ELECTRONIC STRUCTURE AND SPECTROSCOPY IN THE GAS AND

CONDENSED PHASE: METHODOLOGY AND APPLICATIONS.

by

Vitalii Vanovschi

A Dissertation Presented to theFACULTY OF THE GRADUATE SCHOOL

UNIVERSITY OF SOUTHERN CALIFORNIAIn Partial Fulfillment of the

Requirements for the DegreeDOCTOR OF PHILOSOPHY

(CHEMISTRY)

January 2009

Copyright 2009 Vitalii Vanovschi

Acknowledgements

I’d like to thank my scientific adviser and mentor Prof. Anna I. Krylov. Her advice

and support were the critical success factors in the projects presented in this thesis. Her

enthusiasm toward the results I’ve obtained was highly encouraging. I appreciate the

mentoring on the number various topics Anna provided me with. Also, I would like to

thank Anna for the team-building activities she organized: hikes, trips, rock-climbing.

Thanks to the current and the former members of Anna’s group, especially Vadim

Mozhayskiy, Dr. Kadir Diri, Evgeny Epifanovsky, Lucasz Koziol, Dr. Sergey V. Levchenko

and Dr. Piotr A. Pieniazek for their help throught the projects.

I’d like to thank Prof. Curt Witting for the fascinating physical theories I’ve learned

in his class and for mentoring me during the course of studies.

I am grateful to our main collaborator on the effective fragment potential project,

Prof. Lyudmila Slipchenko for answering all the numerous questions I had about this

complicated method. Also I’d like to thank Prof. Mark S. Gordon for organizing a

welcome visit to Iowa State University where, working in the laboratory of the original

EFP inventors, I made a substantial progress on this project.

I’d like to thank Dr. Yihan Shao and Dr. Ryan Steele who helped me to figure out

the nuts and bolts of Q-Chem. Especially I’d like to thank Dr. Christopher Williams for

implementing the multipole integrals, required for the EFP electrostatics and polariza-

tion components.

ii

Table of Contents

Acknowledgements ii

List of Figures v

List of Tables viii

Abstract ix

1 Implementation of the Effective Fragment Potential method 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.5 Exchange-repulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.6 Implementation of EFP in Q-Chem . . . . . . . . . . . . . . . . . . . . 281.7 EFP application: Beyond benzene dimer . . . . . . . . . . . . . . . . . 371.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Reference list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2 Structure, vibrational frequencies, ionization energies, and photoelectronspectrum of the para-benzyne radical anion 572.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.2 Molecular orbital picture and the origin of vibronic interactions in the

para-benzyne anion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.3 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . 652.4 Equilibrium structures, charge distributions, and theD2h→ C2v poten-

tial energy profiles at different levels of theory . . . . . . . . . . . . . . 672.5 Vibrational frequencies and the photoelectron spectrum of para-C6H

−4 . 77

2.6 DFT self-interaction error and vibronic interactions . . . . . . . . . . . 822.7 Calculation of electron affinities of p-benzyne . . . . . . . . . . . . . . 862.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Reference list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

iii

3 Future work 963.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Reference list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Bibliography 104

A GAMESS input for generating EFP parameters of benzene 117

B Geometries of benezene oligomers 119

C Converter of the EFP parameters 122

D Fragment-fragment electrostatic interaction 128

E Buckingham multipoles 130E.1 The conversion from the sperical to the Buckingham multipoles . . . . 130E.2 The conversion from the Cartesian to the Buckingham multipoles . . . . 132

F The Q-Chem input for EFP water cluster 134

G Q-Chem’s keywords affecting the EFP parameters computation 138

iv

List of Figures

1.1 Comparison of continuum (a) and discrete (b) models for a water moleculein the environment of other water molecules. . . . . . . . . . . . . . . 3

1.2 Uracil in aqueous solution. Active region (uracil) is shown with a ballsand sticks model. Effective fragments (water) are depicted by lines. . . 4

1.3 Distributed multipole expansion points for a water molecule. . . . . . . 10

1.4 Sketch of the charge penetration phenomenon for two water molecules. 13

1.5 The shape of the grid (monotonous gray area) for fitting the screeningparameters of a water molecule. . . . . . . . . . . . . . . . . . . . . . 15

1.6 Distributed polarizabilities for a water molecule. . . . . . . . . . . . . 16

1.7 Computing the induced dipoles through the two level self-consistentiterative procedure. HereΨ is the ab initio wavefunction,µ is a setof induced dipoles,FΨ is the force due to theab initio wavefunction andFµ is the force due to the induced dipoles. . . . . . . . . . . . . . . . . 19

1.8 Distributed dispersion points for a water molecule. . . . . . . . . . . . 23

1.9 The exchange-repulsion LMO centroids for a water molecule. . . . . . 27

1.10 Class diagram for the EFP energy part of the implementation. . . . . . 29

1.11 The internal structure of theFragmentParamsclass. . . . . . . . . . . 29

1.12 Representing the orientation with the Euler angles. xyz is the fixed sys-tem, XYZ is the rotated system, N is the intersection between the xy andXY planes called theline of nodes. Transformation between the xyz andXYZ systems is given byα rotation in the xy plane, followed byβ rota-tion in the new y’z’ plane (around the N axis), followed byγ rotation inthe new x”y” plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.13 The rotation matrix corresponding to the Euler anglesα, β, γ. . . . . . 31

v

1.14 The flow of control between Q-Chem and EFP energy module. . . . . . 32

1.15 The format for the$efp paramssection. . . . . . . . . . . . . . . . . . 35

1.16 The format for the$efp fragmentssection. . . . . . . . . . . . . . . . 35

1.17 Sandwitch (S), t-shaped (T) and parallel-displaced (PD) structures ofthe benzene dimer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1.18 The structures of eight benzene trimers. . . . . . . . . . . . . . . . . . 45

1.19 Dependence of the non-additive energy share on the length of benzeneoligomers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.1 Frontier MOs and leading electronic configurations of the two lowestelectronic states atD2h andC2v structures. . . . . . . . . . . . . . . . . 62

2.2 The two lowest electronic states of C6H−4 can be described by: (i) EOM-

IP with the dianion reference; (ii) EOM-EA with the triplet neutral p-benzyne reference; (iii) EOM-SF with the quartet p-benzyne anion ref-erence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.3 D2h structures optimized at different levels of theory. . . . . . . . . . . 68

2.4 CCC angles, bonds lengths calculated by different coupled-cluster meth-ods and B3LYP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.5 The distance between the radical-anion centers calculated by differentcoupled-cluster methods and B3LYP. . . . . . . . . . . . . . . . . . . . 70

2.6 The NBO charge at C1 (top panel) and dipole moments (bottom panel)at the optimizedC2v geometries. The ROHF-CCSD and ROHF-EA-CCSD values are calculated at the geometries optimized by the corre-sponding UHF-based method. . . . . . . . . . . . . . . . . . . . . . . 71

2.7 Energy differences between theC2v and D2h structures (6-311+G**basis). Upper panel: calculated using the UHF-CCSD optimized geome-tries. MCSCF and CASPT2N employ the MCSCF geometries. Lowerpanel: calculated at the respective optimized geometries. . . . . . . . . 73

2.8 The potential energy profiles along theD2h-C2v scans. At theD2h struc-ture, CCSD and CCSD(T) energies are calculated using symmetry-brokenand pure-symmetry UHF references (see section 2.3). . . . . . . . . . . 75

2.9 Displacements along the normal modes upon ionization (D2h structure). 80

vi

2.10 Top: Singlet (solid line) and triplet (dotted line) bands of the electronphotodetachment spectrum forD2h using the CCSD equilibrium geome-tries and frequencies of the anion. Bottom: the experimental (solid line)and the calculated (dotted line) spectra forD2h using the CCSD structureand normal modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

2.11 The experimental (solid line) and the calculated (dotted line) spectra forC2v using the CCSD structure and normal modes. . . . . . . . . . . . . 83

2.12 The experimental (solid line) and the calculated (dotted line) spectra forD2h using the B3LYP structure and normal modes. . . . . . . . . . . . 84

2.13 Energy differences between theD2h andC2v minima (using the CCSDstructures) calculated using different density functionals. . . . . . . . . 87

2.14 Adiabatic electron affinities calculated by energy differences for the sin-glet (top panel) and triplet (bottom panel) state of the para-benzynediradical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

vii

List of Tables

1.1 The geometric parameters and energies of the benzene dimers . . . . . 41

1.2 Dissociation energies of the benzene dimers at different levels of theory. 42

1.3 Total and many-body interaction energies of the benzene trimers (firstgroup). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

1.4 Total and many-body interaction energies of the benzene trimers (sec-ond group). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1.5 Binding energies of benzene oligomers computed with EFP. . . . . . . 49

2.1 Energy differences between theC2v and D2h minima (eV) in the 6-311+G** basis set. Negative values correspond to theC2v minimumbeing lower. ZPEs are not included. . . . . . . . . . . . . . . . . . . . 74

2.2 Vibrational frequencies of the para-benzyne radical anion and the dirad-ical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2.3 Displacements in normal modes upon ionization,A ∗√

amu. . . . . . . 80

2.4 Electron affinities (eV) of the a3B1u and X1Ag states of the para-benzynediradical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

viii

Abstract

Method developments and applications of the condensed phase and gas phase modeling

techniques are presented in two parts of this work.

