compares
exceptionally well to experimental values for the ∆ES-T in the cases of 5 and 10. This
method provides necessary insight and analysis of these highly reactive species.
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CHAPTER VIII
CONCLUDING REMARKS
This dissertation presents the application of several quantum chemical methods to
the description of carbon-functionalized material properties used in cutting edge
technologies. The necessary use of these high-level ab initio methods would inform future
scientists on how best to utilize the discussed materials in future experiments and real-
world applications.
The reactivity of pristine graphene as well as single and double carbon vacancy
defects was assessed toward chemisorption of a hydrogen radical test particle using the
CASSCF and MR-CISD methods with pyrene as a graphene model. While all carbon
centers for each structure formed stable C-H bonds, single carbon vacancy pyrene
possessed the strongest reaction (was most stabilizing) toward the incoming hydrogen
radical thanks to the dangling bond left in the six-membered ring as a result of the defect.
Bond formation with the hydrogen radical at this site was found to involve at least three
closely spaced electronic excited states near the ground state and complicated diabatic
electronic behavior at extended C-H bond distances. The carbon centers comprising the
five-membered ring were found to quite stabilizing as well, though without the same degree
of involvement from the complex manifold of excited states. A similar situation was found
for the five-membered rings of the double vacancy structure. A rigid potential energy scan
was performed to provide a survey of reactivity across the face of each structure confirming
the stability at the carbon centers and finding more repulsive regions at the existing
hydrogens and in the center of rings.
The structure and electronic states of double vacancy graphene with the intrinsic
dopant, silicon, were investigated by way of DFT calculations using pyrene and
circumpyrene as model systems. The silicon defect was characterized by means of NO
occupations, NBO analyses, and MEP surfaces. The dopant produced a non-planar low-
spin ground state with D2 symmetry, with a small energy gap between it and the planar low
spin D2h symmetric structure. The high spin structures were planar and found higher in
energy. The hybridization of the non-planar structures was sp3 while the planar structures
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were sp2d showing d-orbital involvement which is thought to be responsible for opening
the band gap. Comparisons between the B3LYP and CAM-B3LYP density functionals
found similar energies and structures, though the CAM-B3LYP low spin structures
displayed more open-shell character. Natural charge and MEP analyses showed charge
transfer from the silicon center to the bonding carbon atoms. Supporting experimental
results suggest a more complex picture, however, as features of the EEL spectrum indicate
an equilibrium existing between the non-planar and planar structures.
The electronic excited states of a BHJ scheme composed of P3HT, T3, and fullerene
(C60) (electron donor, spacer material, and electron acceptor respectively), were studied
using DFT with consideration to environmental effects. Experimental findings using this
BHJ scheme showed dramatically increased PCE with the terthiophene spacer material.
The excited state spectra calculated for isolated system and using the LR environmental
model did not explain the marked increase in PCE as the local excitons generated in the
bright π→π* of P3HT would decay into local excitons located on the fullerene units,
thereby quenching any generated CT states. The electronic excited state spectra calculate
using the SS environment however, showed a drastically different result as the local
excitons generated in the bright π→π* of P3HT were shown to decay into significantly
stabilized P3HT→C60 and T3→C60 CT states helping to explain the increased PCE. The
increase in PCE was thusly explained as resulting from increased physical separation
caused by the spacer decreasing the Coulomb binding energy and reducing the rate of
charge recombination, and stabilizing the CT states below local C60 excitons increasing the
rate of charge separation.
High-level SOS-MP2 and SOS-ADC(2) calculations were performed on several
oligomer dimers of PPV for the purpose of providing much needed benchmark calculations
of the structure and electronic excited states of the material that is paradigmatic in the
description of EET processes in PAHs. Dimers composed of two, three, and four repeating
phenylenevinylene (PPV2, PPV3, and PPV4 respectively) units were investigated using a
variety of correlation consistent basis sets. As PPV2 is the smallest structure, it was subject
to study with basis sets up cc-pVQZ and aug-cc-pVQZ, showing good agreement with the
smaller cc-pVTZ basis. The conformation was studied comparing the sandwich-stacked
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system with a displaced dimer, showing smaller stacking distances and increased π-π*
interactions in the latter. DFT methods exhibited larger stabilization energies. Two
intensive electronic transitions were found for each dimer using the SOS-ADC(2) method,
where the oscillatory strength increased with increasing chain length. A similar trend was
found for the magnitude of interchain charge transfer as analyzed using omega matrices
and NTO transitions. DFT results calculated a more diverse interchain CT spectra than with
the SOS-ADC(2) method.
The biradical character of trans-diindenoacenes, cis-diindenoacenes, and trans-
indenoindenodi(acenothiophenes) with increasing number of core fused-benzene rings
were investigated using MR-AQCC calculations where the biradical character was
described by the ∆ES-T and NU. For each structure studied, a low spin singlet ground state
was found, and aside from trans-diindenobenzene and trans-
indenoindenodi(acenobenzene), each had significant singlet diradical character. The ∆ES-T
was shown to decrease with increasing number of core rings for each set of structures
correlating to an increase in the NU. There is good agreement to experiment for the
calculated values for ∆ES-T. The cis-diindenoacene structures were remarkable by
comparison as the ∆ES-T was lower and NU larger than the other structures. Increasing NU
correlated with the greater unpaired electron density in the singlet ground state and was
greatest for the carbon bonded to hydrogen in the five-membered ring. While the NU in the
first triplet excited state was relatively unchanged with increasing number of core rings,
the larger structures stabilized the unpaired electron density, thus lowering the energy of
the triplet state and the ∆ES-T gap. In the thiophene-fused structures, the thiophene groups
disrupted the aromaticity and concentrated the unpaired density in the core of the
structures. Consequently, relatively little unpaired electron density extended to the capping
six-membered rings.
