reducing the computational cost of ab initio …/67531/metadc9061/m2/...reducing the computational...

REDUCING THE COMPUTATIONAL COST OF AB INITIO METHODS

Benjamin Mintz, B.S.

Dissertation Prepared for the Degree of

DOCTOR OF PHILOSOPHY

UNIVERSITY OF NORTH TEXAS

August 2008

APPROVED: Angela K. Wilson, Major Professor and

Departmental Coordinator of Graduate Studies

Martin Schwartz, Committee Member Thomas R. Cundari, Committee Member Wes T. Borden, Committee Member Michael G. Richmond, Chair of the Department of

Chemistry Sandra L. Terrell, Dean of the Robert B. Toulouse

School of Graduate Studies

Mintz, Benjamin. Reducing the Computational Cost of Ab Initio Methods. Doctor of

Philosophy (Chemistry), August 2008, 210 pp., 42 tables, 13 illustrations, 293 references.

In recent years, advances in computer technology combined with new ab initio

computational methods have allowed for dramatic improvement in the prediction of energetic

properties. Unfortunately, even with these advances, the extensive computational cost, in

terms of computer time, memory, and disk space of the sophisticated methods required to

achieve chemical accuracy – defined as 1 kcal/mol from reliable experimental data effectively –

limits the size of molecules [i.e. less than 10‐15 non‐hydrogen atoms] that can be studied.

Several schemes were explored to help reduce the computational cost while still maintaining

chemical accuracy. Specifically, a study was performed to assess the accuracy of ccCA to

compute atomization energies, ionization potentials, electron affinities, proton affinities, and

enthalpies of formation for third‐row (Ga‐Kr) containing molecules. Next, truncation of the

correlation consistent basis sets for the hydrogen atom was examined as a possible means to

reduce the computational cost of ab initio methods. It was determined that energetic

properties could be extrapolated to the complete basis set (CBS) limit utilizing a series of

truncated hydrogen basis sets that was within 1 kcal/mol of the extrapolation of the full

correlation consistent basis sets. Basis set truncation for the hydrogen atom was then applied

to ccCA in the development of two reduced basis set composite methods, ccCA(aug) and

ccCA(TB). The effects that the ccCA(aug) and ccCA(TB) methods had upon enthalpies of

formation and the overall percent disk space saved as compared to ccCA was examined for the

hydrogen containing molecules of the G2/97 test suite. Additionally, the Weizmann‐n (Wn)

methods were utilized to compute the several properties for the alkali metal hydroxides as well

as the ground and excited states of the alkali monoxides anion and radicals. Finally, a multi‐

reference variation to the correlation consistent Composite Approach [MR‐ccCA] was presented

and utilized in the computation of the potential energy surfaces for the N2 and C2 molecules.

ii

Copyright 2008

by

Benjamin Mintz

iii

ACKNOWLEDGMENTS

I gratefully acknowledge the many people that have been instrumental throughout the

process of earning this doctoral degree at the University of North Texas (UNT). Foremost, I

extend my greatest gratitude to my advisor Professor Angela K. Wilson, for whom I have the

utmost respect and admiration. Her passion for theoretical chemistry influenced me as an

undergraduate student and continues to inspire me even today. Furthermore, she is a kind and

compassionate advisor who will always be a friend and colleague.

I thank my committee, the Wilson Group graduate students and postdoctoral fellow

Nathan J. DeYonker for their time and comments on this dissertation. I also thank Duane

Gustavus and David Hrovat who worked diligently supporting the great computational

resources at UNT. Support for the work presented in this dissertation was provided by the

National Science Foundation, U.S. Department of Education, National Center for

Supercomputing Applications, UNT’s Faculty Research Grant Program, Academic Computing

Services, and Center for Scientific Computing and Modeling (CASCaM). I thank the collaborators

who contributed to a small portion of the computational work in this dissertation: Nathan J.

DeYonker (Ch. 3); Kristin P. Lennox, Sage Driskell and Amy Shah (Ch. 4); and Levi Howard (Ch.7).

Finally, I would like to thank my family and friends for their constant support throughout the

long and arduous doctoral process, especially my mother (Pamela K. Mintz), sister (Stacy K.

Mintz), and my spectacularly wonderful wife Christina Mintz.

iv

TABLE OF CONTENTS

ACKNOWLEDGMENTS ..................................................................................................................... iii

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

LIST OF ILLUSTRATIONS.................................................................................................................. xii

CHAPTERS

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

2. GENERAL METHODOLOGY .................................................................................................. 5

2.1. Methods ............................................................................................................... 5 2.2. Basis Sets ........................................................................................................... 10 2.3. Correlation Consistent Basis Sets ...................................................................... 16

3. PERFORMANCE OF THE CORRELATION CONSISTENT COMPOSITE APPROACH (ccCA) FOR THIRD‐ROW (Ga‐Kr) MOLECULES ................................................................... 18

3.1. Introduction ....................................................................................................... 18 3.2. Methodology ..................................................................................................... 20 3.3. Results and Discussion ....................................................................................... 25

3.3.1. The G3/05 Test Suite .................................................................... 25 3.3.2. Geometries ................................................................................... 26 3.3.3. Atomization Energies ................................................................... 29 3.3.4. Enthalpies of Formation ............................................................... 33 3.3.5. Ionization Energies ....................................................................... 35 3.3.6. Electron and Proton Affinities ...................................................... 37 3.3.7. Overall Performance of ccCA ....................................................... 38

3.4. Conclusions ........................................................................................................ 40

4. TRUNCATION OF THE CORRELATION CONSISTENT BASIS SETS FOR THE HYDROGEN ATOM: A VIABLE MEANS TO REDUCE COMPUTATIONAL COST .................... 42

4.1. Introduction ....................................................................................................... 42 4.2. Methodology ..................................................................................................... 45

4.2.1. 4‐Point Extrapolation Scheme ..................................................... 45 4.2.2. 2‐ and 3‐Point Extrapolation Schemes ......................................... 48

4.3. Results and Discussion – 4‐Point Extrapolation Scheme ................................... 50

v

4.3.1. Geometries ................................................................................... 50 4.3.2. Atomization Energies ................................................................... 56 4.3.3. Complete Basis Set (CBS) Limits ................................................... 60 4.3.4. Ionization Energies ....................................................................... 71 4.3.5. CBS Limits for Ionization Energies ................................................ 73 4.3.6. Percent CPU Time Savings ............................................................ 78

4.4. Results and Discussion – 2‐ and 3‐Point Extrapolation Schemes ...................... 82 4.4.1. Hydrogen Correction: Three‐Point CBS Extrapolations ............... 82 4.4.2. Hydrogen Correction: Two‐Point CBS Extrapolations .................. 88 4.4.3. Impact of Method Choice on the Hydrogen Correction .............. 94 4.4.4. Application to the G3/99 Test Suite ............................................. 96 4.4.5. Percent CPU Time and Disk Space Saved ................................... 110

4.5. Conclusions ...................................................................................................... 117

5. BASIS SET TRUNCATION: APPLICATION TO THE CORRELATION CONSISTENT COMPOSITE APPROACH .................................................................................................. 119

5.1. Introduction ..................................................................................................... 119 5.2. Development of ccCA(aug) .............................................................................. 120 5.3. Development of ccCA(TB) ................................................................................ 121 5.4. Enthalpies of Formation .................................................................................. 129 5.5. % Disk Space Saved .......................................................................................... 134 5.6. Conclusions ...................................................................................................... 136

6. STRUCTURES AND THERMOCHEMISTRY OF THE ALKALI METAL OXIDE RADICALS, ANIONS, AND HYDROXIDES: ASSESSING THE VARIANT WnC METHODS ....................... 138

6.1. Introduction ..................................................................................................... 138 6.2. Methodology ................................................................................................... 145

6.2.1. W1 and W2 Methods ................................................................. 145 6.2.2. W1C and W2C Methods ............................................................. 147 6.2.3. Multi‐Reference W2C Methods ................................................. 148 6.2.4. Geometries and Energies ........................................................... 149

6.3. Results and Discussion ..................................................................................... 150 6.3.1. Extent of Multi‐Reference Character ......................................... 150 6.3.2. Geometries ................................................................................. 153 6.3.3. Enthalpies of Formation ............................................................. 157 6.3.4. Predicted Ground States for MO– and MO•. .............................. 163 6.3.5. State Splitting ............................................................................. 164 6.3.6. Electron Affinities ....................................................................... 167 6.3.7. Gas‐Phase Acidities .................................................................... 169 6.3.8. Bond Dissociation Energies ........................................................ 171

6.4. Conclusions ...................................................................................................... 174

vi

7. DEVELOPMENT OF A MULTI‐REFERENCE CORRELATION CONSISTENT COMPOSITE APPROACH [MR‐ccCA] .................................................................................................... 176

7.1. Introduction ..................................................................................................... 176 7.2. Methodology ................................................................................................... 178 7.3. Potential Energy Surfaces (PES) for Diatomic Molecules ................................ 181

7.3.1. N2 Potential Energy Surface ....................................................... 182 7.3.2. C2 Potential Energy Surface ........................................................ 185

7.4. Conclusions ...................................................................................................... 187

8. SUMMARY ....................................................................................................................... 189

REFERENCES ................................................................................................................................ 195

vii

LIST OF TABLES

Table 3.1. Formal scaling of various ab initio methods. ............................................................. 19

Table 3.2. Geometries determined with B3LYP/cc‐pVTZ computations. All bond lengths (re) are in angstroms, and bond angles (a) are in degrees. Experimental values are also presented where available. .............................................................. 26

Table 3.3. The error in the ccCA‐P and ccCA‐S4 atomization energies (kcal/mol) relative to experiment. The ccCA results include both first‐ and second‐order spin‐orbit coupling (SOC). The errors as compared with experiment for G3 and G4 atomization energies are also listed for comparison. ............................................... 31

Table 3.4. The error as compared with the experimental enthalpies of formation [∆fH (298K)] computed using the ccCA‐P and ccCA‐S4 methods, which include the first‐ and second‐order spin‐orbit coupling (SOC). The errors compare with experiment determined using G3 and G4 are also listed. All values are in kcal/mol. .................................................................................................................... 34

Table 3.5. The error as compared with the experimental ionization energies. The calculations include both the first‐ and second‐order SOC. Results determined with G3 and CCSD(T) are also presented for comparison. All values are in kcal/mol. ............................................................................................... 36

Table 3.6. The errors as compared with experiment for the electron and proton affinities. The computations include both first‐ and second‐order SOC. Results determined with G3 and CCSD(T) are also presented for comparison. All values are in kcal/mol. .......................................................................................... 38

Table 3.7. Mean absolute deviation and mean signed error for all of the computed properties. All values are in kcal/mol. ...................................................................... 40

Table 4.1. Correlation consistent basis sets for hydrogen and the first row (B‐Ne) atoms. The primitive functions are shown in parenthesis, and the contracted functions are shown in brackets. Also shown is the total number of contracted basis functions (N). .................................................................................. 44

Table 4.2. Bond lengths and angles calculated with full cc‐pVnZ basis sets. The errors resulting from the use of truncated basis sets also are shown. These are reported relative to the bond lengths and angles obtained using the full cc‐pVnZ basis sets, and are reported in Å and degrees, respectively. “…” indicates there was no change relative to the full basis set when the truncated basis sets were utilized. ............................................................................ 52

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Table 4.3. Bond lengths and angles of the neutral molecules computed with the full correlation consistent basis sets. The errors resulting from the use of the truncated basis sets are also shown. These are reported relative to the bond lengths and angles obtained using the full correlation consistent basis sets, and are in angstroms and degrees, respectively. “…” means that no change was observed relative to the full basis set. ............................................................... 53

Table 4.4. Bond lengths and angles of the cations computed with the full correlation consistent basis sets. The errors resulting from the use of the truncated basis sets are also shown. These are reported relative to the bond lengths and angles obtained using the full correlation consistent basis sets, and are in angstroms and degrees, respectively. “…” means that no change was observed relative to the full basis set. ....................................................................... 55

Table 4.5. The lowering of the calculated atomization energy per hydrogen atom resulting from the truncation of the correlation consistent basis sets. Energy reductions are reported in kcal/mol.......................................................................... 57

Table 4.6. Atomization energies calculated with full cc‐pVnZ basis sets. The errors resulting from the use of the truncated basis sets are also shown. These are reported relative to the energies obtained using the full cc‐pVnZ basis sets, and are in kcal/mol. ................................................................................................... 59

Table 4.7. CBS limits determined using atomization energies obtained from a series of four calculations, and the error obtained upon utilizing the truncated basis set series [cc(full) – cc(f,–x,–y,–z)]. See text for description of the notation. The CBS limits are reported in kcal/mol. ................................................................... 62

Table 4.8. Ionization energies computed using the full correlation consistent basis sets, and the errors arising from the use of the truncated basis sets. The errors resulting from the use of the truncated basis sets are reported relative to the ionization energies obtained using the full correlation consistent basis sets. “…” represents no change with respect to the full basis set. Ionization energies are given in eV. ............................................................................................ 72

Table 4.9. CBS limits for the ionization energies reported in eV for the full correlation consistent basis set series, and the error, as compared to cc(full), resulting from utilizing the truncated basis set series. Both the Feller exponential and the mixed exponential/Gaussian extrapolations are shown. The mean absolute deviation (MAD) is also provided. .............................................................. 74

Table 4.10. Percent CPU time saved for the reduced basis set as compared with the full basis set computation. ............................................................................................... 80

ix

Table 4.11. Percent CPU time saved for the reduced basis set computations as compared to the full basis set computation. The CPU savings includes the time saved for the overall computations, including Hartree‐Fock and CCSD(T). ..................................................................................................................... 81

Table 4.12. Atomization energies for methane through decane determined with MP2 combined with the full cc‐pVnZ basis sets. The error upon truncation of the hydrogen basis set is also shown. All values are in kcal/mol. .................................. 83

Table 4.13. CBS limits for the atomization energies of methane through decane utilizing the three‐point extrapolation formula Eqn. 4.2, and the error relative to these limits for the truncated basis set series. The mean MAD between the cc(full) and truncated basis set series is also shown. All values are in kcal/mol. .................................................................................................................... 84

Table 4.14. Absolute deviation per hydrogen between the atomization energy (kcal/mol) CBS limits determined with the cc(full) and truncated basis set series for the hydrocarbon series methane through decane. ................................... 85

Table 4.15. Atomization energy CBS limits (kcal/mol) determined from MP2 energies found by using the full cc‐pVnZ [where n = D, T, and Q] basis set series [cc(full)] for the molecules methane though decane. The error as compared to the cc(full) limit is shown for truncated basis set series limits when the hydrogen corrections is included. The MAD is also provided. .................................. 87

Table 4.16. CBS limits for the atomization energies found utilizing both a cc‐pVDZ‐cc‐pVTZ and a cc‐pVTZ‐cc‐pVQZ two‐point extrapolation utilizing Eqn. 4.4. The error as compared to the full basis sets when the truncated hydrogen basis sets are used to extrapolate to the CBS limit is also provided. The error is shown both with and without the empirical hydrogen correction. All values are in kcal/mol. ............................................................................................... 92

Table 4.17. The deviation per hydrogen of the CBS limit for atomization energies for the truncated basis sets as compared to the CBS limit for the full basis sets computed with four different levels of theory. The hydrogen corrections for each level of theory are also shown. All values are in kcal/mol. .............................. 95

Table 4.18. Mean absolute deviation between the full and truncated basis sets for the two and three point extrapolations for the atomization energies of the G3‐99 test suite. The correction per hydrogen for each basis set extrapolation is also shown. All values are kcal/mol. .......................................................................... 98

Table 4.19. Atomization energy CBS limits (kcal/mol) determined from MP2 energies found by using the full cc‐pVnZ [where n = D, T, and Q] basis set series

x

[cc(full)]. The error compared to the cc(full) associated with utilizing the truncated basis set series extrapolations both with and without the hydrogen corrections is also shown. The f indicates that the full cc‐pVDZ basis set is used for the hydrogen atom, and the ‐1 or ‐2 indicates the number of basis functions removed from the cc‐pVTZ and cc‐pVQZ basis sets for hydrogen. [i.e. cc(f,‐1,‐1) indicates that the full cc‐pVDZ, cc‐pVTZ (‐1d ), and cc‐pVQZ (‐1f ) basis sets for the hydrogen atom were used] ............................. 99

Table 4.20. CBS limits for the atomization energies found utilizing Eqn. 4.4 using the full cc‐pVDZ and cc‐VTZ. The error as compared to the full basis sets when the truncated hydrogen basis sets are used to extrapolate to the CBS limit is also provided. The error is shown with and without the empirical hydrogen correction. All values are in kcal/mol. .................................................................... 103

Table 4.21. CBS limits for the atomization energies found utilizing Eqn. 4.4 using the full cc‐pVTZ and cc‐pVQZ basis sets. The error as compared to the full basis sets when the truncated hydrogen basis sets are used to extrapolate to the CBS limit is also provided. The truncated extrapolations are labeled A‐F for the cc(TZ,QZ‐1f), cc(TZ,QZ‐1f2d), cc(TZ,QZ‐1f2d1d), cc(TZ‐1d,QZ‐1f), cc(TZ‐1d,QZ‐1f2d), and cc(TZ‐1d,QZ‐1f2d1d), respectively. The error is shown with and without the empirical hydrogen correction. All values are in kcal/mol. ................ 106

Table 4.22. Percent CPU time saved for the hydrogen containing molecules of the G3/99 test suite computed with MP2 combined with the full and truncated cc‐pVnZ basis sets. ................................................................................................................. 111

Table 4.23. Percent hard disk space saved for the hydrogen containing molecules of the G3/99 test suite computed with MP2 combined with the full and truncated cc‐pVnZ basis sets. ................................................................................................... 114

Table 5.1. Correction per hydrogen and the origin correction for each of the calculations utilizing the correlation consistent basis sets if ccCA(TB). All values are mEh. ........................................................................................................ 126

Table 5.2. Differences between total energies computed with the full and with the truncated basis sets. The energy differences are shown both with and without the hydrogen and origin corrections. All values are in mEh. ..................... 128

Table 5.3. Enthalpies of formation (kcal/mol) computed with ccCA, and the errors as compared to ccCA arising due to the use of ccCA(aug) and ccCA(TB) are presented. Experimental values are also provided. ............................................... 130

xi

Table 5.4. Average percent disk space saved for each step in ccCA(aug) and ccCA(TB) as compared to ccCA(full). The overall average percent disk space saved for ccCA(aug) and ccCA(TB) is also reported. ................................................................ 135

Table 6.1. Orbitals included in the correlation space defined as relaxed valence (rv) for non‐metals and relaxed inner valence (riv) for alkali metals.a ................................ 144

Table 6.2. T1 Diagnostic and Percent SCF Contribution to the Total Atomization Energy (kJ/mol) for various geometries. ............................................................................. 152

Table 6.3. Optimized Geometries (Å) for Alkali Metal Monoxide Anions (MO–), Radicals (MO•) and Hydroxides (MOH). ................................................................................ 155

Table 6.4. Enthalpies of Formation (298 K, kJ/mol) for Alkali Metal Monoxide Anions (MO–), Radicals (MO•) and Hydroxides (MOH). ....................................................... 160

Table 6.5. Splitting of States (0 K, kJ/mol) for Alkali Metal Monoxide Anions (MO–) and Radicals (MO•). ........................................................................................................ 166

Table 6.6. Electron Affinities (0 K, kJ/mol) for the Metal Monoxide Radicals (MO•). .............. 168

Table 6.7. Gas‐Phase Acidities (0 K, kJ/mol) for Alkali Metal Hydroxides (MOH). ................... 170

Table 6.8. Bond dissociation energies (0K, kJ/mol) for the alkali metal hydroxides. .............. 173

xii

LIST OF ILLUSTRATIONS

Figure 2.1. The radial wavefunction for the 1s atomic orbital. ................................................... 12

Figure 2.2. Plot of STO, GTO, and contracted GTO. ..................................................................... 15

Figure 2.3. Systematic construction of the correlation consistent basis sets. ............................ 17

Figure 4.1. Complete basis set limit extrapolations of atomization energy for H2 using both the full and truncated cc‐pVnZ basis sets. The curves show the CBS fits, and the horizontal lines represent the atomization energy arising from each of the subsequent basis truncations as shown in the text boxes. Energies are reported in kcal/mol. ................................................................................................. 68

Figure 4.2. Complete basis set limit extrapolation for the atomization energies (AE) of (a) CH4; (b) NH3; (c) H2O; (d) HF; (e) HCN using both the full and truncated cc‐pVnZ basis sets. The curve shows the CBS extrapolation utilizing the formula proposed by Feller (Eqn. 4.1). The horizontal lines represent the AE arising from using each of the basis sets that were discussed in Section 4.2. AE are reported in kcal/mol. ................................................................................................. 69

Figure 4.3. Plots of the error between the CBS limits determined with the full and the (a) cc(f,–1,–1), (b) cc(f,–1,–2), and (c) cc(f,–1,–3) basis set series as the number of hydrogen atoms increase in the hydrocarbon series methane through decane. ......................................................................................................... 86

Figure 4.4. Plots of the error between the CBS limits determined with the full and the (a) cc(DZ,TZ–1d), (b) cc(TZ,QZ–1f), (c) cc(TZ,QZ–1f2d), (d) cc(TZ,QZ–1f2d1d), (e) cc(TZ–1d,QZ–1f), (f) cc(TZ–1d,QZ–1f2d), (g) and cc(TZ–1d,QZ–1f2d1d) basis set series as the number of hydrogen atoms increase in the hydrocarbon series methane through decane. ......................................................... 89

Figure 5.1. Linear plot of the difference between the total energy (mEh) computed with the full and truncated basis sets versus the number of hydrogen atoms in the linear alkane series methane through hexane. a) MP2/CBS limit computed with Eqn. 5.1, b) MP2/CBS limit compute with Eqn. 5.2, c) MP2/cc‐pVTZ, d) CCSD(T)/cc‐pVTZ, e) MP2/aug'‐cc‐pVTZ, f) MP2(FC1)/aug'‐cc‐pCVTZ, g) MP2/cc‐pVTZ‐DK ................................................................................. 124

Figure 7.1. Two molecular orbitals of the methylene molecule. One configuration shows a pair of electrons occupying the px orbital of the carbon atom, and the other configuration shows a pair of electrons occupying the py orbital of the carbon atom. This analysis was explained by Schmidt and Gordon.275 .................. 177

xiii

Figure 7.2. The potential energy surface for the N2 molecule computed with ccCA. Also, the individual computations required to compute the ccCA total energy are provided. .................................................................................................................. 183

Figure 7.3. The potential energy surface for the N2 molecule computed with MR‐ccCA. Also, the individual computations required to compute the MR‐ccCA total energy are provided. ............................................................................................... 184

Figure 7.4. Potential energy surface for the ground state N2 molecule computed with the ccCA and MR‐ccCA methods. ............................................................................ 185

Figure 7.5. The potential energy surface for the ground X1Σg+ state for the C2 molecule

computed with the MR‐ccCA and ccCA methods. Also displayed is the PES for the excited B1Πg state of C2 computed with the MR‐ccCA method. ................. 187

1

CHAPTER 1

INTRODUCTION

In computational chemistry, the approximate solution to the Schrödinger equation1‐6

allows for the prediction of chemical properties including geometries, energetic properties (i.e.

atomization energies, ionization energies, proton affinities, and electron affinities), and

vibrational frequencies. In recent years, advances in computer technology combined with new

ab initio methods have allowed for dramatic improvement in the prediction of energetic

properties, enabling “chemical accuracy” – defined as ± 1‐2 kcal/mol from reliable experiment –

to be achieved. This has greatly enhanced the predictive powers of computational chemistry in

areas ranging from the interpretation of experiment, understanding of species that can be

difficult to study experimentally [e.g. reactive excited states], and chemical reactivity, to name

just a few.7‐10 Unfortunately, even with these advances, the extensive computational cost, in

terms of computer time, memory, and disk space of the sophisticated methods required to

achieve chemical accuracy effectively limits the size of molecules [i.e. less than 10‐15 non‐

hydrogen atoms] that can be studied.

To date, there have been many developments that have resulted in less computationally

expensive methods while still allowing a high level of accuracy to be achieved, which allow

larger molecules to be studied. Some examples include localized treatment of electron

correlation,11‐35 parallel code implementation to divide the computational cost across multiple

2

CPUs,36‐52 and hybrid/composite methods including ONIOM,53‐58 Gaussian‐n (Gn) methods,59‐77

Weizmann‐n (Wn) methods,78‐83 the high accuracy extrapolated ab initio thermochemistry

(HEAT) method,84‐86 complete‐basis‐set (CBS‐n) methods,87‐94 and the newly developed

correlation consistent composite approach (ccCA).95‐101 The first part of this dissertation focuses

upon the use of composite methods – approaches which combine a series of less

computationally expensive calculations to replicate results that would typically only be

achieved with much more sophisticated and computationally costly calculations. The

development and application of composite methods has proven to be useful because they

provide a means for larger molecules [i.e. comprised of ~15‐30 non‐hydrogen atoms] to be

routinely studied.69,70,72,73,96,97 Specifically, the correlation consistent composite approach

(ccCA) developed by DeYonker, Wilson, and Cundari is discussed in Chapter 3. The ccCA method

was previously benchmarked for a wide variety of molecular systems that include first‐ (B‐Ne)

and second‐row (Al‐Ar) main group atoms,95,96 s‐block atoms,97,101 transition metal complexes,99

and transition states.100 Recently, the basis sets required for ccCA to compute properties for

molecules containing third‐row (Ga‐Kr) atoms was developed,102 and Chapter 3 provides a

detailed study of the performance that ccCA achieves in the computation of molecular

properties for the third‐row (Ga‐Kr) containing molecules of the G3/05 test suite.

While composite methods, such as ccCA, have been shown to be a useful means to reduce

the computational cost of ab initio methods, Chapter 4 discusses an alternate means to reduce

the computational scaling by reducing the number of basis functions contained in the basis set.

Specifically, Chapter 4 focuses upon a unique family of basis sets known as the correlation

3

consistent basis sets, which were developed by Dunning and co‐workers.102‐114 A detailed

description of the correlation consistent basis sets is provided in Section 2.3, but in summary,

these basis sets are useful because properties computed with these basis sets converge

monotonically to an asymptotic limit known as the complete basis set (CBS) limit.115‐118 This

monotonic convergence is important because when the correlation consistent basis sets are

combined with a method, such as coupled cluster theory including single, double, and

quasipertubative triple excitations [CCSD(T)],119‐123 CBS limits can generally achieve chemical

accuracy.124,125 The goal of the research discussed in Chapter 4 was to systematically reduce the

number of basis functions utilized in the correlation consistent basis sets for the hydrogen atom

while still preserving the monotonic convergent behavior that is key to the utility of these basis

sets. The implementation of basis set truncation within ccCA is then discussed in Chapter 5 as a

means to further reduce the computational scaling of ccCA.

In Chapter 6 the application of another family of composite methods is discussed.

Specifically, the Wn methods, developed by Martin et al. (Wn methods),78‐83 are applied to the

ground state for the alkali metal hydroxides (MOH) and the ground and excited states of the

monoxide radicals (MO•) and anions (MO–) [where M = Li, Na, and K]. The Wn methods were

chosen because of their ability to provide sub‐chemical accuracy (<0.25 kcal/mol), albeit at a

substantial computational expense. This high level of accuracy is required for studying the alkali

metal species due to the existence of low‐lying excited states (experimentally known to be on

the order of < 0.5–2.0 kcal/mol above the ground state).126,127 Also, multi‐reference

formulations of the Wn method are currently available,82 which is useful because several of the

4

molecules of interest are known to possess a significant amount of multi‐reference

character.128,129 The final chapter focuses on the development and application of ccCA to

compute properties for molecules that possess a significant amount of multi‐reference

character.

5

CHAPTER 2

GENERAL METHODOLOGY

2.1. Methods

In its simplest form the Schrödinger equation1‐6 can be written as,

ĤΨ EΨ 2.1

where Ψ represents the wave function and Ĥ is the Hamiltonian operator that operates on the

wave function to give the total energy eigenvalue (E) and the eigenvector (Ψ). The Hamiltonian

operator for a system with N electrons and M nuclei has the form,

Ĥ = 12 i

2

N

i=1

ZAriA

N

i=1

M

A=1

1rij

N

j>i

N

i=1

12MA

A2

M

A=1

ZAZBrAB

M

B>A

M

A=1

2.2

where ZA and ZB are the nuclear charges for atoms A and B, respectively, riA is the distance that

ith electron is from atom A, rij is the distance that the ith electron is from the jth electron, rAB is

the distance that atom A is from atom B, MA is the ratio of the mass of nucleus A to the mass of

an electron, and i2 and A

2 refer to differentiation with respect to the coordinates of the ith

electron and Ath nucleus, respectively. The first term in Eqn. 2.2 represents the kinetic energy

of the electrons, the second term is the attractive columbic energy of the nuclear‐electron

interaction, the third term is the repulsive potential energy of the electron‐electron interaction,

6

the fourth term is the kinetic energy of the nuclei, and the final term is the repulsive potential

energy arising from the nuclear‐nuclear interaction.

Unfortunately, the Schrödinger equation is too complex to solve exactly for all but the

simplest of chemical systems, which entail atoms or molecules containing only one electron.

Therefore, approximations must be introduced to solve this equation for many electron

systems. One such approximation that is inherent to all of the methods utilized in this

dissertation is the Born‐Oppenheimer approximation,130,131 which assumes that the motions of

the nuclei are much slower relative to the motion the electrons, and thus, these motions can be

decoupled and solved separately giving rise to the electronic Schrödinger equation as shown

below,

ĤelecΦelec elecΦelec 2.3

Ĥelec ‐12 i

2

N

i=1

‐ ZAriA

M

A=1

N

i=1

1rij

N

j>i

N

i=1

2.4

where Ĥelec only includes the electronic terms from Eqn. 2.2, and Фelec is the electronic wave

function. The electronic energy elec for a molecule with fixed nuclei, therefore, can be

determined by solving the electronic Schrödinger equation in the presence of fixed nuclei. The

total energy for molecules, however, must also include the nuclear repulsion, and so the total

energy for a molecule is

total elec ZAZBrAB

M

B>A

.

M

A=1

2.5

7

The Born‐Oppenheimer approximation is usually a good approximation but can begin to

break down, for example, if there is more than one solution to the electronic Schrödinger

equation. One example of this breakdown is an avoided crossing – the point at which two

potential energy surfaces come close together but do not cross because they have the same

symmetry (See e.g. Ref. 132 page 56). The error introduced by the Born‐Oppenheimer

approximation, however, is generally small. For example, a study performed by Rusic et al.

showed that the magnitude of the error introduced by the Born‐Oppenheimer approximation

was only 0.1 kcal/mol in the computed atomization energy of water.131 Overall, the Born‐

Oppenheimer approximation is key to all of the methods utilized throughout, and its use should

have minimal impact on the computed properties.

The simplest ab initio method is the Hartree‐Fock (HF) method,133‐138 which was initially

proposed by Hartree in 1927 when he suggested using the mean‐field approximation. In this

approach, the Coulombic electron‐electron interaction from Eqn. 2.4 1 rij⁄ is approximated by

computing the interaction of each individual electron with the average potential field of all of

the other electrons. It was later shown that Slater determinants – a determinant of one‐particle

orbitals – could be used in Hartrees method, which effectively reduces the many‐electron

problem into a one‐electron problem. The key to the use of Slater determinants is that they

satisfy the antisymmetry or Pauli Exclusion Principle, which states that the wave function must

be antisymmetric with respect to the interchange of two electrons.

The HF method is proven to recover about 99% of the total energy for molecular systems,

and the remaining 1 % can largely be attributed to the Coulombic electron‐electron interaction

8

approximation. The error introduced by the mean‐field approximation is defined as the

correlation energy and is defined as,

Ecorr Eexact ‐ EHF 2.6

where EHF is the HF energy, Eexact is the exact energy, and Ecorr is the electron correlation energy.

Even though the correlation energy only represents ~1% of the total energy, it can often be

required even for a qualitatively correct computation of molecular properties. For example,

Feller and Peterson found that the mean absolute deviation (MAD) of 66 atomization energies

for molecules containing first‐ (Li‐Ne) and second‐row (Na‐Ar) atoms, compared to experiment,

was on the order of 60 kcal/mol when the HF method was utilized.139

The need to account for electron correlation has, therefore, spurred the development of

post‐HF methods, often called electron correlation methods, which were initially based on the

HF single Slater determinant. One of the simplest post‐HF methods to be developed is known as

the configuration interaction (CI) method, which generates the exact wave function Φ for any

state of a system based on the ground state HF single determinant wave function Ψ by the

following equation,

|Φ c0|Ψ0 car |Ψar

ra

cabrs |Ψab

rs

a<b r<s

cabcrst |Ψabc

rst a<b<cr<s<t

2.7

where the first term in Eqn. 2.7 is simply the ground state HF wave function (Ψ0) multiplied by a

coefficient (c0), the second term is all single excitations resulting from an electron being excited

from an occupied orbital (a) into an unoccupied orbital (r) multiplied by the coefficient , the

9

third term represents all double excitations resulting from the excitation of two electrons out of

occupied orbitals into unoccupied orbitals, the fourth term includes all triple excitations, and so

forth. An exact solution to the electronic Schrödinger equation for many electron systems

utilizing the CI approach requires that all possible excitations are included in Eqn. 2.7 [i.e. full CI

(FCI)]. The use of FCI, however, is only feasible for small molecular systems (i.e. 2‐4 non‐

hydrogen atoms).140‐145 Therefore, Eqn. 2.7 is generally truncated at a specific excitation level,

which gives rise to a series of CI methods [i.e. CI including all single excitations (CIS), CI including

all single and double excitations (CISD), CI including all single, double, and triple excitations

(CISDT), …].

For the studies presented in this dissertation, electron correlation is included through the

use of second‐order Møller‐Plesset perturbation [MP2]146‐148 theory and coupled cluster theory

including single, double, and quasiperturbative triple excitation [CCSD(T)].119‐123 The idea behind

perturbation theory is that the approximate solution differs only a small amount from the exact

solution. Specifically, the approximate Hamiltonian (Ĥ0) and wave function (Ψ0) can be

perturbed a small amount to produce a new Hamiltonian (Ĥ′) and wave function (Ψ′) that is an

improvement over Ĥ0 and Ψ0. When Ĥ0 is taken to be a sum of Fock operators [i.e. the mean‐

field approximation is applied to Eqn. 2.4], the resulting theory is known as Møller‐Plesset

perturbation [See Ref. 132 and 148 for a more in‐depth discussion of Møller‐Plesset

perturbation theory]. In coupled cluster (CC) theory electron correlation is introduced with the

use of a cluster operator as shown below,

eT = 1 + T1 + T2 + 1

2T12

+ T3 + T2T1 + 1

6T13

+ … 2.8

10

where eT is a general cluster operator that includes all excitations for a give reference wave

function. The first term on the right‐hand side of the equation comes from the reference wave

function, the second term T1 includes all single excitations, the third term T2+1

2T12 includes

the product of two single excitations 1

2T12 and a double excitation T2 , and so forth. As with

CI, the use of all excitation within CC theory is required for an exact solution to the Schrödinger

equation, which is impractical. Therefore, Eqn. 2.8 is usually truncated at a given excitation

level, which gives several CC methods [i.e. CC theory including all single excitations (CCS); CC

theory including all single and double excitations (CCSD); CC theory including all single, double,

and triple excitations (CCSDT); …]. A complete description of CC theory is not within the scope

of this dissertation, but a more in‐depth discussion is given in Refs. 119‐123, 132, and 149. In

summary, the utilization of post‐HF methods to include the effects of electron correlation is

vital for all of the methods utilized throughout this dissertation.

2.2. Basis Sets

Every ab initio computation has two primary sources of errors. The first type of error is

the intrinsic error, which is due to the chosen method [i.e. HF, MP2, CCSD(T), etc.]. The second

type of error is introduced by approximations made to the wave function (Ψ). To begin to

understand the basis set approximation, the exact solution to the electronic Schrödinger

equation for the hydrogen atom can be separated into a product of a radial Rn,l(r) and angular

Yl,ml components to give the following form for the wavefunction,150

11

Ψnlm r,θ, Rn,l r Yl,mlθ, 2.9

where n, l, and m are the principle, angular, and magnetic quantum numbers, respectively, r is a

distance from the nucleus, θ is the colatitudes angle, and is the azimuthal angle. (For a more

complete description see Ref. 150 page 349) The angular component Yl,ml is known as the

spherical harmonic functions, which describe the orbital types [e.g. s, p, d, …]. The radial

component Rn,l(r) has the general form,

Rn,l r = polynomial in r × exponential decay 2.10

where the polynomial in r is known as the associated Laguerre polynomial. (Expressions for

several radial wave functions are given in ref. 150) Unfortunately, due to the 1 r⁄ term in Eqn.

2.2, the Schrödinger equation is not separable for systems that contain more than one electron,

and the wave function must, therefore, be approximated for many electron systems. In 1951

Roothaan suggested using linear combinations of a complete set of known functions called

basis functions, which allows the HF wave function to be determined utilizing matrix algebra.151

There are several functions that could be utilized as a basis function, but in general, the

basis functions that are often used are an attempt to approximate the exact wave function for

the hydrogen atom. Figure 2.1 shows a plot of the radial wave function for the 1s atomic orbital

for which the basis function will attempt to approximate. The first feature that should be noted

from Figure 2.1 is that the exact wave function for the hydrogen 1s atomic orbital comes to a

point when r=0. This means that the hydrogen wave function is discontinuous at the origin [i.e.

the derivative of Eqn. 2.9 does not equal zero when r=0], which is known as a cusp condition.

Figure 2.1 also shows that Eqn. 2.9 decays exponentially with respect to r. The correct

description of these two features is essential for any basis function that is utilized in the

approximation of Eqn. 2.9.

Figure 2.1. The radial wavefunction for the 1s atomic orbital.

0.0

0.5

1.0

1.5

2.0

2.5

0 1 2 3 4 5

In practice, approximating Eqn 2.9 is accomplished with two types of basis functions

known as Slater type orbitals (STO)152 and Gaussian type orbitals (GTO).153 STOs have the

functional form,

χlmnζSTO r,θ, NlmnζYlml

θ, rn‐1e‐ζr 2.11

12

where l, m, and n are the same quantum numbers as before, Nlmnζ is a normalization factor and

ζ is a parameter that describes how quickly the basis function decays. STOs are exact in that

they correctly describe both the cusp condition of Eqn 2.9 as well as the exponential decay with

respect to r. However, STOs are an approximate solution to Eqn. 2.9 because the complex radial

component Rn,l(r) is replaced by the simpler rn‐1e‐ζr term, which removes the proper

13

description of radial nodes.154 To approximate the radial nodes, linear combinations of STOs are

utilized. For an exact description of the wave function, an infinite number of STOs are required,

which is impractical. However, finite numbers of STOs can be used, which converge rapidly as

the number of basis functions increases because STOs correctly describe the boundary

conditions for Eqn 2.9 [i.e. cusp condition and exponential decay with respect to r].

Unfortunately, the use of STOs to approximate the exact wave function for many electron

systems is impractical because the simple analytic solutions needed to solve the complex

integrals [e.g. if χr(1), χs(1), χt(2), χu(2) are STOs that are each centered on four different nuclei,

then a four‐centered two electron integral is generated when solving the Schrödinger equation]

are currently unknown, which means that the integrals must be solved numerically. This makes

the use of STOs very expensive and is therefore generally limited to atomic and diatomic

systems.

The complexities inherent to STOs was circumvented by Boys who suggested using

Gaussian type orbitals (GTOs),153 which have the following form,

χijkζGTO x,y,z,r Nijkζxb

i ybj zb

k e‐ζr2 2.12

where Nijkζ is a normalization factor; i, j, and k are nonnegative integers; ζ is again an exponent

that describes how quickly the orbital decays; x, y, and z are cartesian coordinates with the

origin at nucleus b; and r is some distance from the origin. Unfortunately, GTOs do not possess

the correct boundary conditions for Eqn. 2.9, as shown in Figure 2.2. That is, GTOs do not

describe the cusp condition near the nucleus nor do they decay exponentially with respect to r.

14

Even with these drawbacks, GTOs have many attributes that have made them widely accepted.

The most important reason is that the complex integrals [i.e. three‐ and four‐ centered two

electron integrals] can be reduced to simpler integrals [two‐centered two electron integrals] for

which simple analytic solutions have been derived, which greatly simplifies the computations

for many electron systems. Additionally, the boundary conditions of Eqn. 2.9 can be

approximated by contracting several GTOs to approximate a single STO, which is shown

pictorially in Figure 2.2 and mathematically below,

χlmnζSTO r,θ, ciχijkζ

GTO x,y,z,ri

2.13

where ci is a set of fixed coefficients. Each individual χijkζGTO x,y,z,r is referred to as a primitive

GTO and the linear combination of GTOs is referred to as a contracted GTO (CGTO). Linear

combinations of these approximate STOs [i.e. CGTOs] are then utilized to approximate the

radial nodes. Unfortunately, to provide an exact description of the cusp condition and

exponential decay inherent to STOs, an infinite number of primitive GTOs must be utilized,

which is impractical. Therefore, an incomplete or finite basis set of CGTOs is commonly used.

Due to the use of a finite basis set, the wave function is not exact, and the resulting error is

termed the incomplete basis set error.

Figure 2.2. Plot of STO, GTO, and contracted GTO.

0.0

0.5

1.0

1.5

2.0

2.5

‐5 ‐4 ‐3 ‐2 ‐1 0 1 2 3 4 5r

GTO

STO

CGTO

15

16

2.3. Correlation Consistent Basis Sets

The basis sets utilized throughout this dissertation are the correlation consistent basis sets

denoted cc‐pVnZ [where n = D, T, Q, 5, etc.].102‐114,155 The cc‐pVnZ basis sets were developed by

Dunning and co‐workers and are a unique family of basis sets that systematic increase in the

number of CGTOs as the level of the basis set increases, as shown in Figure 2.3. The foundation

of the cc‐pVnZ basis sets is a set of s and p functions that are optimized at the Hartree‐Fock (HF)

level of theory. For first‐row (Li‐Ne) atoms, an additional set of s, p, and d functions are added

to this foundation to create a (3s2p1d) basis set, which corresponds to the cc‐pVDZ basis set.

The additional s, p, and d shell of angular momentum functions are optimized using the CISD

method. The cc‐pVnZ basis sets are unique in that the additional s, p, and d functions are

chosen to be included in one shell because they all contributed similar amounts of correlation

energy. The cc‐pVTZ basis set is then developed in a similar fashion. Specifically, a minimal basis

set is first determined with HF to which two shells of angular momentum functions are included

that are optimized with CISD. The first shell added to the minimal basis set includes an s, p, and

d function, while the second shell includes an s, p, d and f function, which gives a (4s3p2d1f)

basis set. The first set of s, p, and d functions included in the construction of the cc‐pVTZ basis

set all contribute similar amounts of correlation energy. Additionally, the second set of s, p, d,

and f functions all contribute similar amounts of correlation energy. Larger basis sets can be

constructed by subsequently adding more shells of angular momentum functions. As a result of

this building up of correlated angular momentum functions, the correlation energy is

systematically recovered as the basis set size is increased from the double‐ζ to triple‐ζ to the

quadruple‐ζ basis set, etc., and the total energy of a molecular system approaches a limit. This

limit is known as the complete, or saturated basis set (CBS) limit – the point at which the error

due to the use of a finite basis set is effectively eliminated. Properties (i.e. geometries and

energetic) computed with the cc‐pVnZ basis sets approach this limit in such a systematic way

that a number of extrapolation formulas have been proposed, which allow the CBS limit to be

approximated. It has been shown that determining the CBS limit with an ab initio method, such

as CCSD(T), can enable energetic properties to be computed within 1‐2 kcal/mol of

experimental data.124,155,156 Furthermore, by reaching the CBS limit and, thus, eliminating the

basis set error, the errors associated with only the chosen ab initio method can be better

understood.

