UNCERTAINTIES OF MOLECULARSTRUCTURAL PARAMETERS

IAEA-ITAMP joint workshop on UncertaintyAssessment for Atomic and Molecular Data

Cambridge, MA, USA, July 7-9, 2014

Attila G. CsászárLaboratory of Molecular Structure and Dynamics &

MTA-ELTE Research Group on Complex Chemical SystemsEötvös University, Budapest, Hungary

OUTLINEIntroduction and motivation

The quantum chemical approach to re

Empirical approaches to re

The spectroscopic approach to re

Different structural typesUncertainties in experiment and theoryvia examples

Summary

Sources of uncertainties in structural parameters

(a) Many different experimental and first‐principles (QC) techniques

(b) Many different structure definitions

(c) Intrinsic uncertainties of the methods

quantum chemistry spectroscopy diffraction

empirical NMR

rotation X‐rays

electrons(GED)

neutrons

semi‐experimental (SE)

DFTab initio (BO)

J. Demaison

GED Spectroscopy ab initio (exl. nucl. motion)

ra B0 fijk re

rm semi‐exp ab initioexp.r,T

comparison

Principal interrelationships

Principal structural types

• equilibrium structures: reBO, read, reSE, reexp

• average structuresposition averages: rz = r,0; r,T (av. nuclear position in ground vibr. state)distance (and angle) averages: rg,T; ra,T; <r2>T1/2; <r ‐2>T‐1/2; <r3>T1/3; <r ‐3>T‐1/3

• mass‐dependent structures: rm; rc; rmρ; rm(1); rm(2)

• empirical structures: r0; rs; rs variants (ps‐Kr)

Accurate (equilibrium) structuresof gas-phase molecules

• Traditional debate (experiment):MW vs. GED

• Newer debate (experiment vs. theory): MW and GED vs. ab initio

• End of debate:experiment AND theory

Variability of structural parameters

• relatively large for angles(accurate determination of angles is usually slightly easier than that of distances)

• relatively small for bonded distances• 1.060 Å < re(C‐H) < 1.120 Å, variability is only about 0.06 Å• 1.153 Å < re(C=O) < 1.206 Å, variability is only about 0.05 Å(1.153 Å for OCSe, 1.160 Å for CO2, 1.205 Å for H2CO, 1.206 Å for CH3CHO)

• 1.10 Å < re(CO single, double, triple) < 1.43 Å, variability is about 0.33 Å(1.1056 Å for HCO+ and 1.4338 Å for HCOOCH3)

Structures of semirigid molecules:the experimental/empirical path

H. D. Rudolph, J. Demaison, A. G. Császár, J. Phys. Chem. A 2013, 117, 12969-12982.

Structures of semirigid molecules:experimental or theoretical paths

H. D. Rudolph, J. Demaison, A. G. Császár, J. Phys. Chem. A 2013, 117, 12969-12982.

Structures of semirigid molecules:joint experimental-theoretical path

H. D. Rudolph, J. Demaison, A. G. Császár, J. Phys. Chem. A 2013, 117, 12969-12982.

I. The quantum chemical approachto equilibrium structures:

reBO, re

ad

Theory: quantum chemistry• Quantum particles: nuclei and electrons• Full treatment is still impractical for all but the simplest many‐body systems

• Often it is sufficient to solve the time‐independentSchrödinger equation (TISE)

• Introduction of the Born‐Oppenheimer (BO) approximation results in electronic structure and nuclear motion theories

• No practical analytic solutions: variational and perturbative treatments

Number of independent internalcoordinate force constants

terms linear XY2 bent XY2 XY3Z X6Y6

Linear re 1 2 3 2

Quadratic fij 3 4 12 34

Cubic fijk 3 6 38 237

Quartic fijkl 6 9 102 1890

Quintic fijklm 6 12 249 12031

Sextic fijklmn 10 16 562 69 448

A. G. Császár, J. Phys. Chem. A 1992, 96, 7898-7904.

Motivations of the Focal-PointAnalysis (FPA) Approach

Get the right result for the right reason for polyatomicand polyelectronic systems.

Attach error bars to theoretical predictions (excellenthandle on uncertainties).

Consider small physical effects tacitly neglected in most quantum chemical studies, such as core correlation, relativistic effects, and corrections to the Born-Oppenheimer approximation.

Approach spectroscopic accuracy (1 cm-1) as opposed to chemical (1 kcal mol-1) or calibration (1 kJ mol-1) accuracy in predictions of relative energies and spectra.