The study of graphene functionalization and defect formation represents an exciting
field of computational study. To this end, as one hydrogenation defect has been discussed
already, the addition of a second hydrogen radical is being studied using molecular
dynamics simulations and circumcoronene model where the potential energy is determined
using DFT. The calculations are being performed in order to determine the energy transfer
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process upon bond formation, the significance of H2 abstraction pathways, and the
probability and location of the second C-H formation at room temperature. Molecular
dynamics simulations are also being carried out in a similar fashion to determine the
dynamic dissociation energy required to remove a single carbon atom from pristine
graphene using the circumcoronene model generating the single carbon vacancy that has
been investigated previously.
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APPENDIX A
ADDITIONAL TABLES AND FIGURES
NOT INCLUDED IN CHAPTER III
Table A.1 Relative energy differences ΔErel (eV) of the local energy minima in the
electronic ground state computed for the different structures of pyrene-1C+Hr using
different theoretical methods and the 6-31G* approach.
ΔErel
CASSCF MR-CISD MR-CISD+Q
C1 0.637 0.960 1.393
C2 0.230 0.681 1.244
C3 0.855 1.163 1.474
C4 2.752 3.179 3.729
C5 1.627 1.809 2.032
C6 3.107 3.638 4.049
C7 2.517 2.700 2.816
C8 3.542 3.930 4.246
C15 0.000a 0.000b 0.000c aTotal energy = -574.277274 Hartree, bTotal energy = -575.610208 Hartree, cTotal energy
= -576.244082 Hartree, d The SA4-CASSCF(5,5) optimized geometry was used.
Table A.2 Relative energy differences ΔErel (eV) of the local energy minima in the
electronic ground state computed for the different structures of pyrene-2C+Hr using
different theoretical methods and the 6-31G* approach.
ΔErel
CASSCF MR-CISD MR-CISD+Q
C1 0.881 0.829 0.767
C2 0.000a 0.000b 0.000c
C3 0.612 0.735 0.806
C4 1.059 1.054 1.047 aTotal energy = -536.452180 Hartree, bTotal energy = -537.716100 Hartree, cTotal energy
= -538.283379 Hartree
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Table A.3 Relative energy differences ΔErel (eV) of the local energy minima in the
electronic ground state computed for the different structures of pyrene+Hr using different
theoretical methods and the 6-31G* basis set.
ΔErel
CASSCF MR-CISD MR-CISD+Q
C1 0.647 0.634 0.578
C2 0.137 0.080 0.000c
C3 0.630 0.649 0.559
C4 0.000a 0.000b 0.018
C16 0.820 0.814 0.691 aTotal energy = -612.336622 Hartree, bTotal energy = -613.718399 Hartree, cTotal energy
= -614.381588 Hartree
Hr bonded to Carbon 1
31a’ 32a’ 33a’ 22a” 23a”
Figure A.1 Plots of the active orbitals of the optimized ground states of pyrene+Hr using
the CASSCF(5,5)/6-31G* approach.
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Hr optimized on Carbon 15
29a’ 30a’ 31a’ 21a” 22a”
Figure A.2 Plots of the active orbitals of the optimized ground states of pyrene-1C+Hr
using the SA4-CASSCF(5,5)/6-31G* approach.
Hr optimized on Carbon 1
27a’ 28a’ 29a’ 20a” 21a”
Figure A.3 Plots of the active orbitals of the optimized ground states of pyrene-2C+Hr
using the SA4-CASSCF(5,5)/6-31G* approach.
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3 2Aa
4 2Ab
Figure A.4 Potential energy scans in terms of the location of Hr for the 3 2A and 4 2A states
of pyrene-2C+Hr using the SA4-CASSCF(5,5)/6-31G* approach. The atom Hr is located
in a plane parallel and 1 Å above the plane defined by the frozen carbon atoms. Lowest
energies: a -574.149530 Hartree, b -574.117820 Hartree. The energy (color scale) is given
in eV, relative to the ground state lowest minimum at -536.385046 Hartree.
Figure A.5 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is
bonded to C2 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum energy = -
574.277274 Hartree
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Figure A.6 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is
bonded to C3 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum energy = -
574.277274 Hartree
Figure A.7 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is
bonded to C4 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum energy = -
574.277274 Hartree
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Figure A.8 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is
bonded to C5 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum energy = -
574.277274 Hartree
Figure A.9 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is
bonded to C6 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum energy = -
574.277274 Hartree
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Figure A.10 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is
bonded to C7 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum energy = -
574.277274 Hartree
Figure A.11 Dissociation curve of the lowest four 2A states of pyrene-1C+Hr when Hr is
bonded to C8 using the SA4-CASSCF(5,5)/6-31G* approach. Minimum energy = -
574.277274 Hartree
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APPENDIX B
ADDITIONAL TABLES AND FIGURES
NOT INCLUDED IN CHAPTER IV
Table B.1 NBO bonding character analysis of pyrene-2C+Si using the CAM-B3LYP/6-
311G(2d,1p) method. The carbon atom discussed is the carbon bonded to silicon.
Structure Bonding Character
CAM-B3LYP/6-311G(2d,1p)
P4 (D2) αa: 0.53(sp2.98d0.03)Si+0.85(sp2.97d0.01)C
βa: 0.53(sp2.98d1.03)Si+0.85(sp2.98 d0.01)C
P5 (D2) αa: 0.53(sp2.98d0.03)Si+0.85(sp3.00d0.01)C
βa: 0.53(sp2.98d0.03)Si+0.85(sp3.00 d0.01)C
P6 (D2) 0.54(sp2.98d0.02)Si+0.84(sp3.20d0.01)C
P7 (D2h) α: 0.45(sp2.01d1.01)Si+0.89(sp2.80d0.01)C
β: 0.46(sp1.99d0.99)Si+0.89(sp2.78d0.01)C
P8 (D2h) 0.45(sp2.00d1.00)Si+0.89(sp2.92d0.01)C
P9 (D2) α: 0.48(sp2.38d0.62)Si+0.88(sp2.39d0.01)C
β: 0.48(sp2.92d0.09)Si+0.87(sp2.45d0.01)C
P10 (D2h) α: 0.46(sp2.00d1.00)Si+0.89(sp2.41d0.01)C
β: 0.45(sp2.00d1.00)Si+0.89(sp2.69d0.01)C
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s
sp3.58
p
sp3.58
p
sp3.58
p
sp3.58
Figure B.1 NBO plots for circumpyrene-2C+Si using the B3LYP/6-311G(2d,1p) approach
(CP2 D2h). Isovalue = ±0.02 e/Bohr3
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P1 (D2) P2 (D2h) P3 (D2h)
Figure B.2 Natural charges of carbon and silicon in units of e of pyrene-2C+Si using the
B3LYP/6-311G(2d,1p) method.
