DEVELOPMENT AND APPLICATION OF EFFECTIVE QUANTUM CHEMICAL

STRATEGIES

By

Prajay Patel

A DISSERTATION

Michigan State University

in partial fulfillment of the requirements

Submitted to

for the degree of

Chemistry – Doctor of Philosophy

2019

ABSTRACT

DEVELOPMENT AND APPLICATION OF EFFECTIVE QUANTUM CHEMICAL

STRATEGIES

By

Prajay Patel

Within the field of computational chemistry, one of the greatest challenges is predicting
thermodynamic properties such as enthalpies of formation and interaction energies to understand
chemical phenomena throughout the periodic table. To predict these properties at a quantitative
level, high-level electronic structure methods, primarily ab initio methods, are used. These
methods are not utilized as often when increasing molecule size due to the significant
computational resources (disk space, memory, CPU time) required. Therefore, effective quantum
chemical schemes that take advantage of numerous cost-effective methods are needed and this
dissertation showcases their development and application towards main group and transition metal
thermochemistry.

In this dissertation, the pKa of late transition metal hydrides, which are important intermediates
in catalytic reactions, were predicted with electronic structure methods including density functional
theory (DFT) and ab initio methods. Insight into the thermochemistry and binding behavior of
these hydrides is key to understanding metal-ligand behavior for inorganic and organometallic
complexes.

To utilize ab initio methods for high accuracy thermochemistry and circumvent their high
computational cost, ab initio composite strategies, such as the correlation consistent Composite
Approach (ccCA), were developed.
In an effort to expand the size limitations of composite
methodologies, ccCA was combined with the domain-based local pair natural orbital (DLPNO)
methods. Denoted as DLPNO-ccCA, this method was developed for main group thermochemistry
and targeted one of the largest molecules examined with composite methodologies.
This
methodology was expanded to key reaction types in organometallic chemistry, such as olefin
insertion in hydroformylation, the largest volume homogeneous chemical reaction in chemical

industry for chemical production, and metal-ligand dissociation. To investigate the vibrational
behavior of chemical systems found in the interstellar medium, ccCA was used to generate
potential energy surfaces (PESs) characterizing vibrational motion to predict anharmonic
frequencies in tandem with vibrational self-consistent field (VSCF) and post-VSCF theory so that
there is a reduction in the computational cost associated with generating accurate PESs for
anharmonic mode-mode couplings as well as calculating contributions from anharmonic
corrections to the potential.

While ab initio methods are critical for attaining quality thermochemical predictions,
addressing polyatomic molecules of increasing size and complexity, electronic structure methods
like DFT are utilized due to the relative computational cost of DFT compared to ab initio
methods. Applications in this dissertation include the modeling of the frontier orbitals of zinc
porphyrin-fullerene supramolecular dyads with DFT to exhibit intramolecular charge transfer and
the prediction of the binding energies for several drug-like molecules to polymer-based host
compounds that display a binding pocket, which models protein-drug binding interactions, as part
of the Statistical Assessment of the Modeling of Proteins and Ligands (SAMPL) blind prediction
competition.

Copyright by
PRAJAY PATEL
2019

This dissertation is dedicated to my family, who supported me on my five-year mission in the final

frontier of formal education.

v

ACKNOWLEDGMENTS

I am truly grateful for everyone who has supported me throughout my academic career. Firstly,
I would like to thank Dr. Angela K. Wilson for her guidance over the years. I would like to thank
the Wilson group, past and current members, for their support throughout the years for insightful
discussions and some fun group bonding activities including but not limited to Becky, John P.,
Andrew, Kameron, John D., Michael, Zainab, Lucas, Thomas, Yiğitcan, Timothé, Hailey, and
Lenin. More so, I would like to thank Dr. Jiaqi Wang for training me when I started on my first
project, Dr. Inga Ulusoy for guidance on several projects throughout my graduate career as well as
Joseph Chung and Max Bowman, who were high school students that worked in the Wilson group
and contributed to one of the projects. I would like to acknowledge my committee at Michigan
State University, Ned Jackson, Gary Blanchard, and Ben Levine as well as my former committee
at the University of North Texas, Martin Schwartz, Lee Slaughter, and Tom Cundari. I do want to
mention Martin Schwartz, who passed away late 2018. His happiness and attitude towards teaching
and university life is one that I hope to exude in my career.

I would like to thank all my friends. This includes but is certainly not limited to Alex, Ben,
Brooke, Bryan, Carlos, Colton, Danielle, Donny, Eric, Erin B., Erin H., Kat, Karla, Kristin, Matt,
Neal, Paul, and Whitney. For all the time we have spent together, whether it was for dinner,
undergraduate and/or graduate classes, exercise, or annual gaming tournaments, those moments
are something to cherish and I hope there are many more to come.

I want to thank my personal trainers Brian, Becky, Leah, and Lexi as well as my boxing partner
Janet for keeping me accountable for my physical fitness and for fostering a positive and motivating
gym environment during my time in Michigan. The time spent definitely helped me adjust to life
in Michigan after moving from Texas.

Thank you to my large extended family and family friends for all of the support from across
the U.S. even if how I explained what I do was not the easiest to understand. I am getting better at
that skill everyday, in part of my numerous explanations. This includes every holiday season and

vi

occasional vacation/reunion...or for Indian people...weddings. To keep this list succinct: Adam,
Arjun, Arti, Chandani, Chris, Dhaval, Jeff, Justin, Katrina, Kelli, Kim, Kyle, Mishaun, Nikki,
Nisha, Paayal, Priya, and Robert.
I do want to specifically acknowledge my two cousins and
grandmother who passed away in 2018. I am grateful to have had the time and memories spent
with them, even with language barriers and how often I managed to see them. It is unfortunate
when life is taken early; however, celebrating life and not taking it for granted is something I hope
to do moving forward.

Finally, I thank my father, Mukesh, mother, Shruti, and sister, Shivani, for keeping me grounded
with our weekly conversations and providing perspective on life in general. While I never been the
best with words, there are not enough to describe how much love and support my family has given
me my whole life, including my decision to pursue graduate school. For that, I am forever grateful.

vii

TABLE OF CONTENTS

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LIST OF TABLES .
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LIST OF FIGURES .
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LIST OF SCHEMES .
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KEY TO SYMBOLS AND ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . xxii
1
CHAPTER 1
5

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INTRODUCTION .
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REFERENCES .

xi

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2.2.5 ONIOM .

CHAPTER 2 THEORETICAL BACKGROUND . . . . . . . . . . . . . . . . . . . . . .

8
2.1 Ab initio methods .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Cost-Saving Wavefunction-based Methods . . . . . . . . . . . . . . . . . . . . . . 13
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 Local Methods
2.2.2 Resolution of the Identity Approximation . . . . . . . . . . . . . . . . . . 16
2.2.3 Domain-Based Local Pair Natural Orbital Methods . . . . . . . . . . . . . 17
2.2.4 Composite Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.4.1 Correlation Consistent Composite Approach . . . . . . . . . . . 22
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Basis Sets .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4.1 Correlation Consistent Basis Sets . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.2 Effective Core Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.3 Auxiliary Basis Sets
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.3.1 AutoAux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
. . . . . . . . . . . . . . . . . . . . . . . . 33
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
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2.6 Vibrational Self Consistent Field Theory . . . . . . . . . . . . . . . . . . . . . . . 37
REFERENCES .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.4.4 Basis Set Superposition Error
Implicit Solvation Models .
2.5.1 COSMO .
2.5.2
2.5.3

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PCM/C-PCM . .
SMD .
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2.5

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CHAPTER 3 PREDICTION OF PKA OF LATE TRANSITION METAL HYDRIDES

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Introduction .

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3.1
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3.2 Theoretical Methods
.
3.3 Results and Discussion .

VIA A QM/QM APPROACH . . . . . . . . . . . . . . . . . . . . . . . . . 63
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
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3.3.1 Utility of DFT in the Real System . . . . . . . . . . . . . . . . . . . . . . 73
3.3.2 Utility of DFT in the Model Layer
. . . . . . . . . . . . . . . . . . . . . . 77
Impact of Exact Exchange on the Accuracy of DFT . . . . . . . . . . . . . 82
3.3.3

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viii

3.3.4

3.3.5
3.3.6
3.3.7
3.3.8 Comparison of Different Methodologies . . . . . . . . . . . . . . . . . .

Impact of Adding Grimme’s Empirical Dispersion Correction on the
. 84
Accuracy of DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Impact on the Choice of Basis Set
. . . . . . . . . . . . . . . . . . . . . . 86
Impact of Cavity Models on Implicit Solvation Models . . . . . . . . . . . 88
Impact of the Expansion of the Size of Model System . . . . . . . . . . . . 91
. 92
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
.
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3.4 Conclusions . .
APPENDIX . .
REFERENCES .

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Introduction .

4.1
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4.2 Computational Methods .
4.3 Results and Discussion .

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CHAPTER 4 UTILIZATION OF THE DOMAIN-BASED LOCAL PAIR NATURAL
ORBITAL METHODS WITHIN THE CORRELATION CONSISTENT
COMPOSITE APPROACH . . . . . . . . . . . . . . . . . . . . . . . . . . 108
. 108
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. 111
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. . . . . . . . . . . . . . . . . . 115
4.3.1 Energetic Properties for the Molecule Set
4.3.2 CPU Timing .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.3.3 Enthalpies and Timing for Linear Alkanes . . . . . . . . . . . . . . . . . . 127
4.3.4 Applications of DLPNO-ccCA . . . . . . . . . . . . . . . . . . . . . . . . 130
. 132
. 134
. 138

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4.4 Conclusions .
APPENDIX .
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REFERENCES .

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CHAPTER 5 COMPUTATIONAL CHEMISTRY CONSIDERATIONS IN

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Introduction .

CATALYSIS: REGIOSELECTIVITY AND METAL-LIGAND
DISSOCIATION .
5.1
.
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5.2 Computational Methods .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
. 149
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. 153
5.2.1 Computational methods for hydroformylation . . . . . . . . . . . . . . . . 153
5.2.2 Computational methods for ligand dissociation . . . . . . . . . . . . . . . 155
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.3.1 Regioselectivity in hydroformylation . . . . . . . . . . . . . . . . . . . . . 156
5.3.2 Metal-ligand dissociation in organometallics . . . . . . . . . . . . . . . . . 162
. 163
. 164

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5.4 Conclusions .
REFERENCES .
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5.3 Results and Discussion .

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CHAPTER 6 VIBRATIONAL POTENTIAL ENERGY SURFACES WITH THE
CORRELATION CONSISTENT COMPOSITE APPROACH AND
DENSITY FUNCTIONAL THEORY . . . . . . . . . . . . . . . . . . . . . 171
. 171
. 174
6.2.1 DFT Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.2.2
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ccCA Calculations
6.3 Results and Discussion .

6.1
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6.2 Computational Methods .

Introduction .

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ix

. .
6.3.1 Diatomics . .
6.3.2 H2O, CO2, NH3 .
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6.3.3 Hydrocarbons .
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6.3.4 Aminophenol
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6.4 Conclusions .
APPENDIX .
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REFERENCES .

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. 176
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CHAPTER 7 CHARGE STABILIZATION OF HIGH POTENTIAL ZINC

PORPHYRIN-FULLERENE VIA AXIAL LIGATION OF
TETRATHIAFULVALENE . . . . . . . . . . . . . . . . . . . . . . . . . . 205
. 205
. 206
. 208

7.1
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7.2 Computational Contributions and Analysis . . . . . . . . . . . . . . . . . . . . .
REFERENCES .
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Introduction .

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CHAPTER 8 SAMPL6 HOST-GUEST CHALLENGE: BINDING FREE ENERGIES

8.1
8.2 Methods .

Introduction .
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8.3 Results .

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VIA A MULTISTEP APPROACH . . . . . . . . . . . . . . . . . . . . . . . 213
. 213
. 216
8.2.1
System Preparation and Simulation Protocol . . . . . . . . . . . . . . . . . 216
8.2.2 Quantum Mechanical Calculations . . . . . . . . . . . . . . . . . . . . . . 219
. 220
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8.3.1 CB8 .
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8.3.2 OA .
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8.3.3 TEMOA .
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8.3.4 Quantum Mechanical Calculations . . . . . . . . . . . . . . . . . . . . . . 227
. 234
Submission Analysis
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
Impact of Truncated Basis Sets . . . . . . . . . . . . . . . . . . . . . . . . 235
Impact of the Extrapolation Scheme B-parameter
. . . . . . . . . . . . . . 237
Impact of Representative Geometries . . . . . . . . . . . . . . . . . . . . . 238
. 240
. 242
. 245

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8.4.1
8.4.2
8.4.3
8.4.4

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8.5 Conclusions .
APPENDIX .
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REFERENCES .

8.4 Discussion .

CHAPTER 9 CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . 251
. 255

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REFERENCES .

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x

LIST OF TABLES

Table 2.1: Summary of ccCA-TM and rp-ccCA steps.

. . . . . . . . . . . . . . . . . . . . 25

Table 3.1: Summary of the density functionals utilized.

. . . . . . . . . . . . . . . . . . . 71

Table 3.2: Theoretical methods for the description of real and model systems within the
two-layer ONIOM scheme using C-PCM, COSMO, and SMD for utility of
DFT in the real layer.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

.

.

Table 3.3: Theoretical methods for the description of real and model systems within the
two-layer ONIOM scheme using C-PCM, COSMO, and SMD for utility of
DFT in the model layer. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Table 3.4: MADs in pKa values of GGA, M-GGA, H-GGA, and HM-GGA Types of
Functionals for Comparison of DFT and DFT-D3 Relative to Experiment with
SMD.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

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Table 3.5: MADs in pKa values relative to experiment for four functionals when changing

the basis set used for the model layer.

. . . . . . . . . . . . . . . . . . . . . . . 87

Table 3.6: MADs of five cavity models in pKa values relative to experiment using the
ONIOM(PBE, M06-L, B3LYP, and M06/aug-cc-pVTZ:B97-D/LANL2DZ)
scheme.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

. .

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.

.

Table 3.7: MADs in pKa values relative to experiment of three expansions of model

system of TM hydrides with SMD. . . . . . . . . . . . . . . . . . . . . . . . . . 92

Table 3.8: Predicted

for

pKa values

are ONIOM
(B3LYP-D3/aug-cc-pVTZ:B97-D3/ SDD), B97-D3/SDD, B3LYP-D3/SDD,
ONIOM(B3LYP/aug-cc-pVTZ:HF/LANL2DZ),
and
ONIOM(CCSD(T)/aug-cc-pVTZ :B97-D3/SDD), respectively. . . . . . . . . . . 94

Schemes A-E, which

Table 3.9: Summary of the basis sets utilized. . . . . . . . . . . . . . . . . . . . . . . . . . 97

Table 3.10: MADs in pKa values of GGA, M-GGA, H-GGA, HM-GGA, and DH-GGA
functionals within low-level methods with solvation models relative to
experiment, with respect to central TM atoms of the TM Hydrides. All of the
results are from calculations with ONIOM(B97-D, M06-L, B3LYP, and M06/
aug-cc-pVTZ:DFT/LANL2DZ) scheme. . . . . . . . . . . . . . . . . . . . . . . 97

xi

Table 3.11: MADs in pKa values of GGA, M-GGA, H-GGA, HM-GGA, and DH-GGA
functionals within low-level methods with solvation models relative to
experiment, with respect to ligands of the TM hydrides. All of the results are
from calculations with ONIOM(B97-D,M06-L, B3LYP,
and M06/
aug-cc-pVTZ:DFT/LANL2DZ) scheme. . . . . . . . . . . . . . . . . . . . . . . 98

Table 3.12: MADs in pKa values of GGA, M-GGA, H-GGA, HM-GGA, and DH-GGA
functionals within high-level methods with solvation models relative to
experiment, with respect to central TM atoms of the TM Hydrides. All of the
results are from calculations with ONIOM(DFT/aug-cc-pVTZ:B97-D,M06L,
and B3LYP/LANL2DZ) scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Table 4.1: Summary of the different variants of ccCA utilized in Chapter 4.

. . . . . . . . . 113

Table 4.2: Summary of the approximations, methods, and auxiliary basis sets (ABS)

utilized in this work for SCF and post-HF calculations.

. . . . . . . . . . . . . . 114

Table 4.3: Slope, intercept, and R2 of the calculated and experimental ∆Hf. The mean
signed deviation (MSD), mean absolute deviation (MAD), standard deviation
(STDEV), and maximum (MAX) deviation for four variants of ccCA based on
the Peterson (P), Schwartz-3 (S3), and Schwartz-4 (S4) extrapolation schemes.
The P and S3 extrapolated values are averaged for PS3. Triple and quadruple-ζ
level basis sets (TQ) were used for all two-point extrapolations. All deviations
are in kcal mol−1.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

.

.

Table 4.4: Mean signed deviation (MSD), mean absolute deviation (MAD), standard
deviation (STDEV), and maximum (MAX) deviation for all schemes. All
deviations are in kcal mol−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Table 4.5: Percent CPU time savings for the three schemes of ABS implementation within
DLPNO-ccCA and RI-ccCA relative to ccCA. The mean percent difference
from ccCA, the most efficient (MAX), and the least efficient (MIN) percent
CPU time savings relative to ccCA timings are shown. All timing studies were
done with ORCA. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

.

.

.

Table 4.6: Deviations in kcal mol−1 from experimental ∆Hf for linear alkanes (CnH2n+2
1 ≤ n ≤ 8) using the atomization approach and using isodesmic approaches
(shown in parentheses). .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Table 4.7: Percent CPU time savings for RI-ccCA and DLPNO-ccCA (FB) relative to
ccCA for linear alkanes (CnH2n+2 1 ≤ n ≤ 8). All timing studies were done
with ORCA. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

. .

. .

.

Table 4.8:

Interactions energies of select examples from the S66 and L7 molecule sets.
All interaction energies are in kcal mol−1. . . . . . . . . . . . . . . . . . . . . . 131

xii

Table 4.9: Component breakdown of the DLPNO-ccCA calculated interaction energies
from the S66 and L7 datasets with counterpoise corrections included. All
interaction energies are in kcal mol−1. . . . . . . . . . . . . . . . . . . . . . . . 132

Table 4.10: Molecule list used for full set calculations. . . . . . . . . . . . . . . . . . . . . . 135

Table 4.11: MP2/CBS counterpoise-corrected interaction energies

calculated for
molecules in the S66 data set used to compare DLPNO-ccCA interaction
energies from Reference 126. All interaction energies are in kcal mol−1.

. . . . 136

Table 4.12: MP2/CBS counterpoise-corrected interaction energies calculated for the
coronene dimer used to compare DLPNO-ccCA interaction energies from
Reference 127. All interaction energies are in kcal mol−1.

. . . . . . . . . . . . 136

Table 4.13: DFT-D3/def2-QZVPP

energies
calculated for the coronene dimer used to compare DLPNO-ccCA interaction
energies from Reference 127. All interaction energies are in kcal mol−1.

non-counterpoise-corrected

interaction

. . . . 136

Table 5.1: A summary of the effect of ∆∆E‡ in kcal mol−1 on the linear-to-branched

ratio (l:b) ratio for hydroformylation. . . . . . . . . . . . . . . . . . . . . . . . . 152

Table 5.2: Comparison of several density functionals to linear-to-branched ratios from

experiment for ee-[Rh(H)(CO)(L)(olefin)] complexes. . . . . . . . . . . . . . .
Table 5.3: Comparison of the approximate ∆∆E‡s based on the calculated l:b ratios for
Experimental ∆∆E‡s are an

ee-
approximation of experimental l:b ratios. All ∆∆E‡s are in kcal mol−1. . . . . . 159

[Rh(H)(CO)(L)(olefin)] complexes.

. 158

Table 5.4: Results using DLPNO methods to predict the linear-to-branched ratio for ee-

[Rh(H)(CO)(DIPHOS)(propene)].

. . . . . . . . . . . . . . . . . . . . . . . . . 161

Table 5.5: Comparison of the gas-phase ligand dissociation energy of H2O from the Pt
complex calculated with DLPNO-rp-ccCA and RI-DFT-D3/aug-cc-pVnZ. All
energies are in kcal mol−1 and are BSSE-corrected. . . . . . . . . . . . . . . . . 163

Table 6.1: Percent CPU Time relative to CCSD(T,full)/aug-cc-pCV5Z to generate all 17

grid points of the PEC for select diatomics.

. . . . . . . . . . . . . . . . . . .

. 180

Table 6.2: Calculated frequencies

in cm−1 for B3LYP/aug-cc-pVTZ, ccCA, and

CCSD(T, full)/aug-cc-pCV5Z for diatomics in Table 6.1.

. . . . . . . . . . . . . 181

Table 6.3: VCIPSI-PT2 frequencies using a combination of TPSS and ccCA for single
mode and vibrational mode-mode coupling potentials. The use of PECs/PESs
is denoted as single:coupled.

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. 186

xiii

Table 6.4: Vibrational frequencies predicted with VCIPSI-PT2 for selected vibrations of

cis-3-aminophenol and trans-3-aminophenol.

. . . . . . . . . . . . . . . . . . . 187

Table 6.5: Calculated frequencies of diatomic and small polyatomic molecules in

cm−1 obtained with ccCA potentials.

. . . . . . . . . . . . . . . . . . . . . . . 193

Table 6.6: Calculated frequencies of diatomics in cm−1 obtained with TPSS/cc-pVnZ

and B3LYP/cc-pVnZ potentials.

. . . . . . . . . . . . . . . . . . . . . . . . . . 194

Table 6.7: Calculated

frequencies

of

selected

diatomics

in

cm−1 with

TPSS/aug-cc-pVnZ and B3LYP/aug-cc-pVnZ potentials. . . . . . . . . . . . . . 195

Table 6.8: Calculated vibrational frequencies for H2O and CO2 in cm−1 utilizing

VCIPSI-PT2 with TPSS/cc-pVnZ potentials.

. . . . . . . . . . . . . . . . . . . 195

Table 6.9: Calculated vibrational frequencies for H2O and CO2 in cm−1 utilizing

VCIPSI-PT2 with B3LYP/cc-pVnZ potentials.

. . . . . . . . . . . . . . . . . . 196

Table 6.10: Calculated vibrational frequencies for H2O and CO2 in cm−1 utilizing

VCIPSI-PT2 with TPSS/aug-cc-pVnZ potentials.

. . . . . . . . . . . . . . . . . 196

Table 6.11: Calculated vibrational frequencies for H2O and CO2 in cm−1 utilizing

VCIPSI-PT2 with B3LYP/aug-cc-pVnZ potentials.

. . . . . . . . . . . . . . . . 196

Table 6.12: Calculated vibrational frequencies for NH3 in cm−1 utilizing VCIPSI-PT2

with both TPSS and B3LYP potentials with the VTZ and aVTZ basis sets. . . . . 197

Table 8.1: Binding free energies for the CB8 host-guest complexes.

. . . . . . . . . . . .

. 222

Table 8.2: Binding free energies for the OA host-guest complexes. . . . . . . . . . . . . .

. 224

Table 8.3: Binding free energies for the TEMOA host-guest complexes.
Table 8.4: Binding free energies for the CB8 complexes in kcal mol−1 with schemes
involving not using the RI approximation, and changing the dielectric constant
of the implicit solvent with the truncated correlation consistent basis sets for
hydrogen.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

. . . . . . . . . . . 226

. .

.

.

.

. .

Table 8.5: Binding free energies for the CB8 complexes in kcal mol−1 with schemes
involving not using the RI approximation, changing the dielectric constant of
the implicit solvent, and two options for basis set choice when extrapolating to
the Kohn-Sham limit. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

Table 8.6: Predicted binding energies for OA and TEMOA using MMPBSA and RI-

B3PW91 after the removal of mean signed error (MSE).

. . . . . . . . . . . .

. 235

xiv

Table 8.7: Predicted binding energies when using different values for B in Equation 8.1

for two-point extrapolations using cc-pVDZ and cc-pVTZ with RI-B3PW91-D3. 238

Table 8.8: Fitting parameter values obtained when using Jensen’s extrapolation scheme
for each component in calculating the binding energy (Equation 8.1). The host
and guest are counterpoise-corrected before the extrapolation was performed.

.

. 243

xv

LIST OF FIGURES

Figure 3.1: From left to right, the compounds are TM(depe)2, TM(depp)2, TM(PNP)2.
(a) The model system (bolded) within the ONIOM-1 QM/QM partitioning
scheme for TM hydrides with the TM atom (Ni, Pd, and Pt) and four
phosphorous atoms in the layer using the high-level method. (b) ONIOM-2:
The QM/QM partitioning scheme for TM hydrides with all the atoms within
the chelate rings in the layer using the high-level method.
(c) ONIOM-3:
The QM/QM partitioning scheme for TM hydrides with all except for the
very outside methyl group in the layer using the high-level method.

. . . . . . . 70

Figure 3.2: MADs in pKa values for the density functionals within low-level methods
relative to experiment. All of the results are from calculations with
ONIOM(PBE,M06L,B3LYP,M06/aug-cc-pVTZ:DFT/LANL2DZ) scheme.
The results of using the four functionals in the model layer are averaged for
the molecule set. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. .

. 75

Figure 3.3: MADs in pKa values for five types of density functionals, GGA, M-GGA,
H-GGA, HM-GGA, and DH-GGA functionals, within low-level methods
relative to experiment. All of the results are from calculations with
ONIOM(PBE,M06L,B3LYP,M06/aug-cc-pVTZ:DFT/LANL2DZ) scheme.
The results of using the four functionals in the model layer are averaged for
the molecule set. . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Figure 3.4: MADs in pKa values for fourteen GGA, M-GGA, H-GGA, HM-GGA, and
DH-GGA functionals within high-level methods relative to experiment. All
of the results are from calculations with ONIOM(DFT/aug-cc-pVTZ:B97-
D,M06L,B3LYP/LANL2DZ) scheme. The MADs in pKa values for the three
functionals in the real layer are averaged for the molecule set.

. . . . . . . . . . 79

Figure 3.5: MADs in pKa values for five types of density functionals, GGA, M-GGA,
H-GGA, HM-GGA, and DH-GGA functionals, within high-level methods
relative to experiment. All of the results are from calculations with
ONIOM(DFT/aug-cc-pVTZ:B97-D,M06L,B3LYP/LANL2DZ) scheme.

. . . . 80

Figure 3.6: MADs of PBE0 vs. percentage of exact exchange where (a) the average
MAD for each metal center and (b) the average MAD for each ligand. All of
the
from ONIOM(PBE0/aug-cc-pVTZ:B97-D/LANL2DZ)
scheme with SMD.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

results

are

xvi

Figure 3.7: MADs in pKa values of DFT and DFT-D3 with SMD relative to experiment,
with respect to central TM atoms and ligand size of TM hydrides. The results
are from calculations involving the ONIOM(DFT(-D3)/aug-cc-pVTZ:B97-
D3/LANL2DZ) scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Figure 3.8: MADs of DFT vs. DFT-D3 with SMD for the functionals in the model layer,
i.e. ONIOM(DFT(-D3)/aug-cc-pVTZ:B97-D3/LANL2DZ). The MADs are
averages of the full molecule set.

. . . . . . . . . . . . . . . . . . . . . . . . . 86

Figure 3.9: Mean absolute deviation (MAD) in pKa values when utilizing different basis
sets relative to experiment, with respect to central TM atoms and ligand
size of TM hydrides where (a) the cc-pVnZ and aug-cc-pVnZ (n=D,T) are
considered for the model layer (HL method) and (b) LANL2DZ and SDD
ECPs are considered for the real layer (LL method).

. . . . . . . . . . . . . . . 88

Figure 3.10: Impact of radii models on (a) C-PCM, (b) COSMO, and (c) SMD. The
default cavities for C-PCM, COSMO, and SMD are UFF, Klamt, and SMD-
Coulomb, respectively. The average MADs are results from calculation with
the ONIOM (PBE, M06L, B3LYP, M06/ aug-cc-pVTZ : B97D/ LANL2DZ)
scheme and then categorized by metal and ligand.

. . . . . . . . . . . . . . . . 90

Figure 3.11: Comparison

and

calculated

of

the

experimental

via
methodological choices represented by their calculated values and the dotted
trend lines. The dashed black line denotes the 1:1 correspondence between
experiment and calculated pKa values.
Schemes A-E are ONIOM
(B3LYP-D3/ aug-cc-pVTZ : B97-D3/ SDD), B97-D3/ SDD, B3LYP-D3/
SDD, ONIOM (B3LYP/ aug-cc-pVTZ : HF/ LANL2DZ), and ONIOM
(CCSD(T)/ aug-cc-pVTZ : B97-D3/ SDD). . . . . . . . . . . . . . . . . . . .

pKa values

. 93

Figure 4.1: Differences in electronic energies (mEh) using Pipek-Mezey (PM) and
Foster-Boys (FB) localization schemes using the def2/JK ABS within
DLPNO-MP2 for complete basis set extrapolation using a combined
Peterson-Schwartz-3 extrapolation scheme (PS3(TQ)). Included subsets are
based on the presence of certain elements (hydrocarbons, halogenated,
chalcogenated, pnictogenated, and Period 3) and electronic features
(aromatic, carbonyl, multiple bonds) as well as the full molecule set. Points
within the dashed lines represent differences less than 0.1 mEh. The box
plots depict the distribution of data within each subset where the band in the
middle represents the median of the data and data points shown as black
circles are more than 3 standard deviations from the median.

. . . . . . . . . . 117

xvii

Figure 4.2: Differences in electronic energies (mEh) between the Pipek-Mezey (PM) and
Foster-Boys (FB) localization methods for all three schemes within DLPNO-
CCSD(T) for the same subsets in Figure 4.1. The dashed lines represent
differences of less than 0.1 mEh. The box plots depict the distribution of
data within each subset where the band in the middle represents the median
of the data and data points shown as black circles are more than 3 standard
deviations from the median. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Figure 4.3: Differences

between

energies

electronic

the CCSD(T)

the
DLPNO-CCSD(T) electronic energies in mEh using the (a) Pipek-Mezey
(PM) and (b) Foster-Boys (FB) localization methods for all three schemes
for the subsets of the full molecule set shown in Figure 4.1. The dashed lines
represent differences less than 0.1 mEh. The box plots depict the distribution
of data within each subset where the band in the middle represents the
median of the data and data points shown as black circles are more than 3
standard deviations from the median.

and

. . . . . . . . . . . . . . . . . . . . . . . 119

Figure 4.4: CPU time of each individual step within (a) ccCA, (b) RI-ccCA, and (c)
DLPNO-ccCA for selected species from the molecule set. The Other category
represents the timing of the MP2/aug-cc-pVDZ, MP2/aug-cc-pVTZ, MP2/cc-
pVTZ, and MP2/cc-pVTZ-DK calculations as these calculations use a small
percentage of the total CPU time. All timing calculations were done with the
ORCA software package.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Figure 4.5: CPU time ratios of DLPNO-ccCA (FB) to (a) ccCA and (b) RI-ccCA. The
ratios for Scheme 1 (blue circle), Scheme 2 (black x), and Scheme 3 (green
triangle) are shown on a log-log scale. All timing was done with C1 symmetry
enforced and done in ORCA.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Figure 4.6: CPU time for ccCA, RI-ccCA, and all 3 schemes for DLPNO-ccCA for the

linear alkanes.

.

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Figure 4.7: Deviations in ∆Hf for ccCA, RI-ccCA, and all 3 schemes for DLPNO-ccCA

for the linear alkanes using the isodesmic approach.

. . . . . . . . . . . . . . . 137

Figure 5.1: Hydroformylation reaction converting olefins to linear and branched

aldehydes via a Rh catalyst.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Figure 5.2: A model of the two reaction pathways for hydroformylation where ∆E

and
‡
are the reaction barriers for forming the linear and branched product,
∆E
b
respectively. The energy difference between the two reaction barriers is
denoted as ∆∆E‡. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

.

.

‡
l

xviii

Figure 5.3: Computationally determined 3D structures of ee-[Rh(H)(CO)(DIPHOS)
(propene)] catalyst complex (top) and the dissociation reaction of H2O from
the cationic (diimine)(aquo)PtII complex (bottom). . . . . . . . . . . . . . . . . 155

Figure 6.1: Mean absolute deviation (MAD) of vibrational frequencies for diatomics
using TPSS/VTZ (blue), B3LYP/VTZ (green), TPSS/aVTZ (purple),
B3LYP/aVTZ (red), and ccCA-S4 (black).

. . . . . . . . . . . . . . . . . . . . 178

Figure 6.2: MAD of vibrational frequencies for H2O, CO2, and NH3 using TPSS/VnZ
(blue), B3LYP/VnZ (green), TPSS/aVnZ (purple), B3LYP/aVnZ (red), and
ccCA-S4 (black). For H2O and CO2, n = ∞. For NH3, n=T.

. . . . . . . . . . 182

Figure 6.3: MAD of vibrational frequencies for C2H2, C2H4, and C2H6 using TPSS/VTZ
(blue), B3LYP/VTZ (green), TPSS/aVTZ (purple), B3LYP/aVTZ (red), and
ccCA-S4 (black).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

.

.

Figure 6.4:

Infrared spectra for cis-3-aminophenol
(top) and trans-3-aminophenol
(bottom) obtained with VCIPSI-PT2 frequencies with ccCA potentials and
B3LYP/cc-pVTZ harmonic frequencies scaled by 1.0066. All intensities are
from the harmonic frequency calculations. A Lorentz broadening of 20
cm−1 was applied. The experimental frequencies and relative intensities
from Ref 8 are shown for comparison.

. . . . . . . . . . . . . . . . . . . . .

. 189

Figure 6.5: Single mode potential energy curves for vibrational modes 8 (left) and 10
(right) of ethene (C=C and C-H symmetric stretches) generated with ccCA
(black) and TPSS/VTZ (red).

. . . . . . . . . . . . . . . . . . . . . . . . . .

. 197

Figure 6.6: Vibrational coupling map for ethene (left) and ethane (right). The vibrational
mode couplings shown in black indicate strongly coupled modes that were
used for all FASTVCI approaches using ccCA.

. . . . . . . . . . . . . . . . . . 198

Figure 6.7: Vibrational

coupling map for

and trans-
3-aminophenol (right). The vibrational mode couplings shown in black
indicate strongly coupled modes that were used for all FASTVCI approaches
using ccCA.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

cis-3-aminophenol

(left)

. .

. .

Figure 7.1: MO6-2X/6-31G* molecular electrostatic potential maps, and the frontier
HOMO and LUMO of the optimized structures of (a) (F15P)Zn-C60 dyad
and (b) C60-(F15P)Zn:Py-phTTF triad. The isovalue used for the MO
depictions was 0.02 while the density value used was 0.0004.

. . . . . . . . . . 207

Figure 8.1: Guest molecules for the cucurbit[8]uril (CB8) host.

. . . . . . . . . . . . . . . 217

Figure 8.2: Guest molecules for the octa-acid (OA) and tetramethyl octa-acid (TEMOA)

hosts. .

.

.

.

.

.

.

.

.

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

xix

Figure 8.3: Host molecules: cucurbit[8]uril (CB8), octa-acid (OA), and tetramethyl octa-

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

acid (TEMOA).

.

Figure 8.4: Structures of the CB8 guest molecules inside the binding pocket generated

from the clustering analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

Figure 8.5: Structures of the OA guest molecules inside the binding pocket generated

from the clustering analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

Figure 8.6: Structures of the TEMOA guest molecules inside the binding pocket generated

from the clustering analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

Figure 8.7: Plots for calculated v. experimental results in kcal mol−1 for (a) CB8 (b)
OA, and (c) TEMOA for MMPBSA (blue), RI-B3PW91-D3 (black), and RI-
B3PW91 (green). The dashed lines in each corresponding color refers to the
best fit line where the statistical outlier (OA-G2) is removed for (b) and (c).
The dashed gray line is the y = x line. . . . . . . . . . . . . . . . . . . . . . . . 232

Figure 8.8: Error plots from experimental results in kcal mol−1 for (a) CB8 (b) OA, and
(c) TEMOA for MMPBSA (blue), RI-B3PW91-D3 (black), and RI-B3PW91
(green). .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

. 233

Figure 8.9: Plots for the correlations calculated after the mean signed errors are removed
from the results in Tables 8.1-8.3 versus experimental results in kcal mol−1 for
(a) OA, and (b) TEMOA for MMPBSA (blue), RI-B3W91-D3 (black). The
dashed lines in each corresponding color refers to the best fit line where the
statistical outlier (OA-G2) for RI-B3PW91-D3 is removed for (a). The dashed
gray line corresponds to the y=x line. . . . . . . . . . . . . . . . . . . . . . . . 244

xx

LIST OF SCHEMES

Scheme 3.1: The direct thermodynamic scheme

. . . . . . . . . . . . . . . . . . . . . . . 72

xxi

KEY TO SYMBOLS AND ABBREVIATIONS

AO atomic orbital
aug-cc-pCVnZ augmented correlation consistent polarized core-valence n-tuple ζ basis set
aug-cc-pVnZ augmented correlation consistent polarized valence n-tuple ζ basis set
aug-cc-pVnZ-PP aug-cc-pVnZ with psuedopotential
CBS complete basis set
CB8 cucurbit[8]uril
cc-pCVnZ correlation consistent polarized core-valence n-tuple ζ basis set
cc-pVnZ correlation consistent polarized valence n-tuple ζ basis set
cc-pVnZ-PP cc-pVnZ with Psuedopotential
ccCA correlation consistent composite approach
CCSD(T) coupled cluster singles-and-doubles with perturbative triples correction
COSMO conductor-like screening model
C-PCM conductor-like polarizable continuum model
DFT density functional theory
DLPNO domain-based local pair natural orbital
∆Hf enthalpy of formation
HF Hartree-Fock
HOMO highest occupied molecular orbital
LUMO lowest unoccupied molecular orbital
MAD mean absolute deviation
MAE mean absolute error
MD molecular dynamics
MM molecular mechanics
MMPBSA molecular mechanics energy combined with Poisson-Boltzmann surface area

continuum solvation method

xxii

MO molecular orbital
MPPT Møller-Plesset perturbation theory
MP2 Møller-Plesset second-order perturbation method
n-tuple n-multiple basis functions (n = 2, 3, 4, etc.)
OA octa-acid
PAO projected atomic orbitals
PCM polarizable continuum model
PES potential energy surface
PNO pair natural orbital
QM quantum mechanics
r2 correlation coefficient
RI resolution-of-the-identity
RMSE root mean square error
SAMPL statistical assessment of the modeling of proteins and ligands
SCF self-consistent field
SMD solvation model based on density
τ Kendall’s tau
τcrit critical Kendall’s tau
TEMOA tetramethyl octa-acid
TM transition metal
VCI vibrational configuration interaction
VPT2 vibrational second-order perturbation theory
VSCF vibrational self-consistent field
ZPE zero point energy

xxiii

CHAPTER 1

INTRODUCTION

Computational chemistry emerged as a field of chemistry even before the digital age began.
Seminal work of chemists, mathematicians, and physicists such as Erwin Schrödinger and Douglas
R. Hartree, as well as Chemistry Nobel Laureates John Pople and Walter Kohn, throughout the
20th century constructed the foundation of computational chemistry even before the digital age
began.1–10 With the advent of computers, computational chemistry has become so important
because it can provide valuable insight into chemical processes and properties that are difficult to
measure experimentally and rationale for mechanistic features within known chemical reactions.
There are many sub-fields of computational chemistry which can be categorized based upon their
theoretical foundation such as ab initio methods, density functional theory (DFT), semiempirical
methods, and molecular dynamics. These approaches are used to investigate the vast numbers of
chemical systems ranging from atoms to proteins and semiconductor materials like graphene and
TiO2. Among the more rigorous of these methods are ab initio approaches, or methods based on
first principles, which focus on modeling the electronic structure of atoms and molecules but often
at a high computational cost (memory, disk space, CPU time) relative to DFT and semiempirical
methods.

As computing power doubled every two years, following Moore’s Law observations, the usage
and development of electronic structure methods to account for the rapid increase in computing
power continues to grow, though the complexity of the mathematical approximations that still need
to be made on modern computing resources inhibits that growth. For ab initio methods, numerous
approximations and methods have been developed to reduce the computational cost and predict
thermodynamic properties within the same range of error as their canonical counterparts.15–24

In this dissertation, a focus is upon the development and integration of a number of strategies
–called ab initio composite strategies– to enable cost reduction in the prediction of thermodynamic
properties for both main group and transition metal species. As described in Chapter 3, the pKas of

1

several Group 10 transition metal hydrides were predicted utilizing multilevel approaches due to
the molecule size (Section 2.2.5) with DFT and ab initio methods, which provides insight into
models needed for the prediction of thermodynamic properties of transition metal hydrides and
other inorganic complexes that contain sterically bulky ligands.

While density functional methods are more commonly used, particularly for molecules of
increasing size and complexity based on their low computational scaling, computations using ab
initio methods can serve as an effective gauge for computational thermodynamic predictions. For
example, ab initio composite strategies (see Section 2.2.4), which utilizes a combination of lower
cost ab initio methods to effectively model a higher cost ab initio method at a fraction of the
computational cost, can be used to reduce the cost associated with prediction of thermochemical
properties. One such composite approach developed in the Wilson group is the correlation consistent
Composite Approach, or ccCA, which has successfully been applied for main group and transition
metal thermochemistry, predicting thermochemical properties like enthalpies of formation, pKas,
and bond dissociation energies within main group chemical accuracy (1 kcal mol−1) and transition
metal chemical accuracy (3 kcal mol−1) on average.25–29

The ccCA variant described in this dissertation in Chapter 4 utilizes the domain-based local pair
natural orbital (DLPNO) methods within the ccCA framework for main group thermochemistry,
denoted as DLPNO-ccCA, to expand the size limitations of ab initio composite methodologies. To
evaluate the efficacy of DLPNO-ccCA for main group thermochemistry, the electronic energies and
enthalpies of formation generated using ccCA, RI-ccCA, which uses the resolution-of-the-identity
(RI) approximation within MP2 to reduce the computational cost, and DLPNO-ccCA are compared.
DLPNO-ccCA was utilized for linear alkanes up to octane as well as molecular dimers exhibiting
noncovalent interactions such as hydrogen bonding and dispersion including the coronene dimer (72
atoms), which is one of the largest molecule targeted with a composite approach to date. Therefore,
showing the effectiveness of composite strategies and their usefulness for larger main group organic
species.

DLPNO-ccCA was also used for transition metal species and organometallic complexes, as

2

shown in Chapter 5. This approach draws on the ccCA variants for the 3d and 4d transition metal-
containing species, ccCA-TM and rp-ccCA, to predict enthalpies of formation, gas phase ligand
dissociation energies, and the regioselectivity of hydroformylation, the largest volume homogeneous
chemical reaction in industry for chemical production. This study shows the applicability of ab
initio composite methods for computational catalysis, which is typically analyzed with density
functional methods.

Vibrational spectroscopy is an important approach for characterizing the structural and
dynamical properties of molecules. Theoretical methods used for vibrational spectroscopy are
often restricted to scaling frequencies within the harmonic approximation or utilizing potential
energy surfaces (PES) generated with computationally costly ab initio methods that characterize
vibrational motion.
In Chapter 6, the correlation consistent Composite Approach (ccCA) and
density functional theory (DFT) have been used to generate PES for polyatomic molecules (2-15
atoms). Frequencies, dipole moments, and infrared absorbance intensities are predicted in tandem
with vibrational self-consistent field (VSCF) and post-VSCF theory to reduce the computational
cost associated with generating PESs for anharmonic mode-mode couplings, calculating
contributions from anharmonic corrections to the potential, and predicting vibrational frequencies
within several cm−1.

Additional research described in this dissertation includes a collaborative effort with Francis
D’Souza at the University of North Texas (Chapter 7) to achieve the goal of artificial photosynthesis
by modeling the donor-acceptor ability of zinc porphyrin-fullerene dyads and triads with DFT
through modeling the frontier orbitals.

In Chapter 8, a combination of molecular dynamics, molecular mechanics, and density
functional methods were used for the sixth Statistical Assessments of the Modeling of Proteins
and Ligands (SAMPL) blind prediction challenge for host-guest binding.
In this challenge,
participants are expected to predict binding affinities and other properties for small molecules
within a host system. The SAMPL challenge allows for the comparison of methods for binding
affinity prediction by using statistical tools and modeling methods that can be essential for

3

host-guest systems. Empirical dispersion corrections, the RI approximation, and truncated basis
sets were utilized to probe how electronic structure approaches that reduce the computational cost
contribute to predicting binding affinities, which provides insight into favorable quantum chemical
strategies for host-guest binding affinities.