In the first part, the theory of the effective fragment potential (EFP) method designed

for accurate modeling of the molecular properties of the extended systems is introduced.

The implementation of EFP developed in this work within the Q-Chem electronic struc-

ture package is discussed in details. The code is applied to the studies ofπ − π inter-

actions in benzene oligomers. The primary interest of these studies are the many-body

and total non-additiveπ − π interactions. A detailed comparison of EFP withab initio

theories is presented for the set of eight benzene trimers. Three-body intermolecular

interaction energies at the EFP level of theory are within 0.04 kcal/mol from the fullab

initio results. To elucidate the asymptomatic behavior of the total non-additive binding

energy, three types of linear benzene oligomers with 10, 20, 30 and 40 monomers are

modeled using our new EFP implementation. The result of this investigation is that with

the increase in the size of linear benzene clusters, the share of the total non-binding

energy approaches a constant and does not exceed 4% for the structures considered in

the study. This result indicates that there are no substantial non-local effects in the bind-

ing mechanism of linear benzene clusters.

In the second part, the equilibrium structure, vibrational frequencies, and ioniza-

tion energies of the para-benzyne radical anion are characterized by coupled-cluster

ix

and equation-of-motion methods. Vibronic interactions with the low-lying excited state

result in a flat potential energy surface along the coupling mode and even in lower-

symmetryC2v structures. Additional complications arise due to the Hartree-Fock insta-

bilities and near-instabilities. The magnitude of vibronic interactions was characterized

by geometrical parameters, charge localization patterns and energy differences between

theD2h andC2v structures. The observed trends suggest that theC2v minimum predicted

by several theoretical methods is an artifact of an incomplete correlation treatment. The

comparison between the calculated and experimental spectrum confirmedD2h struc-

ture of the anion, as well as accuracy of the coupled-cluster and spin-flip geometries,

frequencies and normal modes of the anion and the diradical. Density functional calcu-

lations (B3LYP) yielded only aD2h minimum, however, the quality of the structure and

vibrational frequencies is poor, as follows from the comparison to the high-level wave

function calculations and the calculated spectrum. The analysis of charge localization

patterns and the performance of different functionals revealed that B3LYP underesti-

mates the magnitude of vibronic interactions due to self-interaction error.

x

Chapter 1

Implementation of the Effective

Fragment Potential method

1.1 Introduction

Condensed phase encompasses the majority of matter on the Earth. Not surprisingly,

many molecular systems that are significant due to their biological, electrical or mechan-

ical properties also exist in the condensed phase. Properties of individual molecules in

such systems are influenced by the environment. Therefore, intermolecular interactions

play crucial role in chemical and physical properties of the condensed phase.

There are three distinct classes of methods for modeling the properties of the

extended systems at molecular level:ab initio (QM), classical (MM) and hybrid

(QM/MM) approaches.

The ab initio methods treat a molecule of interest together with the neighboring

molecules as one system at the same level ofab initio theory. The benefit of this

approach is that a high accuracy can be achieved, with a proper choice of theab ini-

tio method. However, computational costs of manyab initio methods scale steeply with

the size of the system [for instance, asN7 for CCSD(T)1], which makes the fullab initio

treatment of the environmental effects impractical.

New directions in the development of theab initio methods with more favorable

computational costs are:

1

• Linear scaling variants ofab initio methods2 are based on local and/or density

fitting (resolution of identity) approximations. At present, they are available for a

few ab initio methods (HF3, DFT4, MP25, CC6).

• Separation techniques such as semi-classical divide and conquer7 and frag-

ment molecular orbitals8 decompose the molecular system into fragments (e.g.,

molecules), which are described by individual and thus localized wavefunctions.

The wavefunctions for each fragment are computed in the field of the frozen wave-

functions of all other fragments until self-consistency is achieved.

• Plane-wave DFT represents the wavefunction in terms of the plane waves9. It

is suitable for systems with periodic boundary conditions (such as metals and

crystals) and features linear scaling of the computational effort.

On the other side of the theory spectrum, the whole molecular system can be mod-

eled using molecular mechanics, that is classically using empirical force fields10–17.

The advantage of this approach is its efficiency - very large systems with many

thousands of atoms can be modeled. However, molecular mechanics is incapable of

modeling chemical reactions or molecular properties depending on the electronic struc-

ture. Furthermore description of the intermolecular interactions is typically limited to

parametrized pairwise potentials and thus lacks accuracy.

In recent years, a number of hybrid methods combining the accuracy ofab initio

and the efficiency of molecular mechanics by treating different parts of the system with

different levels of theory has been developed18–21.

The hybrid methods can be subcategorized into continuum and discrete ones

(Fig. 1.1). The continuum methods consider the environment to be uniform, as the

name suggests. Its effects on an individual molecule is represented with a potential that

depends only on the shape of the cavity the molecule allocates and the bulk properties

2

of the matter. The simplicity of continuum models makes them computationally effi-

cient, but limit their accuracy. The discrete methods treat individual molecules of the

environment explicitly, for instance by including all the neighbor molecules within a

certain radius in the model. Unlike the fullyab initio methods, the molecules of the

environment are described classically, i.e., by a molecular mechanics (MM) force field.

The discrete methods are superior in the sense that their accuracy can be systematically

improved by (1) increasing the number of neighboring molecules included in the model

and by (2) using a higher level theory to describe them. Overall, the continuum models

are preferred when accurate description of the bulk is essential. For example, a shift in

the absorption spectrum of a molecule in a solvent can be well reproduced by a con-

tinuum model. They break down when the environment is non-uniform (e.g., enzyme

active center) or when relaxation is important due to the strong interaction between

solvent and solute. Such cases call for discrete methods. However, to describe bulk

behavior the discrete methods need to be coupled with a sampling technique (molecular

dynamics or Monte Carlo).

Figure 1.1: Comparison of continuum (a) and discrete (b) models for a water moleculein the environment of other water molecules.

3

Depending on the way QM and MM parts are integrated with each other, the dis-

crete QM/MM methods can be categorized as mechanical embedding and electrostatics

embedding18. Mechanical embedding computes the MM parameters from the QM wave-

function and then treats the whole system classically. Although mechanical embedding

is simple and computationally cheap, the large drawback is that the perturbation of the

QM wavefunction due to MM part is ignored. Electrical embedding explicitly accounts

for the presence of the MM part by including the perturbative potential from MM into

the Hamiltonian of the QM system. Thus, it provides better description of the electronic

structure of the QM part, at the price of the increased computational complexity.

One of the promising hybrid methods based on the electrostatic embedding is the

effective fragment potential (EFP) approach22–33.

In EFP the molecular system is decomposed into an active region and a number of

effective fragments. An example of such decomposition for uracil (active region) in

water (effective fragments) is demonstrated in Fig. 1.2.

Figure 1.2: Uracil in aqueous solution. Active region (uracil) is shown with a balls andsticks model. Effective fragments (water) are depicted by lines.

The active region is a part of the molecular system that requires accurate description,

for instance, because it participates in a chemical reaction or its electronic properties are

4

of interest. Therefore, the active region is modeled with anab initio method. The effec-

tive fragments are the spectator molecules and thus do not require high accuracy for their

internal description. However, since the interaction between the active region and the

effective fragments affects the properties of the active region, this interaction should also

be modeled with reasonable accuracy. An effective fragment in EFP is described semi-

classically by a set of multipoles (with screening coefficients), polarizability points,

dispersion points and exchange-repulsion parameters.

The interaction between the effective fragments and the active region is accounted

for by adding a one-electron term to the Hamiltonian of the active region:

H ′ = H + 〈p |V | q〉 , (1.1)

whereV is the potential due to the effective fragments.

The expression for the inter-fragment interaction energy is derived by applying

perturbation theory to the wavefunctions of two interacting molecules, as detailed in

Ref. 53. Within this approach, the total energy for fragments A and B is:

Etotal = E0 + E1 + E2 + ...

E0 = EA + EB

E1 = 〈00 |T | 00〉

E2 = −∑mn

〈00 |T |mn〉 〈mn |T | 00〉E0

mn − E000

,

(1.2)

whereEA andEB are energies of the individual isolated fragments,En are the pertur-

bative corrections,T is the electrostatics operator,E0mn is the energy of the excited state

mn for the two non-interacting fragments.

5

The first order perturbative correctionE1 is the electrostatic interaction. In EFP,

it is described by a distributed multipole expansion (section 1.2). The second order

perturbative correction consists of polarization and dispersion terms. The polarization

component of EFP is accounted for by distributed polarizabilities (section 1.3). The

dispersion is represented similarly by a set of distributed dispersion points (section 1.4).

These terms are derived from the exactab initio expressions and, therefore, their accu-

racy can be systematically improved. The higher order terms in perturbative expansion

are expected to have smaller values and are neglected in the current EFP formalism.