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P4 (D2) P5 (D2) P6 (D2)
P7 (D2h) P8 (D2h) P9 (D2)
P10 (D2h)
Figure B.3 Natural charges of carbon and silicon in units of e of pyrene-2C+Si using the
CAM-B3LYP/6-311G(2d,1p) method.
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P1 (D2) P2 (D2h)
P3 (D2h)
Figure B.4 Molecular electrostatic potential plot for pyrene-2C+Si mapped onto the
electron density isosurface with an isovalue of 0.0004 e/Bohr3 using the B3LYP/6-
311G(2d,1p) method.
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P4 (D2) P5 (D2)
P6 (D2) P7 (D2h)
P8 (D2h) P9 (D2)
P10 (D2h)
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Figure B.5 Molecular electrostatic potential plot for pyrene-2C+Si mapped onto the
electron density isosurface with an isovalue of 0.0004 e/Bohr3 using the CAM-B3LYP/6-
311G(2d,1p) method.
Figure B.6 Unprocessed EEL spectrum of the Si-C4 site.
90 100 110 120 130
6.5
7.0
7.5
8.0
8.5
10
5 c
ou
nts
Energy loss (eV)
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Flat Corrugated
Figure B.7 First two unoccupied states immediately above the Fermi level spanning about
1 eV for the flat and corrugated Si/graphene structures from the periodic DFT calculations.
Electron density isovalue is 0.277 e/Bohr3.
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APPENDIX C
ADDITIONAL TABLES AND FIGURES
NOT INCLUDED IN CHAPTER V
Table C.1 Excitation energies Ω of the isolated standard C60-T3-P3HT system using the
CAM-B3LYP/6-31G approach
Exc. Ω(eV) f CT Direction of
CT
Location of
exciton
S1 2.69 2.08 0.00 P3HT
S2 2.83 0.00 0.97 P3HTC60
S3 2.84 0.00 1.00 P3HTC60
S4 2.85 0.00 0.99 P3HTC60
S5 2.87 0.00 0.99 T3C60
S6 2.87 0.00 0.99 T3C60
S7 2.93 0.00 0.99 T3C60
S8 3.41 1.18 0.01 T3
S9 3.47 0.10 0.00 P3HT
S10 3.63 0.00 0.97 P3HTC60
S11 3.63 0.00 1.00 P3HTC60
S12 3.65 0.00 0.99 P3HTC60
S13 3.92 0.02 0.00 P3HT
S14 3.92 0.02 0.94 T3C60
S15 3.93 0.01 0.94 T3C60
S16 4.10 0.00 0.98 P3HTC60
S17 4.11 0.00 0.98 P3HTC60
S18 4.12 0.00 0.99 P3HTC60
S19 4.14 0.02 0.95 T3C60
S20 4.16 0.00 0.99 T3C60
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Table C.2 Excitation energies Ω of the isolated standard C60-T3-P3HT system using the
CAM-B3LYP/6-31G* approach
Exc. Ω (eV) f CT Direction of
CT
Location of
exciton
S1 2.62 2.04 0.00 P3HT
S2 2.91 0.00 0.97 P3HTC60
S3 2.91 0.00 1.00 P3HTC60
S4 2.92 0.00 0.99 P3HTC60
S5 2.93 0.00 0.99 T3C60
S6 2.94 0.00 0.99 T3C60
S7 2.99 0.00 0.99 T3C60
S8 3.33 1.16 0.01 T3
S9 3.39 0.10 0.00 P3HT
S10 3.70 0.00 0.97 P3HTC60
S11 3.70 0.00 1.00 P3HTC60
S12 3.71 0.00 0.99 P3HTC60
S13 3.87 0.02 0.00 P3HT
S14 3.99 0.02 0.94 T3C60
S15 3.99 0.03 0.94 T3C60
S16 4.20 0.03 0.95 T3C60
S17 4.23 0.00 0.98 P3HTC60
S18 4.24 0.00 0.98 P3HTC60
S19 4.25 0.00 0.99 P3HTC60
S20 4.27 0.00 0.99 T3C60
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Table C.3 Lowest excitation energies Ω of the isolated components P3HT, T3 and C60 using
the CAM-B3LYP/6-31G* approach.