Figure 2.3. Systematic construction of the correlation consistent basis sets.

cc‐pVQZ

cc‐pVTZ

cc‐pVDZ

HF sp

+1s1p1d

+1s1p1d1f

+1s1p1d1f1g

17

18

CHAPTER 3

PERFORMANCE OF THE CORRELATION CONSISTENT COMPOSITE APPROACH (ccCA) FOR THIRD‐

ROW (Ga‐Kr) MOLECULES†

3.1. Introduction

A primary goal of quantum chemistry has long been to obtain predictions of energetic

properties of chemical systems to within “chemical accuracy,” commonly defined as 1‐2

kcal/mol. Advances made throughout the history of computational chemistry have helped to

make this goal attainable for a number of molecular systems. However, a major stumbling block

remains: the methods that are capable of calculating properties within chemical accuracy can

only be applied to a very small number of molecules, typically to those with fewer than 10‐15

atoms heavier than hydrogen.157 The inability to study large systems using quantitative

methods can be attributed to the high computational cost arising from the large N‐scaling

(where N is the number of basis functions in the basis set) of ab initio methods. The formal

scaling of a variety of ab initio methods is given in Table 3.1. This, combined with the slow

convergence of the basis set expansion to a limit, which is due to the difficulty of satisfying the

† This chapter was partially adapted from the publication of J. Chem. Theory and Comput., Vol 4, N. J. DeYonker, B. Mintz, T. R. Cundari, and A. K. Wilson, “Application of the Correlation consistent composite approach (ccCA) to Third‐Row (Ga‐Kr) Molecules”, Pages 328‐334, Copyright (2008), with permission from American Chemical Society.

19

inter‐electronic cusp condition, as described in Chapter 2, has severely limited the application

of ab initio methods to model larger molecules.

Table 3.1. Formal scaling of various ab initio methods.

Method Scaling

HF N4

MP2 N5

MP4 N7

CCSD N6

CCSD(T)a N6 + N7

CCSDT N7

a) The CCSD part scales as N6 and the (T) scales as N7

To make advanced ab initio methods more tractable to larger molecules, a number of

approaches can be used to reduce the computational cost. Several of these methods were

discussed in Chapter 1 and include (a) hybrid and composite methods such as ONIOM,53‐58

Gaussian‐n methods,59‐77 Weizmann‐n methods,78‐83 HEAT method,84‐86 and the correlation

consistent composite approach (ccCA)95‐101; (b) localization schemes;11‐35 and (c) parallel

methods.36‐52

This chapter describes a study that assesses the performance of ccCA for the computation

of thermochemical properties (i.e. dissociation energies, enthalpies of formation, ionization

energies, electron affinities, and proton affinities) for molecules containing third‐row (Ga‐Kr)

atoms. A goal of ccCA is for it to be a method that can be utilized across the periodic table, and

thus far, ccCA has already proven to be an efficient method that can be utilized to compute

thermochemical properties for molecules containing first‐ (Li‐Ne) and second‐ (Na‐Ar) row

20

atoms to within 1 kcal/mol of reliable experimental data.95‐97,100,101 Also, ccCA was recently

applied to transition metal species achieving an accuracy of ~3.4 kcal/mol. (Note, “transition

metal chemical accuracy” is defined as ± 3 kcal/mol.99 )

Previously, the basis sets required to compute properties for molecules that contain third‐

row (K‐Kr) atoms were not fully developed. However, core‐valence basis sets for Ga‐Kr were

recently developed,102 which now enables molecules containing these atoms to be studied. The

basis sets for K and Ca are currently under development by Peterson,158 so this chapter only

focuses on molecules containing the third‐row atoms Ga‐Kr. This work was done in

collaboration with Nathan D. DeYonker (a postdoctoral fellow in Angela K. Wilson’s research

group) who helped in the collecting and tabulation of data.

3.2. Methodology

The goal of any composite method is to approximate the results of a larger/more

expensive method with several smaller/less expensive methods. This is accomplished by

assuming that the results from the smaller/less expensive methods can be combined in an

additive manner to achieve an accuracy similar to that of the more expensive method.61 The

correlation consistent composite approach is an MP2 based composite method that utilizes the

CBS limit obtained from MP2/aug‐cc‐pVnZ [where n = D (2), T(3), and Q(4)] as a reference

energy. To this reference energy, corrections are added to account for electron correlation

beyond MP2, and include core‐valence electronic interactions, relativistic effects, atomic spin‐

orbit interactions, and zero‐point energy (ZPE) corrections.

21

The first step in ccCA is the geometry optimization and frequency computation, which are

performed using B3LYP combined with the cc‐pVTZ basis set. The B3LYP/cc‐pVTZ geometries

are then used in all subsequent ccCA steps. It should be noted that for all second‐row (Al‐Ar)

atoms, the tight d basis sets, developed by Wilson et al.,105 are utilized for all of the ccCA steps

[e.g. cc‐pV(T+d)Z is used instead of cc‐pVTZ]. These basis sets include an additional tight‐d basis

function to correct for deficiencies in the original cc‐pVnZ basis sets for computing molecular

properties, especially for sulfur containing systems. The use of the cc‐pV(T + d)Z is essential in

achieving an accuracy of 1 kcal/mol and is recommended for all computations containing

second‐row atoms.

There are several extrapolation formulas that can be utilized to determine the CBS limits,

but in a previous ccCA study, it was determined that two extrapolation formulas provided the

best agreement with experiment.96 The first is a mixed Gaussian/exponential formula proposed

by Peterson et al.159 and is shown below,

En E0 P Ae‐ n‐1 Be‐ n‐1 2 3.1

where En is the total energy computed at the nth basis set level, n is the cardinal number of the

basis set (i.e. D=2, T=3, …), E0(P) is the total energy at the CBS limit computed with Eqn. 3.1, and

A and B are parameters that are determined in the extrapolation. The second formula used in

this study is based upon a two‐point 1/(lmax^4) scheme proposed by Schwartz160 using the aug‐

cc‐pVTZ and aug‐cc‐pVQZ total energies and has the following form,161‐164

22

En E0 S4 C

lmax 124 3.2

where E0(S4) is the total energy at the CBS limit computed with Eqn. 3.2, C is a parameter

determined in the extrapolation, and lmax is the highest angular momentum function included in

the basis set. For all of the molecules included in this study, lmax simply corresponds to the

cardinal number. However, this rule does not apply for all atoms, so care must be taken when

extending ccCA to molecules containing atoms such as transition metals, as discussed by

DeYonker et al.99

After the MP2/CBS reference energy is determined, a series of additive corrections are

then computed. The first correction accounts for higher‐order electron correlation that is not

completely described by MP2 and is computed by taking the difference between the

MP2/cc‐pVTZ and CCSD(T)/cc‐pVTZ total energies. This correction is labeled ∆E(CC) and is

expressed as,

∆E CC E[CCSD(T)/cc‐pVTZ] ‐ E[MP2/cc‐pVTZ]. 3.3

The next correction is the scalar relativistic correction. The scalar relativistic correction

accounts for the inability of the non‐relativistic Schrödinger equation (Eqn. 2.2) to account for

the effects of relativity. The assumptions is that Eqn. 2.2 breaks down as the velocity of the

inner electrons approach the speed of light, which is most notable as the nuclear charge

increases. The scalar relativistic correction computed in this study is obtained from frozen‐core

MP2 computations utilizing the cc‐pVTZ‐DK basis set39 and the spin‐free, one‐electron Douglas‐

23

Kroll‐Hess (DKH) Hamiltonian.165‐167 The MP2 relativistic correction is labeled ∆E(DK) and is

formulated as,

∆E DK E[MP2/cc‐pVTZ‐DK] ‐ E[MP2/cc‐pVTZ]. 3.4

The final correction is to account for core‐valence correlation effects. For most chemical

systems, the bonding between atoms is generally well described utilizing only the valence

electrons, so the inner electrons are normally frozen to help reduce the computational cost.

The neglect of the core‐valence electron interactions usually has a negligible effect on

molecular properties. However, there are some molecules for which core‐valence electron‐

electron interaction must be accounted for in order to achieve the target accuracy of 1

kcal/mol. For example, Shepler et al. found that the core‐valence effects contributed 0.75

kcal/mol to the dissociation energy of I2.168 The core‐valence correction in ccCA is computed by

taking the difference between MP2/aug‐cc‐pVTZ and MP2(FC1)/aug‐cc‐pCVTZ as shown below,

∆E CV E[MP2(FC1)/aug‐cc‐pCVTZ] ‐ E[MP2/aug‐cc‐pVTZ]. 3.5

MP2(FC1) refers to the inclusion of the core‐valence electrons in the correlation space which

corresponds to the following; a) for first‐row (Li‐Ne) atoms the 1s electrons are added to the

correlation space, b) for second‐row (Na‐Ar) atoms the 2s and 2p electrons are added to the

correlation space, and c) for third‐row (Ga‐Kr) atoms the 3s and 3p electrons are added to the

correlation space.

24

Another correction is the zero‐point energy (ZPE) correction, which is determined from

B3LYP/cc‐pVTZ frequency computations. The ZPE energy is required because the computed

energy is determined at the bottom of the potential well. Whereas, the lowest energy level that

is physically observed experimentally is where ν = 0, which corresponds to a zero‐point

vibrational energy of E0 = (1/2)ħω (see Ref. 150 for a detail description). Therefore, to compare

with experimental data, the ZPE should be included in the total energy.

To summarize, ccCA utilizes a reference energy that is computed as the MP2/CBS limit

found from either Eqn. 3.1 or Eqn. 3.2. To this reference energy the higher‐order effects are

included in an additive manner to give the following ccCA total energy,

E ccCA E MP2 aug‐cc‐pV∞Z⁄ ∆E CC ∆E DK ∆E CV ZPE. 3.6

The ccCA energy that uses the MP2/CBS limit computed from Eqn. 3.1 is referred to as ccCA‐P,

and the ccCA energy that utilized MP2/CBS limit computed from Eqn. 3.2 is referred to as ccCA‐

S4.

The last correction included in ccCA is the spin‐orbit coupling (SOC), which arises because

the spin (S) and angular momentum (L) quantum numbers couple. This coupling causes

degenerate orbitals to split, which leads to a lowering of the total energy. In the initial

development of ccCA, only the atomic SOC was included. However, for molecules containing

third‐row (Ga‐Kr) atoms it was shown that first‐order molecular SOC [i.e. open‐shell molecules

in which spatially and spin‐degenerate states split (e.g. 2Π)] corrections can alter energies by up

to 4.1 kcal/mol, and that even second‐order molecular SOC [i.e. the second‐order SOC apply for

two cases that include 1) spin‐degenerate states that lower in energy due to spin‐orbit effects

25

(e.g. 3Σ) and 2) open and closed shell molecules in which spin‐orbit effects cause the ground

state energy to lower (e.g. 2Σ, 1Σ, …)] can affect atomization energies by up to 0.8 kcal/mol, as

shown in a study by Blaudeau and Curtiss.74,169 Due to the magnitude of the SOC for both first‐

and second‐order effects, SOC are described in ccCA utilizing values computed in previous

theoretical studies and are labeled ∆(SOa) and ∆(SOm) for atoms and molecules,

respectively.74,169

All computations performed in this chapter are computed with the Gaussian03 program

package.170

3.3. Results and Discussion

3.3.1. The G3/05 Test Suite

In earlier studies, ccCA was tested on the G3/99 training set, which includes 376 first‐ and

second‐row atomic and molecular properties.67 This training set was chosen due to its large

number and diversity of molecules. The set includes atomization energies, ionization energies,

electron affinities, proton affinities, and enthalpies of formation from experiment with

uncertainties of less than 1 kcal/mol providing a stringent test set for any computational

method. More recently, the G3/99 test set was expanded to form the G3/05,74 which includes

molecules containing third‐row (K, Ca, and Ga‐Kr) atoms. The third‐row molecules in the G3/05

test set are used in this study to assess the performance of ccCA in the calculation of third‐row

molecular properties. However, K and Ca have not been included as the necessary basis sets do

not exist. Development of the correlation consistent basis set required for ccCA is currently in

26

progress for K and Ca,158 and when developed and tested, ccCA will be utilized to study these

species. In all, the G3/05 test set includes 51 properties for molecules containing third‐row (Ga‐

Kr)_atoms including; 19 atomization energies (D0), 11 enthalpies of formation (∆Hf), 15

ionizations potentials (IP), 4 electron affinities (EA), and 2 proton affinities (PA).

3.3.2. Geometries

The geometries computed with B3LYP/cc‐pVTZ are presented in Table 3.1. Experimental

values, when available, are also provided for comparison. While a detailed analysis of the

B3LYP/cc‐pVTZ computed geometries is not the primary focus of this project, the results are

presented here for completeness. In general, the computed values are within 0.01 Å for bond

lengths and 1.0° for bond angles from experiment. A more detailed analysis of the

performance of B3LYP for geometries as well as several other molecular properties for third‐

row (Ga‐Kr) molecules was reported by Yockel, Mintz, and Wilson.124

Table 3.2. Geometries determined with B3LYP/cc‐pVTZ computations. All bond lengths (re) are in angstroms, and bond angles (a) are in degrees. Experimental values are also presented where available.

Molecule Ground State Parameter B3LYP/cc‐pVTZ Expt.a

GeH4 1A1 re (Ge‐H) 1.534 1.514b

AsH 3Σ— re (As‐H) 1.535 1.535

AsH+ 2Π re (As‐H) 1.539 ‐‐‐

AsH2 2B1 re (As‐H) 1.530 1.518c

a (H‐As‐H) 90.9 90.7c

(table is continued on next page)

27

(Table 3.2 continued)

Molecule Ground State Parameter B3LYP/cc‐pVTZ Expt.a

AsH2+ 1A1 re (As‐H) 1.533 ‐‐‐

a (H‐As‐H) 90.9 ‐‐‐

AsH3 1A1 re (As‐H) 1.525 1.511c

a (H‐As‐H) 92.1 92.1c

SeH 2Π re (Se‐H) 1.475 1.475

SeH+ 3Σ— re (Se‐H) 1.496 ‐‐‐

SeH‐ 1Σ+ re (Se‐H) 1.480 ‐‐‐

SeH2 1A1 re (Se‐H) 1.471 1.460c

a (H‐Se‐H) 91.3 90.6c

SeH2+ 2B1 re (Se‐H) 1.489 ‐‐‐

a (H‐Se‐H) 91.6 ‐‐‐

HBr 1Σ+ re (H‐Br) 1.424 1.414

HBr+ 2Π re (H‐Br) 1.460 1.448

As2 1Σg

+ re (As‐As) 2.105 2.103

BBr 1Σ+ re (B‐Br) 1.903 1.888

Br2 1Σg

+ re (Br‐Br) 2.315 2.281

Br2+ 1Πg re (Br‐Br) 2.222 ‐‐‐

BrCl 1Σ+ re (Br‐Cl) 2.165 2.136

BrF 1Σ+ re (Br‐F) 1.774 1.759

BrF+ 2Π re (Br‐F) 1.691 ‐‐‐


28


Molecule Ground State Parameter Molecule Ground State

NaBr 1Σ+ re (Na‐Br) 2.521 ‐‐‐

NaBr+ 2Π re (Na‐Br) 2.922 ‐‐‐

BrO 2Π re (Br‐O) 1.732 1.717

BrO‐ 1Σ+ re (Br‐O) 1.839 1.814d

HOBr 1A’ re (H‐O) 0.966 0.961e

re (O‐Br) 1.845 1.834e

a (H‐O‐Br) 103.2 102.3e

HOBr+ 2A” re (H‐O) 0.968 ‐‐‐

re (O‐Br) 1.735 ‐‐‐

a (H‐O‐Br) 109.8 ‐‐‐

GaCl 1Σ+ re (Ga‐Cl) 2.242 2.202

GeO 1Σ+ re (Ge‐O) 1.629 1.625

GeS2 1Σg

+ re (Ge‐S) 2.011 ‐‐‐

CH3Br 1A1 re (C‐H) 1.083 1.082f

re (C‐Br) 1.960 1.934f

a (H‐C‐Br) 111.2 111.2f

CH4Br 1A’ re (C‐Br) 2.023 ‐‐‐

re (Br‐H1) 1.443 ‐‐‐

re (C‐H2) 1.085 ‐‐‐


29


Molecule Ground State Parameter Molecule Ground State

CH4Br 1A’ re (CH3) 1.083 ‐‐‐

a (H1‐Br‐C) 97.4 ‐‐‐

a (H2‐C‐Br) 101.9 ‐‐‐

a (H3‐C‐Br) 105.3 ‐‐‐

a (H4‐C‐H3) 115.4 ‐‐‐

KrF2 1Σg

+ re (Kr‐F) 1.891 1.875c

a) From Ref. 171 unless otherwise noted b) From Ref. 172 c) From Ref. 173 d) From Ref. 174 e) From Ref. 175 f) From Ref. 176

3.3.3. Atomization Energies

Table 3.3 presents the 19 atomization energies (D0) of the G3/05 test suite. As compared

with experiment, when only first‐order SOC are included, 13 of the 19 atomization energies

were within 1 kcal/mol of the experimental values, and only two contain errors larger than 2

kcal/mol. These molecules were GeH4 and GeO which had errors of 2.7 and 2.1 kcal/mol,

respectively, for ccCA‐P, and 2.8 and 2.1 kcal/mol for ccCA‐S4. Overall, the largest difference

between atomization energies computed with ccCA‐P and ccCA‐S4 is 0.2 kcal/mol, which was

computed for CH3Br.

30

For comparison, results determined from G3 and G4 computations were also

examined.69,74 The performance of G3 is similar to that of ccCA, with 12 of the of the computed

atomization energies being within 1 kcal/mol of experiment, and only two with errors larger

than 2 kcal/mol. These molecules are GeH4 and NaBr, for which the G3 computed atomization

energies were in error by –2.5 and –2.1 kcal/mol, respectively. When utilizing the G4 method,

seven of the computed atomization energies were closer to experiment as compared to G3,

with the largest improvements occurring for NaBr and GaCl where the G4 computed

atomization energies were closer to experiment by 1.4 and 0.9 kcal/mol, respectively, as

compared with G3.

It should be noted that both G3 and G4 contain an experimentally derived parameter

known as the higher level correction (HLC).67,69 This correction was recently shown to account

for ~ 60 kcal/mol of the atomization energies as described by G3 for linear alkanes up to

octane.96 Use of experimental parameters, such as the HLC, is undesirable because deviation

from the test set of molecules used to create the parameter can lead to significant errors. One

example that shows the HLC breakdown is provided by a study performed by Schulz et al. who

found that the standard G2 method performed poorly in determining thermochemical

properties for the alkali metal and alkaline earth metal oxides and hydroxides (M2O, MOH

where M = Li, Na, and K; M'O, M'(OH)2 where M' = Be, Mg, and Ca), with errors greater than 25

kcal/mol.177 The correlation consistent composite approach, however, contains no such

parameter while achieving accuracies similar to G3 and G4 for atomization energies of third‐

row molecules. Additionally, in a recent study, ccCA achieved accuracies of ~1 kcal/mol for

31

alkali and alkaline metal containing molecules similar to the molecules studied by Sullivan et

al.97,101

Inclusion of second‐order SOC in the ccCA formulation improved twelve atomization

energies as compared to when only first‐order SOC were included, while four computed

atomization energies had larger errors as compared to experiment after second‐order SOC was

included. Overall, second‐order SOC improved the atomization energies computed with ccCA

for molecules containing third‐row (Ga‐Kr) atoms. The mean absolute deviation (MAD) as

compared to experiment for ccCA‐P and ccCA‐S4 when only first‐order SOC are included was

0.9 kcal/mol and reduced to 0.8 kcal/mol when second‐order SOC was included.

Table 3.3. The error in the ccCA‐P and ccCA‐S4 atomization energies (kcal/mol) relative to experiment. The ccCA results include both first‐ and second‐order spin‐orbit coupling (SOC). The errors as compared with experiment for G3 and G4 atomization energies are also listed for comparison.

1st order SOC 2nd order SOC

Expt ccCA‐P ccCA‐S4 ccCA‐P ccCA‐S4 G3a G4b

GeH4 → Ge + 4H 270.5a –2.7 –2.8 –2.7 –2.7 –2.5 –2.5

AsH → As + H 64.6b 1.6 1.6 1.5 1.5 –0.1 0.7

AsH2 → As + 2H 131.2b –0.7 –0.7 –0.5 –0.6 –0.8 –0.7

AsH3 → As + 3H 206b 0.3 0.3 0.6 0.6 1.4 1.3

SeH → Se + H 74.3c –0.7 –0.7 –0.2 –0.2 –1.1 –0.7

SeH2 → Se + 2H 153.2c –0.4 –0.4 –0.2 –0.2 0.9 1.1


32




GaCl → Ga + Cl 109.9d 0.4 0.4 0.4 0.4 –1.5 –0.6

GeO → Ge + O 155.2d –2.1 –2.1 –2.0 –2.0 –1.6 –1.0

As2 → 2As 91.3e –1.1 –1.0 –0.5 –0.4 –0.4 –0.4

BrCl → Br + Cl 51.5d –0.1 –0.2 0.1 0.1 0.3 0.5

BrF → Br + F 58.9d –0.2 –0.2 0.1 0.1 0.3 –0.4

BrO → Br + O 55.3d 0.6 0.6 0.9 0.9 0.1 0.3

Br2 → 2Br 45.4d –0.5 –0.6 –0.6 –0.6 –0.1 0.9

BBr → B + Br 103.5e 1.2 1.2 1.5 1.5 0.7 1.5

NaBr → Na + Br 86.2e –0.4 –0.3 –0.1 0.0 –2.1 –0.7

CH3Br → Br + C + 3H 358.2f –0.7 –0.9 –0.4 –0.6 –0.3 0.4

GeS2 → Ge + 2S 191.7g 1.5 1.6 1.6 1.7 –1.9 –1.2

KrF2 → Kr + 2F 21.9d 0.2 0.2 0.3 0.2 –0.6 –1.7

MAD 0.9 0.9 0.8 0.8 0.9 0.9

a) From Ref. 74 b) From Ref. 69 c) From Refs. 178,179 d) From Ref. 180 e) From Ref. 181 f) From Ref. 182 g) From Ref. 171 h) From Ref. 183 i) From Ref. 184

33

3.3.4. Enthalpies of Formation

The deviation from experiment for the ccCA computed enthalpies of formation [∆Hf

(298K)] are presented in Table 3.3. In general, enthalpies of formation are difficult to compute

to a high degree of accuracy, which make them a stringent thermochemical test for any ab

initio method. As shown in Table 3.3 for ccCA‐P, when only first‐order SOC are included, only

four computed enthalpies have errors, as compared to experiment, of less than 1 kcal/mol.

These errors were 0.8, 0.9, 0.1, and 0.6 kcal/mol as observed for the CF3Br, C3H7Br, C3H7Br, and

CHF2Br molecules, respectively. Overall, ccCA‐S4 resulted in a mean absolute deviation (MAD),

as compared to experiment, which was 0.3 kcal/mol larger than the MAD computed with ccCA‐

P. The most notable differences between ccCA‐P and ccCA‐S4 was observed for C6H13Br and

C5H8Br2 where the error as compared with experiment increased by 0.7 and 0.5 kcal/mol,

respectively, upon going from the ccCA‐P to the ccCA‐S4 method. When second‐order SOC are

included in ccCA, errors in ten of the eleven enthalpies of formation were reduced. The only

molecule for which the error as compared with experiment increased after second‐order SOC

are included was C6H5Br, for which the error increased by only 0.3 kcal/mol for both ccCA‐P and

ccCA‐S4 methods. Overall, the inclusion of molecular second‐order SOC are important in the

computation of enthalpies of formation for third‐row (Ga‐Kr) containing molecules in the G3/05

test suite.

As compared to G3, the ccCA‐P methods performed better for seven of the eleven

molecules when only first‐order SOC are included. When second‐order SOC are included in

ccCA‐P, nine of the eleven molecules improved with an average 1.1 kcal/mol shift toward

34

experiment. The ccCA‐S4 method also performed better than G3 for six of the eleven

molecules when only first‐order SOC are included, and eight of the eleven molecules when

second‐order SOC are included.

G4 is the newest formulation of the Gaussian‐n methods and contains a six parameter

optimized HLC as compared to G3’s four parameter HLC. It is, therefore, unsurprising that G4 is

a significant improvement over G3. G4 also performed better than ccCA when only first‐order

SOC were included. However, when second‐order SOC are included, the MAD as compared to

experiment is only 1.2 and 1.4 kcal/mol for ccCA‐P and ccCA‐S4, respectively, which is only 0.3

and 0.5 kcal/mol higher than the 0.9 kcal/mol MAD determined for G4. Overall, when first‐

order SOC are included, both ccCA methods outperform G3, and when second‐order SOC are

included, ccCA‐P was comparable to the G4 method in the computation of enthalpies of

formation for molecules containing third‐row atoms.

Table 3.4. The error as compared with the experimental enthalpies of formation [∆fH (298K)] computed using the ccCA‐P and ccCA‐S4 methods, which include the first‐ and second‐order spin‐orbit coupling (SOC). The errors compare with experiment determined using G3 and G4 are also listed. All values are in kcal/mol.


Molecule Expt.a ccCA‐P ccCA‐S4 ccCA‐P ccCA‐S4 G3b G4c

CF3Br ‐155.0 ± 0.7 0.8 1.1 0.5 0.7 2.3 0.4

CCl3Br ‐10.0 ± 0.4 1.2 1.2 0.9 0.9 2.9 1.4

C2H3Br 18.9 ± 0.4 1.8 2.0 1.5 1.7 2.0 1.4

C2H5Br ‐14.8 ± 0.4 1.6 1.9 1.3 1.6 1.2 0.3


35



Molecule Expt.a ccCA‐P ccCA‐S4 ccCA‐P ccCA‐S4 G3b G4c

C3H7Br ‐23.8 ± 0.8 0.9 1.3 0.6 1.0 0.6 ‐0.5

C6H5Br 25.2 ± 1.0 0.1 0.5 ‐0.2 0.2 1.5 1.2

C6H13Br ‐35.4 ± 0.4 1.9 2.6 1.6 2.2 1.2 0.3

C3H6Br2 ‐17.1 ± 0.3 2.8 3.1 2.2 2.5 2.7 1.0

CHF2Br ‐101.6 ± 0.2 0.6 0.7 0.2 0.4 1.2 ‐0.4

COBr2 ‐27.1 ± 0.1 2.1 2.0 1.4 1.4 2.7 1.2

C5H8Br2 ‐13.1 ± 0.4 2.9 3.4 2.3 2.8 3.2 1.9

MAD 1.5 1.8 1.2 1.4 2.0 0.9

a) From Ref. 185 b) From Ref. 74 c) From Ref. 69

3.3.5. Ionization Energies

The experimental ionization energies (IE) and the error computed with the various ccCA,

G3, and G4 methods are presented in Table 3.5. When first‐order SOC are included in ccCA, the

error compared to experiment was less than one kcal/mol for all but four of the molecules,

which include SeH (–1.3 and –1.2 kcal/mol), SeH2 (–1.2 and –1.1 kcal/mol), HOBr (–1.6 and –1.5

kcal/mol), and NaBr (–5.2 and –5.0 kcal/mol) for the ccCA‐P and ccCA‐S4 methods, respectively.

It should be noted that the IE of NaBr computed with G3 and G4 were also in error of –4.9 and

–4.7 kcal/mol, respectively, as compared to experiment. Overall, the MAD for the ccCA

36

methods, as shown in Table 3.5, was 0.8 kcal/mol, which was the same as the MAD computed

for G3, and slightly better than the MAD computed for G4 (0.9 kcal/mol).

Table 3.5. The error as compared with the experimental ionization energies. The calculations include both the first‐ and second‐order SOC. Results determined with G3 and CCSD(T) are also presented for comparison. All values are in kcal/mol.



Ga → Ga+ 138.3c –0.1 –0.1 –0.2 –0.1 –0.2 –0.6

Ge → Ge+ 182.2c 0.0 –0.1 –0.2 –0.2 –0.1 –0.2

As → As+ 225.7c –0.5 –0.5 –0.7 –0.8 –0.4 –0.3

Se → Se+ 224.9c 0.5 0.6 0.7 0.7 1.0 –0.7

Br → Br+ 272.4c –0.3 –0.2 0.3 0.4 0.5 –0.4

Kr → Kr+ 322.9c 0.5 0.7 0.0 0.2 1.3 1.2

AsH → AsH+ 222.3d 0.1 0.1 –0.3 –0.3 –1.0 –0.5

AsH2 → AsH2+ 217.8d 0.3 0.3 0.4 0.4 –0.8 –0.6

SeH → SeH+ 227.0e –1.3 –1.2 –0.4 –0.4 0.1 –0.6

SeH2 → SeH2+ 228.0e –1.2 –1.1 –1.1 –1.1 –0.3 –0.4

HBr → HBr+ 268.9f –0.3 –0.2 –0.7 –0.6 0.8 1.0

Br2 → Br2+ 242.6f –0.1 0.1 –0.8 –0.6 –0.2 0.4

HOBr → HOBr+ 245.3g –1.6 –1.5 –1.6 –1.5 –0.4 –0.2

BrF → BrF+ 271.7f –0.1 0.1 –0.1 0.1 0.7 1.4


37




NaBr → NaBr+ 191.6f –5.2 –5.0 –5.2 –5.0 –4.9 –4.7

MAD 0.8 0.8 0.8 0.8 0.8 0.9

a) From Ref. 74 b) From Ref. 69 c) From Ref. 186 d) From Ref. 180 e) From Ref. 181 f) From Ref. 171 g) From Ref. 187

3.3.6. Electron and Proton Affinities

The error as compared with experiment for the electron affinities (EA) and proton

affinities (PA) are presented in Table 3.6 for ccCA, G3, and G4. Overall, ccCA is within one

kcal/mol of the experimental EAs provided in Table 3.6. The one exception was the EA for BrO,

for which the deviation from experiment was –1.6 and –1.5 kcal/mol computed with the ccCA‐P

and ccCA‐S4 methods, respectively, when only first‐order SOC are included. When second‐

order SOC are included, the deviation from experiment for the EA of BrO was reduced to 0.9

kcal/mol. Overall, when second‐order SOC are included, all molecules were within one kcal/mol

of experiment with the exception of Br–, for which the error was computed to be –1.4 and –1.6

kcal/mol for the ccCA‐P and ccCA‐S4, respectively.

38

Table 3.6. The errors as compared with experiment for the electron and proton affinities. The computations include both first‐ and second‐order SOC. Results determined with G3 and CCSD(T) are also presented for comparison. All values are in kcal/mol.



Electron Affinities

Ge → Ge – 28.43 0.0 0.0 0.1 0.1 –0.5 –0.7

Br → Br – 77.6 0.1 0.3 0.4 0.6 –0.5 –0.7

BrO → BrO – 54.4 –1.6 –1.5 0.9 0.9 –1.3 –1.5

SeH → SeH – 51.0 –0.5 –0.5 –0.5 –0.5 –0.4 –0.5

Proton Affinities

Br‐ → HBr 322.6 –1.0 –1.2 –1.4 –1.6 –0.3 0.2

BrCH3 → BrCH4+ 157.3 0.1 0.0 0.1 0.0 0.4 0.2

a) From Ref. 74 b) From Ref. 69 c) From Ref. 186 d) From Ref. 188 e) From Ref. 174 f) From Ref. 189

3.3.7. Overall Performance of ccCA

For the Ga‐Kr containing species in the G3/05 training set, ccCA can reliably predict

energetic properties to within chemical accuracy. The mean signed deviation (MSD), reported in

Table 3.7, when only first‐order SOC are included was –0.02 kcal/mol for ccCA‐P and 0.07

kcal/mol for ccCA‐S4, indicating almost no overall bias in the reliability of the ccCA. When only

39

first‐order SOC are included, the MAD for ccCA‐P was 0.95 kcal/mol, and for ccCA‐S4 was 1.00

kcal/mol.

For the 51 atomic and molecular properties computed in the training set, ccCA‐P is an

improvement of 0.12 kcal/mol as compared to the G3 model chemistry, which had a MAD of

1.07 kcal/mol. Also, ccCA outperformed the CBS‐n methods.87‐94 [Note, the enthalpies of

formation were not computed with the CBS‐n methods.]190 Comparing the 40 energetic

properties previously studied with CBS‐n (19 D0, 15 IPs, 4 EAs, and 2 PAs), the ccCA‐P and ccCA‐

S4 MADs were 0.79 and 0.78 kcal/mol, respectively, while the best CBS‐n method, CBSQB3,

resulted in a MAD of 1.12 kcal/mol. Additionally, previous CCSD(T)/cc‐pV∞Z studies performed

by Yockel, Mintz, and Wilson were also performed for the same 40 energetic properties,124,125

and interestingly, the ccCA model chemistry have a lower MAD for this set, as compared to

most CCSD(T)/aug‐cc‐pV∞Z and CCSD(T)/aug‐cc‐pV∞Z‐PP methods.

Including second‐order SOC effects reduced the MAD of ccCA‐P from 0.95 to 0.88

kcal/mol and ccCA‐S4 from 1.00 to 0.92 kcal/mol, which resulted in similar performance as

compared with the G4 model chemistry. By including second‐order SOC, 25 of the ccCA‐P

deviations are reduced by more than 0.10 kcal/mol, while the errors are increased for 18

species. Overall, atomic second‐order SOC improve the reliability of the properties computed

for the third‐row molecules of the G3/05 training set.

40

Table 3.7. Mean absolute deviation and mean signed error for all of the computed properties. All values are in kcal/mol.


ccCA‐P ccCA‐S4 ccCA‐P ccCA‐S4 G3a G4b

MAD 0.95 1.00 0.88 0.92 1.07 0.86

MSD –0.02 0.07 –0.03 0.05 0.11 –0.01

a) From Ref. 74 b) From Ref. 69

3.4. Conclusions

The correlation consistent composite approach was utilized to determine 51 atomic and

molecular properties for the Ga‐Kr species in the G3/05 training set. The ccCA energies were

compared to experiment as well as other widely used model chemistries. Overall, when only

first‐order SOC are included, ccCA performed better than G3, with a MAD compared to

experiment of 0.95 and 1.00 kcal/mol for ccCA‐P and ccCA‐S4, respectively, as compared to the

MAD of 1.07 kcal/mol computed with G3. Spin‐orbit coupling is a physical phenomenon that

cannot be ignored when computing thermodynamic and energetic properties of species that

contain third‐row (Ga‐Kr) atoms. The addition of second‐order spin‐orbit corrections results in a

substantial improvement in the accuracy of ccCA, as the MAD reduces from 0.95 and 1.00

kcal/mol, when only first‐order SOC were included, to 0.88 and 0.92 kcal/mol for ccCA‐P and

ccCA‐S4, respectively, when second‐order SOC were included. Additionally, the core‐valence

additive correction to the ccCA energy is essential for a proper description of the third‐row

energetic properties contained in the G3/05 training set. In fact, the MP2‐based ccCA method

41

was found to perform better than CCSD(T)/cc‐pV∞Z results due to the better modeling of the

core‐valence electron correlation in ccCA method, which were not accounted for in the

published coupled cluster CBS results. For third‐row p‐block molecules, ccCA is a viable

alternative to other model chemistries that rely on semiempirical corrections to the correlation

energy. Furthermore, the modeling of heavier p‐block elements is an important step in the

development of a pan‐periodic table composite method that is capable of yielding accurate

thermodynamics.

42

CHAPTER 4

TRUNCATION OF THE CORRELATION CONSISTENT BASIS SETS FOR THE HYDROGEN ATOM: A

VIABLE MEANS TO REDUCE COMPUTATIONAL COST‡

4.1. Introduction

Even with advances in computer technology and computational methodologies, the

prediction of energetic properties to within chemical accuracy is still typically limited to

chemical systems that contain less than 10‐15 non‐hydrogen atoms.157 The goal of the research

presented in this chapter was to accomplish a systematic reduction in basis set size while

minimizing the impact on the overall accuracy of a given ab initio method. There are a number

of studies to date which have aimed to reduce the basis set size as a means to enable an

increase in the size of molecules that can be studied. For example, a study performed by Curtiss

et al. examined computational cost reduction for quadratic configuration interaction including

single, double, and iterative triple excitations [QCISD(T)] paired with the Pople‐type 6‐31G(d,p)

‡ This chapter was adapted from the following publications 1) J. Chem. Phys., Vol. 121, B. Mintz, K. P. Lennox, and A. K. Wilson, “Truncation of the Correlation Consistent Basis Sets: An Effective Approach to the Reduction of Computational Cost?”, Pages 5629‐5634, Copyright (2004), reused with permission from American Institute of Physics , 2) J. Chem. Phys., Vol. 122, B. Mintz and A. K. Wilson, “Truncation of the Correlation Consistent Basis Sets: Extension to Third‐Row (Ga‐Kr) Molecules”, Pages 134106/1‐10, Copyright (2005), reused with permission from American Institute of Physics, and 3) Int. J. Quantum Chem., Vol. 107, B. Mintz, S. Driskell, A. Shah, and A. K. Wilson, “Truncation of the Correlation Consistent Basis Sets: Application to Extended Systems”, Pages 3077‐3088, Copyright (2007), with permission from Wiley Periodicals, Inc.

43

basis set.75 Curtiss showed that removing the p polarization function from the basis set for the

hydrogen atom affected the mean absolute deviation (MAD) computed for the G2 test suite by

only 0.05 kcal/mol.75 However, this same level of accuracy was not observed when the d

polarization function was removed from the basis set for non‐hydrogen atoms. Specifically,

Curtiss found that the MAD for the G2 test suite increased by more than 2 kcal/mol. Earlier

work performed by del Bene, showed that the diffuse s functions for the hydrogen atom in

basis sets such as 6‐31++G(d,p) and 6‐311++G(2d,2p) had little effect on hydrogen bond

energies and proton affinities.191 Another study by del Bene found that omitting diffuse

functions from Dunning’s correlation consistent basis sets for the hydrogen atom had an effect

of less than 0.70 kcal/mol on the proton affinities of NH3, H2O, and HF.192

The focus of the present study is on the systematic reduction of the correlation consistent

basis sets developed by Dunning and co‐workers [cc‐pVnZ; where n = D(2), T(3), Q(4), 5].102‐

114,155 As discussed in Section 2.3, the correlation consistent basis sets systematically build up

the number of contracted GTOs as the level of basis set increases [i.e. functions that account

for similar amounts of correlation energy are added together in shells of angular momentum

functions]. As a result of this construction, the correlation energy is systematically recovered as

the basis set size is increased (i.e. cc‐pVDZ, cc‐pVTZ,…), and the total energy approaches the

CBS limit.

One of the drawbacks of the correlation consistent basis sets, however, is their steep

increase in the number of basis functions (N) upon increasing the size of the basis set. As

shown in Table 4.1, increasing the correlation consistent basis set from the cc‐pVDZ to the

44

cc‐pVTZ level nearly doubles N for first‐row atoms and almost triples N for the hydrogen atom.

A similar increase is observed upon increasing the basis set from cc‐pVTZ to cc‐pVQZ, as well as

from cc‐pVQZ to cc‐pV5Z. Due to this steep increase in the number of basis functions,

computing the CBS limit with a series of cc‐pVnZ basis sets to achieve chemical accuracy is

generally limited to molecular systems that contain less than 10‐20 non‐hydrogen atoms.

Reducing the number of higher angular momentum functions for the cc‐pVnZ basis sets as a

means to lower the computational cost (N‐scaling) of ab initio methods can provide the ability

to increase the size of molecular systems that can be studied. Accomplishing this reduction,

however, is a daunting task because removal of the higher angular momentum functions can

potentially destroy the systematic construction of the cc‐pVnZ basis sets, and, thus, impact the

ability to determine the CBS limit.

Table 4.1. Correlation consistent basis sets for hydrogen and the first row (B‐Ne) atoms. The primitive functions are shown in parenthesis, and the contracted functions are shown in brackets. Also shown is the total number of contracted basis functions (N).

cc‐pVnZ Hydrogen N First‐Row (B‐Ne) Atoms N

n = D (4s 1p)/[2s 1p] 5 (9s 4p 1d)/[3s 2p 1d] 14

n = T (5s 2p 1d)/[3s 2p 1d] 14 (10s 5p 2d 1f)/[4s 3p 2d 1f] 30

n = Q (6s 3p 2d 1f)/[4s 3p 2d 1f] 30 (12s 6p 3d 2f 1g)/[5s 4p 3d 2f 1g] 55

n = 5 (8s 4p 3d 2f 1g)/[5s 4p 3d 2f 1g] 55 (14s 8p 4d 3f 2g 1h)/[6s 5p 4d 3f 2g 1h] 91

Specifically, basis set truncation studies are presented which focus on the affect that

hydrogen basis set truncation had upon geometries, atomization energies (AE), and ionization

energies (IE) for a series of first‐ (B‐Ne) and third‐ (Ga‐Kr) row molecules.193,194 The ability to

45

extrapolate AE and IE computed with the truncated basis sets to a CBS limit that is comparable

to the full basis set extrapolation was also examined. The study of the molecules containing

first‐row (B‐Ne) atoms was performed with the assistance of Kristin P. Lennox (TAMS student)§.

A subsequent study expanded upon the initial basis set truncation work to examine a larger test

suite with molecules that are more substantial in size, the largest molecule being decane.195

The follow‐up study was performed with the assistance of Sage Driskell (TAMS student)§ and

Amy Shah (TAMS student)§ who helped with the computations for Section 4.4. The study

presented in Section 4.4 focus predominantly upon the utilization of a truncated basis set series

to determine the CBS limits for atomization energies for this much greater range of molecular

species.

4.2. Methodology

4.2.1. 4‐Point Extrapolation Scheme

Geometry optimizations were performed on a test set of first‐row (B‐Ne) containing

molecules [H2, CH4, NH3, H2O, HF, and HCN] and third‐row containing molecules [GeH4, AsH,

AsH+, AsH2, AsH2+, AsH3, SeH, SeH

+, SeH2, SeH2+, HBr, HBr+, HOBr, and HOBr+]. Geometry

optimizations were performed using CCSD(T) combined with two series of basis sets that

include: (1) the full cc‐pVnZ sets [where n = D(2), T(3), Q(4), and 5], which provide a useful

§ The Texas Academy of Mathematics and Science (TAMS) is a unique residential program for high school‐aged Texas students who are high achievers and interested in mathematics and science.

46

benchmark for (2); and (2) a truncated series of cc‐pVnZ sets for the hydrogen atom [truncation

is described below] and the full cc‐pVnZ basis sets for the non‐hydrogen atoms. While there are

a number of approaches that could be taken in basis set reduction, all of the following basis set

truncation studies utilized the systematic truncations as described in the following paragraph.