With the wide range of applications, possible applications, and development presented in
this dissertation, including biochemical processes, astrochemistry, artificial photosynthesis, and
organometallic catalysis, the size and complexity of the molecules examined presents the challenges
and successes of modeling organic, inorganic, and organometallic complexes effectively with DFT
and ab initio composite strategies.

4

REFERENCES

5

REFERENCES

[1] Schrödinger, E. Quantisierung als Eigenwertproblem. Ann. Phys. 1926, 384, 361–376.
[2] Schrödinger, E. An undulatory theory of the mechanics of atoms and molecules. Phys. Rev.

1926, 28, 1049–1070.

[3] Hartree, D. R. The Wave Mechanics of an Atom with a non-Coulomb Central Field. Part III.
Term Values and Intensities in Series in Optical Spectra. Math. Proc. Cambridge Philos. Soc.
1928, 24, 426–437.

[4] Fock, V. Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems.

Zeitschrift für Phys. 1930, 61, 126–148.

[5] Lennard-Jones, J.; Pople, J. A. The Molecular Orbital Theory of Chemical Valency. IV. The
Significance of Equivalent Orbitals. Proc. R. Soc. A Math. Phys. Eng. Sci. 1950, 202, 166–180.
[6] Hohenberg, P.; Kohn, W. Inhomogeneous Electron Gas. Phys. Rev. 1964, 136, B864–B871.
[7] Kohn, W.; Sham, L. J. Self-consistent equations including exchange and correlation effects.

Phys. Rev. 1965, 140, A1133–A1138.

[8] Honig, B.; Karplus, M. Implications of torsional potential of retinal isomers for visual

excitation. Nature 1971, 229, 558–560.

[9] Warshel, A.; Karplus, M. Calculation of ground and excited state potential surfaces of
conjugated molecules. I. Formulation and parametrization. J. Am. Chem. Soc. 1972, 94,
5612–5625.

[10] Warshel, A.; Karplus, M. Calculation of ππ* Excited State Conformations and Vibronic

Structure of Retinal and Related Molecules. J. Am. Chem. Soc. 1974, 96, 5677–5689.

[11] Tsuzuki, S.; Uchimaru, T.; Tanabe, K. Ab Initio Calculations of Intermolecular Interaction

Potentials of Corannulene Dimer. J. Phys. Chem. A 1998, 102, 740–743.

[12] Kobayashi, R. A CCSD(T) Study of the Relative Stabilities of Cytosine Tautomers. J. Phys.

Chem. A 1998, 102, 10813–10817.

[13] Fox, S. J.; Dziedzic, J.; Fox, T.; Tautermann, C. S.; Skylaris, C.-K. Density functional theory
calculations on entire proteins for free energies of binding: Application to a model polar
binding site. Proteins Struct. Funct. Bioinforma. 2014, 82, 3335–3346.

[14] Riplinger, C.; Sandhoefer, B.; Hansen, A.; Neese, F. Natural triple excitations in local coupled

cluster calculations with pair natural orbitals. J. Chem. Phys. 2013, 139, 134101.

[15] Kendall, R. A.; Früchtl, H. A. The impact of the resolution of the identity approximate integral
method on modern ab initio algorithm development. Theor. Chem. Acc. 1997, 97, 158–163.

6

[16] Almlöf, J. Elimination of energy denominators in Møller-Plesset perturbation theory by a

Laplace transform approach. Chem. Phys. Lett. 1991, 181, 319–320.

[17] Häser, M.; Almlöf, J. Laplace transform techniques in Møller-Plesset perturbation theory. J.

Chem. Phys. 1992, 96, 489–494.

[18] Wilson, A. K.; Almlöf, J. Møller-Plesset correlation energies in a localized orbital basis using

a Laplace transform technique. Theor. Chem. Acc. 1997, 95, 49–62.

[19] Lambrecht, D. S.; Doser, B.; Ochsenfeld, C. Rigorous integral screening for electron

correlation methods. J. Chem. Phys. 2005, 123, 184102.

[20] Schütz, M.; Hetzer, G.; Werner, H.-J. Low-order scaling local electron correlation methods.

I. Linear scaling local MP2. J. Chem. Phys. 1999, 111, 5691–5705.

[21] Sæbø, S.; Tong, W.; Pulay, P. Efficient elimination of basis set superposition errors by the
local correlation method: Accurate ab initio studies of the water dimer. J. Chem. Phys. 1993,
98, 2170–2175.

[22] Handy, N. C.; Carter, S. Large vibrational variational calculations using ‘multimode’ and an

iterative diagonalization technique. Mol. Phys. 2004, 102, 2201–2205.

[23] Scribano, Y.; Benoit, D. M. Iterative active-space selection for vibrational configuration
interaction calculations using a reduced-coupling VSCF basis. Chem. Phys. Lett. 2008, 458,
384–387.

[24] Raghavachari, K.; Trucks, G. W.; Pople, J. A.; Head-Gordon, M. A fifth-order perturbation

comparison of electron correlation theories. Chem. Phys. Lett. 1989, 157, 479–483.

[25] DeYonker, N. J.; Cundari, T. R.; Wilson, A. K. The correlation consistent composite approach

(ccCA): An alternative to the Gaussian-n methods. J. Chem. Phys. 2006, 124, 114104.

[26] DeYonker, N. J.; Wilson, B. R.; Pierpont, A. W.; Cundari, T. R.; Wilson, A. K. Towards the
intrinsic error of the correlation consistent Composite Approach (ccCA). Mol. Phys. 2009,
107, 1107–1121.

[27] Laury, M. L.; DeYonker, N. J.; Jiang, W.; Wilson, A. K. A pseudopotential-based composite
method: The relativistic pseudopotential correlation consistent composite approach for
molecules containing 4d transition metals (Y-Cd). J. Chem. Phys. 2011, 135, 214103.

[28] Jiang, W.; DeYonker, N. J.; Determan, J. J.; Wilson, A. K. Toward accurate theoretical
thermochemistry of first row transition metal complexes. J. Phys. Chem. A 2012, 116, 870–
885.

[29] Riojas, A. G.; Wilson, A. K. Solv-ccCA: Implicit solvation and the correlation consistent
composite approach for the determination of pKa. J. Chem. Theory Comput. 2014, 10, 1500–
1510.

7

CHAPTER 2

THEORETICAL BACKGROUND

The fundamental equation of quantum mechanics is the time-independent Schrödinger

equation,1

ˆHΨ = EΨ

(2.1)

in which finding an approximate solution is an integral part of computational chemistry. The
Hamiltonian operator, ˆH, operates on the wavefunction describing the system of interest, Ψ, and
returns an energy eigenvalue, E, for the wavefunction, which is an eigenfunction by definition. The
Hamiltonian ( ˆH) is the total energy operator that describes the interactions of the N electrons and
the M nuclei with nuclear charge Z via the kinetic energy of the electrons (i, j) and nuclei (A, B),
the nuclei-electron interactions at a distance riA, the electron-electron interactions at a distance rij,
and the nuclei-nuclei interactions at a distance RAB, as shown in Equation 2.2.

N(cid:88)

i=1

i − M(cid:88)

A=1

∇2

1
2

ˆH =

∇2

1

2MA

A − N(cid:88)

i=1

ZA
riA

− N(cid:88)

N(cid:88)

1
rij

+

N(cid:88)

N(cid:88)

i=1

j>1

A=1

B>A

ZAZB
RAB

(2.2)

The difficulty with multi-electron systems is the inadequate description of the electron-electron
interactions, which implies that the Schrödinger equation is only exactly solvable for one-electron
systems. Therefore, approximations must be made to account for the electron-electron interactions
present in chemical systems.

An important approximation is the Born-Oppenheimer approximation,2 which assumes that
the nuclei are stationary relative to the electrons in a system since the nuclei are much heavier
than electrons and that electrons move faster than nuclei giving the appearance of stationary nuclei.
Through this approximation, the Hamiltonian can be reduced to the electronic Hamiltonian,3
Equation 2.3.

ˆHelec = − N(cid:88)

i − N(cid:88)

∇2

1
2

i=1

− N(cid:88)

N(cid:88)

i=1

j>1

ZA
riA

1
rij

(2.3)

i=1

8

Equation 2.3 represents the motion of N electrons in a field of M nuclei. The kinetic energy
term for nuclei is approximated as zero. The nuclei-nuclei Coulombic energy term is a constant
term, not integrated through all space and is thus removed from Equation 2.2. Using the electronic
Hamiltonian results in the electronic Schrödinger equation, Equation 2.4.

ˆHelecΨelec = EelecΨelec

(2.4)

The electronic wavefunction (Ψelec) is dependent only on the electron spatial coordinates;
however, electrons have a spin component that is included in the overall wavefunction.3 Since the
wavefunction should not be solely described by neither spatial nor spin components, a suitable
principle that combines both descriptions of the electronic wavefunction is required. Starting with
the antisymmetry principle, the electronic wavefunction must change signs with respect to electron
exchange of both the spatial and spin coordinates, Equation 2.5.3

Ψ(cid:0)(cid:126)x1, . . . , (cid:126)xi, . . . , (cid:126)xj, . . . , (cid:126)xN

(cid:1) = −Ψ(cid:0)(cid:126)x1, . . . , (cid:126)xj, . . . , (cid:126)xi, . . . , (cid:126)xN

(cid:1)

(2.5)

Equation 2.5 shows an antisymmetric wavefunction with respect to the coordinate vectors for
N electrons. The Hartree product, a many-electron wavefunction that considers non-interacting
electrons, is shown in Equation 2.6.3

ΨHP ((cid:126)x1, (cid:126)x2, . . . , (cid:126)xN ) = χi ((cid:126)x1) χj((cid:126)x2)··· χk((cid:126)xN )

(2.6)

Equation 2.6 shows the Hartree product where χ is the k spin orbitals for N electrons and their
respective spatial and spin coordinates. While the Hartree product does not satisfy the antisymmetry
principle outlined in Equation 2.5, a linear combination of these Hartree products is generated to
satisfy this principle. The generalized form of this linear combination is known as a Slater

9

determinant, shown in Equation 2.7a.

Ψ ((cid:126)x1, (cid:126)x2, . . . , (cid:126)xN ) =

1√
N !

=

1√
N !

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

N !(cid:88)

i

χ1 ((cid:126)x1) χ2 ((cid:126)x1)

χ1 ((cid:126)x2) χ2 ((cid:126)x2)

...

...

χ1 ((cid:126)xN ) χ2 ((cid:126)xN )

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

··· χk ((cid:126)x1)
··· χk ((cid:126)x2)
...
··· χk ((cid:126)xN )

...

(−1)pn Piχ1((cid:126)x1)··· χk ((cid:126)xN )

(2.7a)

(2.7b)

1√
N !

Equation 2.7a illustrates a Slater determinant where N is the number of electrons, χ is the k
spin orbitals where k is equal to 2N for each position x, the columns are electron orbitals, and the
factor is a normalization constant, the (−1)pn
rows represent electrons. In Equation 2.7b, the
represents the parity of the ith term, and P is a permutation operator acting on the Hartree product,
Equation 2.6. Slater determinants take advantage of the antisymmetry principle by representing a
multi-electron wavefunction in the form of a determinant. Useful properties of a Slater determinant
include the exchange of any two rows or columns, or interchanging two electrons, resulting in a
change of sign of the determinant. Also, any two rows or columns that are identical, indicating two
electrons with the same spin orbital, results in the wavefunction to be zero. Determinants can be
used to satisfy the antisymmetry principle and the Pauli exclusion principle, respectively.

Other approximations used to solve the Schrödinger equation in molecular quantum chemistry
include the combination of a theoretical method and basis set. This work focuses on two major
classes of methods that are used to approximate the Schrödinger equation, wavefunction methods,
or ab initio methods, and Density Functional Theory (DFT).

2.1 Ab initio methods

For ab initio methods, the fundamental approximation is the Hartree-Fock (HF) approximation,
which averages the effects of electron-electron interactions through an average potential νHF (i).4,5
The Fock operator is an effective one-electron operator, shown in Equation 2.8.

f (i) = −1
2

∇2

ZA
riA

+νHF (i)

(2.8)

i − M(cid:88)

A=1

10

In Equation 2.8, νHF (i) is the average potential of electron i in the field of the other electrons in
the system. Using the Fock operator reduces the multi-electron Schrödinger equation to numerous
one-electron equations. Roothaan-Hall equations are utilized to make the HF equations into a
matrix form as shown in Equation 2.9.

FC = εSC

(2.9)

In Equation 2.9, F is the Fock matrix, S is the overlap matrix, C is the coefficient matrix, and ε is
the orbital energy obtained from applying the Fock operator to a wavefunction. The elements of
these matrices represent integrals involving basis functions defined by the linear combination of
atomic orbitals, or LCAO, approximation, shown in Equation 2.10.

Ψi =

Cµiφµ i = 1, 2, . . . , K

(2.10)

Equation 2.10 shows the LCAO approximation through K basis functions, denoted by φ,
representing the wavefunction of electron i. The elements of C are the coefficients Cµi from
Equation 2.10. The elements of F are shown in Equation 2.11.

Fµν =

φ∗
µ (1) f (1) φν (1) d(cid:126)r1

(2.11)

K(cid:88)

µ=1

(cid:90)

(cid:90)

Equation 2.11 is the matrix representation of the Fock operator, f (1), with a set of basis functions
φµ.3 The elements of S are shown in Equation 2.12.

Sµν =

φ∗
µ (1) φν (1) d(cid:126)r1

(2.12)

To solve these equations, an initial guess is proposed for the density matrix P, defined as C*C,
based on diagonalizing S, which in turn is used to generate F. Orthogonalizing and diagonalizing
F results in a new guess for C and thus P. This procedure is iterated until the change in energy
and the change in the density matrix is negligible, and thus is called the self-consistent field (SCF)
procedure used to solve for the HF energy.3 The electron correlation energy is defined as the
difference between the exact energy and the HF-calculated energy, as shown in Equation 2.13.

Ecorr = Eexact − EHF

11

(2.13)

While the correlation energy is a small percentage of the total electronic energy, it can have
a large magnitude.6 Therefore, inclusion of the correlation energy is essential for the accurate
prediction of chemical and physical properties. Post-HF methods recover the correlation energy
not accounted by the HF method by adding excited determinants. One such method is many-
body perturbation theory (MBPT), which utilizes a perturbation expansion with the Hartree-Fock
Hamiltonian as the zeroth-order Hamiltonian.3 Using the Rayleigh-Schrödinger expansion of a
generalized Hamiltonian, Equation 2.14, Møller and Plesset developed Møller-Plesset perturbation
theory (MPPT).

ˆH = ˆH0 + λV

(2.14)

In Equation 2.14, is the corrected Hamiltonian, is the sum of one-electron Fock operators, λ is
a dimensionless parameter between 0 and 1, and is the perturbation.3,7 The nth-order electronic
energy is termed as the MPn methods. The MPPT Hamiltonian uses the Fock operator as the
zeroth-order Hamiltonian (MP0), which double-counts electron repulsion when using the HF
wavefunction. The MP1 energy eliminates one set of electron-electron interactions through using
the operator. Therefore, the HF energy is the sum of the MP0 and MP1 energies.3,6,8 The first
perturbation that accounts for corrections beyond HF is MP2, which adds a second-order correction
and provides a size-extensive correction at a low cost, Equation 2.15.

E

(2)
1 =

V1nVn1
1 − E
(0)

(0)
n

E

(2.15)

Equation 2.15 uses the operator from Equation 2.14 on all doubly excited determinants.
Further expansions via MPPT are not used as often due to the high scaling and computational cost
even though the corrections recover more correlation energy.6 Another approach to recovering
correlation energy is the coupled-cluster method that employs a cluster operator, eT , to the HF
reference wavefunction.9 The cluster operator is defined as the sum of the single ( ˆT1), double
( ˆT2), triple ( ˆT3),. . . , N-tuple ( ˆTN) excitation operator for N electrons, shown in Equation 2.16.

ˆT = ˆT1 + ˆT2 + ˆT3 + . . . + ˆTN

(2.16)

12

The operator, Equation 2.16, generates all ith excited Slater determinants when operated on the
HF reference wavefunction. The coupled cluster wavefunction is the resulting wavefunction after
operating on the HF reference wavefunction with the exponential of the cluster operator, Equation
2.17.

ˆT = 1 + ˆT 1 +

e

ˆT2 +

+

ˆT3 + ˆT2 ˆT1 +

+ . . .

(2.17)

(cid:19)

1
6

ˆT 3
1

(cid:18)

(cid:19)

(cid:18)

1
2

ˆT 2
1

For the coupled-cluster methods, Equation 2.11 utilizes the Taylor expansion of the exponential
of the cluster operator. This Taylor expansion generates the multiplicative terms ( ˆT2 ˆT1) and product
) that help account for size inconsistency problems that occur in configuration interaction,
terms ( ˆT 2
1
or CI, methods.8–10 CI methods are defined by specifying configurations of the spin orbitals
that construct each Slater determinant in reference to the HF wavefunction.3 By generating all
possible excitations for N electrons, full CI is achieved, which is the exact answer to the electronic
Schrödinger equation within a given basis set. Using the same excitation operator defined in
Equation 2.16, CI wavefunctions can be generated using (1+ ˆT ) using intermediate normalization
rather than as the excitation operator on the HF wavefunction to produce excited determinants.8 One
of the more popular coupled cluster methods is CCSD(T), which uses coupled cluster with single
excitations, double excitations, and perturbative triple excitations.11 The scaling of a particular
method is a key descriptor for determining computational cost. The Hartree-Fock method scales
as N 4, which relies on the relative size of the system, i.e. number of basis functions N.8 Post-HF
methods, such as MP2 and CCSD(T), have a larger scaling due to the inclusion of correlation
energy and more complex operators. MP2 scales at N 5 and CCSD(T) scales iteratively at N 7.8

2.2 Cost-Saving Wavefunction-based Methods

With increasing molecule size and thus an increasing number of basis functions,
wavefunction-based methods have decreased practicality. This section outlines approaches, e.g.
local correlation methods, composite methods, and multilevel approaches,
that reduce the
computational cost associated with (CPU time, memory, disk space) higher level ab initio
methodologies.

13

2.2.1 Local Methods

The molecular orbitals –canonical MOs– generated from the diagonalization of the Fock matrix
are characteristically delocalized, even for smaller molecular systems. The short-range effect of
dynamic correlation has a dependence on distance of r-6 like dispersion energy.12 When using
canonical MOs to describe electron correlation, its short-range aspect cannot be properly exploited,
both in terms of gaining a more qualitatively correct picture of the electron correlation relative
to localized MOs and reducing the high computational cost of ab initio methods.13 Hence, the
development of local correlation methods.

The concept of localized MOs was first introduced by Lennard-Jones, Pople, and Hall.14–18
Since then, localization techniques19–23 and local correlation methods24–47 have been developed
to utilize localized occupied MOs to localize the dynamic correlation. Localization imposes a
mathematical constraint of maximum insensitivity for changes in distant nuclear charges, which
allows orbitals to be localized around covalent bonds and atomic lone pairs. Localized MOs
are generated through exploiting the invariance of the Hartree-Fock wave function with respect to
orthogonal transformations, and are popular in their application to occupied orbitals.19–23,48 For the
occupied orbitals, the Foster-Boys (FB)19,20 and the Pipek-Mezey (PM)23 localization techniques
are more widely used compared to the Edmiston-Ruedenberg method.21–23,49 Both the Foster-Boys
and Pipek-Mezey localization techniques scales as N 3 (N is the number of basis functions) since
both approaches calculate one-electron dipole integrals and no two-electron integrals, whereas the
Edmiston-Reudenberg scheme scales as N 5, which is caused by the calculation of two-electron
integrals.

The Foster-Boys localization approach minimizes the spatial extension of the MOs

N(cid:88)

fFB[φ] =

|(cid:104)φiφi|r1 − r2|φiφi(cid:105)|2

(2.18)

or equivalently maximizing the sum of squares between the distances of orbital centroids from the

i

14

origin of the coordinate system.19,20

fFB[φ] =

N(cid:88)

i

|(cid:104)φi|r|φi(cid:105)|2

(2.19)

The Pipek-Mezey localization approach uses the operator expectation value definition of the gross
Mulliken orbital population to define the localized orbitals

N(cid:88)

n(cid:88)

N(cid:88)

n(cid:88)

fPM =

(QA

ii)2 =

i=1

A=1

i=1

A=1

where the sum of atomic centers A and QA
ii
The Hermitian operator ˆPµ is defined as

[(cid:104)φi| ˆPµ|φi(cid:105)]2

(2.20)

is the population of orbital |p(cid:105) on atomic center A.23

where

1
2

(|(cid:101)µ(cid:105)(cid:104)µ| + |µ(cid:105)(cid:104)(cid:101)µ|)
(cid:88)

(S−1)νµ|ν(cid:105)

ˆPµ = ˆP

†
µ =

|(cid:101)µ(cid:105) =

(2.21)

(2.22)

ν

where S is the overlap matrix (Equation 2.12) and {|(cid:101)µ(cid:105)} are biorthonormal to the atomic basis

functions {|µ(cid:105)}. The localized MOs for Pipek-Mezey localization are obtained through maximizing
Equation 2.20.23 The Pipek-Mezey localization method is the use of Mulliken population analysis,
which suffers from unphysical behavior, i.e. yielding occupation numbers for individual Mulliken
charges that are larger than 1 or less than 0.23,48 The unphysical behavior is caused by overlap
populations that occur from a non-orthogonal AO basis. An alternative Pipek-Mezey scheme
utilizing the Löwdin population analysis has been developed to account for such deficiencies.50
However, regardless of which population analysis method is used, the Pipek-Mezey localization
properly separates σ-π bonds unlike the Foster-Boys localization approach.

For local correlation methods, the corresponding virtual orbitals are spanned by a set of projected
atomic orbitals (PAOs) or pair natural orbitals (PNOs),24,25,51,52 but have been spanned by localized
virtual Hartree-Fock orbitals.53,54

15

2.2.2 Resolution of the Identity Approximation

The resolution of the identity (RI) approximation enables four-center two-electron repulsion
integrals to be expressed as two- and three-centered electron repulsion integrals, reducing the
computational cost of calculating the electron repulsion integrals from O(ζ12) to approximately
O(ζ9).55–57 The RI approximation involves the insertion of an approximate resolution of the
identity into the Hilbert space of two interacting charge densities ρ and ˜ρ, where

ρij = ij

for products of molecular orbitals i, j, and

˜ρij =

(cid:88)

cij,P P

for a linear combination of auxiliary basis functions P with coefficients cij,P .56

(2.23a)

(2.23b)

Through a minimization of the residual density, ρ-˜ρ, the four-center integrals are approximated
using Equation 2.24, where the sum is over all functions within a fitted auxiliary basis set (see
Section 2.4.3).

(ia|jb) = (ia|ˆ1|jb) ≈(cid:88)

(ia|P )(P|Q)−1(Q|jb)

(2.24)

P Q

In Equation 2.24, (ai|P ) and (P|Q) are the three- and two-center electron repulsion integrals,
respectively, a, b denote virtual molecular orbitals, i, j denote occupied molecular orbitals, and
P, Q denote auxiliary basis functions.58 The RI method has been implemented in post-HF methods
as well, reducing the cost of MP2 calculations while recovering a similar amount of correlation
energy.56 The RI approximation serves a key role in the integral generation for the domain-based
local pair natural orbital methods (see section 2.2.3).

In a standard implementation of the RI approximation, the coulomb and exchange integrals are
fitted to the auxiliary basis set. However, within the RI framework, variants that create auxiliary
basis sets for the coulomb and exchange integrals individually have been developed. Variants
included in Chapters 4 and 5 primarily focuses on the contributions from Neese et al., who developed
a split-J approximation59 and the chain of spheres exchange approximation (COSX)60,61 for the

16

coulomb and exchange portion of the Fock matrix, respectively. The Split RI-J algorithm was a
modification of the RI approximation for the Coulomb interaction based on removing redundancies
in calculating the Coulomb matrix. The COSX approximation utilizes a semi-numerical integration
similar to Friesner’s pseudo-spectral method to construct small ‘chains’ of shells of basis functions
with contributions to the exchange matrix above a certain threshold. The combination of the split
RI-J and COSX, or RIJCOSX, is mainly used on molecular systems exceeding 50 atoms including
open-shell transition metal species due to computational efficiency. The RIJCOSX approximation
for the HF wavefunction was considered for 20 closed shell reactions and 9 reactions and yielded a
mean absolute error of 0.19 kcal mol−1 with a maximum absolute error of 0.64 kcal mol−1. The
RIJCOSX wavefunction has been combined with the RI-MP2 method resulting in errors ranging
from 0.01 to 1.2 mEh for organic systems comprised of 30-57 atoms.60 Because of its demonstrated
utility, the RIJCOSX approximation is considered alongside the standard RI approximation for SCF
energies.

2.2.3 Domain-Based Local Pair Natural Orbital Methods

The

domain-based

local

orbital

pair

natural

(DLPNO) methods,62–66

primarily
DLPNO-CCSD(T), have been shown to result in a reduced computational cost relative to the cost
of CCSD(T) for transition metal-based catalysts and larger organic systems such as complex
hydrocarbons and fullerenes while yielding electronic energies within 1 kcal mol−1 from
CCSD(T) electronic energies.67–71 Recent developments of DLPNO methodology utilize sparse
maps that take advantage of the sparsity of data by reducing the number of matrix elements stored
and omitting terms in sums that are smaller than a predefined tolerance to achieve linear scaling
with respect
to N basis functions in all major computational steps including the integral
transformation and storage,
(PNO)
construction.65,66,72 These are improvements to the original version of DLPNO-CCSD(T) that
was developed by Riplinger et al,62,63 which included terms that scaled by N 2 in the screening of
electron pairs. DLPNO-MP2 recovers approximately 99.9% of the RI-MP2 correlation energy64

and the pair natural orbital

the

initial guess,

17

and DLPNO-CCSD(T) recovers 92.2% of the canonical triples correction63 and greater than
99.6% of the canonical CCSD(T) correlation energy.63,65

The DLPNO methods use a single determinant reference wave function with the occupied
molecular orbitals (MOs) localized. The Fock matrix is then constructed followed by the
determination of the projected atomic orbitals (PAOs). The refined electron pair prescreening uses
differential overlap integrals (DOI)

(cid:115)(cid:90)

DOIik =

|fi (r)|2 |gk (r)|2 dr

(2.25)

where fi and gk are basis functions (square integrable one-electron functions), which are
calculated via numerical integration techniques to achieve linear scaling.64 Domains are selected
via normalized DOI between localized MOs and PAOs. This allows both the occupied and
unoccupied spaces to be taken into account for domain definition as well as ensuring that all PAOs
that have a significant differential overlap with occupied orbital i are included in the correlation
domain.64 This refined prescreening method eliminates fewer electron pairs than the previous
implementation based on multipole estimates and pair correlation energies.
The integral
transformation from atomic orbital (AO) basis three-index integrals (µν|K) to (i˜µ|K) follows,
where µ, ν, λ, σ represent AOs, K, L refer to ABS, i, j, k, l denotes localized MOs, and ˜µ,˜ν label
˜µ˜ν = (i˜µ|j ˜ν), are then constructed to find the
the PAOs. The local exchange operators, Kij
semi-canonical local MP2 (SC-LMP2) or local MP2 (LMP2) guess amplitudes

˜µ˜ν = −
T ij

Kij
˜µ˜ν

ε˜µ + ε˜ν − Fii − Fjj

(2.26)

where ε˜µ and ε˜ν are energies of quasi-canonical, non-redundant PAOs, and Fii and Fjj are diagonal
Fock matrix elements in terms of localized orbitals.21,22 These guess amplitudes generate the pair
density

Dij = Tij ˜Tij+

+ Tij+ ˜Tij

(2.27)

from which the PNOs are approximated via diagonalization.

Beyond these steps, the DLPNO-MP2 and DLPNO-CCSD(T) methods separate as the DLPNO-
CCSD(T) method uses SC-LMP2 as a crude guess for which electron pairs (ij) to include in a

18

subsequent SC-LMP2 calculation and coupled cluster iterations whereas DLPNO-MP2 uses the
LMP2 framework to calculate total energies in one iteration. For DLPNO-CCSD(T), an estimate of
the pair correlation energy εij is computed from the electron pairs that survived the prescreening,
which are separated into three classes based on the dimensionless parameters TCutP airs and
TCutP airsM P 2. The first class consists of pairs that are included in CCSD calculations (strong
pairs), the second class contains pairs for MP2 that are kept for the triples correction (weak pairs),
and the third class includes pairs that are not considered further. Pairs in the first two classes are
used in a more accurate SC-LMP2 calculation. Pair correlation energy estimates from the third class
are added to the SC-LMP2 energy to approximate the error introduced by local approximations.65
The scaling reduction from N 7 to N 5 of DLPNO-CCSD(T) lies in the contribution of orbital
triples to the triples energy.63 Orbital triples (ijk) only contribute to the triples energy if the three
pairs (ij), (ik) and (jk) survive the pair selection process. The domain for (ijk) is the union of
the individual orbital domains (i), (j), and (k). Like the generation of PNOs, the triples natural
orbitals (TNOs) are computed from diagonalizing the average of the pair densities for all three
pairs (Equation 2.27) generated from a local, non-redundant PAO basis formed from redundant
PAOs that span the triples domain. The integrals for TNOs are calculated via the RI approximation
through transformations from the redundant PAO basis to the TNO basis for subsets of three-index
electron repulsion integrals. The singles and doubles amplitudes are projected into the TNO basis
through the triples/pair overlap.
Intermediates that enter the triples calculation as well as the
actual contribution of a given triple are calculated by the canonical (T) correction.63 (The original
publications provide more details about DLPNO methods and their development.60,63–65,73,74)

2.2.4 Composite Methods

Ab initio composite methods have been around for a long time and are effective approaches to
reduce computational cost.
Ab initio composite strategies are those that emulate a more
computationally demanding method by adding up contributions that describe aspects of modeling
like relativity, spin-orbit, and contributions from electrons beneath the
quantum chemistry,

19

valence shell (sub-valence electrons), all of which increase the computational cost of ab initio
methods. This additive approach can save greater than 90% of the total CPU time. In the case of
CO, using a composite strategy only took 2 minutes whereas using the effective ab initio method a
composite strategy portrays took over 8 hours!

Popularized by John Pople in 1989, ab initio composite methods targeted main group
thermochemistry with the goal of estimating energies yielded by ab initio methods that would
require significant computational resources efficiently.75 Throughout the 1990s, other composite
approaches and improvements of Pople’s composite strategies
targeting chemical and
spectroscopic accuracy, which is defined as 1 kcal mol−1 and 1 kJ mol−1, respectively, appeared
and targeted atoms and small polyatomic molecules with more than six non-hydrogen
atoms.72,76–85 In the early 2000s, simultaneous developments of composite strategies targeting
chemical86–90 and spectroscopic91–95 accuracy emerged, primarily for applications in main
group thermochemistry. With the present computing power, composite methodologies have
targeted molecules as large as buckminsterfullerene (C60).96

Though ab initio composite methods have primarily targeted main group thermochemistry
due to the abundance of well-established reliable experiments, there have been limited studies
in the development and application of ab initio composite methods towards transition metal and
f-block thermochemistry as reliable experimental data is sparse.97–107 As well, the chemistry in
this region of the periodic table becomes increasingly more complex due to the d and f electrons,
which contribute to low-lying excited states, i.e. excited states that are close in energy to the
ground state electronic configuration, as well as the significant effects of relativity and spin-orbit
coupling. When formulating composite strategies for transition metal and f-block thermochemistry,
all of these factors require the usage of relativistic basis sets and pseudopotentials (see Section
2.4.2), multireference methods that accounts for low-lying excited states, and/or a relativistic
Hamiltonian that includes spin-orbit coupling and scalar relativity, and if needed, higher orders
of relativity. Therefore, the increase in chemical complexity down the periodic table leads to the
development of variants with respect to composite methodologies implemented for main group

20

thermochemistry.97–107

Ab initio composite approaches are utilized to predict thermochemical properties like ∆Hf,
ionization potentials, and bond dissociation energies (BDE) within chemical accuracy (1 kcal
mol−1) and spectroscopic accuracy (1 kJ mol−1) reliably throughout the periodic table at a
computational cost significantly less than the effective level of theory. Based on their success for
predicting thermochemical properties for main group, transition metal, and to an extent f-block
molecules, ab initio composite strategies could be used as a gateway to understand reaction
mechanisms, design catalysts, and characterize heavier elements utilizing wavefunction-based
methods.

Ab initio composite approaches are comprised of a reference energy and additive corrections
the inclusion of
to the reference energy that account for effects like relativity, spin-orbit,
interactions between sub-valence and valence electrons, and the energy contributions from
numerous simultaneously excited electrons. For the reference, reliable electronic structures
(electron configuration and geometry) and scaling vibrational contributions to the energy are
needed.108 Some composite methods utilize higher cost methods to correct for anharmonicity
through perturbative methods.109,110

Using the optimized structures, single point calculations are done to obtain the reference energy.
This is generally attained through one to three single point calculations involving an ab initio method
such as MP2, MP4, or CCSD(T) (see Section 2.1) and increasingly larger basis sets (see Section
2.4). While composite methods have utilized one single calculation to obtain a reference energy,
most will utilize two or three calculations where the method is the same and the basis sets increase
in size. The intention is to find the energy attained with an infinitely large description of the
electron space. This is also known as the complete basis set (CBS) limit. Composite schemes that
target the CBS limit for the reference energy typically utilize the correlation consistent basis sets
to extrapolate to the CBS limit via analytic formulas based on basis set size111,112 and maximum
angular momentum.113,114

Additive corrections are then included to supplement the reference energy in a methodical

21

fashion. These corrections to the energy are arising from scalar relativity, spin orbit (from
experimental atomic values), and interactions between valence and sub-valence electrons. When
these corrections are added to the reference energy, the total electronic energy is obtained at a
much lower computational cost than a calculation for the effective level of theory a composite
method achieves.

Many ab initio composite methods have been developed targeting either chemical accuracy or
spectroscopic accuracy. Composite methodologies that target chemical accuracy include the
Gaussian-n (Gn) methods,75–79,89,90,115 Complete Basis Set (CBS-X) method,72,80–84,86,88 and
the correlation consistent Composite Approach (ccCA).87,97,116–122 Classes of composite
methodologies targeting spectroscopic accuracy for thermochemical properties of diatomics and
very small polyatomic molecules (roughly 3-10 atoms) due to the high computational cost of the
methods involved include the Weizmann-n (Wn) methods,85,94,95,123–125 High Accuracy
Extrapolated Ab initio Thermochemistry (HEAT),92,126,127 Feller-Peterson-Dixon (FPD)
method,128,129 and the focal point analysis method.130–135

Composite methodologies that target chemical accuracy are more efficient that those that target
spectroscopic accuracy and allow larger molecules to be studied.136–143 There are many variants of
composite methodologies including ones modified to describe aqueous phase chemistry or expand
to larger molecules.

2.2.4.1 Correlation Consistent Composite Approach

The correlation consistent Composite Approach (ccCA) was created in 2006 by Wilson and
co-workers as an alternative to the Gn methods.87 While successful for s-block and p-block
thermochemistry in the first four periods,144–146 methodological adjustments were made to the
ccCA formulation in 2009, which included scaling vibrational contributions and options for the
extrapolation scheme for the CBS limit.117

The formulation of ccCA is

EccCA = Eref + ∆ECC + ∆ECV + ∆EDK + ∆ESO + ∆EZP E

(2.28)

22

where Eref is obtained at the MP2/CBS level by combining CBS extrapolations for the SCF
energies and MP2 correlation energies with the aug-cc-pVNZ basis sets. The SCF energy is
extrapolated with the Feller147,148 two-point extrapolation scheme

E (n) = EHF/CBS + Be−Cn

(2.29)

where n indicates the ζ-level of the basis set, E(n) is the energy at the nth ζ-level, EHF/CBS
represents the Hartree-Fock electronic energy at the CBS limit, B is a fitting parameter, and 1.63
is used for C.111 To extrapolate the MP2 correlation energies, previous ccCA studies considered
several different extrapolation schemes, including Peterson’s three-point extrapolation scheme149

E (n) = EMP2/CBS + Be−(n−1) + Ce−(n−1)2

(2.30)

where EMP2/CBS represents the electronic energy at the CBS limit, and B and C are fitting
parameters. The Peterson (P) three-point extrapolation uses the double-, triple-, and quadruple-ζ
correlation consistent basis sets. Other extrapolation schemes used in this work include inverse
cubic and quartic equations, commonly referred to as the Schwartz-3 (S3) and Schwartz-4 (S4)
two-point extrapolation schemes, respectively.112,113,150–153

E (lmax) = EMP2/CBS +

E (lmax) = EMP2/CBS +

(cid:18)

lmax +

1
2

B

(lmax)3

B

(cid:19)4

(2.31)

(2.32)

In Equations 2.31 and 2.32, lmax is the highest angular momentum function included in the
basis set, which differs for main group and transition metals. Both of the S3 and S4 two-point
extrapolation schemes use the lmax of the triple- and quadruple-ζ level basis sets, denoted as S3(TQ)
and S4(TQ). Since the S3 scheme tends to overestimate the energy at the CBS limit due to a slower
convergence rate and the Peterson scheme tends to underestimate the energy at the CBS limit due
to more rapid convergence, the average of both schemes, denoted PS3(TQ), is considered.117,154
The core-core (CC) correlation (∆ECC) accounts for higher levels of correlation beyond the MP2

23

level by using CCSD(T) at the cc-pVTZ level.

∆ECC = E [CCSD(T)/cc-pVTZ] − E [MP2/cc-pVTZ]

(2.33)

The core-valence (CV) correction accounts for the n and n-1 orbital shells, where n ≥ 2. This
correction accounts for the interactions between valence and sub-valence electrons whereas the
other composite steps only include valence-valence interactions. The FC1 notation indicates the
inclusion of the n-1 orbital shell.

∆ECV = E [MP2(FC1)/aug-cc-pCVTZ] − E [MP2/aug-cc-pVTZ]

(2.34)

The scalar relativistic correction uses the second-order spin-free Douglas Kroll Hess Hamiltonian
to account for scalar relativistic effects.155–157

∆EDK = E [MP2/cc-pVTZ-DK] − E [MP2/cc-pVTZ]

(2.35)

Experimental spin-orbit corrections for atoms are applied from tables provided by Moore.158

This formalism is used as the base model and is altered to accommodate the need for relativistic
corrections and effective core potentials for transition metals. For ccCA-TM,116 developed for 3d
transition metals, the modifications from ccCA include the use of scalar relativistic basis sets and
the use of CCSD(T) and an augmented double-ζ core-valence basis set. For rp-ccCA,97 developed
for 4d transition metals, effective core potentials (ECPs) are used in all steps of the ccCA-TM
formulation. Variants where the fundamental aspects of ccCA remains the same but the steps are
modified have been developed to adapt to chemical problems as well, such as organic acid/base
chemistry (Solv-ccCA),122 active-site chemistry (ONIOM-ccCA, ONIOM-rp-ccCA),119,159 and
modeling open-shell organic species, such as radicals (MR-ccCA, ccCA-CC(2,3)).118,120 For
example, in Solv-ccCA, all methodological steps within ccCA remain the same except for including
an implicit solvent model to describe long-range solvent effects.

24

Table 2.1: Summary of ccCA-TM and rp-ccCA steps.

Geometry
Optimization

Eref

Extrapolations

MP2/CBS

Extrapolations

∆CC

∆CV

∆DK
∆SO
ZPE

ccCA-TM

B3LYP/cc-pVTZ-DK
HF/aug-cc-pVTZ-DK
HF/aug-cc-pVQZ-DK

Equation 2.29

MP2/aug-cc-pVDZ-DK
MP2/aug-cc-pVTZ-DK
MP2/aug-cc-pVQZ-DK
Equations 2.30 -2.32

rp-ccCA

B3LYP/cc-pVTZ-PP
HF/aug-cc-pVTZ-PP
HF/aug-cc-pVQZ-PP

Equation 2.29

MP2/aug-cc-pVDZ-PP
MP2/aug-cc-pVTZ-PP
MP2/aug-cc-pVQZ-PP
Equations 2.30 -2.32

CCSD(T)/cc-pVTZ-DK

- MP2/cc-pVTZ-DK

CCSD(T,FC1)/aug-cc-pCVDZ-DK
– CCSD(T)/aug-cc-pCVDZ-DK

CCSD(T)/cc-pVTZ-PP

- MP2/cc-pVTZ-PP

CCSD(T,FC1)/aug-cc-pCVDZ-PP
– CCSD(T)/aug-cc-pCVDZ-PP

Included in previous steps
Experimental atomic values

Vibrational ZPE scaled by 0.989

Included in previous steps
Experimental atomic values

Vibrational ZPE scaled by 0.989

in terms of

Numerous routes have been utilized to reduce the computational cost associated with ccCA to
expand the range of molecules
size that can be examined with this
approach.119,121,159–161 RI-ccCA and ccCA-F12 implemented mathematical approximations to
mitigate the cost of calculating four-center-two-electron repulsion integrals and using the
aug-cc-pVQZ basis set, which both are major contributions to the overall computational cost of
ccCA.121,161

ccCA and its adaptations are suitable for applications targeting chemical accuracy for chemical
systems ranging from atoms and diatomics to organometallic complexes and biomolecules. Some
of these applications are presented in Chapters 4-6.

2.2.5 ONIOM

Multilayer methods provide additional routes to reduce computaitonal cost.

For these
approaches, a molecular syste is divided into multiple layers and each layer is treated with a

25

different theoretical approach. This enables the chemistry of greatest interest to be targeted with a
high-level method, while the overall molecular system is treated with a more approximate, albeit
more efficient, approach. One of the earlier uses of multilayer methods combined quantum
mechanical (QM) methods with molecular mechanics (MM) force fields to measure the torsional
potential energy surface of the retinal molecule.162 The use of this hybrid methodology was
extended to describe ground and excited-state potential energy surfaces in tandem with a
Pariser-Parr-Pople SCF-CI method163,164 for π electrons and empirical functions for σ electrons,
respectively.165,166 This method was later generalized into the QM/MM method, which includes
a model system and the real system.167 The model system describes the chemically significant
portion of the system and uses QM methods for higher accuracy, whereas the real system is
described by a less accurate but more computationally efficient MM force field. The total energy
of the whole system is shown in Equation 2.36.

EQM/MM = EQM + EMM + EQM-MM

(2.36)

Equation 2.36 is an additive scheme168 combining the energy of the two systems, EQM and
EMM, and the energy of the interaction between the two systems, EQM-MM. In contrast to this
additive scheme employed for the QM/MM method, our Own N-Integrated molecular Orbital
molecular Mechanics, or ONIOM169–178 method is an extrapolative scheme that can utilize a
QM/QM or a QM/MM scheme. The development of the ONIOM methodology started with the
development of an alternative QM/MM scheme known as IMOMM, or Integrated Molecular Orbital
+ Molecular Mechanics, shown in Equation 2.37.169

EIMOMM = EONIOM2(QM:MM) = EQM,model + EMM,real − EMM,model

(2.37)

In Equation 2.37, the total energy of this extrapolative scheme is evaluated as the MM method
for the model system is subtracted from the sum of the energies obtained through the QM method
for the model system and the MM method for the real system. The main difference between
EQM/MM and EIMOMM is that the subtractive operation for EIMOMM removes the doubly-counted
MM contributions to the total energy in Equation 2.36.169,179 The IMOMM scheme was extended

26

to QM/QM systems in the Integrated Molecular Orbital + Molecular Orbital formalism, which is
denoted as IMOMO or ONIOM2(QM1:QM2).172 The total energy of the system is calculated in
the same manner as the IMOMM method except for the use of a second QM method replacing the
MM force field, shown in Equation 2.38.

EIMOMO = EONIOM2(QM1:QM2) = EQM1,model + EQM2,real − EQM2,model

(2.38)

ONIOM is not limited to two layer systems. A combination of the IMOMM and the IMOMO
methods yield a three-layer ONIOM method denoted as ONIOM3(QM1:QM2:MM) as utilized in
Equation 2.39.175

EONIOM3(QM1:QM2:MM) = EQM1,model + EQM2,intermediate − EQM2,model
+EMM,real − EMM,intermediate

(2.39)

For ONIOM3, three layers, model, intermediate, and real with a different level of theory used to
describe each layer. In general, the high level method, QM1, is an ab initio method, the intermediate
level method, QM2, is a DFT method, and the real level method is a MM force field. Based on the
formulations for ONIOM2 and ONIOM3, the ONIOM method can be generalized to an arbitrary
n-layer n-level method, Equation 2.40.

n(cid:88)

E[level(i),model (n + 1 - i)] − n(cid:88)

EONIOMn =

E[level(i),model(n + 2 - i)]

(2.40)

i=1

i=2

The n=2 (ONIOM2) and n=3 (ONIOM3) forms of n-layer ONIOM are most commonly used,
as n ≥ 3 approaches become impractical. Overall, the ONIOM method is most commonly used for
large biological macromolecules,162,165,166 transition metal complexes,180,181 and organometallic
catalysts.182–185

2.3 Density Functional Theory

Density functional theory (DFT) originates from the Hohenberg-Kohn theorems.186,187 In
1964, an existence proof showed that the charge density (ρ[r]) determines the electronic properties
of the ground state including energy. DFT utilizes the electron density as a variable to approximate

27

the solution to the Schrödinger equation. Analogous to the Roothaan-Hall Equations in the Hartree-
Fock formalism, the DFT equivalent –the Kohn-Sham equations– were derived by Kohn and Sham
in the early 1960s.186,187 The DFT energy is shown in Equation 2.41.