Exc. Ω (eV) f
P3HT
S1 2.62 2.10
S2 3.39 0.00
S3 3.87 0.02
S4 4.06 0.14
S5 4.49 0.01
S6 4.63 0.00
S7 4.84 0.04
S8 4.86 0.00
S9 4.89 0.00
S10 4.93 0.01
T3
S1 3.30 1.09
S2 4.54 0.00
S3 4.91 0.00
S4 5.07 0.00
S5 5.13 0.00
S6 5.15 0.01
S7 5.24 0.01
S8 5.30 0.06
S9 5.41 0.00
S10 5.43 0.02
C60
S1 2.47 0.00
S2 2.47 0.00
S3 2.47 0.00
S4 2.53 0.00
S5 2.53 0.00
S6 2.53 0.00
S7 2.56 0.00
S8 2.56 0.00
S9 2.56 0.00
S10 2.57 0.00
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Table C.4 Excitation energies Ω of the standard C60-T3-P3HT system the using the CAM-
B3LYP/6-31G* approach in the LR environment
Exc. Ω (eV) f CT Direction of
CT
Location of
exciton
S1 2.53 2.22 0.00 P3HT
S2 3.12 0.00 0.97 P3HTC60
S3 3.12 0.00 1.00 P3HTC60
S4 3.13 0.00 0.99 P3HTC60
S5 3.22 0.60 0.57 T3C60
S6 3.23 0.14 0.90 T3C60
S7 3.24 0.48 0.56 T3C60
S8 3.27 0.10 0.00 P3HT
S9 3.29 0.02 0.99 T3C60
S10 3.87 0.01 0.00 P3HT
S11 3.91 0.00 0.97 P3HTC60
S12 3.91 0.00 1.00 P3HTC60
S13 3.92 0.00 0.99 P3HTC60
S14 3.98 0.19 0.00 P3HT
S15 4.28 0.01 0.85 T3C60
S16 4.29 0.13 0.86 T3C60
S17 4.37 0.00 1.00 P3HTT3
S18 4.45 0.00 0.97 P3HTC60
S19 4.45 0.00 0.99 P3HTC60
S20 4.46 0.00 0.99 P3HTC60
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Table C.5 Excitation energies Ω of the standard C60-T3-P3HT system the using the CAM-
B3LYP/6-31G approach in the LR environment
Exc. Ω (eV) f CT Direction of
CT
Location of
exciton
S1 2.60 2.26 0.0 P3HT
S2 3.08 0.00 0.97 P3HTC60
S3 3.08 0.00 1.00 P3HTC60
S4 3.09 0.00 0.98 P3HTC60
S5 3.20 0.01 0.99 T3C60
S6 3.21 0.01 0.99 T3C60
S7 3.27 0.02 0.99 T3C60
S8 3.32 1.21 0.01 T3
S9 3.35 0.09 0.00 P3HT
S10 3.87 0.00 0.97 P3HTC60
S11 3.88 0.00 1.00 P3HTC60
S12 3.89 0.00 0.98 P3HTC60
S13 3.92 0.01 0.00 P3HT
S14 4.08 0.20 0.00 P3HT
S15 4.28 0.05 0.91 T3C60
S16 4.28 0.04 0.91 T3C60
S17 4.35 0.00 1.00 P3HTC60
S18 4.35 0.00 0.98 P3HTC60
S19 4.36 0.00 0.99 P3HTC60
S20 4.40 0.00 1.00 P3HTT3
Table C.6 Excitation energies Ω of the standard C60-T3-P3HT system using the CAM-
B3LYP/6-31G* approach in the SS environment
Exc. Ω (eV) f Direction of
CT
Location of
Exciton
S1 1.29 0.00 P3HTC60
S2 1.29 0.00 P3HTC60
S3 1.29 0.00 P3HTC60
S4 1.29 0.00 P3HTC60
S5 1.88 0.00 T3C60
S6 1.88 0.00 T3C60
S7 1.88 0.00 T3C60
S8 1.88 0.00 T3C60
S9 1.90 0.00 T3C60
S10 2.63 2.04 P3HT
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Table C.7 Excitation energies Ω of the standard C60-T3-P3HT system using the CAM-
B3LYP/6-31G approach in the SS environment
Exc. Ω (eV) f CT Direction of
CT
Location of
Exciton
S1 1.23 0.00 1.00 P3HTC60
S2 1.23 0.00 1.00 P3HTC60
S3 1.24 0.00 1.00 P3HTC60
S4 1.84 0.00 1.00 T3C60
S5 1.84 0.00 1.00 T3C60
S6 1.86 0.00 1.00 T3C60
S7 2.06 0.00 1.00 P3HTC60
S8 2.06 0.00 1.00 P3HTC60
S9 2.07 0.00 1.00 P3HTC60
S10 2.70 2.07 0.00 P3HT
Table C.8 Excitation energies Ω of the isolated standard C60-T3-P3HT system using the
CAM-B3LYP/6-31G* approach (only core orbitals were frozen)
Exc. Ω (eV) f CT Direction of
CT
Location of
exciton
S1 2.47 0.00 0.01 C60
S2 2.47 0.00 0.00 C60
S3 2.48 0.00 0.01 C60
S4 2.52 0.00 0.01 C60
S5 2.53 0.00 0.00 C60
S6 2.53 0.00 0.01 C60
S7 2.56 0.00 0.01 C60
S8 2.56 0.00 0.01 C60
S9 2.57 0.00 0.00 C60
S10 2.57 0.00 0.00 C60
S11 2.62 2.03 0.00 P3HT
S12 2.84 0.00 0.01 C60
S13 2.85 0.00 0.01 C60
S14 2.85 0.00 0.01 C60
S15 2.86 0.00 0.01 C60
S16 2.86 0.00 0.01 C60
S17 2.91 0.00 1.00 P3HTC60
S18 2.91 0.00 1.00 P3HTC60
S19 2.92 0.00 1.00 P3HTC60
S20 2.93 0.00 1.00 T3C60
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Table C.9 Excitation energies Ω of the isolated extended (C60)3-(T3)3-P3HT system using
the CAM-B3LYP/6-31G* approach
Exc. Ω (eV) f CT Direction of
CT
Location of
exciton
1 1A 2.33 0.00 1.00 (T3)3(C60)3
2 1A 2.37 0.00 1.00 (T3)3(C60)3
3 1A 2.38 0.00 1.00 (T3)3(C60)3
4 1A 2.56 0.01 0.99 (T3)3(C60)3
5 1A 2.59 0.00 1.00 (T3)3(C60)3
6 1A 2.62 0.00 0.99 (T3)3(C60)3