One of the reasons of the EFP success29 is that some of the higher order terms are

believed to cancel out each other, thus omitting them does not affect the total inter-

action energy substantially. For example, charge-transfer and exchange-induction are

similar in magnitude, but have the opposite signs and thus cancel out each other. The

exchange-dispersion partially cancels out with higher order dispersion terms (such as

induced dipole-induced quadrupole) that are also omitted in the EFP formalism29.

Perturbative treatment of intermolecular interactions described above works well for

large distances, but breaks down when molecules are close to each other and their elec-

tronic densities overlap. To account for this effect, EFP includes a charge-penetration

correction (section 1.2) to the electrostatics and the exchange-repulsion component (sec-

tion 1.5) accounting for the Pauli exclusion principle. Both terms are also derived from

the exactab initio expressions. Thus, EFP intermolecular interaction consists of four

components: electrostatics (with charge-penetration correction), polarization, disper-

sion and exchange-repulsion.

The interaction energy between the effective fragments is computed as:

Eef−ef = Eelec + Epol + Edisp + Eex−rep (1.3)

6

Since the active region is described by anab initio method, the electrostatics and

polarization interactions with the effective fragments are accounted for by additional

one-electron terms in the Hamiltonian:

H ′ = H +⟨p∣∣V elec + V pol

∣∣ q⟩ (1.4)

In the present EFP model, dispersion and exchange-repulsion interactions between

active region and the effective fragments are treated similarly to the fragment-fragment

interactions as additive corrections to the total energy.

One of the advantages of EFP is that all of its terms has been derived from the exact

ab initio expressions. Thus, the parameters necessary to describe an effective fragment

can be obtained from the wavefunction of an isolated molecule. The parameters are

computed only once for every fragment type and then can be reused for all consequent

computations. This makes EFP method very efficient. Its computational cost isO(N2)

for the system withN effective fragments, plus the cost of theab initio computation for

the active region, which does not depend on the number of fragments.

Another advantage of EFP is that, unlike standard QM/MM approaches, it is a polar-

ized force field. Therefore, the effective fragments not only polarize the active region,

but also are polarized by it (polarized embedding), as well as by each other. Therefore,

EFP describes polarization effects more accurately than standard QM/MM approaches.

The third advantage is that the accuracy of EFP can be systematically improved

by introducing higher order terms and cross terms in the perturbative expansion (e.g.

exchange-dispersion), as well as higher order terms in the expansion of the individual

components (e.g. hexapole-hexapole term for electrostatics, or induced dipole-induced

quadrupole term for dispersion).

7

The first implementation of EFP appeared in the GAMESS34 quantum chemistry

package, which is developed by the original authors of the EFP method22. By now

EFP has been successfully applied for treating the environmental effects in a number

of molecular systems35–41. In addition, due to the accurate description of the inter-

fragment interactions, EFP has been successfully used to study the relative stabilities of

the molecular clusters and the dissociation energy profiles27,28,42–46.

The primary goal of this work was to implement the EFP method in Q-Chem47. The

benefits of such implementation are: (1) the extended availability of the EFP method, (2)

the ability to generate EFP fragments parameters with a higher level methods available

in Q-Chem (such as CCSD) and (3) creating the base for future development of EFP in

which the active region can be modeled with a correlatedab initio theorya.

The second goal was to apply the developed EFP implementation to the benzene

clusters. These systems are interesting due to theirπ-π stacking interactions, which are

similar in nature to those in DNA48.

The individual components of EFP are described in details in the four following

sections. Section 1.6 describes the developed implementation of EFP in Q-Chem. Its

application to benzene trimers and higher oligomers is presented in section 1.7. The last

section of the chapter presents summary of the results and conclusion.

aThe present GAMESS implementation only enables a non-correlated description of the active region(HF or DFT).

8

1.2 Electrostatics

Electrostatic component of the EFP energy accounts for Coulomb interactions. For

molecular systems with hydrogen bonds or for strongly polar molecules, electrostatics

has the leading contribution to total EFP energy49.

EFP electrostatics was devised to provide an accurate, yet compact and computa-

tionally efficient description of the molecular electrostatic properties22. Buckingham49

has shown that the Coulomb potential of a molecule can be expressed using multipole

expansion at a single spacial pointk:

V eleck (x) =

qk

rkx

+

x,y,z∑a

µkaFa(rkx) +

1

3

x,y,z∑a,b

ΘkabFab(rkx)−

1

15

x,y,z∑a,b,c

ΩkabcFabc(rkx), (1.5)

whereq, µ, Θ, andΩ are net charge, dipole, quadrupole, and octopole, respectively, and

Fa, Fab, andFabc are the electric field, field gradient, and field second derivatives, due

to the effective fragment. This expression converges to the exact electrostatic poten-

tial when an infinite number of terms are included. However, it is known to converge

slowly50 and thus requires a large number of terms to achieve a reasonable accuracy,

even if such expansions are assigned to the individual atoms in the molecule, as in Mul-

liken population analysis51. That makes a single point or atomic based representations

of the molecular electrostatic potential impractical.

In his work on distributed multipoles analysis50,52–54, Stone has shown that a better

description of the Coulomb potential can be obtained by using many points as centers

of the multipole expansions. In fact, for a system with a wavefunction expressed in the

basis ofN gaussians, the Coulomb potential can be described exactly with a finite num-

ber of multipoles located atN2 points corresponding to the gaussian products centers55.

However, this representation becomes inefficient for systems with a large basis.

9

Numerical tests conducted on a number of systems have demonstrated that an accu-

rate representation of the electrostatic potential can be achieved by using multipoles

expansion around atomic centers and bond midpoints (that is, the points known to have

high electronic density) and truncating multipoles expansions after the octopoles50,52.

This representation has been adopted by EFP22,25.

Thus, electrostatic potential of an effective fragment is specified by charges, dipoles,

quadrupoles and octopoles placed at atomic centers and bond midpoints of a molecule.

An example of such expansion for a water molecule is shown in Fig. 1.3.

Figure 1.3: Distributed multipole expansion points for a water molecule.

The energy of electrostatic interaction for a system of effective fragments can be

expressed as:

Eelec =1

2

∑A

∑B 6=A

(∑k∈A

∑l∈B

Eeleckl +

∑I∈A

∑J∈B

EelecIJ +

∑I∈A

∑l∈B

EelecIl

), (1.6)

whereA andB are the fragments,k andl are the multipole expansion points,I andJ

are the nuclei.

10

The electrostatic interaction between two expansion points is computed using the

following equation22:

Eeleckl = Ech−ch

kl + Ech−dipkl + Ech−quad

kl +

Ech−octkl + Edip−dip

kl + Edip−quadkl + Equad−quad

kl

(1.7)

The specific equations for charge-charge (D.1), charge-dipole (D.2), charge-quadrupole

(D.3), charge-octopole (D.4), dipole-dipole (D.5), dipole-quadrupole (D.6) and

quadrupole-quadrupole (D.7) are provided in Appendix D.

Nuclei-multipole and nuclei-nuclei terms are given by equations (1.8) and (1.9)

respectively22.

EelecIl = Enuc−ch

Il + Enuc−dipIl + Enuc−quad

Il + Enuc−octIl (1.8)

EelecIJ =

ZIZJ

RIJ

(1.9)

Equations for nuclei-charge (D.8), nuclei-dipole (D.9), nuclei-quadrupole (D.10) and

nuclei-octopole (D.11) terms can be found in Appendix D.

Electrostatic interaction between an effective fragment and theab initio part is

described by introducing perturbation toab initio Hamiltonian:

H = H0 + V (1.10)

The perturbation is a one-electron term computed as a sum of contributions from all the

expansion points and nuclei of all the effective fragments:

V =∑

p

∑q

∑A

(∑k∈A

dpq

⟨p∣∣V elec

k

∣∣ q⟩+∑I∈A

⟨p

∣∣∣∣ZI

R

∣∣∣∣ q⟩)

, (1.11)

11

whereA is an effective fragment,I is a nucleus,p andq are atomic orbitals of theab

initio part,ZI is nucleus charge,dpq is an element of the atomic density matrix andV eleck

is electrostatic potential from an expansion point.

A one-electron contribution to the Hamiltonian from an individual expansion point

consists of 4 terms, each of which originates from the electrostatic potential of the cor-

responding multipole22:

⟨p|V elec

k |q⟩

=

⟨p

∣∣∣∣qk

R

∣∣∣∣ q⟩+

⟨p

∣∣∣∣∑x,y,za µk

aa

R3

∣∣∣∣ q⟩+

⟨p

∣∣∣∣∣∑x,y,z

a,b Θkab(3ab−R2δab)

3R5

∣∣∣∣∣ q⟩

+

⟨p

∣∣∣∣∣∑x,y,z

a,b,c Ωkabc(5abc−R2(aδbc + bδac + cδab))

5R7

∣∣∣∣∣ q⟩

,

(1.12)

whereR, a, b andc are the distance and distance components between the electron and

the expansion pointk.