E[ρ] = Ts[ρ] + Vne[ρ] + J[ρ] + Exc[ρ]

(2.41)

Equation 2.41 is dependent on the kinetic energy (Ts) of non-interacting electrons, the energy
term for nuclear-electron interactions (Vne), electron-electron repulsion interactions (J), and the
exchange-correlation energy term (Exc). In principle, the exact form of the exchange-correlation
functional makes DFT an exact and ab initio method; however, the exact form of the exchange-
correlation functional is not known based on the inhomogeneity of the charge density.188 Therefore,
the implementation of DFT is the development of functionals that approximate the exchange-
correlation functional. Density functionals are sorted into a hierarchy based on the complexity of
the functional. As defined by Perdew as the “Jacob’s ladder” for DFT, the tiers of functionals from
least to most complex are the local spin density approximation (LDA), the generalized gradient
approximation (GGA), meta-GGA, hybrid-GGA, hybrid-meta GGA, and double hybrid GGA.189
The local spin density approximation (LDA or LSDA) is based on the uniform electron gas
model and was first introduced by Kohn and Sham.187 LDA uses the exchange for the uniform
electron gas to create a functional solely dependent on the spin density.190

(cid:90)

ELDA

XC =

ρ(r)εXC [ρ]r

(2.42)

Equation 2.42 represents the exchange-correlation (XC) energy for LDA functionals, which is
dependent on the single particle density ρ(r) and the XC energy per particle, εXC [ρ(r)]. LDA is
known to be more effective at describing solid state lattice parameters than more complex DFT
functionals due to the similarity of metallic systems to the homogeneous electron gas.188,191,192
GGA functionals incorporate the gradients of the spin densities in the expression for exchange-
correlation energy and are therefore a correction to the LDA, shown in Equation 2.43.

(cid:90)

EGGA

XC =

XC (n↑(r), n↓(r),|∇n↑(r)|,|∇n↓(r)|)d3r
eGGA

(2.43)

28

Meta-GGA functionals, which decrease the amount of self-interaction error introduced by GGA
functionals, use the Laplacian of the spin densities as two additional variables and the kinetic
energy densities, τ, as shown in Equation 2.44.193,194

(cid:90)

EM-GGA

XC

=

XC (n↑(r), n↓(r),|∇n↑(r)|,|∇n↓(r)|,|∇2n↑(r)|,|∇2n↓(r)|, τ↑, τ↓)d3r
eGGA

(2.44)
LDA, GGA, and meta-GGA functionals are referred to as local functionals because the electronic
energy density at a single point is dependent on the behavior of the density in proximity to that
point.193,195–197

Hybrid functionals combine the GGA exchange correlation functional with the exact exchange
defined in the Hartree-Fock method using the Kohn-Sham orbitals in order to address the
shortcomings of the self-exchange of DFT functionals as shown in Equation 2.45.

XC = a(EX,exact − EGGA
Ehybrid

X ) + EGGA
XC

(2.45)

Equation 2.45 is applied to both GGA and meta-GGA functionals and thus named hybrid-GGA and
hybrid-meta-GGA functionals, respectively. Double hybrid functionals utilize the PT2 correlation
energy into the correlation functional.198,199 Due to the addition of a percentage of exact exchange
into the functional, hybrid-GGA, hybrid-meta-GGA, and double-hybrid functionals are referred to
as non-local functionals.

Based on the “Jacob’s ladder” model for DFT by Perdew,189 for each rung of the ladder,
additional "factors" are appended to the functionals of the rung below, as illustrated from Equations
2.42-2.45. As a result, the increasing complexity of functionals progressing up Jacob’s ladder
implies an assumption of greater accuracy. However, greater accuracy cannot be presumed, i.e.
local functionals may be more effective at describing a system than non-local functionals. Therefore,
the rational choice of DFT functionals should be determined by carefully considering the calibration
of DFT functionals with experiments or high accuracy ab initio methods for a particular application.
DFT is able to yield results for thermodynamic properties comparable to post-HF methods
at a reduced computational cost as DFT scales at approximately N 4 or N 5 depending on the

29

complexity of the functional where N is the number of basis functions. DFT has the inability to
properly account for the weak interactions due to dispersion forces that arises from local exchange-
correlation, and systems with long-range interactions–dissociation of radials and other charged
odd-electron systems, and self-interaction error. Also, the exchange-correlation functional is
local, which is unsuitable for charge transfer reactions. Attempts have been made to overcome
the inability to accurately describe long-range interactions through dispersion-corrected density
functionals (DFT-D methods),200,201 which use a semi-empirical parameterization to correct for
the lack of dispersion, and the double-hybrid-GGA functionals.

2.4 Basis Sets

A basis set consists of mathematical functions that are used to describe the electronic
wavefunction. Gaussian basis functions, shown in Equation 2.46, are the most common functions
used in basis sets.

φ(ζ, r) = N e−ζr2

(2.46)

For the gaussian-type orbital (GTO) or a Gaussian primitive (Equation 2.46), N is the normalization
constant, ζ is the exponent and r is the electron-nucleus distance. Gaussian-type functions were
chosen since the product of two GTOs is another GTO, which greatly simplifies calculating the
four-center two-electron repulsion integrals – the most computationally expensive step in the SCF
procedure (Section 2.1).

A linear combination of these gaussian primitives (Equation 2.10) minimizes the number of
basis functions needed for an accurate representation of the MOs. Gaussian basis sets are designed
in hierarchies of increasing size (ζ-level). While increasing the ζ-level of a basis set increases
the computational cost, a systematic way to obtain higher quality results is attained. Basis sets
commonly utilized in electronic structure calculations are atom-centered and energy-optimized,
i.e. the exponents are optimized to minimize the electronic energy, thus allowing a more widely
applicable basis set.

Two popular styles of basis sets include the Pople-style basis sets developed by Pople202–205

30

and the correlation consistent basis sets developed by Dunning and co-workers.206–215

2.4.1 Correlation Consistent Basis Sets

The correlation consistent basis sets are referred to as correlation consistent, polarized, valence,
n-ζ, or cc-pVnZ where n = double-ζ (DZ), triple-ζ (TZ), quadruple-ζ (QZ), etc.
level of basis
set. The correlation consistent basis sets can be augmented through the addition of low-exponent
diffuse functions, noted as the aug-cc-pVnZ basis sets. The correlation consistent family of basis
sets also includes cc-pCVnZ basis sets that account for the correlation energy from the interaction of
core-core and core-valence electrons as well as the valence-valence correlation energy,209,213 and
the cc-pVnZ-DK set accounts for scalar relativistic effects and is implemented for main group, 3d
transition metal, and lanthanide atoms.212,214–216 The cc-pV(n+d)Z basis sets were developed for
second-row atoms (Al-Ar) through the inclusion of an additional tight-d function and reoptimization
of the d-function in the basis set to address deficiency in the original correlation consistent basis
sets.211

For ab initio methods, one of the main advantages to these basis sets is their unique construction,
which enables the extrapolation of some properties like energies, to the complete basis set (CBS)
limit,147 which eliminates the basis set incompleteness error. At the CBS limit, the electronic
energy is not changed by the addition of extra basis functions since the basis set completely spans
the space of molecular orbitals, making an infinite or complete basis set.

2.4.2 Effective Core Potentials

When describing chemical systems with elements beyond the first-row transition metals, many
basis functions are required to define all of the electrons, which causes a significant increase in
the computational time needed relative to earlier main group species. In addition, any basis set
that describes these TM systems needs to account for the effects of relativity that can manifest
in elements beyond the first-row transition metals. Therefore, the concept of the effective core
potential (ECP), or pseudopotential (PP), was developed.217 An ECP portrays the core electrons

31

with a potential that is fitted from relativistic calculations and treats the remainder of the electrons
explicitly, which reduces the computational cost relative to their all-electron counterparts and
generally has a negligible effect on accuracy.8 cc-pVnZ-PP is a form of basis sets that have been
developed that pair ECPs with correlation consistent basis sets for the valence space.214,215

2.4.3 Auxiliary Basis Sets

Auxiliary basis sets (ABS) were designed to offset the increase in computational cost arising
from the calculation of four-center two-electron repulsion integrals in methods such as MP2 (i.e.
RI-MP2) by using the resolution of the identity (RI) approximation. To provide details, ABS can
be obtained through fitting procedures involving the coulomb integrals (basis/J or J-fit), or both
the coulomb and exchange integrals (basis/JK or JK-fit), as discussed below.57,218,219 For a J-fit
auxiliary basis set, the coefficients are fitted to a linear combination of three center (ij|a) and two-
center coulomb integrals whereas the K-fit auxiliary basis set is obtained through the difference
between the exact exchange and the approximate exchange generated.

The auxiliary basis sets for MP2 were optimized by minimizing the quantity

(cid:88)

iajb

δRI =

1
4

((cid:104)ab||ij(cid:105) − (cid:104)ab||ij(cid:105)RI )2


a − 
i + 
b − 
j

(2.47)

with respect to the auxiliary basis set exponents, where (cid:104)ab||ij(cid:105) = (ai|bj) − (aj|bi). The auxiliary
basis sets are constructed so that the number of auxiliary basis functions are not greater than four
times the number of basis functions in the standard basis set, as this could negate the advantage
gained for computational cost reduction. Also, the quantity in Equation 2.47 must be less than
10−6 when divided by EMP2, and |EMP2 − ERI-MP2| must be less than 20 µEh for auxiliary basis
sets in reproducing MP2 energies, but at a fraction of the cost.57,218,219

2.4.3.1 AutoAux

For lower parts of the periodic table, there are many atoms for which optimized auxiliary
basis sets are not available. For instance, auxiliary basis sets for cc-pCVnZ basis sets optimized

32

for ab initio correlated methods are not available. To expand the availability of auxiliary basis
sets, Stoychev et al.220 developed a generation scheme called AutoAux within the ORCA software
package.221 Their scheme was used to generate ABS for def2-SVP, def2-TZVP, def2-QZVPP, and
cc-pwCVnZ where n = D, T, Q, and 5, for H-Rn. They calculated both absolute and relative
energies via several reaction sets. For the cc-pwCVnZ basis sets, the average RI error was within
175 µEh relative to absolute HF and MP2 energies calculated with the AutoAux feature. While
AutoAux is useful for generating auxiliary basis sets on-the-fly in a calculation, these sets are often
twice the size of optimized auxiliary basis sets but can still benefit from the RI approximation. The
AutoAux scheme is utilized for the transition metal species in Chapter 5.

2.4.4 Basis Set Superposition Error

Basis set superposition error (BSSE) arises for interaction energies of molecular complexes via
an improved description of each fragment in the presence of the basis set of the other fragment.
The interaction energy (∆EAB) between two molecular fragments A and B is

∆EAB = EAB

AB (AB) − EA

A (A) − EB

B (B)

(2.48)

To overcome this error, when describing the energy of fragment A, the presence of a ghost fragment
B, i.e. the inclusion of the basis functions of fragment B without the atoms of fragment B present,
serves to counterbalance the effect that basis set B has on fragment A in the calculation of complex
AB.152 The counterpoise-corrected interaction energy is

∆ECP

AB = EAB

AB (AB) − EAB

A (AB) − EAB

B (AB)

(2.49)

is the energy of complex AB calculated with the basis set for AB, EAB

A (AB) and
where EAB
AB
B (AB) are the energies of fragments A and B, respectively, calculated with the basis set for AB.
EAB
When substituting Equation 2.49 into Equation 2.48, the counterpoise correction to the interaction
energy is obtained.

AB − ∆EAB =(cid:0)EA

A (AB)(cid:1) +(cid:0)EB

A (A) − EAB

∆ECP

corr = ∆ECP

B (AB)(cid:1)

B (B) − EAB

(2.50)

33

B (B) > EAB

B (AB).6

A (AB) and EB

Therefore, for variational wavefunctions, the counterpoise correction is always positive since
A (A) > EAB
EA
Depending on the nature of the interaction, molecular interaction energies vary considerably
in magnitude. Interaction energies range from 100-500 mEh for covalent bonds to 50-500 µEh for
dispersion-bound complexes.152 While BSSE is present in all electronic structure calculations, the
effects of BSSE are more prevalent for weakly-bound interactions, i.e. van der Waal interactions.

2.5 Implicit Solvation Models

As numerous chemical reactions are performed in solution, appropriate computational models
are needed to characterize solute-solvent interactions and describe other properties such as charge
distribution and solvation free energies. Two types of models that are used to incorporate the
effects of solvation explicit solvation models where all of the solvent molecules are explicitly
represented in the calculation and implicit solvation models that represents the solvent molecules
as a continuum. While explicit models can describe short range solute-solvent interactions, these
models are computationally expensive as they require 100-1000 solvent molecules for a single QM
calculation. Using implicit solvation models yields a lower computational cost relative to using
explicit solvent molecules but neglect detailed descriptions of the solute-solvent interactions.

Implicit solvation models are based on the approximation of a liquid medium as a dielectric
unstructured fluid through the use of a quantum mechanical description of the solute.
Implicit
solvation models provide an extension of the Born and Onsager models previously used to
describe fundamental properties of solutions.222,223 The general formulation of the solute-solvent
system223,224 in implicit models is that the solute is represented by a permanent point dipole µ
and a polarizability α with a radius a, and the solvent molecules are modeled as the average of the
charge distribution represented as a continuum dielectric medium with a fixed dielectric value (ε).
The Poisson equation, shown in Equation 2.51, is used to define the electrostatic potential as a
function of charge density.10

∇ε(r) ∗ ∇φ(r) = −4πρ(r)

(2.51)

34

In Equation 2.51, the solute is described explicitly within a cavity of vacuum while the solvent is
described implicitly via charge distribution.

The initial shapes of the vacuum cavity are spheres and ellipsoids, for which the Poisson
equation was solved analytically by Born222 and Onsager,223 respectively. The Born solvation
model creates a single point charge inside a spherical cavity.222 The Onsager model calculates the
dipole moment of the solute by using the point-dipole approximation, and thus, is only applicable
to molecules with dipole moments.223 When describing multipolar systems, the reaction field has
a poor description if the molecule is not spherical since the Onsager model uses an elliptical or
near-spherical cavity. Therefore, arbitrary cavities that use the overlap of atomic spheres defined
by their van der Waals radii and utilize a numerical solution to the Poisson equation are essential
for the development of accurate QM solvation models. A procedure used for calculations with a
solvation model is the self-consistent reaction field, which originates from using the solutions to
the Poisson equation as a perturbation to the gas phase Hamiltonian used for ab initio (Section 2.1)
and density functional methods (Section 2.3), as previously discussed.

Most implicit models are parameterized to describe aqueous solvation free energies at room
temperature; SM8,225 SMD,226 and COSMO-RS227 are solvation models that also describe non-
aqueous solvents and elevated temperatures. Implicit models are used extensively in pKa studies
due to the pKa depending on the solvation free energy.228,229

2.5.1 COSMO

The COnductor-like Screening MOdel (COSMO) was developed by Klamt in 1993 and is
based on the screening in conductors, which are infinitely strong dielectrics.230 This approach uses
arbitrarily-shaped cavities and a boundary element method to describe apparent surface charges that
define the same electrostatic potential as a numerical solution to the Poisson equation. COSMO is
used to calculate the energies of a molecule within a dielectric medium. The dielectric screening
energies for a given geometry scale with the dielectric permittivity of ε of the screening medium

35

as shown in Equation 2.52; x is 0.5 for the COSMO model.

ε − 1
ε + x

where 0 ≤ x ≤ 2

(2.52)

This is due to the response of a conductor to a solute charge distribution compared to the response
from a dielectric medium.10 The COSMO approach allows the calculation of analytical gradients
within an arbitrarily-shaped cavity. Therefore, geometry optimizations in the solvation phase are
practical as numerical gradients, which increase the computational cost, are typically used; however,
one of the difficulties includes finding the optimum parameters such as a set of van der Waal radii
to create the solvent accessible surface.230

2.5.2 PCM/C-PCM

The Polarizable Continuum Model (PCM) was developed in 1981 and employs an apparent
surface charge on the cavity surface.231,232 When using PCM and its variants, arbitrary cavity
shapes are used, unlike Onsager models; this provides better electronic energy results through a
more realistic description of the solute in solution. The basic PCM definition, Equation 2.53,
utilizes a continuous surface charge, σ(s), with the gradient on the internal (in) part of the surface
to describe the apparent surface charge distribution.231–233

σ(s) =

ε − 1
4πε

∂
∂(cid:126)n

(Vm + Vσ)in

(2.53)

In Equation 2.53, ε is the dielectric constant, (cid:126)n indicates the unit vector perpendicular to the cavity
surface pointing outward, VM, is the potential generated by the charge distribution, and Vσ is the
potential over the whole space generated by the polarization of the dielectric medium.

Unfortunately, using arbitrarily-shaped cavities are rather expensive because of the requirement
of numerical solutions for the derivatives and gradients. The difficulties with these models are
the sharp edges created by the overlapping spheres on the solvent accessible surface. Therefore,
the surface is smoothed by other spheres not centered on atoms to simulate the solvent excluded
surface. The C-PCM234 formulation was adapted in 1998 based on COSMO and used a conductor-
like setting within the PCM model. As described with COSMO, the solvent was treated as a

36

conductor and the polarizability of the system becomes zero, which decreases the complexity of
solving the Poisson equation. C-PCM utilizes the same equation used by PCM, Equation 2.53,
with the key assumption of using the scaling factor, Equation 2.52, to describe the polarization
charges such that the Gauss law is obeyed; in reference to Equation 2.52, x is 0 for the C-PCM
model.234 The C-PCM model is amongst the most widespread implicit solvation models used for
studies on organometallic systems,180,235–237 and the development of hybrid QM/QM schemes for
solvation.176,238

2.5.3 SMD

The universal implicit solvation model SMD226 was developed by Truhlar where the full solute
electron density is used without defining partial atomic charges. This density-based model separates
the solvation free energy into two components: the bulk electrostatic energy calculated through the
integral equation formalism PCM model, which replaces the molecular electric field on the surface
with the electrostatic potential; and the cavity-dispersion-solvent-structure component. The first
component uses the SCRF treatment and the solution for the nonhomogeneous Poisson equation,
Equation 2.54.

∇(ε∇φ) = −4πρf

(2.54)

The second component is the contribution arising from short-range solute-solvent interactions
in the first solvation shell. SMD describes the solvent accessible surface via a superposition of
nuclear-centered spheres with intrinsic Coulomb radii. SMD focuses on the standard solvation
free energies and was parameterized using 2821 solvation data points including free energies in 90
non-aqueous solvents and water.

2.6 Vibrational Self Consistent Field Theory

Vibrational spectroscopy is a useful approach to characterize intermolecular interactions for
reaction pathways and vibrational motion.
In the theoretical treatment of vibrational spectra,
accurate potential energy surfaces (PES) are necessary to describe nuclear dynamics, reaction

37

dynamics, and quantum rate constants.239–241 However for vibrational calculations, electronic
structure methods are often restricted to the harmonic oscillator approximation since the vibrational
Hamiltonian can be partitioned into a set of one-dimensional harmonic oscillators using normal
mode coordinates within the harmonic oscillator approximation.242 The errors inherent in both the
harmonic oscillator approximation and electronic structure methods accumulate to yield deviations
over 100 cm−1 for vibrational frequencies in some cases although the individual contributions of
the harmonic approximation and electronic structure method to the error are unknown.

Computationally, the harmonic oscillator approximation is conceptually simpler than fully
anharmonic calculations but results in a loss of accuracy for vibrational properties. One way to
correct for anharmonicity in molecular vibrations is to apply an empirical scaling factor to
harmonic vibrations, commonly called a frequency scaling factor.108 The scaling factor is
determined through a least-squares fitting to corresponding experimental frequencies; thus, this
approach is an underlying potential for addressing computing observables of the anharmonic PES.
However, scaling factors for DFT are approximately 1.00 ± 0.05 whereas those for ab initio
methods are lower (0.95 ± 0.05),108 implying that DFT yields more accurate vibrational
frequencies with the harmonic approximation and introduces uncertainty into which aspects of
DFT contribute to predicting vibrational frequencies.

Directly calculating the anharmonic PESs for vibrations provides better insight about removing
uncertainty arising from both the harmonic approximation and the use fo common global frequency
scaling factors. One of the ab initio methods developed for anharmonic vibrational spectroscopy is
vibrational self consistent field (VSCF) theory, which was developed in the late 1970s.243–247 The
vibrational Schrödinger equation with mass-weighted normal coordinates Qi,

 Ψn (Q1, . . . , QN ) = EN Ψn (Q1, . . . , QN )

(2.55)

−1

2

N(cid:88)

j=1

∂2
∂Q2
j

+ V (Q1, . . . , QN )

where V is the potential energy function of the system, n is the state number, and N is the number
of vibrational degrees of freedom (normal modes), utilizes the Born-Oppenheimer approximation
and neglects rotational coupling effects to vibration. VSCF theory is similar to the Hartree-Fock

38

theory (see Section 2.1) since each vibrational mode is characterized in the mean field of the other
vibrational motions. Unlike in Hartree-Fock theory, the total wavefunction of VSCF approximation
is a product of single mode wavefunctions akin to a Hartree product

Ψ(Q1, . . . , QN ) =

(n)
ψ
i

(Qi)

(2.56)

i=1

where the single mode wavefunctions ψ

are called the modals and QN are mass-weighted
normal coordinates since vibrations are distinguishable. Error due to introducing the separability
approximation depends on the coordinate system used.248

(n)
i

Using a variational principle for the ansatz in Equation 2.56 leads to the single mode VSCF

N(cid:89)

(cid:34)

equation

∂2
∂Q2
i
where the mean effective potential V

− 1
2

(cid:42) N(cid:89)

j(cid:54)=i

(n)
+ V
i

(Qi)

(Qi)

(n)
i

(Qi) for mode Qi is given by

(n)
i

(n)
i ψ

(n)
i = ε
ψ

(cid:35)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N(cid:89)
(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)V (Q1, . . . , QN )
(cid:88)

j(cid:54)=i

(2)
ij (Qi, Qj) +
V

(n)
V
i

(Qi) =

ψ

(n)
j

(Qj)

(n)
ψ
j

(Qj)

To examine the full potential V (Q1, Q2, . . . , QN ), the potential can be expanded via a multimode
expansion

(cid:88)

(cid:88)

V (Q1, Q2, . . . , QN ) =

(1)
V
i

(Qi) +

i

ij

ijk

where V

(1)
i

(Qi) are the single-mode diagonal terms

(1)
V
i

(Qi) = V (0, . . . , Qi, . . . , 0)

(2.57)

(2.58)

(cid:43)

(3)

ijk (Qi, Qj, Qk) + . . . (2.59)
V

the pair-wise interactions W

(2)

ij (Qi, Qj) from the expansion of V (Q1, Q2, . . . , QN ) are
ij (Qi, Qj) − V

(Qi) − V

(Qj)

(1)
j

(1)
i

(2)

W

(2)
ij (Qi, Qj) = V

= V (0, . . . , Qi, . . . , Qj, . . . , 0) − V

(1)
i

(Qi) − V

(1)
j

(Qj)

39

(2.60)

(2.61a)

(2.61b)

and so forth with higher order expansions. N-order expansions of the potential are not feasible for
N larger than six since the integration over the potential is a N-1 dimensional integral. Therefore,
the expanded potential is usually truncated in terms of a quartic or sextic force field.239

Equations 2.57 and 2.58 are solved self-consistently for the single mode wavefunctions, energies,
and effective potentials. Several methods can be applied for the solution of Equation 2.57 to get
both the ground and excited VSCF states of the system. Due to this approximation, the total energy
is given by

En =

(n)

i + (n − 1)

ε

ψ

(n)
j

(Qj)

(n)
ψ
j

(Qj)

(2.62)

(cid:42) N(cid:89)

j(cid:54)=i

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)V (Q1, . . . , QN )

(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N(cid:89)

j(cid:54)=i

(cid:43)

N(cid:88)

i=1

The major computational difficulty is due to the evaluation of multidimensional integrals inherent in
Equations 2.57, 2.58, and 2.62, especially for large systems, and that depends on the mathematical
form of the potential. Hence, the choice of potential plays a key role in the VSCF approximation.
Since VSCF describes the effect of a singular vibrational mode in the mean field of all other
vibrational modes as the Hartree-Fock method does for electrons,
the effects of correlation
between modes need to be described. For post Hartree-Fock methods such as MP2 and CI, there
are complementary vibrational equivalents that correlate vibrational motion. For example, for
perturbation theory, the full vibrational Hamiltonian is written in the form

H = HSCF,(n) + ∆V (Q1, . . . , QN )

(2.63)

where HSCF,(n) is the Hamiltonian used in Equation 2.57 and the equation for ∆V is given by

∆V (Q1, . . . , QN ) = V (Q1, . . . , QN ) − N(cid:88)

(n)
V
i

(Qi)

(2.64)

i=1

where ∆V represents all correlation effects between vibrational modes. Considering the pair-wise
approximation terms in Equations 2.61a, ∆V can be rewritten as

∆V (Q1, . . . , QN ) =

(1)
V
i

(Qi) +

(n)
V
i

(Qi)

(2.65)

N(cid:88)

i=1

(cid:88)

(cid:88)

j

i>j

ij (Qi, Qj) − N(cid:88)

(2)

W

i=1

where the potential at the minimum is taken at zero, leaving diagonal terms and pair-wise terms
as shown in Equations 2.60 and 2.61b. Methods that account for correlation effects of vibrational
modes are known as post-VSCF methods.

40

Post-VSCF methods include VSCF-PT2, which is a second-order perturbation to account for
correlation effects between vibrational modes, as well as vibrational coupled cluster (VCC) and
vibrational configuration interaction (VCI) methods, and a combination of VCI with perturbatively
selected interactions (VCIPSI-PT2).239,249–254 The idea is that ∆V , which is the difference between
the true Hamiltonian and the VSCF Hamiltonian, must be small as VSCF is a good approximation.
VCI yields the best possible results variationally given the basis set limits.239,250 Analogous to
CI, every possible contribution of a complete set of functions is considered and thus full VCI
with an infinite basis set is the exact solution to the vibrational time independent Schrödinger
equation (Equation 2.55) given the constraints (BO approximation and neglecting rotational effects
on vibration).

For N normal modes, there are N(N-1)/2 coupling potentials. Each coupling potential is
computed with electronic structure methods on a grid of Ngrid × Ngrid points (Ngrid = 16 in
Chapter 6). For example, C6H6, which has 30 normal modes, would require 111,360 single point
calculations for all 435 pair-wise coupling potentials assuming Ngrid = 16. This requires
additional approximations such as the vibrational configuration interaction with perturbation
selected interactions (VCIPSI) algorithm developed by Scribano and Benoit254 to iteratively
select the VCI active space based on previous implementations of this algorithm for ab initio
electronic structure calculations255 and VCI methods.251–253 The active space is treated
variationally and then increased iteratively using a vibrational Møller-Plesset barycentric (VMPB)
partition scheme to improve the representation of the complete VCI wavefunction.
The
VCIPSI-PT2 method utilizes the final VMPB correction in the VCI active space. Implementation
of this algorithm led to a savings of 70-80% while only deviating from VCI by approximately 0.01
cm−1 for all vibrations of CH4 when using MP2/aug-cc-pVTZ for generating the vibrational PES
and a savings of 85% and a deviation of 0.07 cm−1 for the OH stretching frequency of benzoic
acid while using 0.49% of the disk space that VCI used for the same vibration.254

Other measures to reduce the computational cost includes screening weakly coupled pair-wise
coupling interactions via a threshold established from calculating the coupling strength (Equation

41

2.66), which can be calculated with only the VSCF potential.256,257

ξ(qi, qj) =

1

N 2grid

ni=1

nj =1

Ngrid(cid:88)

Ngrid(cid:88)

(k)

|V
ij (ni, nj)|

(2.66)

By removing non-essential vibrational coupling elements from the potential, a fast-VSCF
approach is attained. This can greatly reduce the computational cost to generate fully anharmonic
PESs for polyatomic molecules of increasing size and complexity.

The use of VSCF and VCI methods are pertinent in Chapter 6, as these methods are used to
analyze anharmonic PESs to predict anharmonic vibrations for diatomic and polyatomic molecules.

42

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43

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62

CHAPTER 3

PREDICTION OF pKa OF LATE TRANSITION METAL HYDRIDES VIA A QM/QM

APPROACH

3.1 Introduction

Transition metal

(TM) hydrides are important

intermediates in many catalytic and
stoichiometric processes such as hydrogenation and hydroformylation.1–9 As numerous
organometallic catalytic reactions include hydride transfers, characterizing metal-ligand binding
properties is vital to understanding how these catalysts work. One such thermodynamic property
for TM hydrides is the pKa. Although the pKa values of a number of TM hydrides have been
measured experimentally, experimental characterization of pKa is not accessible for all TM
hydrides. Therefore, with computational approaches, such as density functional theory (DFT),
geometries, spectroscopic constants, and energetics, and thermodynamic properties such as pKas,
Gibbs free energies, and enthalpies of formation, become an important route to predict various
molecular and thermodynamic properties in the absence of experimental measurements.10–17

The development of density functionals has motivated their wide application, and, in 2001,
Perdew proposed the Jacob’s ladder analogy to classify density functionals into primary rungs that
present the hierarchy of density approximations.18 This is explained in Section 2.3. Essentially, the
inherent complexity of functional class increases with higher rungs in Jacob’s ladder; however, the
accuracy of a functional is not necessarily dependent on its complexity. Therefore, in determining
choice of density functional, calibration of density functional approaches with data from experiment
or high accuracy wavefunction-based calculations such as CCSD(T) should be done.19 Though
CCSD(T) is often considered as the “gold standard” of quantum chemistry, it is not computationally
affordable (memory, disk space, CPU time) for routine calculations of many TM complexes, which
often are bound to numerous large ligands.20–24

For the prediction of thermodynamic properties of TM-containing complexes with sterically

63

hindering ligands, such as for TM hydrides, DFT is often considered, as it is readily used for
molecules of increasing size and complexity.25 In a study by Tekarli et al., the gas-phase enthalpies
of formation (∆Hf) of 19 3d TM-containing species were calculated to assess the performance
of 44 density functionals paired with cc-pVTZ and cc-pVQZ basis sets.26Among the considered
functionals, the B97-1, and PBE1KCIS functionals resulted in the lowest mean absolute deviations
(MADs) relative to experiment. A similar study was done by Laury et al. for the ∆Hf’s of 30 4d
species, considering the utility of 22 density functionals. Of the functionals considered, B2GP-
PLYP and mPW2-PLYP yielded the lowest MAD from experiment.27 Riley and Merz28 examined
the performance of 12 functionals with the 6-31G* and TZVP basis sets for the calculation of
∆Hf’s of 94 TM species. TPSS1KCIS in combination with the TZVP basis set resulted in the
lowest MAD from experiment in their study. Wang et al.’s study of TM atom mediated Cβ-O bond
cleavage of the β-O-4 linkage of lignin used density functionals to compare the binding, activation,
and reaction enthalpies with respect to CR-CCSD(T).29 They found that the property that yielded
the lowest MADs from CR-CCSD(T) fluctuated depending on functional choice as well as choice
of 3d, 4d, or 5d metal. Overall, the lowest average deviation from CR-CCSD(T) for predicting the
reaction energetics was provided by PBE0. These gas phase studies demonstrate that functional
choice should be strongly based upon the molecular systems of interest, considering the TM, as
well as the number and types of ligands, and the property of interest for TM species.

Since the ligands for bulky TM hydrides predominately consist of main group atoms,
approaches that have been useful for main group thermochemistry should be considered in
identifying approaches that may be effective for the description of TM hydrides. In a study by
Goerigk and Grimme,30 a thorough benchmark of 47 density functionals from the GMTKN30
database for general main group thermochemistry recommended functionals including the GGA
B97-D3 and the meta-GGA, oTPSS-D3. PW6B95 was identified as the most robust hybrid in their
study. However, in comparing their results for main group species to the aforementioned TM
complexes,26–28 functionals that are optimal for each TM may not perform well for the ligands.

In the development of the MN15-L density functional, Yu et al. ranked 48 density functionals

64

based on their performance for 33 molecular databases.31 This study showed that B97-1, which
performed well for the thermochemistry of 3d TM-containing compounds,26 ranked 3 out of the 48
chosen functionals for SR-MGM-BE9, which examines single-reference main-group metal bond
energies, but ranked 29 for πTC13, which examines thermochemistry of hydrocarbon π systems.
The opposite trend is shown for M11-L, which ranked 14 for πTC13 but 45 for SR-MGM-BE9.
The rank changes of B97-1 and M11-L for the 2 databases emphasize that some density functionals
are good for TM chemistry but poor at describing main group ligands, and vice versa.

As ligand complexity increases, solvation shells of the complex as well as the electronic and
steric effects of ligands should be considered alongside the chemically important region with the
metal center. However, a single functional may not portray all aspects of increasingly complex
systems useful for homogeneous catalysis effectively. Thus, the main goal of this study is to develop
a scheme that accounts for an optimal method choice for the metal and an optimal method choice
for the ligand.

For systems containing numerous non-hydrogen atoms, the use of cost-effective multilayer
fragmentation approaches such as ONIOM,32–36 Molecules-in-Molecules,37 and the Molecular
Tailoring Approach38 can provide a framework for such a combination. However, as the ONIOM
method (Section 2.2.5) has been commonly applied to transition metal complexes and homogeneous
catalysis,39–41 whereas other fragmentation approaches are often utilized for biomolecules and
water clusters,37,42,43 ONIOM is used in this study. However, because of the size of many
TM hydrides, it can be costly to use high level theoretical methods (e.g. CCSD(T)) to directly
model them, and even within an ONIOM scheme, the size of the model layer can also limit the
application of high level theoretical methods in the model layer, making them impractical. Thus,
while a combination of a higher level (HL) method and a lower level (LL) method demonstrates a
traditional use of ONIOM [i.e., ONIOM(HL:LL)], such a layering scheme can also be utilized to
consider the strengths of methods in a metal and non-metal partitioning of a molecule, as is done
in this chapter.

As compared with the number of gas phase computational studies on TM species, far fewer

65

studies have been reported on the solvent effects on TM compounds. Such studies are important,
as many TM reactions are carried out in a solvated phase, including TM hydride-mediated
catalysis. For the solvated phase, the pKa exhibits the strongest effects of solvation relative to their
gas phase analogs due to the charge separation of the species involved. Previous studies by Liptak
and Shields44–47 and others48–52 have examined the use of both direct and relative
thermodynamic schemes for pKa calculations. These studies show that direct thermodynamic
schemes for calculating pKas of unknown acids have excellent agreement with experiment with
reduced computational cost over relative schemes. Therefore, the direct thermodynamic scheme,
shown in Scheme 3.1, will be used for this work.

Implicit solvation models (see Section 2.5) are often utilized for practical computations of bulk
TM species.53–57 For implicit solvent models, the choice of a cavity model, which defines the
shape and size of the cavity occupied by a solute species in the solvent, has been shown to have
an impact on the prediction of the pKa of organic acids using DFT.58–60 For instance, in the study
of the aqueous solvation free energies of 10 organic species calculated with seven cavities (UAKS,
UAHF, UAHF, Bondi, Pauling, UA0, and UFF) using the B3LYP/6-31+G(d) method with the
C-PCM solvation model, UAKS and UAHF resulted in the lowest MADs relative to experiment
in comparison to the other considered cavity models.58 Also, a systematic study of solvation
free energy and pKa values of monoprotic, diprotic, and triprotic acids based on DFT(B3LYP,
PBE, BVP86, and M05-2X)/aug-cc-pVTZ methods combined with the C-PCM and SMD solvation
models showed that the Pauling cavity in combination with M05-2X resulted in the lowest deviation
among the UFF, UAKS, Pauling, and Klamt cavity models.60 Though the prediction of pKa values
has been shown to be related to the choice of cavity model, studies showing the utility of density
functionals in terms of the choice of cavity models for TM-containing species are limited.61

In a study by Qi et al.,62 using CCSD(T) with an insufficient basis set, such as LANL2DZ+p, to
calculate the model layer of TM hydrides was found to fail dramatically in describing TM hydrides,
while an improvement of the basis set achieved better results. However, further improving the basis
set will make CCSD(T) impractical in the treatment of the model layer. Thus, they tried to use

66

density functionals to describe the whole systems with a high-level basis set to describe the model
layer and low level basis sets to describe the rest of region, which yields much better results than
CCSD(T) with a low-level basis set. Therefore, DFT can perform well in calculating properties
of TM hydrides and the choice of basis set is more important than the choice of method (e.g.,
CCSD(T) vs density functionals).

As shown above,26–28,30 TM (model layer) and main group elements (the main component of
TM hydrides) can be described well with multiple density functionals. Therefore, instead of using
the same density functionals to describe the whole systems, it is worth examining if the combination
of different density functionals in ONIOM will provide better description for TM hydrides systems.
To assess the appropriateness of density functionals combined with several levels of basis sets
within the ONIOM scheme for TM hydrides in solvated phase, comprehensive studies must be
carried out where a much wider variety of functionals are considered.

In this chapter, to address the ability of electronic structure methods to describe the pKas of
TM hydrides, density functionals utilized in partnership with basis sets of at least triple-ζ quality
are investigated, including ONIOM(DFT:DFT) schemes.63 As well, to consider TM chemistry
in solution, the impact of solvent model (SMD, COSMO, and C-PCM) and the degree to which
the several cavity models affect the determination of the pKa values of TM hydrides are analyzed
in this study. The influence of the addition of exact exchange and dispersion corrections is
considered. As shown in the above examples26,27,64 and several other studies,65–67 the choice
of basis set and the size of the molecules28,64 can have an impact on the utility of density
functionals; therefore, an understanding of the influence of basis set choice and size of the model
layer within the ONIOM scheme also is assessed for several basis sets. This investigation provides
insight about the selection of computational methods for TM hydrides that can be applied to
investigate other thermodynamic properties of catalysts for many important chemical reactions,
such as hydrogenation and hydroformylation.

67

3.2 Theoretical Methods

The two layer ONIOM scheme was used with a variety of density functionals and several
basis sets to determine pKa values of Group 10 TM hydrides ([HNi(depe)2]+, [HNi(depp)2]+,
[HNi(PNP)2]+, [HPd(depe)2]+, [HPd(depp)2]+, [HPd(PNP)2]+, [HPt(depe)2]+, [HPt(PNP)2]+).
All calculations were performed using the GAUSSIAN 09 software package.68 For all considered
TM hydrides, geometry optimizations and frequency calculations (using vibrational ZPE scaled by
0.9890)69 were performed using B3LYP/cc-pVTZ in both the gas phase and acetonitrile solvent
to replicate experimental conditions. Acetonitrile solvent systems were treated using the C-PCM,
COSMO, and SMD continuum solvation models.53–57 All stationary points were verified to be true
minima, with no imaginary frequencies. The thermochemical corrections from B3LYP/cc-pVTZ
frequency calculations were added to the single point energies to obtain gas phase and solvation
free energies at 298 K.

Subsequently, single-point calculations were performed with the two-layer ONIOM method
presented in Section 2.2.5 with Equation 2.38.32–36 Since choosing how to partition the molecular
systems into layers can have a significant impact upon the calculated energies, several core regions
have been considered: (a) the metal atom and four phosphorous atoms; (b) the metal, phosphorous
atoms, and the chelate rings; and, (c) all atoms except for the terminal methyl group. The results from
these expansions are defined in this study as ONIOM-1, ONIOM-2, and ONIOM-3, respectively,
and are shown in Figure 3.1. The ONIOM-1 scheme is primarily used due to computational cost.
To evaluate the impact of the DFT approaches used within ONIOM for the prediction of
pKas of TM hydrides, the following DFT methods were utilized (summarized in Table 3.1), listed
by functional class: (a) Generalized Gradient Approximation (GGA): BLYP,70,71 PBE,72 and
B97-D73; (b) meta-GGA (M-GGA): M06L,16 BB95,74,= and TPSS75; (c) hybrid GGA (H-
GGA): PBE0,72,76,77 B3LYP,70,71,78 and B3P8671,79; (d) hybrid-meta GGA (HM-GGA): M06,16
M06HF80; and, (e) double hybrid GGA (DH-GGA): B2PLYP81 based on their utilization for these
types of compounds. Additionally, Grimme’s empirical dispersion correction (D3)82 was added
to several density functionals selected from GGA, M-GGA, H-GGA, and HM-GGA functionals,

68

to evaluate the effect of a dispersion correction on the accuracy of predictions of pKas of the
TM hydrides. To evaluate the impact of the percentage of exact exchange, the percentage of exact
exchange for PBE0 was varied from 0% to 80% in intervals of 5% since PBE0 includes no empirical
parameters that may affect the utility of DFT; hence, avoiding interference from other empirical
parameters.

69

Figure 3.1: From left to right, the compounds are TM(depe)2, TM(depp)2, TM(PNP)2. (a) The
model system (bolded) within the ONIOM-1 QM/QM partitioning scheme for TM hydrides with
the TM atom (Ni, Pd, and Pt) and four phosphorous atoms in the layer using the high-level method.
(b) ONIOM-2: The QM/QM partitioning scheme for TM hydrides with all the atoms within the
chelate rings in the layer using the high-level method. (c) ONIOM-3: The QM/QM partitioning
scheme for TM hydrides with all except for the very outside methyl group in the layer using the
high-level method.

70

Table 3.1: Summary of the density functionals utilized.

BLYP70,71
PBE72
B97-D73
M06L16
BB9574
TPSS75

Type
GGAa
GGAa
GGAa
M-GGAb
M-GGAb
M-GGAb

%HF
0%
0%
0%
0%
0%
0%

Exchange/Correlation
Becke88/Perdew86/Lee-Yang-Parr
Perdew-Burke-Ernzerhof/ Perdew-Burke-Ernzerhof
B97-D/B97-D
M06L/M06L
Becke88/Perdew86/Becke95
Tao-Perdew-Staroverov-Scuseria/Tao-Perdew-
Staroverov-Scuseria

25% Perdew-Burke-Ernzerhof/Perdew-Burke-Ernzerhof
20% Becke88/Perdew86/Lee-Yang-Parr
20% Becke88/Perdew86
27% M06/M06
52% M05-2X/M05-2X
54% M06-2X/M06-2X
100% M06HF/M06HF
50% Becke88/Perdew86/Lee-Yang-Parr

H-GGAc
H-GGAc
H-GGAc
HM-GGAd
HM-GGAd
HM-GGAd
HM-GGAd
DH-GGAe

PBE072,76,77
B3LYP70,71,78
B3P8671,79
M0616
M05-2X83
M06-2X16
M06HF80
B2PLYP81
aGGA (generalized-gradient approximation)
bM-GGA (meta GGA)
cH-GGA (hybrid GGA)
dHM-GGA (hybrid meta GGA)
eDH-GGA (double hybrid GGA)

For the lower level within the ONIOM calculations, the relativistic effective core potential (ECP)
and valence double-ζ basis set of Hay and Wadt (LANL2DZ)84 as well as the Stuttgart/Dresden
(SDD)85–87 relativistic ECP and valence triple-ζ basis set were considered. For LANL2DZ and
SDD, 10, 28, and 60 electrons were frozen for Ni, Pd, and Pt, respectively. For the high-level
method, cc-pVDZ, cc-pVTZ, aug-cc-pVDZ, and aug-cc-pVTZ were used.88–91 For Ni species,
correlation consistent basis sets with the one-particle Douglas-Kroll-Hess Hamiltonian for scalar
relativistic effects were applied (e.g., aug-cc-pVTZ-DK) for all atoms.92 For Pd and Pt species,
the small-core relativistic pseudopotential basis sets (e.g., aug-cc-pVTZ-PP) were used for Pd and
Pt while the all-electron basis sets (e.g. aug-cc-pVTZ) were used for main group atoms since the
pseudopotentials are incompatible with the DKH Hamiltonian and are constructed to account for
relativistic effects of the heavy atom.90,91,93 In the following sections, the terms DK for Ni and PP

71

for Pd and Pt are dropped for clarity from the selected basis set notations.