A. G. Császár, W. D. Allen, H. F. Schaefer, J. Chem. Phys. 1998, 108, 9751-9764.

The Focal-Point Analysis (FPA) approachuse of a family of basis sets which systematically

approaches completeness (e.g., (aug-)cc-p(C)VnZ)applications of low levels of theory with prodigious basis sets (typically direct RHF and MP2 computations with up to thousands of basis functions)higher-order (valence) correlation (HOC) treatments [these days FCI, CCSDTQ, CCSDT, and CCSD(T)] with the largest possible basis setslayout of a two-dimensional extrapolation grid based on an assumed additivity of correlation incrementseschewal of empirical corrections/extrapolationsaddition of “small” correction terms (CC, Rel, DBOC)

A. G. Császár, W. D. Allen, H. F. Schaefer, J. Chem. Phys. 1998, 108, 9751-9764.

Electron correlation treatment

Hamilto

nian

One

-par

ticle

bas

is s

et

Exac

tan

swer

DZHF

TZHF

QZHF

5ZHF

Z(CBS)

HF

DZCCSD

DZCCSD(T)

DZFull-CI

DZCCSDT

(MP2, CISD) (MP4, CISDTQ)

CBSFull-CI

Rubik’s cube of BO electronic structure theory

Diatomic paradigms

• W. D. Allen and A. G. Császár, J. Chem. Phys. 98, 2983(1993). 

• A. L. L. East, W. D. Allen, and A. G. Császár, inStructures and conformations of non‐rigid molecules, Kluwer: Dordrecht, 1993.

• A. G. Császár and W. D. Allen, J. Chem. Phys. 104,2746 (1996).

Percent error curves for the RHF/DZP electronic energy derivatives

___________________________________________________________________ W. D. Allen and A. G. Császár, J. Chem. Phys. 98, 2983 (1993).

Choice of reference structure

Theory: incremental corrections to the equilibrium geometry of H2

16O

Level re/Å e/degreesaug‐cc‐pV6Z ICMRCI_FC 0.958 705 104.413CBS ICMRCI_FC 0.000 085 +0.010Core correlation 0.000 986 +0.134Relativistic (Breit) +0.000 164 0.074QED (1‐e Lamb) 0.000 002 +0.003DBOC +0.000 040    +0.015Final 0.957 837 104.500

A. G. Császár, G. Czakó, T. Furtenbacher, J. Tennyson, V. Szalay, S.V. Shirin, N. F. Zobov, O. L. Polyansky, J. Chem. Phys. 2005, 122, 214305.

II. The mass-dependent („empirical”) approach to equilibrium structures:

rm

“Empirical” structuresGuiding principle: I0 = Ie + ɛ, where I0 are rotational constantscorresponding to the ground vibrational state of an isotopologue

choice of ɛ symbol comment

0 r0 LSQ fitting

constant rs Costain, 1958; no LSQ

g g1/2 rm

(1) 1979; for each principal axis g

g g1/2+dg

∏ 1/(2N-2) rm(2) Watson, 1999

III. The semiexperimental approachto equilibrium structures:

reSE

Traditional (semi)experimental approach(inverse, top down)

Determine (preferentially experimentally)rovibrational energy levels(ground state) effective rotational constantsvibration-rotation interaction constantscentrifugal and electronic corrections equilibrium geometry

Recommended first-principles approach(direct, bottom up)

Determine (preferentially fully ab initio) adiabatic potential energy hypersurface equilibrium geometryChecking the accuracy of re:rotational energy levelsvibrationally averaged spectroscopic constants

νννν EckartBBB

)2/1( vv ee BB

Be B0 idi

2i1

3N6

What we need

i B0 B(vi 1)

ith fundamental statei =1, …3N ‐ 6

Rotational constants ≤ 3 (per isotopologue)

Spectra = m isotopic species x 3N ‐ 5 vibrational states

Semiexperimental structure: principles

Spectroscopy (MW or IR) B0 (or I0)

Ab initio (or DFT) cubic force field i (or )

some important advantages:• small ( < 1%): high accuracy is ″not needed”• easy for isotopologues• fast

Semiexperimental structure: practice

Possible to study large molecules

• Proline (and other amino acids)• 17 atoms• 45 parameters• no. of cubic force constants: 62 835• W. D. Allen, E. Czinki, A. G. Császár, Chem. Eur. J.  2004, 10, 4512

• tropinone• 23 atoms• 34 parameters• J. Demaison,… J. Phys. Chem. A 2012

Semiexperimental structure: problems• accuracy of the ground‐state rotational constants• isotopic species: difficult or time consuming to produce• presence of small coordinates• strong Coriolis and/or anharmonic resonances• large rotation of the principal axes• accuracy of the force field

• form of the potential• H atoms: large amplitude motion or …

• ill‐conditioning (solution: mixed regression)• …

)1(0 ii vBB

Rotational constants (in cm-1) of D2

16O: “experiment” vs. theoryLABEL EXPT. #1 VAR DIFF.