Specifically, the basis set truncation entailed the systematic reduction in the number of

higher angular momentum [i.e. d and above] functions within the hydrogen basis set. The first

basis set level that was truncated was the triple‐ζ level, which was truncated by the elimination

of the d function (–1d). At the quadruple‐ζ level, a series of truncations were performed, which

include: (a) removal of the f function (–1f ); (b) removal of the f function and the outermost d

function (–1f 2d) [the outermost d function is the function with the smallest ζ exponent]; and

(c) the removal of all higher angular momentum functions (–1f 2d1d). Finally, the quintuple‐ζ

basis set was truncated by (a) removal of the g function (–1g); (b) removal of the g function and

the outermost f function (–1g 2f) [again the outermost function refers to the function with the

smallest ζ exponent]; (c) the removal of all g and f functions (–1g2f1f); (d) the removal of all g

and f functions as well as the outermost d function (–1g2f1f3d); (e) the removal of all g and f

functions as well as the outermost and second outermost d function (–1g2f1f3d2d); and finally

(e) the removal of all higher angular momentum functions (–1g2f1f3d2d1d). Atomization

energies, bond lengths, and bond angles for both series of basis sets were obtained for all first‐

and third‐row containing molecules, while ionization energies were computed for the third‐row

containing molecules. Zero‐point corrections were determined at the cc‐pVTZ level and are

included in all of the computed molecular properties. Also due to the large spin‐orbit splitting

47

that was previously observed for the third‐row (Ga‐Kr) atoms and molecules,60,169 atomic and

molecular spin‐orbit corrections are included, which were obtained from configuration

interaction calculations performed by Blaudeau and Curtiss.60,169

A key property of the correlation consistent basis sets is the systematic nature of the basis

sets, which enables extrapolation to the CBS limit for structural and energetic properties upon

increasing the basis set size. Although there are several possible extrapolation formulas

available, for this study two formulas were chosen to be studied. The first is the widely used

and established Feller extrapolation scheme and is shown below,196

De(n) De ∞ Ae‐Bn 4.1

where De(n) is the dissociation energy at the nth ζ‐level, n is the cardinal number of the basis

set, De(∞) is the atomization energy at the CBS limit, and A and B are parameters determined in

the extrapolation. The second extrapolation formula utilized for this study was a mixed

exponential/Gaussian formula given by Eqn. 3.1. For first‐row containing molecules, the

extrapolations were performed on the atomization energies, whereas for the third‐row

containing molecules, the extrapolations were performed on the total energies of the atoms

and molecules, separately. The atomization and ionization energy CBS limits for the third‐row

containing molecules were then computed using the total energy CBS limits. Extrapolations

that utilize either atomization energies or total energy are both widely utilized in the

literature,124,125,155 and the utility of these two schemes to extrapolate the truncated basis sets

to the CBS limit was examined. All calculations were carried out using the Molpro or Gaussian

program packages.160,187

48

4.2.2. 2‐ and 3‐Point Extrapolation Schemes

Geometry optimizations were performed utilizing B3LYP/6‐31G(d) for the molecules

methane through decane and the hydrogen‐containing molecules of the G3/99 test suite, which

include 1,3‐difluorobenzene (C6H4F), 1,4‐difluorobenzene (C6H4F), pyrazine (C4H4N2), 2,5‐

dihydrothiophene (C4H6S), 2‐methyl‐thiophene (C5H6S), 3‐methyl pentane (C6H14), acetic

anhydride (C6H14), aniline (C6H5NH2), 1,1‐dimethoxy ethane (C4H10O2), acetyl acetylene (C4H4O),

crotonaldehyde (C4H6O), isobutene nitrile (C4H7N), isobutanal (C4H8O), 1,4‐dioxane (C4H8O2),

1,2‐dicyano ethane (C4H4N2), chlorobenzene (C6H5Cl), di‐isopropyl ether (C6H14O), diethyl

disulfide (C4H10S2), diethyl ether (C4H10O), diethyl ketone (C5H10O), isopropyl acetate (C5H10O2),

methyl ethyl ketone (C4H8O), n‐methyl pyrrole (C5H7N), piperidine (cyc‐C5H10NH),

tetrahydropyran (C5H10O), tetrahydropyrrole (C4H8NH), tetrahydrothiophene (C4H8S),

tetrahydrothiopyran (C5H10S), tetramethylsilane (C4H12Si), methyl acetate (C3H6O2), azulene

(C10H8), benzoquinone (C6H4O2), cyclooctatetraene (C8H8), cyclopentanone (C5H8O), dimethyl

sulfone (C2H6O2S), divinyl ether (C4H6O), n‐butyl chloride (C4H9Cl), naphthalene (C10H8), phenol

(C6H5OH), pyrimidine (C4H4N2), t‐butyl chloride (C4H9Cl), t‐butanethiol (C4H9SH), t‐butanol

(C4H9OH), t‐butyl methyl ether (C5H12O), t‐butylamine (C4H9NH2), tetrahydrofuran (C4H8O),

toluene (C6H5CH3), cyclopentane (C5H10), cyclohexane (C6H12), isoprene (C5H8), 1,3‐

cyclohexadiene (C6H8), 1,4‐cyclohexadiene (C6H8), fluorobenzene (C6H5F), methyl allene (C4H6),

neopentane (C5H12), and nitro‐S‐butane (C4H9NO2).197 Møller–Plesset second order perturbation

theory (MP2)146‐148 single‐point computations were then performed utilizing the B3LYP/cc‐pVTZ

49

geometries to obtain total energies. For all second‐row atoms (Al‐Ar), the tight d correlation

consistent basis sets [cc‐pV(n + d)Z]105 were utilized (see Section 3.2).

The CBS limit was determined using both the full cc‐pVnZ basis set series and all

combinations of truncated basis set series [i.e. truncated basis sets that include 1) removal of

the d function from the cc‐pVTZ basis set, 2) removal of the g function from the cc‐pVQZ basis

set, 3) removal of the g and outermost d function, and 4) removal the g and both d functions

from the cc‐pVQZ basis set]. In this study two‐ and three‐point extrapolation schemes were

utilized. The two‐point scheme was developed based on studies performed by Schwartz,

Kutzelnigg and Morgan (Eqn. 3.2),160,198,199 and is based on an inverse power relationship with

respect to the highest angular momentum (lmax) represented in the basis sets. The extrapolation

formula has the following form,161‐164

E lmax =ECBS+B

(lmax)3 . 4.2

A simple analytic form for Eqn. 4.3 was given by Halkier et al.,161 which has the form,

ECBS E2 lmax 2

3 ‐ E1 lmax 13

lmax 23 ‐ lmax 1

3 4.4

where ECBS is the energy at the CBS limit, E1 and E2 are the energies resulting from two

calculations each utilizing a difference basis set level with E1 being the energy computed with

the smaller of the two basis sets [e.g. The cc‐pVTZ basis set is used to compute E1, and the cc‐

pVQZ basis set is used to computed E2.], and lmax1 and lmax2 are the highest angular momentum

function for each of the two basis sets [e.g. For first‐row (Li‐Ne) molecules, lmax1 = 3 for the cc‐

50

pVTZ basis set, and lmax2 = 4 for the cc‐pVQZ basis set]. The three‐point scheme that was utilized

is the mixed Gaussian/exponential (Eqn. 3.1).159

The extrapolation to the CBS limit was done for the total energies of the atoms and

molecules separately, and these limits were used to determine the atomization energies at the

CBS limit. All B3LYP computations were performed with the Gaussian03 program,170 while all

MP2 computations were performed using the Molpro program.200 All computations for the

G3/99 molecules were performed on an IBM pSeries 690 supercomputer (“copper”) located at

the National Center for Supercomputing Applications. All cc‐pVDZ and cc‐pVTZ computations

utilized four processors, while all cc‐pVQZ computations used eight processors.

4.3. Results and Discussion – 4‐Point Extrapolation Scheme

4.3.1. Geometries

CCSD(T) optimized structures for the first‐row (B‐Ne) containing molecules are reported in

Table 4.2. As shown in this table, basis set truncation for the hydrogen atom had minimal

impact (±0.001Å) upon bond lengths, and in most cases, there was no fluctuation in bond

length, which is shown with “…“. At most, there was a slight (0.01°–0.04°) and (0.06°–0.12°)

change in bond angle, depending upon the choice of truncated basis set used, which was

observed for NH3 and H2O, respectively. Overall, the effect of truncation upon the optimized

structures for the first‐row (B‐Ne) containing molecules was minimal.

CCSD(T) optimized structures for the third‐row (Ga‐Kr) containing molecules are provided

in Table 4.2 and Table 4.3 provides the structures of the neutral species, while Table 4.4

51

provides the structures for the cationic species. For the neutral molecules, the largest

deviations in the bond length with respect to the full basis set for the third‐row molecules was

0.003 Å as observed for AsH, AsH2, AsH3, SeH, SeH2, and HBr after removal of the d function

from the cc‐pVTZ basis set. For the cationic species, the maximum deviation for the third‐row

molecules was 0.004 Å computed for the HBr+ bond length after the d function was removed

from the cc‐pVTZ basis set. The largest deviation in the bond angle with respect to the full basis

set for both the neutral and cationic third‐row molecules was 0.19°, which was the case for

both the HOBr and HOBr+ molecules after all of the higher angular momentum functions were

removed from the cc‐pVQZ basis set. Overall, the effect on the computed structures for the

third‐row molecules was minimal when a truncated hydrogen basis set is used.

52

Table 4.2. Bond lengths and angles calculated with full cc‐pVnZ basis sets. The errors resulting from the use of truncated basis sets also are shown. These are reported relative to the bond lengths and angles obtained using the full cc‐pVnZ basis sets, and are reported in Å and degrees, respectively. “…” indicates there was no change relative to the full basis set when the truncated basis sets were utilized.

H2 CH4 NH3 H2O HF HCN r(H‐H) r(C‐H) r(N‐H) a(H‐N‐H) r(O‐H) a(H‐O‐H) r(H‐F) r(C‐H) r(C‐N)

cc‐pVDZ 0.761 1.104 1.020 107.68 0.966 101.91 0.920 1.083 1.175

cc‐pVTZ 0.742 1.089 1.010 108.39 0.959 103.58 0.917 1.067 1.160 ‐1d … … … 0.01 0.001 0.12 0.001 0.001 …

cc‐pVQZ 0.742 1.088 1.009 108.56 0.958 104.12 0.916 1.067 1.156 ‐1f … … … 0.01 … 0.06 0.001 … … ‐1f 2d –0.001 … … 0.03 … 0.14 … … … ‐1f 2d 1d –0.001 … … 0.04 … 0.19 0.001 … …

cc‐pV5Z 0.741 1.088 1.009 108.66 0.958 104.37 0.917 1.067 1.156 ‐1g … … … … … … … … … ‐1g 2f … … … … … … … … … ‐1g2f 1f … … … 0.01 … 0.02 … … … ‐1g2f 1f 3d … … … 0.01 … 0.06 … … … ‐1g2f 1f 3d 2d –0.001 … … 0.02 … 0.10 … … … ‐1g2f 1f 3d 2d 1d … … … 0.02 … 0.12 … … …

Expt.a 0.741 1.091 1.012 106.67 0.958b 104.44 0.917 1.066c 1.153c

a) From Ref. 201 unless otherwise noted b) From Ref. 202 c) From Ref. 203

53

Table 4.3. Bond lengths and angles of the neutral molecules computed with the full correlation consistent basis sets. The errors resulting from the use of the truncated basis sets are also shown. These are reported relative to the bond lengths and angles obtained using the full correlation consistent basis sets, and are in angstroms and degrees, respectively. “…” means that no change was observed relative to the full basis set.

GeH4 AsH AsH2 AsH3

Basis Set r(Ge‐H) a(H‐Ge‐H) r(As‐H) r(As‐H) a(H‐As‐H) r(As‐H) a(H‐As‐H) cc‐pVDZ 1.543 109.47 1.542 1.534 91.09 1.529 92.58

cc‐pVTZ 1.542 109.47 1.535 1.529 91.18 1.524 92.58 ‐1d 0.002 … 0.003 0.003 –0.02 0.003 –0.04

cc‐pVQZ 1.541 109.47 1.533 1.528 91.16 1.523 92.56 ‐1f … … 0.001 … … 0.001 … ‐1f 1d –0.001 … … … 0.08 … 0.06 ‐1f 1d 2d 0.001 … 0.002 0.001 0.08 0.002 0.04

cc‐pV5Z 1.541 109.47 1.533 1.528 91.15 1.523 92.58 ‐1g … … 0.001 … 0.00 0.001 –0.03 ‐1g 2f … … 0.001 … 0.03 0.000 –0.01 ‐1g 2f 1f … … 0.001 … 0.02 0.001 –0.02 ‐1g 2f 1f 3d … … 0.001 … 0.06 … 0.01 ‐1g 2f 1f 3d 2d … … 0.001 –0.001 0.09 … 0.05 ‐1g 2f 1f 3d 2d 1d … … 0.001 … 0.09 0.001 0.04

Expt.a 1.514b … 1.535 1.518c 90.7c 1.511c 92.1c


54

(Table 4.3 continued) SeH SeH2 HBr HOBr

Basis Set r(Se‐H) r(Se‐H) a(H‐Se‐H) r(H‐Br) r(H‐O) r(O‐Br) a(H‐O‐Br) cc‐pVDZ 1.478 1.473 91.34 1.426 0.974 1.888 100.82

cc‐pVTZ 1.473 1.468 91.26 1.420 0.965 1.840 102.27 ‐1d 0.003 0.003 … 0.003 0.001 … 0.17

cc‐pVQZ 1.472 1.468 91.27 1.421 0.964 1.833 102.93 ‐1f 0.001 0.001 0.01 0.001 … … 0.04 ‐1f 1d 0.001 … 0.11 … … … 0.16 ‐1f 1d 2d 0.002 0.002 0.11 0.002 … … 0.19

cc‐pV5Z 1.472 1.468 91.28 1.421 0.964 1.830 103.18 ‐1g 0.001 … … … … … 0.01 ‐1g 2f … … 0.03 … … … 0.05 ‐1g 2f 1f 0.001 0.001 0.03 0.001 … … 0.06 ‐1g 2f 1f 3d 0.001 … 0.05 0.001 … … 0.06 ‐1g 2f 1f 3d 2d … … 0.09 … … ‐0.001 0.10 ‐1g 2f 1f 3d 2d 1d 0.001 0.001 0.09 0.001 … ‐0.001 0.11

Expt.a 1.475 1.46c 90.6c 1.414 0.961 1.834 102.3d

a) From Ref. 171 unless otherwise noted b) From Ref. 172 c) From Ref. 173 d) From Ref. 175

55

Table 4.4. Bond lengths and angles of the cations computed with the full correlation consistent basis sets. The errors resulting from the use of the truncated basis sets are also shown. These are reported relative to the bond lengths and angles obtained using the full correlation consistent basis sets, and are in angstroms and degrees, respectively. “…” means that no change was observed relative to the full basis set.

AsH+ AsH2+ SeH+ SeH2

+ HBr+ HOBr+ Basis Set r(As‐H) r(As‐H) a(H‐As‐H) r(Se‐H) r(Se‐H) a(H‐Se‐H) r(H‐Br) r(H‐O) r(O‐Br) a(H‐O‐Br)

cc‐pVDZ 1.538 1.532 91.87 1.496 1.488 92.05 1.457 0.993 1.770 107.60

cc‐pVTZ 1.534 1.529 91.83 1.491 1.485 91.82 1.452 0.984 1.729 108.47

‐1d 0.003 0.002 –0.04 0.003 0.002 –0.04 0.004 0.001 … 0.19

cc‐pVQZ 1.533 1.528 91.77 1.490 1.484 91.80 1.453 0.983 1.724 108.97 ‐1f 0.001 0.001 … 0.002 0.001 0.01 0.001 … … 0.04

‐1f 1d … 0.001 … 0.001 … 0.08 0.001 … … 0.14

‐1f 1d 2d 0.002 0.001 … 0.003 0.002 0.08 0.003 … … 0.19

cc‐pV5Z 1.533 1.528 91.76 1.491 1.484 91.81 1.453 0.983 1.721 109.09 ‐1g 0.001 … … … … 0.00 0.001 … … 0.01

‐1g 2f … … 0.02 … … 0.02 0.001 … … 0.04

‐1g 2f 1f 0.001 … 0.02 … 0.001 0.02 0.001 … … 0.05

‐1g 2f 1f 3d 0.001 … 0.04 … 0.001 0.03 0.001 … … 0.06

‐1g 2f 1f 3d 2d 0.001 … 0.08 … … 0.07 0.001 … … 0.10

‐1g 2f 1f 3d 2d 1d 0.001 0.001 0.08 0.001 0.001 0.07 0.002 … … 0.11

56

4.3.2. Atomization Energies

Table 4.5 shows the reduction in atomization energy per hydrogen atom arising from

truncation of the hydrogen basis set as compared with the atomization energy determined

using the full, standard correlation consistent basis set. At the quintuple‐ζ level, the average

reduction for the removal of the 1g and outermost f (2f) basis functions was 0.06 and 0.10

kcal/mol per hydrogen, respectively. The truncation of the cc‐pVTZ basis set resulted in a

lowering of energy per hydrogen that ranged from 0.41 to 1.62 kcal/mol for the H2 and HBr

molecules, respectively, which demonstrates that the 1d function in the raw computation of

atomization energies with the cc‐pVTZ basis set is very important and should not simply be

removed from the hydrogen basis set. Overall, the energy contributions of the higher angular

momentum functions to the atomization energy grew at a consistent rate with respect to the

increase in electronegativity of the neighboring non‐hydrogen atom. For example, the lowering

in the atomization energy per hydrogen due to the removal of the d function from the cc‐pVTZ

basis set was found to be 0.84, 1.08, 1.22, and 1.38 kcal/mol for CH4, NH3, H2O, and HF,

respectively. This observed trend was attributed to the increasing need for higher angular

momentum functions to describe the highly polar character of molecules such as HF, as noted

in work done by Martin and Taylor.204

57

Table 4.5. The lowering of the calculated atomization energy per hydrogen atom resulting from the truncation of the correlation consistent basis sets. Energy reductions are reported in kcal/mol.

Basis Set H2 CH4 NH3 H2O HF HCN GeH4 AsH AsH2 AsH3 SeH SeH2 HBr Avg.

cc‐pVTZ

‐1d 0.41 0.84 1.08 1.22 1.38 0.82 1.01 1.16 1.17 1.17 1.40 1.41 1.62 1.13

cc‐pVQZ

‐1f 0.07 0.21 0.27 0.34 0.37 0.18 0.20 0.25 0.25 0.25 0.33 0.33 0.41 0.27

‐1f 2d 0.19 0.18 0.26 0.29 0.34 0.15 0.35 0.44 0.43 0.41 0.43 0.42 0.45 0.33

‐1f 2d 1d 0.34 0.38 0.46 0.54 0.60 0.31 0.34 0.38 0.38 0.38 0.43 0.43 0.49 0.42

cc‐pV5Z

‐1g 0.02 0.06 0.07 0.09 0.11 0.05 0.04 0.05 0.05 0.05 0.07 0.07 0.08 0.06

‐1g 2f 0.03 0.06 0.08 0.10 0.12 0.05 0.09 0.12 0.12 0.12 0.14 0.14 0.16 0.10

‐1g2f 1f 0.06 0.09 0.11 0.13 0.14 0.08 0.08 0.09 0.09 0.09 0.11 0.11 0.12 0.10

‐1g2f 1f 3d 0.06 0.05 0.07 0.09 0.09 0.03 0.07 0.10 0.10 0.09 0.10 0.09 0.10 0.08

‐1g2f 1f 3d 2d 0.24 0.11 0.13 0.14 0.15 0.07 0.24 0.27 0.27 0.26 0.27 0.27 0.27 0.21

‐1g2f 1f 3d 2d 1d 0.23 0.17 0.19 0.21 0.23 0.13 0.17 0.18 0.18 0.18 0.20 0.20 0.21 0.19

58

In Table 4.6, atomization energies are shown for each of the first‐row (B‐Ne) and third‐

row (Ga‐Kr) containing molecules as determined with both the full and truncated basis sets. For

the cc‐pV5Z basis set, both the expensive 1g and outermost 2f basis functions can be removed

without significant loss in accuracy. Specifically, the mean absolute deviation (MAD) for the

truncated cc‐pV5Z basis set in which the 1g and 2f functions were removed was less than 0.32

kcal/mol, as compared to the full cc‐pV5Z atomization energies. On the other hand, the

removal of the 1d function from the cc‐pVTZ basis set resulted in much more substantial errors

as compared to the atomization energies computed using the full cc‐pVTZ basis set, with the

deviation from the full basis set ranging in size from –0.82 kcal/mol to –4.04 kcal/mol for HCN

and GeH4, respectively. The results in Table 4.6 for the cc‐pVTZ basis set reiterate the

significant effect that basis set truncation has upon the raw atomization energies. It should be

stressed, however, that the goal for this study was to reduce the computational cost with

minimal impact upon the computed properties. Obviously, many of truncated basis sets utilized

in Table 4.6 are not suitable to achieve chemical accuracy, and therefore, are not

recommended for single‐calculation predictions of molecular properties, such as atomization

energies. However, Section 4.3.3 focuses upon CBS extrapolations utilizing the truncated basis

sets to determine whether a suitable approach can be defined.

59

Table 4.6. Atomization energies calculated with full cc‐pVnZ basis sets. The errors resulting from the use of the truncated basis sets are also shown. These are reported relative to the energies obtained using the full cc‐pVnZ basis sets, and are in kcal/mol.

Basis Set H2 CH4 NH3 H2O HF HCN GeH4 AsH AsH2 AsH3 SeH SeH2 HBr MAD

cc‐pVDZ 97.30 367.30 244.93 195.32 120.35 274.29 255.31 55.99 118.29 186.88 67.47 138.82 79.36

cc‐pVTZ 102.07 384.85 266.04 211.53 130.94 292.20 267.44 60.92 127.26 198.88 71.90 146.74 83.72

‐1d –0.93 –3.36 –3.24 –2.44 –1.38 –0.82 –4.04 –1.16 –2.33 –3.50 –1.40 –2.82 –1.62 2.23

cc‐pVQZ 102.82 388.90 272.02 216.36 133.95 298.51 271.26 62.61 130.34 203.02 73.65 149.9 85.45

‐1f –0.15 –0.85 –0.82 –0.69 –0.37 –0.18 –0.78 –0.25 –0.50 –0.75 –0.33 –0.66 –0.41 0.52‐1f 2d –0.52 –1.56 –1.59 –1.26 –0.71 –0.33 –2.17 –0.69 –1.36 –1.98 –0.76 –1.51 –0.86 1.18‐1f 2d 1d –1.20 –3.10 –2.95 –2.33 –1.31 –0.64 –3.54 –1.07 –2.12 –3.12 –1.19 –2.38 –1.35 2.02

cc‐pV5Z 103.02 390.01 274.05 218.00 134.94 300.49 272.09 63.08 131.20 204.15 74.12 150.73 85.91

‐1g –0.04 –0.24 –0.22 –0.18 –0.11 –0.02 –0.17 –0.05 –0.10 –0.15 –0.07 –0.13 –0.08 0.12‐1g 2f –0.10 –0.46 –0.45 –0.39 –0.22 –0.11 –0.54 –0.18 –0.35 –0.51 –0.21 –0.42 –0.24 0.32‐1g2f 1f –0.22 –0.83 –0.78 –0.64 –0.36 –0.18 –0.86 –0.27 –0.53 –0.78 –0.32 –0.63 –0.37 0.52‐1g2f 1f 3d –0.34 –1.03 –1.00 –0.82 –0.46 –0.22 –1.12 –0.37 –0.72 –1.05 –0.42 –0.82 –0.47 0.68‐1g2f 1f 3d 2d –0.81 –1.47 –1.39 –1.11 –0.60 –0.29 –2.06 –0.64 –1.26 –1.84 –0.68 –1.36 –0.74 1.10

‐1g2f 1f 3d 2d 1d –1.28 –2.17 –1.96 –1.53 –0.84 –0.43 –2.74 –0.82 –1.62 –2.38 –0.88 –1.75 –0.95 1.49

Expt.a 103.28b 392.50 276.70 219.35 135.20 301.80 270.50c 64.60d 131.10d 206.00d 74.30e 153.20e 86.50f

a) From Ref. 201 unless otherwise noted b) From Ref. 171 c) From Refs. 179 and 178 d) From Ref. 180 e) From Ref. 181 f) From Ref. 182

60

4.3.3. Complete Basis Set (CBS) Limits

An important aspect in the study of basis set truncation was to preserve the systematic

and convergent behavior of the correlation consistent basis sets, which was assessed by

obtaining CBS limits for the atomization energies utilizing both the full and truncated basis sets.

Table 4.7 shows CBS limits computed with the full correlation consistent basis sets using Eqn.

4.1 and Eqn. 4.2 as well as the error that arises upon utilizing a series of truncated basis set.

The notations to denote the four‐point extrapolations are described by cc(full) for the

extrapolation of the full basis set series, and by cc(f,–x,–y,–z) [where f refers to the full cc‐pVDZ

basis set, while –x, –y, and –z refer to the number of functions removed from the cc‐pVTZ, cc‐

pVQZ, and cc‐pV5Z basis sets, respectively]. For example, the cc(f,–1,–2,–3) extrapolation

scheme utilizes the cc‐pVDZ, cc‐pVTZ(–1d), cc‐pVQZ (–1f 2d), and cc‐pV5Z(–1d 2f 1f ) truncated

basis sets, while the cc(f,–1,–2,–4) extrapolation scheme utilizes the cc‐pVDZ, cc‐pVTZ(–1d), cc‐

pVQZ(–1f 2d), and cc‐pV5Z (–1g 2f 1f 3d) truncated basis sets.

It should be noted that, although, many of the extrapolation schemes that utilize

properties computed with various truncated basis sets in the determination of the CBS limits

resulted in mean absolute deviation (MAD) within 1 kcal/mol of the cc(full) CBS limits. However,

when choosing a truncation scheme, the convergent behavior of the basis sets series is as

important as the overall accuracy of the extrapolation because if the basis set series loose this

convergent behavior, there can be no conclusions drawn upon the accuracy of the

extrapolation. Based on both the accuracy, which was compared to the cc(full) CBS limits, and

the convergent behavior of the truncated basis set series, four truncated basis set series were

61

determined to be useful, which are cc(f,–1,–1,–3), cc(f,–1,–1,–4), cc(f,–1,–2,–3), and

cc(f,–1,–2,–4).

As shown in Table 4.7, when the Feller extrapolation formula (Eqn. 4.1) was utilized, the

CBS limits computed with the cc(f,–1,–1,–3), cc(f,–1,–1,–4), cc(f,–1,–2,–3), and cc(f,–1,–2,–4)

extrapolations resulted in errors as compared to cc(full) of less than 1 kcal/mol with the only

exception being –1.05 kcal/mol, which was observed for GeH4 when the cc(f,–1,–2,–4)

extrapolation was used. Overall, the Feller extrapolation resulted in a MAD as compared to

cc(full) of 0.31, 0.30, 0.41, and 0.41 kcal/mol for the cc(f,–1,–1,–3), cc(f,–1,–1,–4),

cc(f,–1,–2,–3), and cc(f,–1,–2,–4) truncated basis set series, respectively.

The mixed Gaussian/Exponential extrapolation formula (Eqn. 3.1) was found to be less

sensitive to basis set truncation as compared to the Feller extrapolation formula (Eqn. 4.1). This

was seen by comparing the largest error computed with the Feller extrapolation, which was

–1.05 kcal/mol, to the largest error computed with the mixed extrapolation, which was only

–0.63 kcal/mol observed for GeH4 when the cc(f,–1,–2,–4) scheme was utilized. The MAD as

compared the cc(full) was found to be only 0.07, 0.08, 0.23, and 0.34 for the cc(f,–1,–1,–3),

cc(f,–1,–1,–4), cc(f,–1,–2,–3), and cc(f,–1,–2,–4) truncated basis set series, respectively, when

Eqn. 3.1 was utilized to compute the CBS limits.

62

Table 4.7. CBS limits determined using atomization energies obtained from a series of four calculations, and the error obtained upon utilizing the truncated basis set series [cc(full) – cc(f,–x,–y,–z)]. See text for description of the notation. The CBS limits are reported in kcal/mol.

Basis Set Series H2 HF NH3 H2O CH4 HCN GeH4 AsH AsH2 AsH3 SeH SeH HBr MAD

Feller Extrapolations

cc(full) 103.02 135.28 274.72 218.62 390.26 301.66 272.08 63.04 131.17 204.15 74.19 150.89 86.06

cc(f,–0,–1,–1) –0.10 –0.27 –0.56 –0.46 –0.59 –0.09 –0.49 –0.15 –0.31 –0.47 –0.21 –0.41 –0.27 0.34

cc(f,–0,–1,–2) –0.14 –0.39 –0.80 –0.68 ‐0.77 –0.18 –0.73 –0.23 –0.47 –0.71 –0.31 –0.61 –0.38 0.49

cc(f,–0,–1,–3) –0.23 –0.52 –1.11 –0.93 –1.08 –0.27 –0.94 –0.29 –0.59 –0.89 –0.38 –0.77 –0.48 0.65

cc(f,–0,–1,–4) –0.30 –0.61 –1.32 –1.10 –1.23 –0.32 –1.10 –0.35 –0.71 –1.07 –0.45 –0.90 –0.48 0.76

cc(f,–0,–1,–5) –0.58 –0.74 –1.67 –1.37 –1.57 –0.40 –1.69 –0.52 –1.06 –1.58 –0.63 –1.27 –0.75 1.06

cc(f,–0,–1,–6) –0.82 –0.95 –2.17 –1.75 –2.06 –0.54 –2.10 –0.63 –1.29 –1.93 –0.76 –1.54 –0.91 1.34

cc(f,–0,–2,–1) –0.29 –0.41 –0.89 –0.70 –0.92 –0.14 –1.19 –0.37 –0.74 –1.08 –0.42 –0.83 –0.48 0.65

cc(f,–0,–2,–2) –0.33 –0.53 –1.13 –0.92 –1.11 –0.24 –1.43 –0.45 –0.90 –1.32 –0.52 –1.03 –0.60 0.81

cc(f,–0,–2,–3) –0.42 –0.67 –1.45 –1.18 –1.41 –0.33 –1.63 –0.51 –1.01 –1.50 –0.59 –1.18 –0.69 0.97

cc(f,–0,–2,–4) –0.49 –0.76 –1.66 –1.35 –1.57 –0.37 –1.80 –0.57 –1.14 –1.67 –0.66 –1.31 –0.77 1.09

cc(f,–0,–2,–5) –0.77 –0.89 –2.02 –1.63 –1.91 ‐0.45 –2.38 –0.74 –1.48 –2.18 –0.84 –1.68 –0.97 1.38


63



cc(f,–0,–2,–6) –0.97 –1.10 –2.52 –2.01 –2.40 ‐0.60 –2.79 –0.85 –1.71 –2.53 –0.98 –1.95 –1.12 1.66

cc(f,–0,–3,–1) –0.65 –0.66 –1.47 –1.14 –1.66 ‐0.26 –1.87 –0.56 –1.12 –1.65 –0.63 –1.25 –0.72 1.05

cc(f,–0,–3,–2) –0.70 –0.79 –1.72 –1.37 –1.85 ‐0.35 –2.11 –0.64 –1.28 –1.89 –0.73 –1.45 –0.84 1.21

cc(f,–0,–3,–3) –0.78 –0.93 –2.05 –1.63 –2.16 ‐0.44 –2.32 –0.70 –1.40 –2.07 –0.80 –1.60 –0.93 1.37

cc(f,–0,–3,–4) –0.85 –1.02 –2.26 –1.82 –2.33 ‐0.49 –2.48 –0.76 –1.52 –2.25 –0.87 –1.74 –1.01 1.49

cc(f,–0,–3,–5) –1.05 –1.16 –2.64 –2.10 –2.67 ‐0.57 –3.07 –0.93 –1.86 –2.76 –1.05 –2.11 –1.20 1.78

cc(f,–0,–3,–6) –1.20 –1.37 –3.15 –2.50 –3.17 ‐0.72 –3.48 –1.04 –2.09 –3.10 –1.19 –2.38 –1.36 2.06

cc(f,–1,–1,–1) 0.29 0.67 1.79 1.42 1.39 0.51 0.32 0.02 0.12 0.28 0.08 0.27 0.17 0.56

cc(f,–1,–1,–2) 0.22 0.50 1.44 1.10 1.09 0.41 0.04 –0.06 –0.05 0.02 –0.02 0.05 0.04 0.39

cc(f,–1,–1,–3) 0.09 0.32 0.98 0.74 0.61 0.31 –0.19 –0.12 –0.18 –0.18 –0.10 –0.11 –0.05 0.31

cc(f,–1,–1,–4) –0.04 0.20 0.68 0.49 0.36 0.26 –0.38 –0.19 –0.31 –0.38 –0.17 –0.25 –0.13 0.30

cc(f,–1,–1,–5) –0.46 0.02 0.17 0.09 –0.16 0.17 –1.06 –0.37 –0.68 –0.95 –0.36 –0.66 –0.34 0.42

cc(f,–1,–1,–6) –0.79 –0.26 –0.54 –0.45 –0.92 0.01 –1.53 –0.48 –0.93 –1.34 –0.50 –0.95 –0.50 0.71

cc(f,–1,–2,–1) 0.15 0.58 1.62 1.30 1.21 0.47 –0.34 –0.20 –0.29 0.28 –0.13 –0.13 –0.04 0.52


64



cc(f,–1,–2,–2) 0.08 0.41 1.25 0.97 0.89 0.36 –0.62 –0.28 –0.47 0.02 –0.23 –0.35 –0.17 0.47

cc(f,–1,–2,–3) –0.06 0.22 0.78 0.59 0.39 0.26 –0.86 –0.34 –0.59 –0.18 –0.31 –0.51 –0.26 0.41

cc(f,–1,–2,–4) –0.19 0.10 0.47 0.33 0.13 0.21 –1.05 –0.41 –0.73 –0.38 –0.38 –0.66 –0.35 0.41

cc(f,–1,–2,–5) –0.62 –0.10 –0.06 –0.07 –0.41 0.12 –1.73 –0.58 –1.10 –0.95 –0.57 –1.06 –0.55 0.61

cc(f,–1,–2,–6) –0.97 –0.38 –0.80 –0.63 –1.19 –0.04 –2.20 –0.70 –1.35 –1.34 –0.71 –1.35 –0.71 0.95

cc(f,–1,–3,–1) –0.06 0.43 1.36 1.13 0.86 0.37 –1.00 –0.39 –0.67 –0.86 –0.34 –0.54 –0.28 0.64

cc(f,–1,–3,–2) –0.15 0.25 0.96 0.77 0.51 0.27 –1.28 –0.47 –0.84 –1.12 –0.44 –0.76 –0.40 0.63

cc(f,–1,–3,–3) –0.31 0.05 0.45 0.36 –0.05 0.16 –1.52 –0.53 –0.97 –1.33 –0.52 –0.93 –0.50 0.59

cc(f,–1,–3,–4) –0.46 –0.08 0.12 0.08 –0.33 0.11 –1.71 –0.59 –1.10 –1.52 –0.59 –1.07 –0.58 0.64

cc(f,–1,–3,–5) –0.93 –0.28 –0.45 –0.36 –0.93 0.02 –2.39 –0.77 –1.47 –2.10 –0.78 –1.47 –0.78 0.98

cc(f,–1,–3,–6) –1.28 –0.58 –1.23 –0.96 –1.77 –0.14 –2.86 –0.89 –1.72 –2.49 –0.92 –1.77 –0.95 1.35

Mixed Gaussian/Exponential Extrapolation

cc(full) 103.15 390.83 275.28 219.01 135.56 301.88 273.01 63.98 132.43 205.63 74.54 151.50 86.32

cc(f,–0,–1,–1) –0.11 –0.61 –0.58 –0.48 –0.27 –0.10 –0.52 –0.16 –0.33 –0.49 –0.22 –0.43 –0.27 0.35


65



cc(f,–0,–1,–2) –0.16 –0.79 –0.77 –0.66 –0.36 –0.18 –0.83 –0.26 –0.53 –0.79 –0.34 –0.67 –0.40 0.52

cc(f,–0,–1,–3) –0.26 –1.10 –1.05 –0.87 –0.48 –0.24 –1.10 –0.34 –0.68 –1.02 –0.43 –0.85 –0.51 0.69

cc(f,–0,–1,–4) –0.36 –1.27 –1.23 –1.02 –0.56 –0.27 –1.32 –0.42 –0.84 –1.24 –0.51 –1.01 –0.59 0.82

cc(f,–0,–1,–5) –0.75 –1.64 –1.56 –1.26 –0.68 –0.33 –2.11 –0.65 –1.29 –1.90 –0.74 –1.46 –0.82 1.17

cc(f,–0,–1,–6) –1.14 –2.23 –2.04 –1.61 –0.88 –0.45 –2.67 –0.80 –1.60 –2.36 –0.90 –1.79 –0.82 1.48

cc(f,–0,–2,–1) –0.28 –0.95 –0.95 –0.76 –0.43 –0.18 –1.19 –0.37 –0.74 ‐–1.08 –0.43 –0.84 –0.49 0.67

cc(f,–0,–2,–2) –0.33 –1.14 –1.14 –0.93 –0.53 –0.25 –1.50 –0.47 –0.94 –1.37 –0.55 –1.08 –0.62 0.83

cc(f,–0,–2,–3) –0.43 –1.45 –1.42 –1.14 –0.64 –0.31 –1.77 –0.55 –1.09 –1.60 –0.64 –1.26 –0.72 1.00

cc(f,–0,–2,–4) –0.53 –1.61 –1.60 –1.29 –0.73 –0.34 –1.99 –0.63 –1.25 –1.83 –0.72 –1.42 –0.81 1.14

cc(f,–0,–2,–5) –0.93 –1.98 –1.93 –1.54 ‐0.84 –0.40 –2.78 –0.86 –1.70 –2.49 –0.94 –1.87 –1.03 1.48

cc(f,–0,–2,–6) –1.32 –2.57 –2.41 –1.89 ‐1.05 –0.52 –3.34 –1.01 –2.01 –2.95 –1.11 –2.20 –1.21 1.81

cc(f,–0,–3,–1) –0.61 –1.69 –1.60 –1.27 ‐0.72 –0.32 –1.84 –0.55 –1.10 –1.63 –0.63 –1.26 –0.72 1.07

cc(f,–0,–3,–2) –0.66 –1.88 –1.80 –1.45 –0.81 –0.40 –2.16 –0.66 –1.31 –1.93 –0.75 –1.50 –0.85 1.24

cc(f,–0,–3,–3) –0.76 –2.19 –2.07 –1.66 –0.93 –0.46 –2.43 –0.73 –1.46 –2.16 –0.84 –1.68 –0.96 1.41

cc(f,–0,–3,–4) –0.86 –2.35 –2.26 –1.81 –1.02 –0.49 –2.64 –0.82 –1.62 –2.38 –0.92 –1.84 –1.04 1.54

cc(f,–0,–3,–5) –1.26 –2.72 –2.58 –2.05 –1.13 –0.55 –3.43 –1.05 –2.07 –3.04 –1.15 –2.28 –1.27 1.89


66



cc(f,–0,–3,–6) –1.65 –3.31 –3.06 –2.40 –1.33 –0.67 –4.00 –1.20 –2.38 –3.50 –1.32 –2.61 –1.45 2.22

cc(f,–1,–1,–1) 0.21 0.52 0.51 0.34 0.19 0.17 0.84 0.23 0.46 0.69 0.25 0.51 0.27 0.40

cc(f,–1,–1,–2) 0.16 0.33 0.31 0.16 0.10 0.10 0.52 0.13 0.25 0.39 0.13 0.27 0.14 0.23

cc(f,–1,–1,–3) 0.06 0.02 0.04 –0.05 –0.02 0.04 0.26 0.05 0.10 0.16 0.04 0.09 0.04 0.07

cc(f,–1,–1,–4) –0.05 –0.15 –0.15 –0.20 –0.10 0.00 0.04 –0.03 –0.06 –0.06 –0.04 –0.07 –0.05 0.08

cc(f,–1,–1,–5) –0.44 –0.51 –0.47 –0.44 –0.22 –0.05 –0.75 –0.26 –0.51 –0.73 –0.27 –0.51 –0.27 0.42

cc(f,–1,–1,–6) –0.83 –1.10 –0.95 –0.80 –0.42 –0.17 –1.32 –0.41 –0.82 –1.18 –0.43 –0.84 –0.45 0.75

cc(f,–1,–2,–1) 0.03 0.18 0.14 0.06 0.03 0.10 0.17 0.02 0.05 0.69 0.04 0.10 0.06 0.13

cc(f,–1,–2,–2) –0.02 –0.01 –0.06 –0.11 –0.06 0.02 –0.15 –0.09 –0.16 0.39 –0.08 –0.13 –0.08 0.10

cc(f,–1,–2,–3) –0.12 –0.32 –0.33 –0.32 –0.18 –0.03 –0.41 –0.16 –0.31 0.16 –0.17 –0.31 –0.18 0.23

cc(f,–1,–2,–4) –0.22 –0.49 –0.52 –0.47 –0.26 –0.07 –0.63 –0.25 –0.47 –0.06 –0.25 –0.47 –0.27 0.34

cc(f,–1,–2,–5) –0.62 –0.86 –0.84 –0.72 –0.38 –0.13 –1.42 –0.47 –0.92 –0.73 –0.47 –0.92 –0.49 0.69

cc(f,–1,–2,–6) –1.01 –1.44 –1.32 –1.07 –0.58 –0.24 –1.99 –0.63 –1.23 –1.18 –0.64 –1.25 –0.67 1.02

cc(f,–1,–3,–1) –0.30 –0.57 –0.52 –0.45 –0.26 –0.05 –0.49 –0.16 –0.32 –0.46 –0.16 –0.31 –0.18 0.33

cc(f,–1,–3,–2) –0.35 –0.75 –0.71 –0.63 –0.35 –0.13 –0.80 –0.27 –0.53 –0.75 –0.28 –0.55 –0.31 0.49

cc(f,–1,–3,–3) –0.45 –1.06 –0.99 –0.84 –0.47 –0.18 –1.07 –0.34 –0.68 –0.98 –0.37 –0.73 –0.41 0.66


67



cc(f,–1,–3,–4) –0.55 –1.23 –1.17 –0.99 –0.55 –0.22 –1.29 –0.43 –0.84 –1.20 –0.46 –0.89 –0.50 0.79

cc(f,–1,–3,–5) –0.94 –1.60 –1.50 –1.23 –0.67 –0.28 –2.08 –0.66 –1.29 –1.87 –0.68 –1.34 –0.73 1.14

cc(f,–1,–3,–6) –1.34 –2.18 –1.97 –1.58 –0.87 –0.39 –2.64 –0.81 –1.59 –2.32 –0.85 –1.67 –0.90 1.47

Expt. 103.28a 392.50b 276.70b 219.35b 135.20b 301.80b 270.50c 64.60d 131.10d 206.00d 74.30e 153.20e 86.50f

a) From Ref. 171 b) From Ref. 201 c) From Refs. 179 and 178 d) From Ref. 180 e) From Ref. 181 f) From Ref. 182

Figure 4.1. Complete basis set limit extrapolations of atomization energy for H2 using both the full and truncated cc‐pVnZ basis sets. The curves show the CBS fits, and the horizontal lines represent the atomization energy arising from each of the subsequent basis truncations as shown in the text boxes. Energies are reported in kcal/mol.

2 3 4 5 6

97

98

99

100

101

102

103

cc (full) cc (f,-1,-2,-3) cc (f,-1,-2,-4)

full-1g-1g 2f-1g 2f 1f-1g 2f 1f 3d-1g 2f 1f 3d 2d-1g 2f 1f 3d 2d 1d

full-1f-1f 2d-1f 2d 1d

full -1d

AE

(kca

l/mol

)

cc-pVnZ

Figure 4.1 shows the curves for the extrapolation of the full series of the correlation

consistent basis sets and for the cc(f,–1,–2,–3) and cc(f,–1,–2,–4) for H2 when the Feller

extrapolation formula was utilized. The cc(f,–1,–1,–3) and cc(f,–1,–1,–4) schemes had similar

curves. As shown, the curves resulting from the truncated basis sets converge more slowly than

that of the full basis sets. However, the CBS limits computed with the cc(f,–1,–2,–3) and

cc(f,–1,–2,–4) truncated basis set series were still within 0.06 and 0.19 kcal/mol from the CBS

limit determined for the full series. Figure 4.2 (a–e) provides similar plots for the remaining

first‐row (B‐Ne) containing molecules CH4, NH3, H2O, HF, and HCN. For HCN, there is very little

68

fluctuation between the curves for the full basis set series, and the two truncated basis set

series. The largest difference in CBS limits for the cc(f,–1,–2,–3) and cc(f,–1,–2,–4) scheme as

compared with the extrapolation of the full basis set series was observed for the cc(f,–1,–2,–3)

scheme for H2O, which differs by 0.59 kcal/mol. The behavior for third‐row (Ga‐Kr) containing

molecules were similar to the first‐row molecules presented in Figure 4.2. Extrapolation curves

determined with the mixed Gaussian/exponential formula were found to be similar to the Feller

extrapolation formula.

Figure 4.2. Complete basis set limit extrapolation for the atomization energies (AE) of (a) CH4; (b) NH3; (c) H2O; (d) HF; (e) HCN using both the full and truncated cc‐pVnZ basis sets. The curve shows the CBS extrapolation utilizing the formula proposed by Feller (Eqn. 4.1). The horizontal lines represent the AE arising from using each of the basis sets that were discussed in Section 4.2. AE are reported in kcal/mol.