Three implicit solvation models, SMD,53 COSMO,54 and C-PCM,55–57 were employed to
include solvent effects in the single point calculations. The UA0, UAKS, Pauling, Bondi and
default cavities (UFF for C-PCM, Klamt for COSMO, and Coulomb-SMD for SMD) were applied.

Scheme 3.1: The direct thermodynamic scheme

The direct thermodynamic scheme for calculating pKas of unknown acids shown in Scheme
3.1 has been used mainly due to its demonstrated utility46,48–52,94 and was used in this study with
the value -4.39 kcal mol−1 for the gas phase free energy of a proton, ∆Ggas(H+), derived using
the Sackur-Tetrode equation.95 For the value of the experimental solvation phase free energy of the
proton in acetonitrile, ∆Gsolv(H+), -260.2 kcal mol−1 has been recommended96 and was used in
this study. Thus, the solvation free energy (∆Gsol) can be calculated using the following equations
(Eq. 3.1-3.5).

∆Gsol = ∆Ggas + ∆∆Gsolv

∆Ggas = Ggas(LnM ) + Ggas(H+) − Ggas(HLnM +)

∆∆Gsolv = ∆Gsolv(LnM ) + ∆Gsolv(H+) − ∆Gsolv(HLnM +)

∆Gsolv(LnM ) = Esolv(LnM ) − Egas(LnM )

∆Gsolv(HLnM +) = Esolv(HLnM +) − Egas(HLnM +)

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

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The pKa values related to free energies of solvation were calculated as

pKa =

∆Gsolv
2.303RT

(3.6)

All of the calculated gas phase free energies in units atm were converted to molar units and the
solvation phase free energies were calculated using [(Esoln + Gnes) − Egas], as defined in the
parametrization of continuum solvent models.47,97 An error of 1.36 kcal mol−1 in ∆Gsolv results
in a deviation of 1 pKa unit. Ho and Coote reported that a direct thermodynamic cycle can be
expected to depart from experiment by 3.5 pKa units.98

3.3 Results and Discussion

The considered molecules are grouped based on central TM atoms (Ni, Pd, and Pt) and the
ligands (depe, depp, and PNP) in order to evaluate the impact of the selected density functionals,
basis sets, cavities, solvation models, and the expansion in size of the high-level region within
ONIOM on the calculated pKas of TM hydrides. Mean absolute deviations (MADs) with respect
to experimental data99–102 are reported. Since [HPt(depp)2]+ does not have readily available
experimental data for pKa due to the highly reactive nature of Pt complexes, a net equation (Eq.
3.7) of the thermochemical cycle103,104 relating hydricities, pKas , and redox potentials was used to
calculate a proposed pKa based on experimental redox potentials and hydricities.100 From Equation
3.7, the proposed pKa for [HPt(depp)2]+ is 28.3.

∆GH− = 1.37(pKa) + 46.1E◦(II/0) + 79.6 kcal mol−1

(3.7)

None of the considered TM hydrides showed significant structural changes in the gas or solvation
phases; therefore, the solvation phase structures obtained with C-PCM were used for the single
point calculations based on computational cost.

3.3.1 Utility of DFT in the Real System

The fourteen density functionals (Table 3.1) were chosen for the real layer and PBE, M06-L,
B3LYP, and M06 were chosen for the model layer. PBE, M06-L, B3LYP, and M06 were chosen to

73

showcase the tiers of functional complexity in the model layer. A summary of method and basis
set choice for this section is provided in Table 3.2. Using C-PCM for [HNi(depp)2]+ and TPSS for
the real layer, the MAD when using PBE, M06-L, B3LYP, and M06 for the model layer was 11.7,
7.4, 6.2, and 9.0 pKa units, respectively. When using B97-D for the real layer, the MAD when
using PBE, M06-L, B3LYP, and M06 for the model layer were 9.7, 5.4, 4.2, and 7.0 pKa units,
respectively. Similarly, using C-PCM for [HPd(depe)2]+, the MAD when using PBE, M06-L,
B3LYP, and M06 for the model layer were 10.8, 7.3, 6.6, and 11.9 pKa units, respectively, using
TPSS in the real layer and 8.8, 5.3, 4.6, 9.9 pKa units, respectively, using B97-D in the real layer.
Since the MADs varied significantly based on functional choice in the model layer, the MADs
from PBE, M06-L, B3LYP, and M06 are averaged to eliminate bias of functional complexity for
the model layer. Therefore, for [HNi(depp)2]+ and [HPd(depe)2]+, the average MAD is 8.6 and
9.2 pKa units for TPSS, and 6.6 and 7.2 pKa units for B97-D. Averaging the MADs for the model
layers and for the molecule set allowed the choice for the real layer to be compared more readily.

Table 3.2: Theoretical methods for the description of real and model systems within the two-layer
ONIOM scheme using C-PCM, COSMO, and SMD for utility of DFT in the real layer.

Method

1
2
3
4

Model systema

PBE/aug-cc-pVTZ
M06L/aug-cc-pVTZ
B3LYP/aug-cc-pVTZ
M06/aug-cc-pVTZ

Real systemb

DFT
DFT
DFT
DFT

aaug-cc-pVTZ (main group atoms for Pd, Pt species), aug-cc-pVTZ-DK (Ni species), aug-cc-pVTZ-PP (Pd and Pt).
bDFT functionals are listed in Table 3.1. LANL2DZ is used as the basis set for the real system.

For the molecule set, the average MAD in the pKa from experiment is provided in Figure 3.2,
where the considered density functional approach for the low level of the ONIOM approach has
been varied. Among the functionals considered, B97-D performed best with MADs of 5.5, 2.7,
and 2.3 pKa units for C-PCM, COSMO, and SMD, respectively, followed by B3LYP (6.3, 3.4, 2.9
pKa units), and M06-L (7.2, 4.5, 3.8 pKa units). Except for B97-D, B3LYP, and M06L, all other
GGA, M-GGA and H-GGA functionals performed similarly regarding each solvation model with
MAD values of about 7.9, 5.0, and 4.3 pKa units for C-PCM, COSMO, and SMD, respectively.

74

The functional with the highest MAD is M06-2X with MAD values of 10.5, 7.7, and 7.1 pKa units
for C-PCM, COSMO, and SMD, respectively. Among the three selected solvation models, SMD
provided the best comparison with experimental pKa data while C-PCM yielded the highest MADs
for all fourteen considered density functionals.

Figure 3.2: MADs in pKa values for the density functionals within low-level methods relative to
experiment. All of the results are from calculations with ONIOM(PBE,M06L,B3LYP,M06/aug-
cc-pVTZ:DFT/LANL2DZ) scheme. The results of using the four functionals in the model layer
are averaged for the molecule set.

It is worth noting that using separate functionals for the core and real layers provided lower
MADs than when the same functional is used for both layers within the ONIOM scheme regarding
each solvation model.
For instance, with SMD, ONIOM (B97-D/ aug-cc-pVTZ : PBE/
LANL2DZ), ONIOM (B97-D/ aug-cc-pVTZ : M06L/ LANL2DZ), ONIOM (B97-D/

75

aug-cc-pVTZ : B3LYP/ LANL2DZ), and ONIOM (B97-D/ aug-cc-pVTZ : M06/ LANL2DZ)
yielded MADs lower than ONIOM (PBE/ aug-cc-pVTZ : PBE/ LANL2DZ), ONIOM (M06L/
aug-cc-pVTZ : M06L/ LANL2DZ), ONIOM (B3LYP/ aug-cc-pVTZ : B3LYP/ LANL2DZ), and
ONIOM(M06/ aug-cc-pVTZ : M06/ LANL2DZ) by 2.4, 1.6, 1.4, and 4.1 pKa units, respectively.
This shows that a mixed basis set approach may not be advantageous for TM hydride systems.

Figure 3.3: MADs in pKa values for five types of density functionals, GGA, M-GGA, H-GGA, HM-
GGA, and DH-GGA functionals, within low-level methods relative to experiment. All of the results
are from calculations with ONIOM(PBE,M06L,B3LYP,M06/aug-cc-pVTZ:DFT/LANL2DZ)
scheme. The results of using the four functionals in the model layer are averaged for the molecule
set.

The utility of the types of density functionals at modeling pKas is shown in Figure 3.3. The
GGA (7.1, 4.2, and 3.6 pKa units for C-PCM, COSMO, and SMD, respectively) and H-GGA

76

(7.3, 4.4, and 3.8 pKa units for C-PCM, COSMO, and SMD, respectively) functionals produced
similar MADs, which were better than all the other types of functionals regardless of solvation
method. In contrast, DH-GGAs performed the worst with MADs of 9.8, 6.9, and 6.0 pKa units
for C-PCM, COSMO, and SMD, respectively, which indicates that the addition of a fraction of the
PT2 correlation energy is a disadvantage for the description of pKas of TM hydrides. Compared
with HM-GGAs, M-GGA functionals, which do not include exact exchange, yielded lower MADs
for all three solvation models. Therefore, exact exchange is not necessary for the description of
the real system. COSMO and SMD performed similarly (5.3 and 4.7 pKa units, respectively) and
resulted in MADs ∼3 pKa units lower than that from C-PCM (8.1 pKa units).

The comparison of functional types of is considered with respect to central TM atoms (Table
3.10) and ligand systems (Table 3.11) employing each of the three solvation models. For the
Ni species, the MADs increased with increasing functional complexity, except for DH-GGAs
for all three solvation models. For Pd and Pt species, H-GGAs yielded the lowest MADs in
comparison to other types of functionals while DH-GGAs always performed the worst for all three
solvation models. Moving from Ni to Pt, the MADs of non-local exchange functionals (H-GGA,
HM-GGA, and DH-GGA) decrease, which indicates that non-local exchange in functionals can
describe TM hydrides with heavier central TM atoms better than those with lighter central TM
atoms. Considering the overall MADs of different types of functionals, the increase in MADs upon
inclusion of exact exchange is more significant for M-GGA functionals (HM-GGA) than it is for the
GGA functionals (H-GGA). As shown in Figure 3.1, the size of the considered ligands increases in
the order of depe, depp, and PNP. Similar MAD was found for each type of functional between all
three solvation models as the size of the ligand increased.

3.3.2 Utility of DFT in the Model Layer

As seen in the previous section, the fluctuation caused by the choice of the four density
functionals in describing the model system of the ONIOM scheme implies that functional choice
for both the model and real layer are factors in calculating pKa values; therefore, the section

77

focuses on the utility of the density functionals for the model layer while keeping the functionals
chosen for the real layer constant. Table 3.3 summarizes the combination of density functionals as
ONIOM schemes designed to measure the influence of the fourteen considered density functionals
combined with the aug-cc-pVTZ basis set in the description of model layers of the TM hydrides
(Figure 3.1a). The real systems (Figure 3.1a) were treated with three density functionals (B97-D,
M06L, and B3LYP) paired with the LANL2DZ basis set, which were selected based on their better
performance as low-level methods shown in the previous section. The rationale for averaging the
MADs from the three selected real system methods is to eliminate bias from the functional chosen for
the real layer and gauge the utility of density functionals in the model layer, as done in the previous
section for the real layer. The MADs for each high-level method, which are based upon deviations of
the calculated pKa values of the TM hydrides from experimental data for each functional using the
C-PCM, COSMO, and SMD, are reported in Figure 3.4. For C-PCM and COSMO, the three best-
performing functionals were B3LYP, M05-2X, and M06-HF, with B3LYP and M06-HF resulting
in the lowest average MADs with C-PCM and COSMO, respectively. For SMD, B97-D, TPSS, and
M05-2X yielded the same average MAD value of 2.1 pKa units. Therefore, unlike the consistency
for density functionals that were found to perform best in describing the real systems among the
solvation models, the utility of density functionals in describing the model layer depended on the
selection of the solvation model.

78

Figure 3.4: MADs in pKa values for fourteen GGA, M-GGA, H-GGA, HM-GGA, and DH-
GGA functionals within high-level methods relative to experiment. All of the results are from
calculations with ONIOM(DFT/aug-cc-pVTZ:B97-D,M06L,B3LYP/LANL2DZ) scheme. The
MADs in pKa values for the three functionals in the real layer are averaged for the molecule
set.

The most accurate pKa values were yielded by different density functionals for each solvation
model (MAD of 2.0 pKa units by B3LYP with C-PCM, 1.9 pKa units by M06-HF with COSMO,
and 2.1 pKa units by B97-D, TPSS, and M05-2X with SMD). PBE resulted in the largest difference
from experimental data with MADs of 6.7, 6.5 and 5.5 pKa units for C-PCM, COSMO, and SMD,
respectively. BB95 and M06 also performed considerably worse than other considered functionals
(except PBE), which resulted in the same MADs of 6.2 pKa units for C-PCM and 5.0 pKa units for
SMD, and similar MADs of about 5.6 pKa units for COSMO.

79

Figure 3.5: MADs in pKa values for five types of density functionals, GGA, M-GGA, H-GGA,
HM-GGA, and DH-GGA functionals, within high-level methods relative to experiment. All of the
results are from calculations with ONIOM(DFT/aug-cc-pVTZ:B97-D,M06L,B3LYP/LANL2DZ)
scheme.

The utility of types of density functionals is shown with the three solvation models in Figure
3.5. The H-GGA functionals provided the most comparable pKa values to the experimental data
with MADs of 3.2, 2.8, and 2.5 pKa units for C-PCM, COSMO, and SMD, respectively, while
the DH-GGA functionals resulted in the highest MADs of 5.2, 4.6, and 4.0 pKa units for C-PCM,
COSMO, and SMD, respectively. The large MAD of DH-GGAs infers that the addition of a fraction
of the PT2 correlation energy should not be considered for the accurate description of the model
layer of TM hydrides. For all three solvation models, GGA and M-GGA functionals yielded larger
MADs than H-GGA and HM-GGA, which indicates that inclusion of exact exchange is necessary

80

to describe the model layer of TM hydrides more appropriately. This lowering of the MADs by
including exact exchange in SMD was less obvious then for C-PCM and COSMO.

Table 3.3: Theoretical methods for the description of real and model systems within the two-layer
ONIOM scheme using C-PCM, COSMO, and SMD for utility of DFT in the model layer.

Method

1
2
3

Model systema

DFT/aug-cc-pVTZ
DFT/aug-cc-pVTZ
DFT/aug-cc-pVTZ

Real systemb

B97-D
M06-L
B3LYP

aDFT functionals are listed in Table 3.1. aug-cc-pVTZ (main group atoms for Pd, Pt species), aug-cc-pVTZ-DK (Ni species),
aug-cc-pVTZ-PP (Pd and Pt).
bLANL2DZ is used as the basis set for the real system.

The types of functionals were compared with respect to central TM atoms (Table 3.12) to
assess if their ability to describe the model layer was determined by their performance on the
description of metal center. The MADs for all types of functionals decrease from lighter to heavier
metal for all three solvation models (Table 3.12). For Ni species, the M-GGA functionals yielded
the lowest MADs of 3.7, 3.4, and 2.6 pKa units with C-PCM, COSMO, and SMD, respectively.
The H-GGA functionals performed the best for C-PCM and COSMO with MADs of 3.2 and 2.9
pKa units, respectively. The GGA, M-GGA, and H-GGA functionals resulted in similar MADs of
about 2.5 pKa units with SMD for Pd species. For Pt species, the HM-GGA functionals produced
comparable MADs of about 1.8 pKa units for COSMO and SMD that were lower than for other
types of functionals. The DH-GGA functional resulted in the largest MADs for all considered metal
species with all three solvation models. The model layer is described better by H-GGA functionals
than GGA functionals. Thus, following the same conclusion based on the overall performance of
functional type, the reduction in MADs for H-GGA functionals from GGA functionals is more
significant for TM hydrides with lighter central TM atoms than for those with heavier central TM
atoms.

81

3.3.3

Impact of Exact Exchange on the Accuracy of DFT

Although there was no systematic trend found between the percentage of exact exchange and
the accuracy of Minnesota functionals for the prediction of the pKas of TM hydrides (Figure 3.4),
H-GGA and HM-GGA functionals showed improvement in predicting pKa values than GGA and
M-GGA functionals when applied to the model layer. Therefore, some light might be still shed
on the impact of exact exchange by investigating if the implementation of other functionals can
be systematically improved as a function of the percentage of exact exchange. PBE0, which has
25% exact exchange included, did improve the accuracy of the local PBE without exact exchange.
Additionally, PBE includes no empirical parameters that may affect the utility of DFT. Therefore,
using the PBE0 functional to examine the impact of exact exchange on the calculation of pKas for
TM hydrides with density functionals can avoid interference from other empirical parameters.
The percentage of exact exchange varied from 0 to 80% in intervals of 5%. The MADs with
respect to central TM atoms and size of ligands of TM hydrides were taken into account with the
ONIOM(PBE0/aug-cc-pVTZ:B97-D/LANL2DZ) scheme and SMD. B97-D was selected due to
its most comparable results to the experimental data and the SMD solvation model was used since
it resulted in lower MADs than either C-PCM or COSMO.

82

Figure 3.6: MADs of PBE0 vs. percentage of exact exchange where (a) the average MAD
for each metal center and (b) the average MAD for each ligand. All of the results are from
ONIOM(PBE0/aug-cc-pVTZ:B97-D/LANL2DZ) scheme with SMD.

As shown in Figure 3.6, 50% exact exchange was preferred when the ligand (depe, depp, and
PNP) is constant and the central atoms changes. For Ni species, the minima all laid at 50%. The
MAD curves of Pd and Pt species were significantly flatter than those for the Ni species. For the
Pd species, all values between 40 and 80% yielded roughly comparable results with the greatest
deviation being 0.6 pKa units. The Pt species had the minima at 65%. For the overall MADs of
the considered species, the minimum can be found at 40% exact exchange. Therefore, the amount
of exact exchange needed is dependent on the choice of TM and independent of the ligands.

83

3.3.4

Impact of Adding Grimme’s Empirical Dispersion Correction on the Accuracy of DFT

Figure 3.7: MADs in pKa values of DFT and DFT-D3 with SMD relative to experiment, with
respect to central TM atoms and ligand size of TM hydrides. The results are from calculations
involving the ONIOM(DFT(-D3)/aug-cc-pVTZ:B97-D3/LANL2DZ) scheme.

The results of the impact of DFT for the model and real layers indicated that the dispersion-
corrected functional, B97-D, was amongst the best functionals in describing both the real and model
layers in the QM/QM scheme for TM hydrides with SMD due to having the lowest MADs with
respect to experimental pKa values. Therefore, it is of interest to evaluate the influence of adding the
Grimme’s empirical (D3) dispersion correction on both LL and HL methods in ONIOM(DFT/aug-
cc-pVTZ:B97-D/LANL2DZ) schemes. The B97-D/LANL2DZ method and basis set combination
for the real layer was applied in this section due to its superior performance relative to other
methods with SMD. The non-dispersion corrected density functionals, BLYP and PBE from the
GGAs, M06L and TPSS from the M-GGAs, PBE0 and B3LYP from the H-GGAs, and M05-2X and
M06-2X from the HM-GGAs, were selected from four types of density functionals to describe the

84

model layer of the TM hydrides. To determine the impact of adding the dispersion correction, the
overall performance of the functionals with and without the dispersion correction was considered
with respect to central TM atoms as well as ligand sizes in the TM hydrides (Figure 3.7). All
results are averaged by functional tier in Table 3.4. The values in Table 3.4 are averaged in Figure
3.7 to clearly define the trend when using dispersion-corrected functionals. Although DFT-D3
methods resulted in lower MADs for all considered species, the improvement by adding dispersion
correction varies as shown in Figure 3.8 The reductions in the MADs were more significant for the
lighter central TM atoms and for TM hydrides with larger sized ligands than for heavier central TM
atoms and for TM hydrides with smaller sized ligands. The comparison of different types of density
functionals with and without the dispersion correction is shown in Table ??. For the functionals
with non-local exchange functionals, the addition of Grimme’s dispersion correction reduced the
MADs more significantly than for functionals with local exchange functionals.

Table 3.4: MADs in pKa values of GGA, M-GGA, H-GGA, and HM-GGA Types of Functionals
for Comparison of DFT and DFT-D3 Relative to Experiment with SMD.

DFT

DFT-D3

GGA
4.1
2.7

M-GGA

H-GGA

HM-GGA

1.9
1.3

2.4
2.0

2.8
2.9

85

Figure 3.8: MADs of DFT vs. DFT-D3 with SMD for the functionals in the model layer,
i.e. ONIOM(DFT(-D3)/aug-cc-pVTZ:B97-D3/LANL2DZ). The MADs are averages of the full
molecule set.

3.3.5

Impact on the Choice of Basis Set

It is well-known that the chosen basis set will also affect the accuracy of calculated properties in
addition to the selected density functional. Therefore, the influence of the basis set on the accuracy
of calculated pKas was assessed with two double-ζ and two triple-ζ quality correlation consistent
basis sets with select density functionals as the HL method, and LANL2DZ and SDD with select
density functionals as the LL method. The aug-cc-pVTZ basis set was utilized for the HL methods
when comparing the MADs of LANL2DZ and SDD basis sets for LL methods and LANL2DZ was
applied for low-level methods for the comparison of the considered correlation consistent basis sets
for high-level methods.

86

The selected density functionals for the high-level method include B97-D, TPSS, B3LYP, and
M05-2X since these functionals yield similar MADs of about 2.3 pKa units and perform better than
the other considered density functionals with respect to experimental pKa values in investigating
the impact of functional choice on the model layer. Only B97-D is applied for low-level methods
since B97-D yields a lower MAD of 2.3 pKa units with SMD than other considered functionals in
investigating the impact of functional choice on the real layer. All calculations in this section used
SMD based on the solvation model’s performance in previous sections.

Table 3.5 shows the dependence of the four selected functionals upon the quality of correlation
consistent basis set for the high-level methods. B97-D and B3LYP only resulted in a small reduction
in MAD of 0.2 pKa units when the basis set quality was increased from aug-cc-pVDZ to aug-cc-
pVTZ while showed a reduction of MADs (more than 0.5 pKa units) upon improving the basis set
from cc-pVDZ to cc-pVTZ.

Table 3.5: MADs in pKa values relative to experiment for four functionals when changing the basis
set used for the model layer.

aug-cc-pVDZa

aug-cc-pVTZa

cc-pVDZa

cc-pVTZa

B97-D
TPSS
B3LYP
M05-2X

3.0
1.5
1.1
1.8

2.8
1.5
0.9
1.9

4.0
1.4
1.6
1.9

2.7
3.3
1.1
1.9

a(aug-)cc-pVnZ-DK was considered for Ni species and (aug-)cc-pVnZ-PP was considered for Pd and Pt species.

As shown in Figure 3.9, the accuracy of the basis set displayed a dependence on the central
TM atoms of the TM hydrides, where cc-pVDZ and cc-pVTZ yielded similar pKa values for Ni
and Pt species while cc-pVDZ performed better than cc-pVTZ for Pd. Similarly, aug-cc-pVDZ
outperformed aug-cc-pVTZ for Pd species but yielded higher MADs than aug-cc-pVTZ for Pt
species. In contrast, the accuracy of the basis sets was not affected by the ligand sizes of the TM
hydrides, as both double-ζ and triple-ζ basis sets, with or without the diffuse functions, consistently
resulted in similar MADs. Both considered double- and triple-ζ correlation consistent basis sets
provided a more accurate description of the model layer of the TM hydrides by including diffuse

87

functions, except for the Ni species.

Figure 3.9: Mean absolute deviation (MAD) in pKa values when utilizing different basis sets
relative to experiment, with respect to central TM atoms and ligand size of TM hydrides where (a)
the cc-pVnZ and aug-cc-pVnZ (n=D,T) are considered for the model layer (HL method) and (b)
LANL2DZ and SDD ECPs are considered for the real layer (LL method).

For the low-level methods, SDD performed better than LANL2DZ with respect to central TM
atoms and ligand sizes of the TM hydrides, except for Ni species (Figure 3.9). The MADs of both
LANL2DZ and SDD decreased as the central TM atoms of TM hydrides becomes heavier.

3.3.6

Impact of Cavity Models on Implicit Solvation Models

The calculated pKa values were also compared to the experimental data from the viewpoint of the
cavities used in computing the C-PCM, COSMO, and SMD reaction fields with the ONIOM(PBE,
M06L, B3LYP, and M06/aug-cc-pVTZ:B97-D/LANL2DZ) scheme used in previous sections to
eliminate functional bias in choice of the HL method. Five cavity models, Pauling, Bondi, UA0,
UAKS, and the default cavity for each solvation model within the GAUSSIAN09 package (UFF for
C-PCM, Klamt for COSMO, and SMD-Coulomb for SMD) were applied to determine the effect

88

of the atomic radii used to build a cavity in the solvent (acetonitrile) on the predicted pKa values
of the TM hydrides. For C-PCM, the Pauling cavity generated the lowest average MAD of 3.4
pKa units while UA0 resulted in the largest MAD of 5.3 pKa units for the full molecule set. For
both COSMO and SMD, the average MAD of the full molecule set yielded the lowest MADs of
3.0 and 2.3 pKa units, respectively, with the GAUSSIAN09 default cavity as shown in both Figure
3.10 and Table 3.6, and the highest average MADs with the UA0 cavity with 5.3 and 5.1 pKa units,
respectively.

89

Figure 3.10: Impact of radii models on (a) C-PCM, (b) COSMO, and (c) SMD. The default cavities
for C-PCM, COSMO, and SMD are UFF, Klamt, and SMD-Coulomb, respectively. The average
MADs are results from calculation with the ONIOM (PBE, M06L, B3LYP, M06/ aug-cc-pVTZ :
B97D/ LANL2DZ) scheme and then categorized by metal and ligand.

90

Table 3.6: MADs of five cavity models in pKa values relative to experiment using the ONIOM(PBE,
M06-L, B3LYP, and M06/aug-cc-pVTZ:B97-D/LANL2DZ) scheme.

Ni
Pd
Pt
depe
depp
PNP
Overall

Ni
Pd
Pt
depe
depp
PNP
Overall

Ni
Pd
Pt
depe
depp
PNP
Overall

Pauling

3.5
3.9
2.8
3.8
3.0
3.4
3.4

Pauling

3.5
3.9
2.0
3.0
3.0
3.4
3.1

Pauling

3.1
3.4
1.6
2.7
2.5
2.9
2.7

Bondi
3.8
4.0
2.9
4.0
3.2
3.6
3.6

Bondi
3.8
4.0
2.1
3.2
3.2
3.6
3.3

Bondi
3.5
3.5
1.8
2.9
2.8
3.1
2.9

C-PCM
UA0
5.8
6.4
3.7
5.8
5.5
4.8
5.3

COSMO

UA0
5.8
6.5
3.7
5.8
5.5
4.8
5.3
SMD
UA0
5.8
6.1
3.6
5.6
5.3
4.5
5.1

UAKS

3.6
4.0
3.0
4.5
3.0
3.2
3.6

UAKS

3.6
4.0
2.2
3.7
3.0
3.2
3.3

UAKS

3.4
3.4
1.8
3.4
2.6
2.7
2.9

Default

5.8
4.0
2.0
4.3
3.5
4.0
3.9

Default

3.1
3.9
1.9
3.0
2.9
3.0
3.0

Default

2.6
2.8
1.4
2.4
2.1
2.2
2.3

3.3.7

Impact of the Expansion of the Size of Model System

To examine the influence of the size of the model system on the utility of density functionals to
predict pKa values of TM hydrides, four functionals, B97-D, TPSS, B3LYP, and M05-2X were used
due to their better agreement with experimental data when used as the HL methods in previous
sections. The ONIOM-1, ONIOM-2, and ONIOM-3 models are depicted in Figure 3.1. The
ONIOM-1 model used the metal and atoms bound directly to the metal as the high level. The
ONIOM-2 model increases the size of the model layer from ONIOM-1 by including the chelating

91

ring connecting the phosphorous atoms. The ONIOM-3 model increases the size of the model
layer in ONIOM-2 by including a methyl group attached to the phosphorous atoms. As shown in
Table 3.7, among the four functionals, only B97-D showed improvement when the size of model
system was expanded from ONIOM-1 to ONIOM-3, while the MADs of the other three functionals
increased. The largest deviation of the MADs between the four functionals are 0.5, 1.8, and 2.7
pKa units for ONIOM-1, ONIOM-2, and ONIOM-3, respectively. The accuracy of the calculated
pKa values of the TM hydrides showed a larger dependence on the selection of density functionals
when a larger sized model system was utilized.

Table 3.7: MADs in pKa values relative to experiment of three expansions of model system of TM
hydrides with SMD.

ONIOM Scheme

ONIOM-1

ONIOM-2

ONIOM-3

B97-D/aug-cc-pVTZ:B97-D/LANL2DZ
TPSS/aug-cc-pVTZ:B97-D/LANL2DZ
B3LYP/aug-cc-pVTZ:B97-D/LANL2DZ
M05-2X/aug-cc-pVTZ:B97-D/LANL2DZ

1.4
1.2
0.9
1.0

2.8
0.9
1.2
1.2

3.2
0.5
1.5
1.3

3.3.8 Comparison of Different Methodologies

Combining the results from all previous sections, the proposed methodology for these systems is
B3LYP-D3/aug-cc-pVTZ:B97-D3/SDD (Scheme A). Shown in Figure 3.11, this proposed scheme
is compared to four other methodological choices: B97-D3/SDD, B3LYP-D3/SDD, B3LYP/aug-
cc-pVTZ:HF/LANL2DZ, and CCSD(T)/aug-cc-pVTZ:B97-D3/SDD, which are Schemes B, C, D,
and E, respectively. Schemes B and C outline the use of a single density functional and Schemes D
and E outline the use of ab initio methods implemented for both the LL and HL method, respectively.
Table 3.8 shows the MADs for each scheme and the average MAD for each scheme presented in
Figure 3.11. The performance of each methodology is compared via the average MAD for the
molecule set. Scheme A had the lowest MAD of 0.6 pKa units while Scheme B had the highest
MAD of 5.5 pKa units.

92

Figure 3.11: Comparison of the experimental and calculated pKa values via methodological choices
represented by their calculated values and the dotted trend lines. The dashed black line denotes
the 1:1 correspondence between experiment and calculated pKa values. Schemes A-E are ONIOM
(B3LYP-D3/ aug-cc-pVTZ : B97-D3/ SDD), B97-D3/ SDD, B3LYP-D3/ SDD, ONIOM (B3LYP/
aug-cc-pVTZ : HF/ LANL2DZ), and ONIOM (CCSD(T)/ aug-cc-pVTZ : B97-D3/ SDD).

93

Predicted pKa values

for Schemes A-E, which are ONIOM (B3LYP-
Table 3.8:
D3/aug-cc-pVTZ:B97-D3/ SDD), B97-D3/SDD, B3LYP-D3/SDD, ONIOM(B3LYP/aug-cc-
pVTZ:HF/LANL2DZ), and ONIOM(CCSD(T)/aug-cc-pVTZ :B97-D3/SDD), respectively.

Scheme A Scheme B Scheme C Scheme D Scheme E

[HNi(depe)2]+
[HNi(depp)2]+
[HNi(PNP)2]+
[HPd(depe)2]+
[HPd(depp)2]+
[HPd(PNP)2]+
[HPt(depe)2]+
[HPt(depp)2]+
[HPt(PNP)2]+
MAD
aObtained through Equation 3.7.

23.6
22.4
23.6
23.3
24.4
21.8
30.1
27.9
27.8
0.6

23.3
22.8
24.1
26.2
26.6
24.4
34.1
32.4
32.1
5.5

29.9
29.8
20.9
26.3
27.8
25.5
34.4
32.8
32.6
4.4

20.5
21.1
22.7
21.7
22.7
20.1
28.6
26.8
27.0
1.4

20.6
19.6
20.9
18.9
21.4
19.2
27.8
26.5
27.3
2.3

Exp
23.8
23.3
22.2
23.2
22.9
22.1
29.7
28.3a
27.6

Schemes B and C were formulated to present how the functionals chosen for Scheme A perform
without the use of ONIOM. The average MAD for Scheme B is approximately 1.1 pKa units higher
than for Scheme C (4.4 pKa units). Both Schemes B and C overestimated the pKas, thus showing
that with the SDD basis set, DFT overestimates the pKas of these TM hydrides. The decrease
in MAD while increasing the complexity of the functional from GGA to H-GGA supports the
results from Section 3.3 where exact exchange is necessary for the correct chemical description
of the metal center. For Scheme A, which uses a hybrid functional to describe the model layer
and a local functional to describe the real system, the quality of the basis set used (aug-cc-pVTZ)
at the metal center and the cancellation of inherent DFT errors due to the extrapolative ONIOM
method explains why Scheme A has the closest correspondence to experiment. Schemes D and
E were chosen to examine wavefunction methods for both the LL and HL method. The average
MAD for Scheme D (1.4 pKa units) is approximately 0.9 pKa units lower than for Scheme E (2.3
pKa units). Using wavefunction methods underestimated the pKas for all molecules examined
except for [HNi(PNP)2]+ for Scheme D. In this case, using DFT was advantageous to describe the
metal center and directly bound atoms over CCSD(T).

94

3.4 Conclusions

This study provides insight into density functionals, solvation models, basis sets, cavity models,
and model layer size that are needed to examine the chemical properties of TM hydrides. Of the
three solvation models considered, the SMD solvation model resulted in lower MADs for predicting
pKa values of TM hydrides than the other two models (COSMO and C-PCM) in comparison to the
experimental data. For the high- and low- level methods within the QM/QM ONIOM scheme, B97-
D yielded the lowest MADs with B97-D, TPSS, and M05-2X with SMD resulted in lower MADs.
The improvement gained including the DFT dispersion correction was more significant for TM
hydrides with lighter central TM atoms and bulkier ligands. Therefore, dispersion is recommended
for these systems. Generally, the triple-ζ basis sets provided lower MADs than the double-ζ basis
sets for the high-level method, while SDD yielded more comparable pKa values to the experimental
data than LANL2DZ for the low-level method. Among the considered cavity models for SMD
(Pauling, Bondi, UA0, UAKS, and SMD-Coulomb), the default cavity (SMD-Coulomb), yielded
the lowest MADs.

For the selection of ONIOM layers, increasing the number of atoms increases the MAD for
all functionals utilized except for B97-D. Thus, the ONIOM-1 scheme (consisting of the metal
atom and immediately bound atoms) is recommended. Using ab initio methods underestimated
the pKa while the use of a single functional largely overestimated the pKa. Therefore, the ONIOM
scheme (B3LYP-D3/aug-cc-pVTZ:B97-D3/SDD) with SMD can be considered as a computational
method to obtain a reliable description of Group 10 TM hydrides, which can serve as a guide for
the calibration of bulkier TM hydrides.

95

APPENDIX

96

Table 3.9: Summary of the basis sets utilized.

Real system

Model system

Ni Species

SDD

LANL2DZ

Pd and Pt Species

SDD

LANL2DZ

cc-pVDZ-DK

aug-cc-pVDZ-DK

cc-pVTZ-DK

aug-cc-pVTZ-DK

cc-pVDZ-PP

aug-cc-pVDZ-PP

cc-pVTZ-PP

aug-cc-pVTZ-PP

Table 3.10: MADs in pKa values of GGA, M-GGA, H-GGA, HM-GGA, and DH-GGA functionals
within low-level methods with solvation models relative to experiment, with respect to central TM
atoms of the TM Hydrides. All of the results are from calculations with ONIOM(B97-D, M06-L,
B3LYP, and M06/ aug-cc-pVTZ:DFT/LANL2DZ) scheme.

C-PCM

COSMO

SMD

Overall-MADa

Central
TM Atom

GGA

M-GGA

H-GGA

HM-GGA DH-GGA

Ni
Pd
Pt
Ni
Pd
Pt
Ni
Pd
Pt
Ni
Pd
Pt

7.2
8.0
6.1
4.5
5.2
3.1
3.8
4.2
2.9
5.2
5.8
4.0

8.0
8.8
6.9
5.5
6.1
3.8
4.8
5.1
3.6
6.1
6.7
4.8

8.2
7.7
5.9
5.5
4.9
2.8
4.9
4
2.7
6.2
5.5
3.8

10.7
9.7
7.6
8.1
6.5
4.5
7.5
5.6
4.5
8.8
7.3
5.5

10.6
10.3
8.5
7.8
7.5
5.2
6.9
6.4
4.9
8.4
8.1
6.2

aAverage results of C-PCM, COSMO, and SMD solvation models

97

Table 3.11: MADs in pKa values of GGA, M-GGA, H-GGA, HM-GGA, and DH-GGA functionals
within low-level methods with solvation models relative to experiment, with respect to ligands of
the TM hydrides. All of the results are from calculations with ONIOM(B97-D,M06-L, B3LYP,
and M06/ aug-cc-pVTZ:DFT/LANL2DZ) scheme.

C-PCM

COSMO

SMD

overall-MADa

Ligand
depe
depp
PNP
depe
depp
PNP
depe
depp
PNP
depe
depp
PNP

GGA
7.2
7.1
7.0
4.1
4.1
4.5
3.8
3.5
3.7
5.0
4.9
5.1

aAverage results of C-PCM, COSMO, and SMD

M-GGA

H-GGA

HM-GGA DH-GGA

7.7
8.1
7.9
4.7
5.2
5.5
4.3
4.6
4.6
5.6
6.0
6.0

7.4
7.3
7.1
4.2
4.3
4.6
3.9
3.8
3.8
5.2
5.1
5.2

9.2
9.3
9.5
6.1
6.3
6.6
5.9
5.9
5.8
7.1
7.2
7.3

10.0
9.9
9.5
6.8
6.8
7.0
6.3
6.0
5.9
7.7
7.6
7.5

Table 3.12: MADs in pKa values of GGA, M-GGA, H-GGA, HM-GGA, and DH-GGA functionals
within high-level methods with solvation models relative to experiment, with respect to central TM
atoms of the TM Hydrides. All of the results are from calculations with ONIOM(DFT/aug-cc-
pVTZ:B97-D,M06L, and B3LYP/LANL2DZ) scheme.

C-PCM

COSMO

SMD

Overall-MADa

Central
TM Atom

GGA

M-GGA

H-GGA

HM-GGA DH-GGA

Ni
Pd
Pt
Ni
Pd
Pt
Ni
Pd
Pt
Ni
Pd
Pt

7.0
3.7
3.0
6.8
3.7
2.6
5.9
2.7
2.2
6.6
3.4
2.6

3.7
3.7
2.9
3.4
3.4
2.2
2.6
2.6
2.1
3.2
3.2
2.4

3.9
3.2
2.6
3.5
2.9
2.1
2.9
2.5
2.2
3.4
2.9
2.3

4.8
4.1
2.8
4.9
4.1
1.7
5.1
3.5
1.9
4.9
3.9
2.1

8.9
6.2
4.2
7.5
5.9
3.3
6.7
4.9
3.2
7.7
5.7
3.6

aAverage results of C-PCM, COSMO, and SMD solvation models

98

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CHAPTER 4

UTILIZATION OF THE DOMAIN-BASED LOCAL PAIR NATURAL ORBITAL

METHODS WITHIN THE CORRELATION CONSISTENT COMPOSITE APPROACH

4.1 Introduction

Over the years, numerous approaches have been developed to try to reduce the computational
cost associated with high-level ab initio methods while maintaining similar accuracy. These
approaches include but are not limited to ab initio composite methods (see Section 2.2.4),1–31 and
local ab initio correlated methods (see Section 2.2.1).32–65

The combination of these approaches should, in principle, expand the range of molecules that can
be targeted with composite methodologies since composite methods are often limited by molecule
size but achieve a high level of accuracy for well-established reliable experiments and local methods
reduce the CPU time while reproducing electronic energies analogous to canonical molecular orbital
methods. Therefore, the premise of this work is to develop a composite methodology that utilizes
local methods to reduce the computational cost while retaining the same level of accuracy as the
canonical composite method.

While the correlation consistent Composite Approach (ccCA) results in a reduction in
computational cost and is comparable in accuracy relative to its target level of theory for main
group species,31 CCSD(T,FC1)/aug-cc-pCV∞Z-DK, additional reduction of computational cost
is desired to facilitate the description of chemical systems of increasing size. For the methodology,
a number of options are available, targeting one or more of the steps of the composite approach.
Approaches include RI-ccCA,25 which utilizes the resolution-of-the-identity (RI) approximation
for the MP2 steps within ccCA, and ccCA-F12,26 which uses explicitly correlated methods for all
steps within ccCA. When using RI-CCSD(T) and CCSD(T)-F12 for RI-ccCA and ccCA-F12,
respectively, neither coupled cluster approach reproduced the same energies as CCSD(T), and
thus led to a decrease in performance of RI-ccCA and ccCA-F12 relative to ccCA. Another

108

[CR-CCSD(T)] within ccCA
example is the use of completely renormalized CCSD(T)
(CR-ccCA), which can be beneficial in situations that might otherwise require multireference
wavefunction treatment, such as MR-ccCA, and which resulted in a reduction in the MAD from
experiment for open-shell species which was less than that of ccCA.23,66

Another route used to reduce the computational cost of ccCA is via Morokuma’s Our own
N-layered Integrated molecular Orbitals and molecular Mechanics (ONIOM) framework,67 such
as in ONIOM-ccCA28 and rp-ccCA-ONIOM,29. While these implementations have been useful,
they are not immune to common multilayer method challenges including judicious choice of model
layer and method combinations (e.g. QM/QM), as well as reliance on error cancellation to obtain
favorable results.68

While there have been approaches to reduce the computational cost of ccCA and other composite
methods, one of the primary factors in the SCF step is the calculation of four-center two-electron
integrals, which formally scales as N 4/8. As mentioned earlier, the RI approximation utilized
within ccCA can mitigate this computational bottleneck successfully for the MP2 step of ccCA by
approximating four-center two-electron integrals as a linear combination of three-center or two-
center two-electron integrals through a projection operator using an auxiliary basis set (ABS). The
use of an ABS reduces the scaling, and thus cost, of the four-center two-electron integrals from
O(K4) to approximately O(K2m) where K is the number of basis functions and m is the number
of auxiliary basis functions, where m < K2 so that an ABS has enough flexibility to adapt to any
Coulomb potential. In practice, different ABS are constructed and used for SCF and correlated
integrals.25,69

Alternatives to using RI methods for mitigating the computational bottleneck of four-center
two-electron integrals include local methods, which have been extensively developed to localize
dynamic correlation.32–65 Although canonical orbitals are characteristically delocalized, a localized
description of the occupied orbitals is important for dynamic correlation since dynamic correlation
for nonmetallic systems is a short-range effect with a dependence on distance of r-6 like dispersion
energy.70

109

The domain-based local pair natural orbital (DLPNO) methods,55–58,61 primarily DLPNO-
CCSD(T), have been shown to result in reduced computational cost relative to the cost of CCSD(T)
for transition metal-based catalysts and larger organic systems such as complex hydrocarbons
and fullerenes, and with comparable accuracy.71–75 (The original publications and Section 2.2.3
provide more details about DLPNO methods and their development.52–54,56–58). To reduce the
costs of DLPNO methods, the DLPNO methods have been paired with the Foster-Boys (FB)76,77
and the Pipek-Mezey (PM)78 techniques to localize occupied MOs.78,79 Both localization schemes
have been described in Section 2.2.1. Numerous studies have utilized each of these localization
approaches to successfully reduce computational cost. Pipek-Mezey localization has been used to
improve the accuracy and efficiency of quantum embedding:80 Foster-Boys localization has been
paired with an explicitly correlated HF approach for the quantum treatment of protons.81 Both
localization approaches have been used in the development of a linear scaling implementation of
the direct random-phase approximation.82

To illustrate, an advantage of methods such as DLPNO is that large energetic differences that
can arise from the localization tails generated from orthogonal localized MOS are effectively
truncated through integral transformation, enabling the screening of less important contributions
to energies.83–85

Localization methods have also been utilized within composite approaches. A previous study
by Montgomery et al. utilized the use of the Pipek-Mezey population localization technique based
within the CBS-QB3 composite method,8,78 yielding a mean absolute deviation (MAD) of 1.10
kcal mol−1 for the G2/97 molecule set for heats of formation, which was comparable to G3 and
G3(MP2) with MADs of 0.94 kcal mol−1 and 1.24 kcal mol−1, respectively; thus, demonstrating
the utility of localized MOs within composite methods.