Electrostatics parameters

The parametersq, µ, Θ, Ω of the effective fragment electrostatics need to be computed

only once for every fragment type. These multipoles are obtained from anab initio

electronic density of the individual molecule using DMA (distributed multipoles analy-

sis) procedure described by Stone50,52. First, a molecular wavefunction expressed in a

basis ofN gaussians is computed with anab initio method. Then spherical multipoles

expansions are calculated at the centers of the gaussian basis function products. These

expansions are finite and include multipoles up toL1 + L2 order whereL1 andL2 are

angular moments of the two gaussians. At the next step, these multipoles are trans-

lated to the EFP expansion points (atomic centers and bond midpoints) as described

in Ref. 50 and truncated after the4th term. Thus obtained set of spherical multipoles

12

is then converted to traceless Buckingham multipoles using the equations provided in

Appendix E.1.

Charge penetration

Model of EFP electrostatics described above works well when electronic densities of the

effective fragments do not overlap. However, for many molecular systems the overlap

is substantial and is known as charge-penetration effect. A sketch demonstrating charge

penetration for two water molecules is shown in Fig. 1.4

Figure 1.4: Sketch of the charge penetration phenomenon for two water molecules.

The magnitude of this effect is commonly around 15% of the total electrostatic

energy and for some systems is as large as 200%27. It was demonstrated in Ref. 24 that

charge penetration can be efficiently accounted for by introducing a screening function

as a prefactor to the Coulomb potential:

V elecµk (x) →

(1− e−αµkrx−µk

)V elec

µk (1.13)

13

This screening is applied only to the potential originating from charges representing

electronic density, and not to the potential of higher multipoles, which represents elec-

tronic density variations27. Therefore only the charge-charge term is modified to account

for the charge-penetration24:

Ech−chkl =

(1− (1 + αkRkl

2)e−αkRkl

)qkql

rklif αk = αl(

1− α2l

α2l−α2

ke−αkRkl − α2

k

α2k−α2

le−αlRkl

)qkql

rklif αk 6= αl

(1.14)

Electrostatics screening parameters

Parameters a and b of the screening function are computed using a fitting procedure

described in Ref. 24 once for every type of the fragment. This procedure optimizes the

screening parameters so that the sum of differences between theab initio electrostatic

potential and the EFP electrostatic potential with screening is minimized:

∆ =∑

p

[V Elec

ab initio(p)− V ElecEFP (p)

], (1.15)

where the sum is taken over all the pointsp of the grid. The grid is constructed such that

it includes the points with substantial electronic density and excludes the points close to

nuclei and bond midpoints, which are inaccessible to multipoles of another fragments.

A projection of such grid for a water molecule is shown in Fig. 1.5.

1.3 Polarization

Polarization accounts for the intramolecular charge redistribution under the influence of

an external electric field. It is the major component of many-body interactions, which

14

Figure 1.5: The shape of the grid (monotonous gray area) for fitting the screening param-eters of a water molecule.

are important for proper description of cooperative molecular behavior. Since polariza-

tion cannot be parametrized by two-body potentials, it is often disregarded in molecular

mechanics models. However, for hydrogen bonded systems, up to 20% of the total

intermolecular interaction energy comes from polarization22.

The exact expression for the polarization energy appears as the second-order term of

a long-range perturbation theory53:

Epol = −∑n6=0

〈00|V |0n〉 〈0n|V |00〉En − E0

, (1.16)

wheren is the excited electronic state andV is the perturbing electrostatic potential.

Expanding electrostatic potentialV in(

1R

)nseries in equation (1.16) and then com-

bining terms with the same electric field derivatives yields the following equation:

Epol = −1

2αabFaFb −

1

6βabcFaFbFc −

1

3Aa,bcFaFbc

−1

6Bab,dcFaFbFcd −

1

6Cab,dcFabFcd + ...,

(1.17)

15

whereFa andFab are electric field derivatives,α is the dipole polarizability tensor,β is

the dipole hyperpolarizability,A, B andC are quadrupole polarizability tensors.

In order to achieve a better convergence, EFP uses distributed polarizabilities placed

at the centers of valence LMOs. This allows one to truncate the expansion (1.17) after

the first term and still retain a reasonable accuracy22. It should be noted that unlike the

total molecular polarizability tensor, which is symmetric, the distributed polarizability

tensors are asymmetric and thus all of 9 tensor components are required. Polarizability

points for a water molecule are shown in Fig. 1.6.

Figure 1.6: Distributed polarizabilities for a water molecule.

Let us discuss how polarization energy is computed for an individual effective frag-

ment in an external force field. First, induced dipoles for every polarizability point are

computed:

µk = αkFk, (1.18)

whereµk, αk andFk are the induced dipole, the polarizability tensor and the external

force field at point k.

16

Then induced dipoles are used to compute the polarization energy:

Epol =∑

k

µkFk (1.19)

Now, let us consider a system with many effective fragments and anab initio part.

In this case, the electric field acting on an effective fragment depends on: multipoles

and nuclei of other effective fragments, induced dipoles of other effective fragments and

electronic density and nuclei of theab initio part:

Fk =∑B 6=A

(∑i∈B

Fmulti (k) +

∑I∈B

F nucI (k) +

∑j∈B

F indj (k)

)

+F ab initio elec(k) + F ab initio nuc(k),

(1.20)

whereFk is the total electric field at polarizability pointk of the effective fragmentA, j

is a polarizability point,i is a multipole expansion point,I is a nucleus,Fmulti (k) is the

field due to the multipolei, F nucI (k) is the field from the nucleusI, F ind

j (k) is the field

due to the induced dipolej, F ab initio elec(k) andF ab initio nuc(k) are the forces from the

electron density and nuclei of theab initio part.

The dependence of the total electric field on other induced dipoles andab initio

density gives rise to two complications in computing the polarization energy discussed

below.

The first complication arises from the fact that an induced dipole depends on the

values of all other induced dipoles in all other fragments in according with equations

(1.18) and (1.20). Therefore, they must be computed self-consistently.

17

Another complication originates from the dependence of the induced dipoles on the

ab initio electron density, which, in turn, is affected by the field created by these induced

dipoles through a one electron contribution to the Hamiltonian:

H ind =∑

A

∑k∈A

∑x,y,za (µa

k + µak)a

R3, (1.21)

whereR, a, b andc are the distance and distance components between the electron and

the polarizability pointk, µak andµa

k are components of the induced dipole and conju-

gated induced dipole, respectively. Thus, when theab initio part is present, conjugated

induced dipoles should also be computed:

µk = αTk Fk, (1.22)

whereαTk andFk are the transposed polarizability tensor and the external force field at

the polarization point k. In the following discussion it is implied that the conjugated

induced dipoles are always computed together with the regular induced dipoles.

As a result of these self-consistency requirements, the total polarization energy is

computed with a two level iterative procedure illustrated in Fig. 1.7. The objective of the

higher level is to converge the wavefunction. The lower level is tasked with converging

the induced dipoles for a given fixed wavefunction.

In the beginning, the induced dipoles are initialized to 0 and the electronic density is

obtained through a guess procedure (such as a generalized Wolfsberg-Helmholtz proce-

dure56).

At every iteration of the higher level, the wavefunction is updated based on the values

of the induced dipoles on the previous step. Then the force from this new wavefunction

acting on the polarizability points is recomputed. The convergence criteria is that the

18

Is μ converged?

μ = 0Ψ = guess()

Compute Ψ

Compute FΨ

Compute Fμ

Compute μ

No

Is Ψ converged?

Yes

Exit

No Yes

Figure 1.7: Computing the induced dipoles through the two level self-consistent iterativeprocedure. HereΨ is theab initio wavefunction,µ is a set of induced dipoles,FΨ is theforce due to theab initio wavefunction andFµ is the force due to the induced dipoles.

wavefunction parameters (molecular orbital coefficients) are within a predefined range

from their values on the previous iteration.

At the lower level, the values of the induced dipoles are computed at every iteration

based on the electric force from theab initio part (which does not change during the

lower-level iterations), as well as the forces from multipoles, nuclei and the force from

the induced dipoles of the previous lower-level iteration. The convergence criteria is that

the difference between the new induced dipoles and those from the previous iteration is

within a predefined threshold.

Thus, when the lower-level iterative procedure exits the induced dipoles are self-

consistent and are consistent with a fixedab initio wavefunction. Convergence of the

19

higher-level iterative procedure results in the induced dipoles and theab initio wave-

function that are self-consistent and consistent with each other. If the molecular system

does not contain theab initio part, then only the lower-level procedure is required.

Once the values of the induced dipoles are obtained, the total energy of the system

can be computed as22:

E = Eother − 1

2

∑A

∑k∈A

µk(Fmultk + F nuc

k ) +1

2

∑A

∑k∈A

(µk + µk)Fab initio eleck , (1.23)

whereEother is the sum of energies of theab initio part and other components of EFP,

A is an effective fragment,k is a polarizability point,µk andµk are induced dipoles and

conjugated induced dipoles,Fmultk is the force due to the multipole expansion points of

other fragments,F nuck is the force due to the all other nuclei,F ab initio elec

k is the force due

to theab initio electronic density (F ab initio eleck =

⟨Ψ∣∣∣f elec

k

∣∣∣Ψ⟩).