2 3 4 5 6365

370

375

380

385

390

a) CH4

cc (full) cc (f,-1,-2,-3) cc (f,-1,-2,-4)

AE (k

cal/m

ol)

cc-pVnZ

2 3 4 5 6

245

250

255

260

265

270

275

b) NH3

cc (full) cc (f,-1,-2,-3) cc (f,-1,-2,-4)

AE

(kca

l/mol

)

cc-pVnZ

69

2 3 4 5 6

195

200

205

210

215

220c) H2O

AE (k

cal/m

ol)

cc-pVnZ

cc (full) cc (f,-1,-2,-3) cc (f,-1,-2,-4)

2 3 4 5 6

120

122

124

126

128

130

132

134

136d) HF

AE

(kca

l/mol

)

cc-pVnZ

cc (full) cc (f,-1,-2,-3) cc (f,-1,-2,-4)

2 3 4 5 6270

275

280

285

290

295

300

e) HCN

AE (k

cal/m

ol)

cc-pVnZ

cc (full) cc (f,-1,-2,-3) cc (f,-1,-2,-4)

70

71

4.3.4. Ionization Energies

Ionization energies are provided in Table 4.8. As shown in the table, truncation of the

basis sets had very little impact upon the ionization energies. [Note, chemical accuracy for IE is

0.043 eV] For example, removal of the 1g function from the quintuple‐ζ level had no effect on

the ionization energy, and removal of the 1f function from the quadruple and quintuple‐ζ levels

showed no more than a 0.002 eV deviation from the energies obtained using the full basis sets.

The largest deviations from the full sets was observed for SeH2, with differences of –0.010,

–0.014, and –0.011 eV for the double‐, triple‐, and quadruple‐ζ basis set levels, respectively,

observed after all of the higher angular momentum functions were removed. The largest MAD,

as compared to the full basis sets, was 0.008 eV and was noted at the quadruple‐ζ level after

removal of all of the higher angular momentum functions. Overall, basis set truncation did not

affect the computed ionization energies.

72

Table 4.8. Ionization energies computed using the full correlation consistent basis sets, and the errors arising from the use of the truncated basis sets. The errors resulting from the use of the truncated basis sets are reported relative to the ionization energies obtained using the full correlation consistent basis sets. “…” represents no change with respect to the full basis set. Ionization energies are given in eV.

Basis Set AsH → AsH+ AsH2 → AsH2+ SeH → SeH+ SeH2 → SeH2

+ HBr → HBr+ HOBr → HOBr+

cc‐pVDZ 9.351 9.137 9.318 9.415 11.158 10.287

cc‐pVTZ 9.524 9.320 9.688 9.736 11.424 10.475 ‐1d ‐

‐

‐

‐ ‐

0.001 ‐0.002 ‐0.005 ‐0.010 ‐0.008 ‐0.007

cc‐pVQZ 9.567 9.370 9.815 9.846 11.563 10.631 ‐1f … … ‐0.001 ‐0.002 ‐0.002 ‐0.002 ‐1f 2d 0.003 ‐0.006 ‐0.006 ‐0.010 ‐0.007 ‐0.005 ‐1f 2d 1d 0.004 ‐0.007 ‐0.008 ‐0.014 ‐0.010 ‐0.007

cc‐pV5Z 9.581 9.385 9.855 9.881 11.607 10.685 ‐1g … … … … … … ‐1g 2f 0.001 ‐0.001 ‐0.001 ‐0.002 ‐0.002 ‐0.001 ‐1g 2f 1f 0.001 ‐0.001 ‐0.002 ‐0.003 ‐0.002 ‐0.002 ‐1g 2f 1f 3d ‐0.002 ‐0.004 ‐0.003 ‐0.006 ‐0.004 ‐0.003 ‐1g 2f 1f 3d 2d ‐0.004 ‐0.007 ‐0.005 ‐0.009 ‐0.006 ‐0.005 ‐1g 2f 1f 3d 2d 1d ‐0.004 ‐0.008 ‐0.006 ‐0.011 ‐0.007 ‐0.006

Expt. 9.641a 9.443a 9.845b 9.886b 11.660c 10.638d

a) From Ref. 180 b) From Ref. 181 c) From Ref. 205 d) From Ref. 187

73

4.3.5. CBS Limits for Ionization Energies

CBS limits computed for the ionization energies (IE) and the MAD for the truncated basis

set extrapolations as compared to the CBS limit determined using the full basis set series are

provided in Table 4.9. Although any combination of truncated basis sets seems to be useful, the

focus here will be on the four combinations that were chosen in Section 4.4.3 [i.e.

cc(f,–1,–1,–3), cc(f,–1,–1,–4), cc(f,–1,–2,–3), and cc(f,–1,–2,–4)],.

When using the Feller exponential formula, the MAD computed for the cc(f,–1,–1,–3),

cc(f,–1,–1,–4), cc(f,–1,–2,–3), and cc(f,–1,–2,–4) CBS limits were 0.002, 0.001, 0.001, and 0.002

eV, respectively. The maximum deviation as compared to the full CBS limit was –0.004 eV

computed for both AsH2 and SeH2 using the cc(–1,–2,–4) truncated basis set series. When the

mixed exponential/Gaussian formula was used the MAD was determined to be 0.000, 0.002,

0.002, and 0.005 eV for the cc(f,–1,–1,–3), cc(f,–1,–1,–4), cc(f,–1,–2,–3), and cc(f,–1,–2,–4) CBS

limits, respectively. The largest error compared to the full CBS limit was –0.007 eV, which was

found for SeH2 when using the cc(f,–1,–2,–4) truncated basis set series. Overall, utilizing the

cc(f,–1,–1,–3), cc(f,–1,–1,–4), cc(f,–1,–2,–3), and cc(f,–1,–2,–4) truncated basis set series had

small effects on the ionization energy CBS limits.

74

Table 4.9. CBS limits for the ionization energies reported in eV for the full correlation consistent basis set series, and the error, as compared to cc(full), resulting from utilizing the truncated basis set series. Both the Feller exponential and the mixed exponential/Gaussian extrapolations are shown. The mean absolute deviation (MAD) is also provided.

Basis Set Series AsH AsH2 SeH SeH2 HBr HOBr MAD

Feller Extrapolation

cc(full) 9.580 9.385 9.857 9.885 11.618 10.714

cc(f,–0,–1,–1) 0.000 –0.001 0.000 –0.001 –0.001 0.028 0.005

cc(f,–0,–1,–2) –0.001 –0.002 –0.001 –0.003 –0.003 0.027 0.006

cc(f,–0,–1,–3) –0.001 –0.002 –0.002 –0.004 –0.003 0.026 0.006

cc(f,–0,–1,–4) –0.002 –0.004 –0.003 –0.006 –0.001 0.025 0.007

cc(f,–0,–1,–5) –0.003 –0.006 –0.005 –0.009 –0.007 0.024 0.009

cc(f,–0,–1,–6) –0.003 –0.007 –0.006 –0.011 –0.008 0.023 0.009

cc(f,–0,–2,–1) –0.002 –0.004 –0.003 –0.005 –0.004 0.027 0.007

cc(f,–0,–2,–2) –0.002 –0.005 –0.004 –0.007 –0.005 0.026 0.008

cc(f,–0,–2,–3) –0.002 –0.005 –0.004 –0.008 –0.006 0.025 0.008

cc(f,–0,–2,–4) –0.003 –0.007 –0.005 –0.010 –0.007 0.024 0.010

cc(f,–0,–2,–5) –0.004 –0.009 –0.007 –0.013 –0.009 0.022 0.011

cc(f,–0,–2,–6) –0.005 –0.010 –0.008 –0.015 –0.010 0.021 0.011

cc(f,–0,–3,–1) –0.002 –0.004 –0.003 –0.007 –0.005 0.026 0.008

cc(f,–0,–3,–2) –0.002 –0.005 –0.004 –0.008 –0.006 0.025 0.009

cc(f,–0,–3,–3) –0.002 –0.005 –0.005 –0.009 –0.007 0.024 0.009

cc(f,–0,–3,–4) –0.003 –0.007 –0.006 –0.012 –0.008 0.023 0.010


75



cc(f,–0,–3,–5) –0.004 –0.009 –0.008 –0.015 –0.010 0.022 0.011

cc(f,–0,–3,–6) –0.005 –0.010 –0.009 –0.016 –0.012 0.020 0.012

cc(f,–1,–1,–1) 0.001 0.001 0.003 0.005 0.004 0.033 0.008

cc(f,–1,–1,–2) 0.000 0.000 0.002 0.003 0.003 0.032 0.007

cc(f,–1,–1,–3) 0.000 0.000 0.002 0.003 0.002 0.031 0.006

cc(f,–1,–1,–4) –0.001 –0.002 0.000 0.000 0.000 0.030 0.006

cc(f,–1,–1,–5) –0.002 –0.004 –0.001 –0.003 –0.002 0.028 0.007

cc(f,–1,–1,–6) –0.002 –0.005 –0.002 –0.005 –0.003 0.027 0.007

cc(f,–1,–2,–1) –0.001 –0.002 0.001 0.002 0.002 0.032 0.006

cc(f,–1,–2,–2) –0.001 –0.003 0.000 0.000 0.000 0.030 0.006

cc(f,–1,–2,–3) –0.001 –0.003 0.000 –0.001 0.000 0.030 0.006

cc(f,–1,–2,–4) –0.002 –0.005 –0.002 –0.004 –0.002 0.028 0.007

cc(f,–1,–2,–5) –0.003 –0.007 –0.003 –0.007 –0.004 0.027 0.009

cc(f,–1,–2,–6) –0.004 –0.008 –0.004 –0.009 –0.005 0.026 0.009

cc(f,–1,–3,–1) –0.001 –0.002 0.000 0.000 0.000 0.031 0.006

cc(f,–1,–3,–2) –0.001 –0.003 –0.001 –0.002 –0.001 0.030 0.006

cc(f,–1,–3,–3) –0.001 –0.003 –0.001 –0.003 –0.002 0.029 0.007

cc(f,–1,–3,–4) –0.002 –0.006 –0.003 –0.005 –0.003 0.028 0.008

cc(f,–1,–3,–5) –0.003 –0.008 –0.004 –0.008 –0.005 0.026 0.009

cc(f,–1,–3,–6) ‐0.004 –0.008 –0.005 –0.010 –0.007 0.025 0.010


76



Mixed exponential/Gaussian Extrapolation

cc(full) 9.590 9.396 9.883 9.905 11.640 10.723

cc(f,–0,–1,–1) 0.000 0.000 –0.001 –0.001 –0.002 0.028 0.005

cc(f,–0,–1,–2) –0.001 –0.001 –0.002 –0.003 –0.003 0.027 0.006

cc(f,–0,–1,–3) –0.001 –0.001 –0.002 –0.003 –0.003 0.027 0.006

cc(f,–0,–1,–4) –0.002 –0.004 –0.003 –0.006 –0.005 0.026 0.008

cc(f,–0,–1,–5) –0.003 –0.006 –0.005 –0.009 –0.006 0.024 0.009

cc(f,–0,–1,–6) –0.003 –0.007 –0.006 –0.010 0.000 0.024 0.008

cc(f,–0,–2,–1) –0.002 –0.003 –0.003 –0.005 –0.004 0.026 0.007

cc(f,–0,–2,–2) –0.002 –0.004 –0.004 –0.007 –0.005 0.026 0.008

cc(f,–0,–2,–3) –0.002 –0.004 –0.004 –0.007 –0.006 0.025 0.008

cc(f,–0,–2,–4) –0.004 –0.007 –0.006 –0.010 –0.007 0.024 0.010

cc(f,–0,–2,–5) –0.005 –0.009 –0.007 –0.013 –0.009 0.023 0.011

cc(f,–0,–2,–6) –0.005 –0.010 –0.008 –0.014 –0.010 0.022 0.011

cc(f,–0,–3,–1) –0.002 –0.004 –0.004 –0.007 –0.006 0.025 0.008

cc(f,–0,–3,–2) –0.002 –0.005 –0.005 –0.008 –0.007 0.024 0.009

cc(f,–0,–3,–3) –0.002 –0.005 –0.005 –0.009 –0.007 0.024 0.009

cc(f,–0,–3,–4) –0.004 –0.007 –0.007 –0.012 –0.009 0.023 0.010

cc(f,–0,–3,–5) –0.005 –0.009 –0.008 –0.014 –0.010 0.022 0.011

cc(f,–0,–3,–6) –0.005 –0.010 –0.009 –0.016 –0.011 0.021 0.012


77



cc(f,–1,–1,–1) 0.000 0.000 0.001 0.002 0.001 0.030 0.006

cc(f,–1,–1,–2) 0.000 –0.001 0.000 0.000 0.000 0.029 0.005

cc(f,–1,–1,–3) 0.000 –0.001 0.000 0.000 –0.001 0.029 0.005

cc(f,–1,–1,–4) –0.002 –0.003 –0.002 –0.003 –0.002 0.028 0.007

cc(f,–1,–1,–5) –0.003 –0.006 –0.003 –0.006 –0.004 0.027 0.008

cc(f,–1,–1,–6) –0.003 –0.006 –0.004 –0.007 –0.005 0.026 0.008

cc(f,–1,–2,–1) –0.001 –0.003 –0.001 –0.002 –0.001 0.029 0.006

cc(f,–1,–2,–2) –0.002 –0.004 0.001 –0.004 –0.003 0.028 0.007

cc(f,–1,–2,–3) –0.002 –0.004 0.000 –0.004 –0.003 0.027 0.007

cc(f,–1,–2,–4) –0.003 –0.006 0.004 –0.007 –0.004 0.026 0.009

cc(f,–1,–2,–5) –0.004 –0.009 –0.005 –0.010 –0.006 0.025 0.010

cc(f,–1,–2,–6) –0.005 –0.009 –0.006 –0.011 –0.007 0.024 0.010

cc(f,–1,–3,–1) –0.001 –0.003 –0.002 –0.003 –0.003 0.028 0.007

cc(f,–1,–3,–2) –0.002 –0.004 –0.003 –0.005 –0.004 0.027 0.008

cc(f,–1,–3,–3) –0.002 –0.004 –0.003 –0.006 –0.005 0.026 0.008

cc(f,–1,–3,–4) –0.003 –0.007 –0.005 –0.008 –0.006 0.025 0.009

cc(f,–1,–3,–5) –0.005 –0.009 –0.006 –0.011 –0.008 0.024 0.010

cc(f,–1,–3,–6) –0.005 –0.009 –0.007 –0.012 –0.009 0.023 0.011

78

4.3.6. Percent CPU Time Savings

The use of the truncated basis sets resulted in a substantial decrease in the percentage of

time required for each calculation, as shown in Table 4.10 for the first‐row (B‐Ne) containing

molecules. For all six of the first‐row molecules, there was an average decrease of 43% in time

required relative to the full basis set calculations when the d function was removed from the cc‐

pVTZ basis set. Similarly, the removal of the f function from the cc‐pVQZ basis set resulted in an

average CPU time savings of 40%, while the elimination of the g function from the cc‐pV5Z basis

set resulted in a 31% decrease in CPU time as compared to the full basis set computations. This

decrease in time was even more substantial as additional functions were stripped from the

quadruple‐ and quintuple‐ζ level basis sets. When the higher angular momentum functions

were removed entirely, there was an average of 70% and 83% cost savings achieved for the cc‐

pVQZ and cc‐pV5Z basis sets, respectively. The average CPU time saved was less significant for

the third‐row (Ga‐Kr) containing molecules, as reported in Table 4.11. However, there were still

significant time savings observed. When all of the higher angular momentum functions were

removed, basis set truncation resulted in a CPU time savings of 26.6%, 55.7%, and 71.3% for the

cc‐pVTZ, cc‐pVQZ, and cc‐pV5Z, respectively, for the third‐row molecules.

Unfortunately, basis set truncation had a large effect on raw atomization energies when

all of the higher angular momentum functions were removed. However, a systematic

convergence for a number of truncated basis sets to the CBS limit proved to give energetic

properties that were comparable to the CBS limit obtained using the full basis sets. The four

combinations of truncated basis sets that were chosen in the previous sections all had the 1d

79

function removed from the triple‐ζ level, which provided an average CPU time savings of 42.9

and 26.6% for the first‐ and third‐row molecules, respectively. Two of the combinations,

cc(f,–1,–1,–3) and cc(f,–1,–1,–4) had the 1f function removed from the quadruple‐ζ level, which

also resulted in a savings of 40.0 and 26.6% for the first‐ and third‐row molecules, respectively.

The other two combinations, cc(f,–1,–2,–3) and cc(f,–1,–2,–4), had the 1f and 2d function

removed from the quadruple‐ζ level basis set, resulting in a savings of 57.8 and 43.7% for the

first‐ and third‐row molecules, respectively. Truncating the quintuple‐ζ level provided the

highest time savings of 64.9 and 52.2% for the removal of the 1g and both f functions [−1g2f1f]

for the first‐ and third‐row molecules, respectively. Removal of the 1g, both f functions, and

outermost d functions [−1g2f1f3d] resulted in average time savings of 73.3 and 60.0% for the

first‐ and third‐row molecules, respectively. Overall, the cc(f,−1,−1,−3),cc(f,−1,−1,−4),

cc(f,−1,−2,−3) and cc(f,−1,−2,−4) provided the best combination of accuracy and time savings as

compared with the use of the cc(full) extrapolation.

Table 4.10. Percent CPU time saved for the reduced basis set as compared with the full basis set computation.

Basis Set H2 CH4 NH3 H2O HF HCN Average

cc‐pVTZ ‐1d 33.3 67.4 57.6 50.7 28.4 20.2 42.9

cc‐pVQZ ‐1f 56.1 51.0 46.6 40.1 27.1 18.8 40.0 ‐1f 1d 71.4 75.9 66.8 59.6 44.7 28.6 57.8 ‐1f 1d 2d 81.2 87.9 82.1 73.6 58.1 38.3 70.2

cc‐pV5Z ‐1g 46.6 25.5 38.8 32.9 23.4 18.0 30.9 ‐1g 2f 75.0 55.4 62.0 56.5 41.3 28.2 53.1 ‐1g 2f 1f 85.4 72.0 75.1 68.7 52.4 36.0 64.9 ‐1g 2f 1f 3d 93.1 82.4 82.4 77.3 62.9 41.6 73.3 ‐1g 2f 1f 3d 2d 96.0 89.0 87.9 83.2 69.6 47.6 78.9 ‐1g 2f 1f 3d 2d 1d 97.5 93.0 91.8 87.2 74.3 50.9 82.5

80

81

Table 4.11. Percent CPU time saved for the reduced basis set computations as compared to the full basis set computation. The CPU savings includes the time saved for the overall computations, including Hartree‐Fock and CCSD(T).

Basis Set AsH AsH+ AsH2 AsH2+ AsH3 GeH4 HBr HBr+ SeH SeH+ SeH2 SeH2

+ HOBr HOBr+ Avg.

cc‐pVTZ

‐1d 5.7 23.8 41.4 30.2 49.7 60.1 17.5 9.4 22.4 17.1 27.7 24.2 22.3 21.0 26.6

cc‐pVQZ

‐1f 23.0 20.3 37.1 27.0 41.0 44.1 16.8 22.4 22.1 22.0 32.7 32.7 18.1 13.6 26.6

‐1f2d 35.1 31.4 55.5 55.2 61.0 67.3 39.0 33.8 46.0 29.0 55.2 52.4 25.9 24.4 43.7

‐1f2d1d 43.4 45.7 68.2 63.3 77.3 81.9 45.9 48.4 53.0 47.1 65.3 62.1 37.9 35.9 55.4

cc‐pV5Z

‐1g 18.8 17.7 27.6 28.9 23.7 38.8 22.0 22.6 24.5 25.6 29.0 30.9 13.3 12.6 24.0

‐1g2f 34.9 26.6 46.5 42.1 54.2 58.3 35.7 30.2 40.6 33.6 46.3 49.0 23.1 26.0 39.1

‐1g2f1f 48.6 46.3 59.2 57.4 66.9 73.8 46.5 40.9 52.3 52.2 58.9 63.1 31.5 33.3 52.2

‐1g2f1f3d 56.2 54.0 70.4 66.2 76.2 82.1 53.8 54.3 55.2 58.7 69.8 69.4 37.0 37.2 60.0

‐1g2f1f3d2d 64.5 60.5 74.3 74.0 82.5 87.5 61.0 60.7 64.6 65.9 75.6 78.2 44.6 42.1 66.9

‐1g2f1f3d2d1d 68.8 66.2 80.6 79.7 87.3 91.4 66.3 65.3 67.7 68.2 80.2 82.0 47.8 46.8 71.3

All calculations were run on a single Pentium IV, 3 GHz processor.

82

4.4. Results and Discussion – 2‐ and 3‐Point Extrapolation Schemes

4.4.1. Hydrogen Correction: Three‐Point CBS Extrapolations

The initial series of molecules that were examined in this study include the hydrocarbons

methane through decane. These molecules were chosen due to their systematic increase in the

number of hydrogen atoms as the size of the molecules increases, which was useful to gauge

the impact that hydrogen basis set truncation had upon molecular properties. Table 4.12

reports the atomization energies that were determined with the full cc‐pVnZ basis sets [where

n = D, T, and Q], as well as the error upon truncation of the cc‐pVnZ basis sets for the hydrogen

atom. As shown, removing all of the higher angular momentum functions from the cc‐pVTZ and

cc‐pVQZ basis sets resulted in errors that were greater than 20 kcal/mol. For example, the

error in the atomization energy for decane was found to be –22.66 and –23.65 kcal/mol when

all of the higher angular momentum functions were removed from the cc‐pVTZ and cc‐pVQZ

basis sets, respectively. This reiterates the findings from Section 4.3 which proved that removal

of the higher angular momentum functions from the cc‐pVnZ basis sets for the hydrogen atom

can have large effects on raw atomization energies and is not recommended as a means to

determine raw atomization energies. However, it was the goal here to determine if the CBS

limit obtained by utilizing the full cc‐pVnZ basis sets could be reproduced with a series of

truncated hydrogen basis sets.

83

Table 4.12. Atomization energies for methane through decane determined with MP2 combined with the full cc‐pVnZ basis sets. The error upon truncation of the hydrogen basis set is also shown. All values are in kcal/mol.

cc‐pVDZ cc‐pVTZ cc‐pVQZ

Molecule full full ‐1d full ‐1f ‐1f2d ‐1f2d1d

Methane 363.61 381.88 –3.80 386.38 –1.21 –2.15 –4.17

Ethane 619.83 650.49 –5.97 658.61 –1.88 –3.30 –6.38

Propane 882.37 925.51 –8.02 937.26 –2.53 –4.41 –8.52

Butane 1142.04 1197.79 –10.13 1213.21 –3.18 –5.55 –10.69

Pentane 1402.49 1470.80 –12.22 1489.88 –3.83 –6.68 –12.84

Hexane 1667.45 1748.16 –14.28 1770.89 –4.48 –7.80 –15.00

Heptane 1925.47 2018.72 –16.36 2045.12 –5.12 –8.92 –17.16

Octane 2184.46 2290.41 –18.48 2320.47 –5.77 –10.07 –19.32

Nonane 2445.03 2563.55 –20.57 2597.27 –6.42 –11.20 –21.48

Decane 2705.71 2836.80 –22.66 2874.19 –7.07 –12.34 –23.65

The CBS limits determined for the atomization energies of methane through decane

utilizing Eqn. 4.2 are given in Table 4.13. The error in the truncated CBS limits as compared to

the CBS limits obtained by extrapolation of the atomization energies computed with the full

basis sets is also provided in Table 4.13. Notations to denote the three‐point extrapolations

are described by cc(full) for the extrapolation of the full basis set series, and by cc(f,–x,–y) for

the truncated basis set series, where f refers to the full cc‐pVDZ basis set, while –x and –y refer

to the number of functions removed from the cc‐pVTZ and cc‐pVQZ basis sets, respectively. To

illustrate, cc(f,–1,–2) indicates that the full cc‐pVDZ basis set was utilized, while the d function

84

was removed from the cc‐pVTZ basis set, and the f and outermost d functions were removed

from the cc‐pVQZ basis set.

Table 4.13. CBS limits for the atomization energies of methane through decane utilizing the three‐point extrapolation formula Eqn. 4.2, and the error relative to these limits for the truncated basis set series. The mean MAD between the cc(full) and truncated basis set series is also shown. All values are in kcal/mol.

Molecule cc(full) cc(f,–1,–1) cc(f,–1,–2) cc(f,–1,–3)

Methane 388.78 0.67 ‐0.90 ‐4.30

Ethane 663.04 1.09 ‐1.29 ‐6.45

Propane 943.71 1.46 ‐1.69 ‐8.58

Butane 1221.70 1.88 ‐2.10 ‐10.71

Pentane 1500.41 2.28 ‐2.50 ‐12.84

Hexane 1783.46 2.65 ‐2.91 ‐14.99

Heptane 2059.73 3.05 ‐3.32 ‐17.12

Octane 2337.12 3.48 ‐3.73 ‐19.25

Nonane 2615.96 3.88 ‐4.14 ‐21.39

Decane 2894.92 4.28 ‐4.56 ‐23.53

MAD 2.47 2.71 13.92

As shown in Table 4.13, the extrapolation of the truncated basis sets can lead to large

errors as compared to cc(full), with a MAD as large as 13.92 kcal/mol, which was observed for

the cc(f,–1,–3) extrapolation. These errors, fortunately, increase systematically as the number

of hydrogen atoms in the molecules increase. Table 4.14 shows the absolute deviation per

hydrogen between the full and truncated CBS limits for the three‐point extrapolation of the

85

atomization energies of methane through decane. As shown in Table 4.14, the absolute

deviation per hydrogen is relatively constant for a given truncated basis set series, which was

found to be 0.19, 0.21, and 1.07 kcal/mol for the cc(f,–1,–1), cc(f,–1,–2), and cc(f,–1,–3) series,

respectively. Since this basis set error is highly systematic for each truncated basis set series,

the basis set error observed upon truncation of the hydrogen basis set can be reduced by a

correction that depends on both the choice of truncated basis set series and the number of

hydrogen atoms that are present in the molecule. The empirical hydrogen corrections were

derived from the error between the cc(full) and truncated basis set series as shown in Table

4.3. From these plots, the correction per hydrogen was found to be –0.20, 0.20, and 1.07

kcal/mol for the cc(f,–1,–1), cc(f,–1,–2), and cc(f,–1,–3) basis set series, respectively.

Table 4.14. Absolute deviation per hydrogen between the atomization energy (kcal/mol) CBS limits determined with the cc(full) and truncated basis set series for the hydrocarbon series methane through decane.

# of Hydrogens cc(f,–1,–1) cc(f,–1,–2) cc(f,–1,–3)

4 0.17 0.22 1.07

6 0.18 0.21 1.07

8 0.18 0.21 1.07

10 0.19 0.21 1.07

12 0.19 0.21 1.07

14 0.19 0.21 1.07

16 0.19 0.21 1.07

18 0.19 0.21 1.07

20 0.19 0.21 1.07

22 0.19 0.21 1.07

Figure 4.3. Plots of the error between the CBS limits determined with the full and the (a) cc(f,–1,–1), (b) cc(f,–1,–2), and (c) cc(f,–1,–3) basis set series as the number of hydrogen atoms increase in the hydrocarbon series methane through decane.

y = -0.199x + 0.127R² = 0.999

‐4.5

‐4.0

‐3.5

‐3.0

‐2.5

‐2.0

‐1.5

‐1.0

‐0.5

0.0

0 10 20 30

Error a

s compared to cc (full)

# of Hydrogens

(a) cc(f,‐1,‐1)

y = 0.203x + 0.063R² = 0.999

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

0 10 20 3

Error a


# of Hydrogens

(b) cc(f,‐1,‐2)

0

y = 1.068x + 0.031R² = 1.000

0.0

5.0

10.0

15.0

20.0

25.0

0 10 20 3

Error a


# of Hydrogens

(c) cc(f,‐1,‐3)

0

86

87

Table 4.15 lists the CBS limits determined with Eqn. 4.2 combined with the atomization

energies computed with the full basis set series. The deviation from this limit when the

truncated basis set series were used to determine the CBS limit is also reported. The hydrogen

correction factor is included in the values reported in Table 4.15 for the truncated basis set

series. Without the hydrogen correction for methane through decane, the MAD is 2.47, 2.71

and 13.92 kcal/mol for cc(f,–1,–1), cc(f,–1,–2), and cc(f,–1,–3), respectively, as shown in Table

4.13. However, as shown in Table 4.15, after the hydrogen correction is included, the MAD was

reduced to 0.13, 0.06, and 0.03 kcal/mol for the cc(f,–1,–1), cc(f,–1,–2), and cc(f,–1,–3),

respectively. This shows that the hydrogen corrections successfully reduce the error introduced

by hydrogen basis set truncation for the cc(f,–1,–1), cc(f,–1,–2), and cc(f,–1,–3) extrapolation

schemes.

Table 4.15. Atomization energy CBS limits (kcal/mol) determined from MP2 energies found by using the full cc‐pVnZ [where n = D, T, and Q] basis set series [cc(full)] for the molecules methane though decane. The error as compared to the cc(full) limit is shown for truncated basis set series limits when the hydrogen corrections is included. The MAD is also provided.

Molecule cc(full) cc(f,–1,–1)a cc(f,–1,–2)b cc(f,–1,–3)c

Methane 388.78 –0.13 –0.08 –0.03

Ethane 663.04 –0.11 –0.06 –0.04

Propane 943.71 –0.14 –0.05 –0.04

Butane 1221.7 –0.12 –0.06 –0.03

Pentane 1500.41 –0.12 –0.05 –0.02

Hexane 1783.46 –0.15 –0.06 –0.04

Heptane 2059.73 –0.15 –0.06 –0.04


88


Molecule cc(full) cc(f,–1,–1)a cc(f,–1,–2)b cc(f,–1,–3)c

Octane 2337.12 –0.12 –0.06 –0.02

Nonane 2615.96 –0.12 –0.06 –0.03

Decane 2894.92 –0.12 –0.07 –0.04

MAD 0.13 0.06 0.03

4.4.2. Hydrogen Correction: Two‐Point CBS Extrapolations

The two‐point extrapolations utilizing Eqn. 4.4 resulted in the same systematic increase in

error between the cc(full) and truncated basis set extrapolations. Empirical hydrogen

corrections were determined by plotting the error between the cc(full) and truncated CBS limits

versus the number of hydrogen atoms that are in the molecules methane through decane, and

these plots are provided in Figure 4.4. To distinguish from the extrapolations that use the cc‐

pVDZ and cc‐pVTZ basis sets from the extrapolations that utilized the cc‐pVTZ and cc‐pVQZ

basis sets, the two‐point schemes are denoted as cc(DZ,TZ) for the extrapolation of the full cc‐

pVDZ and cc‐pVTZ basis set results, and cc(TZ,QZ) for the extrapolation of the full cc‐pVTZ and

cc‐pVQZ basis set results. The truncated basis set extrapolation are designated with a minus

sign to denote which functions were removed [e.g., cc(TZ–1d,QZ–1f2d) defines an extrapolation

of the cc‐pVTZ and cc‐pVQZ in which the d function was removed from the cc‐pVTZ basis set

and the f and outermost d function were removed from the cc‐pVQZ basis set]. The empirical

hydrogen correction per hydrogen atom was determined to be 1.49, 0.56, 0.98, 1.87, –0.20,

0.22, and 1.11 kcal/mol for the cc(DZ,TZ–1d), cc(TZ,QZ–1f), cc(TZ,QZ–1f2d), cc(TZ,QZ–1f2d1d),

cc(TZ–1d,QZ–1f), cc(TZ–1d,QZ–1f2d), and cc(TZ–1d,QZ–1f2d1d) basis set extrapolations,

respectively.

Figure 4.4. Plots of the error between the CBS limits determined with the full and the (a) cc(DZ,TZ–1d), (b) cc(TZ,QZ–1f), (c) cc(TZ,QZ–1f2d), (d) cc(TZ,QZ–1f2d1d), (e) cc(TZ–1d,QZ–1f), (f) cc(TZ–1d,QZ–1f2d), (g) and cc(TZ–1d,QZ–1f2d1d) basis set series as the number of hydrogen atoms increase in the hydrocarbon series methane through decane.

y = 1.485x - 0.490R² = 1.000

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

0 10 20 30

Error a


# of Hydrogens

(a) cc(DZ,TZ ‐1d)

y = 0.561x - 0.122R² = 1.000

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0 10 20 3

Error a


# of Hydrogens

(b) cc(TZ,QZ ‐1f)

0

y = 0.977x - 0.188R² = 1.000

0.0

5.0

10.0

15.0

20.0

25.0

0 10 20 30

Error a


# of Hydrogens

(c) cc(TZ,QZ ‐1f 2d)

y = 1.869x - 0.221R² = 1.000

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

45.0

0 10 20 3

Erro

r as c

ompa

red

to c

c (f

ull)

# of Hydrogens

(d) cc(TZ,QZ ‐1f 2d 1d)

0

89

The CBS limits determined by the two‐point extrapolations utilizing Eqn. 4.4 are shown in

Table 4.16. The first observation that can be made from Table 4.16 is that the extrapolated

atomization energies can deviate substantially between the cc(DZ,TZ) and cc(TZ,QZ), with

differences as large as 9.49 kcal/mol which occured for decane. In earlier work by Halkier et

al.,161 the authors cautioned that the use of the cc‐pVDZ basis set results in a two‐point CBS‐

y = -0.201x + 0.129R² = 0.999

‐5.0

‐4.5

‐4.0

‐3.5

‐3.0

‐2.5

‐2.0

‐1.5

‐1.0

‐0.5

0.0

0 10 20 30

Error a


# of Hydrogens

(e) cc(TZ -1d,QZ -1f)

y = 0.214x + 0.064R² = 0.999

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 10 20

Error a


# of Hydrogens

(f) cc(TZ ‐1d,QZ ‐1f 2d)

30

y = 1.106x + 0.030R² = 1.000

0.0

5.0

10.0

15.0

20.0

25.0

30.0

0 10 20 3

Error a


# of Hydrogens

(g) cc(TZ ‐1d,QZ ‐1f 2d 1d)

0

90

91

fitting procedure could be problematic, as a limit near the true CBS limit may not be found.

However, a study by Truhlar noted that a two‐point extrapolation utilizing the cc‐pVDZ and cc‐

pVTZ basis set results provided bond energies for Ne, HF, and H2O that were closer to

experiment than using a single cc‐pV6Z computation, and at less than 1% of the cost. Therefore,

the utility of a two‐point extrapolation which included the cc‐pVDZ and cc‐pVTZ results is

discussed here because the use of the smaller basis sets could prove to be useful when

calculations with the larger basis sets are not feasible.

The next observation from Table 4.16 is that the deviations between the full basis set

extrapolations and the truncated basis set extrapolations is extremely large when the empirical

hydrogen correction was not included, with MAD as large as 32.20 kcal/mol for the

cc(DZ,TZ–1d) extrapolation for decane. However, when the correction was included the

deviation between the full basis set extrapolation and the truncated basis set extrapolation was

reduced to well below 0.50 kcal/mol, with the largest MAD being 0.49 kcal/mol observed for

the cc(DZ,TZ–1d) basis set extrapolation for decane.

92

Table 4.16. CBS limits for the atomization energies found utilizing both a cc‐pVDZ‐cc‐pVTZ and a cc‐pVTZ‐cc‐pVQZ two‐point extrapolation utilizing Eqn. 4.4. The error as compared to the full basis sets when the truncated hydrogen basis sets are used to extrapolate to the CBS limit is also provided. The error is shown both with and without the empirical hydrogen correction. All values are in kcal/mol.

Molecule CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22 MAD

cc(DZ,TZ) 389.57 663.40 943.67 1221.26 1499.57 1782.14 2057.99 2335.02 2613.45 2891.99

cc(DZ,TZ‐1d)

w/ out correction –5.39 –8.48 –11.40 –14.39 –17.36 –20.29 –23.25 –26.26 –29.23 –32.20 18.83

w/ correctiona 0.55 0.43 0.49 0.47 0.47 0.52 0.53 0.49 0.49 0.49 0.49

cc(TZ,QZ) 389.66 664.54 945.83 1224.46 1503.80 1787.48 2064.38 2342.40 2621.88 2901.48

cc(TZ,QZ‐1f)

w/ out correction –2.10 –3.26 –4.38 –5.50 –6.62 –7.74 –8.86 –9.98 –11.10 –12.22 7.18

w/ correctionb 0.15 0.11 0.11 0.11 0.12 0.11 0.11 0.12 0.12 0.12 0.12

cc(TZ,QZ‐1f2d)

w/ out correction –3.71 –5.71 –7.63 –9.60 –11.55 –13.48 –15.44 –17.42 –19.38 –21.34 12.53

w/ correctionc 0.20 0.16 0.20 0.18 0.19 0.21 0.21 0.19 0.18 0.17 0.19

cc(TZ,QZ‐1f2d1d)

w/ out correction –7.22 –11.03 –14.74 –18.48 –22.22 –25.94 –29.68 –33.42 –37.16 –40.91 24.08

w/ correctiond 0.26 0.18 0.21 0.21 0.21 0.22 0.23 0.22 0.22 0.20 0.22

(table continued on next page)

93


Molecule CH4 C2H6 C3H8 C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22 MAD

cc(TZ‐1d,QZ‐1f)

w/ out correction 0.67 1.10 1.48 1.89 2.30 2.68 3.08 3.51 3.91 4.31 2.49

w/ correctione –0.14 –0.11 –0.14 –0.13 –0.13 –0.15 –0.16 –0.13 –0.13 –0.13 0.14

cc(TZ‐1d,QZ‐1f2d)

w/ out correctionf –0.94 –1.35 –1.78 –2.21 –2.63 –3.07 –3.50 –3.93 –4.37 –4.81 2.86

w/ correction –0.08 –0.06 –0.06 –0.06 –0.05 –0.06 –0.06 –0.06 –0.07 –0.08 0.06

cc(TZ‐1d,QZ‐1f2d1d)

–

–8.89

–11.09 w/ out correction 4.45 –6.68 –13.30 –15.52 –17.74 –19.94 –22.15 –24.38 14.41 w/ correctiong –0.03 –0.04 –0.04 –0.03 –0.03 –0.04 –0.04 –0.03 –0.03 –0.04 0.03

a) Correction per hydrogen for cc(DZ,TZ‐1d) is 1.49 kcal/mol. b) Correction per hydrogen for cc(TZ,QZ‐1f) is 0.56 kcal/mol. c) Correction per hydrogen for cc(TZ,QZ‐1f 2d) is 0.98 kcal/mol. d) Correction per hydrogen for cc(TZ,QZ‐1f 2d 1d) is 1.87 kcal/mol. e) Correction per hydrogen for cc(TZ‐1d,QZ‐1f) is ‐0.20 kcal/mol. f) Correction per hydrogen for cc(TZ‐1d,QZ‐1f 2d) is 0.22 kcal/mol. g) Correction per hydrogen for cc(TZ‐1d,QZ‐1f 2d 1d) is 1.11 kcal/mol.

94

4.4.3. Impact of Method Choice on the Hydrogen Correction

This section reports the impact that modifying the choice of optimization and/or single‐

point calculations had upon the hydrogen correction factor for atomization energies. In the

earlier part of this study, the MP2/cc‐pVnZ//B3LYP/6‐31G(d) level of theory was utilized.

However, in this portion of the study, four different levels of theory were examined including

(1) MP2/cc‐pVnZ//MP2/6‐31G(d); (2) MP2/cc‐pVnZ//MP2/cc‐pVTZ; (3) MP2/cc‐

pVnZ//B3LYP/cc‐pVTZ; and (4) CCSD(T)/ccpVnZ//MP2/6‐31G(d) [where n = D(2), T(3), and Q(4)].

Although the hydrocarbon series methane through decane was originally used to compute the

hydrogen correction, the larger heptane through decane molecules were omitted in this section

due to the high computational cost that the CCSD(T) method would require. Also, it was found

that the removal of these larger molecules had no effect on the hydrogen corrections

computed utilizing the MP2/cc‐pVnZ//B3LYP/6‐31G(d) level of theory.

Table 4.17 shows the errors per hydrogen for the three‐point CBS limits computed with

the truncated basis sets as compared to the CBS limits determined with the full basis sets. As

shown, the hydrogen correction is unaffected when the optimization method is changed. This is

demonstrated from a basis set perspective by replacing the 6‐31G(d) basis set with the cc‐pVTZ

basis set, and it is also shown from a method perspective by replacing B3LYP with MP2. No

matter which method was used in the optimization, the hydrogen correction was determined

to be –0.20, 0.20, and 1.07 kcal/mol for the cc(f,–1,–1), cc(f,–1,–2), and cc(f,–1,–3), respectively.

This consistency, however, was not maintained when the single‐point computation was

changed from MP2 to CCSD(T). The hydrogen corrections computed for the CCSD(T)/cc‐

95

pVnZ//B3LYP/6‐31G(d) level of theory were determined to be –0.28, 0.04, and 0.69 kcal/mol for

cc(f,–1,–1), cc(f,–1,–2), and cc(f,–1,–3), respectively.

Table 4.17. The deviation per hydrogen of the CBS limit for atomization energies for the truncated basis sets as compared to the CBS limit for the full basis sets computed with four different levels of theory. The hydrogen corrections for each level of theory are also shown. All values are in kcal/mol.

MP2//MP2/6‐31G(d) cc(f,–1,–1) cc(f,–1,–2) cc(f,–1,–3)

Methane 0.17 –0.22 –1.07 Ethane 0.18 –0.21 –1.07 Propane 0.18 –0.21 –1.07 Butane 0.19 –0.21 –1.07 Pentane 0.19 –0.21 –1.07 Hexane 0.19 –0.21 –1.07

Hydrogen Correction 0.20 –0.20 –1.07

MP2//MP2/cc‐pVTZ cc(f,–1,–1) cc(f,–1,–2) cc(f,–1,–3)



MP2//B3LYP/cc‐pVTZ cc(f,–1,–1) cc(f,–1,–2) cc(f,–1,–3)

Methane 0.17 –0.22 –1.08 Ethane 0.18 –0.21 –1.08 Propane 0.18 –0.21 –1.07 Butane 0.19 –0.21 –1.07


96


MP2//B3LYP/cc‐pVTZ cc(f,–1,–1) cc(f,–1,–2) cc(f,–1,–3)

Pentane 0.19 –0.21 –1.07 Hexane 0.19 –0.20 –1.07


CCSD(T)//B3LYP/6‐31G(d) cc(f,–1,–1) cc(f,–1,–2) cc(f,–1,–3)



4.4.4. Application to the G3/99 Test Suite

The G3/99 test suite consists of the G2/97 test suite as well as 75 additional molecules,

including hydrocarbons as large as naphthalene (C10H8). The focus of this section was the

application of the two‐ and three‐point schemes developed in the previous section towards the

hydrogen‐containing molecules taken from the molecules that were added to the G2/97 test

set to form the G3/99 test suite. The CBS limit for the atomization energies computed with

cc(full) as well as the error resulting from the utilization of cc(f,–1,–1), cc(f,–1,–2), and

cc(f,–1,–3) truncated schemes computed utilizing the mixed Gaussian/exponential

extrapolation formula is included in Figure 4.19. As the molecules n‐pentane through n‐octane

were studied in Sections 4.4.1 and 4.42, they are excluded from this section, which results in a

test set of 56 molecules. Also, the CBS limit for the atomization energies computed cc(DZ‐TZ)

97

and cc(TZ‐QZ) basis set schemes computed with Eqn. 4.4 are provided in Table 4.20 and Table

4.21, respectively. The errors associated in utilizing the truncated basis set in the two‐point

extrapolations are also shown in Table 4.20 and Table 4.21. While these tables are useful in

studying basis set truncation, the discussion here focuses upon the MAD between the full and

truncated basis set extrapolations.