As both approaches are widely used, each of the localization schemes will be considered in the
incorporation of the DLPNO methods within the ccCA framework. As there have been successes
utilizing the DLPNO methods to reproduce RI-MP2 and CCSD(T) correlation energies as well as
implementing localization schemes in a composite framework, the goal of this work is to incorporate

110

the DLPNO methods within the ccCA framework to reduce computational cost with little or no
impact upon the accuracy. The accuracy and relative CPU timing for the calculation of enthalpies
of formation using DLPNO-ccCA is compared against ccCA and RI-ccCA to elucidate the efficacy
of the DLPNO methods.

4.2 Computational Methods

A set of 119 closed shell systems from the G2/97 molecule set, including both first and second
row atoms (listed in the Appendix), was used to investigate the enthalpies of formation (∆Hf).86
All calculations were done with the ORCA 4.0 program.87,88 Geometry optimizations were done
at the B3LYP89 level with the cc-pVTZ90 basis set. All calculations that include Al-Cl (3p) were
done using the recommended version of the correlation consistent basis set, the cc-pV(T+d)Z basis
set.91 Energies were converged to 10-6 Eh and gradients were converged to 10-4 Eh/bohr for the
geometry optimization. The vibrational ZPE and vibrational contribution to internal energy were
scaled by 0.989 at 298.15 K and 1 atm to account for anharmonicity.24 Thermal corrections to
enthalpy were calculated at 298.15 K and 1 atm. Experimental spin-orbit corrections for atoms
were applied from tables provided by Moore.92 The formulation of ccCA is described in section
2.2.4.1 and variants used in this chapter are shown in Table 4.1.

To supplement

the experimental ∆Hf from the NIST-JANAF thermochemical

tables,93
experimental ∆Hf for LiH, Li2, LiF, Na2, and NaCl were obtained based on the work of
Cioslowski et al.94 Also, as detailed theoretical studies95–98 on COF2, F2CCF2, and CH2CHCl
suggest that the experimental ∆Hf were likely in error,95 the values for ∆Hf used in this work for
COF2, F2CCF2, and CH2CHCl are -145.6 ± 1.0, -160.8 ± 0.8, and 5.0 ± 1.0 kcal mol−1,
respectively, which were obtained via ab initio calculations.96–99 The values for the atomic
enthalpies of formation at 0 K for C and H used in this work were based on the work by Tasi et
al.100 The values for atomic enthalpies of formation for B, Si, and Al atoms were adopted from
Karton et al.101 A UHF reference was used for O3. ∆Hf were calculated using the total
atomization approach, which uses open-shell variants for the atoms.

111

SCF energies were converged to 10-8 Eh in all single point energy calculations. The thresholds
for DLPNO-MP2 were set to TCutDO = 5.0 * 10-3, TCutPNO = 10-9, and TCutMKN = 10-3. For
DLPNO-CCSD(T) calculations, TCutPairs = 10-5 Eh, TCutPNO = 10-7, and TCutMKN = 10-4.102
These thresholds were established as the TightPNO setting in ORCA.88,102 For DLPNO-CCSD(T),
the Foster-Boys (FB)76,77 and Pipek-Mezey (PM)78 localization schemes were used within ORCA
to localize the occupied orbitals after the SCF energy was calculated.

Table 4.2 shows a summary of the approximations and the auxiliary basis sets (ABS) used in
this work. The AutoAux103 feature within ORCA was implemented to generate Li and Na ABS
for correlated methods. The basis sets utilized in this work were the cc-pVnZ and aug-cc-pVnZ
basis sets and the cc-pV(n + d)Z and aug-cc-pV(n + d)Z basis sets for Na-Cl, where n = D,
T, Q.90,104–107 The ABS for coulomb-fitting (RI-J),108,109 coulomb-exchange fitting (RI-JK),109
and correlated methods (RI-C)110,111 are denoted as basis/J, basis/JK, and basis/C, respectively,
in this work. The implementation of the ABS was done in three schemes. Scheme 1 only
utilizes the correlation consistent ABS for correlated methods. Scheme 2 utilizes the correlation
consistent ABS for correlated methods and the def2/JK ABS for the SCF energy using the RI-JK
approximation. Scheme 3 uses the correlation consistent ABS for correlated methods and either
the RIJCOSX112 or RI-JK113 approximation for the SCF energy with the appropriate def2 ABS.
The RIJCOSX approximation is used for RI-MP2 and DLPNO-CCSD(T) (uses the def2/J ABS),
and the RI-JK approximation is used for DLPNO-MP2 (uses the def2/JK ABS).57,112 The def2
ABS were chosen based on their availability throughout the periodic table.

112

Table 4.1: Summary of the different variants of ccCA utilized in Chapter 4.

ccCA

B3LYP/cc-pVTZ
HF/aug-cc-pVTZ
HF/aug-cc-pVQZ
HF/aug-cc-pV∞Z

Equation 2.29

RI-ccCA

B3LYP/cc-pVTZ
HF/aug-cc-pVTZ
HF/aug-cc-pVQZ
HF/aug-cc-pV∞Z

Equation 2.29

DLPNO-ccCA
B3LYP/cc-pVTZ
HF/aug-cc-pVTZ
HF/aug-cc-pVQZ
HF/aug-cc-pV∞Z

Equation 2.29

MP2/aug-cc-pVDZ
MP2/aug-cc-pVTZ
MP2/aug-cc-pVQZ
MP2/aug-cc-pV∞Z
Equations 2.30 - 2.32

RI-MP2/aug-cc-pVDZ
RI-MP2/aug-cc-pVTZ
RI-MP2/aug-cc-pVQZ
RI-MP2/aug-cc-pV∞Z
Equations 2.30 - 2.32

DLPNO-MP2/aug-cc-pVDZ
DLPNO-MP2/aug-cc-pVTZ
DLPNO-MP2/aug-cc-pVQZ
DLPNO-MP2/aug-cc-pV∞Z

Equations 2.30 - 2.32

MP2(FC1)/aug-cc-pCVTZ

RI-MP2(FC1)/aug-cc-pCVTZ DLPNO-MP2(FC1)/aug-cc-pCVTZ

Geometry Optimization

Eref

MP2/CBS

∆CC

∆CV

∆DK
∆SO
ZPE

CCSD(T)/cc-pVTZ

-MP2/cc-pVTZ

– MP2/aug-cc-pVTZ
MP2/cc-pVTZ-DK
– MP2/cc-pVTZ

CCSD(T)/cc-pVTZ
- RI-MP2/cc-pVTZ

DLPNO-CCSD(T)a/cc-pVTZ

- DLPNO-MP2/cc-pVTZ

– RI-MP2/aug-cc-pVTZ
RI-MP2/cc-pVTZ-DK
– RI-MP2/cc-pVTZ

– DLPNO-MP2/aug-cc-pVTZ
DLPNO-MP2/cc-pVTZ-DK
– DLPNO-MP2/cc-pVTZ
Experimental atomic values

Experimental atomic values

Experimental atomic values

Vibrational ZPE scaled by 0.989 Vibrational ZPE scaled by 0.989 Vibrational ZPE scaled by 0.989

aThe Pipek-Mezey (PM) and Foster-Boys (FB) localization schemes were considered for orbital localization for DLPNO-CCSD(T) whereas for DLPNO-MP2, only the FB
localization was used.

113

Table 4.2: Summary of the approximations, methods, and auxiliary basis sets (ABS) utilized in
this work for SCF and post-HF calculations.

SCF
RI Approximations

ABS

Methods using RI

ABS

Post-HF

Scheme 1

–

–

Scheme 2

RI-JK

def2/JK

Scheme 3

RIJCOSXb

RI-JKc

def2/J
def2/JK

RI-MP2

DLPNO-MP2

DLPNO-CCSD(T)

RI-MP2

DLPNO-MP2

DLPNO-CCSD(T)

RI-MP2

DLPNO-MP2

DLPNO-CCSD(T)

aug-cc-pVnZ/Ca

cc-pVTZ/C

aug-cc-pwCVTZ/C

aug-cc-pVnZ/Ca

cc-pVTZ/C

aug-cc-pwCVTZ/C

aug-cc-pVnZ/Ca

cc-pVTZ/C

aug-cc-pwCVTZ/C

an = D, T, Q.
bRIJCOSX is used for calculations involving HF + RI-MP2 and HF + DLPNO-CCSD(T) with the def2/J ABS.
cRI-JK is used for calculations involving HF + DLPNO-MP2 with the def2/JK ABS.

CPU timing studies were done in serial (single core) on a localDell OptiPlex 390 computer
with 16 GB of DDR3 memoryto consider the efficiency of DLPNO-ccCA relative to ccCA and
RI-ccCA using ORCA.87,88 All energies were calculated without the use of symmetry. The usage
of these methods for the molecule set was considered for the first timing study. The use of
ABS for the SCF step (Schemes 2 and 3) were timed for DLPNO-ccCA and RI-ccCA. Linear
alkanes (CnH2n+2 for n=1-8) were considered in a second CPU timing study to assess the effect
of systematically increasing the molecule size on the CPU time and energetics within the ccCA
framework. ∆Hf for linear alkanes are computed with the atomization approach and with isodesmic
schemes for CnH2n+2 for n = 3-8 since using isodesmic schemes have been shown to reduce the
error for computed ∆Hf for linear alkanes.114 The isodesmic schemes are shown in Table 4.6.
For RI-ccCA, only the Scheme 1 implementation of ABS was investigated. For DLPNO-ccCA,
Schemes 1-3 were considered. The Foster-Boys (FB) localization scheme, the default scheme in
ORCA, was used for all DLPNO-ccCA timing calculations.

114

4.3 Results and Discussion

4.3.1 Energetic Properties for the Molecule Set

The ∆Hf’s for each molecule of the set were calculated using ccCA, RI-ccCA, and
DLPNO-ccCA. Four schemes for the extrapolation of the reference energy within ccCA, Peterson
(P), Schwartz-3 (S3), Schwartz-4 (S4), and Peterson-Schwartz-3 (PS3) extrapolation schemes
were considered, and denoted as ccCA-P, ccCA-S3(TQ), ccCA-S4(TQ), and ccCA-PS3(TQ),
respectively.115–122 These approaches are compared in Table 4.3. The ccCA-PS3(TQ) shows the
lowest mean absolute deviation (MAD) for ∆Hf’s of0.94kcal mol−1 and the lowest magnitude for
mean signed deviation (MSD) ∆Hf’s of-0.20kcal mol−1 compared to all other extrapolation
schemes considered. Based on the MAD for ccCA-PS3(TQ), this is the extrapolation scheme
utilized for RI-ccCA and DLPNO-ccCA in this work. The PS3(TQ) moniker is removed from the
name for conciseness.

Table 4.3: Slope, intercept, and R2 of the calculated and experimental ∆Hf. The mean signed
deviation (MSD), mean absolute deviation (MAD), standard deviation (STDEV), and maximum
(MAX) deviation for four variants of ccCA based on the Peterson (P), Schwartz-3 (S3), and
Schwartz-4 (S4) extrapolation schemes. The P and S3 extrapolated values are averaged for PS3.
Triple and quadruple-ζ level basis sets (TQ) were used for all two-point extrapolations. All
deviations are in kcal mol−1.

Slope
Intercept

R2

MSD
MAD
STDEV
MAX

ccCA-P
1.0006
0.7283
0.9999

-0.71
1.10
1.27
4.10

ccCA-S3(TQ)

0.9999
-0.3192
0.9999

ccCA-S4(TQ)

1.0000
0.7210
0.9999

ccCA-PS3(TQ)

1.0003
0.2046
0.9999

0.32
1.00
1.14
2.66

-0.72
1.11
1.28
4.14

-0.20
0.94
1.18
3.38

The effect of using the Pipek-Mezey and Foster-Boys localization techniques are demonstrated
and
in box plots
DLPNO-CCSD(T)/cc-pVTZ electronic energies (Figures 4.2 and 4.3). When using the def2/JK

for DLPNO-MP2/aug-cc-pV∞Z electronic

energies

(Figure 4.1),

115

ABS for the SCF energies in combination with MOs generated with Foster-Boys localization,
lower electronic energies were generated than a pairing with MOs generated with Pipek-Mezey
localization for DLPNO-MP2/aug-cc-pVnZ (n = D, T, Q). At the complete basis set (CBS) limit,
MOs generated with the Pipek-Mezey localization method yielded nearly identical electronic
energies to MOs generated with Foster-Boys localization (approximately a 0.006 mEh difference)
for all of the molecule subsets shown in Figure 4.1. However, for DLPNO-CCSD(T), the effect of
implementing thresholds such as electron pair screening, domain selection, and PNO generation,
on using different
changed the final
DLPNO-CCSD(T) electronic energies within ±2 mEh as shown in Figure 4.2, which can affect
the total DLPNO-ccCA energies by ±1.4 kcal mol−1.

localization schemes

for

the occupied MOs,

116

Figure 4.1: Differences in electronic energies (mEh) using Pipek-Mezey (PM) and Foster-Boys
(FB) localization schemes using the def2/JK ABS within DLPNO-MP2 for complete basis set
extrapolation using a combined Peterson-Schwartz-3 extrapolation scheme (PS3(TQ)). Included
subsets are based on the presence of certain elements (hydrocarbons, halogenated, chalcogenated,
pnictogenated, and Period 3) and electronic features (aromatic, carbonyl, multiple bonds) as well as
the full molecule set. Points within the dashed lines represent differences less than 0.1 mEh. The
box plots depict the distribution of data within each subset where the band in the middle represents
the median of the data and data points shown as black circles are more than 3 standard deviations
from the median.

117

Figure 4.2: Differences in electronic energies (mEh) between the Pipek-Mezey (PM) and Foster-
Boys (FB) localization methods for all three schemes within DLPNO-CCSD(T) for the same subsets
in Figure 4.1. The dashed lines represent differences of less than 0.1 mEh. The box plots depict the
distribution of data within each subset where the band in the middle represents the median of the
data and data points shown as black circles are more than 3 standard deviations from the median.

118

Figure 4.3: Differences between the CCSD(T) electronic energies and the DLPNO-CCSD(T)
electronic energies in mEh using the (a) Pipek-Mezey (PM) and (b) Foster-Boys (FB) localization
methods for all three schemes for the subsets of the full molecule set shown in Figure 4.1. The
dashed lines represent differences less than 0.1 mEh. The box plots depict the distribution of data
within each subset where the band in the middle represents the median of the data and data points
shown as black circles are more than 3 standard deviations from the median.

119

Figure 4.3 depicts the difference between canonical CCSD(T) electronic energies and DLPNO-
CCSD(T) electronic energies generated using localized MOs from the Pipek-Mezey (Figure 4.3a)
and Foster-Boys (Figure 4.3b) schemes. As shown in Figure 4.3a, the effect of the DLPNO-
CCSD(T) truncation and screening parameters on localized MOs from Pipek-Mezey localization
was that DLPNO-CCSD(T) electronic energies were lower than canonical CCSD(T) electronic
energies. In Figure 4.3b, using the DLPNO-CCSD(T) truncation and screening parameters on the
Foster-Boys localized MOs resulted in higher electronic energies for DLPNO-CCSD(T) relative
to CCSD(T) electronic energies, more noticeably for Schemes 2 and 3, since there are more
positive outliers in the box plots for the subsets of the full molecule set separated by atom type
and functional group. Therefore, depending on which scheme is implemented, the choice of initial
localization technique, Foster-Boys or Pipek-Mezey, has a significant effect on the final DLPNO-
CCSD(T) electronic energies based on the effect of implementing different thresholds defined
within DLPNO-CCSD(T) on localized MOs.

Out of the 119 molecules in this molecule set, only 26 exhibited a negligible difference in
the MADs (< 0.01 kcal mol−1) between the two localization schemes using Scheme 1 and ABS
(for correlated methods only). These molecules include those with an even charge distribution
such as alkanes since the difference in electronic energy between the two localization schemes was
within 1 mEh for the hydrocarbons data subset, as shown in the boxplot for Scheme 1 in Figure
4.2. For Scheme 2 (ABS for correlated methods and RI-JK approximation for SCF), 23 molecules
resulted in a negligible difference in the MADs between both localization schemes. Notable cases
where deviations decreased more than 0.5 kcal mol−1 when using the Pipek-Mezey localization
scheme relative to the Foster-Boys localization scheme include cyclic aromatic systems, halogenated
systems, and molecules characterized with triple bonds. This aligns with the known issues with
Boys localization for ring systems,123 which includes the formation of degenerate bonding MOs
instead of a σ-π separation for systems with multiple bonds and aromatic ring systems. The largest
differences in MAD was 3.31 kcal mol−1 when using the Foster-Boys localization scheme and
1.91 kcal mol−1 when using the Pipek-Mezey localization scheme for O3. The use of the RI-JK

120

approximation for SCF (Scheme 2) led to lower electronic energies when using the Pipek-Mezey
localization scheme relative to the Foster-Boys localization scheme as shown in Figure 4.2.

When using the RIJCOSX approximation and def2/J ABS (Scheme 3), only 46 molecules
yielded a lower MAD for ∆Hf when using the Pipek-Mezey localization scheme and only 22
showed a negligible difference in MAD (< 0.01 kcal mol−1). The largest difference in MAD
was 3.16 kcal mol−1 when using Foster-Boys localization and 1.76 kcal mol−1 when using Pipek-
Mezey localization for O3 for Scheme 3. With the implementation of ABS, the number of molecules
that favored the use of the Pipek-Mezey localization scheme for DLPNO-CCSD(T) decreased from
65 molecules with Scheme 1 to 50 molecules for Scheme 2 and 46 molecules for Scheme 3.
This suggests that Pipek-Mezey localization may not be as useful when used with the RI-JK and
RIJCOSX approximations for the SCF orbitals even though Pipek-Mezey localization has been
proven useful for; (a) cyclic aromatic rings; (b) halogenated compounds; and, (c) molecules such
as P2, which are characterized by triple bonds.

Table 4.4 shows the MSD, MAD, the standard deviation of MADs, and the maximum
deviation from experimental ∆Hf for all variations paired with auxiliary basis functions.Overall,
DLPNO-ccCA shows good agreement with ccCA for experimental ∆Hf since for Scheme 1 (ABS
for correlated methods), DLPNO-ccCA (PM), DLPNO-ccCA (FB), and RI-ccCA lowered the
average MAD relative to ccCA by 0.01, 0.01, and 0.04 kcal mol−1, respectively. The maximum
deviation from experiment for DLPNO-ccCA(PM), DLPNO-ccCA(FB), and RI-ccCA was2.73
kcal mol−1 for NCCN, 3.77 kcal mol−1 for O3, and 4.25 kcal mol−1 for O3 respectively. Using
ABS for SCF energies (Scheme 2) lowered the MAD for DLPNO-ccCA (PM), DLPNO-ccCA
(FB), and RI-ccCA by 0.08, 0.00, and 0.24 kcal mol−1, respectively. The maximum deviation
from experiment for DLPNO-ccCA (PM), DLPNO-ccCA (FB), and RI-ccCA was2.37, 3.77, and
2.53kcal mol−1, respectively, forSi2H6, O3, and H2CCHCN. The use of RIJCOSX for SCF
calculations for DLPNO-CCSD(T) and RI-MP2 calculations (Scheme 3) lowered the average
MAD for DLPNO-ccCA (PM) and DLPNO-ccCA (FB) to 0.92 kcal mol−1 and 0.94 kcal mol−1,
respectively, but increased the average MAD for RI-ccCA to 1.12 kcal mol−1. The largest

121

deviation from experiment for DLPNO-ccCA (PM) and DLPNO-ccCA (FB) was2.69 kcal
mol−1 for Si2H6 and 3.77 kcal mol−1 for O3, respectively. For RI-ccCA, the largest deviation
from experiment was 4.52 kcal mol−1 for SiCl4. Therefore, the recommended implementation of
ABS for DLPNO-ccCA is the use of RIJCOSX for SCF calculations within DLPNO-CCSD(T)
and RI-JK for SCF calculations within DLPNO-MP2 calculations (Scheme 3) in conjunction with
Pipek-Mezey localization.

122

Table 4.4: Mean signed deviation (MSD), mean absolute deviation (MAD), standard deviation (STDEV), and maximum (MAX) deviation
for all schemes. All deviations are in kcal mol−1.

DLPNO-ccCA (PM)

Scheme

Scheme

Scheme

MSD
MAD
STDEV
MAX

1
0.21
0.95
1.13
2.73

2
0.40
0.91
1.03
2.37

3
0.52
0.92
0.98
2.69

DLPNO-ccCA (FB)

Scheme

2
0.09
0.98
1.21
3.77

Scheme

3
0.21
0.94
1.15
3.77

Scheme

1

-0.35
0.92
1.16
4.25

RI-ccCA
Scheme

2
0.09
0.75
0.93
2.53

Scheme

3
0.52
1.09
1.23
4.52

Scheme

1
0.38
0.98
1.14
3.76

123

4.3.2 CPU Timing

To give insight about the performance of the ccCA variants in terms of time, the total CPU time
was measured as the sum of the CPU times of the single point calculations that are included in the
Scheme 1 methodology (Table 4.2). Fifteen molecules from the molecule set were selected based
on their varying size to demonstrate the performance and potential bottlenecks of the ccCA variants.
This subset included CH3Cl, NF3, PF3, H3CH2COCH3, cyclic and linear alkenes, and alkanes
(bicyclo[1.1.0]butane, cyclobutane, cyclobutene, isobutene, trans-butane, isobutane, spiropentane),
and cyclic aromatic molecules (furan, thiophene, benzene, pyridine). The CPU times were taken
as percentages of the total CPU time and show that the bottleneck step is the MP2/aug-cc-pVQZ
step. For DLPNO-ccCA, the DLPNO-MP2/aug-cc-pVQZ uses 72.7% of the total CPU time,
MP2(FC1)/aug-cc-pCVTZ uses 10.1% of the total CPU time, DLPNO-CCSD(T)/cc-pVTZ uses
5.8% of the total CPU time, and the other steps require 11.4% of the total CPU time as shown in
Figure 4.4. This is consistent with RI-ccCA assessed in the same fashion. For Schemes 2 and 3,
the use of ABS for the SCF energy, while decreasing the total CPU time, does not change the ratio
of CPU time savings significantly.

The CPU timings for the full 119 molecule set are shown in Table 4.5, which displays the
mean, largest, and smallest percent CPU time savings for RI-ccCA and DLPNO-ccCA relative to
ccCA, and Figure 4.5, which depicts the CPU time savings for DLPNO-ccCA compared to ccCA
(Figure 4.5a) and RI-ccCA (Figure 4.5b). In Scheme 1 (ABS for correlated methods), the percent
difference in CPU time between DLPNO-ccCA and ccCA averaged 29.1% but 32.8% for RI-ccCA
and ccCA, indicating that RI-ccCA is slightly more efficient, overall, than DLPNO-ccCA when
using Scheme 1, as shown in Table 4.5. The use of ABS for the SCF energy and correlated methods
(Scheme 2) drastically increased the percent CPU time savings to approximately 87.5% and 92.5%
for DLPNO-ccCA and RI-ccCA, respectively, relative to use of ABS for correlated methods only
(Scheme 1). The changes in percent CPU savings is due to applying the RI approximation to both
the SCF and correlation energy calculation energy.

124

Figure 4.4: CPU time of each individual step within (a) ccCA, (b) RI-ccCA, and (c) DLPNO-
ccCA for selected species from the molecule set. The Other category represents the timing of
the MP2/aug-cc-pVDZ, MP2/aug-cc-pVTZ, MP2/cc-pVTZ, and MP2/cc-pVTZ-DK calculations
as these calculations use a small percentage of the total CPU time. All timing calculations were
done with the ORCA software package.

125

Table 4.5: Percent CPU time savings for the three schemes of ABS implementation within DLPNO-
ccCA and RI-ccCA relative to ccCA. The mean percent difference from ccCA, the most efficient
(MAX), and the least efficient (MIN) percent CPU time savings relative to ccCA timings are shown.
All timing studies were done with ORCA.

DLPNO-ccCA (FB) (%)

Scheme

Scheme

MEAN
MAX
MIN

1
29.1
40.2
-4.0

2
87.3
95.3
27.9

Scheme

3
86.5
94.8
21.0

Scheme

1
32.6
38.0
10.9

RI-ccCA (%)

Scheme

2
92.2
96.0
34.7

Scheme

3
83.6
95.9
34.7

RI-ccCA, which shows 32.8% CPU time savings relative to ccCA, is slightly more efficient
than DLPNO-ccCA, which shows 29.1% CPU time savings relative to ccCA, when using Scheme
1. However, with increasing molecule size and depending on the RI approximation that was used,
DLPNO-ccCA is more efficient than RI-ccCA. This is shown especially for Scheme 3, where the
use of RIJCOSX for SCF within RI-MP2 and DLPNO-CCSD(T) increased the CPU time for RI-
ccCA relative to DLPNO-ccCA. Based on Figure 4.5a and Table 4.5, the RIJCOSX approximation
slightly increases the CPU time of DLPNO-ccCA relative to using the RI-JK approximation for
the SCF step within DLPNO-CCSD(T), but decreased the MAD when using DLPNO-ccCA to
calculate the ∆Hf. The increase in CPU time of RIJCOSX relative to RI-JK is due to the size
of the molecules since a threshold exists between the efficiency of RIJCOSX versus RI-JK for
molecular size. Therefore, using RI-JK for SCF within DLPNO-MP2 and RIJCOSX for SCF
within DLPNO-CCSD(T) for DLPNO-ccCA is recommended for smaller systems.

126

Figure 4.5: CPU time ratios of DLPNO-ccCA (FB) to (a) ccCA and (b) RI-ccCA. The ratios for
Scheme 1 (blue circle), Scheme 2 (black x), and Scheme 3 (green triangle) are shown on a log-log
scale. All timing was done with C1 symmetry enforced and done in ORCA.

4.3.3 Enthalpies and Timing for Linear Alkanes

Deviations from experimental ∆Hf are shown in Table 4.6. Only the Foster-Boys localization
scheme was used for linear alkanes since using Pipek-Mezey localization did not significantly change
the final DLPNO-CCSD(T) energy. The trend of increasing deviation with increasing number of
carbon atoms is consistent with previous ccCA studies and common with many other methods
when using the atomization approach.5,19,25 DLPNO-ccCA yields a smaller deviation when ccCA
overestimates the ∆Hf, as shown for CnH2n+2 for n ≥ 4 in Table 4.6, and a larger deviation when
ccCA underestimates the ∆Hf, as shown for CnH2n+2 for n ≤ 3 in Table 4.6. This is due to
the contribution of DLPNO-CCSD(T) as both ccCA and RI-ccCA, which use CCSD(T), follows
the same trend for the rate of increase in deviation from experiment when using the atomization
approach for ∆Hf. Also, the chosen thresholds for the PNOs allow for noncovalent interactions,
which are present in (CnH2n+2 for n ≥ 3, to be better characterized. The percentage of electron
pairs that are screened out of the calculation in the pre-screening process is a potential source of
error in the prediction of ∆Hf for smaller molecules.

127

Table 4.6: Deviations in kcal mol−1 from experimental ∆Hf for linear alkanes (CnH2n+2 1 ≤ n ≤
8) using the atomization approach and using isodesmic approaches (shown in parentheses).

RI-ccCA
Scheme

1

-18.28
-20.76

DLPNO-ccCA (FB)

Scheme

3

-18.62
-20.67

Scheme

1

-18.47
-21.13

Scheme

2

-18.45
-21.08

-25.72
(-24.64)

-29.00
(-30.40)

-33.68
(-34.64)

-38.41
(-39.40)

-43.16
(-44.29)

-47.90
(-49.16)

-26.21
(-24.58)

-29.57
(-30.45)

-34.32
(-34.54)

-39.15
(-39.27)

-43.98
(-44.19)

-48.79
(-49.06)

-26.32
(-24.76)

-29.46
(-30.59)

-34.21
(-34.56)

-39.01
(-39.12)

-43.81
(-44.19)

-48.60
(-49.07)

-26.05
(-25.50)

-29.22
(-30.41)

-34.30
(-35.48)

-38.94
(-39.73)

-43.64
(-44.67)

-48.33
(-49.55)

Exp

ccCA

CH4
C2H6

-17.90±0.10
-20.03±0.07

C3H8a

-25.02±0.12

C4H10b

-30.31±0.14

C5H12c

-35.11±0.19

C6H14d

-39.89±0.19

C7H16e

-44.78±0.18

-18.26
-20.72

-25.66
(-24.64)

-28.93
(-30.40)

-33.59
(-34.64)

-38.30
(-39.40)

-43.05
(-44.30)

-47.82
(-49.22)

-49.90±0.31

C8H18f
aIsodesmic Reaction: 2 C2H6 → C3H8 + CH4
bIsodesmic Reaction: C5H12 + C2H6 → C4H10 + C3H8
cIsodesmic Reaction: C4H10 + C2H6 → C5H12 + CH4
dIsodesmic Reaction: C4H10 + C3H8 → C6H14 + CH4
eIsodesmic Reaction: C6H14 + C2H6 → C7H16 + CH4
fIsodesmic Reaction: C7H16 + C2H6 → C8H18 + CH4

The timing results are shown in Table 4.7. Even when using Scheme 1, the time savings
associated with increasing the number of carbon atoms in the linear chain is evident for DLPNO-
ccCA in comparison to ccCA and RI-ccCA, largely due to DLPNO-CCSD(T). The CPU time
savings for RI-ccCA and DLPNO-ccCA for methane, 36.7% and 30.9%, respectively, and for
ethane, 37.8% and 36.1%, respectively, show that RI-ccCA is more efficient than DLPNO-ccCA
for smaller molecules; however, starting with propane (n = 3), DLPNO-ccCA is more efficient than
RI-ccCA with CPU time savings of 39.6% and 38.2%, respectively, and the percent CPU time

128

saving monotonically increases with increasing carbon atoms for DLPNO-ccCA.

Table 4.7: Percent CPU time savings for RI-ccCA and DLPNO-ccCA (FB) relative to ccCA for
linear alkanes (CnH2n+2 1 ≤ n ≤ 8). All timing studies were done with ORCA.

RI-ccCA
Scheme 1

36.7
37.8
38.2
37.1
35.7
33.7
30.1
35.3

CH4
C2H6
C3H8
C4H10
C5H12
C6H14
C7H16
C8H18

Scheme 1

30.9
36.1
39.6
41.6
45.8
50.5
56.6
68.7

DLPNO-ccCA (FB)

Scheme 2

88.3
93.1
94.8
95.1
95.4
95.7
96.2
97.2

Scheme 3

87.3
92.5
94.4
94.8
95.1
95.5
96.0
97.0

For DLPNO-ccCA, Scheme 2 and 3 protocols were implemented to examine further effects of
cost savings for larger systems relative to those in the molecule set. When using ABS, the deviations
in ∆Hf (Table 4.6) varies depending on which RI approximation was used for the SCF portion of
the DLPNO-CCSD(T) calculation. When using RI-JK (Scheme 2), the trend in deviation from
experimental ∆Hf remained the same from using ABS for correlated methods only (Scheme 1)
but with a slightly higher predicted ∆Hf. Apart from methane, the use of RIJCOSX (Scheme 3)
caused the prediction of enthalpy of formation to be lower in magnitude, which caused the deviation
for linear alkanes to lie between DLPNO-ccCA when using ABS for correlated methods (Scheme
1) and ccCA results. Isodesmic approaches are used for larger linear alkanes as these have been
shown to reduce the error without increasing the cost.112 The isodesmic schemes are shown in
Table 4.6. When using the isodesmic approaches, the deviations associated with ∆Hf are reduced
by 0.3 to 1.3 kcal mol−1 relative to using the atomization approach for ∆Hf. Regardless of which
approach was used, i.e. atomization approach or isodesmic schemes, the calculated ∆Hf generated
with all schemes of DLPNO-ccCA yields deviations in agreement with calculated ∆Hf generated
with ccCA and RI-ccCA.

As shown in Table 4.7, the percent CPU time savings drastically increases when using ABS for
SCF calculations. For methane, the percent CPU time savings increased from 30.9% using ABS

129

only for correlated methods (Scheme 1) to 88.3% using RI-JK for SCF and ABS for correlated
methods (Scheme 2) and 87.3% using RIJCOSX for SCF and ABS for correlated methods (Scheme
3). For octane, the percent CPU time savings increased from 68.7% using Scheme 1 to 97.2%
and 97.0% using Scheme 2 and 3, respectively. As with Scheme 1 ABS implementation for
DLPNO-ccCA, the percent CPU time savings monotonically increase as the number of carbon
atoms increase. Employing RIJCOSX with DLPNO-CCSD(T) caused a slight decrease in percent
CPU time savings for all linear alkanes assessed. However, this difference decreases with increasing
number of carbons, inferring that RIJCOSX is beneficial for larger molecules.

4.3.4 Applications of DLPNO-ccCA

The proposed DLPNO-ccCA (Pipek-Mezey localization and using RIJCOSX with
DLPNO-CCSD(T)) has been applied tothe S66and the L7 (coronene dimer) data sets124–127
These datasets target long-range weakly bound systems and are calibrated to CCSD(T)/CBS
interaction energies. For S66, examples were picked from the three subcategories of the dataset
for presentation, hydrogen-bound molecules (water dimer), dispersion-dominated interactions
(stacked uracil dimer), and a combination of both ( CH3NH2-Peptide, T-shaped benzene dimer).
The L7 dataset
targets larger noncovalent complexes predominantly exhibiting dispersion
interactions. All calculated interaction energies are counterpoise-corrected. Comparing the
effectiveness of DLPNO-ccCA interaction energies relative to CCSD(T)/CBS interaction energies
provides a computational cost-effective way to haveab initio data present for these larger
molecular systems and serves as a potential gauge for DFT and other scaling/cost-reduction
methods. DLPNO-ccCA calculated interaction energies were compared against the interaction
energies generated with CCSD(T)/CBS and an average of MP2/CBS, MP2C/CBS, MP2.5/CBS,
SCS-MP2/CBS, SCS(MI)-MP2/CBS given the optimized structures from the original publication
of the S66 molecule set.126

For all cases except for the uracil stacked dimer, interaction energies calculated with DLPNO-
ccCA yielded smaller deviations from the CCSD(T)/CBS interaction energies than mean MP2/CBS

130

interaction energies. For the T-shaped benzene dimer, the MAD from the reference interaction
energy was 0.03 kcal mol−1 and 0.17 kcal mol−1 using DLPNO-ccCA and MP2/CBS, respectively,
whereas the MAD from the reference interaction energy for the uracil stacked dimer was 0.34 kcal
mol−1 and 0.27 kcal mol−1 using DLPNO-ccCA and MP2/CBS, respectively.

For the coronene dimer, the interaction energy calculated with DLPNO-ccCA deviates from
the QCISD/CBS and DFT-D3/def2-QZVP reference interaction energies by 3.62 kcal mol−1 and
6.44 kcal mol−1, respectively. The DFT-D3/def2-QZVP presented in Table4.8is an average of
calculated interaction energies with the B3LYP-D3, BLYP-D3, TPSS-D3, PW6D95-D3, and
M06-2X-D3 functionals in tandem with the def2/QZVP basis set.127 This shows that
DLPNO-ccCA has
that exhibit primarily
dispersion-dominated interactions between molecules due to the truncation parameters for
screening orbital pairs and triples. This holds since DLPNO-ccCA yields lower MADs than
MP2/CBS for CCSD(T)/CBS interaction energies for complexes that exhibited both long-distance
hydrogen bonding and dispersion interactions.

some issues when dealing with complexes

Interactions energies of select examples from the S66 and L7 molecule sets. All

Table 4.8:
interaction energies are in kcal mol−1.
DLPNO-ccCA

S66
(H2O)2

-4.88
-9.48
-5.31
-2.69

CCSD(T)/CBS (∆a(TQ)Z)

Uracil dimer (S)
CH3NH2-Peptide
Benzene dimer (T)

MP2/CBSa
-4.85±0.17
-9.56±0.92
-5.15±0.38
-3.05±0.38
DLPNO-ccCA QCISD(T)/CBS DFT-D3/def2-QZVPb MP2/CBSa
-26.32±6.56
aThe MP2/CBS value presented is the average of the counterpoise-corrected MP2.5/CBS, MP2C/CBS, MP2/CBS, SCS(MI)-
MP2/CBS, SCS-MP2/CBS interaction energies. 125,126 FL
bThe DFT-D3 value presented is an average of the B3LYP-D3, BLYP-D3, TPSS-D3, PW6D95-D3, and M06-2X-D3 functionals
used in combination with the def2-QZVP basis set with no counterpoise correction included. 127 FL

Coronene dimer

-27.98

-24.36

-4.86
-9.82
-5.42
-2.58

L7

-21.54±1.28

Analyzing the individual components of DLPNO-ccCA for each of the dimers investigated from
the S66 and L7 data sets yielded insight into necessary electronic contributions towards calculating
interaction energies. As shown in Table 4.9, for the molecules from the S66 data set, the primary

131

contribution towards interaction energies is the inclusion of core-core correlation effects as an
additive correction to the DLPNO-MP2/CBS reference energy. While the core-valence correlation
effects affected the total interaction energy by less than 0.1 kcal mol−1 for the dimers in S66,
core-valence correlation effects increased the total interaction energy by 4.05 kcal mol−1 for the
coronene dimer. This shows that core-valence interactions are necessary when considering larger
dispersion-dominated molecules.

Table 4.9: Component breakdown of the DLPNO-ccCA calculated interaction energies from the
S66 and L7 datasets with counterpoise corrections included. All interaction energies are in kcal
mol−1.

S66
(H2O)2

Uracil dimer (S)
CH3NH2-Peptide
Benzene dimer (T)

L7

Coronene Dimer

DLPNO-MP2/CBS ∆CC ∆CV ∆DK

DLPNO-ccCA

-5.03
-11.03
-5.53
-3.71

0.21
1.60
0.26
0.88

-0.07
-0.06
-0.04
-0.02

0.01
0.002
0.003
-0.003

-4.88
-9.49
-5.31
-2.85

-48.29

16.27

4.05

-0.01

-27.98

Since DLPNO-ccCA was applied to the coronene dimer, the ∆Hf of coronene can be calculated
as well. With DLPNO-ccCA, the computed ∆Hf was 69.3 kcal mol−1 with the atomization
approach and the experimental ∆Hf is 70.5±2.7 kcal mol−1.128 This shows good agreement with
calculating the ∆Hf for larger main group organic species with ab initio composite strategies.

4.4 Conclusions

A new formulation of ccCA, DLPNO-ccCA, incorporating the DLPNO methods has been
developed and used to determine the ∆Hf for 119 molecules of the first and second row main group
from the G2/97 molecule set, a set of 8 linear alkanes, the S66 dataset, and the coronene dimer. The
Foster-Boys and Pipek-Mezey localization schemes followed by integral screening have been used
to aid in reducing the computational cost of the DLPNO-ccCA approach. It was found that by choice
of localization method in one step of the DLPNO-ccCA approach, the DLPNO-CCSD(T) step, can

132

result in an impact in the MAD from experiment in the enthalpy of formation by as much as 1.4
kcal mol−1, whereas for the MAD in the DLPNO-MP2/aug-cc-pV∞Z step, the MAD is impacted
by only 0.004 kcal mol−1. For smaller molecules, using localized MOs generated by the Pipek-
Mezey localization yielded a lower MAD overall compared to using localized MOs generated by the
Foster-Boys localization when ABS are implemented for SCF calculations. Overall, Pipek-Mezey
localization of occupied MOs yielded lower MADs for ∆Hf for cyclic aromatic rings, halogenated
compounds, and molecules characterized with a triple bond whereas there was no significant
numerical difference in the MADs for ∆Hf when using localization techniques for alkanes.

The use of RIJCOSX for the SCF step within DLPNO-CCSD(T) and RI-JK for the SCF step
within DLPNO-MP2 is recommended for DLPNO-ccCA paired with Pipek-Mezey localization.
DLPNO-ccCA reduces the computational cost compared to ccCA. RI-ccCA tends to save more
CPU time (92.5%) for smaller molecules than DLPNO-ccCA when using RI-JK for SCF energies
(87.5%). However, DLPNO-ccCA tends to result in more CPU time savings (86.7%) than RI-
ccCA (84.3%) with the use of RIJCOSX for RI-MP2 and with increasing molecule size within the
molecule set. When using DLPNO-ccCA for methane, ethane and propane, the deviation from
experimental ∆Hf increased relative to ccCA, but decreased relative to ccCA for molecules with n
≥ 4 where n is the number of carbon atoms. The percent cost savings in CPU time from utilizing
the DLPNO methods for linear alkanes range from approximately 88% to 97% with increasing
number of carbon atoms when using ABS for both SCF and correlated methods.

In summary, DLPNO-ccCA reduced the computational cost associated with ccCA by
approximately 87% while maintaining an overall MAD of no more than 1 kcal mol−1 from
reliable experiment and ab initio calculations for ∆Hf of main group complexes. More so than
RI-ccCA, DLPNO-ccCA significantly reduces the computational cost of ccCA for the larger
molecules in the molecule set, and thus allows access to investigate thermodynamic properties for
larger molecules with the same level of accuracy of ccCA.

133

APPENDIX

134

APPENDIX

Table 4.10: Molecule list used for full set calculations.

C4H6 (cyclobutene)
C4H8 (cyclobutane)
C4H8 (isobutene)

C4H10 (trans – butane)

C4H10 (isobutane)
C5H8 (spiropentane)

C6H6 (benzene)
C4H4S (thiophene)

H2CSCH2 (thiooxirane)

(CH2)2O (oxirane)
OCHCHO (glyoxal)
NCCN (cyanogen)

H2CCHCl
H2C –– CHCN
H3CONO
H3CSiH3

CH3CH2CH2Cl

HCOOCH3
H3CCONH2
H2CCH2NH

C4H6 (2-butyne)

C4H6 (methylene cyclopropane)
C4H6 (bicyclo[1.1.0 ] butane)

C3H4 (propyne)
C3H4 (allene)

C3H4 (cyclopropene)

C3H6 (propene)

C3H6 (cyclopropane)

C3H8 (propane)

C5H5N (pyridine)
C4H4O (furan)
H3CH2COCH3

C4H6 (trans – 1,3 – butadiene)

H2CCO
C4H4NH
H3CCOH

F2CCF2 (D2h)
H3CCH2OH
H3COCH3
(CH3)3N
(CH3)2SO
H3CCH2SH
H3CSCH3
H2CCHF
H3CCH2Cl
(CH3)2NH
H3CCH2NH2
H3CCOCH3
CH3COOH
CH3CFO
CH3C(O)Cl
HCOOH

(CH3)2CHOH

+)

Si2H6
CH3Cl
H3CSH
HOCl
SO2
BF3
BCl3
AlF3
AlCl3
CF4
CCl4
OCS (1
Σ
CS2
COF2
SiF4
SiCl4
NNO
ClNO
NF3
PF3
O3
F2O
ClF3
H2

Cl2CCCl2
CF3CN
CH2F2
CHF3
CH2Cl2
CHCl3
H3CNH2
H3CCN
H3CNO2

LiH

CH2 (1A1)

CH4
NH3
H2O
HF
SiH2
SiH4
PH3
SH2
HCl
Li2
LiF
C2H2
C2H4
C2H6
HCN
CO
H2CO
CH3OH

H2NNH2

HOOH

N2

O2

F2

CO2 (Dh)

Na2
P2
Cl2
NaCl
SiO
CS
ClF

135

Table 4.11: MP2/CBS counterpoise-corrected interaction energies calculated for molecules in
the S66 data set used to compare DLPNO-ccCA interaction energies from Reference 126. All
interaction energies are in kcal mol−1.

(H2O)2

Uracil dimer (S)
CH3NH2-Peptide
Benzene dimer

MP2

-4.96
-11.14
-5.53
-3.75

MP2.5

MP2C

SCS-MP2

-4.93
-9.47
-5.32
-3.05

-4.97
-9.37
-5.41
-2.96

-4.51
-8.25
-4.45
-2.58

SCS(MI)-

MP2
-4.91
-9.56
-5.04
-2.90

Table 4.12: MP2/CBS counterpoise-corrected interaction energies calculated for the coronene
dimer used to compare DLPNO-ccCA interaction energies from Reference 127. All interaction
energies are in kcal mol−1.

MP2

-38.98

MP2.5

-22.80

MP2C

-20.88

SCS-MP2

-27.53

SCS(MI)-

MP2
-31.71

coronene
dimer

Table 4.13: DFT-D3/def2-QZVPP non-counterpoise-corrected interaction energies calculated for
the coronene dimer used to compare DLPNO-ccCA interaction energies from Reference 127. All
interaction energies are in kcal mol−1.