In equation (1.23), the second term is the total polarization energy of all the effective

fragments. However, the second and the third term do not add up to the total polarization

energy because a part of the later is implicitly included in theab initio energy (due to

the effect of the induced dipoles on the wavefunction).

Computational cost of the polarization component consists of the cost of computing

the induced dipoles and the energy. The latter is a trivial step with the cost ofO(N2) for

N effective fragments. Thus, the former defines the overall computational cost.

At the higher level, the number of iterationsh is similar to that for theab initio

system without EFP, while at the lower level the convergence is achieved within bounded

number of iterations. Every lower-level iteration scales asO(N2), thus, the overall

computational cost isO(hN2).

If system does not have theab initio part, then only the lower-level iterative proce-

dure is required and computational cost isO(N2).

20

Polarization parameters

The polarization parameters in the EFP model are LMOs and polarizability tensors.

Both can be obtained from anab initio calculation once for every type of the fragment

and then reused in all consequent calculations. LMOs are obtained with the Boys local-

ization procedure based on theab initio electronic density57,58.

Polarizabilities are obtained based on the definition (1.24) as a derivative of the

dipole moment (µ) w.r.t. electric field (F ) using the finite differences method.

αab =∂µa

∂Fb

, (1.24)

In this approach, a dipole moment of an LMO is first computed without an electric field.

Then, by means of 3 additionalab initio computations, the LMO and its dipole moment

are computed in the presence of a small electric field acting along one of the coordinate

axis. Based on LMO dipole moments, the polarizability tensor can be computed as:

αab =µa(Fb)− µa(0)

Fb

(1.25)

1.4 Dispersion

For many molecular systems dispersion is the main attractive component of the inter-

molecular interaction energy. An accurate description of dispersion interaction is a diffi-

cult task even forab initio methods — inclusion of the electron correlation at high level

is required. An additional challenge for EFP is that its dispersion description must be

computationally efficient.

21

The exact expression (1.26) for the dispersion energy appears in the second order

term of a long-range perturbation theory.

Edisp = −∑n6=0

∑m6=0

〈00|V |mn〉 〈mn|V |00〉EA

m + EBn − EA

0 − EB0

, (1.26)

whereA andB are two interaction molecules,m andn are their excited electronic states

andV is the perturbing electrostatic potential.

By expanding electrostatic potentialV in(

1R

)nseries dispersion energy can be

expressed in the so called London series:

Edisp =C6

R6+

C8

R8+

C10

R10+ ..., (1.27)

where theCn coefficients correspond to induced dipole—induced dipole (C6), induced

dipole—induced quadrupole (C8), induced quadrupole—induced quadrupole and

induced dipole—induced octopole (C10) etc. interactions, Thus, dispersion energy can

be alternatively expressed as a sum:

Edisp = Eind dip−ind dip + Eind dip−ind quad + Eind quad−ind quad + Eind dip−ind oct + ...

(1.28)

A better convergence can be obtained by using a distributed dispersion model with

expansion points located at localized molecular orbitals centroids59. Expansion points

for a water molecule are demonstrated in Fig. 1.8.

When using distributed dispersion, a reasonable accuracy can be achieved by trun-

cating the expansion after the first term and approximating the rest as1/3 of the this

22

Figure 1.8: Distributed dispersion points for a water molecule.

term59. Induced dipole-induced dipole interaction energy between two induced dipoles

k and l can be computed by using the following equation:

Eind dip−ind dipkl =

x,y,z∑abcd

T klabT

klcd

∫ ∞

0

αkac(iν)αl

bd(iν)dν, (1.29)

whereαkac andαl

bd are dynamic polarizability tensors,iν is the the imaginary frequency,

T klab andT kl

cd are the electrostatic tensors of the second rank, i.e.,T klab =

3ab−R2klδab

R5kl

.

In general, the interaction energy described by (1.29) is anisotropic, distance depen-

dent and computationally expensive26. The approximation used by EFP includes only

the trace of polarizability tensor, which is anisotropic and distance independent.

The total dispersion energy for a system of effective fragments is:

Edisp =4

3

∑A

∑B

∑k∈A

∑l∈B

Ckl6

R6kl

, (1.30)

23

whereA andB are effective fragments,k andl are LMOs,Ckl6 is intermolecular disper-

sion coefficient, andR6kl is the distance between two LMO centroids.

Intermolecular dispersion coefficients are computed using the 12 point Gauss-

Legendre integration formula:

Ckl6 =

12∑i=1

wi2ν0

(1− ti)2αk(iνi)α

l(iνi), (1.31)

whereαk(iνi) andαl(iνi) are1/3 of the traces of the corresponding polarizability ten-

sors.

Dispersion damping

For small distances between effective fragments dispersion coefficients must be cor-

rected to take into account the charge penetration effect. In the original EFP model, the

Tang—Toennies formula60 with parameterb = 1.5 is used:

Ckl6 →

(1− e−bR

∞∑k=0

(bR)k

k!

)Ckl

6 (1.32)

However, recent studies revealed that this formula over-damps the dispersion and a more

appropriate value forb lies between2 and3 and is system dependent29. As an alternative,

Ref. 29 suggested to use a damping function dependent on the electronic density overlap

Skl between LMOsk andl:

Ckl6 →

(1− |Skl|2(1− 2ln|Skl|+ 2ln2|Skl|)

)Ckl

6 (1.33)

24

Dispersion parameters

As follows from equations (1.30) and (1.31), the dispersion parameters for an effective

fragment are the coordinates of LMO centroids and sets of dynamic polarizability ten-

sor traces for 12 frequencies for each LMO. These parameters can be computed once

for every type of an effective fragment and then reused in EFP calculations. Dynamic

polarizabilities can be computed using dynamic time-dependent Hartree-Fock (TDHF)

or dynamic time-dependent density functional theory (TDDFT) as described in Ref. 26.

1.5 Exchange-repulsion

Exchange-repulsion originates from the Pauli exclusion principle, which states that for

two identical fermions the wavefunction must be anti-symmetric. Inab initio methods

such as Hartree-Fock and DFT, it is accounted for by the exchange operator. In classical

force fields, exchange-repulsion is introduced as a positive (repulsive) term, e.g.,(

1r

)12in the Lennard-Jones potential. However, since the Pauli exclusion principle is intrin-

sically quantum, its classical descriptions are not accurate. EFP uses a wavefunction-

based formalism to account for the electron exchange. Thus exchange-repulsion is the

only non-classical component of EFP and the only one that is repulsive.

The EFP exchange-repulsion energy term has been derived from the exact equation

for the exchange-repulsion energy23,61:

Eexch =

⟨ΨAΨB|AHAB|ΨAΨB|

⟩⟨ΨAΨB|AΨAΨB

⟩ − 〈ΨAΨA|VAB|ΨBΨB〉 − EA − EB, (1.34)

25

whereA andB are two molecules described byΨA andΨB wavefunctions,VAB is the

intermolecular Coulomb interaction, andA is the antisymmetrization operator:

A = P0 − P1 + P2 − P3 + ..., (1.35)

wherePi is a permutation ofi electron pairs consisting of one electron from each

molecule.

Truncating sequence (1.35) after the second term, applying an infinite basis set

approximation and a spherical gaussian overlap approximation as shown in Ref. 61

leads to the expression for the exchange-repulsion energy in terms of localized molecu-

lar orbitals:

Eexch =1

2

∑A

∑B 6=A

∑i∈A

∑j∈B

Eexchij (1.36)

Eexchij = −4

√−2ln|Sij|

π

S2ij

Rij

−2Sij

(∑k∈A

FAikSkj +

∑l∈B

FBjl Sil − 2Tij

)

+2S2ij

(∑J∈B

−ZJR−1iJ + 2

∑l∈B

R−1il

∑I∈A

−ZIR−1Ij + 2

∑k∈A

R−1kj −R−1

ij

),

(1.37)

whereA andB are the effective fragments,i, j, k andl are LMOs,I andJ are nuclei,

S andT are intermolecular overlap and kinetic energy integrals,F is the Fock matrix

element.

The positions of LMO centroids for a water molecule are shown in Fig. 1.9.

Formula for theEexchij involves intermolecular overlap and kinetic energy integrals,

which are expensive to compute. In addition, since equation (1.37) is derived within an

26

Figure 1.9: The exchange-repulsion LMO centroids for a water molecule.

infinite basis set approximation, it requires a reasonably large basis set to be accurate.

These factors make exchange-repulsion the most computationally expensive part of the

EFP energy calculations of moderately sized systems.

Large systems require additional considerations. Equation (1.36) contains a sum

over all the fragment pairs and its computational cost formally scales asO(N2) with

the number of effective fragmentsN . However, exchange-repulsion is a short range

interaction —- the overlap and kinetic energy integrals of LMOs vanish very fast with

the interfragment distance. Therefore, by employing a distance based screening, the

number of overlap and kinetic energy integrals scales asO(N). As a result, for large

systems exchange-repulsion becomes less computationally expensive than long range

components of EFP (such as Coulomb interactions).