Table 4.18 shows the MAD between the full and truncated basis sets, both before and

after the hydrogen correction is included. As shown, before the hydrogen correction is included

the MAD is large, with only the cc(f,–1,–1), cc(f,–1,–2) cc(TZ–1d, QZ–1f), and cc(TZ–1d, QZ–1f2d)

basis set extrapolations being less than 2 kcal/mol. The largest MAD is 14.89 kcal/mol, which

was found for the cc(TZ,QZ–1f2d1d) extrapolation. The corrections that were developed in the

previous section allow for the removal of these large errors, as indicated in Table 4.18. After the

hydrogen correction is included in the atomization energy CBS limits, the MAD are all reduced

to below 0.30 kcal/mol. Table 4.18 suggests that the hydrogen truncation schemes presented in

this study work extremely well, with the largest MADs as compared to the full extrapolations

being 0.29 kcal/mol, which was found for the cc(DZ,TZ–1d) extrapolation. It is interesting to

note that the removal of all d functions and higher for the hydrogen basis set can be achieved

with less than a 0.20 kcal/mol deviation in the CBS atomization energy as compared with that

arising from the full hydrogen basis sets. For example, the MAD as compared to cc(full) for all of

the molecules tested from the G3/99 test suite is 0.15 kcal/mol for both the cc(f,–1,–3) and

cc(TZ–1d,QZ–1f2d1d) extrapolations.

98

Table 4.18. Mean absolute deviation between the full and truncated basis sets for the two and three point extrapolations for the atomization energies of the G3‐99 test suite. The correction per hydrogen for each basis set extrapolation is also shown. All values are kcal/mol.

Extrapolation Corrections

per hydrogenMAD w/ out correction

MAD w/ correction

cc(f,–1,–1) –0.20 1.53 0.09

cc(f,–1,–2) 0.20 1.69 0.11

cc(f,–1,–3) 1.07 8.56 0.15

cc(DZ,TZ–1d) 1.49 11.72 0.29

cc(TZ,QZ–1f) 0.56 4.48 0.07

cc(TZ,QZ–1f2d) 0.98 7.79 0.15

cc(TZ,QZ–1f2d1d) 1.87 14.89 0.24

cc(TZ–1d,QZ–1f) –0.20 1.54 0.09

cc(TZ–1d,QZ–1f2d) 0.22 1.78 0.14

cc(TZ–1d,QZ–‐1f2d1d) 1.11 8.87 0.15

99

Table 4.19. Atomization energy CBS limits (kcal/mol) determined from MP2 energies found by using the full cc‐pV‐nZ [where n = D, T, and Q] basis set series [cc(full)]. The error compared to the cc(full) associated with utilizing the truncated basis set series extrapolations both with and without the hydrogen corrections is also shown. The f indicates that the full cc‐pVDZ basis set is used for the hydrogen atom, and the ‐1 or ‐2 indicates the number of basis functions removed from the cc‐pVTZ and cc‐pVQZ basis sets for hydrogen. [i.e. cc(f,‐1,‐1) indicates that the full cc‐pVDZ, cc‐pVTZ (‐1d ), and cc‐pVQZ (‐1f ) basis sets for the hydrogen atom were used]

cc(full) cc(f,‐1,‐1) cc(f,‐1‐,2) cc(f,‐1,‐3)

Molecule w/ corr.w/out corr. w/out corr. w/ corr. w/out corr. w/ corr.

1,3‐difluorobenzene 1378.71 0.78 –0.02 –0.73 0.07 –4.05 0.23

1,4‐difluorobenzene 1378.15 0.77 –0.03 –0.74 0.06 –4.05 0.23

pyrazine 1090.08 0.77 –0.03 –0.80 0.00 –4.21 0.07

2,5‐dihydrothiophene 1047.83 1.17 –0.03 –1.32 –0.12 –6.45 –0.03

2‐methyl thiophene 1229.97 1.11 –0.09 –1.26 –0.06 –6.33 0.09

3‐methyl pentane 1781.60 2.64 –0.16 –2.95 –0.15 –14.99 –0.01

acetic anhydride 1335.44 1.05 –0.15 –1.23 –0.03 –6.25 0.17

aniline 1502.08 1.43 0.03 –1.45 –0.05 –7.47 0.02

1,1‐dimethoxy ethane 1417.60 1.83 –0.17 –2.31 –0.31 –10.96 –0.26

acetyl acetylene 950.27 0.74 –0.06 –0.73 0.07 –3.92 0.36

crotonaldehyde 1088.39 1.11 –0.09 –1.23 –0.03 –6.31 0.11

isobutene nitrile 1159.14 1.27 –0.13 –1.42 –0.02 –7.37 0.12

isobutanal 1212.41 1.50 –0.10 –1.68 –0.08 –8.51 0.05


100




1,4‐dioxane 1301.92 1.53 –0.07 –1.79 –0.19 –8.72 –0.16

1,2‐dicyano ethane 1088.95 0.73 –0.07 –0.73 0.07 –4.04 0.24

chlorobenzene 1328.60 0.97 –0.03 –0.93 0.07 –5.13 0.22

di‐isopropyl ether 1882.62 2.63 –0.17 –3.03 –0.23 –15.04 –0.06

diethyl disulfide 1349.59 1.84 –0.16 –2.25 –0.25 –10.85 –0.15

diethyl ether 1318.69 1.88 –0.12 –2.18 –0.18 –10.85 –0.15

diethyl ketone 1497.87 1.85 –0.15 –2.09 –0.09 –10.62 0.08

isopropyl acetate 1616.99 1.83 –0.17 –2.13 –0.13 –10.63 0.07

methyl ethyl ketone 1218.72 1.46 –0.14 –1.68 –0.08 –8.47 0.09

N‐methyl pyrrole 1321.28 1.31 –0.09 –1.46 –0.06 –7.40 0.09

piperidine 1545.82 2.19 –0.01 –2.28 –0.08 –11.82 –0.05

tetrahydropyran 1487.83 1.93 –0.07 –2.06 –0.06 –10.68 0.02

tetrahydropyrrole 1262.52 1.82 0.02 –1.93 –0.13 –9.79 –0.16

tetrahydrothiophene 1173.42 1.56 –0.04 –1.70 –0.10 –8.58 –0.02

tetrahydrothiopyran 1454.48 1.93 –0.07 –2.11 –0.11 –10.70 0.00

tetramethylsilane 1454.32 2.19 –0.21 –2.88 –0.48 –13.03 –0.19

methyl acetate 1052.44 1.05 –0.15 –1.31 –0.11 –6.41 0.01


101




azulene 2095.71 1.55 –0.05 –1.53 0.07 –8.25 0.31

benzoquinone 1413.89 0.75 –0.05 –0.70 0.10 –3.99 0.29

cyclooctatetraene 1729.02 1.60 0.00 –1.53 0.07 –8.27 0.29

cyclopentanone 1384.90 1.54 –0.06 –1.54 0.06 –8.31 0.25

dimethyl sulfone 940.15 1.08 –0.12 –1.34 –0.14 –6.37 0.05

divinyl ether 1064.62 1.12 –0.08 –1.22 –0.02 –6.27 0.15

n‐butyl chloride 1213.34 1.69 –0.11 –1.91 –0.11 –9.67 –0.04

naphthalene 2129.61 1.54 –0.06 –1.48 0.12 –8.21 0.35

phenol 1444.97 1.18 –0.02 –1.28 –0.08 –6.49 –0.07

pyrimidine 1093.47 0.76 –0.04 –0.80 0.00 –4.20 0.08

t‐butyl chloride 1221.06 1.64 –0.16 –1.95 –0.15 –9.61 0.02

t‐butanethiol 1289.76 1.92 –0.08 –2.40 –0.40 –11.11 –0.41

t‐butanol 1334.44 1.88 –0.12 –2.31 –0.31 –11.03 –0.33

t‐butyl methyl ether 1600.55 2.21 –0.19 –2.64 –0.24 –12.94 –0.10

t‐butylamine 1388.49 2.12 –0.08 –2.52 –0.32 –12.10 –0.33

tetrahydrofuran 1204.85 1.56 –0.04 –1.68 –0.08 –8.61 –0.05

toluene 1618.35 1.52 –0.08 –1.57 0.03 –8.34 0.22


102




cyclopentane 1382.35 1.96 –0.04 –2.01 –0.01 –10.68 0.02

cyclohexane 1659.66 2.35 –0.05 –2.37 0.03 –12.66 0.18

isoprene 1252.15 1.48 –0.12 –1.65 –0.05 –8.42 0.14

1,3‐cyclohexadiene 1422.18 1.57 –0.03 –1.55 0.05 –8.35 0.21

1,4‐cyclohexadiene 1421.75 1.57 –0.03 –1.53 0.07 –8.32 0.24

fluorobenzene 1356.07 0.97 –0.03 –0.92 0.08 –5.11 0.24

methyl allene 957.30 1.10 –0.10 –1.19 0.01 –6.22 0.20

neopentane 1506.77 2.19 –0.21 –2.56 –0.16 –12.83 0.01

nitro‐s‐butane 1435.16 1.67 –0.13 –1.88 –0.08 –9.55 0.08

103

Table 4.20. CBS limits for the atomization energies found utilizing Eqn. 4.4 using the full cc‐pVDZ and cc‐VTZ. The error as compared to the full basis sets when the truncated hydrogen basis sets are used to extrapolate to the CBS limit is also provided. The error is shown with and without the empirical hydrogen correction. All values are in kcal/mol.

cc(DZ,TZ) cc(DZ,TZ‐1d)

Molecule w/out corr. w/ corr.

1,3‐difluorobenzene 1374.86 –5.90 0.05

1,4‐difluorobenzene 1374.25 –5.89 0.05

pyrazine 1082.94 –6.00 –0.05

2,5‐dihydrothiophene 1044.30 –8.92 0.00

2‐methyl thiophene 1224.83 –8.67 0.24

3‐methyl pentane 1780.39 –20.20 0.61

acetic anhydride 1330.04 –8.35 0.57

aniline 1495.97 –10.68 –0.28

1,1‐dimethoxy ethane 1416.17 –14.39 0.47

acetyl acetylene 946.81 –5.45 0.49

crotonaldehyde 1084.99 –8.66 0.26

isobutene nitrile 1155.59 –9.90 0.50

isobutanal 1209.64 –11.53 0.36

1,4‐dioxane 1299.24 –11.84 0.04

1,2‐dicyano ethane 1082.39 –5.66 0.29

chlorobenzene 1322.89 –7.40 0.03

di‐isopropyl ether 1881.13 –20.16 0.64

diethyl disulfide 1346.27 –14.41 0.45

diethyl ether 1317.72 –14.50 0.36

diethyl ketone 1494.73 –14.32 0.54

isopropyl acetate 1613.57 –14.22 0.64

methyl ethyl ketone 1215.84 –11.36 0.53


104




N‐methyl pyrrole 1316.31 –10.11 0.29

piperidine 1542.04 –16.46 –0.11

tetrahydropyran 1485.50 –14.75 0.11

tetrahydropyrrole 1259.08 –13.60 –0.23

tetrahydrothiophene 1170.60 –11.84 0.04

tetrahydrothiopyran 1450.97 –14.77 0.09

tetramethylsilane 1454.26 –17.07 0.77

methyl acetate 1049.57 –8.42 0.50

azulene 2088.05 –11.79 0.10

benzoquinone 1406.12 –5.79 0.16

cyclooctatetraene 1723.63 –11.93 –0.05

cyclopentanone 1380.78 –11.69 0.20

dimethyl sulfone 941.06 –8.46 0.45

divinyl ether 1062.70 –8.67 0.25

n‐butyl chloride 1211.38 –13.07 0.30

naphthalene 2122.12 –11.80 0.09

phenol 1439.76 –9.17 –0.26

pyrimidine 1086.39 –5.97 –0.02

t‐butyl chloride 1218.79 –12.76 0.61

t‐butanethiol 1287.13 –14.93 –0.07

t‐butanol 1332.72 –14.63 0.23

t‐butyl methyl ether 1599.29 –17.12 0.71

t‐butylamine 1385.88 –16.14 0.21

tetrahydrofuran 1202.95 –11.86 0.03

toluene 1613.82 –11.64 0.25

cyclopentane 1380.63 –14.80 0.06


105




cyclohexane 1657.46 –17.68 0.15

isoprene 1250.15 –11.44 0.45

1,3‐cyclohexadiene 1418.52 –11.82 0.07

1,4‐cyclohexadiene 1418.13 –11.85 0.04

fluorobenzene 1351.85 –7.38 0.05

methyl allene 955.85 –8.46 0.46

neopentane 1505.85 –17.00 0.83

nitro‐s‐butane 1429.57 –12.91 0.46

Table 4.21. CBS limits for the atomization energies found utilizing Eqn. 4.4 using the full cc‐pVTZ and cc‐pVQZ basis sets. The error as compared to the full basis sets when the truncated hydrogen basis sets are used to extrapolate to the CBS limit is also provided. The truncated extrapolations are labeled A‐F for the cc(TZ,QZ‐1f), cc(TZ,QZ‐1f2d), cc(TZ,QZ‐1f2d1d), cc(TZ‐1d,QZ‐1f), cc(TZ‐1d,QZ‐1f2d), and cc(TZ‐1d,QZ‐1f2d1d), respectively. The error is shown with and without the empirical hydrogen correction. All values are in kcal/mol.

cc(TZ,QZ) A B C D E F

w/

corr. w/out corr.

w/out corr.

w/ corr.

w/out corr.

w/ corr.

w/ out corr.

w/ corr.

w/out corr.

w/

corr.

w/out corr.

w/ Molecule

corr.

1,3‐difluorobenzene 1382.33 –2.25 0.00 –3.80 0.11 –7.22 0.26 0.78 –0.02 –0.78 0.08 –4.19 0.23

1,4‐difluorobenzene 1381.77 –2.25 0.00 –3.80 0.11 –7.22 0.26 0.78 –0.03 –0.78 0.08 –4.19 0.23

pyrazine 1093.29 –2.30 –0.06 –3.93 –0.01 –7.44 0.04 0.77 –0.03 –0.85 0.07 –4.36 0.06

2,5‐dihydrothiophene 1050.60 –3.40 –0.03 –5.97 –0.10 –11.26 –0.05 1.18 –0.03 –1.39 –0.07 –6.68 –0.05

2‐methyl thiophene 1233.18 –3.33 0.03 –5.78 0.09 –11.01 0.20 1.12 –0.09 –1.33 –0.07 –6.56 0.08

3‐methyl pentane 1785.63 –7.71 0.15 –13.47 0.22 –25.90 0.26 2.67 –0.16 –3.10 –0.24 –15.53 –0.05

acetic anhydride 1339.02 –3.23 0.14 –5.58 0.29 –10.76 0.45 1.06 –0.15 –1.30 –0.15 –6.47 0.16

aniline 1506.09 –4.04 –0.11 –7.01 –0.17 –13.22 –0.14 1.45 0.04 –1.52 0.09 –7.74 0.01

1,1‐dimethoxy ethane 1421.38 –5.54 0.07 –9.82 –0.04 –18.74 –0.05 1.84 –0.17 –2.43 –0.35 –11.35 –0.29

acetyl acetylene 952.83 –2.05 0.19 –3.57 0.34 –6.86 0.62 0.75 –0.06 –0.77 –0.11 –4.06 0.36

crotonaldehyde 1091.14 –3.33 0.04 –5.74 0.13 –10.98 0.23 1.11 –0.10 –1.29 –0.04 –6.54 0.10

isobutene nitrile 1162.06 –3.80 0.12 –6.58 0.26 –12.72 0.37 1.28 –0.13 –1.50 –0.12 –7.63 0.11

isobutanal 1215.37 –4.41 0.08 –7.69 0.13 –14.74 0.21 1.51 –0.10 –1.77 –0.14 –8.82 0.03


106

107



Molecule w/

corr. w/out corr.

w/out corr.

w/ corr.

w/out corr.

w/ corr.

w/ out corr.

w/ corr.

w/out corr.

w/

corr.

w/out corr.

w/

corr.

1,4‐dioxane 1305.51 –4.54 –0.05 –7.96 –0.14 –15.11 –0.16 1.54 –0.07 –1.88 –0.11 –9.03 –0.18

1,2‐dicyano ethane 1092.02 –2.17 0.07 –3.68 0.24 –7.09 0.39 0.73 –0.07 –0.77 0.02 –4.18 0.24

chlorobenzene 1331.93 –2.82 –0.01 –4.78 0.10 –9.12 0.23 0.98 –0.03 –0.98 0.10 –5.32 0.21

di‐isopropyl ether 1887.16 –7.70 0.16 –13.54 0.15 –25.93 0.24 2.66 –0.17 –3.18 –0.34 –15.58 –0.09

diethyl disulfide 1353.29 –5.54 0.07 –9.77 0.01 –18.64 0.05 1.86 –0.16 –2.37 –0.29 –11.24 –0.18

diethyl ether 1321.99 –5.55 0.06 –9.74 0.04 –18.68 0.01 1.90 –0.12 –2.29 –0.21 –11.24 –0.18

diethyl ketone 1501.43 –5.49 0.13 –9.55 0.23 –18.35 0.34 1.87 –0.15 –2.20 –0.18 –11.00 0.06

isopropyl acetate 1621.06 –5.46 0.15 –9.54 0.23 –18.31 0.38 1.84 –0.17 –2.24 –0.25 –11.01 0.05

methyl ethyl ketone 1221.68 –4.36 0.13 –7.60 0.23 –14.60 0.35 1.47 –0.14 –1.76 –0.17 –8.77 0.08

N‐methyl pyrrole 1324.80 –3.87 0.06 –6.73 0.12 –12.86 0.23 1.32 –0.09 –1.54 –0.09 –7.67 0.08

piperidine 1549.71 –6.24 –0.07 –10.85 –0.09 –20.70 –0.14 2.21 –0.01 –2.40 0.03 –12.25 –0.08

tetrahydropyran 1491.50 –5.63 –0.01 –9.75 0.03 –18.64 0.05 1.95 –0.07 –2.18 –0.01 –11.07 –0.01

tetrahydropyrrole 1265.82 –5.15 –0.10 –9.01 –0.21 –17.12 –0.30 1.84 0.02 –2.03 0.00 –10.14 –0.19

tetrahydrothiophene 1176.38 –4.51 –0.02 –7.87 –0.05 –14.97 –0.02 1.57 –0.04 –1.79 –0.05 –8.89 –0.04

tetrahydrothiopyran 1458.06 –5.64 –0.03 –9.80 –0.02 –18.66 0.03 1.95 –0.07 –2.22 –0.04 –11.08 –0.02

tetramethylsilane 1457.94 –6.56 0.18 –11.79 –0.05 –22.26 0.17 2.21 –0.21 –3.02 –0.62 –13.49 –0.22


108



Molecule w/

corr. w/out corr.

w/out corr.

w/ corr.

w/out corr.

w/ corr.

w/ out corr.

w/ corr.

w/out corr.

w/

corr.

w/out corr.

w/

corr.

methyl acetate 1055.27 –3.27 0.10 –5.70 0.17 –10.96 0.25 1.06 –0.15 –1.38 –0.19 –6.64 –0.01

azulene 2100.85 –4.50 –0.01 –7.67 0.15 –14.60 0.35 1.56 –0.06 –1.62 0.11 –8.54 0.30

benzoquinone 1417.63 –2.22 0.03 –3.71 0.20 –7.10 0.37 0.75 –0.05 –0.74 0.09 –4.13 0.29

cyclooctatetraene 1733.31 –4.51 –0.02 –7.74 0.08 –14.70 0.25 1.62 0.00 –1.61 0.13 –8.57 0.28

cyclopentanone 1388.27 –4.44 0.05 –7.62 0.20 –14.61 0.34 1.56 –0.06 –1.62 0.05 –8.61 0.24

dimethyl sulfone 944.02 –3.26 0.11 –5.76 0.11 –10.94 0.28 1.09 –0.12 –1.41 –0.23 –6.59 0.04

divinyl ether 1067.50 –3.33 0.04 –5.73 0.13 –10.95 0.27 1.12 –0.09 –1.28 –0.04 –6.50 0.14

n‐butyl chloride 1216.25 –5.00 0.05 –8.73 0.08 –16.73 0.09 1.71 –0.11 –2.01 –0.13 –10.01 –0.06

naphthalene 2134.72 –4.51 –0.02 –7.63 0.20 –14.56 0.39 1.55 –0.06 –1.57 0.17 –8.50 0.35

phenol 1448.69 –3.52 –0.16 –6.06 –0.19 –11.43 –0.22 1.19 –0.02 –1.35 0.09 –6.72 –0.09

pyrimidine 1096.67 –2.29 –0.05 –3.90 0.01 –7.41 0.06 0.77 –0.04 –0.84 0.07 –4.35 0.07

t‐butyl chloride 1223.96 –4.90 0.15 –8.60 0.20 –16.51 0.31 1.66 –0.16 –2.05 –0.27 –9.95 0.00

t‐butanethiol 1292.89 –5.73 –0.12 –10.18 –0.40 –19.17 –0.48 1.93 –0.09 –2.52 –0.25 –11.51 –0.45

t‐butanol 1337.76 –5.62 0.00 –9.94 –0.16 –18.94 –0.25 1.90 –0.12 –2.43 –0.27 –11.43 –0.37

t‐butyl methyl ether 1604.48 –6.56 0.17 –11.57 0.16 –22.20 0.23 2.23 –0.19 –2.78 –0.37 –13.40 –0.13

t‐butylamine 1392.03 –6.15 0.02 –10.94 –0.18 –20.81 –0.26 2.14 –0.08 –2.65 –0.31 –12.53 –0.36


109



Molecule w/

corr. w/out corr.

w/out corr.

w/ corr.

w/out corr.

w/ corr.

w/ out corr.

w/ corr.

w/out corr.

w/

corr.

w/out corr.

w/

corr.

tetrahydrofuran 1207.91 –4.52 –0.03 –7.86 –0.03 –15.00 –0.05 1.57 –0.04 –1.77 –0.02 –8.91 –0.07

toluene 1622.20 –4.45 0.04 –7.63 0.19 –14.61 0.34 1.53 –0.08 –1.65 0.02 –8.64 0.21

cyclopentane 1385.56 –5.62 –0.01 –9.72 0.06 –18.67 0.02 1.98 –0.04 –2.12 0.04 –11.07 –0.01

cyclohexane 1663.43 –6.71 0.03 –11.58 0.16 –22.19 0.24 2.37 –0.05 –2.50 0.05 –13.11 0.16

isoprene 1255.16 –4.38 0.11 –7.61 0.22 –14.59 0.36 1.49 –0.12 –1.73 –0.12 –8.72 0.13

1,3‐cyclohexadiene 1425.58 –4.48 0.01 –7.71 0.12 –14.72 0.23 1.59 –0.03 –1.64 0.08 –8.65 0.20

1,4‐cyclohexadiene 1425.17 –4.50 –0.01 –7.70 0.12 –14.70 0.25 1.58 –0.03 –1.62 0.11 –8.62 0.23

fluorobenzene 1359.47 –2.81 0.00 –4.77 0.12 –9.09 0.26 0.98 –0.03 –0.98 0.10 –5.30 0.23

methyl allene 959.69 –3.24 0.13 –5.60 0.26 –10.79 0.43 1.11 –0.10 –1.26 –0.10 –6.44 0.19

neopentane 1510.16 –6.53 0.21 –11.42 0.32 –22.02 0.41 2.20 –0.22 –2.69 –0.32 –13.29 –0.01

nitro‐s‐butane 1439.23 –4.94 0.11 –8.61 0.19 –16.52 0.30 1.69 –0.13 –1.98 –0.15 –9.89 0.06

110

4.4.5. Percent CPU Time and Disk Space Saved

The percent CPU time savings upon truncating the basis set for hydrogen is provided in

Table 4.22. As shown, hydrogen basis set truncation resulted in substantial time savings with

the removal of the d function from the cc‐pVTZ basis set providing, on average, a 24.0% CPU

time savings. The removal of the f function from the cc‐pVQZ basis set resulted in a 25.4% time

savings, while the removal of the d functions resulted in 44.3% and 53.2% time savings for the

outer and inner d functions, respectively.

The amount of disk space that was saved after basis set truncation is also shown in Table

4.23. The disk space is less affected by hydrogen basis set truncation as compared to CPU time

savings, but there are still large disk space savings that are seen upon truncation. As shown in

Table 4.23, removal of the d function from the cc‐pVTZ basis set resulted in an average of 24.0%

disk space reduction, while the truncation of the cc‐pVQZ basis set resulted in average disk

space savings of 18.3%, 30.5%, and 41.0% after removal of the f, outermost d, and innermost d

functions, respectively.

111

Table 4.22. Percent CPU time saved for the hydrogen containing molecules of the G3/99 test suite computed with MP2 combined with the full and truncated cc‐pVnZ basis sets.

Molecular cc‐pVTZ cc‐pVQZ

Molecule Formula (‐1d) (‐1f) (‐1f2d) (‐1f2d1d)

1,3‐difluorobenzene C6H4F 16.9 15.8 25.2 35.4


pyrazine C4H4N2 11.7 14.2 21.9 30.8

2,5‐dihydrothiophene C4H6S 18.7 27.5 45.7 55.1

2‐methyl thiophene C5H6S 19.9 11.8 35.8 45.0

3‐methyl pentane C6H14 42.6 39.2 59.5 72.0

acetic anhydride C4H6O3 19.2 17.5 35.6 48.4

aniline C6H5NH2 20.3 22.3 46.6 55.4

1,1‐dimethoxy ethane C4H10O2 22.1 27.9 49.9 61.4

acetyl acetylene C4H4O 14.9 12.5 28.6 33.8

crotonaldehyde C4H6O 5.0 21.8 39.9 49.4

isobutene nitrile C4H7N 21.1 27.0 48.2 58.3

isobutanal C4H8O 16.0 30.1 48.5 58.5

1,4‐dioxane C4H8O2 22.5 29.0 46.7 57.0

1,2‐dicyano ethane C4H4N2 26.8 14.4 34.4 39.6

chlorobenzene C6H5Cl 15.7 17.3 34.4 41.7

di‐isopropyl ether C6H14O 35.9 35.5 57.5 68.1

diethyl disulfide C4H10S2 29.5 29.2 52.0 60.3

diethyl ether C4H10O 8.6 31.8 55.2 62.1

diethyl ketone C5H10O 23.5 29.3 54.6 59.9

isopropyl acetate C5H10O2 19.6 25.1 51.1 56.9

methyl ethyl ketone C4H8O 28.8 27.0 52.0 59.3

n‐methyl pyrrole C5H7N 19.8 23.4 44.0 50.2

piperidine cyc‐C5H10NH 31.5 35.1 57.0 65.5


112




tetrahydropyran C5H10O 33.5 34.0 53.1 62.2

tetrahydropyrrole C4H8NH 22.2 26.4 49.9 58.7

tetrahydrothiophene C4H8S 23.2 31.5 51.2 57.8

tetrahydrothiopyran C5H10S 21.5 27.2 50.9 59.3

tetramethylsilane C4H12Si 25.9 35.0 57.3 64.6

methyl acetate C3H6O2 15.1 20.6 39.5 43.7

azulene C10H8 16.0 19.6 35.9 43.0

benzoquinone C6H4O2 14.7 13.2 26.8 33.1

cyclooctatetraene C8H8 32.5 24.3 41.8 51.5

cyclopentanone C5H8O 39.1 29.9 47.1 61.6

dimethyl sulfone C2H6O2S 13.2 22.7 39.4 45.8

divinyl ether C4H6O 24.5 26.0 41.5 49.3

n‐butyl chloride C4H9Cl 29.6 30.5 49.8 58.8

naphthalene C10H8 26.2 21.0 36.8 44.9

phenol C6H5OH 28.1 22.6 37.6 45.9

pyrimidine C4H4N2 27.5 15.6 30.4 35.6

t‐butyl chloride C4H9Cl 33.9 33.1 48.6 62.4

t‐butanethiol C4H9SH 35.0 34.4 54.1 66.6

t‐butanol C4H9OH 31.6 35.6 56.9 66.5

t‐butyl methyl ether C5H12O 39.2 31.1 53.1 63.1

t‐butylamine C4H9NH2 32.0 33.3 57.6 66.9

tetrahydrofuran C4H8O 27.5 33.4 52.3 58.5

toluene C6H5CH3 26.0 23.8 43.8 50.9

cyclopentane C5H10 13.7 23.7 51.6 63.0

cyclohexane C6H12 17.4 6.7 36.2 59.9

isoprene C5H8 24.4 28.2 44.0 55.3


113




1,3‐cyclohexadiene C6H8 21.6 28.1 40.7 52.0


fluorobenzene C6H5F 19.5 23.9 34.0 40.1

methyl allene C4H6 47.6 24.1 38.4 46.3

neopentane C5H12 39.9 38.3 53.2 64.0

nitro‐s‐butane C4H9NO2 24.4 25.3 43.5 51.1

Average 24.0 25.4 44.3 53.2

114

Table 4.23. Percent hard disk space saved for the hydrogen containing molecules of the G3/99 test suite computed with MP2 combined with the full and truncated cc‐pVnZ basis sets.





pyrazine C4H4N2 13.5 11.7 19.2 26.8

2,5‐dihydrothiophene C4H6S 22.3 16.7 28.1 38.2

2‐methyl thiophene C5H6S 19.8 15.1 25.4 34.6

3‐methyl pentane C6H14 32.9 23.7 39.2 52.5

acetic anhydride C4H6O3 20.9 14.0 23.7 32.3

aniline C6H5NH2 22.7 15.1 25.7 35.1

1,1‐dimethoxy ethane C4H10O2 28.1 20.5 34.2 46.1

acetyl acetylene C4H4O 16.7 13.2 22.1 30.5

crotonaldehyde C4H6O 23.2 17.4 28.8 39.6

isobutene nitrile C4H7N 25.5 19.1 31.4 42.8

isobutanal C4H8O 27.2 20.4 33.9 45.7

1,4‐dioxane C4H8O2 24.6 18.6 30.7 41.6

1,2‐dicyano ethane C4H4N2 14.7 11.7 19.8 27.2

chlorobenzene C6H5Cl 15.6 11.7 19.8 27.6

di‐isopropyl ethe C6H14O 31.0 22.5 37.1 49.9

diethyl disulfide C4H10S2 27.0 20.4 33.6 45.5

diethyl ether C4H10O 29.9 22.5 37.2 49.8

diethyl ketone C5H10O 27.4 20.2 33.9 45.5

isopropyl acetate C5H10O2 25.6 18.9 31.9 42.9

methyl ethyl ketone C4H8O 29.5 20.5 33.8 45.7

N‐methyl pyrrole C5H7N 23.1 17.2 28.6 38.9

piperidine cyc‐C5H10NH 31.4 21.8 35.9 48.4


115




tetrahydropyran C5H10O 29.9 20.6 34.1 46.2

tetrahydropyrrole C4H8NH 28.8 21.1 35.1 47.3

tetrahydrothiophene C4H8S 26.6 20.2 33.2 45.0

tetrahydrothiopyran C5H10S 29.1 20.4 33.7 45.7

tetramethylsilane C4H12Si 32.4 23.8 39.0 52.4

methyl acetate C3H6O2 21.5 16.3 28.2 37.9

azulene C10H8 17.3 12.6 21.5 29.5

benzoquinone C6H4O2 12.6 9.6 16.3 22.6

cyclooctatetraene C8H8 21.1 15.3 26.2 35.6

cyclopentanone C5H8O 24.6 18.1 30.4 41.1

dimethyl sulfone C2H6O2S 20.7 16.3 27.9 37.4

divinyl ether C4H6O 22.0 16.7 27.8 37.8

n‐butyl chloride C4H9Cl 28.3 21.1 34.8 47.0

naphthalene C10H8 17.5 12.8 22.1 30.2

phenol C6H5OH 19.9 13.4 22.8 31.3

pyrimidine C4H4N2 15.1 11.0 19.5 26.2

t‐butyl chloride C4H9Cl 28.2 20.9 35.0 47.0

t‐butanethiol C4H9SH 29.8 22.1 36.8 49.1

t‐butanol C4H9OH 30.5 22.3 37.1 49.8

t‐butyl methyl ether C5H12O 30.9 22.8 37.3 50.1

t‐butylamine C4H9NH2 32.5 23.4 38.6 51.8

tetrahydrofuran C4H8O 27.1 20.1 33.5 45.1

toluene C6H5CH3 24.7 16.5 27.9 37.9

cyclopentane C5H10 20.0 33.5 45.9 59.4

cyclohexane C6H12 24.2 32.8 59.3 70.7

isoprene C5H8 27.2 20.1 33.3 45.0


116






fluorobenzene C6H5F 12.9 13.8 21.0 24.9

methyl allene C4H6 25.6 19.1 31.6 43.0

neopentane C5H12 33.2 24.1 39.9 53.2

nitro‐s‐butane C4H9NO2 24.8 18.3 30.2 41.1

Average 24.0 18.3 30.5 41.0

117

4.5. Conclusions

The systematic truncation of the correlation consistent basis sets as a means to reduce

computational cost has been examined. Using a selected series of truncated basis sets on

hydrogen enabled smooth convergence to the CBS limit, with only a slight fluctuation in overall

accuracy in atomization energy, as compared with the results obtained from the full, standard

correlation consistent basis sets. Additionally, there is essentially no impact upon structures

when these truncated basis sets are used. The use of the truncated sets cc(f,–1,–1,–3),

cc(f,–1,–1,–4), cc(f,–l,–2,–3) and cc(f,–1,–2,–4) were shown to give CBS limits for atomization

and ionization energies that are within 1 kcal/mol to the CBS limits obtained using the full

correlation consistent basis sets for molecules containing first‐ (B‐Ne) and third‐row (Ga‐Kr)

atoms. This was verified by using two different formulas: the first is the well‐established Feller

exponential formula and the second is a mixed exponential/Gaussian formula.

Two‐ and three‐point extrapolations were also investigated for two sets of molecules,

which include methane through decane and the hydrogen‐containing molecules of the G3/99

test suite. Section 4.5 proved that extrapolation of the truncated basis set series resulted in

large errors as compared to extrapolation of the full basis set. However, these errors were

constant relative to the number of hydrogen atoms contained in the molecule as well as the

truncated basis set series used. Corrections for this error were determined, which were utilized

to provide errors less than 0.3 kcal/mol from the CBS limits determined with the full basis sets.

The hydrogen correction was also found to be unaffected by modifications to the method

and/or basis set utilized in the optimization; however, it was effected by the choice of method

118

used for the single point computation. Overall, basis set truncation was shown to be a viable

means to reduce the computational scaling [i.e. CPU time and disk space requirements], while

still taking advantage of the systematic nature of the correlation consistent basis sets in the

extrapolation of molecular properties to the CBS limit.

119

CHAPTER 5

BASIS SET TRUNCATION: APPLICATION TO THE CORRELATION CONSISTENT COMPOSITE

APPROACH

5.1. Introduction

As shown in Chapter 3, the correlation consistent composite approach (ccCA) is a viable

means to help alleviate the high computational cost of ab initio methods while still allowing

chemical accuracy to be achieved. In large part, the ability of ccCA to achieve a high level of

accuracy without the use of experimentally derived parameters can be attributed to the

correlation consistent basis sets. The correlation consistent composite approach takes

advantage of the systematic convergent behavior of the correlation consistent basis sets to

determine the MP2/CBS limit, which is used as the reference energy. A series of additive

corrections are then computed and added to the reference energy, which was fully discussed in

Chapter 3.

A drawback of the correlation consistent basis sets (discussed in Chapter 4) is the

substantial increase in the number of basis functions contained in the basis set as the level of

basis set is increased. For example, as shown in Chapter 4, increasing the basis set size from cc‐

pVDZ to cc‐pVTZ doubles the number of basis functions for first‐row (Li‐Ne) atoms and triples

the number of basis functions for hydrogen. This steep increase in basis set size is a limiting

factor in the application of ccCA to compute properties of larger molecules (e.g. one of the

120

largest molecules studied with ccCA, to date, is Be‐(acac)2, BeC10H14O.97 If the size of the basis

sets that ccCA utilizes could be reduced, while still maintaining chemical accuracy in the

prediction of energetic properties, the size of molecules that could be studied with this

composite method could be increased.

This chapter combines basis set truncation, discussed in Chapter 4, with the correlation

consistent composite approach, discussed in Chapter 3. Implementing basis set truncation

within ccCA requires further investigation into basis set truncation because the impact that

basis set truncation has upon the individual computations as part of ccCA have not been

performed. Also, several of the basis sets utilized in ccCA include a set of diffuse functions,

which needs to be examined. The effect of removing the augmented functions and the higher

angular momentum functions equal to or greater than d from the basis sets for the hydrogen

atom while using the full basis sets on all other atoms has upon enthalpies of formation

computed with ccCA was studied.

5.2. Development of ccCA(aug)

Earlier work performed by del Bene, showed that omitting diffuse functions from

Dunning’s correlation consistent basis sets for the hydrogen atom had less than 0.70 kcal/mol

effect on proton affinities for NH3, H2O, and HF.192 Spurred by del Bene’s study and the success

of the truncated basis set approaches discussed in Chapter 4, a simplified version of ccCA is

proposed [ccCA(aug)] in which the cc‐pVnZ basis sets are utilized rather than the aug‐cc‐pVnZ

basis sets for hydrogen. Thus, the reference MP2 calculations in ccCA are now denoted

121

MP2/aug’‐cc‐pVnZ [where n = D(2), T(3), Q(4); and the aug’ refers to the modified basis set

strategy]. The MP2 total energies were then extrapolated to the complete basis set (CBS) limit

using the mixed Gaussian/exponential formula (Eqn. 3.1) proposed by Peterson et al.159 and

using a two‐point 1/(lmax^4) scheme proposed by Schwartz (Eqn. 3.2)160,198 The ccCA(aug)

reference energy [E0(MP2/CBS’)] is then determined by averaging the CBS limit determined

from Eqn. 3.1 and the limit determined from Eqn. 3.2.

The additive corrections were computed as described in Chapter 3, with the exception of

the core‐valence correlation correction [ΔE(CV)], which is the only other step in ccCA requiring

the augmented correlation consistent basis sets. In this step, the aug’ truncated basis set

scheme will be used in both the MP2/aug’‐cc‐pVnZ and the MP2(FC1)/aug’‐cc‐pVnZ, and the

core‐valence correction computed from these two computations is denoted ∆E(CV’).

The final ccCA(aug) total energy that utilizes the aug’‐cc‐pVnZ basis sets is described by

the following equation,

ccCA aug E0[MP2 CBS'] ∆ CC ∆ D⁄ E E K

∆E CV' ∆ SO ZPE. 5.1

5.3. Development of ccCA(TB)

In Chapter 4, several ways to reduce the basis set size for hydrogen while still maintaining

accuracy within 1 kcal/mol of the results computed with full basis sets were presented. The

truncation scheme developed in Section 4.5 was utilized here to develop a truncated ccCA

scheme, denoted ccCA(TB). The ccCA(aug) method discussed in Section 5.2 serves as the

122

starting point for ccCA(TB). As in ccCA(aug), all augmented function included in the hydrogen

basis sets were removed. For ccCA(TB), further truncation was done by the removal of all basis

functions equal to or greater than d from the hydrogen basis sets. Specifically, the 1d function

was removed from the cc‐pVTZ, cc‐pVTZ‐DK, aug’‐cc‐pVTZ and aug’‐cc‐pCVTZ basis sets while

the 1f, 2d and 1d functions were removed from the aug’‐cc‐pVQZ basis set. As the aug’‐cc‐pVDZ

basis set only contains s and p angular momentum functions, no further truncation was done.

As noted in Section 4.5, computing molecular properties with hydrogen basis sets that do not

include d functions can lead to large errors as compared to results determined with the full

basis sets. Fortunately, this error was found to be a systematic error that could be significantly

reduced with a correction that depended on the number of hydrogen atoms contained in the

molecule. For example, as discussed Section 4.5, a correction per hydrogen of 1.07 kcal/mol

was determined for MP2/CBS limit atomization energies utilizing the cc(f,–1,–3) basis set series

[where f refers to the use of the full cc‐pVDZ, the –1 refers to the use of the cc‐pVTZ basis set

with the 1d function removed, and the –3 refers to the use of the cc‐pVQZ basis set with the 1f,

2d and 1d functions removed].195 However, as these corrections were also found to be method

specific, corrections were needed for each step in ccCA(TB).

It should be noted that the corrections needed for ccCA(TB) are not included in ccCA(aug).

As well, the empirical hydrogen corrections proposed for use in the ccCA(TB) scheme is a well‐

defined correction, which enables the error introduced by the truncation of the basis set for the

hydrogen atom to be reduced, and is developed based upon the difference in energies

computed with the full and truncated basis set. The empirical corrections utilized in ccCA(TB)

123

should not to be confused with a correction such as the HLC utilized in the G2, G3, and G4

methods, which were developed by fitting the molecular properties computed with Gn to

experimental data.63,67‐69,74 An empirical correction that is based on experimental data has been

shown to suffer if the molecule of interest is outside of the test set that was used to develop

the correction.129,177 Also, parameters based on experiment, such, as the HLC would need to be

re‐parameterized if new experimental data is determined, whereas the hydrogen corrections

for ccCA(TB) is solely based on theoretically computed energies and will only need to be

computed once. It should, however, be noted that the hydrogen corrections developed for

ccCA(TB) could potentially be prone to large errors if the hydrogen atom becomes important to

the molecular property of interest, such as for hydrogen bonding or transfer.

The hydrogen corrections were developed by plotting the difference in the total energy

between computations with the full and truncated basis set versus the number of hydrogen

atoms contained in the n‐alkane series methane through hexane, as shown in Figure 5.1. The

corrections per hydrogen atom were determined as the slopes of the linear plots in Figure 5.1,

which were found to be a) –1.279 millihartree (mEh) for the MP2/aug’‐cc‐pVnZ CBS limit [where

n = D(2), T(3), and Q(4)] computed with Eqn. 5.1, b) –1.345 mEh for the MP2/aug’‐cc‐pVnZ CBS

limit [where n = T(3) and Q(4)] computed with Eqn. 5.2, c) –1.674 mEh for MP2/cc‐pVTZ,

d) –1.494 mEh for CCSD(T)/cc‐pVTZ, e) –1.673 mEh for MP2/cc‐pVTZ‐DK, f) –1.839 mEh for

MP2/aug’‐cc‐pVTZ, and g) –1.932 mEh for MP2(FC1)/aug’‐cc‐pVTZ. A summary of these

corrections is provided in Table 3.1.

Figure 5.1. Linear plot of the difference between the total energy (mEh) computed with the full and truncated basis sets versus the number of hydrogen atoms in the linear alkane series methane through hexane. a) MP2/CBS limit computed with Eqn. 5.1, b) MP2/CBS limit compute with Eqn. 5.2, c) MP2/cc‐pVTZ, d) CCSD(T)/cc‐pVTZ, e) MP2/aug'‐cc‐pVTZ, f) MP2(FC1)/aug'‐cc‐pCVTZ, g) MP2/cc‐pVTZ‐DK

124

0

5

10

15

20

4 6 8 10 12 14

∆E (m

E h)

# of hydrogens

E0(P)

f(x)=1.279*x + 0.6346R2=1.000

a)

0

5

10

15

20

25

4 6 8 10 12 14

∆E (m

E h)

# of hydrogens

E0(S4)

f(x)=1.345*x+0.5926R2=1.000

b)

0

5

10

15

20

25

4 6 8 10 12 14

∆E (m

E h)

# of hydrogens

MP2/cc‐pVTZ

f(x)=1.674*x ‐ 0.6078R2=1.000

c)

0

5

10

15

20

25

4 6 8 10 12 14

∆E (m

E h)

# of hydrogens

CCSD(T)/cc‐pVTZ

f(x)=1.494*x ‐ 0.5863R2=1.000

d)

0

5

10

15

20

25

30

4 6 8 10 12 14

∆E (m

E h)

# of hydrogens

MP2/aug'‐cc‐pVTZ

f(x)=1.839*x ‐ 0.4839R2=1.000

e)

0

5

10

15

20

25

30

4 6 8 10 12 14

∆E (m

E h)

# of hydrogens

MP2(FC1)/aug'‐cc‐pCVTZ

f(x)=1.832*x ‐ 0.4630R2=1.000

f)

0

5

10

15

20

25

4 6 8 10 12 14∆E

(mE h)

# of hydrogens

MP2/cc‐pVTZ‐DK

f(x)=1.673*x ‐ 0.6075R2=1.000

g)

A final correction also included to account for the deviation of these linear plots, shown in

Figure 5.1, from the origin. This correction is needed because in theory, when there are no

hydrogen atoms, the only basis sets that would be present are the full correlation consistent

basis sets for the non‐hydrogen atoms. This means that ΔE would be zero when the number of

hydrogen atoms equal zero, and thus, the linear plots in Figure 5.1 should pass through the

origin. This, however, does not occur for any of the plots, as shown by the non‐zero y‐intercept

(i.e. when x=0) for all of the linear equations given in Figure 5.1, and therefore, a correction to

account for this error is needed, which is termed the “origin correction”. The origin correction is

shown in each of the linear equations provided in Figure 5.1. The origin correction was then

subtracted from the total energy of each individual step of ccCA(TB). For example, the y‐

intercept for the CBS limit computed with Eqn. 3.1 [E0(P)] was found to be 0.6346 mEh, which

means that E0(P) computed with the truncated basis sets is too high and should be reduced by

0.6346 mEh. In contrast, the MP2/cc‐pVTZ y‐intercept was computed to be –0.6078 mEh, which

means that the MP2/cc‐pVTZ(–1d) total energy is too low and should be raised by 0.6078 mEh.