B3LYP-D3

BLYP-D3

-23.22

-22.82

TPSS-D3
-21.19

PW6D95-D3 M06-2X-D3

-19.93

-20.55

coronene
dimer

136

Figure 4.6: CPU time for ccCA, RI-ccCA, and all 3 schemes for DLPNO-ccCA for the linear
alkanes.

Figure 4.7: Deviations in ∆Hf for ccCA, RI-ccCA, and all 3 schemes for DLPNO-ccCA for the
linear alkanes using the isodesmic approach.

137

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138

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CHAPTER 5

COMPUTATIONAL CHEMISTRY CONSIDERATIONS IN
CATALYSIS: REGIOSELECTIVITY AND METAL-LIGAND

DISSOCIATION

5.1 Introduction

For the prediction of thermodynamic information (i.e., enthalpies, free energies), reaction
barriers, HOMO-LUMO gaps, and other fundamental properties, density functional theory (DFT)
approaches are commonly used for catalysis. For early main group chemistry (i.e., hydrocarbons),
there are many different density functionals can be used, with very little difference in predicted
property arising from the choice of functional to describe energetics, and with limited exceptions, as
demonstrated by Karton et al.1 However, for transition metal species, the utility of each functional
can vary widely based upon choice of metal, choice of ligand, and property of interest.2–8 To
illustrate, for a set of ~20 3d transition metal species, B3LYP/CEP-31G(d) resulted in errors from
experiment from the predicted enthalpies of formation by ~100 kcal mol−1.2 However, when the
same functional is applied to a different set of the transition metal species —a set that has the
smallest reported experimental uncertainties in the enthalpy of formation-– the error is ~6-7 kcal
mol−1.4,5 So, indeed, extraordinarily large variances can occur depending upon metal and ligand.
For catalysis, where there may be interest in understanding the thermochemistry with much smaller
errors in energy, this magnitude of error may be of limited utility.

Computational approaches have been designed to improve upon the predictions possible by DFT
for transition metal species. With ab initio composite approaches like the correlation consistent
Composite Approach, ccCA, designed in our group,9–13 differences of ~2-3 kcal mol−1, on average,
can be achieved in the prediction of enthalpies of formation for 3d transition metal species.12,13 As
well, ccCA targeted 4d transition metal chemistry by utilizing relativistic pseudopotentials, denoted
as rp-ccCA, to model relativistic contributions from core electrons and yielded differences of ~3
kcal mol−1 from experimental enthalpies of formation.10,13 This is useful, but more costly than

149

DFT approaches. Strategies have evolved that help to reduce the computational cost, which is the
bottleneck in these calculations (e.g., DLPNO-ccCA) while preserving the accuracy or, as a means
to provide quantitative energy predictions for transition metal-based catalytic processes.

So far, these comments have focused upon general trends in the prediction of thermodynamic
properties of molecular systems. However, a question is, what is the utility of computational
approaches for an important industrial process like hydroformylation? More specifically, how
useful are computational approaches, particularly new approaches like DLPNO-ccCA, a form
of ccCA, for important properties like regioselectivity and metal-ligand binding? And, is the
qualitative or quantitative picture impacted by computational method choice?

The mechanism for Rh-based hydroformylation was well-established by Wilkinson in the late
1960s to early 1970s.14 As the largest volume homogeneous chemical reaction conducted in
industry for chemical production, the process converts olefins to aldehydes in a syngas mixture. The
advantage of Rh-based hydroformylation as opposed to Co-based hydroformylation is the favorable
reaction conditions (ambient temperature and pressure). The efficacy of a catalyst designed for
hydroformylation is the ratio of the linear aldehyde to the branched aldehyde (Fig. 5.1), known as
the linear-to-branched ratio. In hydroformylation, the formation of the linear aldehyde is favored
although there are studies targeting asymmetric hydroformylation, i.e.
the production of the
branched aldehyde.15,16 This is measured through the kinetics of the migratory insertion of the
olefin to the catalyst.

Figure 5.1: Hydroformylation reaction converting olefins to linear and branched aldehydes via a
Rh catalyst.

Numerous

computational

the

of
hydroformylation due to its importance in chemical industry.17–26 To account for the size of the
catalysts and the limited computing power at the time, earlier computational studies either

targeted modeling

regioselectivity

studies

have

150

substituted PPh3 ligands with much smaller PH3 ligands or utilized multilevel computational
chemistry methods, such as ONIOM,27,28 to model the bond breaking and formation region with
DFT while relegating the sterically bulky ligands to a computationally more affordable method,
such as molecular mechanics (MM).17–21 While more recent studies also utilize multilevel
approaches for hydroformylation, more rigorous ab initio methodologies are used to model bond
breaking and forming regions and use DFT to model the steric ligands.22,23 Other studies have
only used DFT to model the olefin insertion step as well as the entire Wilkinson catalytic
cycle.23–25,29,30 These studies provide insight into potential electronic contributions of the
sterically bulky ligands as well as the mechanism by identifying the rate-determining step, which
can change based on the type of ligand. Machine learning approaches have recently been
developed to screen potential ligands based on their regioselectivity and is a rising trend in
computational catalysis.26,31

‡
‡
Figure 5.2: A model of the two reaction pathways for hydroformylation where ∆E
and ∆E
b
l
are the reaction barriers for forming the linear and branched product, respectively. The energy
difference between the two reaction barriers is denoted as ∆∆E‡.

The kinetics of hydroformylation is very sensitive energetically, i.e. the differences in energy
between competing pathways (∆∆E‡), illustrated in Fig. 5.2, can be less than 1 kcal mol−1.17
With such small differences in energy for the competing pathways, calculating the correct linear-
to-branched ratio can be difficult to predict with computational methods. For example, as shown in
Table 5.1, if ∆∆E‡ = 0 (reaction barriers are equivalent), the l:b ratio is 50:50. However, lowering
the barrier for the linear product by 1 kcal mol−1 (∆∆E‡ = -1 kcal mol−1) results in a product ratio

151

increase to approximately 84:16 and lowering the barrier for the formation of the linear product by
an additional 2 kcal mol−1 (∆∆E‡ = -3 kcal mol−1) indicates that the reaction highly favors the
linear product (100:0 ratio).
Table 5.1: A summary of the effect of ∆∆E‡ in kcal mol−1 on the linear-to-branched ratio (l:b)
ratio for hydroformylation.

∆∆E‡ (kcal mol−1)

0 to -1
1 to -2
-2 to -3

Linear-to-branched ratio (l:b)

50:50 to 84:16
84:16 to 97:3
97:3 to 100:0

Basically, considering the challenges mentioned earlier about the prediction of thermochemistry
properties for transition metal species, achieving the level of accuracy needed to even predict the
correct product distributions seems unsurmountable. Cancellation of errors that can occur from
comparing energy differences is helpful, though the errors from experiment are not necessarily the
same across a reaction pathway, and, thus, gauging method utility for each problem, considering
metal, ligand, and property, is essential. Thus, in this chapter, the impact of method and basis set
choice – the route to describe the molecular orbitals – are considered to determine the impact of
these choices upon the prediction of linear-to-branched ligand ratio, as well as the ligand dissociation
energy.

Another aspect that is important in catalysis is the description of metal-ligand dissociation, as it
is a primary step in all homogeneous catalytic reactions, e.g. product dissociation from Rh-catalyzed
hydroformylation and solvent interactions with olefin hydrogenation, as well as gas phase ligand
dissociation for organometallic reactions targeting C-H activation. Here, to gain understanding
about the utility of the ab initio composite strategy, DLPNO-ccCA, a cationic (diiimine)(aquo)PtII
complex was examined. This PtII complex was chosen since PtII complexes with ligands containing
aromatic and aliphatic C-H bonds are involved in the oxidative addition of alkanes and have been
a focus in C-H activation studies where the ligand substitution step is rate-determining.32–35

152

5.2 Computational Methods

5.2.1 Computational methods for hydroformylation

DFT and ab initio calculations were done in this study. Several density functionals were
utilized, selecting a number of widely used functionals varying in complexity: B3LYP,36,37
B3P86,36,38 BLYP,37,39 BP86,38,39 PBE,40,41 and PBE0.40–42 (It should be noted that while
increased complexity often means better property predictions, this is not necessarily guaranteed.)
Grimme’s dispersion correction with Becke-Johnson dampening (D3BJ) was included for B3LYP
and PBE0 to correct for long-range intramolecular interactions.43 The Stuttgart/Dresden basis set,
and pseudopotential (SDD) was used for all DFT calculations.44,45 Though it is commonly
believed that a triple-ζ quality basis set is sufficient for DFT calculations, earlier work has
demonstrated that for the predictions of energetic properties of transition metal species,
quadruple-ζ level basis sets can have an impact on the energies, and, thus, this level of basis set
was considered.3,4 As well, this choice of basis set followed earlier work done by Kumar et al.24,
and all structures for the DFT and ab initio calculations were based on this prior work. DFT
calculations in the present work were done with Gaussian16.46

Several ab initio correlated methods also were used including domain-based pair natural orbital
(DLPNO) methods,47–52 DLPNO-MP2 and DLPNO-CCSD(T)), the MP2 and CCSD(T) varieties
of the DLPNO approach. The DLPNO approach enables computational cost reductions from
typical MP2 and CCSD(T) calculations. And, CCSD(T) is of particular interest, as this method
is known for its utility in energy predictions when paired with a high-quality (which typically
means large) basis set. The DLPNO calculations were done with the ORCA program suite.53
Calculations were done using Dunning’s correlation consistent polarized valence-n-ζ (“zeta”) basis
sets (aug-cc-pVnZ, where n=D (double), T (triple), Q (quadruple)), and considering augmented
(aug-cc-pVnZ) and augmented core-valence (aug-cc-pCVnZ) forms of the sets.54,55 For P and
Cl, the recommended tight d versions of the correlation consistent basis sets, denoted as cc-
pV(n + d)Z, aug-cc-pV(n + d)Z, and aug-cc-pCV(n + d)Z were used.55 The correlation consistent

153

pseudopotentials (cc-pVnZ-PP) were used for Rh and Pt atoms.56,57 The correlation consistent
Composite Approach (ccCA) for 4d transition metals was also considered,10 utilizing the DLPNO
methods for the composite steps to reduce the computational resources associated with the size of
the compound, denoted as DLPNO-rp-ccCA.58

To calculate the regioselectivity for hydroformylation, the following equation was used for the

linear-to-branched ratio

l : b = kl : kb =

‡
e(−∆G
l
‡
e(−∆G
b

= e(−∆∆G‡/kT ) ≈ e(−∆∆E‡/kT )

(5.1)

/kT )

/kT )

where ∆∆G‡ is the free energy barrier, ∆∆G‡ is the energy difference between the two
reaction pathways, k is the Boltzmann constant, and T is the temperature. This equation assumes
the olefin insertion step is irreversible. The Rh-catalyst-olefin complex examined with the DLPNO
methods is ee – [Rh(H)(CO)(DIPHOS)(propene)] where the bis-phosphine DIPHOS ligand is in
the equatorial-equatorial (ee) conformation. The ligands examined with DFT include (PPh3)2,
and more structurally complex bis-phosphine ligands, TBDCP, DIOP, and DIPHOS. All ligands
are attached to a [Rh(H)(CO)] backbone as indicated in the Wilkinson catalytic cycle for Rh-
based hydroformylation. Olefins examined with (PPh3)2 include pentene, hexene, heptene, octene,
decene, dodecene, styrene, and vinyl acetate. Propene is coordinated with all bisphosphine ligands.
As the experiments were carried out in toluene, the SMD implicit solvent model59 was used to
mimic the long-range solvent effects of toluene on the Rh catalyst.

The

Rh-catalyst-olefin

complex

examined with

the DLPNO methods

is
in the
ee-[Rh(H)(CO)(DIPHOS)(propene)] where the bis-phosphine DIPHOS ligand is
equatorial-equatorial (ee) conformation. The ligands examined with DFT include (PPh3)2, and
more structurally complex bis-phosphine ligands, TBDCP, DIOP, and DIPHOS. All ligands are
attached to a [Rh(H)(CO)] backbone as indicated in the Wilkinson catalytic cycle for Rh-based
hydroformylation. Olefins examined with (PPh3)2 include pentene, hexene, heptene, octene,
decene, dodecene, styrene, and vinyl acetate. Propene is coordinated with all bisphosphine
ligands.

154

Figure 5.3: Computationally determined 3D structures of ee-[Rh(H)(CO)(DIPHOS) (propene)]
catalyst complex (top) and the dissociation reaction of H2O from the cationic (diimine)(aquo)PtII
complex (bottom).

5.2.2 Computational methods for ligand dissociation

The gas phase ligand dissociation energy was evaluated by the difference between the complex

and the respective fragments.

∆Edissoc = EAB − EA + EB

(5.2)

where EAB is the electronic energy of the complex, EA is the electronic energy of fragment A,
and EB is the electronic energy of fragment B. Ab initio calculations, in particular, are susceptible
to basis set superposition error (BSSE), which can result in overbinding of the ligands.60 BSSE
can occur when there is an imbalance in basis set size for the species considered in determining an
energy difference. The effects have been addressed by conducting calculations on the individual
species in the presence of the basis set associated with the other species. A correction for the BSSE
was applied to all DLPNO calculations.

To study fundamental organometallic reactions that occur in the gas phase, a cationic
(diimine)(aquo)PtII complex prevalent in C-H activation and oxidative addition of alkanes was
chosen (see Fig. 5.3). This molecule was chosen due to computational feasibility based on the
molecule size. The calculated zero point energy (ZPE) of the reaction obtained with a frequency
calculation at the BP86 level and the PBE0 optimized structures were obtained from Weymuth et

155

al.61 This choice of functional for frequency calculations was selected since the ZPE of the
reaction did not significantly change with respect to functional choice.61 PBE0 structures were
utilized based on their success for heavier elements. A few density functionals, PBE0, B3LYP,
and TPSSh62 utilizing cost-saving techniques,
i.e the resolution-of-the-identity or RI
approximation, were paired with the augmented correlation consistent basis
sets and
pseudopotentials of triple- and quadruple-ζ level quality (aug-cc-pVnZ, n=T, Q), as well as
DLPNO-rp-ccCA to determine ligand dissociation energies. All ligand dissociation calculations
were done in the ORCA program suite.

5.3 Results and Discussion

5.3.1 Regioselectivity in hydroformylation

The DFT l:b ratios for all Rh catalysts are shown in Table 5.2. The corresponding ∆∆E‡s for
all DFT results are shown in Table 5.3. The DLPNO results for hydroformylation are shown in
Table 5.4 for the l:b ratios, including the l:b determined for calculations that have been corrected for
BSSE. With DFT, qualitatively correct l:b ratios are obtained for most of the examined catalyst-olefin
complexes as shown in Table 5.2. However, this largely depends on which type of functional is used.
For example, using BLYP, BP86, and PBE generally predicted l:b ratios that are in disagreement with
experiment for (PPh3)2 ligands, particularly for hexene, heptene, octene, dodecene, and styrene,
which produced l:b ratios of 2:98, 26:74, 11:89, 84:16, and 87:13, respectively, for BLYP, and
similar ratios for BP86 and PBE (Table 2). With an increase in complexity in the functionals, i.e.
B3LYP, B3P86, and PBE0, l:b ratios of 67:33 47:53 and 76:24 for B3LYP, 71:29, 67:33, 68:32 for
B3P86, and 75:25, 73:27, 71:29 for PBE0, were predicted for the conversion of heptane, octene, and
dodecane with (PPh3)2 ligands, respectively. And for ee-[Rh(H)(CO)(PPh3)2(pentene)], the linear
product is predicted. However, the inclusion of Grimme’s dispersion correction for B3LYP and
PBE0 predicted l:b ratios that predicted the more favorable produce, in agreement with experiment
for all examined catalyst-olefin complexes with the exception of ee-[Rh(H)(CO) (PPh3)2(decene)]
(l:b ratios of 0:100 and 3:97 for B3LYP-D3 and PBE0-D3, respectively) and ee-[Rh(H)(CO)

156

(DIPHOS)(propene)] (l:b ratios of 17:83 and 11:89 for B3LYP-D3 and PBE0-D3, respectively).
Based on the l:b ratios found, introducing complexity in the functional can but not always improve
property prediction.

157

Table 5.2: Comparison of several density functionals to linear-to-branched ratios from experiment for ee-[Rh(H)(CO)(L)(olefin)]
complexes.

L=(PPh3)2
Pentene
Hexane
Heptane
Octene
Decene
Dodecene
Styrene

Vinyl acetate

L=TBDCP
Propene

L=DIOP
Propene

L=ee-DIPHOS

propene

L=ea-DIPHOS

propene

BLYP

BP86

PBE

B3LYP

B3P86

PBE0

B3LYP-D3

PBE0-D3

83:17
2:98
26:74
11:89
84:16
45:55
87:13
0:100

81:19
3:97
28:72
20:80
93:7
33:67
92:8
0:100

79:21
6:94
33:67
31:69
89:11
41:59
79:21
0:100

95:5
10:90
67:33
47:53
53:47
76:24
83:17
0:100

95:5
19:81
71:29
67:33
69:31
68:32
84:16
0:100

95:5
23:77
75:25
73:27
64:36
71:29
90:10
0:100

100:0
100:0
100:0
100:0
0:100
100:0
0:100
0:100

90:10

88:12

89:11

96:4

95:5

96:4

92:8

99:1
99:1
99:1
100:0
3:97
99:1
1:99
0:100

95:5

99:1

99:1

100:0

100:0

100:0

100:0

100:0

100:0

3:97

3:97

3:97

5:95

5:95

5:95

17:83

11:89

83:17

73:27

71:29

86:14

77:23

76:24

88:12

81:19

Expa

95:5
92:8
86:14
81:19
74:26
87:12
11:89
9:91

92:8

90:10

69:31b

69:31b

aReferences 14,63–69
bThe data references the ea conformer of DIPHOS.

158

Table 5.3: Comparison of the approximate ∆∆E‡s based on the calculated l:b ratios for ee- [Rh(H)(CO)(L)(olefin)] complexes.
Experimental ∆∆E‡s are an approximation of experimental l:b ratios. All ∆∆E‡s are in kcal mol−1.
B3LYP-D3

PBE0-D3

BLYP

BP86

Expa

PBE

B3LYP

B3P86

PBE0

L = (PPh3)2
Pentene
Hexene
Heptene
Octene
Decene
Dodecene
Styrene

Vinyl acetate

L = TBDCP

Propene

L = DIOP
Propene

L = ee-DIPHOS

propene

L = ea-DIPHOS

propene

-0.96
2.44
0.62
1.26
-0.98
0.11
-1.10
6.24

-0.88
2.07
0.56
0.81
-1.49
0.43
-1.44
6.55

-0.80
1.65
0.41
0.47
-1.23
0.21
-0.78
6.59

-1.78
1.29
-0.42
0.06
-0.08
-0.68
-0.94
6.02

-1.74
0.87
-0.53
-0.42
-0.47
-0.43
-1.00
6.35

-1.74
0.71
-0.66
-0.60
-0.34
-0.54
-1.29
6.27

-3.67
-3.93
-3.74
-4.87
3.36
-3.92
5.62
5.06

-1.31

-1.16

-1.23

-1.86

-1.71

-1.92

-1.45

-2.57

-3.12

-3.22

-3.28

-3.76

-4.02

-3.59

2.00

2.10

2.08

1.70

1.80

1.75

0.93

-0.96

-0.58

-0.54

-1.05

-0.72

-0.69

-1.17

aReferences 14,63–69
aThe data references the ea conformer of DIPHOS.

-3.05
-2.70
-2.87
-3.85
2.05
-2.91
2.87
5.42

-1.70

-4.22

1.25

-0.86

-1.74
-1.44
-1.07
-0.85
-0.62
-1.13
1.24
1.36

-1.44

-1.30

-0.47b

-0.47b

159

For ee-[Rh(H)(CO) (PPh3)2(vinyl acetate)], the predicted ∆∆E‡ was ~6 kcal mol−1 for each
functional considered as shown in Table 3, indicating the branched isomer is favored. Overall, the
dispersion-corrected functionals resulted in a lowering of the ∆∆E‡ for ee-[Rh(H)(CO)
(PPh3)2(vinyl acetate)] by approximately 1 kcal mol−1, however, this did not impact the product
distribution. Similarly, for ee[Rh(H)(CO)(DIOP)(propene)], the dispersion correction functionals
resulted in a lowered the predicted ∆∆E‡ by ~0.3 kcal mol−1 and did not impact the product
distribution as the predicted ∆∆E‡ was ~4 kcal mol−1.
for ee-[Rh(H)(CO)
(PPh3)2(styrene)] and ee-[Rh(H)(CO) (PPh3)2(decene)],
the dispersion-corrected functionals
predicted the ∆∆E‡ to be ~6 kcal mol−1 and ~3 kcal mol−1 greater than the ∆∆E‡ predicted
with non-dispersion-corrected functionals. While this change in ∆∆E‡ predicted product ratios of
0:100 and 1:99 for B3LYP-D3 and PBE0-D3, respectively, with styrene as the olefin, with decene
as the olefin, the predicted product ratios were 0:100 and 3:97 for B3LYP-D3 and PBE0-D3,
respectively.

However,

For the DIPHOS ligand, the relative orientation of the Rh-H and Rh-CO bond to the DIPHOS
ligand was a major factor in predicted l:b ratios with DFT. In the ee conformation, all predicted l:b
ratios with DFT predicted the branched product whereas the linear product is predicted for the ea
conformation, in qualitative agreement with experiment. This is to be noted for any calculation.
The small conformation changes from the ee conformation to the ea conformation led to lowering
of the ∆∆E‡ by ~2-3 kcal mol−1 for all functionals, changing the product ratio to favor the linear
product over the branched ratio. This exhibits the high sensitivity of ∆∆E‡, which can greatly
affect product formation ratios with changes as small as a few tenths of a kcal mol−1, as exhibited
by the ∆∆E‡s of -0.54 and -1.05 kcal mol−1 that yielded product ratios of 71:29 and 86:14 for
PBE and B3LYP, respectively. Ergo, based on the observed trends from the DFT calculations, there
remains a need to investigate hydroformylation with electron correlation methods.

160

Table 5.4: Results using DLPNO methods to predict
[Rh(H)(CO)(DIPHOS)(propene)].

the linear-to-branched ratio for ee-

DLPNO-MP2/aug-cc-pVDZ-PP
DLPNO-MP2/aug-cc-pVTZ-PP
DLPNO-MP2/aug-cc-pVQZ-PP

DLPNO-MP2/cc-pVTZ-PP

DLPNO-CCSD(T)/cc-pVTZ-PP

DLPNO-CCSD(T)/aug-cc-pCVDZ-PP

DLPNO-CCSD(T,FC1)/aug-cc-pCVDZ-PP

DLPNO-rp-ccCA

Experimental

l:b
25:75
29:71
16:84
18:82
1:99
100:0
100:0
100:0

l:b (corrected for BSSE)

100:0
100:0
100:0
100:0
100:0
100:0
100:0
100:0

69:31

The data references the equatorial-axial (ea) conformer of DIPHOS.

Here, the ee-[Rh(H)(CO)(DIPHOS)(propene)] catalyst-olefin complex is considered, as DFT
was unable to address the regioselectivity of this reaction correctly in any case. For the DLPNO
methods, the l:b ratio is predicted to favor the branched isomer. When BSSE has been addressed,
the linear isomer is favored. For DLPNO-rp-ccCA, the molecular orbital space is well-described,
and, thus, accounting for BSSE effects is not necessary, and the l:b ratio predicted with DLPNO-
rp-ccCA is 100:0. This is primarily due to the interactions between the electrons from core
orbitals with electrons in valence orbitals as DLPNO-CCSD(T)/aug-cc-pCVDZ-PP and DLPNO-
CCSD(T,FC1)/aug-cc-pCVDZ-PP, which includes sub-valence electron (FC1) excitations within
the molecular orbital space, both favored the linear isomer with product ratios of 100:0. The
results from implementing the DLPNO methods indicate that electronic effects from including core
electrons within the valence basis set are significant in determining ∆∆E‡ given the large magnitude
relative to other calculated ∆∆E‡s with ab initio methods. Even for qualitative predictions,
DLPNO-rp-ccCA is useful without having to correct for BSSE. By utilizing a well-described
molecular orbital space – DLPNO-rp-ccCA does predict the proper regioselectivity; DFT either
does not predict the correct regioselectivity, such as for ee-[Rh(H)(CO) (PPh3)2(styrene)] and ee-
[Rh(H)(CO) (PPh3)2(hexene)], which predicted qualitatively inconsistent product ratios for most
In addition, the regioselectivity is highly sensitive to functional
of the functionals examined.

161

choice, as the performance is not consistent as the ligand type and olefin changes. However, for
the ab initio methods considered, simply improving the description of the molecular orbital space,
either by BSSE correction or by including sub-valence electrons in the molecular orbital space for
interactions yielding qualitative agreement with experiment.

5.3.2 Metal-ligand dissociation in organometallics

and

density

functionals

utilizing

to

several

both

experiment

to experiment.

an error of 1.7 kcal mol−1 relative

The gas phase ligand dissociation energies are shown in Table 5.5 with DLPNO-rp-ccCA
compared
the
resolution-of-the-identity approximation. For gas-phase ligand dissociation, DLPNO-rp-ccCA
yields
When utilizing the
resolution-of-the-identity approximation within DFT calculations, RI-PBE0/aug-cc-pVTZ,
RI-B3LYP/aug-cc-pVTZ, and RI-TPSSh yields dissociation energies of 20.7, 20.2, and 19.6 kcal
mol−1, respectively. However, increasing the quality of the molecular orbital space, i.e. using
aug-cc-pVQZ, increased the error by 0.4, 0.5, and 0.5 kcal mol−1 for RI-PBE0, RI-B3LYP, and
RI-TPSSh, causing concern for utilizing DFT with higher quality basis sets. Regardless of
functional and basis set choice,
the predicted dissociation energy was greater than 5 kcal
mol−1 lower than the experimental value. With the dispersion correction included for
RI-PBE0/aug-cc-pVTZ, the predicted dissociation energy increased to 23.6 kcal mol−1. This
would suggest that accounting for dispersion is necessary for DFT predictions of gas-phase
properties. DLPNO-rp-ccCA calculations yielded favorable results for ligand dissociation energy
in comparison to DFT, but there are factors that can contribute to computationally predicted
dissociation energies.
For example, as density functionals are primarily used to generate
structures for large organometallic complexes, the choice of functional for optimization must be
considered. The predicted dissociation energies can change by a few kcal mol−1 based on slight
structural change (root mean square deviation of ~20 pm) between functionals and by 10’s of kcal
mol−1 for significant structural changes such as ligand reorientation. Also, the basis set choice
can affect the quality of predictions as indicated from the lowering of predicted dissociation

162

energy by increasing basis set quality.

Table 5.5: Comparison of the gas-phase ligand dissociation energy of H2O from the Pt complex
calculated with DLPNO-rp-ccCA and RI-DFT-D3/aug-cc-pVnZ. All energies are in kcal mol−1 and
are BSSE-corrected.

RI-PBE0/aug-cc-pVTZ
RI-B3LYP/aug-cc-pVTZ
RI-TPSSh/aug-cc-pVTZ
RI-PBE0/aug-cc-pVQZ
RI-B3LYP/aug-cc-pVQZ
RI-TPSSh/aug-cc-pVQZ
RI-PBE0-D3/aug-cc-pVTZ

DLPNO-rp-ccCA

Experiment

20.7
20.2
19.6
20.3
19.7
19.1
23.6
24.2

25.9 ± 0.7

5.4 Conclusions

Basically, considering the challenges mentioned earlier about the prediction of thermochemistry
properties for transition metal species, achieving the level of accuracy needed to even predict the
correct product distributions seems unsurmountable. Cancellation of errors that can occur from
comparing energy differences is helpful, though the errors from experiment are not necessarily the
same across a reaction pathway, and, thus, gauging method utility for each problem, considering
metal, ligand, and property, is essential. A typical method choice for the study of transition metal
species is DFT. Unfortunately, there is no “magic” functional to use for all problems. While ab
initio methods like CCSD(T) or composite methods that try to replicate it like ccCA can be quite
useful and are more dependable from system to system and, generally, across a reaction pathway,
they are more costly, and may require additional measures to ensure quality results are obtained
sometimes reaching near saturation of the orbital space (even more costly!). DFT can be very
useful, but properly gauging it is important, as illustrated by this study.

163

REFERENCES

164

REFERENCES

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CHAPTER 6

VIBRATIONAL POTENTIAL ENERGY SURFACES WITH THE
CORRELATION CONSISTENT COMPOSITE APPROACH AND

DENSITY FUNCTIONAL THEORY

6.1 Introduction

Vibrational spectroscopy is one of the most useful techniques available to science due to its
unique window into the structure, dynamical behavior, and bonding properties of molecules.1
Modeling vibrational interactions is critical to understand infrared absorption, mechanisms, and
kinetics of chemical reactions. Calculated frequencies can be utilized to predict thermodynamic
properties like the enthalpy of formation and reaction barriers.2–6 As well, computational techniques
are necessary to substantiate novel experiments where the resolution is insufficient or there are
difficulties in isolating the molecule, i.e. diatomics, amino acids, or even short-lived molecules like
the transition state of a chemical reaction and radicals.7

Computational chemistry techniques can also interpret and assign vibrational features to a
specific molecule or type of motion. The increase of computational power in the digital age allows
for the development of numerous methods for quantum mechanical modeling within the field,
investigating deeper into the electronic structure of many-body systems, such as electronic
properties, potential energies, and anharmonic vibrational frequencies.8–10 Electronic structure
methods have been utilized to investigate vibrational frequencies with potential energy surfaces
(PESs) describing dynamical motion, obtaining frequencies that are within several cm−1 of
experiment.9,11–14 These methods have a caveat, in which the accuracy attained for vibrational
frequencies is exchanged for high computational cost, most prominent being disk space and CPU
time.9,10 This is coupled with the number of grid points needed to generate the requisite PESs,
which can easily exceed tens of thousands or even one million.11,12 Thus, the combination of
electronic structure methods with the vast number of grid points needed for a PES reduces the
feasibility of utilizing electronic structure methods for predicting vibrational properties within

171

several cm−1. This allows for continual development towards creating computational methods
with low computational cost while attaining vibrations predictions within several cm−1 from
well-established reliable experiments. Efficient processes have been developed to reduce the
computational cost while yielding deviations from experimental vibrations by several cm−1 or
deviations less than 1 kcal mol−1 for reaction barriers.11,15–17

The correlation consistent Composite Approach (ccCA)18,19 is considered, as a route to alleviate
the cost of generating potential energy surfaces. ccCA has been utilized to describe PESs. For
example, the multireference wavefunction ccCA (MR-ccCA) was utilized to analyze the potential
energy curve of the torsional rotation of the carbon-carbon double bond in ethylene, predicting
the barrier height of cis-trans isomerism and yielded errors approximately 0.7 kcal mol−1 from
experiment.16 As an alternative to utilizing multireference methods, completely renormalized
coupled cluster (CR-CC(2,3)) was implemented within the ccCA formalism, CR-ccCA(2,3).17 This
method utilizes a single reference completely renormalized coupled cluster that can correctly treat
reaction pathways such as the thermal pericyclic rearrangement of bicyclo[1.1.0]butane to trans-
buta-1,3-diene and chemical species, e.g. diradicals, that would normally require multireference
methods.

In addition to ccCA, DFT can be used to find properties of these electronic many-body
systems at an affordable computational cost relative to the ab initio methods utilized for
vibrational spectroscopy.20 Density functionals have been largely designed for main group
thermochemical properties; however, DFT cannot adequately describe noncovalent interactions,
such as π-π stacking or weak hydrogen bonding, crucial in larger polyatomic molecules and could
be significant when describing weakly-bound ligands and noncovalent interactions between two
molecules.21,22

Anharmonic vibrational frequencies properly characterize vibrational motion more than
harmonic frequencies; however, these calculations can be quite expensive.23,24 While harmonic
frequencies require the second derivative of the potential energy function, anharmonic frequencies
require at least the third or fourth derivative to solve.25 Generally, computational methods are

172

restricted to the harmonic approximation for vibrational frequencies that led to the development of
empirical scale factors that can be tailored for high and low frequencies.24 An analysis taking
anharmonic effects into account would therefore lead to calculated frequencies that are not perfect
overtones and provide a more accurate description of the vibrational behavior.26,27

To account for anharmonicity computationally, computational strategies often include a
perturbative correction, such as VPT2, to the potential. However, vibrational self-consistent field
(VSCF) theory, which was developed in the late 1970s, fully accounts for anharmonicity by
considering the vibrational Schrödinger Equation. Approximations are made which make VSCF
theory analogous to Hartree-Fock theory.28–32 More recent studies utilizing VSCF theory
implement on certain biologically pertinent vibrations for amino acid peptide chains,11,33 which
are not typically targeted with the rigorous ab initio methods utilized for potential energy surfaces
(potentials) of diatomics and small polyatomic molecules like H2O and formaldehyde.12,34,35 In a
study by Roy et al., a VSCF-PT2 approach was utilized with both a B3LYP-D2 potential and a
multilevel HF/MP2 potential to characterize anharmonic vibrational motion of an opioid peptide
[Ala2, Leu5]-leucine enkephalin (ALE).11 They found that the B3LYP and multilevel HF/MP2
potential systematically underestimated and overestimated the experimental frequencies for the
OH and NH stretching modes for each amino acid, respectively, by a few tens of cm−1. The
average of the frequencies compensates for the respective under and overestimation of frequencies
and yielded theoretical predictions within 10 cm−1 of experiment, which was better than the
B3LYP and the multilevel HF/MP2 potentials individually as well as scaled harmonic
calculations, thus showing the efficacy of VSCF theory towards predicting anharmonic vibrations
for systems as large as a pentapeptide.

In this chapter, the correlation consistent Composite Approach (ccCA) and density functional
theory (DFT) have been used to generate potential energy surfaces (PES) for diatomic and small
polyatomic molecules to predict structural and vibrational properties such as frequencies and
infrared absorbance intensities in tandem with vibrational self-consistent field (VSCF) and post-
VSCF theory. Extrapolations schemes for ccCA and functional and basis set choice within DFT

173

were considered.

This was done to determine the efficacy of each method for each method. The combination
of electronic structure methods such as ccCA and DFT with post-VSCF theory aims to reduce the
computational cost associated with generating accurate PESs for anharmonic mode-mode couplings
as well as calculating contributions from anharmonic corrections to the potential.

6.2 Computational Methods

The PVSCF program15,36 was used for vibrational analysis and ORCA 4.0 was used for the
electronic structure calculations necessary to generate the potential energy surfaces.37,38 The
molecule set included 20 molecules: H2, CO, LiH, N2, NO+, OH, NH, HF, BF, O2, SiO, H2O,
CO2, NH3, C2H2, C2H4, C2H6, cis-3-aminophenol, and trans-3-aminophenol. These were
chosen based on the availability of experimental frequencies for these molecules and their
presence in the interstellar medium along with the notion of correlating the results with other
studies that use more computationally demanding post-HF methods for predicting anharmonic
frequencies.22 Experimental vibrational frequencies were obtained from Herzberg, Huber, and
Shimanouchi.39–41 Equilibrium bond lengths for the diatomic molecules were obtained from
CISD/cc-pVTZ
using
RI-B3LYP-D3/aug-cc-pVTZ within the ORCA package. Since the PVSCF program uses the
hessian as an initial guess, a Hessian generated with a more approximate method is sufficient for
the purpose of this study. For polyatomic molecules, B3LYP/cc-pVTZ geometries and hessians
were used in accordance with the ccCA methodology.19

calculations

generated

Hessians

were

and

the

initial

The potentials were generated via an interpolation of 16 grid points by a multimode expansion
using curvilinear coordinates. Potential energy curves (PECs) are generated for diatomics while
surfaces that plot the effect of two different vibrational modes concurrently vibrating, or vibrational
mode coupling, are generated for polyatomic molecules. The extracted potential energies and
dipole moments were then run with the PVSCF program to obtain the anharmonic frequencies
and infrared (IR) intensities of each molecule, respectively. For diatomic molecules, a Fourier

174

Grid Hamiltonian approach was used to calculate the single vibrational frequency.42,43 For all
polyatomic molecules, a vibrational configuration interaction method (VCIPSI-PT2) was used to
analyze the effects of vibrational mode coupling.44 VCIPSI-PT2 utilizes vibrational configuration
interaction with perturbatively selected interactions (VCIPSI), which reduces the computational
cost compared to standard VCI approaches while maintaining the same level of accuracy.15,44

6.2.1 DFT Calculations

Density functionals come in numerous flavors based on the number of parameters and the
operations performed on the electronic density surface to varying degrees of success. For example,
B3LYP45,46 is heavily parameterized for main group thermochemistry while TPSS47 has no
empirical parameters; yet both density functionals are popular and have been reported to yield low
mean absolute errors for main group thermochemistry.48 TPSS and B3LYP were therefore used as
the density functionals in this work.

Dunning’s standard and augmented correlation consistent basis sets from double- to quintuple-ζ
(cc-pVnZ (VnZ) and aug-cc-pVnZ (aVnZ), n= D, T, Q, 5) were used.49 These particular basis sets
were built to systematically increase the types of energy contributions and subsequently, these types
of functions included in the basis set, which leads to a smooth convergence of energetic properties
towards an infinite basis set that would describe all possible space in which electrons exist. The
Feller extrapolation scheme was used since this is a three-point extrapolation scheme, which allows
the extrapolation function to converge to a limit closer to the experimental values, and uses the
exponential form (Equation 2.29). The effects of the Feller extrapolation scheme were examined to
provide insight into energies that would be obtained when using a more computationally demanding
basis set (such as sextuple-ζ or higher). For polyatomic molecules with more than three atoms,
only cc-pVTZ and aug-cc-pVTZ was used.

175

6.2.2

ccCA Calculations

The implementation of ccCA has been described in Section 2.2.4.1. Standard cartesian, or
rectilinear, coordinates were used for diatomics and linear polyatomic molecules (CO2, C2H2).
Curvilinear coordinates were used for nonlinear polyatomic molecules (H2O, aminophenol
isomers). SCF energies were converged to 10−8 Eh in all single point energy calculations.18,19
ccCA electronic energies were used to generate all potential energy curves for singular vibrational
motion and surfaces for mode-mode coupling, i.e.
simultaneous vibrational motion for two
vibrational modes.

For C2H4, C2H6, and aminophenol isomers, DFT calculations were used to generate all
vibrational mode couplings. The number of vibrational mode couplings calculated with ccCA
were decreased via a screening threshold to isolate strongly coupled vibrational modes.23,50
Selected vibrational modes from coupling maps are provided in the Appendix. The coupling
strength is largely independent of the choice of method used to generate the potential energy
surface. This is denoted as a FASTVCI approach in this work.

6.3 Results and Discussion

The calculated frequencies for diatomics, H2O, CO2, and NH3 are shown in the Appendix. The
mean absolute deviation (MAD) was analyzed by basis set, functional, and by number of atoms to
note specific trends or certain occurrences within the calculations. For ccCA potentials, utilizing
different extrapolation schemes did not significantly affect the predicted vibrational frequency as
shown in Table 6.5. Therefore, for conciseness, only the frequencies predicted with ccCA-S4
potentials are presented as ccCA-S4 yielded the lowest errors of all extrapolations schemes utilized.

6.3.1 Diatomics

With DFT potentials at the complete basis set limit, the calculated frequencies yielded a MAD
that ranged from 0 to 149 cm−1 depending on the molecule and functional whereas with ccCA
potentials, the calculated frequencies yielded a MAD that ranged from 0 to 22 cm−1.

176

Examining functional choice, frequencies predicted with B3LYP-generated potential energy
curves (PECs) had performed well compared to TPSS-generated PECs, with having seven molecules
(H2, BF, HF, NH, OH, O2, and SiO) that yielded smaller MAD compared to TPSS. This is indicated
in Figure 6.1 from TPSS yielding lower MADs for LiH, CO, N2, and NO+ relative to B3LYP. For N2,
the predicted frequency generated with TPSS aligned with those predicted with ccCA-S4. This is
plausible, due to B3LYP having parameters fitted. In addition, the deviation values ranged from 33
to 149 cm−1 whereas B3LYP had a larger range of error of 42 to 178 cm−1 when TPSS-generated
potentials yielded lower errors for calculated frequencies than B3LYP-generated PECs. The larger
range is due to the B3LYP/V∞Z PEC for LiH yielding a high error for calculated frequency relative
to experiment. Based on the choice of molecule, functional choice had a larger effect on predicted
vibrational frequency than basis set choice.

177

Figure 6.1: Mean absolute deviation (MAD) of vibrational frequencies for diatomics using
TPSS/VTZ (blue), B3LYP/VTZ (green), TPSS/aVTZ (purple), B3LYP/aVTZ (red), and ccCA-
S4 (black).

When comparing the effects of basis set choice holistically, augmented basis sets tended to
produce lower MADs for experimental frequencies in comparison to non-augmented basis sets.
This is evident as the MAD decreased from 43± 37 and 39± 35 for B3LYP/VnZ and B3LYP/aVnZ,
respectively, and 57 ± 52 and 55 ± 50 for TPSS/VnZ and TPSS/aVnZ, respectively. This could
be due to the extra diffuse function presented in aVnZ basis sets that can better describe a larger
internuclear distance for diatomics. From the supplemental tables (Tables 6.6-6.7), using a larger
basis set could lead to more accurate predictions as the mean error when using VDZ across all
molecules and functionals was 73 ± 10 cm−1 whereas the mean error when using VTZ, VQZ, and
V5Z across all molecules and functionals was 44 ± 11, 42 ± 10, and 41 ± 10 cm−1, respectively.
Although there is an ∼30 cm−1 decrease in error between VDZ and the higher ζ-level basis sets,

178

there is not a consistent lowering of error in calculated frequency with respect to an increase in
basis set size as the error in frequency for all DFT/VTZ, DFT/VQZ and DFT/V5Z potentials were
statistically not different based on the 25% relative uncertainty in deviations across all molecules
and functionals. Therefore, triple-ζ quality basis sets may be useful as a compromise between cost
and accuracy for generating potentials describing vibrational motion for polyatomic molecules.

Diatomics that exhibit covalent triple bonds (CO, NO+, and N2) yielded lower deviations
from experimental frequencies with TPSS (31, 12, 8 cm−1, respectively) while diatomics with
single or double bonds yielded lower deviations from experiment with B3LYP (43, 93, 109 cm−1,
respectively). For the diatomics with covalent triple bonds, the parameterization within B3LYP
and the increase in electron density between atoms may be the cause of the higher deviations in
calculated frequency from experiment obtained with B3LYP-generated PECs opposed to TPSS-
generated PECs. HF, SiO, BF, and OH are molecules for which B3LYP yielded smaller deviations
from experiment, and have a larger electron density around the more electronegative atom, which
indicates that B3LYP is preferred when calculating polar molecules.

Across all diatomics examined, ccCA yielded a MAD of 9 ± 7 cm−1 whereas B3LYP/VnZ
and TPSS/VnZ yielded MADs of 43 ± 37 and 57 ± 52, respectively. This is shown in Figure
6.1. This indicates that using potentials generated with ccCA yield lower deviations for predicted
frequencies than DFT, with the notable exception of SiO, where using B3LYP regardless of basis
set yielded frequencies closer to experiment than ccCA by approximately 20 cm−1. This may be
due to the presence of static correlation as the molecule vibrates. With ccCA, diatomics that have
a larger difference in mass between the two atoms (OH, BF, HF) tended to yield lower deviations
from experiment for calculated frequencies with the exception of NH, which yielded an error of 16
cm−1. PECs for homonuclear diatomics yielded higher deviations with an increase in mass as H2,
N2, and O2, yielded errors of 5, 10, and 11 cm−1, respectively. As well, PECs for LiH and NO+,
heteronuclear diatomics with similar mass between atoms, tended to yield higher errors (8 and 11
cm−1) among the ccCA results although PECs for CO and BF yielded errors of 0 and 2 cm−1,
respectively. This would suggest that unlike DFT, there is no consistent trend between molecular

179

weight or atom type present and frequencies generated with ccCA PECs even though PECs at the
ccCA level of theory generates lower errors across all diatomics compared to DFT PECs.