7 1A 2.67 1.92 0.00 P3HT
8 1A 2.74 0.00 1.00 (T3)3(C60)3
9 1A 2.75 0.00 1.00 (T3)3(C60)3
10 1A 2.76 0.00 1.00 (T3)3(C60)3
11 1A 2.78 0.00 1.00 (T3)3(C60)3
12 1A 2.79 0.00 0.99 (T3)3(C60)3
13 1A 2.80 0.00 1.00 (T3)3(C60)3
14 1A 2.81 0.00 0.98 (T3)3(C60)3
15 1A 2.82 0.00 1.00 (T3)3(C60)3
16 1A 2.82 0.00 1.00 P3HT(C60)3
17 1A 2.83 0.00 1.00 (T3)3(C60)3
18 1A 2.85 0.00 1.00 P3HT(C60)3
19 1A 2.89 0.00 1.00 P3HT(C60)3
20 1A 2.96 0.00 0.97 (T3)3(C60)3
21 1A 2.96 0.00 0.99 (T3)3(C60)3
22 1A 2.97 0.00 0.98 (T3)3(C60)3
23 1A 3.02 0.00 1.00 (T3)3(C60)3
24 1A 3.04 0.00 1.00 (T3)3(C60)3
25 1A 3.05 0.00 1.00 (T3)3(C60)3
26 1A 3.06 0.00 1.00 P3HT(C60)3
27 1A 3.07 0.00 1.00 P3HT(C60)3
28 1A 3.08 0.00 1.00 P3HT(C60)3
29 1A 3.10 0.00 0.10 (T3)3
30 1A 3.14 0.00 1.00 (T3)3(C60)3
31 1A 3.15 0.00 1.00 (T3)3(C60)3
32 1A 3.15 0.00 1.00 P3HT(C60)3
33 1A 3.16 0.00 1.00 P3HT(C60)3
34 1A 3.16 0.00 1.00 (T3)3(C60)3
35 1A 3.17 0.00 1.00 P3HT(C60)3
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Table C.9, Continued
36 1A 3.26 0.07 0.03 (T3)3
37 1A 3.28 0.00 0.99 (T3)3(C60)3
38 1A 3.29 0.00 0.99 (T3)3(C60)3
39 1A 3.29 0.00 0.97 (T3)3(C60)3
40 1A 3.43 1.38 0.03 P3HT/(T3)3
Table C.10 Excitation energies Ω of the isolated extended (C60)3-(T3)3-P3HT system using
the CAM-B3LYP/6-31G approach
Exc. Ω (eV) f CT Direction of
CT
Location of
exciton
S1 2.24 0.00 1.00 (T3)3(C60)3
S2 2.28 0.00 1.00 (T3)3(C60)3
S3 2.30 0.00 1.00 (T3)3(C60)3
S4 2.47 0.01 0.99 (T3)3(C60)3
S5 2.51 0.00 1.00 (T3)3(C60)3
S6 2.54 0.00 1.00 (T3)3(C60)3
S7 2.68 0.00 1.00 (T3)3(C60)3
S8 2.69 0.00 1.00 (T3)3(C60)3
S9 2.70 0.00 1.00 (T3)3(C60)3
S10 2.71 0.00 1.00 (T3)3(C60)3
S11 2.72 0.00 1.00 (T3)3(C60)3
S12 2.73 0.00 1.00 (T3)3(C60)3
S13 2.73 0.00 1.00 P3HT(C60)3
S14 2.74 1.96 0.00 P3HT
S15 2.75 0.00 0.98 (T3)3(C60)3
S16 2.75 0.00 1.00 (T3)3(C60)3
S17 2.76 0.00 1.00 P3HT(C60)3
S18 2.77 0.00 0.97 (T3)3(C60)3
S19 2.79 0.00 1.00 P3HT(C60)3
S20 2.92 0.00 0.98 (T3)3(C60)3
S21 2.92 0.00 1.00 (T3)3(C60)3
S22 2.93 0.00 0.97 (T3)3(C60)3
S23 2.96 0.00 1.00 (T3)3(C60)3
S24 2.97 0.00 1.00 (T3)3(C60)3
S25 2.99 0.00 1.00 (T3)3(C60)3
S26 3.00 0.00 1.00 P3HT(C60)3
S27 3.01 0.00 1.00 P3HT(C60)3
S28 3.02 0.00 1.00 P3HT(C60)3
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Table C.10, Continued
S29 3.07 0.00 1.00 (T3)3(C60)3
S30 3.08 0.00 1.00 (T3)3(C60)3
S31 3.09 0.00 1.00 (T3)3(C60)3
S32 3.10 0.00 1.00 P3HT(C60)3
S33 3.11 0.00 1.00 P3HT(C60)3
S34 3.11 0.00 1.00 P3HT(C60)3
S35 3.15 0.00 0.11 (T3)3
S36 3.23 0.00 0.99 (T3)3(C60)3
S37 3.23 0.00 0.99 (T3)3(C60)3
S38 3.24 0.00 0.97 (T3)3(C60)3
S39 3.32 0.10 0.03 (T3)3
S40 3.49 1.97 0.08 (T3)3
Table C.11 Excitation energies Ω of the extended C60-T3-P3HT system using the CAM-
B3LYP/6-31G approach in the SS environment
Exc. Ω (eV) f CT Direction of
CT
Location of
exciton
S1 1.59 0.00 1.00 P3HT(C60)3
S2 1.59 0.00 1.00 P3HT(C60)3
S3 1.60 0.00 1.00 P3HT(C60)3
S4 1.61 0.00 1.00 P3HT(C60)3
S5 1.77 0.00 1.00 (T3)3(C60)3
S6 1.77a 0.00 1.00 (T3)3(C60)3
S7 1.88 0.00 1.00 (T3)3(C60)3
S8 1.91 0.00 1.00 (T3)3(C60)3
S9 1.91 0.00 1.00 (T3)3(C60)3
S10 1.91 0.00 1.00 (T3)3(C60)3
S11 1.91 0.00 1.00 (T3)3(C60)3
S12 2.10 0.00 1.00 (T3)3(C60)3
S13 2.10 0.00 1.00 (T3)3(C60)3
S14 2.10 0.00 1.00 (T3)3(C60)3
S15 2.12 0.00 1.00 (T3)3(C60)3
S16 2.12 0.00 1.00 (T3)3(C60)3
S17 2.13 0.00 1.00 (T3)3(C60)3
S18 2.13 0.00 1.00 (T3)3(C60)3
S19 2.21b 0.00 1.00 (T3)3(C60)3
S20 2.75 1.92 0.00 P3HT aExcitation energy averaged over two oscillating energies: 1.77 eV and 1.76 eV, bExcitation energy averaged over three oscillating energies: 2.12 eV, 2.21 eV, 2.29 eV
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APPENDIX D
ADDITIONAL TABLES AND FIGURES
NOT INCLUDED IN CHAPTER VI
Table D.1 Interchain distances (Å) and ΔE (kcal/mol) in the stacked PPV2 dimer computed
at MP2 and MP2/BSSE levels with several basis sets using C2h symmetry.