A (2 2 0) 17.456 17.536 -0.080

A (0 2 2) 16.857 16.619 0.238

A (3 0 0) 14.354 14.667 -0.313

A (1 0 2) 14.551 14.261 0.290

B (0 2 2) 7.292 7.352 -0.060

B (3 0 0) 6.956 7.020 -0.064

B (1 0 2) 7.112 7.099 0.087

Rotational constants (in cm-1) of D2

16O: “experiment” vs. theoryLABEL EXPT. #1 VAR DIFF. EXPT. #2 DIFF.

A (2 2 0) 17.456 17.536 -0.080 17.545 0.008

A (0 2 2) 16.857 16.619 0.238 16.631 0.012

A (3 0 0) 14.354 14.667 -0.313 14.677 0.010

A (1 0 2) 14.551 14.261 0.290 14.255 0.006

B (0 2 2) 7.292 7.352 -0.060 7.359 0.007

B (3 0 0) 6.956 7.020 -0.064 7.018 0.002

B (1 0 2) 7.112 7.099 0.087 7.113 0.014

Large rotation of the principal axis system upon isotopic substitution

• important for oblate molecules (A B >> C)• non‐constant systematic error : A  B• see  J. Demaison, A. G. Császár, L. D. Margulés, H. D. Rudolph,

J. Phys. Chem. A 2011, 115, 14078.

Ia Ib angleparent 0.0001 -0.0001 018Osyn +0.0151 -0.0160 41°

example: residuals of HNO3 (mean value: 0.0001)

How about uncertainties?

The structure of HCO+

r(HC) r(CO) method reference1.09725(4) 1.10474(2) expt. Woods 881.0919(5) 1.1058(2) ab initio Sebald 901.09215 1.10545 ab initio Martin 931.0919(9) 1.1053(3) semi-expt. Puzzarini 96

1.0924 1.10558 expt. Dore 03

IV. Different structural types

Vibrationally averaged vs. equilibrium geometries

• effective

• average

• rms

• substitution       rs

2/12 rrv

r

2/12r

223

e 1 r

1er

221

e 1 r

223

e 1 fr

ee / rrr

C-H bond length in cyclopropane

Method value ref. Method value ref.

r0 1.083(3) Butch 1973 r0 1.0774 Endo 1987

ra 1.089(3) Bastiansen 1964 rz 1.080(3)

rg 1.099(2) Yamada 1985 re 1.074(1)

rz 1.084(2) reSE 1.079(1) Gauss 2000

re 1.083(5) reBO 1.079 Demaison 2000

Discrepancies: usual between different methods

Molecule bond re method r

OCCl2 C=O 1.176(2) rs 1.1852(5)

OCSe C=O 1.1533(3) rs 1.1561(2)

GeH3Cl Ge-Cl 2.1447(2) r0 2.14947(5)

Discrepancies:not rare within the “same” method (re)

molecule parameter I II

HCO+ C-H 1.09725(4) 1.0919(9)

FCP C-F 1.272 1.284

H2C=C=C: C=C: 1.291(1) 1.287

PH3 P-H 1.41161(2) 1.4168(2)

V. Enlightening examples concerninguncertainties in experiments and theory

Isotopic effects on the equilibrium geometry of water

Isotopomer re/Å e/degreesH2

16O 0.957 837 104.500H2

17O 0.000 001 +0.0001H2

18O 0.000 001 +0.0003D2

16O 0.000 020 0.010D2

18O 0.000 023 0.009HD16O 0.000 001/ 0.002

0.000 022

A. G. Császár, G. Czakó, T. Furtenbacher, J. Tennyson, V. Szalay, S.V. Shirin, N. F. Zobov, O. L. Polyansky, J. Chem. Phys. 2005, 122, 214305.

Isotopic effects on the equilibrium geometry of water

Isotopomer re/Å e/degreesH2

16O 0.957 837 104.500H2

17O 0.000 001 +0.0001H2

18O 0.000 001 +0.0003D2

16O 0.000 020 0.010D2

18O 0.000 023 0.009HD16O 0.000 001/ 0.002

0.000 022

Conclusion: isotopic effects are large for non‐equilibrium but almost negligible for equilibrium structural parameters.

Conclusion: problems with experimental uncertainties

are ubiquitious.