The CPU time was measured with a Dell OptiPlex 390 with 16 GB DDR3 memory for a
few diatomics to show the approximate cost of generating a PEC of 17 grid points with DFT,
ccCA, and CCSD(T,full)/aug-cc-pCV5Z where full denotes the inclusion of all electrons for first-
row main group diatomics in the correlation space. This is shown in Table 6.1. As expected,
B3LYP calculations are more computationally affordable than ab initio methods but yielded higher
deviations for the molecules used (Table 6.2). Also, generating a PEC at the ccCA level, which
aims to model energies at the CCSD(T,full)/aug-cc-pCV∞Z-DK level, is more affordable than
using CCSD(T,full)/aug-cc-pCV5Z by several hours depending on the molecule size. As shown
in Table 6.1 for example, ccCA yielded a percent CPU time savings of 99.16 % for N2 relative
to CCSD(T,full)/aug-cc-pCV5Z. Interestingly, the use of CCSD(T,full)/aug-cc-pCV5Z to generate
the PES yielded larger absolute errors (10 ± 3) than ccCA (6 ± 4) for these molecules. The higher
absolute errors and large increase in CPU time between CCSD(T,full)/aug-cc-pCV5Z and ccCA
suggests that potentials generated with ccCA are more accurate when using a VSCF approach.

Table 6.1: Percent CPU Time relative to CCSD(T,full)/aug-cc-pCV5Z to generate all 17 grid points
of the PEC for select diatomics.

H2
LiH
CO
N2

B3LYP/aVTZ

98.81
99.89
99.97
99.98

ccCA
90.66
98.82
99.01
99.16

180

Table 6.2: Calculated frequencies in cm−1 for B3LYP/aug-cc-pVTZ, ccCA, and CCSD(T, full)/aug-
cc-pCV5Z for diatomics in Table 6.1.

B3LYP/aVTZ

4189
1372
2186
2420

H2
LiH
CO
N2

MAD ± STD

43 ± 29

ccCA
4157
1368
2143
2340

6 ± 4

CCSD(T,full)/aug-cc-pCV5Z

4155
1348
2128
2323

10 ± 3

Exp
4162.2
1360
2143
2330

6.3.2 H2O, CO2, NH3

For DFT potentials, both functional choice and basis set quality affected the accuracy of
stretching, bending, or
predicted frequencies and the type of vibration that is observed, i.e.
inversion. For ccCA, the range in deviation is primarily due to the type of vibration observed. In
Figure 6.2, the DFT potentials are obtained at the CBS limit (V∞Z and aV∞Z) for H2O and CO2
since there are only 3 and 4 vibrational normal modes, respectively, which leads to the calculation
of 3 and 7 vibrational mode-mode couplings, i.e. the surface describing two vibrations occurring
simultaneously. H2O, CO2, and NH3 yielded a larger range of deviations from experimental
frequencies than the diatomics, wavering from 2 to 294 cm−1 with DFT potentials and from 2
to 57 cm−1 for ccCA potentials. For NH3, the vibrational mode-mode coupling potentials were
generated at the triple-ζ level based on the cost-to-accuracy ratio for the diatomic molecules, the
observation of results from H2O and CO2, and the number of vibrational mode-mode coupling
potentials required for 6 normal modes. All vibrational mode-mode coupling potentials consist of
256 grid points.

181

Figure 6.2: MAD of vibrational frequencies for H2O, CO2, and NH3 using TPSS/VnZ (blue),
B3LYP/VnZ (green), TPSS/aVnZ (purple), B3LYP/aVnZ (red), and ccCA-S4 (black). For H2O
and CO2, n = ∞. For NH3, n=T.

In Figure 6.2, the MAD is the average across all frequencies for a particular molecule for each
method (B3LYP/VnZ, B3LYP/aVnZ, TPSS/VnZ, TPSS/aVnZ, and ccCA). For DFT, the choice
between aVnZ and VnZ altered the curvature of the potentials enough to yield larger variations in
the errors for calculated frequencies between the molecules with the exception of H2O. When aVnZ
was used in H2O, the error across all vibrations was the same as V∞Z, regardless of functional
choice. For CO2, using aV∞Z increased the mean error for all frequencies by 13 cm−1 relative
to using V∞Z for TPSS and decreased the mean error by 15 cm−1 for B3LYP. For NH3, using
aVTZ for the potentials decreased the mean error across all frequencies by ∼50 cm−1 relative to
using VTZ for both TPSS and B3LYP. This indicates that when using DFT to generate potentials
for vibrational calculations, augmented basis sets properly characterize both bending and stretching

182

behavior for small polyatomic species.

Functional choice was a larger factor in terms of general curvature of the potentials as B3LYP-
generated potentials yielded deviations for calculated frequency approximately 20-60 cm−1 lower
than TPSS-generated potentials. For H2O, CO2, and NH3, TPSS potentials inadequately described
both symmetric and asymmetric stretching modes with errors ranging from 70 to 225 cm−1,
whereas B3LYP potentials yielded errors in the range of 3 to 55 cm−1. In comparison, the bending
vibrational modes yielded errors ranging from 19 to 42 cm−1 for TPSS potentials. It is plausible
that in addition to parameterization for main group species, B3LYP potentials produced results
closer to experimental data because B3LYP includes exact exchange. The results infer that B3LYP
is preferred in generating potentials for calculating frequencies of polyatomic molecules when
utilizing DFT to generate potentials for vibrational motion.

With ccCA, the stretching modes for H2O yielded deviations within 2 cm−1 of experiment,
whereas the calculated bending mode was ∼20 cm−1 larger than the experimental frequency. The
difference of 20 cm−1 is most likely due to the weak coupling between the bending and stretching
modes and may be corrected through coupling all three vibrational modes together simultaneously.
For CO2, ccCA potentials did not properly characterize the vibrational motion of the out-of-plane
bending, and stretching normal modes with deviation of 40, 37, and 56 cm−1, respectively. This
may be in part due to the use of a standard cartesian coordinate system for displacing the molecule
in vibration whereas the deviations are generally lower when using a curvilinear coordinate system
as is the case for H2O. For NH3, ccCA potentials utilized for VCIPSI-PT2 predicted the inversion
barrier to within 10 cm−1 and N-H stretching modes within 15 cm−1. This analysis indicates
that when using a curvilinear coordinate system, ccCA potentials yield lower errors for stretching
modes opposed to bending modes, such as for H2O.

6.3.3 Hydrocarbons

This section highlights bonding character and its effect on vibrational potentials generated
with DFT and ccCA through analyzing C2H2, C2H4, and C2H6. Potential energy surfaces were

183

generated with basis set superposition error (BSSE)-corrected energies. With ccCA, only strongly
coupled vibrational modes are considered for C2H4 and C2H6, which is depicted via coupling maps
in the Appendix.

Figure 6.3: MAD of vibrational frequencies for C2H2, C2H4, and C2H6 using TPSS/VTZ (blue),
B3LYP/VTZ (green), TPSS/aVTZ (purple), B3LYP/aVTZ (red), and ccCA-S4 (black).

For the hydrocarbons, there is a noticeable improvement among frequencies predicted with
DFT potentials. TPSS provided a smaller absolute deviation in ethyne, while B3LYP was preferred
for ethene and ethane. The number of C-H bonds largely affected deviations from experimental
frequencies. Findings regarding the basis set choice were consistent with other molecules examined
thus far in that DFT/aVTZ potentials yielded lower errors for calculated frequencies relative to using
DFT/VTZ potentials when using VCIPSI-PT2 to compute the frequencies.

Over all observed frequencies, DFT potentials yield lower MADs from experimental frequencies

184

than ccCA. For C2H2, the difference in magnitude between total MADs of frequencies predicted
with DFT and ccCA potentials was approximately 5 cm−1. Yet for C2H4 and C2H6, this difference
increases to 10-15 cm−1. This is primarily due to the large deviations in the C-H stretching
vibrations around 3000 cm−1 for ccCA potentials as well as the number of strongly coupled
vibrational modes that include a non-IR active symmetric C-H stretching mode used for the
FASTVCI approach. This is also consistent with other composite strategies that utilize perturbative
anharmonic corrections.51 This would suggest that for molecules like C2H4 and C2H6, a softer
potential as generated via DFT for the non-IR active symmetric stretches is more characteristic of
the vibrational motion.

With ccCA potentials, the C-C stretching mode for C2H2, C2H4, and C2H6 yielded errors for
predicted frequencies with VCIPSI-PT2 of 17, 21, and 7 cm−1, respectively, all of which are IR
inactive. This would suggest that ccCA potentials are more adequate for describing vibrations
involving covalent single bonds than double or triple bonds. This is also supported in part by the
low deviations observed for H2O and NH3 and high deviations for CO2 observed for stretching
modes.

To correct for this discrepancy between frequencies generated with DFT potentials and ccCA
potentials, a multilevel approach can be utilized where the coupling elements from ccCA potentials
can be added to the frequencies generated via the uncoupled DFT potentials for each vibration. This
is denoted as DFT:ccCA in this work. To illustrate this concept for ethene, where the shape of the
single mode PECs for both the individual non-IR active C –– C and C – H symmetric stretches differ
between ccCA and DFT (Figure 6.5) , the frequencies are obtained where the mode-mode coupling
elements from ccCA potentials are applied to the single mode PECs generated with TPSS/VTZ.
As shown in Table 6.3, when using TPSS single mode PECs in tandem with ccCA mode-mode
coupling potentials, the error decreases to 24 cm−1 from 57 cm−1 when using ccCA single mode
and mode-mode coupling PECs and potentials, respectively. While the difference between using
TPSS and ccCA mode-mode coupling potentials with single mode PECs was only 1 cm−1, ccCA
mode-mode coupling potentials lowered the predicted frequency relative to using TPSS mode-

185

mode coupling potentials, which lowered the deviation as TPSS potentials overestimated the C-H
stretching modes (modes 9-12 in Table 6.3). Overall, a multilevel approach may be useful when
one method generates a single mode PEC more representative of the vibrational motion indicated
by the predicted frequency and for larger polyatomic systems.

Table 6.3: VCIPSI-PT2 frequencies using a combination of TPSS and ccCA for single mode and
vibrational mode-mode coupling potentials. The use of PECs/PESs is denoted as single:coupled.

Mode

Exp

1
2
3
4
5
6
7
8
9
10
11
12

826
949
943
1023
1236
1342
1444
1623
2989
3026
3103
3106

TPSS:PSS

All
821
976
963
1046
1229
1354
1455
1628
3030
2992
3087
3118

MAD

18

TPSS:TPSS
FASTVCI

ccCA-S4:ccCA-S4

FASTVCI

TPSS:ccCA-S4

FASTVCI

827
988
981
—
1238
1363
1468
1636
3050
3004
3113
3145

25

834
975
979
—
1233
1360
1462
1649
3143
3104
3221
3247

57

827
991
977
—
1239
1363
1467
1652
3043
2998
3105
3132

24

Mode 4 did not strongly couple to any other vibrational mode and hence excluded from FASTVCI calculations.

6.3.4 Aminophenol

For cis-3-aminophenol and trans-3-aminophenol, the NH2 torsion, 318.5 cm−1 and 329 cm−1,
respectively, and OH wagging vibrations, 307 cm−1 and 316 cm−1, respectively were examined.8
Each chosen vibration was coupled to all 38 other normal modes for DFT calculations. With ccCA,
only strongly coupled vibrational modes determined through the DFT calculations were included
in vibrational analysis. Coupling maps depicting strongly coupled vibrational modes are included
in the Appendix.

Calculated frequencies obtained with ccCA potentials yielded a lower deviation for experimental
NH2 torsion and OH wagging vibrational modes than frequencies obtained with DFT potentials.

186

This is shown in Table 6.4. B3LYP/aVTZ yields lower deviations than TPSS/aVTZ for both the NH2
torsion and OH wagging vibrational modes for both cis-3-aminophenol and trans-3-aminophenol.
For ccCA potentials the NH2 torsional mode was better characterized for cis-3-aminophenol with
an error of 5.5 cm−1 and the OH wagging motion was better characterized for trans-3-aminophenol
with an error of 1 cm−1. While the deviations obtained with ccCA potentials are lower than
10 cm−1 for trans-3-aminophenol, this approach can be utilized to spectroscopically differentiate
between aminophenol isomers that differ by the direction of the OH bond relative to the NH2
substituent.

Table 6.4: Vibrational frequencies predicted with VCIPSI-PT2 for selected vibrations of cis-3-
aminophenol and trans-3-aminophenol.

cis-3-aminophenol

NH2 torsion

OH wag

trans-3-aminophenol

NH2 torsion

OH wag

Exp

318.5
307

329
316

B3LYP/aVTZ

TPSS/aVTZ

ccCA-S4

322
300

333
330

345
296

351
325

313
275

322
317

When using a multilevel approach for the aminophenol isomers, utilizing ccCA single mode
PECs and B3LYP/aVTZ mode-mode coupling potentials (ccCA:B3LYP) for all mode-mode
couplings between the NH2 torsion and all other vibrations as well as between the OH wagging
and all other vibrations (75 total mode-mode couplings) yielded lower deviations than if only
ccCA mode-mode coupling potentials were used for cis-3-aminophenol for the few strongly
coupled modes isolated. The deviations for both the NH2 torsion and OH wagging decreased by 2
cm−1. For trans-3-aminophenol, the use of this multilevel approach increased the deviation by 2
cm−1. This may be in part due to how the predicted frequencies using B3LYP/aVTZ potentials
were higher than those predicted with ccCA potentials.

The computed infrared (IR) spectra uses the frequencies generated via VCIPSI-PT2 for
potentials generated with ccCA and DFT. To show the efficacy of the VCIPSI-PT2 predictions, the

187

generated spectra is compared to harmonic B3LYP/cc-pVTZ frequencies scaled by 1.0066 per the
study by Merrick et al.24 The intensities for all frequencies are based on the harmonic calculation.
For cis-3- and trans-3-aminophenol, the most intense peaks from harmonic intensities were near
600 cm−1, whereas the most intense experimental peaks were at 307 and 755 cm−1, respectively.
The computed spectra shows peaks in the 200-300 cm−1 range indicating torsional motion among
the C atoms in the ring. For the NH2 torsion and OH wagging motions for cis-3-aminophenol, the
VCIPSI-PT2 frequencies with ccCA potentials were more closely aligned to experiment than both
the scaled harmonic frequencies and VCIPSI-PT2 frequencies with potentials generated with
B3LYP/aVTZ. Even with intensities generated via the harmonic frequency calculation,
the
frequencies obtained via the ccCA potentials with VCIPSI-PT2 yielded a more accurate
representation of the spectra than the scaled harmonic frequencies. This would suggest that the
computed IR spectra would be more representative of the experimental IR spectra with a full
description of the mode-mode couplings.

188

Figure 6.4: Infrared spectra for cis-3-aminophenol (top) and trans-3-aminophenol (bottom) obtained
with VCIPSI-PT2 frequencies with ccCA potentials and B3LYP/cc-pVTZ harmonic frequencies
scaled by 1.0066. All intensities are from the harmonic frequency calculations. A Lorentz
broadening of 20 cm−1 was applied. The experimental frequencies and relative intensities from
Ref 8 are shown for comparison.

189

6.4 Conclusions

Overall, with ccCA potentials, the mean absolute deviation for calculated frequency from
experiment was lower than with DFT potentials. Functional choice had a more significant effect
on the predicted frequency than basis set for potentials generated with DFT. For diatomics, TPSS
potentials tended to properly characterize molecules that exhibit covalent triple bonds and B3LYP
potentials tended to yield lower absolute errors in frequency from experiment for polar molecules.
This trend held with the small polyatomics and hydrocarbons. When considering timings, ccCA
yielded lower deviations than CCSD(T,full)/aug-cc-pCV5Z with up to 99% CPU time savings for
diatomic molecules.

For H2O, CO2, and NH3, the predicted frequencies between potentials generated with ccCA and
DFT yielded similar errors across all vibrations. DFT predicted the bending behaviors better than
ccCA whereas ccCA predicted the stretching behaviors better than DFT. The use of a curvilinear
coordinate system yielded lower errors relative to using a standard Cartesian coordinate system as
indicated by the deviations observed for H2O and CO2.

For hydrocarbons, DFT characterized the C-H stretching behavior better than ccCA as the
errors for DFT potentials were lower than for ccCA potentials for C-H stretching modes. Trends
observed for all molecules examined indicate that B3LYP/aVTZ is the more favorable DFT method
and basis set combination in terms of generating a PES for vibrational motion when coupled with
VCIPSI-PT2 to compute frequencies. A multilevel approach that utilizes the single mode PECs
with DFT and the coupled vibrational modes generated with ccCA yields lower frequencies than
if only DFT were utilized. This is useful for expanding to larger polyatomic systems and for when
one method generates PECs that yield lower deviations than another, as was the case with the
hydrocarbons.

For aminophenol, the errors obtained with VCIPSI-PT2 were lower than those for scaled
harmonics, indicating the success of utilizing this approach to characterize specific vibrations
for polyatomic systems. The FASTVCI approach of only utilizing strongly coupled vibrational
modes saves computational resources when generating potentials with electronic structure methods.

190

B3LYP potentials serve as a good approximation for both the NH2 torsional mode and OH wagging
mode for both the cis-3-aminophenol and trans-3-aminophenol isomers. For cis-3-aminophenol,
the ccCA potentials yielded lower deviations for the NH2 torsion where the opposite is true for
trans-3-aminophenol.

Overall, ab initio composite strategies and in some cases DFT can be utilized for depicting
vibrational behavior of small polyatomic molecules present in the interstellar medium and can be
used in tandem with post-VSCF theory as a gauge for predicting anharmonic vibrations without the
harmonic frequencies with frequency scaling factors applied and perturbative corrections.

191

APPENDIX

192

APPENDIX

Table 6.5: Calculated frequencies of diatomic and small polyatomic molecules in cm−1 obtained
with ccCA potentials.

ccCA-P

ccCA-S3

ccCA-S4

ccCA-PS3

Exp40

Diatomics
H2
LiH
CO
N2
BF
HF
NH
NO+
OH
O2
SiO

4156
1372
2143
2340
1377
3960
3142
2355
3566
1591
1220

Small Polyatomics

H2O

CO2

NH3

1616
3660
3758

707
676
1296
2292

939
1613
1606
3343
3455
3458

4157
1371
2144
2340
1377
3960
3143
2355
3566
1592
1220

1616
3660
3759

708
676
1297
2293

939
1613
1606
3343
3455
3459

4162
1360
2143
2330
3570
2344
1379
3961
3126
1556
1242

1595
3657
3756

667
667
1333
2349

950
1626
1626
3336
3443
3443

4157
1370
2144
2341
1377
3961
3144
2356
3567
1593
1221

1616
3661
3760

708
676
1297
2293

939
1613
1606
3343
3456
3459

4157
1368
2143
2340
1376
3960
3142
2355
3566
1591
1220

1616
3660
3758

707
675
1296
2292

940
1613
1606
3342
3455
3458

193

Table 6.6: Calculated frequencies of diatomics in cm−1 obtained with TPSS/cc-pVnZ and
B3LYP/cc-pVnZ potentials.

VDZ

TPSS/cc-pVnZ
H2
LiH
CO
N2
BF
HF
NH
NO+
OH
O2
SiO

4193
1333
2100
2343
1240
3774
3051
2344
3371
1509
1151

B3LYP/cc-pVnZ
4148
H2
1347
LiH
CO
2187
2427
N2
1322
BF
3848
HF
NH
3034
NO+
2463
3473
OH
O2
1610
1194
SiO

VTZ

4196
1359
2111
2343
1331
3810
3085
2348
3451
1501
1196

4190
1367
2176
2423
1386
3918
3108
2459
3545
1595
1234

V5Z

4196
1360
2112
2340
1333
3813
3091
2352
3459
1510
1204

4188
1370
2188
2421
1380
3911
3122
2453
3554
1604
1247

V∞Z

Exp40

4191
1366
2111
2342
1332
3812
3092
2352
3460
1510
1200

4189
1538
2185
2423
1378
3905
3129
2453
3555
1603
1240

4162
1360
2143
2330
1379
3961
3126
2344
3570
1556
1242

4162
1360
2143
2330
1379
3961
3126
2344
3570
1556
1242

VQZ

4196
1357
2113
2338
1333
3813
3089
2349
3456
1509
1203

4187
1365
2186
2421
1382
3915
3116
2452
3551
1603
1247

194

Table 6.7: Calculated frequencies of selected diatomics in cm−1 with TPSS/aug-cc-pVnZ and
B3LYP/aug-cc-pVnZ potentials.

aVDZ
Molecule
TPSS/aug-cc-pVnZ
4181
H2
1328
LiH
CO
2086
2335
N2
1240
BF
HF
3774
3051
NH
NO+
2344
3424
OH
O2
1514
1156
SiO

B3LYP/aug-cc-pVnZ
H2
LiH
CO
N2
BF
HF
NH
NO+
OH
O2
SiO

4137
1342
2160
2416
1282
3880
3092
2444
3530
1612
1197

aVTZ

aVQZ

aV5Z

aV∞Z

Exp40

4195
1361
2108
2341
1331
3810
3085
2348
3453
1510
1203

4189
1372
2181
2420
1375
3902
3118
2448
3545
1605
1247

4196
1360
2111
2339
1333
3813
3088
2349
3458
1500
1201

4188
1374
2186
2421
1379
3907
3122
2452
3552
1594
1243

4196
1359
2112
2340
1333
3813
3090
2352
3459
1511
1205

4189
1370
2187
2422
1380
3908
3123
2453
3554
1605
1247

4195
1359
2112
2342
1332
3812
3092
2352
3460
1510
1205

4190
1226
2187
2423
1379
3961
3124
2453
3570
1604
1242

4162
1360
2143
2330
1379
3961
3126
2344
3570
1556
1242

4162
1360
2143
2330
1379
3961
3126
2344
3570
1556
1242

Table 6.8: Calculated vibrational frequencies for H2O and CO2 in cm−1 utilizing VCIPSI-PT2
with TPSS/cc-pVnZ potentials.

Molecule Mode

H2O

CO2

1
2
3

1
2
3
4

VDZ
1614
3515
3612

634
634
1334
2288

VTZ
1615
3540
3630

641
641
1348
2302

VQZ
1614
3546
3635

643
643
1350
2304

195

V5Z
1614
3547
3637

642
642
1350
2303

V∞Z
1614
3548
3638

642
642
1349
2303

Exp41
1595
3657
3756

667
667
1333
2349

Table 6.9: Calculated vibrational frequencies for H2O and CO2 in cm−1 utilizing VCIPSI-PT2
with B3LYP/cc-pVnZ potentials.

Molecule Mode

H2O

CO2

1
2
3
1
2
3
4

VDZ
1643
3576
3670
650
650
1381
2368

VTZ
1613
3635
3723
667
667
1399
2365

VQZ
1605
3641
3730
670
670
1401
2357

V5Z
1599
3645
3734
670
670
1401
2353

V∞Z
1599
3645
3734
669
669
1399
2351

Exp41
1595
3657
3756
667
667
1333
2349

Table 6.10: Calculated vibrational frequencies for H2O and CO2 in cm−1 utilizing VCIPSI-PT2
with TPSS/aug-cc-pVnZ potentials.

Molecule Mode

H2O

CO2

1
2
3
1
2
3
4

aVDZ
1614
3515
3612
591
591
1311
2531

aVTZ
1615
3540
3630
628
628
1319
2466

aVQZ
1614
3546
3635
636
636
1319
2444

aV5Z
1614
3547
3637
636
636
1318
2439

aV∞Z
1614
3548
3638
636
636
1318
2438

Exp41
1595
3657
3756
667
667
1333
2349

Table 6.11: Calculated vibrational frequencies for H2O and CO2 in cm−1 utilizing VCIPSI-PT2
with B3LYP/aug-cc-pVnZ potentials.

Molecule Mode

H2O

CO2

1
2
3
1
2
3
4

aVDZ
1598
3626
3724
660
660
1383
2336

aVTZ
1599
3635
3725
669
669
1398
2349

aVQZ
1599
3642
3731
670
670
1401
2353

aV5Z
1599
3644
3733
669
669
1400
2353

aV∞Z
1599
3644
3733
669
669
1400
2352

Exp41
1595
3657
3756
667
667
1333
2349

196

Figure 6.5: Single mode potential energy curves for vibrational modes 8 (left) and 10 (right) of
ethene (C=C and C-H symmetric stretches) generated with ccCA (black) and TPSS/VTZ (red).

Table 6.12: Calculated vibrational frequencies for NH3 in cm−1 utilizing VCIPSI-PT2 with both
TPSS and B3LYP potentials with the VTZ and aVTZ basis sets.

Mode

1
2
3
4
5
6

B3LYP/VTZ

994
1632
1653
3391
3474
3441

B3LYP/aVTZ

886
1576
1596
3327
3441
3450

TPSS/VTZ

1039
1659
1668
3418
3619
3668

TPSS/aVTZ

960
1598
1659
3230
3357
3373

Exp41
950
1626
1626
3336
3443
3443

197

Figure 6.6: Vibrational coupling map for ethene (left) and ethane (right). The vibrational mode
couplings shown in black indicate strongly coupled modes that were used for all FASTVCI
approaches using ccCA.

Measures to reduce the computational cost includes screening out weakly coupled pair-wise
coupling interactions via a threshold established from calculating the coupling strength (Equation
2.66), which can be calculated with only the VSCF potential.23,50 By removing non-essential
vibrational coupling elements from the potential, a FASTVSCF approach is attained. By utilizing
this approach, the computational time to generate all vibrational potential energy surfaces is reduced
by approximately a factor of 6 (12 out of 78 coupling modes were calculated with ccCA) for ethene
as only the potential energy surfaces for all of the shaded squared were generated with ccCA. For
ethene, modes 9-12 characterize the C-H stretching modes (both symmetric and asymmetric). In
general, symmetric and asymmetric vibrations are strongly coupled largely due to the effect each
type of vibration has on the other.

In contrast to ethene, which exhibited stronger coupling modes for the C-H stretches at
approximately 3000 cm−1, ethane only had one strong coupling mode in this region, which is the
coupling between C-H symmetric and C-H asymmetric stretches. Other modes that were strongly
coupled include the rotational barrier of ethane around the C-C bond (mode 1) and C-C stretching

198

(mode 9). Including the coupling strength screened out 67 vibrational mode, leaving 11 shown in
black in Figure 6.6. This effectively reduced the computational cost of generating the full 2D
surface by approximately 93%.

Figure 6.7: Vibrational coupling map for cis-3-aminophenol (left) and trans- 3-aminophenol (right).
The vibrational mode couplings shown in black indicate strongly coupled modes that were used for
all FASTVCI approaches using ccCA.

For cis-3-aminophenol, only the OH wagging (mode 3) and NH2 torsion (mode 5) were analyzed.
Therefore, all vibrations that coupled to these modes were included. In terms of coupling strength,
only 6 of the 78 mode-mode coupling potentials were analyzed with ccCA, again reducing the cost
by approximately 92%. For trans-3-aminophenol, only the OH wagging (mode 4) and NH2 torsion
(mode 5) were analyzed. Therefore, all vibrations that coupled to these modes were included. In
terms of coupling strength, only 12 of the 78 mode-mode coupling potentials were analyzed with
ccCA, reducing the cost of generating the full 2D surface by approximately 84%.

199

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204

CHAPTER 7

CHARGE STABILIZATION OF HIGH POTENTIAL ZINC
PORPHYRIN-FULLERENE VIA AXIAL LIGATION OF

TETRATHIAFULVALENE

7.1 Introduction

Sustainable production of electricity and fuel using abundant solar photons is one of the most
highly researched topics in modern science.1–12 Often, the design of light energy harvesting
materials follows the concepts developed by Mother Nature in bacterial and green plant
photosynthetic systems.13–15 The primary photochemical events in natural photosynthesis
involves capture capturing and funneling of sun light by a group of well-organized chromophores
called ‘antenna’ systems and promoting electron transfer using the funneled light into the
‘reaction center’ where a cascade of electron transfer events occurs leading to the generation of
long-lived charge separated states. Over the last two to three decades, early photo-events of
natural photosynthesis have been mimicked by building donor-acceptor systems to visualize
energy and electron transfer or a combination of these two events.16–45 One strategy used in
building donor-acceptor systems that are capable of producing high-energy charge separated states
include choosing donors that are difficult to oxidize and acceptors that are difficult to reduce.
Under these conditions, the stored energy in the charge separated state is equivalent to the
potential difference between the oxidation and reduction potentials of the donor and acceptor,
respectively. However, challenges exist to accomplish this goal where the excited state energy
from either the donor or the acceptor may not be sufficient to drive the electron transfer process in
an energetically feasible fashion.1

Collaborators have synthesized a donor-acceptor dyad, (F15P)Zn – C60, capable of generating
1This chapter is reprinted from Obondi, C. O.; Lim, G. N.; Jang, Y.; Patel, P.; Wilson, A. K.;
Poddutoori, P. K.; D’Souza, F. J.Phys. Chem. C 2018, 122, 13636–13647 with permission of the
American Chemical Society.

205

charge separated state carrying an energy of 1.70 eV (see Figure 7.1 for structure of the dyad).46
In that study, the electron donor zinc porphyrin was functionalized with meso-pentafluorophenyl
substituents that made the zinc porphyrin difficult to oxidize by 0.43 eV compared to simple
zinc porphyrins. The singlet excited energy of 1(F15P)Zn· (= 2.21 eV) was sufficient to drive
the electron transfer process. The lifetime of the charge separated state was persistent for about
50-60 ns.
In this chapter, to prolong the lifetime of the charge separated state, collaborators
developed supramolecular triads using a hole transporting tetrathiafulvalene, TTF, linked via the
well-known metal-ligand axial coordination approach.47 Here, the TTF was functionalized with
either pyridine or phenylpyridine coordinating ligands. Supramolecular triad formation including
binding constants and stoichiometry of the complexes were determined by spectroscopic methods.

7.2 Computational Contributions and Analysis

dyad

and

triads

((F15P)Zn – C60)

Computational support provided by the author supplemented the synthesis of
(C60 – (F15P)Zn:Py-TTF

these
supramolecular
and
C60 – (F15P)Zn:Py-phTTF) via modeling the molecular electrostatic potential and frontier
orbitals. The M06-2X/6-31G* method and basis set combination was based on qualitative
modeling indicative of intramolecular charge transfer via the relative location of the HOMO and
LUMO.48–50 M06-2X was chosen based on its implementation for main group and transition
metal thermochemistry.48 The Pople-style 6-31G* basis set was used based on its small size and
availability for all atoms present in this compound.49,50 The molecular electrostatic potential
(MEP) for all of complexes were modeled on a scale of strong attractive potential (red) to a strong
repulsive potential (blue) with respect to a positive test charge. This provides insight into the
potential binding behavior of these systems.

The geometry and electronic structures of the (F15P)Zn – C60 dyad, and C60 – (F15P)Zn:Py-TTF
and C60 – (F15P)Zn:Py-phTTF triads were probed using the hybrid-metal Minnesota functional
M06-2X with 54% exact exchange and the 6-31G* basis set using Gaussian09.48–51 Figure 7.1
depicts molecular electrostatic potential (MEP) maps and frontier HOMO and LUMO for the

206

optimized structures. In the case of the dyads, the frontier HOMO was on the (F15P)Zn and LUMO
on the C60 making them the donor and acceptor sites, respectively. Interestingly, for the triad, the
HOMO was shifted to the TTF site without altering the location of the LUMO, which is attributed
to the easier oxidation of TTF over (F15P)Zn. HOMO-1 occupied the (F15P)Zn for the triads. It
may be pointed out here that the density of the dyad was not affected by the addition of TTF ligand
except at the porphyrin center where the potential was neutral, which indicates that the central
metal is fully coordinated and not likely to bind an additional ligand. The estimated center-to-
center distances between Zn and C60 in the dyad and triads were ~17.5 Å while these distances
between Zn and TTF were ~18.0 and ~17.7 Å, respectively, in the case of C60 – (F15P)Zn:Py-TTF
and C60 – (F15P)Zn:Py-phTTF triads.

Figure 7.1: MO6-2X/6-31G* molecular electrostatic potential maps, and the frontier HOMO and
LUMO of the optimized structures of (a) (F15P)Zn-C60 dyad and (b) C60-(F15P)Zn:Py-phTTF
triad. The isovalue used for the MO depictions was 0.02 while the density value used was 0.0004.

207

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CHAPTER 8

SAMPL6 HOST-GUEST CHALLENGE: BINDING FREE ENERGIES VIA A

MULTISTEP APPROACH

8.1 Introduction

Tremendous advances in technological capabilities have enabled computational approaches to
be applied to discern a broad range of physical, chemical, and biological phenomena across scales in
molecular science.1–6 With emphasis on molecular design, computational approaches have found
great utility towards innovation in drug discovery. Considering the time and cost of the drug
pipeline, from the discovery process to market, in silico biophysical methods serve an important
role in expediting and reducing the cost of the discovery process, facilitating the identification,
optimization, and refinement of potential drug candidates and providing comprehensive insight
into the mechanism of action and structure-property relationships at the atomic level that are
ultimately critical to a drug’s efficacy.1

7–12

In computational strategies towards structure-based design, an important step is the prediction
of probable conformations of a ligand bound to the host. To identify better possible candidate
binding modes, they can be ranked via scoring functions and further evaluated via molecular
simulation and free energy calculations. From free energy calculations, selectivity profiles may be
constructed not only to determine binding affinities but also to provide understanding into how the
ligand recognizes its host.

Because of the complexity that occurs in ligand-bound protein systems, relatively smaller
representative models such as polymer-based host-guest systems are used to assess free energy
methods.13–18 Although host structures selected to represent proteins are typically much smaller
1This chapter is reprinted from Eken, Y.; Patel, P.; Díaz, T.; Jones, M. R.; Wilson, A. K.
SAMPL6 Host – Guest Challenge : Binding Free Energies via a Multistep Approach. J. Comput.
Aided. Mol. Des. 2018, 32 (10), 1097–1115. with permission of the Springer International
Publishing.

213

than proteins, they are large enough to possess a cavity or binding pocket that allows non-covalent
binding of multiple guest molecules. The advantage of using host-guest systems for assessing free
energy methods is that they tend to be more rigid and symmetric than proteins, which results in
fewer conformations that need to be sampled.19–23 Even in the representation of proteins by more
simplistic models, modeling binding free energies for these smaller models is challenging since no
clear "best" computational chemistry approach has been identified; efforts are needed to better
resolve strategies towards predictions of binding free energies. Statistical Assessment of the
Modeling of Proteins and Ligands (SAMPL) blind challenges provide a unique platform to
validate available methods and stimulate the development of new methods for quantitative
predictions.13,16,18,24–26 In these challenges, binding affinities and other physicochemical
properties are predicted, using computational models without
the benefit of insight from
experiment; they are then later compared to unpublished experimental measurements that allow
the comparison of different computational prediction methods.

While classical molecular dynamics (MD) methods are commonly used to investigate
host-guest interactions, molecular mechanics (MM) force fields result in a limited treatment of
effects resulting from polarization, charge transfer, and many body effects which can impact the
description of properties such as binding free energies.9,27–31 To better account for these effects,
quantum mechanical (QM) approaches, which are more costly, are commonly used in drug
discovery research,9,32 and have been used in previous SAMPL competitions.17,33–35 For
example, in the SAMPL5 competition for host-guest binding, Caldarau et al.33 used DFT-D3 and
DLPNO-CCSD(T) to predict the binding energies for octa-acid (OA) host-guest systems. In this
approach, they used TPSS-D3/def2-SVP optimized structures and host structures are constrained
during MD simulations to reduce the flexibility of the host and limit the structural distortions
resulting from the repulsion between the negative charge of the ligands and the large negative
charge of the OA hosts. This approach yielded binding energies approximately 12.0 kcal
mol−1 greater than the experimental binding affinities, with a low correlation coefficient (r2 ≈ 0),
and a statistically insignificant Kendall’s rank correlation coefficient (τ ≤ 0.20) for all attempts for

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the host-guest systems in the SAMPL5 blind challenge due to incorrect representative structures,
not sampling enough conformational binding positions for
ligands, and thermochemical
corrections that yielded up to a 7.2 kcal mol−1 difference depending on the method of choice.
This performance demonstrates the limited sampling capabilities of current QM methods
compared to MD methods, obtained representative structures, as well as thermodynamic and
solvation corrections.

Contrary to this, in the SAMPL4 competition for host-guest binding, Mikulskis et al.35 were
successful with both MM- and QM-based approaches for OA hosts with mean absolute deviations
(MADs) less than 2.0kcal mol−1. Their MM approach, which utilized free energy perturbation
(FEP) calculations, yielded MADs of approximately 1.0 kcal mol−1 while their QM approaches
with DFT-D3 optimized structures yielded MADs of approximately 1.0-2.0 kcal mol−1 depending
on the implementation of a solvent in the calculations, i.e. no solvent, implicit solvent, or a
combined implicit-explicit solvent. However, the combination of FEP and DFT-D3 did not yield
favorable results due to the large difference between the MM and DFT potential energy functions.
Sure et al.34 provided another successful attempt at using DFT-D3 for the SAMPL4 competition
for host-guest binding of a macrocyclic cucurbit[7]uril host by optimizing the geometry at the
TPSS-D3/def2-TZVP level of theory after pre-optimizing possible binding scenarios with the
HF-3c semiempirical method. These optimizations were followed by single point calculations
using PW6B95-D3/def2-QZVP with the g- and f-functions for non-hydrogen and hydrogen atoms
removed, respectively, with the COSMO-RS implicit solvent model, which yielded a MAD of 2.0 ±
0.5kcal mol−1. These two studies highlight that for the SAMPL4 competition, host-guest structure
optimization and higher-level MM-based approaches like FEP can be vital in characterizing correct
binding interactions at the QM level.

In this work, efforts in MD and QM methods are combined to predict binding affinities for
fourteen ligands to a macrocyclic cucurbit[8]uril host19,21,22,36 and eight ligands to two variants of
the OA deep-cavity cavitands.20,23 Using MD simulations to obtain representative structures, MM-
and QM-based methods are utilized to predict binding free energies. Within the QM methods, the

215

use of a resolution-of-the-identity (RI) approximation designed for larger molecules,37 Grimme’s
D3 atom-pairwise dispersion corrections with Becke-Johnson damping,38 and truncated correlation
consistent basis sets for the hydrogen atoms39 are evaluated to probe how different electronic
structure approaches that reduce the computational cost contribute to predicting binding affinities.
Insights into what strategies are more favorable for host guest-binding will help to build a framework
for predicting host-guest binding affinities using QM approaches.

8.2 Methods

8.2.1 System Preparation and Simulation Protocol

The initial structures for the guest molecules are shown in Figures 8.1 and 8.2, and the three
host molecules, cucurbit[8]uril (CB8), octa-acid (OA), and tetramethyl octa-acid (TEMOA), are
shown in Figure 8.3. These molecules were issued with the SAMPL6 challenge dataset were used
to generate the host-guest systems. The CB8 molecule has no formal charge whereas the octa-acids
(OA/TEMOA) have eight deprotonated carboxylic acid groups and thus a formal charge of -8. Even
though OA and TEMOA are water-soluble structurally similar deep-cavity cavitands, the TEMOA
host has four methyl groups in place of four hydrogen atoms present in the OA host located on the
upper rim of the cavitand that enclose the hydrophobic binding pocket.

Initial binding poses of guest molecules binding to the host were generated and refined through
a ∆G scoring function.40–45 Subsequent molecular dynamics simulations were then carried out
in Amber16.7 to relax the host-guest systems in aqueous solution.46 An MM-based approach
(MMPBSA) was used to calculate the binding free energies at the MM-level, which is a standard
level of theory when dealing with drug binding interactions.47 This portion was done by co-authors
Yiğitcan Eken, Thomas Díaz, and Michael Jones.

216

Figure 8.1: Guest molecules for the cucurbit[8]uril (CB8) host.

217

Figure 8.2: Guest molecules for the octa-acid (OA) and tetramethyl octa-acid (TEMOA) hosts.

Figure 8.3: Host molecules: cucurbit[8]uril (CB8), octa-acid (OA), and tetramethyl octa-acid
(TEMOA).

218

8.2.2 Quantum Mechanical Calculations

All quantum mechanical calculations were done by the author. The individual structures
generated from the clustering of MD trajectories, shown in Figures 8.4-8.6, for each host-guest
complex were used for all quantum chemical calculations. The host and guest molecules were
analyzed with the same geometry as from the complex. The thermal corrections for all molecules
were calculated at the HF/6-31G(d) level of theory in Gaussian 16 and the vibrational contributions
were scaled by 0.8953.48 Single point energies were obtained using ORCA 4.049 with the B3PW91
density functional50–52 since B3PW91 has been shown to properly treat long-range covalent
interactions. In the treatment of the exact exchange in the functional, the RIJCOSX approximation37
was used with the def2 auxiliary basis set53 to reduce the computational cost associated with the
number of atoms in the host-guest complex since the RIJCOSX approximation has been shown to be
five times as efficient for molecules of similar size to the host-guest systems. To mimic the aqueous
solution, the SMD implicit solvation model54 was used with water (ε = 78.4) as the implicit solvent.
Grimme’s D3 dispersion correction with Becke-Johnson damping was used to investigate long-range
covalent interactions as the inclusion of D3 dispersion improves intermolecular interaction energies
predicted with DFT.34,38,55,56

The cc-pVnZ57 basis sets were used for all single point calculations (see Section 2.2.2 for
reasons).58–61 Knowing the CBS limit, which removes basis set incompleteness error, the error
for the property of interest, i.e. binding free energy, only corresponds to the intrinsic error of the
chosen QM method. Therefore, to extrapolate to the Kohn-Sham limit for DFT methods, analogous
to the CBS limit for wavefunction-based methods, the cc-pVnZ basis sets were used (n = D, T )
with the following extrapolation scheme proposed by Jensen

E(lmax) = ECBS + A(lmax + 1)e−B

√

ns

(8.1)

where lmax is the maximum angular momentum function in the basis set and ns is the number
of s functions in the basis set.62 The B-parameter was set to 5.5 in agreement with Jensen for
use as a two-point extrapolation scheme. Due to the abundance of weak molecular interactions in

219

biomolecules, the calculated binding energies were counterpoise-corrected before the extrapolations
were performed on each host, guest, and host-guest complex.63,64

Additional electronic structure modeling techniques were applied to the CB8 host-guest systems
to examine the impact of approximations on the binding free energy. Targeting reduction in
computational time, the cc-pVnZ basis sets were truncated via the removal of higher angular
momentum basis functions for hydrogen atoms. This has been shown to reduce the computational
time by approximately 42.9% and 57.8% when removing 1 d function from the cc-pVTZ basis
set, denoted as cc-pVTZ(-1d), and 2 d functions and 1 f function from the cc-pVQZ basis set,
denoted as cc-pVQZ(-1f2d), respectively, and yielded the results closest to the atomization energies
generated with the full basis sets at the complete basis set limit.39

Binding free energies calculated with and without the use of the resolution-of-the-identity (RI)
approximation were examined to gauge how the RI approximation, which leads to a reduction in
CPU time, affects the accuracy. To characterize the ionic strength of the solution used in experiment,
the dielectric constant for the implicit water solvent was also altered from 78.4 for pure water to
76.4 given the concentration of the sodium chloride solution used in the MD simulations and the
experimentally determined relation between the concentration of an ionic solution and the dielectric
constant.65

8.3 Results

The binding free energies submitted as part of the SAMPL6 competition are shown in Tables
8.1-8.3 for CB8, OA, and TEMOA host-guest systems, respectively. For each host-guest complex,
statistical measurements were used to gauge the effectiveness of each of the three methods, which
are MMPBSA, RI-B3PW91-D3, and RI-B3PW91, in predicting experimental binding free energies.
These include the mean absolute error (MAE), the root mean square error (RMSE), Kendall’s Tau
(τ) rank correlation coefficient, which measures how well a method ranked calculated binding free
energies relative to experimental binding free energies where τ values closer to one correspond to
increased qualitative accuracy of the prediction, and the correlation coefficient (r2). To demonstrate

220

there is no correlation in ranking between the calculated binding free energies and the experimental
binding free energies, τ values are compared against τcrit, a cutoff value obtained through a table
of critical values generated by Monte Carlo simulations of a τ distribution, which is similar to the
normal Z distribution used to reject the null hypothesis.66,67

8.3.1 CB8

The binding free energy predictions for the CB8 host with the three methods submitted were
compared to experiment (Table 8.1). The predicted values were significantly more negative than
experimental binding free energies with an MAE of 16.69, 33.58, and 15.54 kcal mol−1 for
MMPBSA, RI-B3PW91-D3, and RI-B3PW91, respectively.

When the binding affinities of the guests to CB8 are ranked from the lowest to the highest
binding affinity, MMPBSA did not correctly rank any of the systems but predicted CB8-G12 to
have a stronger binding affinity relative to the other complexes, which correlates to experiment
well. RI-B3PW91-D3 correctly ranked CB8-G2 as the tenth strongest bound host-guest complex
and predicted that CB8-G12 was more tightly bound relative to the other CB8 host-guest systems.
RI-B3PW91 correctly ranked CB8-G6, CB8-G2, CB8-G1, and CB8-G3 as fifth, tenth, eleventh,
and fourteenth, respectively, while the remaining systems were ranked incorrectly. Unlike both
MMPBSA and RI-B3PW91-D3, RI-B3PW91 predicted CB8-G12 to have a lower binding affinity
relative to the other CB8 host-guest systems.