Exchange-repulsion parameters

The parameters required for equation (1.37) are the Fock matrix elements and the LMOs

(orbital coefficients, basis and centroids). They can be computed once for every type of

27

the fragment by performing a Hartree-Fock calculation on the molecule followed by an

an orbital localization procedure (such as Boys LMO)23.

1.6 Implementation of EFP in Q-Chem

This section describes the new implementation of the EFP method in the Q-Chem elec-

tronic structure package.

As detailed in section 1.1, within EFP formalism the molecular system is decom-

posed into an active region and a number of effective fragments. Since the active region

is treated by anab initio method, EFP requires an interface with a general-purpose quan-

tum chemistry package. Our implementation was developed using C++62 as a module

within the state of-the-art quantum chemistry package Q-Chem47. The standard template

library (STL) was extensively used throughout the implementation.

The developed code consists of the two independent parts: EFP energy and EFP

parameters. The former is responsible for computing energy of the molecular system,

whereas the latter allows one to obtain effective fragment parameters.

Computing EFP energy

The EFP energy part of the implementation allows one to compute total energy of a

system comprised of an optional active region and a number of effective fragments.

The total energy includes contributions from all the EFP components described in the

four previous sections: electrostatics (with charge-penetration correction), polarization,

dispersion, exchange-repulsion.

A simplified high-level class diagram for the EFP energy code is shown in Fig. 1.10

(a dashed arrow represents a dependence). The main classEFP is responsible for man-

aging the interactions between other classes and providing the interface to the rest of

28

the Q-Chem package. ClassesElectrostatics, Polarization, DispersionandExchange-

repulsionencapsulate the functionality required for the corresponding individual energy

components. ClassPolarizationhas the dependence on the classElectrostaticsbecause

induced dipoles depend on the field created by multipoles. The responsibility of the

ConfigReaderclass is reading the effective fragment parameters and configuration of

the molecular system, and storing them in the internal representation using classesFrag-

mentParamsandFragment.

Dispersion

EFP

Electrostatics

Polarization ExchangeRepulsion ConfigReader

FragmentParams Fragment

Figure 1.10: Class diagram for the EFP energy part of the implementation.

FragmentParams

std::vector< double > er_fock_matrix

std::vector< PolarizationPoint >polarizabilities

std::vector< XYZPoint >er_lmos

std::vector< std::vector< double > > er_wavefunction

std::vector< DispersionPoint >dispersions

std::string

er_basis

std::vector< EFPAtom >

atoms

std::vector< MultipolePoint >

multipoles

Figure 1.11: The internal structure of theFragmentParamsclass.

29

The internal structure forFragmentParamsclass is shown in Fig. 1.11. The labels

above the arrows indicate the name of the data member. This class encapsulates all

the parameters for a given EFP fragment type: atoms, multipoles, dispersion points,

polarizabilities and parameters for exchange-repulsion (which start with erprefix).

The classFragmentrepresents an individual effective fragment in the molecular sys-

tem. It stores the fragment type, its position and orientation. The position is specified

in terms of translation in Cartesian coordinates (x, y, z) relative to the origin of theab

initio system. The orientation is specified by the Euler angles (α, β, γ) in terms of the

3 consequent rotations, as explained in Fig. 1.12. The rotation matrix corresponding to

these angles is given in Fig. 1.13.

N

x

y

z

Z

X

Y

α

β

γ

Figure 1.12: Representing the orientation with the Euler angles. xyz is the fixed system,XYZ is the rotated system, N is the intersection between the xy and XY planes calledthe line of nodes. Transformation between the xyz and XYZ systems is given byαrotation in the xy plane, followed byβ rotation in the new y’z’ plane (around the Naxis), followed byγ rotation in the new x”y” plane.

30

R = cosα cosγ − sinα cosβ sinγ −cosα sinγ − sinα cosβ cosγ sinβ sinα

sinα cosγ + cosα cosβ sinγ −sinα sinγ + cosα cosβ cosγ −sinβ cosα

sinβ sinγ sinβ cosγ cosβ

Figure 1.13: The rotation matrix corresponding to the Euler anglesα, β, γ.

The positionx′i, y′i, z

′i of an atomi of an effective fragment can be computed as:

x′i

y′i

z′i

= R

xi

yi

zi

+

x

y

z

, (1.38)

wherexi, yi, zi are the coordinates of the atom in the fragment frame,R is the rotation

matrix defined in Fig. 1.13, andx, y, z are the fragment coordinates.

The flow of control between Q-Chem and the EFP module is presented in Fig. 1.14.

TheReadParamsandReadFragmentscalls instruct the EFP module to read the defini-

tions of the fragment types and the molecular system configuration from the Q-Chem

standard input (discussed in the next subsection). TheInit call checks the effective

fragments parameters and the configuration for consistency, creates and initializes the

instances ofElectrostatics, Polarization, DispersionandExchange-Repulsionclasses.

TheUpdateRepresentationmethod is called every time when the geometry of the molec-

ular system changes. It performs rotation and translation of atoms, multipoles, polariza-

tion tensors, dispersion points and LMOs in accordance with the coordinates and ori-

entation for each individual fragment. TheNuclearEnergyfunction computes the EFP

energy components, which are independent from the wavefunction ofab initio system

(inter-fragment electrostatics, dispersion and exchange-repulsion).

31

If the active region is present, then for every iteration of the SCF procedure Q-Chem

calls theHamiltonianmethod and theWFDependentEnergyfunction. TheHamiltonian

method updates the one-electron term of the active region Hamiltonian in accrodance

with the equations (1.10) and (1.21). After updating the wavefunction accordingly, Q-

Chem callsWFDependentEnergyto compute the wavefunction-dependent part of the

EFP energy (polarization).WFDependentEnergytakes the electronic density of the

active region as a argument. If the active region is absent, Q-Chem callsWFDepen-

dentEnergyonly once with a zero electronic density as an argument.

Finally, thePrintSummarymethod is called to print the individual energy compo-

nents for the inter-fragment EFP energy.

Q-Chem EFP

ReadParams()

ReadFragments()

Init()

UpdateRepresentation()

NuclearEnergy()

Hamiltonian()

WFDependentEnergy()

energy

energy

for every SCF cycle

PrintSummary()

hamiltonian

for every new geometry

Figure 1.14: The flow of control between Q-Chem and EFP energy module.

32

The correctness of the EFP energy implementation in Q-Chem was verified by

performing the energy computation on a few dozens of examples and comparing the

results with the GAMESS implementation developed by the original authors of the EFP

method. The examples used for testing include systems composed of water, ammonia

and benzene molecules in different configurations.

EFP input format

The Q-Chem standard input consists of a number of sections delimited by$keywordand

$end. The mandatory sections are$molecule, which describes the geometry ofab initio

part, and$remsection, which specifies parameters of theab initio computation. Other

sections are specific to different Q-Chem modules.

The EFP code extends the Q-Chem input with 2 new sections:$efp paramsand

$efp fragments, which describe the parameters of the EFP fragment types and define

the molecular system by providing fragments’ positions and orientations, respectively.

In the EFP input sections, all coordinates are in Angstroms, unless Q-Chem’s

INPUT BOHRkeyword is set toTRUE, in which case all the coordinates are in Bohr,

all angles are in radians and other values are in atomic units. All keywords used in the

EFP input section are case insensitive.

Fig. 1.15 presentswhenhe format for the$efp paramssection. It consists of the

parameters’ definitions for one or more fragment types. A definition starts by providing

the name of the fragment type, listing all atoms and their coordinates. It is followed

by the list of all the multipole expansion points (mult). Each multipole expansion point

is defined by its coordinate, multipoles and a damping parameter (DAMP). Multipoles

with different angular momentum are specified on the separate lines in an arbitrary order.

In thier definitionC is charge,D , Q andO are dipole, quadrupole and octupole

components, respectively. All multipoles are optional, but when provided, then all their

33

components should be specified in the order demonstrated in Fig. 1.15. The damping

parameter is optional and should be preceeded by thecdampkeyword.

Next in the fragment definition are the polarization points (pol). They are specified

by the coordinates and components of the polarization tensor P.

The dispersion points (disp) are described similarly by the coordinates and the traces

of the dynamic polarizability tensors Cat 12 different frequencies.

The last subsection describes the parameters required for the exchange-repulsion:

er basis specifies the basis used, erfock provides the Fock matrix, erlmos specifies the

LMO centroids and erwavefunction defines each LMO in terms of the atomic orbitals.

The subsections for different EFP components are optional and can appear in an arbi-

trary order.

Fig. 1.16 presents the format for the$efp fragmentssection. Every effective frag-

ment in the molecular system is defined by the fragment type, followed by its Cartesian

coordinates (X, Y, Z) specifying the translation relative to theab initio system origin

and Euler angles specifying the fragment orientation (α, β, γ). If the orientation is not

provided, the default values of the angles are set to 0. The fragment type should match

one of the fragments in the$efp paramssection.

The active region is specified in the$moleculesection the same way as anab initio

system is specified in Q-Chem47. In order to define a molecular system composed of

the effective fragments only, the keywordEFP FRAGMENTSONLY should be set to

TRUEin the$remsection of the Q-Chem standard inputb.