• degrees of freedom large• no isotopic substitution (e.g., F)• no systematic error

– , electron correlation, resonances, …• independent errors (autocorrelation)• small condition number • problem of weighting

Estimation of the uncertainty from the standard deviations of the fit

Solution strategies

• increase the accuracy of the data• experimental: accuracy of B0, , …• ab initio: accuracy of B0 ‐ Be

• decrease the condition number • increase the variety of data (semi‐exp. !)

• clever isotopic substitution• "mixed" regression (or predicate observations)

Mixed regression

y = X

• weighted least squares

• predicate

S wi yiexp ˆ y i

calc i 2

min

S wi yiexp ˆ y i

calc i 2

w j jpred j

calc 2 min

where wk 1 k2

VI. How to unify data coming frommany different sources?

Equilibrium and effective structures of the H2

16O molecule

r/Å /degrees

BornOppenheimer 0.957 82 104.485Adiabatic (H2

16O) 0.957 85 104.500(D2

16O) 0.957 83 104.490Spectroscopic 0.957 77 104.48Effective (rg) (H2

16O) 0.976 25 103.96(300 K) (D2

16O)            0.971 36 104.03_______________________________________________________ A. G. Császár, G. Czakó, T. Furtenbacher, J. Tennyson, V. Szalay, S. V. Shirin,N. F. Zobov, O. L. Polyansky, J. Chem. Phys. 122, 214305 (2005).G. Czakó, E. Mátyus, A. G. Császár, J. Phys. Chem. A 113, 11665 (2009).

________________________________

_________________________________________________________________________

rg structures of water (T = 293 K)H2

16O D216O

rg(OH)/Å rg(HH)/Å rg(OD)/Å rg(DD)/Årg calc.

(GED)(Spectr.)

0.97566(0.9763)(0.9745)

1.53812(1.567)(1.537)

0.97136(0.9700)(0.9702)

1.53122(1.526)(1.531)

ra calc.(Adiabatic re

0.971380.95785

1.529561.51472

0.967830.95783

1.525161.51460)

rg re 0.01781(0.0182)cubic

0.01353(0.0131)cubic__________________________________________________________

G. Czakó, E. Mátyus, A. G. Császár, J. Phys. Chem. A 113, 11665 (2009).

Temperature dependence ofrg(OH) of water

________________________________________________________

G. Czakó, E. Mátyus, A. G. Császár, J. Phys. Chem. A 113, 11665 (2009).

H216O

rg(OH)/Å rg(HH)/Å0 K

300 K700 K

0.975650.976250.97718

1.538231.538121.53825

1000 K1500 K

0.978130.98048

1.539071.54070

GED parameters, root-mean-square amplitudelg (Å) and anharmonicity parameter Å

T/K lg(OX) lg(XX) (OX) (XX)

0 0.0690 0.1142 7.0 ‐1.7

500 0.0690 0.1145 6.5 ‐1.9

1500 0.0713 0.1280 5.3 ‐24.7

0 0.0586 0.0958 3.8 ‐0.2

500 0.0587 0.0972 3.6 ‐1.2

1500 0.0630 0.1172 3.0 ‐25.0

H2O

D2O

NH3 and ND3 structures (300 K)

_______________________________________________________ I. Szabó, C. Fábri, G. Czakó, E. Mátyus, A.G. Császár, J. Phys. Chem. A 116, 4356 (2012).

A system with tunneling motion: NH3

T/K rg(NH)rovib

rg(NH)vib

ra(NH)rovib

ra(NH)vib

rg(HNH)rovib

rg(HNH)vib

0 1.02962 1.02962 1.02454 1.02454 106.733 106.733

10 1.02966 1.02965 1.02458 1.02457 106.717 106.717

30 1.02969 1.02965 1.02461 1.02457 106.713 106.717

50 1.02972 1.02965 1.02464 1.02457 106.711 106.717

re(NH) = 1.01117 and e(HNH) = 106.362. All distances in angstrom, all angles in degrees.

VII. SummaryIt is possible to obtain very accurate molecular structures by experimental means(spectroscopy or GED) when the number of molecular parameters is not large (notlarger than the number of rotational constants for spectroscopy) but the standarddeviation of the fit is definitely not a good indicator of the accuracy obtained.

It is possible to obtain first‐principles predictions of structural parameters of anykind with reliable uncertainty estimates but this procedure is constrained at the moment to molecules containing no more than about 4 atoms and requireenormous computations is accuracy a specific goal.

Comparison

method difficulty speed accuracy price

first principles difficult slowdifficult

to estimate

expensive

experiment difficult slow expensive

semi-experimental rather easy rather fast rather cheap

Each method has its advantages and weaknesses