125

126

It should be noted that the origin correction was not included in the truncation schemes

discussed in Chapter 4 because the deviation from the origin was found to be negligible (≤0.3

mEh).195 The only exception was observed for the cc(DZ,TZ–1d) [which denotes the use of the

cc‐pVDZ and cc‐pVTZ(–1d) basis sets] truncation scheme, for which a value of 0.7968 mEh was

determined for the origin correction.195 Initial tests to include the origin correction to the

cc(DZ,TZ–1d) truncation scheme found that for the G3/99 molecules studied in Chapter 4, half

of the atomization energies computed with cc(DZ,TZ–1d) improved compared to cc(full) while

the other half of the atomization energies computed with cc(DZ,TZ–1d) were further away from

cc(full), which resulted in a net effect on the MAD of almost zero. The origin correction,

however, is included in the ccCA(TB) formulation for completeness and is suggested for any

future studies that are based upon the work herein. A summary of the origin corrections for all

of the individual steps in ccCA(TB) is provided in Table 5.1.

Table 5.1. Correction per hydrogen and the origin correction for each of the calculations utilizing the correlation consistent basis sets if ccCA(TB). All values are mEh.

Corrections

Per Hydrogen Origin

Correction

E0(P) 1.279 0.6346

E0(S4) 1.345 0.5926

CCSD(T)/cc‐pVTZ 1.494 –0.5863

MP2/cc‐pVTZ 1.674 –0.6078

MP2(FC1)/aug'‐cc‐pCVTZ 1.832 –0.4630

MP2/aug'‐cc‐pVTZ 1.839 –0.4839

MP2/cc‐pVTZ‐DK 1.673 –0.6075

127

The difference between the total energies computed with the full and with the truncated

basis sets for each of the individual computations in ccCA is reported in Table 5.2. As shown in

Table 5.2, when the corrections are not included, the error between the total energies

computed with the full and with the truncated basis sets were large for methane (e.g. 5.356 to

6.809 mEh for the CCSD(T)/cc‐pVTZ and MP2/aug’‐cc‐pCVTZ computations, respectively) and

rises dramatically as the number of hydrogen atoms increase (e.g. for hexane the errors were

18.537 to 25.237 mEh for the E0(P) and MP2/aug’‐cc‐pVTZ computations, respectively). In

contrast, when the corrections are included in the total energies, the deviation in the total

energies computed with the truncated basis set as compared to the full basis sets was small

with the largest error determined for the MP2/aug’‐cc‐pVTZ computation (0.099 mEh).

128

Table 5.2. Differences between total energies computed with the full and with the truncated basis sets. The energy differences are shown both with and without the hydrogen and origin corrections. All values are in mEh.

Molecule E0(P) E0(S4)

MP2(FC1)/

aug’‐cc‐pCVTZ

MP2/

aug’‐cc‐pVTZ

CCSD(T)/

cc‐pVTZ

MP2/

cc‐pVTZ

MP2/

cc‐pVTZ‐DK

Before Corrections

Methane 5.793 6.004 6.809 6.789 5.356 6.053 6.052

Ethane 8.268 8.635 10.600 10.649 8.410 9.468 9.467

Propane 10.848 11.337 14.211 14.249 11.382 12.799 12.797

Butane 13.444 14.061 17.833 17.875 14.353 16.129 16.126

Pentane 16.000 16.746 21.511 21.561 17.325 19.460 19.457

Hexane 18.537 19.413 25.184 25.237 20.297 22.791 22.788

After Corrections

Methane 0.042 0.031 –0.057 –0.083 –0.033 –0.035 –0.032

Ethane –0.040 –0.027 0.071 0.099 0.033 0.032 0.037

Propane –0.019 –0.016 0.018 0.020 0.016 0.015 0.021

Butane 0.020 0.018 –0.024 –0.031 –0.001 –0.004 0.004

Pentane 0.017 0.014 –0.010 –0.023 –0.017 –0.021 –0.011

Hexane –0.004 –0.009 –0.001 –0.025 –0.032 –0.037 –0.027

129

5.4. Enthalpies of Formation

The truncated ccCA schemes were applied to the 104 enthalpies of formation for the

hydrogen containing molecules within the G2/97 test suite, which are listed in Table 5.3. The

enthalpies of formation computed using ccCA and the errors due to the utilization of ccCA(aug)

and ccCA(TB) are reported in Table 5.3. Enthalpy of formation determined from experiment for

the molecules are also shown for comparison. It should be noted that the errors reported in

Table 5.3 for the truncated ccCA methods are deviations compared to the full ccCA method and

not with experiment. The largest deviation between ccCA(aug) and ccCA was observed for

Si2H6, in which there was only a difference 0.4 kcal/mol between the enthalpies computed with

ccCA and ccCA(aug). Overall, the mean absolute deviation (MAD) for the enthalpies computed

with ccCA(aug) as compared with ccCA was only 0.1 kcal/mol. It should be emphasized that

ccCA(aug) is a simple empirical‐free truncation scheme that can be utilized while still

maintaining accuracy comparable to ccCA.

In general, enthalpies computed with ccCA(TB) were within 1 kcal/mol of the ccCA values,

with the exception only six out of the 104 enthalpies, which were the errors for SiH3, SiH4, PH3,

SH2, Si2H6, and CH3SiH3 were determined to be 1.0, 1.3, 1.4, 1.0, 2.2, and 1.1 kcal/mol,

respectively. It is interesting to note that for all but one of these molecules, the enthalpies

computed with ccCA(TB) were closer to experiment than ccCA. For example, the enthalpy of

formation computed with ccCA for SiH3 was 46.8 kcal/mol, whereas the enthalpy of formation

computed with ccCA(TB) for SiH3 was 47.8 kcal/mol, which is 1.0 kcal/mol closer to the

experimental value of 47.9 kcal/mol.206 This assessment, however, does not mean that ccCA(TB)

130

is more accurate than ccCA. The better agreement of ccCA(TB) with experiment is most likely

due to a fortuitous cancellation of error. Overall, the mean absolute deviation (MAD) for

ccCA(TB) as compared to ccCA(full) was found to be only 0.3 kcal/mol.

Table 5.3. Enthalpies of formation (kcal/mol) computed with ccCA, and the errors as compared to ccCA arising due to the use of ccCA(aug) and ccCA(TB) are presented. Experimental values are also provided.

Molecule ccCA ccCA(aug) ccCA(TB) Expt.a

G2‐1 test set

LiH 32.4 0.0 –0.1 33.3

BeH 81.0 –0.1 –0.5 81.7

CH 142.3 0.0 –0.2 142.4

CH2 (3B1) 94.0 0.0 –0.2 93.5

CH2 (1A1) 102.1 0.0 –0.1 102.5

CH3 34.9 0.0 –0.1 35.1

CH4 –18.2 0.0 0.1 –17.9

NH 85.8 0.0 –0.2 85.2

NH2 44.1 0.0 –0.1 44.5

NH3 –11.8 0.0 0.0 –11.0

OH 8.6 0.0 –0.2 8.9

OH2 –59.1 0.0 –0.1 –57.8

FH –66.1 0.0 –0.1 –65.1

SiH2 (1A1) 63.0 0.1 0.8 65.2

SiH2 (3B1) 86.1 0.1 0.5 86.2

SiH3 46.8 0.2 1.0 47.9

SiH4 6.4 0.2 1.3 8.2

PH2 32.4 0.2 0.8 33.1

PH3 1.5 0.2 1.4 1.3


131



SH2 –5.8 0.1 1.0 –4.9

ClH –22.5 0.0 0.4 –22.1

C2H2 55.1 –0.2 –0.9 54.2

C2H4 12.3 0.0 –0.3 12.5

C2H6 –20.8 0.0 0.1 –20.1

HCN 31.3 –0.1 –0.6 31.5

HCO 10.0 0.0 –0.4 10.0

H2CO –26.7 0.0 –0.3 –26.0

H3COH –49.2 0.0 0.0 –48.2

H2NNH2 21.9 –0.1 0.0 22.8

HOOH –33.2 0.0 0.1 –32.5

Si2H6 16.1 0.4 2.2 19.1

CH3Cl –20.2 0.0 0.1 –19.6

H3CSH –6.4 0.1 0.7 –5.5

HOCl –18.9 0.0 –0.1 –17.8

G2‐2 test set

Hydrocarbons

CH3CCH (propyne) 44.8 –0.1 –0.5 44.2

CH2=C=CH2 (allene) 45.7 –0.1 –0.5 45.5

C3H4 (cyclopropene) 68.0 –0.1 –0.4 66.2

CH3CH=CH2 (propylene) 4.5 0.0 –0.2 4.8

C3H6 (cyclopropane) 12.6 0.0 –0.3 12.7

C3H8 (propane) –25.9 0.0 0.1 –25.0

CH2CHCHCH2 (butadiene) 26.8 0.0 –0.4 26.3

C4H6 (2‐butyne) 35.7 0.0 –0.3 34.8

C4H6 (methylene cyclopropane) 46.4 –0.1 –0.5 47.9


132



C4H6 (bicyclobutane) 53.8 0.0 –0.4 51.9

C4H6 (cyclobutene) 38.6 –0.1 –0.3 37.4

C4H8 (cyclobutane) 6.1 0.0 –0.2 6.8

C4H8 (isobutene) –4.3 0.0 –0.2 –4.0

C4H10 (trans butane) –31.1 0.0 0.1 –30.0

C4H10 (isobutane) –32.9 0.0 0.0 –32.1

C5H8 (spiropentane) 44.2 –0.1 –0.5 44.3

C6H6 (benzene) 20.7 0.0 –0.7 19.7

Substituted hydrocarbons

CH2F2 –108.3 –0.1 –0.3 –107.7

CHF3 –166.8 –0.1 –0.3 –166.6

CH2Cl2 –22.8 0.1 0.0 –22.8

CHCl3 –24.3 0.1 –0.1 –24.7

CH3NH2 (methylamine) –6.0 0.0 0.0 –.5

CH3CN (methyl cyanide) 18.0 0.0 –0.4 17.7

CH3NO2 (nitromethane) –18.0 0.0 –0.3 –17.8

CH3ONO (methyl nitrite) –16.2 0.0 –0.1 –15.9

CH3SiH3 (methyl silane) –7.7 0.1 1.1 –7.0

HCOOH (formic acid) –91.1 0.0 –0.3 –90.5

HCOOCH3 (methyl formate) –86.4 0.0 –0.2 –85.0

CH3CONH2 (acetamide) –57.5 –0.1 0.2 –57.0

C2H4NH (aziridine) 29.9 0.0 –0.2 30.2

(CH3)2NH (dimethylamine) –4.9 0.0 0.1 –4.4

CH3CH2NH2 (trans ethylamine) –13.0 –0.1 0.0 –11.3

CH2CO (ketene) –11.4 –0.1 –0.4 –11.4

C2H4O (oxirane) –13.0 0.0 –0.3 –12.6


133



CH3CHO (acetaldehyde) –40.0 –0.1 –0.3 –39.7

HCOCOH (glyoxal) –51.2 –0.1 –0.5 –50.7

CH3CH2OH (ethanol) –57.3 –0.1 0.0 –56.2

CH3OCH3 (dimethylether) –44.9 0.0 0.1 –44.0

C2H4S (thiooxirane) 17.7 0.1 0.0 19.6

(CH3)2SO (dimethyl sulfoxide) –37.1 0.0 0.3 –36.2

C2H5SH (ethanethiol) –12.0 0.1 0.7 –11.1

CH3SCH3 (dimethyl sulfide) –10.0 –0.1 0.4 –9.0

CH2=CHF (vinyl fluoride) –34.3 –0.1 –0.4 –33.2

C2H5Cl (ethyl chloride) –27.4 0.0 0.1 –26.8

CH2=CHCl (vinyl chloride) 5.1 0.0 –0.3 8.9

CH2=CHCN (acrylonitrile) 45.6 –0.1 –0.5 43.2

CH3COCH3 (acetone) –52.2 0.0 –0.3 –51.9

CH3COOH (acetic acid) –103.7 0.0 –0.3 –103.4

CH3COF (acetyl fluoride) –105.3 0.0 –0.4 –105.7

CH3COCl (acetyl chloride) –57.9 0.0 –0.3 –58.0

CH3CH2CH2Cl (propyl chloride) –32.6 0.1 0.1 –31.5

(CH3)2CHOH (isopropanol) –66.5 0.0 0.0 –65.2

C2H5OCH3 (methyl ethyl ether) –53.1 0.0 0.0 –51.7

(CH3)3N (trimethylamine) –6.9 0.0 0.1 –5.7

C4H4O (furan) –7.9 –0.1 –0.5 –8.3

C4H4S (thiophene) 27.3 0.0 –0.4 27.5

C4H5N (pyrrole) 26.1 –0.1 –0.7 25.9

C5H5N (pyridine) 34.4 –0.2 –0.7 33.6

Inorganic hydides

H2 –0.5 0.0 0.1 0.0


134



HS 34.1 0.1 0.3 34.2

Radicals

CCH 138.0 –0.1 –0.6 135.1

C2H3 (2A') 71.9 –0.1 –0.4 71.6

CH3CO (2A') –2.4 0.0 –0.3 –2.5

H2COH (2A) –4.6 0.0 –0.1 –4.1

CH3O (2A') –4.8 0.0 –0.2 5.0

CH3CH2O (2A'') –3.3 –0.1 –0.2 –3.3

CH3S (2A') 29.0 0.0 0.1 29.8

C2H5 (2A') 28.8 0.0 –0.1 28.9

(CH3)2CH (2A') 21.1 –0.1 –0.1 21.5

(CH3)3C (t‐butyl radical) 12.5 0.0 –0.2 12.3

MAD 0.1 0.3

a) Experimental enthalpies of formation were taken from Ref. 206.

5.5. % Disk Space Saved

The average percent disk space saved for each individual truncated computation required

for ccCA(aug) and ccCA(TB) as compared to the full computations required for ccCA are

reported in Table 5.4. Removal of the augmented functions from the hydrogen atoms basis set

provided disk space saving ranging from 4.7 to 21.5 percent for the MP2/aug’‐cc‐pVDZ and

MP2/aug’‐cc‐pVQZ computations, respectively. However, the CCSD(T)/cc‐pVTZ and MP2/cc‐

pVTZ‐DK calculations did not result in any disk space savings within ccCA(aug) because neither

135

of these steps include augmented basis functions. This is unfortunate because the CCSD(T)/cc‐

pVTZ computation is the bottleneck for ccCA and must be reduced to study larger molecules.

Although, if a reduction in the overall computational scaling of ccCA is warranted the use of

ccCA(aug) can reduced the disk space requirements for several steps as compared to ccCA.

When ccCA(TB) is utilized, significant disk space savings as compared to ccCA was

achieved. As shown in Table 5.4, the truncated MP2/aug’‐cc‐pVQZ computation requires on

average 39.8 percent less disk as compared to the full MP2/aug‐cc‐pVQZ. Additionally, the most

expensive step in ccCA is the CCSD(T) step, which required 39.1 percent less disk space for the

ccCA(TB) method as compared to ccCA. As the CCSD(T)/cc‐pVTZ step in ccCA is the bottleneck,

utilization of ccCA(TB) is a viable means to reduce the computational scaling of ccCA, which will

allow larger molecules to be studied.

Table 5.4. Average percent disk space saved for each step in ccCA(aug) and ccCA(TB) as compared to ccCA(full). The overall average percent disk space saved for ccCA(aug) and ccCA(TB) is also reported.

% Disk Space Saved

ccCA(aug)

MP2/aug’‐cc‐pVDZ 4.7

MP2/aug’‐cc‐pVTZ 15.8

MP2/aug’‐cc‐pVQZ 21.5

CCSD(T)/cc‐pVTZ 0.0

MP2(FC1)/aug’‐cc‐pCVTZ 15.4

MP2/cc‐pVTZ‐DK 0.0


136


% Disk Space Saved

ccCA(TB)

MP2/aug’‐cc‐pVDZ 4.7

MP2/aug’‐cc‐pVTZ 23.7

MP2/aug’‐cc‐pVQZ 39.8

CCSD(T)/cc‐pVTZ 39.3

MP2(FC1)/aug’‐cc‐pCVTZ 22.1

MP2/cc‐pVTZ‐DK 9.2

5.6. Conclusions

Basis set truncation schemes were applied to ccCA, and it was shown that removal of the

augmented functions from the correlation consistent basis sets for the hydrogen atom resulted

in only a small impact on the enthalpies of formation computed with ccCA. When angular

momentum functions greater than or equal to d functions were also removed from the

hydrogen basis sets there was a significant effect on the accuracy of ccCA. There error was

shown to be a systematic error based on the number of hydrogen atoms contained in the

system. A correction per hydrogen was developed to account for this systematic basis set error

for each step of ccCA, resulting in the truncated basis set ccCA [ccCA(TB)]. Overall, the MAD for

ccCA(aug) and ccCA(TB) as compared to ccCA was 0.1 and 0.3 kcal/mol for the enthalpies of

formation of the G2/97 test suite of molecules. The ccCA(TB) method was also shown to

provide significant disk space savings as compared to ccCA with average disk space savings of

39.8 and 39.3 percent for the MP2/aug’‐cc‐pVQZ and CCSD(T)/cc‐pVTZ steps, respectively. The

ability to reduce the computational scaling with the use of ccCA(TB) proves to be a useful

137

means to study larger molecules. Overall, both the ccCA(aug) and ccCA(TB) methods were

shown to be viable means to reduce the computational scaling of ccCA, while preserving the

overall accuracy of the composite approach.

138

CHAPTER 6

STRUCTURES AND THERMOCHEMISTRY OF THE ALKALI METAL OXIDE RADICALS, ANIONS, AND

HYDROXIDES: ASSESSING THE VARIANT WnC METHODS

6.1. Introduction

Alkali metal monoxides and hydroxides play an important role in areas ranging from

atmospheric chemistry to high‐temperature combustion.207‐213 For example, the lifetime of

NaO• (A2Σ+) is sufficiently long so as to enable it to serve as the chain carrier for Na (2Π)

production in the mesosphere.214 In terms of high‐temperature combustion, the presence of

alkali metal monoxides and hydroxides (MO• and MOH with M = alkali metal), is a known cause

of corrosion in high‐temperature industrial reactor components, which can lead to industrial

down time.215‐217 These and many other important applications have led to an interest in the

experimental and theoretical characterization of the alkali metal monoxides and hydroxides.

Unfortunately, there is little or no gas‐phase experimental information currently available for

the enthalpies of formation for many of these species, and some of the known values have

large uncertainties, such as the experimental enthalpy of formation for the KO• radical (71.13 ±

42 kJ/mol).218 This limited amount of experimental data has encouraged many computational

studies on the alkali metal monoxides and hydroxides.219‐236

One of the most thoroughly studied and still unresolved problems for the alkali metal

monoxides relates to the ground state of the MO• radicals. Although experimental and

139

theoretical studies have shown that the ground state changes from 2Π to 2Σ+ as the size of the

alkali metal increases,237‐243 agreement has not yet been reached as to where this crossover

occurs. Thus, while both experimental and theoretical studies agree that the ground states of

NaO• and RbO• are 2Π and 2Σ+, respectively,126,219,224,234,239 there is yet no clear consensus about

the ground state of KO•. One of the earliest experimental studies was a magnetic deflection

experiment, which pointed to a 2Σ+ ground state for KO•.238 On the other hand, two other

experiments have suggested that the ground state of KO• is 2Π. These conclusions were based

upon a failure to observe an electron spin resonance (ESR) spectrum, which was suggested to

be consistent with a 2Π state,239 as well as from rotational spectroscopy experiments that

concluded the 2Π state to be 2.4 kJ/mol lower in energy than 2Σ+.244,245

One of the earliest theoretical studies on the alkali metal monoxides was carried out by

Richards and So in 1975.234 They considered NaO•, KO•, and RbO• by means of Hartree‐Fock

calculations and found a 2Π ground state for NaO• and 2Σ+ ground state for KO• and RbO•, in

agreement with the magnetic deflection experiment. In contrast, Allison et al. more recently

reported that the ground state of KO• is 2Π.219 Their conclusion was based on configuration

interaction calculations that included single and double excitations (CISD) and used a valence‐

double‐zeta contraction of a basis set developed by Rappe and Goddard.246 However, it has

been suggested that the basis set used in their study was too small, which could lead to an

incorrect prediction of the ground state.232 Bauschlicher, Partridge, and Dyall also showed that

basis set choice could lead to basis set superposition errors (BSSE) as large as 2–5 kJ/mol for

the 2Π and 2Σ+ states of KO,220 which is greater than the predicted splitting for these states.

140

Allison et al. provided a qualitative explanation for the observed change in the ground

state of the MO• radical as the size of the alkali metal increases,219 based on a balance between

the Pauli repulsive forces and the quadrupole interactions. They noted that the Pauli repulsive

term is smaller for the 2Σ+ state than for the 2Π state, and is proportional to 1/ (where Re is

the M–O bond length). On the other hand, the quadrupole term is repulsive for the 2Σ+ state

but decreases rapidly with increasing metal size because the interactions are proportional to

1/ . Thus, when the metal is small and Re is small, the 2Π state lies lower in energy due to a

dominant repulsive quadrupole term for the 2Σ+ state. However, as the metal increases in size

(leading to a larger Re), the quadrupole interaction term becomes negligible, and the Pauli term,

which favors the 2Σ+ state, dominates.

More recently, Lee et al. carried out a comprehensive study on the KO• radical that

considered the effects of spin‐orbit coupling, core‐valence correlation, and BSSE.228 Coupled

cluster calculations with single, double and quasiperturbative triple excitations [CCSD(T)]

combined with five different basis sets were performed. The largest basis set for K included a

[3s2p] contraction of Huzinaga and Klobukowski’s (24s16p) basis,247 augmented with 9s, 10p,

6d, 5f, 4g, 2h and 2i functions, resulting in a [12s12p6d5f4g2h2i] basis set, while the standard

augmented correlation‐consistent sextuple‐zeta basis set [aug‐cc‐pV6Z] was used for O.110 Lee

et al. concluded that the 2Σ+ state is the ground state for KO• when spin‐orbit coupling is not

considered, and that core‐valence effects and BSSE corrections do not change the ordering of

the 2Π and 2Σ+ states, regardless of the basis set used. A principal objective of their study was

to examine the impact of spin‐orbit coupling, included by means of the Breit‐Pauli operator, on

the potential energy curves for the low‐lying states of KO•.248 They found that when spin‐orbit

coupling is not considered, the 2Π and 1Σ+ potential energy curves cross near their minimum

energies. However, when spin‐orbit coupling is considered, an avoided crossing occurs between

the 2Π and 2Σ+ (actually 2Π1/2 and 2Σ ) potential energy curves. This leads to the lowest

energy state of the KO• radical being the 2Σ state at short R, but the 2Π1/2 state at large R.

The avoided crossing and mixing of the 2Σ and 2Π1/2 states suggest that the use of multi‐

reference methods in the determination of the 2Π and 2Σ+ electronic structures for the KO•

radical may be necessary. Lee et al. suggested the complicated electronic structure for the

lowest energy state as a possible reason why there may be difficulties in interpreting the

microwave spectrum of KO•.

+2/1

+2/1

+2/1

Prior to the study of Lee et al., Bauschlicher et al. examined the ground and low‐lying

states of MO• radicals and MO– anions (where M = Li, Na, and K)221 using multi‐reference

configuration interaction calculations that included single and double excitations, as well as a

size‐consistency correction (MRCI+Q).249 This method was combined with basis sets having the

form of (14s9p5d3f)/[9s7p5d3f], (20s14p3d2f)/[9s7p3d2f], (25s18p4d3f)/[9s9p4d3f], and

(11s6p3d1f)/[5s4p3d1f] for the Li, Na, K, and O atoms, respectively. Bauschlicher et al.

concluded that the ground state of KO• is 2Σ+, with a splitting of 2.2 kJ/mol between the 2Π and

2Σ+ states. Their study also found that the ground state for the anions changes from 3Π to 1Σ+

as the size of the alkali metal increases. While the ground states of LiO– and KO– were found to

be 3Π and 1Σ+, respectively, the study was inconclusive about the ground state of

NaO–.

141

142

These previous studies have shown that the theoretical prediction of energy‐related

properties for the alkali metal monoxides may be problematic and may require the use of multi‐

reference methods in the case of systems that possess large multi‐reference character.

Additionally, the multi‐reference methods should be combined with large basis sets, as small

basis sets were suggested to result in the incorrect ground state for the KO• radical.220

Unfortunately, the use of multi‐reference methods with large basis sets can become

computationally very demanding in terms of memory, disk space, and CPU time requirements.

Therefore, we have investigated the use of high‐level composite approaches in the present

study.

The philosophy of the composite methods is to combine a series of results from

computationally simpler ab initio methods in order to approximate results for a more accurate,

but computationally more expensive approach. There are several types of composite methods

in current use, including the Gaussian approaches of Pople, Curtiss, Raghavachari and co‐

workers (e.g.,G1, G2, G3, G3X and G4),59‐77,197 the complete‐basis‐set methods of Petersson and

co‐workers (e.g., CBS‐QB3),87‐94 the Weizmann methods of Martin and co‐workers [W1, W2, W3

and W4],78‐83 the HEAT method of Stanton and co‐workers,84‐86 and the recent correlation‐

consistent composite approach of Wilson and co‐workers (ccCA).95‐101 The present study

investigates the use of several modified versions of the Wn methods, which were chosen

because of their ability to achieve accuracies of ~1 kJ/mol for thermochemical properties such

as enthalpies of formation, ionization energies, electron affinities and proton affinities.81,83

Additionally, the Wn methods have been adapted to incorporate multi‐reference

143

procedures,250‐252 which may be necessary for some of the systems that will be studied, such as

the KO• radical.228 A detailed discussion of the Wn methods follows in Section 6.2.1.

In previous studies that used composite methods for alkali metal and alkaline earth metal

compounds, it has been found that standard implementations are sometimes not sufficient.

For example, Schulz et al. found that the standard G2 method performed poorly in determining

thermochemical properties for the alkali metal and alkaline earth metal oxides and hydroxides

(M2O, MOH where M = Li, Na, and K; M'O, M'(OH)2 where M' = Be, Mg, and Ca), with errors

greater than 100 kJ/mol.177 They found that by extending the correlation space, using CCSD(T)

instead of QCISD(T), and performing CCSD(T)/6‐311+G(3df,2p) calculations without the additive

approximations, the large errors obtained in the G2 calculations no longer occur. However,

there were still several calculated values that deviated from experiment by 10–30 kJ/mol, such

as the enthalpies of formation for CaO, Be(OH)2, Mg(OH)2, Ca(OH)2, and K2O.

Sullivan et al. explored variants of the G3 and Wn methods.129 They concluded that the

accuracy of the G3 method was optimal when a relaxed‐inner‐valence correlation space for the

metal and a relaxed‐valence correlation space for the oxygen (riv,rv) was used, as found earlier

by Schultz et al. for G2.177 The (riv,rv) active space is shown in Table 6.1. The (riv,rv) correlation

space, however, was found to lead to spectacular failures for the Wn methods. One notable

example is the enthalpy of formation for NaOH, which is listed as –197.76 kJ/mol in the JANAF

tables.218 Using the standard correlation space, the W1 method gave a value of

–187.7 kJ/mol, but the W1 method using (riv,rv) resulted in a value of –286.9 kJ/mol. The

failures in the Wn methods when combined with the (riv,rv) correlation space were found to be

144

associated with the use of the standard Dunning correlation‐consistent basis sets (cc‐pVnZ

where n = D(2), T(3), etc.).105,107,109‐112 These basis sets were not designed to describe the inner‐

valence region that was being relaxed. To compensate for this need, Sullivan et al. used

core‐valence basis sets, cc‐pWCVnZ (denoted WCVnZ, where the W stands for Weizmann, to

distinguish them from the existing cc‐pCVnZ113,114 and cc‐pwCVnZ253 basis sets), developed by

Martin,254 in modified Wn methods denoted WnC. The WnC methods, which use the (riv,rv)

correlation space, showed significant improvements over the Wn (riv,rv) methods. For example,

W1C predicts an enthalpy of formation of –189.6 kJ/mol for NaOH,129 which is a significant

improvement over the W1 (riv,rv) value of –286.9 kJ/mol.

Table 6.1. Orbitals included in the correlation space defined as relaxed valence (rv) for non‐metals and relaxed inner valence (riv) for alkali metals.a

Atom Frozen active

rv: 1s 2s 2p

O 1s

H

riv:

Li 1s 2s 2p

Na 1s 2s 2p 3s 3p

K 1s 2s 2p 3s 3p 4s 4p a) Correlation space used in

Ref. 177 and 129

In the present study, the performance of the WnC methods is assessed for describing

alkali metal monoxide radicals (MO•), anions (MO–) and hydroxides (MOH). The utility of these

145

methods has been compared with prior experiment and theoretical results, and the structures

and molecular properties of MO•, MO– and MOH (M = Li, Na, and K) have been examined.

Specifically, this study investigated (a) the enthalpies of formation for all of these species, (b)

the state splitting for MO• and MO–, (c) the electron affinities for MO•, and (d) the bond

dissociation energies and gas‐phase acidities for MOH.

6.2. Methodology

The W1 and W2 methods81,83 use extrapolation procedures to estimate the CCSD(T)

complete‐basis‐set (CBS) limit in a cost‐effective way. The CCSD(T) CBS limit is then used as a

reference energy to which corrections are added to provide the final W1 and W2 total energies.

These corrections include core correlation, scalar relativistic effects, first‐order spin‐orbit

coupling, and zero‐point vibrational energies. While details of the W1 and W2 methods have

been described previously,81,83 the methods and their variants are summarized briefly below for

completeness.

6.2.1. W1 and W2 Methods

The standard procedures use B3LYP/cc‐pVTZ+1 and CCSD(T)/cc‐pVQZ+1 optimized

geometries for the W1 and W2 procedures, respectively, where the +1 indicates that an

additional tight d function has been added to the Dunning’s cc‐pVnZ basis set for second row

atoms. It should be noted that the +1 basis sets are different from the tight d correlation

consistent basis set [cc‐pV(n+d)Z] developed by Dunning, Peterson, and Wilson for second row

atoms.105 While the use of the Dunning basis sets is highly recommended due to their careful,

146

systematic development and well‐established behavior, Martin’s basis sets were utilized to

maintain consistency with Sullivan’s previous study.129

CCSD single point calculations are then performed using the above geometries combined

with the aug'‐cc‐pV5Z+2d1f basis set, and CCSD(T) calculations are performed with the aug'‐cc‐

pVDZ+2d and aug'‐cc‐pVnZ+2d1f [where n = T(3), Q(4)] basis sets. Again, the +2d and +2d1f

indicate that there are high exponent d and f functions added to the basis set for second row

atoms. The aug' prefix indicates that diffuse functions are not included on hydrogen, Group I, or

Group II metal atoms. The CCSD correlation and triple‐excitation (T) components of the CCSD(T)

total energy and the SCF total energy are all separately extrapolated to obtain CBS limits for

each component.

Both the W1 and W2 methods determine the CBS limit of the SCF component by using a

two‐point extrapolation formula,83

El E∞ B/lmax5 6.1

for the extrapolation of the SCF total energy, where El is the energy of a given basis set [i.e. DZ,

TZ, etc.], lmax is the maximum angular momentum function for a given basis set [d = 2, f = 3, g =

4, etc. for main group atoms], E∞ is the energy at the CBS limit, and B is a parameter

determined during the fit. However, these two methods differ in that the W1 method

determines the SCF energies using the aug'‐cc‐pVTZ+2d1f and aug'‐cc‐pVQZ+2d1f basis sets,

while the W2 method determines the SCF energies using the aug'‐cc‐pVQZ+2d1f and aug'‐cc‐

pV5Z+2d1f basis sets.

The CCSD contribution to the total energy is extrapolated using the formula,83

147

E∞ El El‐El‐1 / lmax

lmax‐1

α

‐ 1 6.2

[where α=3 and 3.22 for the W1 and W2 methods, respectively]. The CCSD contribution to the

energy is found by subtracting the SCF energy from the CCSD energy, and the CCSD

components used in the extrapolation for W1 came from using the aug'‐cc‐pVTZ+2d1f and aug'‐

cc‐pVQZ+2d1f basis sets while the W2 used the aug'‐cc‐pVQZ+2d1f and aug'‐cc‐pV5Z+2d1f basis

sets. Lastly, the (T) contribution is extrapolated using Eqn. 6.2, but the energies used in the (T)

extrapolation were determined with the smaller aug'‐cc‐pVDZ+2d and aug'‐cc‐pVTZ+2d1f basis

sets for the W1 method and aug'‐cc‐pVTZ+2d1f and aug'‐cc‐pVQZ+2d1f basis sets for the W2

method.

Once the CCSD(T) CBS limit is determined, core correlation is then added by means of

taking the difference in CCSD(T)/MTsmall calculations with and without the core electrons

frozen. The MTsmall core correlation basis sets81 consists of Dunning’s uncontracted triple‐ζ

basis set and includes 2d and 1f core functions that are derived from the highest exponent d

and f angular momentum functions, successively multiplied by 3. Lastly, scalar relativistic

corrections are added which are obtained as the ACPF expectation values of the first‐order

Darwin and mass‐velocity operators.255,256 However, the 1s orbital on second‐ and third‐row

atoms are held frozen for all core correlation and scalar relativistic calculations.

6.2.2. W1C and W2C Methods

As noted above, earlier work by Sullivan et al. found that relaxation of the inner‐valence

space on the alkali metals could lead to large errors for the standard Wn methods.129 These

errors were attributed to deficiencies in the cc‐pVnZ basis set description of the core region, so

148

the cc‐pWCVnZ core‐valence basis sets, which help alleviate this deficiency, were developed by

Martin.254 This modification to the Wn methods has been incorporated in the WnC methods.

The W1C procedure uses the same geometries as standard W1 [B3LYP/cc‐pVTZ+1] for

first‐ and second‐row systems, while the geometries for third‐row systems were calculated

using B3LYP/A'WCVTZ, where A'WCVnZ represents the aug'‐cc‐pWCVnZ basis set. Here, aug' is

again used to denote that only the oxygen atom is augmented with a set of diffuse functions.

The W2C procedure calculates all geometries using the A'WCVQZ basis sets for first‐, second‐,

and third‐row systems.

The current study has altered the WnC methods slightly from the previous work by using

more accurate geometries. B3LYP/A'WCV5Z geometries are used for the W1C method, while

CCSD(T)/A'WCV5Z and CCSD(T)/AA'WCV5Z geometries are used for the W2C method, where

AA'WCVnZ denotes the aug‐aug'‐cc‐pWCVnZ basis set, which includes a set of diffuse functions

on the alkali metal. The correlation space used for all of the WnC methods is (riv,rv).

6.2.3. Multi‐Reference W2C Methods

The earlier work of Sullivan et al. obtained the T1 diagnostic231 and the percent SCF

contribution to the total atomization energy,252 to give an indication of the extent of multi‐

reference character. They found that several of the alkali metal oxides indeed possess values

for these indicators that suggest significant multi‐reference character [multi‐reference

character is discussed in Chapter 7], and so they used the multi‐reference versions of the W2C

procedure.250‐252 Therefore, the multi‐reference methods are also utilized in this study.

Specifically, the single‐reference CCSD(T) calculations of W2C are replaced in the multi‐

149

reference variants by averaged coupled‐pair functional (ACPF)257,258 or averaged quadratic

coupled cluster (AQCC)259‐261 calculations. The resultant methods are denoted W2C‐CAS‐ACPF

and W2C‐CAS‐AQCC, respectively.129

For the present work, all of the multi‐reference W2C calculations are based on geometries

calculated using the multi‐reference configuration interaction method including single and

double excitations and a size‐consistency correction [MRCI+Q], combined with the A'WCV5Z

and AA'WCV5Z basis sets, resulting in four varieties of multi‐reference W2C methods, namely

W2C‐CAS‐ACPF//A'WCV5Z, W2C‐CAS‐ACPF//AA'WCV5Z, W2C‐CAS‐AQCC//A'WCV5Z, and

W2C‐CAS‐ACPF//AA'WCV5Z, where //A'WCV5Z and //AA'WCV5Z indicate the use of geometries

optimized at the MRCI+Q level using these basis sets.

All of the multi‐reference methods used in the W2C methods include a correction for size

inconsistency; size inconsistency can arise from the use of a truncated configuration interaction

method [i.e. CISD].262 The ACPF and AQCC methods include a size consistency correction to the

Hamiltonian, while the MRCI+Q calculation includes an a posteriori correction. It has been

shown that, although the ACPF and AQCC methods are not strictly size‐consistent, the error due

to size consistency is negligible for practical applications.261 For all of the multi‐reference

calculations, the (riv,rv) correlation space has been used.

6.2.4. Geometries and Energies

Geometries were determined by first calculating the potential energy curves for each of

the states of MO– and MO• (where M = Li, Na, and K) using the following levels of theory:

B3LYP/A'WCV5Z, CCSD(T)/A'WCV5Z, CCSD(T)/AA'WCV5Z, MRCI+Q/A'WCV5Z and

150

MRCI+Q/AA'WCV5Z. A standard Dunham analysis263 was then performed on a series of nine

points with bond distances ranging from 1.06–2.12 Å, 1.59–2.96 Å, and 1.85–3.18 Å for LiO• and

LiO–, NaO• and NaO–, and KO• and KO–, respectively, separated by 0.05 Å around the minima of

the potential energy curves. A seventh‐order polynomial was used to fit the potential energy

curve to determine the optimal bond distances for the MO– and MO• species. For the alkali

metal hydroxides (MOH), standard geometry optimizations were performed using the above

levels of theory. Frequency calculations were carried out using B3LYP/A'WCV5Z to obtain zero‐

point vibrational energies (ZPVE) and enthalpy temperature corrections (ΔΔHf) for the

determination of the enthalpies of formation at 298 K (ΔHf°). The vibrational frequencies were

scaled by 0.985 in accordance with the standard W1 and W2 procedures.81,83 Enthalpies of

formation were obtained using the atomization method, which is discussed in detail by

Nicolaides et al.264 All calculations were carried out with the Gaussian or Molpro program

packages.170,200

6.3. Results and Discussion

6.3.1. Extent of Multi‐Reference Character

The T1 diagnostic and the percent SCF contribution to the total atomization energy are

presented in Table 6.2. A value greater than 0.02 for the T1 diagnostic or a percent SCF

contribution to the total atomization energy that is smaller than 30 percent has previously been

shown to be indicators that single reference methods could fail due to high multi‐reference

character. Looking at Table 6.2, there are several states for which the T1 diagnostic is slightly

151

larger than 0.02. Some examples include the 3Π and 3Σ+ states for LiO– and KO– for which the T1

diagnostic is 0.021, 0.022, 0.023, and 0.023, respectively. However, the percent SCF

contribution to the total atomization energy for these states is well above 30 percent with 39.8

percent being the lowest, which was determined for the 3Σ+ KO– state with the

CCSD(T)/A'WCV5Z geometry. The most significant multi‐reference character is seen for the 1Σ+

state for LiO–, NaO–, and KO–, for which the percent SCF contribution to the total energy was

found to be –11.5, –60.4, and –54.9 percent, respectively, utilizing CCSD(T)/AA'WCV5Z. It

should be noted, that the T1 diagnostic and the percent SCF contribution to the total

atomization energy presented in Table 6.2 are indications that multi‐reference methods may be

needed, but they do not guarantee that single reference methods will fail. By far, the T1

diagnostic of 0.150 and a percent SCF contribution of –51.7 percent for the 1Σ+ state of KO–

using CCSD(T)/A'WCV5Z suggest that multi‐reference methods should be considered.

152

Table 6.2. T1 Diagnostic and Percent SCF Contribution to the Total Atomization Energy (kJ/mol) for various geometries.

Molecule State % SCF Contribution T1

CCSD(T)/

A'WCV5Z aCCSD(T)/

A'WCV5Z bCCSD(T)/

AA'WCV5Z c CCSD(T)/

A'WCV5Z a

LiO– 1Π 57.6 57.5 58.4 0.016 1Σ+ –11.3 –11.5 –11.5 0.067 3Π 53.9 54.0 54.6 0.021 3Σ+ 48.6 48.7 49.4 0.022

LiO• 2Π 51.2 51.1 51.1 0.018 2Σ+ 45.6 45.5 45.8 0.019

LiOH 65.8 65.8 65.9 0.012

NaO– 1Π 47.6 47.5 47.9 0.016 1Σ+ –60.1 –60.6 –60.4 0.045 3Π 43.5 43.2 43.7 0.020 3Σ+ 39.4 39.1 39.6 0.019

NaO• 2Π 34.9 35.0 35.0 0.018 2Σ+ 29.2 29.3 29.3 0.016

NaOH 61.2 61.3 61.3 0.011

KO– 1Π 43.8 45.7 47.5 0.019 1Σ+ –57.4 –51.7 –54.9 0.150 3Π 42.1 41.4 43.3 0.023 3Σ+ 40.6 39.8 41.8 0.023

KO• 2Π 38.1 37.9 37.9 0.017 2Σ+ 36.7 36.4 36.4 0.013

KOH 62.0 61.8 61.8 0.012 a) Geometries computed with B3LYP/A'WCV5Z b) Geometries computed with CCSD(T)/A'WCV5Z c) Geometries computed with CCSD(T)/AA'WCV5Z

153

6.3.2. Geometries

Optimized structures for the metal oxides and hydroxides are presented in Table 6.3. As

previously mentioned, there is only a small amount of experimental data available for these

systems, but the available data is given for comparison. Bond lengths calculated at the B3LYP,

CCSD(T) and MRCI+Q levels using the A'WCV5Z basis set are generally quite similar, all lying

within ~0.01 Å of each other. However, the bond lengths calculated with B3LYP for the 1Σ+

state of the anions (MO–) show quite large deviations from results obtained with CCSD(T) and

MRCI+Q, which is due to the large multi‐reference character of the 1Σ+ state. Thus, the B3LYP

metal–oxygen bond distances differ from MRCI+Q values by 0.081, 0.040, and 0.074 Å for LiO–,

NaO–, and KO–, respectively. Overall, the mean absolute deviations (MAD) for the

B3LYP/A'WCV5Z geometries, as compared with CCSD(T) and MRCI+Q were both found to be

0.017 Å. It should be noted that the MADs are significantly reduced when the 1Σ+ state is

omitted from the analysis, with MAD values of 0.006Å as compared with both CCSD(T) and

MRCI+Q.

CCSD(T) performed well for the geometries of species for which multi‐reference character

was negligible. However, CCSD(T) showed a large deviation for the 1Σ+ state due to the large

multi‐reference character for this state. In comparing the performance of CCSD(T)/ A'WCV5Z

and CCSD(T)/AA'WCV5Z, we generally find differences in bond lengths that are less than 0.001

Å. However, this is not the case for the KO– anion, for which the AA'WCV5Z basis set gives bond

lengths that are shorter by 0.018, 0.039, 0.017, and 0.018 Å for the 1Π, 1Σ+, 3Π, and 3Σ+ states,

respectively. Overall, the MADs for the CCSD(T) bond lengths, as compared with the MRCI+Q

154

computations, are 0.022 Å for both the A'WCV5Z and AA'WCV5Z basis sets. When the 1Σ+ state

is omitted, the MAD for the CCSD(T)/A'WCV5Z geometries is 0.006 as compared with

MRCI+Q/A'WCV5Z geometries. When the AA'WCV5Z basis set is used, the corresponding MAD

is 0.002 Å.