1-1’ 2-2’ 3-3’ 4-4’ 5-5’ ΔE
MP2
cc-pVDZ 3.58 3.56 3.54 3.61 3.62 -9.13
aug-cc-pVDZ 3.48 3.45 3.44 3.50 3.52 -20.23
cc-pVTZ 3.57 3.55 3.53 3.59 3.60 -11.80
aug-cc-pVTZ 3.57 3.45 3.46 3.53 3.64 -15.10
MP2/BSSE
cc-pVDZ 3.78 3.77 3.77 3.78 3.79 -4.73
cc-pVTZ 3.66 3.64 3.63 3.67 3.68 -8.95
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Tab
le D
.2 I
nte
rchai
n d
ista
nce
s (Å
) an
d Δ
E (
kca
l/m
ol)
in the
stac
ked
PP
V4 d
imer
com
pute
d
at S
OS
-MP
2 a
nd S
OS
-MP
2/B
SS
E l
evel
s w
ith s
ever
al b
asis
set
s usi
ng C
2h s
ym
met
ry.
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Table D.3 Interchain distances (Å) and ΔE (kcal/mol) in the diplaced PPV4 dimer computed
at SOS-MP2 level with several basis sets using C2h symmetry.
1-1’ 2-2’ 3-3’ 4-4’ 5-5’ 6-6’ 7-7’ 8-8’ ΔE
cc-pVTZ 3.57 3.50 3.46 3.45 3.45 3.46 3.50 3.57 -21.17
cc-pVQZa - - - - - - - - -19.56
CBS limit extrapolation:
TQb -18.39 aSingle-point energy at SOS-MP2/cc-pVTZ optimized geometry, bExtrapolation made
using cc-pVTZ and cc-pVQC
Table D.4 Single chain PPV2 vertical excitations using SOS-ADC(2) and SOS-MP2
optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator strength)
Basis State Excitation ΔE (eV) f
cc-pVDZ S1 1 1Bu 4.57 0.51 S2 1 1Ag 4.65 0.00 S3 2 1Bu 4.74 0.72 S4 2 1Ag 6.10 0.00 S5 1 1Bg 7.63 0.00 S6 1 1Au 7.76 0.00 S7 2 1Bg 7.80 0.00 S8 2 1Au 8.22 0.00
cc-pVTZ S1 1 1Bu 4.52 0.75 S2 1 1Ag 4.66 0.00 S3 2 1Bu 4.72 0.43 S4 2 1Ag 6.00 0.00 S5 1 1Bg 7.24 0.00 S6 1 1Au 7.26 0.00 S7 2 1Bg 7.56 0.00 S8 2 1Au 8.04 0.01
cc-pVQZ S1 1 1Bu 4.49 0.77 S2 1 1Ag 4.66 0.00 S3 2 1Bu 4.71 0.40 S4 2 1Ag 5.95 0.00 S5 1 1Au 6.86 0.00 S6 1 1Bg 6.95 0.00
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Table D.4, Continued
S7 2 1Bg 7.26 0.00 S8 2 1Au 7.66 0.02
Table D.5 Stacked PPV2 dimer vertical excitations using SOS-ADC(2) and SOS-MP2
optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator strength, CT –
interchain charge transfer)
Basis State Excitation ΔE (eV) f CT (e)
cc-pVDZ S1 1 1Bg 4.21 0.00 0.17
S2 1 1Au 4.51 0.00 0.05
S3 2 1Bg 4.53 0.00 0.08
S4 1 1Bu 4.58 0.36 0.01
S5 1 1Ag 4.63 0.00 0.01
S6 2 1Bu 4.81 1.82 0.01
S7 2 1Au 5.78 0.00 0.21
S8 2 1Ag 6.11 0.00 0.02
cc-pVTZ S1 1 1Bg 4.10 0.00 0.18
S2 1 1Au 4.50 0.00 0.08
S3 2 1Bg 4.51 0.00 0.08
S4 1 1Bu 4.55 0.74 0.03
S5 1 1Ag 4.65 0.00 0.02
S6 2 1Bu 4.76 1.32 0.02
S7 2 1Au 5.64 0.00 0.22
S8 2 1Ag 6.01 0.00 0.03
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Table D.6 Stacked PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and SOS-
MP2/cc-pVTZ optimized geometry showing the largest contributing hole/electron NTO
pairs (Isovalue = ±0.03 Bohr3)
Excitation % Hole Electron
1 1Bg 82
12
1 1Au 28
26
2 1Bg 40
16
11
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Table D.6, Continued
1 1Bu 41
30
1 1Ag 19
18
16
2 1Bu 46
36
2 1Au 55
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Table D.6, Continued
25
2 1Ag 33
25
14
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Table D.7 Displaced PPV2 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and
SOS-MP2/cc-pVTZ optimized geometry showing the largest contributing hole/electron
NTO pairs (Isovalue = ±0.03 Bohr3)
Excitation % Hole Electron
1 1A 46
45
2 1A 40
29
3 1A 23
19
17
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Table D.7, Continued
13
4 1A 28
21
19
13
5 1A 35
20
16
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Table D.7, Continued
14
6 1A 53
32
7 1A 55
40
8 1A 62
33
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Table D.8 Stacked PPV3 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and SOS-
MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator
strength, CT – interchain charge transfer)
State Excitation ΔE (eV) f CT (e)
S1 1 1Bg 3.47 0.00 0.18
S2 1 1Bu 3.98 3.60 0.02
S3 2 1Bg 4.21 0.00 0.10
S4 2 1Bu 4.38 0.34 0.02
S5 1 1Au 4.39 0.00 0.16
S6 2 1Au 4.57 0.00 0.15
S7 1 1Ag 4.60 0.00 0.04
S8 2 1Ag 5.11 0.00 0.02
Table D.9 Stacked PPV4 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and SOS-
MP2/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy, f – oscillator
strength, CT – interchain charge transfer)