A complete example of the Q-Chem standard input for the system composed of 3

water molecules (one as an active region and two others as effective fragments) is given

in Appendix F.

bSince the$moleculesection is mandatory in Q-Chem, even if the active region is disabled it shouldstill contain at least one atom, for instance, He. In the nutshell, assigningEFP FRAGMENTSONLYkeyword toTRUEsimply disables any interaction between theab initio part and the effective fragments.

34

$efp params

fragment <fragment type 1>

ATOM X Y Z

...

mult X Y Z

[cdamp DAMP]

[C]

[DX DY DZ]

[QXX QXY QXZ QYY QYZ QZZ]

[OXXX OXXY OXXZ OXYY OXYZ OXZZ OYYY OYYZ OYZZ OZZZ]

...

pol X Y Z

PXX PXY PXZ PYX PYY PYZ PZX PZY PZZ

...

disp X Y Z

S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12

...

er basis BASIS

er fock

F1 F2 F3 F4 ...

er lmo

X Y Z

...

er wavefunction

1 C1 C2 C3 C4 ...

2 C1 C2 C3 C4 ...

...

fragment <fragment type 2>

...

$end

Figure 1.15: The format for the$efp paramssection.

$efp fragments

<fragment type> X Y Z [ α β γ]

...

$end

Figure 1.16: The format for the$efp fragmentssection.

Computing EFP parameters

The EFP parameters part of the code computes the electrostatics, charge-penetration and

polarization parameters for an isolated molecule. The parameters need to be computed35

only once for every fragment type and then can be reused in all subsequent EFP energy

computations.

Since the EFP model has been derived from theab initio theory, the first step in

computing the EFP parameters for a molecule is to obtain its electronic density. This

can be achieved by means of anyab initio method available in Q-Chem that supports

explicit electronic density (for instance: HF, DFT, CCSD).

From the electronic density, the electrostatic parameters (multipoles) are computed

using the distributed multipole analysis procedure as described in section 1.2. The mul-

tipole expansion is computed up to octupoles. The points for the expansion are the

nuclei and the bond midpoints.

The screening parameters accounting for the charge-penetration are computed by

the fitting procedure, which minimizes the difference between the exact electrostatic

potential and potential from the multipoles expansion points multiplied by the screening

function, as detailed in the section 1.2. The optimization of screening parameters is

based on the Newton-Raphson algorithm63.

The polarization parameters are computed using the finite difference method based

on 4ab initio computations: one with no external field and 3 with a small (by default

10−5 V/m) electrostatic field along one of the coordinate axis, as described in section

1.3. In each computation, the localized molecular orbitals are obtained with the Boys

localization procedure57,58.

The computation of the dispersion and exchange-repulsion parameters have not yet

been implemented in Q-Chem. They can be obtained from the quantum chemistry pack-

age GAMESS34, using the code developed by the original authors of the EFP method.

Using itsmakefpjob type, all the EFP parameters can be produced in the GAMESS for-

mat. An example of the GAMESS input file generating the EFP parameters for benzene

and the output produced are shown in Appendix A. To facilitate the conversion from

36

the GAMESS format to the Q-Chem format, a converter script (Appendix C) has been

developed in Python64. It takes a file with the GAMESS EFP parameters as a command

line argument and prints out the parameters in the Q-Chem format. The conversions

performed internally by the script are the syntax conversion, unit conversion (from Bohr

to A) and redundancy elimination (replacing a dynamic polarizability tensor with its

trace).

The Q-Chem keywords affecting the EFP parameters computation are provided in

Appendix G.

1.7 EFP application: Beyond benzene dimer

Aromatic π − π interactions have the major influence on the structure and properties

of many biologically important molecular systems65. They govern nucleotide stacking

patterns in DNA and RNA48, control the ternary structures of proteins66, play an impor-

tant role in the intercalation of drugs with DNA67, affect the properties of the techno-

logically important polymers aramids68, impact the crystal structure of many aromatic

compounds69, and stabilize host—guest complexes70,71.

Studying theπ−π interactions is a major challenge for both experiment and theory72.

The experimental investigations are difficult due to the complex environments in which

these interactions occur (proteins, DNA). The weakness of individualπ−π interactions

and the flatness of the PES along the coordinates defining the relative position of the

aromatic rings make it difficult to compute even for small model systems in the gas

phase73–75.

The theoretical studies ofπ−π interactions, as of any non-covalent interactions, are

challenging due to the weakness of these interactions relative to typical covalent interac-

tions. Their proper treatment requires high accuracyab initio methods [e.g. CCSD(T)]

37

and large basis sets with diffuse functions [e.g. aug-cc-pVTZ]76, which both contribute

to the high computational complexity. At present, high accuracyab initio computations

are feasible only for systems with a fewπ − π interacting aromatic rings.

However, in nature several interactingπ−π moieties are usually present, e.g., many

nucleotides are responsible for the secondary secondary structure of DNA48. Therefore,

it is important to understand whether multipleπ−π interactions can be treated as a sum

of the pair-wise interactions between neighboring aromatic rings or there are significant

non-additive many-body interaction components.

Theoretical studies of multipleπ−π interactions have been so far limited due to the

computational complexity reasons discussed above77–80 . An interesting approach was

taken in the work by Tauer and Sherrill78 who studied benzene trimers and tetramers

using MP2 with a specially selected basis set. MP2 is known to overestimate the binding

energies. For instance, MP2 with a large basis set (aug-cc-pVQZ) overestimates the

dissociation energy of the benezene dimer by 70%76. In contrast, small basis sets are

known to underestimate binding energies. Thus, MP2 with a small basis set may benifit

from the cancelation of errors. The authors selected a basis (cc-pVDZ+78) such that the

errors from MP2 and the basis cancel out each other. Binding energies of the different

benzene dimer configurations at the MP2/cc-pVDZ+ level of theory are within 15%

from the best known theoretical values [CCSD(T)/CBS]. This level of theory has been

used to investigate the nature of binding in benzene trimers and tetramers. The goal

was to understand the importance of the total nonadditive and the individual many-

body interaction energies. The authors concluded that the binding energy of trimers

and tetramers can be described as a simple nearest neighbor sum with 10% accuracy,

and that non-additive (many-body and long range two-body) interactions will become

increasingly important for larger clusters.

38

An important question for theπ − π stacked systems is how the share of the non-

additive energy component grows with the increase in the number of aromatic rings in

the cluster: whether it becomes significant in large clusters or does it stay similar to

that of benzene trimers. A related question is if the binding energy of multipleπ − π

interactions can be well approximated by the sum of the pair-wise nearest neighbor

interaction energies. To answer these fundamental questions, one needs to study a series

of the increasingly large aromatic clusters and observe the asymptotic behavior of the

non-additive energy component. Unfortunately, aromatic clusters with more than a few

benzene rings are beyond the reach of high levelab initio theories. Even using the

proposed in Ref. 78 MP2/cc-pVDZ+ method, computing the energy of the cluster with

a few tens of the benzene rings is presently unfeasible.

The approach to the multipleπ − π interactions problem taken in this work is to

employ the effective fragment potential (EFP) — a semi-empirical method designed

specifically to account for the inter-molecular interactions. In recent works27,28,81, EFP

has been used in the extensive studies of benzene dimers. EFP was found to be in

excellent agreement with high-levelab initio theories. For the three conformers of the

benzene dimer shown in Fig. 1.17, the maximum binding energy was within 0.4 kcal/-

mol and the intermolecular separation was within 0.2A from the values obtained with

CCSD(T)/aug-cc-pVQZ. Additionally, the authors found that the EFP energy profiles

(Fig 10.2 and 10.3 in Ref. 28) for the individual types of interactions (electrostatics,

polarization, dispersion and exchange-repulsion) along the intermolecular separation

coordinates were in excellent agreement with the symmetry-adapted perturbation theory

(SAPT). Thus, not only the total binding energies of the benzene dimers are reproduced

well by EFP, but also the individual types of the interactions are properly accounted for.

39

`

S T PD

RR

R1

R2

Figure 1.17: Sandwitch (S), t-shaped (T) and parallel-displaced (PD) structures of thebenzene dimer.

The discussion above indicates that EFP is an excellent candidate for studying the

systems withπ−π interactions. This work uses our new implementation of EFP to inves-

tigate multipleπ − π interactions. Firstly, the EFP binding energies and geometries for

the sandwich (S), t-shaped (T) and parallel-displaced (PD) structures are compared with

those obtained at differentab initio levels of theory, including the best known to date

theoretical estimates [CCST(T)/CBS]. Then, eight different benzene trimers are studied

in order to understand how well EFP accounts for the non-additive energy contributions.

Finally, the asymptotic behavior of multipleπ− π interactions is investigated in 3 types

of benzene oligomers.