155

Table 6.3. Optimized Geometries (Å) for Alkali Metal Monoxide Anions (MO–), Radicals (MO•) and Hydroxides (MOH).

Molecule State B3LYP/ A'WCV5Z

MRCI+Q/ A'WCV5Z

MRCI+Q/ AA'WCV5Z

CCSD(T)/ A'WCV5Z

CCSD(T)/ AA'WCV5Z

Expt.

LiO– 1Π 1.752 1.759 1.749 1.755 1.746

1Σ+ 1.606 1.697 1.687 1.647 1.648

3Π 1.752 1.752 1.747 1.755 1.746

3Σ+ 1.667 1.670 1.660 1.671 1.659

LiO• 2Π 1.684 1.689 1.689 1.689 1.689 1.688d

2Σ+ 1.588 1.591 1.591 1.591 1.591 1.599a

LiOH r(Li–O) 1.580 1.580 1.580 1.580 1.580 1.5816(10)b

r(O–H) 0.948 0.949 0.948 0.949 0.949 0.9691(21)b

NaO– 1Π 2.152 2.142 2.136 2.138 2.134

1Σ+ 2.003 2.045 2.043 1.979 1.977

3Π 2.153 2.144 2.140 2.140 2.134

3Σ+ 2.063 2.048 2.043 2.050 2.044

NaO• 2Π 2.060 2.053 2.054 2.054 2.054 2.052c

2Σ+ 1.956 1.952 1.952 1.952 1.952 1.95 a

NaOH r(Na–O) 1.940 1.940 1.939 1.940 1.939 1.95(2)b

r(O–H) 0.951 0.951 0.952 0.951 0.952

KO– 1Π 2.441 2.443 2.422 2.437 2.420


156


Molecule State B3LYP/ A'WCV5Z

MRCI+Q/ A'WCV5Z

MRCI+Q/ AA'WCV5Z

CCSD(T)/ A'WCV5Z

CCSD(T)/ AA'WCV5Z

Expt.

KO– 3Π 2.444 2.440 2.417 2.438 2.421

3Σ+ 2.288 2.284 2.265 2.281 2.263

KO• 2Π 2.322 2.326 2.326 2.324 2.324 2.321c

2Σ+ 2.168 2.174 2.174 2.171 2.171 2.168c

KOH r(M‐O) 2.203 2.205 2.202 2.202 2.202 2.196(3)b

r(O‐H) 0.955 0.951 0.955 0.955 0.955 0.960(10)b a) From Ref. 205 b) From Ref. 265 c) From Ref. 126 d) From Ref. 127 e) From Ref. 266

157

6.3.3. Enthalpies of Formation

The calculated and experimental enthalpies of formation for the alkali metal hydroxides

are displayed in Table 6.4. In some cases, the experimental values show significant variation

among the different experimental studies and some of these studies have significant

uncertainties. On the basis of high‐level theoretical calculations, Sullivan et al. recommended

values of –239 ± 5, –189 ± 5, and –223 ± 5 kJ/mol for LiOH, NaOH, and KOH, respectively,

calculated as a weighted average of their two best theoretical predictions [W2C//ACQ and

G3[CC](dir,full)].129 The enthalpies of formation found in the present study for the alkali metal

hydroxides lie within these uncertainties with the exception of the W2C‐CAS‐AQCC results for

LiOH, NaOH, and KOH, which are 2.1, 1.0, and 2.0 kJ/mol lower in energy than the values

computed by Sullivan et al., respectively.129

For the present theoretical predictions, a similar approach to Sullivan et al. is taken to

obtained values by averaging results at the three highest levels of theory, namely W2C,

W2C‐CAS‐ACPF, and W2C‐CAS‐AQCC using geometries optimized at the CCSD(T) or MRCI+Q

level with the AA'WCV5Z basis set. The one exception is for the 1Σ+ state, which has been taken

as the average of the W2C‐CAS‐ACPF and W2C‐CAS‐AQCC results (again obtained using

MRCI+Q/AA'WCV5Z optimized geometries). The single‐reference W2C value was omitted in this

case due to the large multi‐reference character in the 1Σ+ state of the MO– anions as reflected,

for example, in the ~20–30 kJ/mol difference between the single‐reference and multi‐reference

W2C results for the enthalpy of formation of the KO– 1Σ+ state.

158

Comparison of the averaged enthalpies of formation with the experimental values shows

that for the 2Π state of NaO• our value of 86.4 kJ/mol lies close to one of the experimental

values (87.4 kJ/mol).218,267 For KO•, the computed enthalpy of formation of 57.0 kJ/mol lies

within 5 kJ/mol of Pedley and Marshall’s value of 59.86 ± 4.16 kJ/mol as well as Lamoreaux and

Hildenbrands value of 61 ± 21 kJ/mol.268,269 It is interesting to note that JANAF lists a value for

the enthalpy of formation for KO• of 71.13 ± 41.9 kJ/mol,218 which is more than 10 kJ/mol

above the computed value (57.0 kJ/mol), and carries a large uncertainty. It seems questionable

whether this is the best choice among the experimental values due to the large uncertainty

associated with the experiment. The deviation between the averaged enthalpy of formation

and the experimental values is largest for the LiO• radical, with all of the experimental values

lying 20–30 kJ/mol higher than the recommended values in Table 6.4. This large deviation

between theory and experiment was previously observed in the determination of the

dissociation energy for the LiO• 2Π radical. Specifically, Lee et al. computed a dissociation

energy of 355.06 kJ/mol by means of a RCCSD(T) computation utilizing a quintuple‐ζ quality

basis set.227 Experimentally, the dissociation energy for LiO• is 336.73 kJ/mol, which was

determined by Hildenbrand.270 This dissociation energy was determined from the enthalpy of

formation that Hildenbrand found for LiO• (69.01 kJ/mol) and is 20‐30 kJ/mol lower in energy

than the enthalpies of formation computed in the current study. The difference between

experiment and theory for the LiO• radical was previously observed by Langhoff who noted that

the experimentally derived 2Π–2Σ+ energy separation (30.68 kJ/mol)126,245 is similar to the

difference between the theoretically and experimentally determined enthalpy of formation.

159

Langhoff suggested the excited A2Σ+ state and not the ground X2Π state was prepared during

the experiment that determined the enthalpy of formation for the LiO• radical.224,225 It is

interesting to note that the recommendation for the A2Σ+ enthalpy of formation (79 kJ/mol) is

within 5 kJ/mol of the experimental enthalpy of formation determined for the X2Π state

(84.10±20.9 kJ/mol).218 It seems reasonable to suggest that the experimental enthalpy of

formation for the LiO X2Π radical may need to be re‐examined.

160

Table 6.4. Enthalpies of Formation (298 K, kJ/mol) for Alkali Metal Monoxide Anions (MO–), Radicals (MO•) and Hydroxides (MOH).

Molecule State W1C//

A'WCV5Z W2C//

A'WCV5Z W2C//

AA'WCV5Z

W2C‐CAS‐ACPF// A'WCV5Z

W2C‐CAS‐ACPF//

AA'WCV5Z

W2C‐CAS‐AQCC// A'WCV5Z

W2C‐CAS‐AQCC//

AA'WCV5ZAverage Experiment

LiO– 1Π 18.7 18.2 16.2 13.1 9.1 12.8 8.9 11.4

1Σ+ 19.2 20.8 20.6 21.0 20.1 24.5 23.7 21.9

3Π 15.0 15.1 11.5 10.9 7.2 10.6 6.8 8.5

3Σ+ 48.0 48.2 44.4 44.3 40.6 43.6 39.8 41.6

LiO• 2Π 51.8 52.4 52.5 47.7 47.7 45.3 45.4 48.5 84.1 0 ± 20.9e 75.31 ± 8.37h 69.01 ± 4.16g

2Σ+ 81.7 82.3 84.6 77.8 77.8 75.5 75.5 79.3

LiOH –241.8 –240.3 –240.2 –244.0 –244.0 –246.1 –246.1 –243.4

–234.30 ± 6.3e –238.1 ±

6i –247.0 ± 3d (–239±5j)

NaO– 1Π 38.4 40.3 38.8 36.7 35.0 37.6 35.9 36.6


161



A'WCV5Z W2C//

A'WCV5Z W2C//

AA'WCV5Z


W2C‐CAS‐ACPF//

AA'WCV5Z


W2C‐CAS‐AQCC//


NaO– 1Σ+ 40.3 42.7 42.1 43.0 42.6 44.0 43.5 43.0

3Π 36.4 37.5 35.9 34.5 33.0 35.2 33.7 34.2

3Σ+ 58.2 59.4 58.0 56.6 55.3 57.0 55.6 56.3

NaO• 2Π 89.3 89.3 89.3 85.7 85.7 84.2 84.2 86.4 83.68e 87.4 ± 4a

2Σ+ 113.5 113.7 113.8 109.8 109.8 108.1 108.1 110.6

NaOH –189.6 –188.1 –188.0 –191.4 –191.3 –193.0 –192.9 –190.8

–197.76 ± 12.6e –191 ± 8i

–186 ± 10c –193 ± 10c (–189 ± 5j)

KO– 1Π 28.1 28.6 23.1 24.1 18.6 24.7 18.9 20.2

1Σ+ –59.8 –107.6 –95.3 19.8 15.3 19.7 15.0 15.1

3Π 26.1 27.2 21.0 23.0 17.6 23.0 17.4 18.7


162



A'WCV5Z W2C//

A'WCV5Z W2C//

AA'WCV5Z


W2C‐CAS‐ACPF//

AA'WCV5Z


W2C‐CAS‐AQCC//


KO– 3Σ+ 27.9 29.2 33.8 25.6 19.5 25.5 19.1 24.1

KO• 2Π 61.6 61.1 61.2 56.2 56.3 53.6 53.7 57.0

71.13 ± 41.9e 65.27 ± 12.55b

61 ± 21h 59.86 ± 4.16g

2Σ+ 59.7 59.4 59.5 55.4 55.5 52.6 52.7 55.9

KOH –226.8 –223.5 –223.4 –227.6 –227.5 –230.1 –230.0 –227.0

–232.0 ± 3e

–231.0c (–223 ± 5j) –229.0 ± 4d –228 ± 5e

a) From Ref. 267 e) From Ref. 218 i) From Ref. 271 b) From Ref. 272 f) From Ref. 270 j) From Ref. Recommended theoretical values c) From Ref. 215 g) From Ref. 268 values from Ref. 129 in parenthesis d) From Ref. 273 h) From Ref. 269

163

6.3.4. Predicted Ground States for MO– and MO•.

Table 6.4 also provides the ordering of the various states. For the radicals, the 2Π state

was both theoretically and experimentally determined to be the ground state for LiO• and

NaO•.237‐243,274 The results presented in Table 6.4 agree with these previous studies with the 2Π

state being 30 and 25 kJ/mol lower in energy than the 2Σ+ state for LiO• and NaO•, respectively.

The KO• ground state, however, has not been resolved. This study finds the 2Σ+ state to be only

1 kJ/mol lower in energy than the 2Π state, which agrees with previous theoretical studies.220,221

To date, there is much less known about the ordering of the metal mono‐oxide anions

(MO–). One of the most detailed studies on these systems is a previous MRCI+Q study

performed by Bauschlicher,221 and the anionic state orderings presented in Table 6.4 are in

agreement with this previous study. For LiO–, the order of the anionic state were determined to

be 3Π < 1Π < 1Σ+ < 3Σ+. For NaO–, Bauschlicher et al. stated that the 3Π and 1Π states were too

close in energy to definitively distinguish which of the two is the ground state. This study also

indicated that NaO– has the same state ordering as LiO–, which is 3Π < 1Π < 1Σ+ < 3Σ+. The

ordering of these states, however, changes for KO– with the 1Σ+ state being the lowest in

energy. It should be noted that the single reference WnC methods fail for the 1Σ+ state due to

the large multi‐reference character of this state with all of the WnC methods predicting a large

negative enthalpy of formation. The multi‐reference methods, however, agree with the

ordering of the anionic states that were previously determined by Bauschlicher,221 which is 1Σ+

< 3Π < 1Π < 3Σ+. It is interesting to note that the lowest and highest KO– are only separated by a

164

5 kJ/mol energy difference, which would make it difficult, both theoretically and

experimentally, to distinguish one electronic state from another.

6.3.5. State Splitting

Splitting between the low‐lying states of the MO• radicals and MO– anions, which have

been calculated with the various WnC and W2C‐CAS methods, have been compared with

available experimental values in Table 6.5. For LiO• and NaO•, all of the computed splitting are

within 1‐2 kJ/mol of the experimental values. Also, all of the methods used to compute the 3Π

→ 3Σ+ state splitting for LiO–, NaO–, and KO– agree to within 1 kJ/mol. For example, the smallest

triplet state splitting for LiO– was 32.98 kJ/mol computed with W2C‐CAS‐AQCC//AA'WCV5Z,

while the largest triplet state splitting was 33.44 kJ/mol computed with

W2C‐CAS‐ACPF//A'WCV5Z. The singlet state splitting, however, shows a much larger variation

between the different methods. For example, the singlet state splitting for LiO– varied from

0.62 to 11.89 kJ/mol for the W1C//A'WCV5Z and W2C‐CAS‐AQCC//A'WCV5Z methods,

respectively.

For the KO• radical, the state splitting determined using the single‐reference WnC

methods resulted in the 2Σ+ state, lying 1.61 to 1.89 kJ/mol lower in energy than the 2Π. The

multi‐reference methods computed smaller state splitting with values ranging from –0.70 to

–0.97 kJ/mol. Again, the general conclusion of this study is that the 2Σ+ and 2Π states of KO• lie

very close in energy. The average of our W2C, W2C‐CAS‐ACPF, and W2C‐CAS‐AQCC results,

determined using geometries optimized with CCSD(T)/AA'WCV5Z or MRCI+Q/AA'WCV5Z,

indicates that the 2Σ+ state lies lower in energy than the 2Π state by 1.0 kJ/mol. This state

165

splitting for KO• is slightly smaller than that reported in previous theoretical studies (e.g. 2.39

kJ/mol and 2.20 kJ/mol)228,229

Table 6.5. Splitting of States (0 K, kJ/mol) for Alkali Metal Monoxide Anions (MO–) and Radicals (MO•).


A'WCV5ZW2C//

A'WCV5ZW2C//

AA'WCV5Z

W2C‐CAS‐ACPF//

A'WCV5Z

W2C‐CAS‐ACPF//

AA'WCV5Z

W2C‐CAS‐AQCC//

A'WCV5Z

W2C‐CAS‐AQCC//

AA'WCV5ZExptl

LiO– 1Π→

1Σ+ 0.62 2.75 4.57 8.09 11.11 11.89 15.05

3Π→

3Σ+ 33.05 33.07 33.00 33.44 33.41 33.03 32.98

LiO• 2Π→

2Σ+ 29.95 29.92 32.13 30.16 30.15 30.20 30.19 30.68a

NaO– 1Π→

1Σ+ 2.12 2.61 3.54 6.14 7.36 6.28 7.47

3Π→

3Σ+ 21.72 21.91 22.07 22.19 22.37 21.77 21.94

2Π→

2Σ+ NaO• 24.31 24.48 24.51 24.20 24.22 23.91 23.93 24.52a

KO– 1Π→

1Σ+ –88.24 –130.04 –112.18 –4.09 –3.04 –4.73 –3.61

3Π→

3Σ+ 1.85 2.20 2.04 2.70

166

a) From Ref. 126 and 127 b) From Ref. 244

1.93 2.63 1.80

KO• 2Π→

2Σ+ –1.89 –1.62 –1.61 –0.72 –0.70 –0.97 –0.92 2.39b

167

6.3.6. Electron Affinities

The calculated electron affinities of the metal monoxide radicals MO• [i.e. the negative of

the enthalpy change for the process MO• + e– → MO–] are presented in Table 6.6. As noted in

Section 6.3.4, the ground states of LiO• and NaO• have been theoretically and experimentally

determined to be 2Π,237‐243,274 so the electron affinities in Table 6.6 were calculated using the 2Π

state as the radical state. However, the ground state for the KO• radical has yet to be resolved,

so Table 6.6 reports the electron affinities calculated with the 2Π state and with the 2Σ+ state as

the radical.

The WnC methods generally agree to within ~ 6 kJ/mol of each other. For LiO•, the

smallest electron affinity for the 3Π state was computed to be 34.9 kJ/mol for

W2C‐CAS‐AQCC//A'WCV5Z, while the largest value was determined to be 41.1 kJ/mol for

W2C//AA'WCV5Z. Unfortunately, there are few prior studies of electron affinities for the alkali

monoxides with which to compare. However, one previous theoretical study determined an

electron affinity for LiO• to be 42.45 kJ/mol by means of quadratic configuration interaction

including single, double and iterative triples excitations [QCISD(T)] combined with the Pople 6‐

311+G(2df) basis set, which compares well with the values presented in Table 6.6.221 The

difference between the NaO• electron affinities computed with the various WnC method was

found to be 4.4 kJ/mol, which was slightly smaller than 6.2 kJ/mol difference found for LiO•.

168

Table 6.6. Electron Affinities (0 K, kJ/mol) for the Metal Monoxide Radicals (MO•).

Radical MO• MO– W1C// A'WCV5Z

W2C// A'WCV5Z

W2C// AA'WCV5Z

W2C-CAS-ACPF// A'WCV5Z

W2C-CAS-ACPF// AA'WCV5Z

W2C-CAS-AQCC// A'WCV5Z

W2C-CAS-AQCC// AA'WCV5Z

LiO• 2Π → 1Π 33.1 34.3 36.4 34.7 38.7 32.6 36.6 1Σ 32.5 31.6 31.8 26.6 27.6 20.7 21.6 3Π 36.9 37.4 41.1 36.9 40.6 34.9 38.7 3Σ 3.8 4.4 8.1 3.4 7.2 1.9 5.7 NaO• 2Π → 1Π 51.1 49.2 50.7 48.7 50.4 46.5 48.2 1Σ 48.9 46.6 47.2 42.6 43.1 40.2 40.7 3Π 53.1 52.1 53.6 51.4 52.9 49.2 50.8 3Σ 31.3 30.2 31.6 29.2 30.6 27.5 28.8

KO• 2Π → 1Π 33.7 39.2 44.7 32.2 37.9 29.1 35.1 1Σ 121.4 168.7 156.4 36.3 41.0 33.9 38.7 3Π 35.7 34.3 40.3 33.4 38.9 30.9 34.1 3Σ 33.9 32.1 38.3 30.7 37.0 28.3 34.1 KO• 2Σ+ → 1Π 31.8 37.6 43.1 31.5 37.2 32.9 37.7 1Σ 119.5 167.1 154.8 35.6 40.3 29.9 35.6 3Π 33.8 32.7 38.7 32.7 38.2 27.3 33.8 3Σ 32.0 30.5 36.7 30.0 36.3 29.1 35.6

169

A previous theoretical value of 50.2 kJ/mol for NaO•,221 again, agrees well with our

computed values, which ranged from 49.2 kJ/mol computed with W2C‐CAS‐AQCC//A'WCV5Z to

53.1 kJ/mol computed with W1C//A'WCV5Z. For KO•, the single reference methods again fail

due to the multi‐reference character of the 1Σ+ state, which is the ground anionic state for KO–.

For the multi‐reference WnC methods, the electron affinities range from 33.9 to 41.0 when the

2Π state is used for the radical state, while the 2Σ+ electron affinity ranges from 29.9 to 40.3

kJ/mol. It is interesting to note that the electron affinity for LiO• and KO• are quite similar with

a difference generally smaller than 1 kJ/mol, while the electron affinity for NaO• is computed to

be ~10 kJ/mol larger.

6.3.7. Gas‐Phase Acidities

The calculated gas‐phase acidities of the alkali metal hydroxides [i.e. the enthalpy changes

for the reaction MOH → MO– + H+] are shown in Table 6.7. For LiO– and NaO–, the lowest

electronic state was determined to be the 3Π, so the discussion of the gas phase acidity for

LiOH and NaOH will focus on lithium and sodium hydroxide losing a proton to produce the 3Π

state. For KO–, the ground electronic state was determined in our work and in earlier work to

be 1Σ+ state,221 which possesses a large degree of multi‐reference character. This means that

the single‐reference methods are inappropriate to compute the transition of KOH to the KO–

1Σ+ electronic state, so we have limited our discussion of the KOH gas phase acidities to the

results found utilizing the multi‐reference W2C methods.

170

Table 6.7. Gas‐Phase Acidities (0 K, kJ/mol) for Alkali Metal Hydroxides (MOH).

MOH MO– W1C//

A'WCV5Z W2C//

A'WCV5Z W2C//

AA'WCV5Z


W2C‐CAS‐ACPF//

AA'WCV5Z


W2C‐CAS‐AQCC//

AA'WCV5Z

LiOH → 1Π 1791.2 1789.3 1787.2 1787.9 1783.9 1789.8 1789.8

1Σ+ 1791.7 1791.9 1791.6 1795.8 1794.9 1801.5 1800.7

3Π 1787.5 1786.2 1782.5 1785.7 1782.0 1787.5 1783.8

3Σ+ 1820.5 1819.2 1815.5 1819.2 1815.4 1820.5 1816.7

NaOH → 1Π 1758.7 1759.2 1757.7 1759.3 1757.6 1761.7 1760.0

1Σ+ 1760.6 1761.5 1761.0 1765.2 1764.7 1767.8 1767.2

3Π 1756.7 1756.3 1754.7 1756.6 1755.1 1759.0 1757.4

3Σ+ 1778.5 1778.2 1776.8 1778.8 1777.5 1780.7 1779.4

KOH → 1Π 1785.7 1776.4 1770.8 1782.6 1776.9 1785.7 1779.7

1Σ+ 1697.7 1646.7 1659.0 1778.3 1773.6 1780.7 1775.8

3Π 1783.6 1781.4 1775.3 1781.4 1775.9 1783.9 1778.2

3Σ+ 1785.4 1783.5 1777.2 1784.1 1777.8 1786.5 1780.0

171

Overall, the difference between the various WnC methods varies by less than 7.0 kJ/mol.

For LiOH, the smallest computed gas phase acidity for LiOH is 1787.2 kJ/mol, which was

computed with W2C‐CAS‐ACPF//AA'WCV5Z, while the largest gas phase acidity is 1786.2 kJ/mol

found with W2C//A'WCV5Z – giving a difference of 5.5 kJ/mol. The difference between the

smallest and largest computed gas phase acidities for NaOH are smaller than LiOH with the

smallest value being 1754.7 for NaOH computed with W2C//AA'WCV5Z and the largest value

being 1759.0 kJ/mol determined with W2C‐CAS‐AQCC//A'WCV5Z – a difference of 4.3 kJ/mol.

For KO–, the gas phase acidities computed for KOH utilizing the multi‐reference W2C‐CAS

methods resulted in a smallest and largest value of 1773.6–1780.7 kJ/mol computed with the

W2C‐CAS‐ACPF//AA'WCV5Z and W2C‐CAS‐AQCC//AA'WCV5Z methods, respectively, which is

difference of 7.1 kJ/mol. The average value for gas phase acidities were also presented

determined which is found in a similar manner as was done for enthalpies of formation [details

in Enthalpy of Formation section]. It was found from these average gas phase acidities that the

loss of a proton for the alkali metal hydroxides occurs in the following energy ordering NaOH <

KOH < LiOH with average values of 1755.7, 1774.7, and 1782.7 kJ/mol, respectively.

6.3.8. Bond Dissociation Energies

The calculated bond dissociation energies (BDEs) [i.e. the enthalpy change for the reaction

MOH → MO• + H•] is presented in Table 6.8. For LiO• and NaO•, the lowest electronic state was

determined to be the 2Π electronic state, and only the BDE in terms of LiOH and NaOH losing a

hydrogen atom to produce the 2Π state is discussed. For KO•, the computed results indicate

that the 2Σ+ state is lower than the 2Π state by only 1 kJ/mol, so to be thorough, the energy

172

difference when the KOH losses a hydrogen atom to produce both the 2Π and 2Σ+ electronic

states will be discussed.

All of the WnC methods predict that NaOH has the weakest bond dissociation energy with

an average of 493.2 kJ/mol. KOH is found to have the next weakest bond. When the KOH

looses a hydrogen atom to produce the 2Π state, the average BDE is found to be 500.1 kJ/mol.

When the KO• 2Σ+ state is produced, the average BDE is 498.9 kJ/mol. LiOH was found to have

the largest BDE with an average value of 508.0 kJ/mol.

173

Table 6.8. Bond dissociation energies (0K, kJ/mol) for the alkali metal hydroxides.

MOH MO•W1C//

A'WCV5Z

W2C//

A'WCV5Z

W2C//

AA'WCV5Z


W2C‐CAS‐ACPF// AA'WCV5Z


W2C‐CAS‐AQCC// AA'WCV5Z

LiOH → LiO• 2Π 509.6 508.7 508.7 507.7 507.7 507.5 507.5

2Σ+ 539.5 538.6 540.8 537.8 537.8 537.7 537.7

NaOH → NaO• 2Π 494.9 493.4 493.4 493.1 493.0 493.2 493.2 2Σ+ 519.1 517.8 517.8 517.2 517.2 517.1 517.1

KOH → KO• 2Π 504.4 500.6 500.6 499.8 499.8 499.8 499.8 2Σ+ 502.5 498.9 498.9 499.0 499.0 498.7 498.8

174

6.4. Conclusions

The structures, enthalpies of formation, state splitting, electron affinities, bond

dissociation energies, and gas‐phase acidities associated with the alkali metal monoxide radicals

(MO•), anions (MO–) and hydroxides (MOH) (M = Li, Na, and K) have been examined using

single‐ and multi‐reference variants of the WnC method, including W1C, W2C, W2C‐CAS‐ACPF,

and W2C‐CAS‐AQCC, based upon geometries optimized with the A'WCV5Z and/or AA'WCV5Z

basis sets. The WnC methods generally provide good agreement with available experimental

data, though there are deviations of 20–30 kJ/mol between the WnC and experimental

enthalpies of formation for the LiO• radical.

In some cases, care needs to be taken because of significant multi‐reference character in

the species examined, estimated by examining the T1 diagnostic and the SCF contribution to the

total atomization energy. On this basis, the 1Σ+ state of the MO– anions possesses the largest

multi‐reference character. As a consequence, use of single‐reference WnC computations for the

1Σ+ state of the MO– anions is unlikely to be reliable. The multi‐reference character is reflected

in the large deviations between results obtained using the single‐reference WnC and multi‐

reference W2C‐CAS methods for the enthalpy of formation of the 1Σ+ state of MO–, the 1Π→1Σ+

electronic state splitting of MO–, the electron affinities of MO•, and the gas‐phase acidities of

MOH.

The use of geometries optimized with augmented basis sets for the metal atom is found

to have a slight impact on several of the calculated molecular properties, most noticeably for

KO–. For example, the bond lengths in the ground and electronic excited state for KO– are ~0.02

175

Å shorter when the diffuse functions are included for the metal, with the most pronounced

difference being 0.038 Å for the 1Σ+ state. This leads to differences of 4–5 kJ/mol in estimates

of the enthalpies of formation of KO–, electron affinities of KO•, and gas‐phase acidities for KOH

obtained with A'WCV5Z and AA'WCV5Z geometries.

For NaO–, the ground electronic state is determined to be the 3Π state. For KO•, the 2Σ+

and 2Π states are found to lie very close in energy. An estimate, obtained by taking the average

of the W2C, W2C‐CAS‐ACPF, and W2C‐CAS‐AQCC energies (obtained with AA'WCV5Z

geometries) yields an energy difference of 1.1 kJ/mol in favor of the 2Σ+ state. Both the gas

phase acidities and the bond dissociation energies for the alkali metal hydroxides is found to

follow the following energy orderings NaOH < KOH < LiOH.

176

CHAPTER 7

DEVELOPMENT OF A MULTI‐REFERENCE CORRELATION CONSISTENT COMPOSITE APPROACH

[MR‐ccCA]

7.1. Introduction

At present, the correlation consistent composite approach (ccCA) only employs methods

that are based on a simple single reference wave function [i.e., corresponding to one Lewis

structure]. However, there are many chemical situations in which a single reference wave

function is inappropriate, such as the case for many transition state structures, reactive

intermediates, excited electronic states, and many transition metal species. One of the simplest

molecules requiring a multi‐reference wave function is methylene [CH2]. As discussed by

Schmidt and Gordon,275 the singlet configuration of linear CH2 requires the mixing of two wave

functions. Figure 7.1 shows that in linear CH2 there are two possible configurations

corresponding to a pair of electrons either occupying the px or the py orbital of the carbon

atom. To provide an accurate description of this system, both configurations must be included,

which is only possible by using a multi‐reference method. This molecule is one of the simplest

cases requiring the use of a more complex wave function. The need for multi‐reference

methods, however, can be even more pronounced in transition metal complexes.276 An

example is the metal‐carbon linkage in alkylidenes, which can be described as: (a) ethylene‐like,

(b) π‐ylide, (c) a coordinated singlet carbine, and (d) a four‐electron donor.276 Because of these

different descriptions, the metal‐carbon bond cannot be described by a simple single‐reference

wave function, such as the methods that are utilized in ccCA. Instead, all of these descriptions

must be allowed to contribute to the wave function, which require a multi‐reference method.

Figure 7.1. Two molecular orbitals of the methylene molecule. One configuration shows a pair of electrons occupying the px orbital of the carbon atom, and the other configuration shows a pair of electrons occupying the py orbital of the carbon atom. This analysis was explained by Schmidt and Gordon.275

177

Configuration 1 Configuration 2

This chapter proposes a multi‐reference correlation consistent composite approach [MR‐

ccCA], in which the single reference methods [i.e., MP2, . . .] that comprise ccCA are replaced

with suitable multi‐reference methods. Initially, the theoretically obvious replacements [e.g.,

CASPT2 for MP2] will be utilized. However, the inherent additive nature of composite methods

does not guarantee that such replacements will still enable the chemical accuracy of ccCA to be

achieved. Key to the development of a MR‐ccCA method is a good understanding of how each

of the smaller computations will impact the overall MR‐ccCA energy. Unfortunately, the

utilization of multi‐reference methods is a particularly daunting task, as multi‐reference

methods require an additional level of complexity – the selection of an active space, which can

substantially affect the results.

178

An active space is a set of orbitals/electrons that are allowed to mix to form the wave

function [e.g. the mixing of the two orbitals in Figure 7.1, which are needed for the correct

wave function for methylene]. For an exact description of a molecular system, all of the orbitals

would need to be included in the active space [i.e. full configuration interaction (FCI)]. However,

the use of all orbitals in the active space is too computationally demanding for all but the

simplest molecular systems. Therefore, a smaller set of orbitals are chosen to be active, while

the rest are effectively kept frozen. This choice of an active space is highly troublesome because

the use of an active space that is correct for one multi‐reference method may not be correct for

another method. This poses a significant challenge to the development of MR‐ccCA, as MR‐

ccCA would encompass more than one type of multi‐reference method. Addressing this

problem will require significant attention and understanding as to the potential sources of

error. Fortunately, there have been efforts to study these types of effects, which are

incorporated into this chapter. Specifically, Sølling et al. developed several G2 and G3 multi‐

reference methods,277 and Martin developed two multi‐reference Wn methods.252 These

previous studies will be used as a guiding tool for the development of MR‐ccCA.

7.2. Methodology

Details of ccCA were discussed in Chapter 3 and will not be repeated, but in summary, the

ccCA total energy is computed by the following formula, (See Section 3.1 for an explanation of

the terms)

E ccCA E0 ccCA ∆E CC ∆E DK ∆E CV ∆E SOa . 7.1

179

The first step in ccCA is the geometry optimization and frequency computation. As the

initial studies presented here focus upon the potential energy surfaces (PES) of two small

diatomic molecules, geometry optimizations and frequencies computations were not

performed, and instead a PES was determined with a series of single‐point computations with a

step size of 0.1 Å between each calculation. However, in general application of MR‐ccCA,

geometry optimization and frequency computations are performed with the complete active

space second order perturbation (CASPT2) method.278

The MR‐ccCA reference energy was determined by series of single‐point computations,

which are performed using CASPT2/aug‐cc‐pVnZ [where n = D(2), T(3), and Q(4)]. The active

space chosen for all steps for the initial MR‐ccCA method was the full valence active space. The

CASPT2 energies were then extrapolated to the complete basis set (CBS) limit utilizing Eqn. 3.1

and Eqn. 3.2. These two CBS limits were subsequently averaged to determine the MR‐ccCA

reference energy, which is denoted E0(MR‐ccCA).

A series of additive corrections were computed and added to the E0(MR‐ccCA) reference

energy. The first correction accounts for higher‐order electron correlation that is not

completely described by CASPT2. Unfortunately, the simple replacement of CCSD(T) utilized in

ccCA with a multi‐reference coupled cluster [MRCC] method is hindered in that MRCC has not

been widely implemented into common ab initio program packages, such as Gaussian and

Molpro.170,200 Therefore, the use of multi‐reference configuration interaction including single

and double excitations and a size‐consistency correction (MRCI+Q) is tested here for the initial

MR‐ccCA development. It should be noted, however, that the use of a multi‐reference coupled

180

cluster method including single, double, and quasiperturbative triple excitations [MR‐CCSD(T)]

is currently being implemented and tested. The multi‐reference higher order correlation energy

correction is denoted ∆E(MRCI+Q) and computed with the following formula,

∆E MRCI+Q E[MRCI+Q/cc‐pVTZ] ‐ E[CASPT2/cc‐pVTZ]. 7.2

Another correction is the scalar relativistic correction, which was obtained from frozen‐

core CASPT2 computations combined with cc‐pVTZ‐DK basis set39 and the spin‐free, one‐

electron Douglas‐Kroll‐Hess (DKH) Hamiltonian.165‐167 The standard CASPT2 relativistic

correction is labeled ∆E(MR‐DK) and is formulated as,

∆E MR‐DK E[CASPT2/cc‐pVTZ‐DK] ‐ E[CASPT2/cc‐pVTZ]. 7.3

The final correction needed is to account for core‐valence correlation effects, which was

computed by taking the difference between CASPT2/aug‐cc‐pVTZ and CASPT2(FC1)/aug‐cc‐

pCVTZ [see Chapter 3 for explanation of MP2(FC1)]. The multi‐reference core‐valence

correction is defined as,

∆E MR‐CV E[CASPT2(FC1)/aug‐cc‐pCVTZ] ‐ E[CASPT2/aug‐cc‐pCVTZ]. 7.4

The final MR‐ccCA method is formulated as,

E MR‐ccCA E0 MR‐ccCA ∆E MRCI+Q ∆ MR‐D E K

∆E MR‐CV ∆E SOa . 7.5

All computations for the MR‐ccCA were performed with the MOLPRO program.170,200

181

7.3. Potential Energy Surfaces (PES) for Diatomic Molecules

The potential energy surface (PES) for molecules can be difficult to compute due to the

inherent problem associated with the changing nature of the wave function when bonds are

broken or formed. It is well known that the restricted Hartree‐Fock (RHF) wave function

includes unphysical ionic terms at the dissociation limit for homolytic bond cleavage, and that

UHF can often be quantitatively incorrect especially in the intermediate bond breaking region,

which often results in unphysical barriers due to unwanted spin states contaminating the wave

function (i.e. spin contamination).279‐293

The errors that arise in using RHF and UHF can be corrected with a post‐HF methods ,

such as full CI.279 However, as FCI is impractical systems larger than 2‐4 atoms, post‐HF methods

truncated at a given excitation level [e.g. CCSD(T)] are commonly utilized, for which correcting a

single‐reference RHF or UHF reference wave functions is difficult. One well‐documented

example is the computation of the potential energy surface (PES) for the N2 molecule.281‐285,288‐

293 As shown by Laidig et al. and Bartlett et al., the use of RHF as a reference wave function for a

series of perturbation computations diverges toward negative infinity at ~1.5Å,285 while the

UHF reference wave function has an erroneous curvature due to the large spin contamination

present in the UHF wave function.282 It was even shown that method such as configuration

interaction including single, double, triple, and quadruple [CISDTQ] does not properly correct

the RHF wave function. Also, the use of coupled cluster theory, which is generally very accurate,

requires at least full quadruple excitations for even a qualitative description of the dissociation

of N2,281,284,289 and it was suggested that quintuple excitations or higher are required for a

182

quantitatively correct PES at bond lengths greater than ~2.5 Å.290 The error associated with the

use of a single‐reference wave function (i.e. RHF and UHF) is known to be remedied if a multi‐

reference method such as multi‐reference coupled cluster theory including single and double

excitations [MR‐CCSD] is utilized.285,288 Therefore, the computation of the PES of the N2

molecule is a good test case for MR‐ccCA.

7.3.1. N2 Potential Energy Surface

The PES of the N2 molecule computed with each individual method utilized within ccCA is

compared in Figure 7.2 along with the ccCA PES. The PES for N2 was determined by a series of

single‐point computations with bond lengths ranging from 0.8 – 3.5 Å with a step size of 0.1 Å.

As shown, all of the MP2 methods utilized in ccCA are well behaved near the minimum.

However, the MP2 methods begin to fail after ~1.7 Å indicted by the divergence of the PES

towards negative infinity. The failure of single‐reference perturbation theory to correct the

RHF and UHF wave function is discussed in previous studies.281 The CCSD(T)/cc‐pVTZ method

performs slightly better than MP2, however, divergence occurs at ~2.0 Å, which again is

reported in the literature.284,289 The failure of these single‐reference methods, as expected,

impacts the PES computed with ccCA. It is interesting to note that the ccCA PES parallels the

CCSD(T)/cc‐pVTZ PES. This qualitative agreement is not unusual because ccCA attempts to

simulate large basis set CCSD(T) results.

Figure 7.2. The potential energy surface for the N2 molecule computed with ccCA. Also, the individual computations required to compute the ccCA total energy are provided.

‐109.70

‐109.60

‐109.50

‐109.40

‐109.30

‐109.20

‐109.10

‐109.00

‐108.90

‐108.80

‐108.70

0.5 1 1.5 2 2.5

Total Ene

rgy (a.u.)

N‐N Distance (Å)

MP2/CBSMP2(FC1)/aug‐cc‐pCVTZMP2/aug‐cc‐pCVTZCCSD(T)/cc‐pVTZMP2/cc‐pVTZMP2/cc‐pVTZ‐DKccCA

183

Figure 7.3. The potential energy surface for the N2 molecule computed with MR‐ccCA. Also, the individual computations required to compute the MR‐ccCA total energy are provided.

‐109.6

‐109.5

‐109.4

‐109.3

‐109.2

‐109.1

‐109.0

‐108.9

0.5 1.5 2.5 3.5

Total Ene

rgy (a.u.)

N‐N Distance (Å)

CASPT2/CBS

CASPT2(FC1)/aug‐cc‐pCVTZ

CASPT2/aug‐cc‐pCVTZ

MRCI+Q/cc‐pVTZ

CASPT2/cc‐pVTZ

CASPT2/cc‐pVTZ‐DK

MR‐ccCA

The PES of the N2 molecule computed with each individual method utilized within MR‐

ccCA is compared in Figure 7.3 along with the ccCA PES. All of the multi‐reference methods

utilized in MR‐ccCA are well behaved both near and far from the minimum, which also occurs

for MR‐ccCA. Overall, the MR‐ccCA method correctly describes the dissociation of the triply

bonded N2 molecule.

The PES computed with both the ccCA and MR‐ccCA methods are shown together in

Figure 7.4. The PES computed with ccCA is well behaved to 2.0 Å, and the two curves are nearly

identical out to this point, with the exception at near‐equilibrium bond distance. The PES

computed with ccCA was found to be slightly lower in energy than that of MR‐ccCA. This

184

difference in energy is due to the use of CCSD(T) in the computation of the higher order

correlation correction in the ccCA as opposed to the use of MRCI+Q in the MR‐ccCA method. It

is expected that the two curves would be nearly identical at the bottom of the potential well if

a multi‐reference CCSD(T) method is incorporated into MR‐ccCA because it was previously

shown that there is little multi‐reference character for N2 near the minium.280,286,287 Overall,

the PES computed with MR‐ccCA behaves correctly past ~2.0 Å, which is not observed for ccCA.

Figure 7.4. Potential energy surface for the ground state N2 molecule computed with the ccCA and MR‐ccCA methods.

‐109.60

‐109.55

‐109.50

‐109.45

‐109.40

‐109.35

‐109.30

‐109.25

‐109.20

‐109.15

0.5 1.5 2.5 3.5

Total Ene

rgy (a.u.)

N‐N distance (Å)

MR‐ccCA

SR‐ccCA

MR‐ccCA (X 1Σg+)

ccCA (X 1Σg+)

7.3.2. C2 Potential Energy Surface

The PES for the ground X1Σg+ state for the C2 molecule computed with both the MR‐ccCA

and ccCA is shown in Figure 7.5. As was observed for N2, the ground state PES computed with

ccCA was found to behave correctly until ~2.2 Å whereas the MR‐ccCA PES provides correct

185

186

behavior out to the dissociation limit. Several difficulties are inherent in the computation of the

ground state PES for C2 due to the presence of a low lying B1Δg excited state, for which the PES

computed with MR‐ccCA is also shown in Figure 7.5. Also, as observed in a study performed by

Abrams and Sherrill,279 the presence of an avoided crossing, which can cause the Born‐

Oppenheimer approximation to fail, at ~1.6 – 1.7 Å added an additional level of complexity.

The ccCA C2 PES is similar to the MR‐ccCA PES to ~1.6 Å with the ccCA PES being slightly

lower in energy near the minimum as compared to the MR‐ccCA PES, which is again due to the

utilization of the CCSD(T) method to compute the higher correlation correction. After ~1.6 Å,

the ccCA PES rises more rapidly than the MR‐ccCA PES, until ~2.2 Å when the ccCA PES begins to

diverge toward negative infinity. This, again, is a well documented error due to the use of the

RHF wave function. The use of UHF is expected to alleviate this error; however, UHF has the

problem of spin contamination, which as shown Bartlett et al. for N2, can lead to erroneous

curvature in the intermediate bond breaking region.282 It is interesting to note that the MR‐

ccCA method closely resembles the FCI results presented by Abrams and Sherrill.279 The MR‐

ccCA PES for C2, however, bends slightly at ~1.6 Å, which is due to the presence of the avoided

crossing as was discussed by Abrams and Sherrill.279 Overall, the MR‐ccCA method provides a

good description for both the ground X1Σg+ and excited B1Δg state for the C2 molecule.

187

‐75.95

‐75.90

‐75.85

‐75.80

‐75.75

‐75.70

‐75.65

‐75.60

0.5 1.5 2.5 3.5

Total Ene

rgy (a.u.)

C‐C distance (Å)

MR‐ccCA (X1Sg+)

MR‐ccCA (B1Dg)

SR‐ccCA (X1Sg+)

MR‐ccCA (X 1Σg+)

MR‐ccCA (B 1Δg)

ccCA (X 1Σg+)

Figure 7.5. The potential energy surface for the ground X1Σg

+ state for the C2 molecule computed with the MR‐ccCA and ccCA methods. Also displayed is the PES for the excited B1Πg state of C2 computed with the MR‐ccCA method.

7.4. Conclusions

A multi‐reference correlation consistent composite approach [MR‐ccCA] has been

developed and tested on the potential energy surfaces of the ground states of N2 and C2 as well

as the excited state of C2. The PES computed with the standard ccCA method were also

presented for N2 and C2 and were compared with MR‐ccCA. Overall, the PES for both molecules

computed with ccCA were well behaved near the minima, but quickly diverged toward negative

infinity. The PES computed with MR‐ccCA was well behaved and was qualitatively comparable

to previous FCI PES for both N2 and C2. MR‐ccCA is, therefore, a viable composite approach that

188

can be utilized in the computation of systems requiring multi‐reference methods. Further

development and testing of MR‐ccCA is currently in progress.