State Excitation ΔE (eV) f CT (e)
S1 1 1Bg 3.20 0.00 0.19
S2 1 1Bu 3.69 5.58 0.02
S3 1 1Au 3.86 0.00 0.21
S4 2 1Bg 4.17 0.00 0.13
S5 2 1Au 4.20 0.00 0.14
S6 1 1Ag 4.31 0.00 0.04
S7 2 1Bu 4.34 0.31 0.23
S8 2 1Ag 4.53 0.00 0.04
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Table D.10 Displaced PPV3 dimer vertical excitations using SOS-ADC(2)/cc-pVTZ and
SOS-MP2/cc-pVTZ optimized geometry showing the largest contributing hole/electron
NTO pairs (Isovalue = ±0.03 Bohr3)
Excitation % Hole Electron
1 1A 50
40
2 1A 55
35
3 1A 38
24
13
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Table D.10, Continued
11
4 1A 31
24
15
14
5 1A 50
30
6 1A 28
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Table D.10, Continued
24
7 1A 26
20
8 1A 16
16
14
12
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Table D.11 Stacked PPV3 dimer vertical excitations using B3LYP-D3/cc-pVTZ and the
B3LYP-D3/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy, f –
oscillator strength, CT – interchain charge transfer)
State Excitation ΔE (eV) f CT (e)
S1 1 1Bg 2.40 0.00 0.50
S2 1 1Bu 2.83 0.04 0.99
S3 2 1Bu 3.28 3.10 0.01
S4 2 1Bg 3.38 0.00 0.50
S5 1 1Au 3.38 0.00 0.50
S6 2 1Au 3.45 0.00 0.50
S7 1 1Ag 3.77 0.00 0.99
S8 3 1Bg 3.79 0.00 0.50
S9 2 1Ag 3.84 0.00 0.08
S10 4 1Bg 3.95 0.00 0.50
S11 3 1Au 3.97 0.00 0.50
S12 3 1Ag 3.99 0.00 0.92
S13 5 1Bg 4.07 0.00 0.50
S14 6 1Bg 4.13 0.00 0.50
S15 4 1Au 4.15 0.00 0.50
S16 3 1Bu 4.18 0.06 0.08
S17 4 1Ag 4.33 0.00 0.05
S18 4 1Bu 4.35 0.01 0.07
S19 5 1Au 4.41 0.00 0.50
S20 7 1Bg 4.44 0.00 0.50
S21 8 1Bg 4.47 0.00 0.50
S22 6 1Au 4.47 0.00 0.50
S23 5 1Bu 4.47 0.02 0.90
S24 5 1Ag 4.48 0.00 0.03
S25 6 1Bu 4.52 0.00 1.00
S26 7 1Bu 4.58 0.01 0.94
S27 6 1Ag 4.58 0.00 0.99
S28 8 1Bu 4.60 0.01 0.83
S29 7 1Ag 4.63 0.00 0.96
S30 9 1Bg 4.71 0.00 0.54
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Table D.12 Stacked PPV3 dimer vertical excitations using CAM-B3LYP-D3/cc-pVTZ and
the CAM-B3LYP-D3/cc-pVTZ optimized geometry (ΔE (eV) – vertical excitation energy,
f – oscillator strength, CT – interchain charge transfer)
State Excitation ΔE (eV) f CT (e)
S1 1 1Bg 3.17 0.00 0.25
S2 1 1Bu 3.73 3.16 0.05
S3 2 1Bu 4.09 0.13 0.96
S4 1 1Au 4.17 0.00 0.28
S5 2 1Bg 4.35 0.00 0.77
S6 3 1Bg 4.47 0.00 0.16
S7 3 1Bu 4.67 0.07 0.02
S8 4 1Bg 4.77 0.00 0.19
S9 2 1Au 4.77 0.00 0.28
S10 1 1Ag 4.78 0.00 0.09
S11 3 1Au 4.88 0.00 0.50
S12 4 1Bu 4.98 0.01 0.01
S13 2 1Ag 5.02 0.00 0.12
S14 3 1Ag 5.03 0.00 0.88
S15 5 1Bg 5.07 0.00 0.50
S16 4 1Ag 5.17 0.00 0.03
S17 6 1Bg 5.27 0.00 0.50
S18 4 1Au 5.33 0.00 0.73
S19 5 1Au 5.39 0.00 0.50
S20 7 1Bg 5.47 0.00 0.50
S21 5 1Bu 5.59 0.34 0.25
S22 5 1Ag 5.67 0.00 0.99
S23 6 1Bu 5.73 0.15 0.40
S24 8 1Bg 5.79 0.00 0.27
S25 7 1Bu 5.84 0.28 0.48
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S1 S2 S3 S4
S5 S6 S7 S8
Figure D.1 Omega matrices for vertical excitations of displaced PPV2 dimer using the SOS-
ADC(2)/cc-pVTZ method with the SOS-MP2/cc-pVTZ optimized geometry. Each
vinylene and phenylene group is a separate fragment: fragments 1-3 belong to one chain,
fragments 4-6 to the other. The y-axis represents the electron and the x-axis represents the
hole. Omega matrix plots calculated using the TheoDORE program package.
S1 S2 S3 S4
S5 S6 S7 S8
Figure D.2 Omega matrices for vertical excitations of stacked PPV4 dimer using the SOS-
ADC(2)/cc-pVTZ method with the SOS-MP2/cc-pVTZ optimized geometry. Each
vinylene and phenylene group is a separate fragment: fragments 1-7 belong to one chain,
fragments 8-14 to the other. The y-axis represents the electron and the x-axis represents the
hole. Omega matrix plots calculated using the TheoDORE program package.
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S1 S2 S3 S4
S5 S6 S7 S8
Figure D.3 Omega matrices for vertical excitations of displaced PPV3 dimer using the SOS-
ADC(2)/cc-pVTZ method with the SOS-MP2/cc-pVTZ optimized geometry. Each
vinylene and phenylene group is a separate fragment: fragments 1-5 belong to one chain,
fragments 6-10 to the other. The y-axis represents the electron and the x-axis represents the
hole. Omega matrix plots calculated using the TheoDORE program package.