221

Table 8.1: Binding free energies for the CB8 host-guest complexes.

Complex
CB8-G0
CB8-G1
CB8-G2
CB8-G3
CB8-G4
CB8-G5
CB8-G6
CB8-G7
CB8-G8
CB8-G9
CB8-G10
CB8-G11
CB8-G12
CB8-G13

MAE
RMSE

τ
r2

Exp

-6.69 ± 0.05
-7.65 ± 0.04
-7.66 ± 0.05
-6.45 ± 0.06
-7.80 ± 0.04
-8.18 ± 0.05
-8.34 ± 0.05
-10.00 ± 0.10
-13.50 ± 0.04
-8.68 ± 0.08
-8.22 ± 0.07
-7.77 ± 0.05
-9.86 ± 0.03
-7.11 ± 0.03

MMPBSA
-29.4 ± 0.3
-31.5 ± 0.3
-25.6 ± 0.3
-34.2 ± 0.5
-30.8 ± 0.3
-18.6 ± 0.3
-19.8 ± 0.2
-17.6 ± 0.4
-30.4 ± 0.2
-19.9 ± 0.5
-19.6 ± 0.3
-17.5 ± 0.4
-31.5 ± 0.4
-25.4 ± 0.3
16.69 ± 0.33a
17.80 ± 0.76b

-0.19
0.00

RI-B3PW91-D3

-49.89
-57.22
-36.86
-44.53
-68.09
-35.92
-31.95
-14.90
-50.34
-37.07
-39.30
-25.75
-62.05
-44.04

RI-B3PW91

6.75
12.70
10.34
26.61
-11.11
2.39
1.26
18.09
4.49
-2.46
0.61
-1.07
15.00
0.17

34.29
36.99
-0.14
0.00

14.88
17.26
0.05
0.00

The mean absolute error (MAE), root mean square error (RMSE), Kendall’s Tau (τ), and r2 are shown. These results correspond
to those submitted for the competition.
aThe uncertainty reported for MAE is the average of the absolute uncertainties.
bThe uncertainty reported for RMSE is the uncertainty of the RMSE with the experimental and calculated uncertainties.

222

Figure 8.4: Structures of the CB8 guest molecules inside the binding pocket generated from the
clustering analysis.

223

8.3.2 OA

Table 8.2: Binding free energies for the OA host-guest complexes.

Complex
OA-G0
OA-G1
OA-G2
OA-G3
OA-G4
OA-G5
OA-G6
OA-G7

MAE
RMSE

τ
r2

Exp

-5.68 ± 0.03
-4.65 ± 0.02
-8.38 ± 0.02
-5.18 ± 0.02
-7.11 ± 0.02
-4.59 ± 0.02
-4.97 ± 0.02
-6.22 ± 0.02

MMPBSA
-12.6 ± 0.2
-11.6 ± 0.1
-18.3 ± 0.2
-10.0 ± 0.2
-17.0 ± 0.2
-9.1 ± 0.2
-11.3 ± 0.2
-11.4 ± 0.1
6.80 ± 0.2a
7.07 ± 0.4b

0.64
0.84

RI-B3PW91-D3

-41.36
-40.67
6.54
-47.94
-48.19
-38.40
-43.19
-47.37

RI-B3PW91

-16.57
-17.15
44.53
-17.62
-13.49
-16.42
-23.31
-23.78

35.46 [38.49]
36.41 [38.52]
0.29 [0.71]
0.44 [0.52]

17.86 [12.85]
22.51 [13.39]
-0.21 [0.05]
0.60 [0.03]

The mean absolute error (MAE), root mean square error (RMSE), Kendall’s Tau (τ), and r2 are shown. Bracketed values indicate
the values after the removal of the statistical outlier (OA-G2). These results correspond to those submitted for the competition.
aThe uncertainty reported for MAE is the average of the absolute uncertainties.
bThe uncertainty reported for RMSE is the uncertainty of the RMSE with the experimental and calculated uncertainties.

The three sets of submitted binding free energy predictions for OA are reported in Table 8.2. All
values predicted using MMPBSA were significantly more negative than experimental measurements
with an MAE of 6.8 ± 0.2kcal mol−1. When ranking the binding affinities of the guest to the
host from lowest to highest binding affinity, MMPBSA correctly placed OA-G2, OA-G4, OA-G6,
OA-G5 as first, second, sixth, and eighth, respectively. The other systems were not ranked correctly;
OA-G0, OA-G1, OA-G7 and OA-G3 ranked third, fourth, fifth, and seventh, respectively, whereas
experimentally ranked fourth, seventh, third, and fifth, respectively.

For RI-B3PW91-D3 and RI-B3PW91, the binding free energy predicted for OA-G2 was
determined as a statistical outlier with 99% confidence, visualized in Figure 8.8, using Dixon’s
Q-Test.68 When the statistical outlier (OA-G2) was excluded from the RI-B3PW91-D3 set, the
MAE, RMSE, Kendall’s Tau (τ), and the correlation coefficient (r2) increased from 35.46 to
38.39kcal mol−1, 36.41 to 38.52kcal mol−1, 0.29 to 0.71, and 0.44 to 0.52, respectively. When
the binding free energy for OA-G2 was excluded from the set of binding free energies obtained

224

with RI-B3PW91, the MAE, RMSE, and r2 decreased from 17.87 to 12.85kcal mol−1, 22.51 to
13.39kcal mol−1, and 0.60 to 0.03, respectively, as shown in Table 8.2.
In Figure 8.7b, the
statistical outlier was removed, which improved and worsened the linear regression model
comparing experiment to RI-B3PW91-D3 and RI-B3PW91, respectively. With the exclusion of
OA-G2, ranking the binding affinities from lowest to highest, RI-B3PW91-D3 correctly ranked
OA-G4, OA-G1, and OA-G5, as first, sixth, and seventh, respectively, while RI-B3PW91 did not
correctly ranked any of the systems.

Figure 8.5: Structures of the OA guest molecules inside the binding pocket generated from the
clustering analysis.

225

8.3.3 TEMOA

Table 8.3: Binding free energies for the TEMOA host-guest complexes.

Exp

-6.06 ± 0.02
-5.97 ± 0.04
-6.81 ± 0.02
-5.60 ± 0.04
-7.79 ± 0.02
-4.16 ± 0.02
-5.40 ± 0.03
-4.13 ± 0.02

Complex

TEMOA-G0
TEMOA-G1
TEMOA-G2
TEMOA-G3
TEMOA-G4
TEMOA-G5
TEMOA-G6
TEMOA-G7

MAE
RMSE

τ
r2

MMPBSA
-12.0 ± 0.2
-11.3 ± 0.2
-19.3 ± 0.2
-8.3 ± 0.2
-19.2 ± 0.3
-6.1 ± 0.2
-10.4 ± 0.2
-6.8 ± 0.3
5.9 ± 0.2a
7.0 ± 0.5b

0.79
0.86

RI-B3PW91-D3

-43.75
-41.98
-51.23
-43.56
-51.98
-37.04
-41.05
-45.98

38.83
39.03
0.57
0.55

RI-B3PW91

-12.80
-10.18
-7.22
-15.29
-12.39
-10.66
-16.94
-10.29

6.23
7.00
-0.14
0.00

The mean absolute error (MAE), root mean square error (RMSE), Kendall’s Tau (τ), and r2 are shown. These results correspond
to those submitted for the competition.
aThe uncertainty reported for MAE is the average of the absolute uncertainties.
bThe uncertainty reported for RMSE is the uncertainty of the RMSE with the experimental and calculated uncertainties.

TEMOA is structurally different from OA because of the substitution of four hydrogens around
the portal to the binding pocket of OA with four methyl groups. While the same guests bound to
TEMOA and OA with similar binding energies, G7 weakly binds to TEMOA relative to the other
guests whereas it binds stronger to OA experimentally. Binding free energy predictions using the
submitted methods for the TEMOA host are reported in Table 8.3. Similar to OA, all three methods
overestimated the binding free energies relative to experiment. RI-B3PW91-D3 overestimated the
binding free energies with an MAE of 38.83kcal mol−1. Of the three methods considered, the
MMPBSA method yielded better binding free energies, both quantitatively (MAE of 5.9 ± 0.2kcal
mol−1) and qualitatively (τ = 0.79), than the QM-based calculations. MMPBSA ranked TEMOA-
G0 and TEMOA-G1 as the third and fourth strongest bound complexes, respectively. Additionally,
MMPBSA predicted that TEMOA-G4 and TEMOA-G2 were the most tightly bound complexes
while TEMOA-G7 and TEMOA-G5 were the most loosely bound complexes. RI-B3PW91-D3
correctly predicted that TEMOA-G4, TEMOA-G2, and TEMOA-G3 were the first, second, and

226

fifth most tightly bound complexes, respectively. Like MMPBSA, RI-B3PW91-D3 predicted that
TEMOA-G5 was a weakly bound host-guest complex relative to the other TEMOA host-guest
systems. RI-B3PW91 correctly predicted TEMOA-G0 as the third strongest bound host-guest
complex and yielded the lowest deviation from experiment (0.41kcal mol−1) for TEMOA-G2.

Figure 8.6: Structures of the TEMOA guest molecules inside the binding pocket generated from
the clustering analysis.

8.3.4 Quantum Mechanical Calculations

The CB8 host-guest systems were used to probe approaches for improving the binding free
energy prediction. Specifically, the effects of 1) utilizing truncated correlation consistent basis sets
as opposed to standard correlation consistent basis sets; 2) utilizing traditional DFT calculations

227

(neglecting the RI approximation); and 3) modifying the dielectric constant used in the continuum
solvation model to reflect the ionic strength of the solution used in experiment were examined.

As shown in Tables 8.1-8.3, for CB8, OA without the statistical outlier (OA-G2), and TEMOA,
the MAE, and RMSE increased by approximately 19.4, 25.5, and 32.6 kcal mol−1 when using
Grimme’s D3 dispersion with RI-B3PW91, respectively, away from experiment. However, when
using Grimme’s D3 dispersion, the τ value decreases from 0.05 to -0.14 for CB8 but increases
from -0.05 to 0.71 when the statistical outlier is removed for OA and increases from -0.14 to 0.57
for TEMOA. This shows the importance of using a dispersion correction for qualitative ranking of
binding affinities.

The binding free energies as a result of utilizing truncated basis sets individually and
extrapolated to the Kohn-Sham limit with a two-point extrapolation using cc-pVDZ and cc-pVTZ
(cc-pV∞Z[D,T]) and a three-point extrapolation to the using cc-pVDZ and truncated triple and
quadruple correlation consistent basis sets, cc-pVTZ(-1d) and cc-pVQZ(-1f2d), denoted as
cc(0,-1,-2), are reported in Tables 8.4 and 8.5, respectively.

228

Table 8.4: Binding free energies for the CB8 complexes in kcal mol−1 with schemes involving not using the RI approximation, and
changing the dielectric constant of the implicit solvent with the truncated correlation consistent basis sets for hydrogen.

Complex
CB8-G0
CB8-G1
CB8-G2
CB8-G3
CB8-G4
CB8-G5
CB8-G6
CB8-G7
CB8-G8
CB8-G9
CB8-G10
CB8-G11
CB8-G12
CB8-G13

Exp

-6.69 ± 0.05
-7.65 ± 0.04
-7.66 ± 0.05
-6.45 ± 0.06
-7.80 ± 0.04
-8.18 ± 0.05
-8.34 ± 0.05
-10.00 ± 0.10
-13.50 ± 0.04
-8.68 ± 0.08
-8.22 ± 0.07
-7.77 ± 0.05
-9.86 ± 0.03
-7.11 ± 0.03

TZ

QZ

B3PW91-D3 (SMD, ε=78.4)
TZ
(-1d)
-49.85
-54.54
-37.32
-45.01
-69.19
-36.17
-31.95
-14.92
-50.61
-37.31
-42.27
-28.63
-62.53
-52.30

(-1f2d)
-49.27
-56.61
-36.39
-44.38
-67.50
-35.53
-31.63
-12.89
-49.89
-36.73
-38.92
-25.37
-61.43
-50.03

-49.91
-57.22
-36.86
-44.54
-68.10
-16.10
-31.96
-14.95
-27.26
-19.22
-15.29
-10.21
-62.08
-51.72

TZ

QZ

RI-B3PW91-D3 (SMD, ε=78.4)
TZ
(-1d)
-49.84 -49.89
-57.21 -57.22
-36.82 -36.86
-44.51 -44.53
-68.07 -68.09
-35.89 -35.92
-31.93 -31.95
-14.90
-14.88
-50.30
-50.34
-37.05 -37.07
-39.28 -39.30
-25.74 -25.75
-61.99 -62.05
-51.73 -44.04

(-1f2d)
-49.25
-56.61
-36.39
-44.38
-67.49
-35.52
-31.62
-12.89
-49.90
-36.71
-38.90
-25.36
-61.40
-50.00

TZ

QZ

RI-B3PW91-D3 (SMD, ε=76.4)
TZ
(-1d)
-49.84 -49.82
-57.21 -57.24
-36.82 -36.87
-44.51 -44.55
-68.07 -68.12
-35.89 -35.95
-31.95 -31.97
-14.91 -14.92
-50.30 -50.36
-37.05 -37.09
-39.28 -39.32
-25.74 -25.80
-61.99 -62.07
-51.74 -51.75

(-1f2d)
-36.26
-56.62
-36.40
-44.40
-67.52
-35.54
-31.64
-12.91
-49.92
-36.74
-38.91
-25.41
-61.41
-50.04

MAE
RMSE

34.29
36.99
-0.14
0.00
The mean absolute error (MAE), root mean square error (RMSE), Kendall’s Tau (τ), and r2 are shown.

27.68
33.79
-0.21
0.07

35.33
37.96
-0.14
0.00

34.81
37.56
-0.12
0.00

34.19
37.03
-0.12
0.00

τ
r2

34.18
37.02
-0.12
0.00

34.81
37.56
-0.12
0.00

34.85
37.6
-0.12
0.00

33.27
36.13
-0.08
0.00

229

Table 8.5: Binding free energies for the CB8 complexes in kcal mol−1 with schemes involving not using the RI approximation, changing
the dielectric constant of the implicit solvent, and two options for basis set choice when extrapolating to the Kohn-Sham limit.

RI-B3PW91-D3 (SMD, ε=78.4)
cc-pV∞Z [D, T]
cc(0,-1,-2)

RI-B3PW91-D3 (SMD, ε=76.4)
cc-pV∞Z [D, T]
cc(0,-1,-2)

B3PW91-D3 (SMD, ε=78.4)
cc-pV∞Z [D, T]
cc(0,-1,-2)

-49.91
-57.22
-36.86
-44.54
-68.10
-16.10
-31.96
-14.95
-27.26
-19.22
-15.29

-47.62
-60.08
-35.25
-43.50
-64.83
-34.89
-31.33
-11.27
-24.47
-36.14
-33.97

-49.89
-57.22
-36.86
-44.53
-68.09
-35.92
-31.95
-14.90
-50.34
-37.07
-39.3

-47.58
-55.85
-36.00
-44.20
-66.23
-35.28
-31.32
-11.30
-49.31
-36.50
-38.63

-49.82
-57.24
-36.87
-44.55
-68.12
-35.95
-31.96
-14.92
-50.36
-37.09
-39.32

-16.15
-55.88
-36.04
-44.26
-66.27
-35.32
-31.34
-11.31
-49.38
-36.58
-38.67

Exp

-6.69 ± 0.05
-7.65 ± 0.04
-7.66 ± 0.05
-6.45 ± 0.06
-7.80 ± 0.04
-8.18 ± 0.05
-8.34 ± 0.05
-10.00 ± 0.10
-13.50 ± 0.04
-8.68 ± 0.08
-8.22 ± 0.07
-7.77 ± 0.05
-9.86 ± 0.03
-7.11 ± 0.03

Complex
CB8-G0
CB8-G1
CB8-G2
CB8-G3
CB8-G4
CB8-G5
CB8-G6
CB8-G7
CB8-G8
CB8-G9
CB8-
G10
CB8-
G11
CB8-
G12
CB8-
G13

MAE
RMSE

τ
r2

-10.21

-62.08

-51.72

27.68
33.79
-0.21
0.07

-20.40

-60.01

-47.18

30.93
34.71
-0.34
0.15

-25.75

-62.05

-44.04

34.29
36.99
-0.14
0.00

-24.88

-60.67

-37.82

32.69
35.56
-0.12
0.00

-25.80

-62.07

-51.75

34.85
37.6
-0.12
0.00

-25.02

-60.70

-40.12

30.65
34.12
-0.01
0.02

These options are cc-pV∞Z [D, T], which use cc-pVDZ and cc-pVTZ to extrapolate to the Kohn-Sham limit, and cc(0,-1,-2), which uses cc-pVDZ, cc-pVTZ(-1d), and cc-pVQZ(-
1f2d) to extrapolate to the Kohn-Sham limit. The binding energies obtained with RI-B3PW91-D3 (SMD, ε=78.4)/cc-pV∞Z [D, T] were submitted. The mean absolute error (MAE),
root mean square error (RMSE), Kendall’s Tau (τ), and r2 are shown.

230

For the CB8 complexes in Table 8.4, using standard DFT (B3PW91-D3) yielded a MAE of
35.33 kcal mol−1 and 34.19 kcal mol−1 with cc-pVTZ(-1d) and cc-pVQZ(-1f2d), respectively,
while RI-DFT (RI-B3PW91-D3) yielded a MAE of 34.81 and 34.18 kcal mol−1 for cc-pVTZ(-1d)
and cc-pVQZ(-1f2d), respectively. When changing ε from 78.4 for pure water to 76.4 to account
for the ionic strength of the solution (RI-B3PW91-D3 (ε=76.4)), all metrics (MAE, RMSE, τ, and
r2) used to gauge the method’s predictive qualities for the binding free energies did not significantly
change with respect to the binding free energies predicted in pure water (RI-B3PW91-D3 (ε=78.4)).
Table 8.5 shows the predicted binding free energies for B3PW91-D3 (ε=78.4), RI-B3PW91-D3
(ε=78.4), and RI-B3PW91-D3 (ε=76.4) at the Kohn-Sham limit using cc-pV∞Z[D,T], a two-point
extrapolation using cc-pVDZ and cc-pVTZ, and cc(0,-1,-2), a three-point extrapolation using cc-
pVDZ, cc-pVTZ(-1d) and cc-pVQZ(-1f2d) for the CB8 complexes. Using the cc(0,-1,-2) basis
set choice for extrapolation, the binding free energies predicted by RI-B3PW91-D3 (ε=78.4) and
RI-B3PW91-D3 (ε=76.4) lowered the MAE by approximately 1.6kcal mol−1, and 4.2kcal mol−1,
respectively, in regards to using the cc-pV∞Z[D,T] scheme.

231

Figure 8.7: Plots for calculated v. experimental results in kcal mol−1 for (a) CB8 (b) OA, and
(c) TEMOA for MMPBSA (blue), RI-B3PW91-D3 (black), and RI-B3PW91 (green). The dashed
lines in each corresponding color refers to the best fit line where the statistical outlier (OA-G2) is
removed for (b) and (c). The dashed gray line is the y = x line.

232

Figure 8.8: Error plots from experimental results in kcal mol−1 for (a) CB8 (b) OA, and (c) TEMOA
for MMPBSA (blue), RI-B3PW91-D3 (black), and RI-B3PW91 (green).

233

8.4 Discussion

8.4.1 Submission Analysis

For the methods submitted to the SAMPL6 competition, using RI-B3PW91-D3 yielded higher
τ values for OA and TEMOA than using RI-B3PW91 for predicting binding free energies. Since
there are eight guests that are bound to OA and TEMOA, τcrit for α=0.05 is 0.57 for 8 data points.
Only MMPBSA correlates with experiment (|τ| > τcrit), as the τ values are 0.64, 0.29, and -0.21 for
MMPBSA, RI-B3PW91-D3, and RI-B3PW91, respectively. However, after removing the statistical
outlier, OA-G2, from the dataset, τ increases from 0.29 to 0.71, which implies that RI-B3PW91-D3
also correlates with experiment. As shown in Table 8.2, RI-B3PW91-D3 ranked the binding free
energies more correctly than MMPBSA when the outlier is excluded. For TEMOA, both MMPBSA
and RI-B3PW91-D3 correlate with experiment with τ values of 0.79 and 0.57, respectively, which
are greater than τcrit.

As shown in Figure 8.7a, there is no correlation between experimental and predicted binding
free energies for the CB8 host-guest systems. This is supported by r2 ≈ 0 and τ values of -0.19,
-0.14, 0.12 for MMPBSA, RI-B3PW91-D3, and RI-B3PW91, respectively, which are smaller in
magnitude than τcrit for α=0.05 for 14 data points, which is 0.36. This also shows an inconsistency
when using Grimme’s dispersion correction, which may be due to the abundance of N and O atoms
present in the CB8 host and empirical descriptors for those atoms. For all sets of the host-guest
systems, RI-B3PW91 had a lower MAE and RMSE than RI-B3PW91-D3 by approximately 19.4-
32.6kcal mol−1, but as a tradeoff, resulted in qualitatively better predictions of the binding affinities
(Figure 8.8). This implies that using a dispersion correction overbinds the guest to the host but is
needed for proper ranking.

To estimate the relative performance of the methods, the mean signed error (MSE) was used
to offset the calculated binding free energies. After the removal of MSE from the MMPBSA and
RI-B3PW91-D3 predicted binding free energies for OA and TEMOA, the MAE and the RMSE
values are recalculated to estimate the performance of methods in relative terms as shown in Table

234

8.6. This correction improved the MAE and RMSE for MMPBSA by 6.8 and 5.9 kcal mol−1 for
OA and TEMOA, respectively. The correction improved RI-B3PW91-D3 MAE and RMSE by
38.39 and 38.83 kcal mol−1 for OA without the OA-G2 outlier and TEMOA, respectively.

Table 8.6: Predicted binding energies for OA and TEMOA using MMPBSA and RI-B3PW91 after
the removal of mean signed error (MSE).

OA

RI-B3PW91-D3

TEMOA

RI-B3PW91-D3

MMPBSA
1.6 ± 0.2a
1.9 ± 0.4b

0.64
0.84

MAE
RMSE
τ
r2
Bracketed values indicate the values after the removal of the statistical outlier (OA-G2).The mean absolute error (MAE) inkcal
mol−1, root mean square error (RMSE) inkcal mol−1, Kendall’s Tau (τ), and r2 are shown.
aThe uncertainty reported for MAE is the average of the absolute uncertainties.
bThe uncertainty reported for RMSE is the uncertainty of the RMSE with the experimental and calculated uncertainties.

11.66 [2.81]
17.87 [3.12]
0.29 [0.71]
0.44 [0.52]

3.49
3.95
0.57
0.55

MMPBSA
3.0 ± 0.2a
3.7 ± 0.5b

0.79
0.86

8.4.2

Impact of Truncated Basis Sets

For the QM calculations, the subset of the CB8 host-guest systems was chosen because the
size of these systems is smaller compared to the octa-acid host-guest systems investigated. While
using the RI approximation, lowering ε from 78.4 for pure water to 76.4 to account for the ionic
strength of the solution increased the MAE by 0.56kcal mol−1. However, altering the dielectric
constant from 78.4 to 76.4 to account for the ionic strength of the solution lowered the MAE from
34.85 to 30.65 kcal mol−1 for the three-point extrapolation with truncated triple-ζ and quadruple-ζ
correlation consistent basis sets, yet for RI-B3PW91-D3 (ε=78.4), the MAE only decreased from
34.29 to 32.69 kcal mol−1 (Table 8.5). Therefore, factors that can change the dielectric constant
should be considered when using implicit solvent models for binding free energy predictions.

The use of the cc(0,-1,-2) basis set scheme lowered the MAE for CB8 complexes by 1.60 kcal
mol−1 relative to using cc-pV∞Z[D,T] (Table 8.5) for RI-B3PW91-D3 (ε=78.4).
In contrast,
when using truncated basis sets and standard basis sets for binding free energies (Table 8.4), the
MAE decreased by 0.51 kcal mol−1 for the CB8 complexes when using cc-pVTZ as opposed
to cc-pVTZ(-1d) for RI-B3PW91-D3 (ε=78.4). The MAE decreased by 0.31 kcal mol−1 when

235

increasing the basis set quality of truncated basis sets for RI-B3PW91-D3 (ε=78.4). Therefore,
within the RI approximation, the decrease in MAE when using cc-pVQZ(-1f2d) highlights the
importance of using higher quality basis sets when extrapolating to the Kohn-Sham limit.

For predictions without the RI approximation, the binding free energies determined using
B3PW91-D3/cc-pVTZ yielded a decrease in the MAE by 7.65 kcal mol−1 relative to B3PW91-
D3/cc-pVTZ(-1d) as shown in Table 8.4. This is believed to be a result from including the four-
center two-electron electron repulsion integrals removed via the RI approximation and the need
for additional polarization when describing interactions with hydrogens between the host and the
guest. This effect also contributes to the increase of 3.25 kcal mol−1 in the MAE between B3PW91-
D3/cc-pV∞Z[D,T] and B3PW91-D3/cc(0,-1,-2). However, as shown in Table 8.5, when employing
truncated basis sets (cc(0,-1,-2)), binding free energy predictions when using RI-B3PW91-D3
(ε=76.4) are more positive and yield a MAE of 0.28 kcal mol−1 lower than B3PW91-D3 (ε=78.4).
This illustrates that within the RI approximation, changing the dielectric constant is as beneficial to
predicting binding free energies as utilizing standard DFT, which is more computationally costly
than RI-DFT.

For the CB8-G6 host-guest complex, which was one of the smaller systems in the set of host-
guest systems, the number of basis functions decreased from 4016 to 3696 with the truncation of
1 d basis function from the cc-pVTZ basis set for hydrogen and decreased from 7640 to 6872 with
the truncation of 1 f and 2 d basis functions from the cc-pVQZ basis set for hydrogen. Since DFT
scales approximately N 3 to N 5 depending on the complexity of the functional where N is the
number of basis functions, truncated basis sets become a practical option for further decreasing the
computational cost while improving the quantitative prediction of binding free energies for these
host-guest systems as truncating 1 d basis function from cc-pVTZ only affected the binding energy
predicted with cc-pVTZ by ≤ 0.06 kcal mol−1 as shown in Table 4 for RI-B3PW91-D3.

236

8.4.3

Impact of the Extrapolation Scheme B-parameter

Another factor that can account for the large deviations between host-guest binding energies
is the parameter used to fit Equation 8.1 for two-point extrapolations. The value of 5.5 proposed
by Jensen for the B-parameter, which was used for atoms and diatomics, caused the extrapolation
curve to converge at a very rapid rate and is reflected in the predictions for the CB8 complexes, as
the binding affinities in Table 8.1 are identical to those predicted with the cc-pVTZ basis set with
the respective method in Table 8.4. Also, when using the three-point extrapolations with truncated
basis sets for the CB8 complexes, the B-parameter yielded an average value of 0.37 (Table 8.8).
Therefore, the value of 0.37 for the B-parameter was applied to two-point extrapolations with cc-
pVDZ and cc-pVTZ to gauge how changing the B-parameter affected the extrapolated binding free
energies (Table 8.7). The results from using 0.37 as the B-parameter in a two-point extrapolation
show that the MAE decreased by 0.84 and 0.42 kcal mol−1 for the CB8 and TEMOA complexes,
respectively. The MAE did not change for the OA complexes. Setting the B-parameter to 0.37 did
not change the τ values for CB8 and OA complexes, however, did increase the τ value from 0.57 to
0.71 for TEMOA.

In addition to applying 0.37 for the B-parameter to predict binding free energies for all host-guest
systems using two-point extrapolations with cc-pVDZ and cc-pVTZ, the value of the B-parameter
was optimized to the value of 0.12 via minimizing the MAE and was applied (Table 8.7). For the
CB8 host-guest systems, shifting the B-parameter from 5.5 to 0.12 had a noticeable impact on the
MAE, which decreased from 34.29 to 29.84 kcal mol−1 for RI-BWPW91-D3. A similar effect was
observed for TEMOA with a decrease in the MAE of 5.07kcal mol−1. There is no notable change
in MAE, RMSE, or τ for the OA complexes with the change in the B-parameter. Furthermore, τ
increases from 0.57 to 0.93 when the B-parameter is changed from 5.5 to 0.12 for TEMOA with RI-
B3PW91-D3, which provides more evidence that dispersion-corrected functionals should be used
for qualitative predictions of binding free energies since |τ| > τcrit. The observed trends imply that
the value of the B-parameter should be reoptimized when using Equation 8.1 for macromolecules.

237

Table 8.7: Predicted binding energies when using different values for B in Equation 8.1 for two-point
extrapolations using cc-pVDZ and cc-pVTZ with RI-B3PW91-D3.

CB8

OA

TEMOA

MAE
RMSE

τ
r2

MAE
RMSE

τ
r2

MAE
RMSE

τ
r2

B=5.5
34.29
36.99
-0.14
0.00

B=0.37
33.45
36.33
-0.14
0.00

B=0.12
29.84
33.34
-0.03
0.00

35.46 [38.39]
36.41 [38.52]
0.29 [0.71]
0.44 [0.52]

35.46 [38.42]
36.43 [38.54]
0.29 [0.71]
0.43 [0.52]

35.43 [38.74]
36.70 [38.86]
0.29 [0.71]
0.43 [0.54]

38.83
39.03
0.57
0.55

38.41
38.60
0.71
0.75

33.76
36.30
0.93
0.58

Bracketed values indicate the values after the removal of the statistical outlier (OA-G2).The mean absolute error (MAE) inkcal
mol−1, root mean square error (RMSE) inkcal mol−1, Kendall’s Tau (τ), and r2 are shown.

Compared to other submissions employing QM methods in the SAMPL6 Host-Guest binding
challenge, our approach yielded quantitatively poorer predictions that may have resulted from the
approximations considered in this work. In our approach, only a single conformational state of
the guest binding to the host system was considered. Additionally, the representative structures of
the individual host-guest systems obtained from clustering the MD trajectories were not optimized
with QM methods and is reflected in our model chemistries.

8.4.4

Impact of Representative Geometries

The representative geometries had a notable impact on the binding free energies. For example,
the orientation of the substituted cyclohexene ring relative to the OA host might be the potential cause
of OA-G2 being a statistical outlier (Figure 8.5). Comparing OA-G2 and TEMOA-G2 in Figures 8.5
and 8.6, where the only difference is the four methyl groups on the host, the structure of the OA-G2
complex has a smaller binding pocket than the TEMOA-G2 complex. While the experimental data
suggests that G2 has a stronger binding affinity towards OA than TEMOA, MMPBSA suggests the

238

opposite. More sampling of representative structures would aid in depicting whether the anomalous
binding behavior of OA-G2 correlates with the positive binding free energies predicted with DFT.
Although the only difference between CB8-G6 and CB8-G7 was the expansion of the ring for
the guest by one CH2 group, the predicted binding affinities for the CB8-G6 and CB8-G7 complexes
differed by approximately 17.0kcal mol−1. This may be due to the binding poses of CB8-G6 and
CB8-G7 complexes, as G6 bound in a perpendicular fashion inside the binding pocket relative
to the host whereas G7 bound in a parallel fashion inside the binding pocket. This would affect
nearby electrostatic interactions and why for B3PW91-D3 (ε=78.4), RI-B3PW91-D3 (ε=78.4), and
RI-B3PW91-D3 (ε=76.4), there was a 3.00 kcal mol−1 difference in the change of binding energies
between CB8-G6 and CB8-G7 when improving basis set quality via the basis set scheme used
for extrapolation (Table 8.5). Ergo, more sampling of chemically relevant structures or enhanced
sampling methods can provide a more robust depiction of the host-guest binding environment.

However, these two methods do not correlate to the CB8 binding free energies since the τ
values are -0.19 and -0.14 for MMPBSA and RI-B3PW91-D3, respectively. This may result from
insufficient sampling as the CB8 guests are larger molecules with higher conformational flexibility.
For example, the size of CB8-G4 does not allow the guest to fit entirely into the binding cavity.
As a result, most of the CB8-G4 molecule is weakly bound to the host from outside of the binding
pocket and only one of the three triethyl amines within the guest can fit into the pocket as shown
in Figure 8.4. Each triethyl amine group could bind to the host from inside the binding cavity,
which would result in alternative binding conformations and affect the overall binding free energy.
To better understand binding free energies of these large structures, more sampling of the different
binding modes is needed to generate weighted averages based on the thermodynamic stability of
predicted poses.

The results for OA and TEMOA systems illustrate that MMPBSA and RI-B3PW91-D3 methods
can be used to qualitatively rank binding energies of small molecules. Among those two methods,
MMPBSA is computationally less expensive, but RI-B3PW91-D3 predicted the relative binding
affinities better for OA and TEMOA host-guest systems. However, the MAE and the corresponding

239

error plots (Figure 8.8) indicate that both methods overestimated the binding free energies. The
MAE reported for the OA and TEMOA complexes state that MMPBSA and RI-B3PW91-D3 predict
overbinding by 6.8 and 35.5kcal mol−1, respectively, for OA complexes and 5.9 and 38.8kcal mol−1,
respectively, for TEMOA complexes. For all systems, the MMPBSA method was the best approach
overall in terms of quantitative predictions.

8.5 Conclusions

When implementing DFT for predicting host-guest binding affinities, the use of Grimme’s D3
dispersion correction was essential for qualitatively predicting the binding free energies for the OA
and TEMOA systems even though the MAE exceeded 35.0 kcal mol−1 for both the OA and TEMOA
systems. When using implicit solvent models, factors that can change the dielectric constant, such
as the ionic strength of the solution, are relevant for predicting binding free energies, as lowering
the dielectric constant lowered the MAE. While RI-B3PW91-D3 reduced the computational cost
relative to B3PW91-D3, B3PW91-D3 yielded a lower MAE. To attain more quantitatively favorable
results, using cc-pVQZ(-1f2d) for hydrogen atoms reduces the computational cost relative to using
cc-pVQZ while simultaneously providing a better standard for extrapolating to the Kohn-Sham
limit than only utilizing cc-pVDZ and cc-pVTZ for extrapolations. Also, truncating 1 d basis
function for hydrogen atoms had a very small effect on predicted binding free energies obtained
with cc-pVTZ, indicating that truncated basis sets are a viable option to reduce the computational
cost while yielding near-identical binding free energies. With the extrapolation scheme utilized, the
B-parameter should be revised for macromolecules since reducing the value of the B-parameter from
the proposed 5.5 to 0.12 reduced the MAE while providing extrapolated binding energies that were
in alignment with those predicted using quadruple-ζ level basis sets. Sampling of different binding
poses becomes pertinent for future investigations as binding orientation in the pocket affected the
predicted binding free energies by approximately 17.0 kcal mol−1 when using RI-B3PW91-D3 for
guests that only differed by one CH2 group.

All methods presented predict overbinding character for these host-guest systems except for

240

RI-B3PW91 for CB8 host-guest systems. MMPBSA and RI-B3PW91-D3 worked well at ranking
binding affinities for smaller guests regardless of the size of the host. The CB8 guest molecules with
a larger van der Waals volume yielded poor prediction of binding free energy due to their higher
conformational flexibility, which can complicate predicting binding poses. To better understand
binding free energies of these large structures, enhanced sampling methods can be used, and
multiple host-guest binding poses can be sampled.

241

APPENDIX

242

APPENDIX

Table 8.8: Fitting parameter values obtained when using Jensen’s extrapolation scheme for each
component in calculating the binding energy (Equation 8.1). The host and guest are counterpoise-
corrected before the extrapolation was performed.

Complex
CB8-G0
CB8-G1
CB8-G2
CB8-G3
CB8-G4
CB8-G5
CB8-G6
CB8-G7
CB8-G8
CB8-G9
CB8-G10
CB8-G11
CB8-G12
CB8-G13

Average

Complex

0.37
0.36
0.36
0.36
0.32
0.38
0.39
0.38
0.37
0.39
0.38
0.39
0.36
0.39

0.37

Host
0.36
0.35
0.36
0.36
0.32
0.38
0.39
0.38
0.37
0.39
0.38
0.39
0.35
0.38

0.37

Guest
0.41
0.37
0.37
0.37
0.34
0.39
0.40
0.40
0.39
0.39
0.38
0.40
0.37
0.40

0.38

243

Figure 8.9: Plots for the correlations calculated after the mean signed errors are removed from the
results in Tables 8.1-8.3 versus experimental results in kcal mol−1 for (a) OA, and (b) TEMOA for
MMPBSA (blue), RI-B3W91-D3 (black). The dashed lines in each corresponding color refers to
the best fit line where the statistical outlier (OA-G2) for RI-B3PW91-D3 is removed for (a). The
dashed gray line corresponds to the y=x line.

244

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CHAPTER 9

CONCLUDING REMARKS

In this dissertation, several quantum chemical strategies have been shown to be effective for
thermodynamic property prediction for main group and transition metal thermochemistry. This
includes utilizing density functional methods predicting the pKas of transition metal hydrides and
utilizing ab initio composite strategies towards main group thermochemistry, vibrational potential
energy surfaces, and organometallic catalysis. Applications include modeling frontier orbitals and
predicting host-guest binding interactions with density functional methods as well.

In Chapter 3, a QM/QM scheme utilizing the ONIOM method was used to predict the pKa of
late transition metal hydrides with bidentate phosphine ligands.
In predicting the pKa of TM
hydrides via the choice of density functional, ab initio method, solvation model, basis set, cavity
model, and model layer size within an ONIOM scheme, the optimal scheme is one that utilizes two
density functionals, one effective at describing the metal center and immediately bound atoms, and
another effective at describing ligands comprised of main group atoms. This was B3LYP-D3/aug-
cc-pVTZ:B97-D3/SDD using the SMD solvation model and default cavity model. Using ab initio
methods underestimated the pKa while the use of a single functional largely overestimated the pKa.
In future studies for these systems, the methodology presented can be expanded to sterically bulkier
bidentate phosphine ligands for Group 10 hydrides utilized for redox potentials to gauge efficacy.
In general studies involving transition metal complexes with sterically bulky ligands, this approach
can be utilized to target functional efficacy for transition metal centers and main group ligands
independently as different tiers of functionals yielded lower deviations from the experimental pKas.
In Chapters 4 and 5, the domain-based local pair natural orbital (DLPNO) methods were
utilized within the correlation consistent Composite Approach (ccCA) and applied to main group
and transition metal thermochemistry. DLPNO-ccCA yielded lower mean absolute deviations than
ccCA for the enthalpies of formation for 119 closed shell main group molecules with a ~87%
CPU time reduction relative to ccCA. As DLPNO-ccCA was implemented for linear alkanes up

251

to octane, the CPU time was reduced by up to 97% for octane relative to ccCA given the lower
scaling of DLPNO-CCSD(T) relative to CCSD(T). DLPNO-ccCA was also successfully applied to
bioorganic complexes from the S66 dataset and the coronene dimer, which marks one of the largest
molecules ever examined with a composite strategy, that exhibit noncovalent interactions.

DLPNO-ccCA can therefore be used to predict thermodynamic properties of organic and
bioorganic molecules typically outside the range of ab initio composite approaches based on
molecule size such as per- and polyfluoroalkyl substances (PFAS). DLPNO-ccCA could also
be utilized to examine thermodynamic properties of drug-like molecules, like pKa or partition
coefficients, as these molecules exhibit multiple hydrogen-bonding sites and more aromatic rings.
Thus, DLPNO-ccCA increases the applicability of ab initio composite strategies for main group
species based on the reduction of computer resources and computational cost.

DLPNO-ccCA was implemented to model organometallic catalysis utilizing the variant of
ccCA targeting 4d transition metal chemistry, rp-ccCA. Denoted as DLPNO-rp-ccCA, this
method was successfully applied towards hydroformylation, which is the largest volume
homogeneous chemical reaction in industry for chemical production, and gas phase ligand
dissociation, which targets modeling metal-ligand interactions with ab initio approaches. A
continuation of this study would include modeling more metal-ligand interactions prevalent in
organometallic catalysis with DLPNO-rp-ccCA and expanding the sample size to include more
catalysts utilized in hydroformylation, where the linear isomer is favored, and asymmetric
hydroformylation, where the branched isomer is favored.
The DLPNO-ccCA variants for
transition metals could also be applied towards 3d transition metals to further increase the
possible molecule space for ab initio composite methodologies.

In Chapter 6, ccCA, B3LYP, and TPSS were utilized to generate potential energy surfaces
that were then used to predict anharmonic vibrational frequencies with vibrational self-consistent
field (VSCF) theory. Overall, with ccCA potentials, the mean absolute deviation for calculated
frequency from experiment was lower than with DFT potentials. With DFT-generated potentials,
functional choice had a more significant effect on the predicted frequency than basis set choice. A

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multilevel approach that utilizes the single mode potential energy curves with DFT and the coupled
vibrational modes generated with ccCA yields lower frequencies than if only DFT were utilized,
which is useful for expanding to larger polyatomic systems. For aminophenol, the errors obtained
with VCIPSI-PT2 were lower than those for scaled harmonics, indicating the success of utilizing
this approach to characterize specific vibrations for polyatomic systems.

Future work on this project could include the investigation of astrochemical molecules with
unusual binding behavior and utilizing different variants of ccCA, such as completely renormalized
ccCA (CR-ccCA(2,3)) to account for bond-breaking behavior occurring in vibrational motion. This
approach can also be implemented to metal-carbonyl stretching as well as uncovering vibrational
behavior not accounted for by the harmonic approximation. Given the large number of electronic
structure calculations involved with generating potential energy surfaces for polyatomic systems,
DLPNO-ccCA could also be considered to investigate vibrational phenomena while reducing the
CPU time relative to ccCA. As well, multilevel approaches can be utilized to investigate the full
anharmonic mode-mode coupling potential energy surfaces to model infrared (IR) spectra that
more closely resemble experimental IR spectra than using the harmonic oscillator approximation.
For Chapter 7, calculations were done to complement synthesis of the zinc porphyrin-fullerene
supramolecular dyad ((F15P)Zn – C60) and the C60 – (F15P)Zn:Py-TTF and C60 – (F15P)Zn:Py-
phTTF triads via modeling the molecular electrostatic potential and frontier orbitals with M06-
2X/6-31G*. For the dyads, the frontier HOMO was on the (F15P)Zn and LUMO on the C60 making
them the donor and acceptor sites, respectively. The HOMO was shifted to the tetrathiafulvalene
site without altering the location of the LUMO for the triads.1 For modeling electronic structure
and frontier orbitals for supramolecular dyads useful in artificial photosynthesis, time-dependent
density functional theory (TDDFT) combined with implicit solvent models can be used to model
UV-Vis absorption spectra to verify observed photochemical phenomena for these systems, such as
the transition at ∼400 nm indicating transitions occurring at the porphyrin.

In Chapter 8, molecular dynamics and DFT methods were used to predict the binding interaction
energies of biological host-guest systems for the sixth Statistical Assessment of Modeling Protein

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and Ligands (SAMPL) blind prediction competition.2 Modeling the host-guest systems with RI-
B3PW91-D3 predicted qualitative ranking of binding affinity to each of the hosts, exhibited by
the Kendall’s tau (τ) statistic while predicting binding energies tens of kcal mol−1 away from
experimental binding interaction energies.
In the future, more orientations could be sampled
and binding poses obtained through molecular dynamics simulations could be optimized with
density functional theory. As well, different density functionals can be utilized to evaluate the
binding interactions of similar systems to provide a gauge for appropriate functionals for host-
guest binding. This would provide a linear regression technique that can be implemented to
predict binding interaction free energies.
In a similar vein, regression-based machine learning
approaches can be used with parameters inputted from molecular dynamics and electronic structure
calculations, opening a new avenue for thermodynamic property prediction.

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REFERENCES

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REFERENCES

[1] Obondi, C. O.; Lim, G. N.; Jang, Y.; Patel, P.; Wilson, A. K.; Poddutoori, P. K.;
D’Souza, F. Charge Stabilization in High-Potential Zinc Porphyrin-Fullerene via Axial Ligation
of Tetrathiafulvalene. J. Phys. Chem. C 2018, 122, 13636–13647.

[2] Eken, Y.; Patel, P.; Díaz, T.; Jones, M. R.; Wilson, A. K. SAMPL6 host-guest challenge: Binding

free energies via a multistep approach. J. Comput. Aided. Mol. Des. 2018, 32, 1097–1115.

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