189

CHAPTER 8

SUMMARY

Several methods were presented to help alleviate the high computational scaling of ab

inito methods. Specifically, the ability of the correlation consistent composite approach to

compute energetic properties [i.e. 19 atomization energies (D0), 11 enthalpies of formation

(∆Hf), 15 ionizations potentials (IP), 4 electron affinities (EA), and 2 proton affinities (PA)] for

molecules containing third‐row (Ga‐Kr) atoms was addressed in Chapter 3. For the 51 energetic

properties the MAD compared to experiment when only first‐order SOC are included was

computed to be 0.95 and 1.00 kcal/mol for ccCA‐S4 and ccCA‐S4, respectively, which was better

than the MAD of 1.07 kcal/mol for the G3 method. When second‐order SOC is included, the

MAD was reduced to 0.88 and 0.92 kcal/mol for ccCA‐P and ccCA‐S4, respectively. The

inclusion of second‐order SOC in ccCA‐P for third‐row energetic properties puts them on par

with the G4 method. Additionally, ccCA was found to outperform CCSD(T)/aug‐cc‐pV∞Z results

as well as CBS‐n model chemistries. Overall, ccCA has been utilized to study molecules

containing first‐ (Li‐Ne), second‐ (Na‐Ar), and third‐row (Ga‐Kr) atoms achieving an accuracy

within 1 kcal/mol of reliable experimental data. The results presented in Chapter 3, however,

show that SOC is important in the accurate prediction of energetic properties for third‐row

molecule and should be included in all future ccCA studies for these systems.

190

The studies presented in Chapter 4 discussed an alternate means of reducing the

computational cost of ab initio methods by reducing the number of basis functions in the

correlation consistent basis for the hydrogen atom. It was found that simply removing functions

from the hydrogen basis set had significant impacts on raw atomization energies, and is not

recommended for single computations. However, the ability to extrapolate a series of

truncated basis set to the CBS limit with a variety of four‐point truncated basis set schemes was

found to be useful. Specifically, the utilization of the cc(f,–1,–1,–3), cc(f,–1,–1,–4),

cc(f,–1,–2,–3), and cc(f,–1,–2,–4) schemes provide CBS limits that were within 1 kcal/mol as

compared to the extrapolation of the full correlation consistent basis sets. Extrapolations that

were based upon two‐ and three‐point truncated basis set schemes were also discussed in

Chapter 4. Basis set truncation utilized with the two‐ and three‐point schemes, however, did

have significant effects on atomization energies, up to –23.65 kcal/mol for decane. However,

this error was found to be highly systematic and dependent on the number of hydrogen atoms

contained in the molecule. Corrections to reduce the error introduced by basis set truncation

were computed based upon energy difference between the full extrapolated total energies and

the truncated extrapolated total energies versus the number of hydrogen atoms contained in a

series of hydrocarbons [i.e. methane through decane]. These corrections were utilized in the

computation of 55 enthalpies of formation for hydrogen containing molecules within the G3/99

test suite. When the corrections were included in the two‐ and three‐point truncated

extrapolations, the MAD was computed to be less than 0.5 kcal/mol as compared to the

extrapolation of the full correlation consistent basis sets. Overall, truncating the correlation

191

consistent basis sets for the hydrogen atom was shown to be a viable means to reduce the

computational cost of ab initio methods while still maintaining chemical accuracy.

Implementation of basis set truncation toward ccCA in the development of two truncated

ccCA schemes was presented in Chapter 5. The first truncated ccCA scheme removed the

augmented functions from the hydrogen basis sets, denoted ccCA(aug). The ccCA(aug)

truncated ccCA method was found to have little impact on the computed enthalpies of

formation taken from the G2/97 test suite, and thus, ccCA(aug) is an empirical‐free truncation

scheme that can be utilized to reduce the computation cost of ccCA. Unfortunately, the

bottleneck in ccCA is the CCSD(T)/cc‐pVTZ computation, which is not truncated in ccCA(aug).

Therefore, the basis set truncation studies presented in Chapter 4 were utilized in the

development of a truncated basis set correlation consistent composite approach [ccCA(TB)].

Unfortunately, the two‐ and three‐point extrapolation schemes developed in Chapter 4

were found to be method dependent and could not be utilized in ccCA, so studies were

performed to determine the effects that basis set truncation had upon each individual step in

ccCA. It was found that basis set truncation had a large effect on the individual ccCA

computations. However, these errors were systematic and depended on the number of

hydrogen atoms contained in the molecule. Corrections for each individual step in ccCA were

developed for ccCA(TB), which were based upon the energy difference (∆E) between ccCA and

ccCA(TB) versus the number of hydrogen atoms contained in a series of hydrocarbons [i.e.

methane through hexane]. An additional correction termed the “origin correction” was also

included in the total energy. The origin correction was needed because the linear plots of ∆E

192

versus the number of hydrogens did not pass through the origin, and a correction to account

for this error was included in the formulation of ccCA(TB). When the hydrogen and origin

corrections were included in ccCA(TB) the MAD for the 55 enthalpies of formation taken from

the G2/97 test suite was computed to be only 0.3 kcal/mol, as compared to ccCA. It was

previously noted in Chapter 5 that the corrections included in ccCA(TB) could be prone to

failure if the hydrogen itself is important to property of interest, such as hydrogen bonding.

Overall, the ccCA(aug) and ccCA(TB) truncated schemes are two viable means to reduce the

computational scaling of ccCA.

In Chapter 6, several WnC methods were utilized to study the alkali metal monoxide

radicals (MO•), anions (MO–) and hydroxides (MOH) [where M = Li, Na, and K]. Specifically, the

structures and molecular properties of MO•, MO– and MOH (M = Li, Na, and K) were investigate

including (a) the enthalpies of formation for all of these species, (b) the state splitting for MO•

and MO–, (c) the electron affinities for MO•, and (d) the bond dissociation energies and gas‐

phase acidities for MOH. It was found that in some cases, care needed to be taken because of

the significant multi‐reference character in the examined species, which was determined by

examining the T1 diagnostic and the percent SCF contribution to the total atomization energy.

Specifically, the 1Σ+ state of the MO– anions had the largest multi‐reference character, for which

the use of single‐reference WnC computations was unlikely to be reliable.

The energetic properties computed with the various WnC methods were compared to the

limited available experimental data. It was found that the properties computed with the WnC

methods were comparable to experiment [i.e. within 3‐5 kJ/mol]. One exception included the

193

enthalpy of formation for the LiO• 2Π ground state, for which the deviation between the theory

and experiment was on the order of ~20 kJ/mol. It was suggested in Chapter 6, as well as

previous theoretical work,224,225 that the experimental enthalpy of formation for LiO• may have

been computed for the low lying 2Σ+, and it was suggested that this value be re‐examined

experimentally. Overall, the Wn methods utilized in Chapter 6 provided good agreement with

the available experimental data.

A multi‐reference correlation consistent composite approach [MR‐ccCA] was previously

developed, and the application of MR‐ccCA in computing the PES was discussed in Chapter 7.

The computation of PES for molecules is known to be difficult due to the change in the multi‐

reference character of the wave function as bonds are broken or formed. A well documented

PES that requires multi‐reference methods is the dissociation of the ground state for the N2

molecule. A study that presented the ground state computed with ccCA and MR‐ccCA was

presented in Chapter 7. Additionally, the PES was computed for each individual computation

that is required for both ccCA and MR‐ccCA was presented. It was found that the computations

required for ccCA behaved well at near‐equilibrium distances, but quickly diverged toward

negative infinity at ~1.7‐2.0 Å. The divergent behavior of these methods impacted the overall

ccCA PES, which also diverged toward negative infinity at ~2.0 Å. The multi‐reference methods

that are utilized within MR‐ccCA, however, were found to be well‐behaved throughout. The

MR‐ccCA was also well‐behaved, which means that the MR‐ccCA method correctly describes

the dissociation energy of the N2 triple bond.

194

Additionally, the PES for the ground state (X1Σg+) of C2 was also computed with both ccCA

and MR‐ccCA. As observed for N2, the ground state PES for C2 was found to be well behaved at

near‐equilibrium bond distances, but diverged toward negative infinity past ~2 Å. The MR‐ccCA

method, however, was well behaved for the computed PES for the ground state of (X1Σg+). Also,

the PES was computed first excited state (B1Δg) of the C2 molecule. Again, the MR‐ccCA was

well behaved throughout the entire computed PES. The MR‐ccCA is proposed to be a useful

multi‐reference composite method that can be utilized in the predictions for molecules that

contain multi‐reference character. Further studies are currently under investigations that

address the ability of MR‐ccCA to compute energetic properties for transition metal, which are

known to be plagued with multi‐reference character.

195

REFERENCES

1 E. Schrödinger, Ann. d. Physik 79, 361 (1926).





6 E. Schrödinger, Die Naturwissenschaften 14, 664 (1926).

7 R. J. Berry, J. Yuan, A. Misra, and P. Marshall, J. Phys. Chem. A 102, 5182 (1998).

8 R. J. Howrwitz, J. S. Francisco, and J. A. Guest, J. Phys. Chem. A 101, 1231 (1997).

9 C. Naulin, M. Costes, Z. Moudden, N. Ghanem, and G. Dorthe, Chem. Phys. 202, 452 (1993).

10 Q. Zhong, L. Poth, J. V. Ford, and A. J. Castleman, Jr., Chem. Phys. Lett. 286, 305 (1998).

11 A. El Azhary, G. Rauhut, P. Pulay, and H.‐J. Werner, J. Chem. Phys. 108, 5185 (1998).

12 C. Hampel and H.‐J. Werner, J. Chem. Phys. 104, 6286 (1996).

13 G. Hetzer, P. Pulay, and H.‐J. Werner, Chem. Phys. Lett. 290, 143 (1998).

14 G. Hetzer, M. Schutz, H. Stoll, and H.‐J. Werner, J. Chem. Phys. 113, 9443 (2000).

15 D. Kats, T. Korona, and M. Schutz, J. Chem. Phys. 125, 104106/1 (2006).

16 L. Maschio, D. Usvyat, F. R. Manby, S. Casassa, C. Pisani, and M. Schutz, Phys. Rev. B: Condens. Matter Mater. Phys. 76, 075101/1 (2007).

17 R. A. Mata and H.‐J. Werner, Mol. Phys. 105, 2753 (2007).

18 R. A. Mata, H.‐J. Werner, and M. Schutz, J. Chem. Phys. 128, 144106/1 (2008).

19 P. Pulay, Chem. Phys. Lett. 100, 151 (1983).

20 G. Rauhut and P. Pulay, Chem. Phys. Lett. 248, 223 (1996).

196

21 G. Rauhut, P. Pulay, and H.‐J. Werner, J. Comput. Chem. 19, 1241 (1998).

22 G. Rauhut and H.‐J. Werner, Phys. Chem. Chem. Phys. 3, 4853 (2001).

23 S. Saebo, Comput. Chem. 7, 63 (2002).

24 S. Saebo and P. Pulay, Chem. Phys. Lett. 113, 13 (1985).

25 S. Saebo and P. Pulay, J. Chem. Phys. 86, 914 (1987).

26 S. Saebo and P. Pulay, J. Chem. Phys. 88, 1884 (1988).

27 S. Saebo and P. Pulay, Annu. Rev. Phys. Chem. 44, 213 (1993).

28 M. Schütz, J. Chem. Phys. 113, 9986 (2000).

29 M. Schütz, J. Chem. Phys. 116, 8772 (2002).

30 M. Schütz, G. Hetzer, and H.‐J. Werner, J. Chem. Phys. 111, 5691 (1999).

31 M. Schütz and H.‐J. Werner, J. Chem. Phys. 114, 661 (2001).

32 M. Schütz and H. J. Werner, Chem. Phys. Lett. 318, 370 (2000).

33 M. Schütz, H.‐J. Werner, R. Lindh, and F. R. Manby, J. Chem. Phys. 121, 737 (2004).

34 H.‐J. Werner and K. Pflueger, Annu. Rep. Comput. Chem. 2, 53 (2006).

35 A. K. Wilson and J. Almlof, Theor. Chim. Acta 95, 49 (1997).

36 J. Baker, K. Wolinski, M. Malagoli, and P. Pulay, Mol. Phys. 102, 2475 (2004).

37 B. Bolding and K. Baldridge, Comput. Phys. Commun. 128, 55 (2000).

38 W. R. Burdick, Y. Saad, L. Kronik, I. Vasiliev, M. Jain, and J. R. Chelikowsky, Comput. Phys. Commun. 156, 22 (2003).

39 W. A. de Jong, R. J. Harrison, and D. A. Dixon, J. Chem. Phys. 114, 48 (2001).

40 Z. Gan, Y. Alexeev, M. S. Gordon, and R. A. Kendall, J. Chem. Phys. 119, 47 (2003).

41 C. M. Goringe, E. Hernandez, M. J. Gillan, and I. J. Bush, Comput. Phys. Commun. 102, 1 (1997).

42 C. Haettig, A. Hellweg, and A. Koehn, Phys. Chem. Chem. Phys. 8, 1159 (2006).

197

43 S. Hirata, J. Phys. Chem. A 107, 9887 (2003).

44 T. Janowski, A. R. Ford, and P. Pulay, J. Chem. Theory Comput. 3, 1368 (2007).

45 J. Juselius and D. Sundholm, J. Chem. Phys. 126, 094101/1 (2007).

46 M. Klene, M. A. Robb, M. J. Frisch, and P. Celani, J. Chem. Phys. 113, 5653 (2000).

47 S. Knecht, H. J. A. Jensen, and T. Fleig, J. Chem. Phys. 128, 014108/1 (2008).

48 I. M. B. Nielsen and C. L. Janssen, J. Chem. Theory Comput. 3, 71 (2007).

49 R. M. Olson, J. L. Bentz, R. A. Kendall, M. W. Schmidt, and M. S. Gordon, J. Chem. Theory Comput. 3, 1312 (2007).

50 P. Stampfuss, W. Wenzel, and H. Keiter, J. Comput. Chem. 20, 1559 (1999).

51 E. F. Valeev and C. L. Janssen, J. Chem. Phys. 121, 1214 (2004).

52 S. C. Watson and E. A. Carter, Comput. Phys. Commun. 128, 67 (2000).

53 S. Dapprich, I. Komiromi, K. S. Byun, K. Morokuma, and M. J. Frisch, Theochem 461‐462, 1 (1999).

54 R. D. J. Froese, S. Humbel, M. Svensson, and K. Morokuma, J. Phys. Chem. A 101, 227 (1997).

55 S. Humbel, S. Sieber, and K. Morokuma, J. Chem. Phys. 105, 1959 (1996).

56 M. Svensson, S. Humbel, R. D. J. Froese, T. Matsubara, S. Sieber, and K. Morokuma, J. Phys. Chem. 100, 19357 (1996).

57 M. Svensson, S. Humbel, and K. Morokuma, J. Chem. Phys. 105, 3654 (1996).

58 T. Vreven and K. Morokuma, J. Comput. Chem. 21, 1419 (2000).

59 A. G. Baboul, L. A. Curtiss, P. C. Redfern, and K. Raghavachari, J. Chem. Phys. 110, 7650 (1999).

60 J.‐P. Blaudeau, M. P. McGrath, L. A. Curtiss, and L. Radom, J. Chem. Phys. 107, 5016 (1997).

61 L. A. Curtiss, J. E. Carpenter, K. Raghavachari, and J. A. Pople, J. Chem. Phys. 96, 9030 (1992).

198

62 L. A. Curtiss, C. Jones, G. W. Trucks, K. Raghavachari, and J. A. Pople, J. Chem. Phys. 93, 2537 (1990).

63 L. A. Curtiss, M. P. McGrath, J.‐P. Blaudeau, N. E. Davis, r. C. Binning, Jr., and L. Radom, J. Chem. Phys. 103, 6104 (1995).

64 L. A. Curtiss, K. Raghavachari, and J. A. Pople, J. Chem. Phys. 98, 1293 (1993).

65 L. A. Curtiss, K. Raghavachari, and J. A. Pople, J. Chem. Phys. 103, 4192 (1995).

66 L. A. Curtiss, K. Raghavachari, P. C. Redfern, A. G. Baboul, and J. A. Pople, Chem. Phys. Lett. 314, 101 (1999).

67 L. A. Curtiss, K. Raghavachari, P. C. Redfern, V. Rassolov, and J. A. Pople, J. Chem. Phys. 109, 7764 (1998).

68 L. A. Curtiss, K. Raghavachari, G. W. Trucks, and J. A. Pople, J. Chem. Phys. 94, 7221 (1991).

69 L. A. Curtiss, P. C. Redfern, and K. Raghavachari, J. Chem. Phys. 126, 084108/1 (2007).


71 L. A. Curtiss, P. C. Redfern, K. Raghavachari, and J. A. Pople, J. Chem. Phys. 114, 108 (2001).

72 L. A. Curtiss, P. C. Redfern, K. Raghavachari, and J. A. Pople, Chem. Phys. Lett. 359, 390 (2002).

73 L. A. Curtiss, P. C. Redfern, K. Raghavachari, V. Rassolov, and J. A. Pople, J. Chem. Phys. 110, 4703 (1999).

74 L. A. Curtiss, P. C. Redfern, V. Rassolov, G. Kedziora, and J. A. Pople, J. Chem. Phys. 114, 9287 (2001).

75 L. A. Curtiss, P. C. Redfern, B. J. Smith, and L. Radom, J. Chem. Phys. 104, 5148 (1996).

76 J. A. Pople, M. Head‐Gordon, D. J. Fox, K. Raghavachari, and L. A. Curtiss, J. Chem. Phys. 90, 5622 (1989).

77 P. C. Redfern, J. P. Blaudeau, and L. A. Curtiss, J. Phys. Chem. A 101, 8701 (1997).

199

78 A. D. Boese, M. Oren, O. Atasoylu, J. M. L. Martin, M. Kallay, and J. Gauss, J. Chem. Phys. 120, 4129 (2004).

79 A. Karton and J. M. L. Martin, Mol. Phys. 105, 2499 (2007).

80 A. Karton, E. Rabinovich, J. M. L. Martin, and B. Ruscic, J. Chem. Phys. 125, 144108/1 (2006).

81 J. M. L. Martin and G. de Oliveira, J. Chem. Phys. 111, 1843 (1999).

82 J. M. L. Martin and S. Parthiban, Understanding Chem. React. 22, 31 (2001).

83 S. Parthiban and J. M. L. Martin, J. Chem. Phys. 114, 6014 (2001).

84 Y. J. Bomble, J. Vazquez, M. Kallay, C. Michauk, P. G. Szalay, A. G. Csaszar, J. Gauss, and J. F. Stanton, J. Chem. Phys. 125, 064108/1 (2006).

85 M. E. Harding, J. Vazquez, B. Ruscic, A. K. Wilson, J. Gauss, and J. F. Stanton, J. Chem. Phys. 128, 114111/1 (2008).

86 A. Tajti, P. G. Szalay, A. G. Csaszar, M. Kallay, J. Gauss, E. F. Valeev, B. A. Flowers, J. Vazquez, and J. F. Stanton, J. Chem. Phys. 121, 11599 (2004).

87 J. A. Montgomery, Jr., M. J. Frisch, J. W. Ochterski, and G. A. Petersson, J. Chem. Phys. 110, 2822 (1999).

88 J. A. Montgomery, Jr., M. J. Frisch, J. W. Ochterski, and G. A. Petersson, J. Chem. Phys. 112, 6532 (2000).

89 J. A. Montgomery, Jr., J. W. Ochterski, and G. A. Petersson, J. Chem. Phys. 101, 5900 (1994).

90 J. W. Ochterski, G. A. Petersson, and J. A. Montgomery, Jr., J. Chem. Phys. 104, 2598 (1996).

91 G. A. Petersson and M. A. Al‐Laham, J. Chem. Phys. 94, 6081 (1991).

92 G. A. Petersson, A. Bennett, T. G. Tensfeldt, M. A. Al‐Laham, W. A. Shirley, and J. Mantzaris, J. Chem. Phys. 89, 2193 (1988).

93 G. A. Petersson, T. G. Tensfeldt, and J. A. Montgomery, Jr., J. Chem. Phys. 94, 6091 (1991).

94 G. P. F. Wood, L. Radom, G. A. Petersson, E. C. Barnes, M. J. Frisch, and J. A. Montgomery, Jr., J. Chem. Phys. 125, 094106/1 (2006).

200

95 N. J. DeYonker, T. R. Cundari, and A. K. Wilson, J. Chem. Phys. 124, 114104/1 (2006).

96 N. J. DeYonker, T. Grimes, S. Yockel, A. Dinescu, B. Mintz, T. R. Cundari, and A. K. Wilson, J. Chem. Phys. 125, 104111/1 (2006).

97 N. J. DeYonker, D. S. Ho, A. K. Wilson, and T. R. Cundari, J. Phys. Chem. A 111, 10776 (2007).

98 N. J. DeYonker, B. Mintz, T. R. Cundari, and A. K. Wilson, J. Chem. Theory Comput. 4, 328 (2008).

99 N. J. DeYonker, K. A. Peterson, G. Steyl, A. K. Wilson, and T. R. Cundari, J. Phys. Chem. A 111, 11269 (2007).

100 T. V. Grimes, A. K. Wilson, N. J. DeYonker, and T. R. Cundari, J. Chem. Phys. 127, 154117/1 (2007).

101 D. S. Ho, N. J. DeYonker, A. K. Wilson, and T. R. Cundari, J. Phys. Chem. A 110, 9767 (2006).

102 N. J. DeYonker, K. A. Peterson, and A. K. Wilson, J. Phys. Chem. A 111, 11383 (2007).

103 N. B. Balabanov and K. A. Peterson, J. Chem. Phys. 123, 064107/1 (2005).

104 T. H. Dunning, Jr., J. Chem. Phys. 90, 1007 (1989).

105 T. H. Dunning, Jr., K. A. Peterson, and A. K. Wilson, J. Chem. Phys. 114, 9244 (2001).

106 K. A. Peterson, Annu. Rep. Comput. Chem. 3, 195 (2007).

107 K. A. Peterson, T. B. Adler, and H.‐J. Werner, J. Chem. Phys. 128, 084102/1 (2008).

108 K. A. Peterson and C. Puzzarini, Theor. Chem. Acc. 114, 283 (2005).

109 T. Van Mourik and T. H. Dunning, Jr., Int. J. Quantum Chem. 76, 205 (2000).

110 A. K. Wilson, T. van Mourik, and T. H. Dunning, Jr., Theochem 388, 339 (1996).

111 A. K. Wilson, D. E. Woon, K. A. Peterson, and T. H. Dunning, Jr., J. Chem. Phys. 110, 7667 (1999).

201

112 D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 98, 1358 (1993).



115 D. Bakowies, J. Chem. Phys. 127, 084105/1 (2007).

116 G. A. Petersson and M. J. Frisch, J. Phys. Chem. A 104, 2183 (2000).

117 P. S. Sagrillo, F. E. Jorge, P. L. Barbieri, and P. A. Fantin, J. Braz. Chem. Soc. 18, 1448 (2007).

118 T. G. Williams, N. J. DeYonker, and A. K. Wilson, J. Chem. Phys. 128, 044101/1 (2008).

119 J. Cizek, J. Chem. Phys. 45, 4256 (1966).

120 J. Cizek, Adv. Chem. Phys. 14, 35 (1966).

121 J. Gauss, W. J. Lauderdale, J. F. Stanton, J. D. Watts, and R. J. Bartlett, Chem. Phys. Lett. 182, 207 (1991).

122 J. Gauss, J. F. Stanton, and R. J. Bartlett, J. Chem. Phys. 95, 2623 (1991).

123 J. D. Watts, J. Gauss, and R. J. Bartlett, J. Chem. Phys. 98, 8718 (1993).

124 S. Yockel, B. Mintz, and A. K. Wilson, J. Chem. Phys. 121, 60 (2004).

125 S. Yockel and A. K. Wilson, J. Chem. Phys. 122, 174310/1 (2005).

126 C. Yamada, M. Fujitake, and E. Hirota, J. Chem. Phys. 90, 3033 (1989).

127 C. Yamada, M. Fujitake, and E. Hirota, J. Chem. Phys. 91, 137 (1989).

128 A. Shulz, B. J. Smith, and L. Radom, J. Phys. Chem. A 103, 7522 (1999).

129 M. B. Sullivan, M. A. Iron, P. C. Redfern, J. M. L. Martin, and L. Radom, J. Phys. Chem. A 107, 5617 (2003).

130 M. Born and R. Oppenheimer, Ann. Phys. (Berlin, Ger.) 84, 457 (1927).

131 B. Ruscic, A. F. Wagner, L. B. Harding, R. L. Asher, D. Feller, D. A. Dixon, K. A. Peterson, Y. Song, X. Qian, C.‐Y. Ng, J. Liu, W. Chen, and D. W. Schwenke, J. Phys. Chem. A 106, 2727 (2002).

202

132 F. Jensen, Introduction to Computational Chemistry. (John Wiley & Sons, England, 1999).

133 V. Fock, Z. Phys. 62, 795 (1930).

134 D. R. Hartree, Proc. Cambridge Phil. Soc. 24, 426 (1928).



137 J. C. Slater, Phys. Rev. 32, 39 (1928).


139 D. Feller and K. A. Peterson, J. Chem. Phys. 108, 154 (1998).

140 G. L. Bendazzoli, B. Deguilhem, S. Evangelisti, F. X. Gadea, T. Leininger, and A. Monari, Chem. Phys. 348, 83 (2008).

141 A. D. Bochevarov and R. A. Friesner, J. Chem. Phys. 128, 034102/1 (2008).

142 J. M. Junquera‐Hernandez, J. Sanchez‐Marin, G. L. Bendazzoli, and S. Evangelisti, J. Chem. Phys. 121, 7103 (2004).

143 J. M. Junquera‐Hernandez, J. Sanchez‐Marin, G. L. Bendazzoli, and S. Evangelisti, J. Chem. Phys. 120, 8405 (2004).

144 A. Monari, J. Pitarch‐Ruiz, G. L. Bendazzoli, S. Evangelisti, and J. Sanchez‐Marin, J. Chem. Theory Comput. 4, 404 (2008).

145 C. D. Sherrill and P. Piecuch, J. Chem. Phys. 122, 124104/1 (2005).

146 M. J. Frisch, M. Head‐Gordon, and J. A. Pople, Chem. Phys. Lett. 166, 281 (1990).

147 M. Head‐Gordon, J. A. Pople, and M. J. Frisch, Chem. Phys. Lett. 153, 503 (1988).

148 C. Møller and M. S. Plesset, Phys. Rev. 46, 618 (1934).

149 T. D. Crawford and H. F. Schaefer, III, Rev. Comp. Chem. 14, 33 (2000).

150 P. Atkins and J. de Paula, Physical Chemistry, Seventh Edition. (W.H. Freeman and Company, New York, 2002).

151 C. C. J. Roothaan, Rev. Mod. Phys. 23, 69 (1951).

203


153 S. F. Boys, Proc. R. Soc. London, Ser. A 200, 542 (1950).

154 T. H. Dunning, Jr., K. A. Peterson, and D. E. Woon, in Encyclopedia of Computational Chemistry, in Section "Correlated Calculations on Molecules with the Correlation Consistent Basis Sets" (John Wiley & Sons Inc., New Jersey, 2008).

155 S. Yockel and A. K. Wilson, Theor. Chem. Acc. 120, 119 (2008).

156 D. Feller and D. A. Dixon, J. Chem. Phys. 115, 3484 (2001).


158 K. A. Peterson, research in progress.

159 K. A. Peterson, D. E. Woon, and T. H. Dunning, Jr., J. Chem. Phys. 100, 7410 (1994).

160 C. Schwartz, Phys. Rev. 126, 1015 (1962).

161 A. Halkier, T. Helgaker, P. Jorgensen, W. Klopper, H. Koch, J. Olsen, and A. K. Wilson, Chem. Phys. Lett. 286, 243 (1998).

162 T. Helgaker, W. Klopper, H. Koch, and J. Noga, J. Chem. Phys. 106, 9639 (1997).

163 J. M. L. Martin, Chem. Phys. Lett. 259, 669 (1996).

164 J. M. L. Martin and T. J. Lee, Chem. Phys. Lett. 258, 136 (1996).

165 B. A. Hess, Phys. Rev. A 33, 3742 (1986).

166 B. A. Hess, Phys. Rev. A 32, 756 (1985).

167 M. Douglas and N. M. Kroll, Ann. Phys. 82, 89 (1974).

168 B. C. Shepler, N. B. Balabanov, and K. A. Peterson, J. Phys. Chem. A 109, 10363 (2005).

169 J. P. Blaudeau and L. A. Curtiss, Int. J. Quantum Chem. 61, 943 (1997).

170 Gaussian 03, Revision C.02, M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. Montgomery, J. A., T. Vreven, K. N. Kudin, J. C. Burant, J. M. Millam, S. S. Iyengar, J. Tomasi, V. Barone, B. Mennucci,

204

M. Cossi, G. Scalmani, N. Rega, G. A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, K. J. E., H. P. Hratchian, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, P. Y. Ayala, K. Morokuma, G. A. Voth, P. Salvador, J. J. Dannenberg, V. G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain, O. Farkas, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. V. Ortiz, Q. Cui, A. G. Baboul, S. Clifford, J. Cioslowski, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. L. Martin, D. J. Fox, T. Keith, M. A. Al‐Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, C. Gonzalez, and J. A. Pople, Gaussian, Inc., Wallingford CT, 2004.

171 K. Huber and G. Herzberg, Molecular Spectra and Molecular Structure 4. (Van Nostrand, Princeton, 1979).

172 K. Ohno, H. HMatsumura, Y. Endo, and E. Hirota, J. Mol. Spectrosc. 118, 1 (1986).

173 J. H. Callomon, E. Hirota, K. Kuchitsu, W. J. Lafferty, A. G. Maki, and C. S. Pote, "Structure Data of Free Polyatomic Molecules", in Landolt‐Bornstein, New Series, Group II (Springer, Berlin, 1976), Vol. 7.

174 M. K. Gilles, M. L. Polak, and W. C. Lineberger, J. Chem. Phys. 96, 8012 (1992).

175 Y. Koga, H. Takeo, S. Kondo, M. Sugie, C. Matsumura, G. A. McRae, and E. A. Cohen, J. Mol. Spectrosc. 138, 467 (1989).

176 G. Graner, J. Mol. Spectrosc. 90, 394 (1981).

177 A. Schulz, B. J. Smith, and L. Radom, J. Phys. Chem. A 103, 7522 (1999).

178 S. R. Gunn and L. G. Green, J. Phys. Chem. 65, 779 (1961).

179 B. Ruscic, M. Schwarz, and J. Berkowitz, J. Chem. Phys. 92, 1865 (1990).

180 J. Berkowitz, J. Chem. Phys. 89, 7065 (1988).

181 S. T. Gibson, J. P. Green, and J. Berkowitz, J. Chem. Phys. 85, 4815 (1986).

182 L. V. Gurvich, I. V. Veyts, and C. B. Alcock, Thermodynamic Properties of Individual Substances, 4th ed. (Hemisphere, New York, 1989).

183 S. A. Kudchadker and A. P. Kudchadker, J. Phys. Chem. Ref. Data 4, 457 (1975).

184 P. A. G. O'Hare and L. A. Curtiss, J. Chem. Thermodynamics 27, 643 (1995).

205

185 J. B. Pedley, R. D. Naylor, and S. P. Kirby, Thermochemical Data of Organic Compounds, 2nd ed. (Chapman and Hall, New York, 1986).

186 C. E. Moore, U.S. Patent No. 1 (1952).

187 B. Ruscic and J. Berkowitz, J. Chem. Phys. 101, 7795 (1994).

188 S. Lias, J. F. Bartmess, J. L. Holmes, R. D. Levin, and W. G. Mallard, J. Phys. Chem. Ref. Data 17, Suppl. No 1 (1998).

189 M. N. Glukhovtsev, J. E. Szulejko, T. B. McMahon, J. W. Gauld, A. P. Scott, B. J. Smith, A. Pross, and L. Radom, J. Phys. Chem. 98, 13099 (1994).

190 V. Ramakrishna and B. J. Duke, J. Chem. Phys. 118, 6137 (2003).

191 J. E. Del Bene, J. Phys. Chem. 92, 2874 (1988).

192 J. E. Del Bene, J. Phys. Chem. 97, 107 (1993).

193 B. Mintz, K. P. Lennox, and A. K. Wilson, J. Chem. Phys. 121, 5629 (2004).

194 B. Mintz and A. K. Wilson, J. Chem. Phys. 122, 134106/1 (2005).

195 B. Mintz, S. Driskell, A. Shah, and A. K. Wilson, Int. J. Quantum Chem. 107, 3077 (2007).

196 D. Feller, J. Chem. Phys. 96, 6104 (1992).

197 L. A. Curtiss, K. Raghavachari, P. C. Redfern, and J. A. Pople, J. Chem. Phys. 112, 7374 (2000).

198 C. Schwartz, Methods in Computational Physics. (Academic Press, New York, 1963).

199 W. Kutzelnigg and J. D. Morgan, J. Chem. Phys. 96, 4484 (1992).

200 MOLPRO, version 2006.1, a package of ab initio programs, H. J. Werner, P. J. Knowles, R. Lindh, F. R. Manby, M. Schütz, P. Celani, T. Korona, G. Rauhut, R. D. Amos, A. Bernhardsson, A. Berning, D. L. Cooper, J. O. Deegan, A. J. Dobbyn, F. Eckert, C. Hampel, G. Hetzer, A. W. Lloyd, S. J. McNicholas, W. Meyer, M. E. Mura, A. Nicklass, P. Palmieri, R. Pitzer, U. Schumann, H. Stoll, A. J. Stone, R. Tarroni, and T. Thorsteinsson, see http://www.molpro.net.

201 M. W. Chase, Jr., C. A. Davies, J. R. Downey, Jr., D. J. Frurip, R. A. McDonald, and A. N. Syverud, JANAF Thermochemical Tables, 4th ed. (American Chemical

http://www.molpro.net/

206

Society and American Institute of Physics for the National Bureau of Standards, New York, 1985).

202 P. Jensen, J. Mol. Spectrosc. 128, 478 (1988).

203 C. C. Costain, J. Chem. Phys. 38, 864 (1958).

204 J. M. L. Martin and P. R. Taylor, Chem. Phys. Lett. 225, 473 (1994).

205 K. P. Huber and G. Herzberg, Constants of Diatomic Molecules. (National Institute of Standards and Technology, Gaithersburg MD, 2005).

206 L. A. Curtiss, K. Raghavachari, P. C. Redfern, and J. A. Pople, J. Chem. Phys. 106, 1063 (1997).

207 E. E. Ferguson, Geohpys. Res. Lett. 5, 1035 (1978).

208 C. E. Kolb and J. B. Elgin, Nature 263, 488 (1976).

209 S. C. Liu and G. C. Reid, Geohpys. Res. Lett. 6, 283 (1979).

210 E. Murad and W. Swider, Geohpys. Res. Lett. 6, 929 (1979).

211 G. A. Osborn, Fuel 71, 131 (1992).

212 F. S. Rowland and Y. Makide, Geohpys. Res. Lett. 9, 473 (1982).

213 L. A. Scandrett and R. J. Clift, Inst. Energy 57, 391 (1984).

214 S. Joo, D. R. Worsnop, C. E. Kolb, S. K. Kim, and D. R. Herschbach, J. Phys. Chem. A 103, 3193 (1999).

215 R. J. M. Konings and E. H. P. Cordfunke, J. Nucl. Mater. 167, 251 (1989).

216 W. Lin, G. Krusholm, K. Dam‐Johansen, E. Musahl, and L. Bank, Proccedings of the International Conference on Fluidized Bed Combustion 14, 831 (1997).

217 M. Ohman, A. Nordin, B. J. Skrivars, R. Backman, and M. Hupa, Energy Fuels 14, 169 (2000).

218 M. W. Chase, Jr., C. A. Davies, J. R. Downey, Jr., D. J. Frurip, R. A. McDonald, and A. N. Syverud, NIST‐JANAF Tables (4th ed.); J. Phys. Chem. Ref. Data, Suppl.1. (1998).

219 J. N. Allison, R. J. Cave, and W. A. Goddard III, J. Phys. Chem. 88, 1262 (1984).

207

220 C. W. Bauschlicher Jr., H. Partridge, and K. G. Dyall, Chem. Phys. Lett. 199, 225 (1992).

221 C. W. Bauschlicher Jr., H. Partridge, and L. G. M. Pettersson, J. Chem. Phys. 99, 3654 (1993).

222 A. I. Boldyrev, J. Simons, and P. R. Schleyer, J. Chem. Phys. 99, 8793 (1993).

223 P. Burk, K. Sillar, and I. A. Koppel, Theochem 543, 223 (2001).

224 S. R. Langhoff, C. W. Bauschlicher Jr., and H. Partridge, J. Chem. Phys. 84, 4474 (1986).

225 S. R. Langhoff, H. Partridge, and C. W. Bauschlicher Jr., Chem. Phys. 153, 1 (1991).

226 E. P. F. Lee, P. Soldán, and T. G. Wright, Chem. Phys. Lett. 295, 354 (1998).

227 E. P. F. Lee, P. Soldán, and T. G. Wright, Chem. Phys. Lett. 347, 481 (2001).

228 E. P. F. Lee, P. Soldán, and T. G. Wright, J. Chem. Phys. 117, 8241 (2002).

229 E. P. F. Lee and T. G. Wright, J. Phys. Chem. 106, 8903 (2002).

230 E. P. F. Lee, T. G. Wright, and J. M. Dyke, Mol. Phys. 77, 501 (1992).

231 T. J. Lee and P. R. Taylor, Int. J. Quantum Chem. Symp. 23, 199 (1989).

232 L. Serrano‐Andrés and A. S. de Merás, Theochem 254, 229 (1992).

233 L. Serrano‐Andrés, A. Sánchez de Merás, R. Pou‐Amérigo, and I. Nebot‐Gil, Chem. Phys. 162, 321 (1992).

234 S. P. So and W. G. Richards, Chem. Phys. Lett. 32, 227 (1975).

235 P. Soldán, E. P. F. Lee, S. D. Gamblin, and T. G. Wright, Phys. Chem. Chem. Phys. 1, 4947 (1999).

236 P. Soldán, E. P. F. Lee, and T. G. Wright, J. Phys. Chem. A. 102, 9040 (1998).

237 S. M. Freund, E. Herbst, R. P. Mariella Jr., and W. Klemperer, J. Chem. Phys. 56, 1467 (1972).

238 R. R. Herm and D. R. Herschbach, J. Chem. Phys. 52, 5783 (1970).

239 D. M. Lindsay, D. R. Herschbach, and A. L. Kwiram, J. Chem. Phys. 60, 315 (1974).

208

240 P. A. G. O'Hare and A. C. Wahl, J. Chem. Phys. 56, 4516 (1972).

241 J. Pfeifer and J. L. Gole, J. Chem. Phys. 80, 565 (1984).

242 R. C. Spiker Jr. and L. Andrews, J. Chem. Phys. 58, 713 (1973).

243 M. Yoshimine, J. Chem. Phys. 57, 1108 (1972).

244 E. Hirota, Bull. Chem. Soc. Jpn. 61, 1 (1995).

245 C. Yamada, S. Yamamoto, and E. Hirota, unpublished.

246 A. K. Rappé and W. A. Goddard III, J. Phys. Chem., to be submitted.

247 S. Huzinaga and M. Klobukowski, Chem. Phys. Lett. 212, 260 (1993).

248 A. Berning, M. Schweizer, H.‐J. Werner, P. J. Knowles, and P. Palmieri, Mol. Phys. 98, 1823 (2000).

249 M. P. Davidson and R. M. Silver, Chem. Phys. Lett. 52, 403 (1977).

250 J. M. L. Martin, Chem. Phys. Lett. 303, 399 (1999).

251 J. M. L. Martin, Spectrochim. Acta A 57, 875 (2001).

252 J. M. L. Martin and S. Parthiban, Quantum Mechanicl Prediction of Thermochemical Data. (Kluwer Academic, Dordrecht, The Netherlands, 2001).

253 K. A. Peterson and T. H. Dunning Jr., J. Chem. Phys. 117, 10548 (2002).

254 M. A. Iron, M. Oren, and J. M. L. Martin, Mol. Phys. 101, 1345 (2003).

255 R. D. Cowan and D. C. Griffin, J. Opt. Soc. Am. 66, 1010 (1976).

256 J. M. L. Martin, J. Phys. Chem. 87, 750 (1983).

257 R. J. Gdanitz, J. Quantum Chem. 85, 281 (2001).

258 R. J. Gdanitz and R. Ahlrichs, Chem. Phys. Lett. 143, 413 (1988).

259 P. G. Szalay, J. Chem. Phys. 100, 6288 (1996).

260 P. G. Szalay and R. J. Bartlett, Chem. Phys. Lett. 214, 481 (1993).

261 P. G. Szalay and R. J. Bartlett, J. Chem. Phys. 103, 3600 (1995).

209

262 R. Ahlrichs, H. Lischka, V. Staemmler, and W. Kutzelnigg, J. Chem. Phys. 62, 1225 (1975).

263 J. L. Dunham, Phys. Rev. 41, 713 (1932).

264 A. Nicolaides, A. Rauk, M. N. Glukhovtsev, and L. Radom, J. Phys. Chem. 100, 17460 (1996).

265 K. Kuchitsu, Landolt‐Bornstein Structure for Free Polyatomic Molecules, New Series Group II. (Berlin‐Heidelberg, 1998).

266 C. W. Bauschlicher, Jr., H. Partridge, and L. G. M. Pettersson, J. Chem. Phys. 99, 3654 (1993).

267 M. Steinberg and K. Schofield, J. Chem. Phys. 94, 3901 (1991).

268 R. H. Lamoreaux and D. L. Hildenbrand, J. Phys. Chem. Ref. Data 13, 151 (1984).

269 J. B. Pedley and E. M. Marshall, J. Phys. Chem. Ref. Data 12, 967 (1983).

270 D. L. Hildenbrand, J. Chem. Phys. 57, 4556 (1972).

271 L. V. Gurvich, G. A. Bergman, L. N. Gorokhov, V. S. Iorisch, V. Y. Leonidov, and V. S. Yungman, J. Phys. Chem. Ref. Data 25, 1211 (1996).

272 T. C. Ehlert, High Temp. Sci. 9, 237 (1977).

273 M. Farber and R. D. Srivastave, High Temp.‐High Press. 20, 119 (1988).

274 J. R. Woodward, J. S. Hayden, and J. L. Gole, Chem. Phys. 134, 395 (1989).

275 M. W. Schmidt and M. S. Gordon, Annu. Rev. Phys. Chem. 49, 233 (1998).

276 M. S. Gordon and T. R. Cundari, Coor. Chem. Rev. 87, 87 (1996).

277 T. I. Sølling, D. M. Smith, L. Radom, M. A. Freitag, and M. S. Gordon, J. Chem. Phys. 115, 8758 (2001).

278 B. Mintz and A. K. Wilson, research in progress.

279 M. L. Abrams and C. D. Sherrill, J. Chem. Phys. 121, 9211 (2004).

280 C. Angeli, B. Bories, A. Cavallini, and R. Cimiraglia, J. Chem. Phys. 121, 054108 (2006).

281 R. J. Bartlett and G. D. Purvis, Int. J. Quantum Chem. 14, 561 (1978).

210

282 R. J. Bartlett and G. D. Purvis, Phys. Scr. 21, 255 (1980).

283 G. K. Chan, M. Kallay, and J. Gauss, J. Chem. Phys. 121, 6110 (2004).

284 K. Kowalski and P. Piecuch, J. Chem. Phys. 113, 5644 (2000).

285 W. D. Laidig, P. Saxe, and R. J. Bartlett, J. Chem. Phys. 86, 887 (1987).

286 H. Larsen, J. Olsen, P. Jørgensen, and O. Christiansen, J. Chem. Phys. 113, 6677 (2000).

287 H. Larsen, J. Olsen, P. Jørgensen, and O. Christiansen, J. Chem. Phys. 114, 10985 (2001).

288 X. Li and J. Paldus, Chem. Phys. Lett. 286, 145 (1998).

289 M. Musiał and R. J. Bartlett, J. Chem. Phys. 122, 224102 (2005).

290 M. Musiał, S. A. Kucharski, and R. J. Bartlett, Chem. Phys. Lett. 320, 542 (2000).

291 P. Piecuch, A. E. Kondo, V. Spirko, and J. Pladus, J. Chem. Phys. 104, 4699 (1996).

292 P. E. M. Siegbahn, Int. J. Quantum Chem. 23, 1869 (1983).

293 T. V. Voorhis and M. Head‐Gordon, J. Chem. Phys. 112, 5633 (2000).

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