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APPENDIX E
ADDITIONAL TABLES AND FIGURES
NOT INCLUDED IN CHAPTER VII
Table E.1 Active orbitals for each MR-AQCC reference space
Reference Space CAS AUX
1
MR-AQCC(CAS(4,4))a 5au, 6au, 5bg, 6bg
MR-AQCC(CAS(4,4)) 5au, 6au, 5bg, 6bg
MR-AQCC
(CAS(10,7)/AUX(3))
3au, 4au, 5au,
6au, 4bg, 5bg, 6bg 7au, 7bg, 8bg
MR-AQCC
(CAS(14,9)/AUX(3))
3au, 4au, 5au,
6au, 2bg, 3bg,
4bg, 5bg, 6bg
7au, 7bg, 8bg
2
MR-AQCC(CAS(4,4)) 6au, 7au, 6bg, 7bg
MR-AQCC
(CAS(12,8)/AUX(4))
4au, 5au, 6au,
7au, 4bg, 5bg,
6bg, 7bg
8au, 9au, 8bg, 9bg
3
MR-AQCC(CAS(4,4)) 7au, 8au, 7bg, 8bg
4
MR-AQCC(CAS(4,4)) 8au, 9au, 8bg, 9bg
5
MR-AQCC(CAS(4,4)) 5b2, 6b2, 5a2, 6a2
6
MR-AQCC(CAS(4,4)) 7b2, 8b2, 5a2, 6a2
7
MR-AQCC(CAS(4,4)) 7b2, 8b2, 7a2, 8a2
8
MR-AQCC(CAS(4,4)) 9b2, 10b2,
7a2, 8a2
9
MR-AQCC(CAS(4,4)) 8au, 9au, 8bg, 9bg
MR-AQCC
(CAS(10,7)/AUX(3))
6au, 7au, 8au,
9au, 7bg, 8bg, 9bg 10au, 10bg, 11bg
10
MR-AQCC(CAS(4,4)) 9au, 10au,
9bg, 10bg
11
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Table E.1, Continued
MR-AQCC(CAS(4,4)) 10au, 11au,
10bg, 11bg
12
MR-AQCC(CAS(4,4)) 11au, 12au,
11bg, 12bg
aonly core orbitals are frozen
Table E.2 NU value (e) of trans-diindenoacene (1-4) singlet 1 1Ag state and vertically
excited 1 3Bu state using the MR-AQCC(CAS(4,4))/6-31G* method.
System State NU (e)
1 1 1Ag aNU 0.547
bNU(H-L) 0.230
1 3Bu aNU 2.376
bNU(H-L) 1.991
2 1 1Ag aNU
1.095
bNU(H-L) 0.672
1 3Bu aNU
2.455
bNU(H-L) 1.997
3 1 1Ag aNU
1.602
bNU(H-L) 1.048
1 3Bu aNU
2.563
bNU(H-L) 1.998
4 1 1Ag aNU
2.178
bNU(H-L) 1.456
1 3Bu aNU
2.722
bNU(H-L) 1.998
aSum over all NO occupations, bSum over only HONO-LUNO pair
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Table E.3 NU value (e) of cis-diindenoacene (5-8) singlet 1 1A1 state and vertically excited
1 3B1 state using the MR-AQCC(CAS(4,4))/6-31G* method.
System State NU (e)
5 1 1A1 aNU 1.414
bNU(H-L) 1.083
1 3B1 aNU 2.345
bNU(H-L) 1.999
6 1 1A1 aNU
1.923
bNU(H-L) 1.506
1 3B1 aNU
2.429
bNU(H-L) 2.000
7 1 1A1 aNU
2.280
bNU(H-L) 1.764
1 3B1 aNU
2.541
bNU(H-L) 2.000
8 1 1A1 aNU
2.550
bNU(H-L) 1.902
1 3B1 aNU
2.703
bNU(H-L) 2.000
aSum over all NO occupations, bSum over only HONO-LUNO pair
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Table E.4 NU value (e) of trans-indenoindenodi(acenothiophene) (9-12) singlet 1 1Ag state
and vertically excited 1 3Bu state using the MR-AQCC(CAS(4,4))/6-31G*(6-311G*)S
method.
System State NU (e)
9 1 1Ag aNU 0.784
bNU(H-L) 0.362
1 3Bu aNU 2.461
bNU(H-L) 1.997
10 1 1Ag aNU
1.505
bNU(H-L) 0.979
1 3Bu aNU
2.552
bNU(H-L) 1.999
11 1 1Ag aNU
2.088
bNU(H-L) 1.451
1 3Bu aNU
2.670
bNU(H-L) 1.999
12 1 1Ag aNU
2.482
bNU(H-L) 1.623
1 3Bu aNU
2.834
bNU(H-L) 1.999
aSum over all NO occupations, bSum over only HONO-LUNO pair
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Figure E.1 Quinoid Kekule and biradical resonance forms of trans-diindenoacenes (1-4),
cis-diindenoacenes (5-8), and trans-indenoindenodi(acenothiophenes) (9-12).
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Figure E.2 Total unpaired electron density (sum over all NOs, Table E.2), in red, of the
vertically excited 1 3Bu state of trans-diindenobenzene (1-4) using the MR-
AQCC(CAS(4,4))/6-31G* method. (isovalue=0.004 e/Bohr3). HOMA index is shown in
the center of each ring.
Figure E.3 Total unpaired electron density (sum over all NOs, Table E.3), in red, of the
vertically excited 1 3B1 state of cis-diindenobenzene (5-8) using the MR-
AQCC(CAS(4,4))/6-31G* method. (isovalue=0.004 e/Bohr3). HOMA index is shown in
the center of each ring.
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Figure E.4 Total unpaired electron density (sum over all NOs, Table E.4), in red, of the
vertically excited 1 3Bu state of trans-indenoindenodi(acenothiophene) (9-12) using the
MR-AQCC(CAS(4,4))/6-31G*(6-311G*)S method. (isovalue=0.004 e/Bohr3). HOMA
index is shown in the center of each ring.