Benzene dimers

The parameters for benzene used throughout this work are those recommended in

Ref. 28. The geometry of the monomer [d(C-C)=1.3942, d(H-H)=1.0823] computed

at the MP2/aug-cc-pVTZ level of theory is from Ref. 82. The effective fragment

potential for the benzene monomer has been generated with GAMESS34 at the HF/6-

311G++(3df,2p) level of theory. As explained in Ref. 26, a basis set of such high qual-

ity is necessary in order to generate accurate dispersion parameters. The input file used

40

for computing EFP parameters is provided in Appendix A. The parameters have been

converted to the Q-Chem format using the script provided in Appendix C.

The benzene dimer structures shown in Fig. 1.17 where constructed analogously to

Ref. 82. The geometries and energies (table 1.1) of 3 benzene dimers corresponding

to the maximum binding at the EFP level of theory were found by performing scans

along the degrees of the freedom shown in Fig. 1.17 with a resolution of 0.1A. At these

geometries, we verified that our implementation computes the total binding energies

and individual energy components correctly by comparing with an analogous GAMESS

computation. The geometries and maximum binding energies we found are only slightly

different (0.1A and 0.15 kcal/mol, respectively) from those obtained in Ref. 28. The

discrepancy is due to the different techniques of the electrostatics screening.c

Table 1.1: The geometric parameters and energies of the benzene dimers

S T PD

Geometric parameters,A R = 4.0 R = 5.2 R1 = 3.9 , R2 = 1.2

Energy, kcal/mol -1.96 -2.38 -2.18

Table 1.2 compares the energies of the 3 benzene dimer structures at different levels

of theory. The best known theoretical estimates are the CCSD(T)/CBS values, which

will serve as a benchmark for our comparison. We can see that CCSD(T)/aug-cc-pVQZ

gives the results within 0.15 kcal/mol from the CBS result. Thus, aug-cc-pVQZ can be

considered sufficient. However, the MP2 results with this basis set are significantly off

(error up to 70%), which is attributed to the overestimation of electron correlation by

MP2 in the benzene dimer83. The method (MP2/aug-cc-pVDZ+) proposed in Ref. 78

cIn Ref. 28 in addition to the screening of charges, which our implementation also performs, the higherorder multipoles (dipoles, quadrupoles, octupoles) were also screened. However, as it was demonstratedin Ref. 27, the accuracy of charge-charge only and charge-charge plus high-order electrostatic screeningsis similar. Therefore, we omitted screening of the higher order multipoles in our implementation.

41

is within 0.4 kcal/mol due to the cancellation of errors. Our EFP results are within 0.6

kcal/mol and thus are of accuracy comparable to MP2/aug-cc-pVDZ+.

Table 1.2: Dissociation energies of the benzene dimers at different levels of theory.

Methoda S T PD

MP2/aug-cc-pVDZ* -2.83 -3.00 -4.12

MP2/aug-cc-pVTZ -3.25 -3.44 -4.65

MP2/aug-cc-pVQZ* -3.35 -3.48 -4.73

MP2/aug-cc-pVQZ -3.37 -3.54 -4.79

MP2/cc-pVDZ+ -1.87 -2.35 -2.84

CCSD(T)/cc-pVDZ* -1.33 -2.24 -2.22

Est. CCSD(T)/aug-cc-pVQZ* -1.70 -2.61 -2.63

Est. CCSD(T)/CBS -1.81 -2.74 -2.78

EFP -1.96 -2.38 -2.18

a The results for theab initio methods are quoted from Ref. 76

Benzene trimers

Using the geometric parameters obtained for dimers (table 1.1), linear benzene trimers

shown in Fig. 1.18 have been constructed. The cyclic structure was obtained by mini-

mizing the binding energy along the intermolecular separation and rotation in the plane

connecting the molecular centers. The minimum corresponds to the molecular sepa-

ratioin of 5.3A, tilted 15 from the perpendicular. The geometries of the trimers are

provided in Appendix B. These trimer structures are analogous to the trimers studied in

Ref. 78 by MP2.

Binding energies for the trimers along with their breakdown into the components are

shown in tables 1.3 and 1.4.E2 is total two-body interaction energy:

E2 = E2(12) + E2(13) + E2(23), (1.39)

42

Table 1.3: Total and many-body interaction energies of the benzene trimers (first group).

S PD T1 T2

EFP

E2(12) -1.96 -2.18 -2.38 -2.40

E2(13) 0.01 0.01 0.02 0.00

E2(23) -1.96 -2.18 -2.40 -2.38

E2 -3.91 -4.35 -4.77 -4.79

E3 0.02 -0.02 0.13 0.06

Etotal -3.89 -4.38 -4.64 -4.73

Etotal(dimer) -3.92 -4.36 -4.76 -4.76

Enon−add 0.03 -0.02 0.12 0.03

∆ 0.77% -0.37% 2.66% 0.57%

MP2/cc-pVDZ+

E3 0.03 0.00 0.08 0.06

Etotal -3.83 -5.88 -4.64 -4.73

Etotal(dimer) -3.74 -5.68 -4.70 -4.70

Enon−add -0.09 -0.20 0.06 -0.03

ECPnon−add 0.03 -0.06 0.1 0.03

∆ 0.78% -1.02% 2.16% 0.63 %

CCSD(T)/cc-pVDZ+

E3 0.04 0.01 0.07

Etotal -0.90 -1.84 -3.14

Etotal(dimer) -0.86 -1.72 -3.20

Enon−add -0.04 -0.12 0.06

ECPnon−add 0.06 0.00 0.10

∆ 6.67% 0.00% 3.18%

E2(NM) - two body interaction between benzenesN and M , E2 - total two bodyinteraction,E3 - three-body interaction,Etotal - total energy of trimer,Etotal(dimer) -total energy of trimer estimated form dimer,Enon−add - non-additive energy component,ECP

non−add - non-additive energy component with counterpoise correction,∆ - share ofthe non-additive energy.

43

Table 1.4: Total and many-body interaction energies of the benzene trimers (secondgroup).

C PD-S S-T PD-T

EFP

E2(12) -2.19 -1.96 -1.96 -2.38

E2(13) -2.19 0.00 -0.12 -0.11

E2(23) -2.19 -2.18 -2.38 -2.18

E2 -6.57 -4.14 -4.46 -4.67

E3 -0.24 0.01 -0.07 -0.06

Etotal -6.81 -4.12 -4.52 -4.73

Etotal(dimer) -6.57 -4.14 -4.34 -4.56

Enon−add -0.25 0.02 -0.18 -0.17

∆ -3.63% 0.43% -4.05% -3.66%

MP2/cc-pVDZ+

E3 -0.33 0.02 -0.03 0.00

Etotal -7.88 -4.86 -4.49 -5.42

Etotal(dimer) -7.32 -4.71 -4.22 -5.19

Enon−add -0.56 -0.15 -0.27 -0.23

ECPnon−add -0.32 -0.01 -0.15 -0.13

∆ -4.06% -0.21% -3.34% -2.40%

CCSD(T)/cc-pVDZ+

E3 -0.25

Etotal -5.09

Etotal(dimer) -4.62

Enon−add -0.47

ECPnon−add -0.26

∆ -5.11%

E2(NM) - two body interaction between benzenesN and M , E2 - total two bodyinteraction,E3 - three-body interaction,Etotal - total energy of trimer,Etotal(dimer) -total energy of trimer estimated form dimer,Enon−add - non-additive energy component,ECP

non−add - non-additive energy component with counterpoise correction,∆ - share ofthe non-additive energy.

44

S T1 T2 PD

C PD-T PD-S S-T

Figure 1.18: The structures of eight benzene trimers.

whereE2(NM) is binding energy for the dimers, created by removing one benzene from

the trimer and computed in the full basis set of the trimer.E3 is three-body interaction

energy computed as:

E3 = Etotal − E2 (1.40)

Etotal(dimer) is total binding energy estimated as the sum of nearest neighbor pair-wise

binding energies.Enon−add andECPnon−add are the non-additive energy components with-

out and with the counterpoise correction:

Enon−add = Etotal − Etotal(dimer)

ECPnon−add = Etotal − ECP

total(dimer)

(1.41)

45

∆ is the share of the non-additive interaction energy which EFP computes as:

∆ =Enon−add

Etotal

(1.42)

andab initio methods as:

∆ =ECP

non−add

Etotal

(1.43)

To evaluate how well EFP describes multipleπ − π interactions, it is necessary to

conduct a comparison withab initio methods. The quality of the description of the

two-body interactions (between two benzene rings) has been investigated in the exten-

sive studies of the benzene dimers energy profiles in Ref. 28. The conclusion was that

EFP results are in good agreement with the high levelab initio theories [CCSD(T) and

SAPT]. Three-body interactions for EFP, MP2/cc-pVDZ+ and CCSD(T)/cc-pVDZ+ are

compared in tables 1.3 and 1.4. The first important observation is that theab initio

E3 values quoted from Ref. 78 were computed using the counterpoise correction to

account for the basis set superposition error, which results from the basis set incom-

pleteness. Three-body interaction energies were calculated by subtracting from total

binding energy of the trimer energies of the 3 possible benzene dimers computed in the

full set basis of the trimer.

All EFP many-body interactions are due to the polarization component. Thus, itsE3

term is three-body polarization energy. From tables 1.3 and 1.4, we can see that three-

body interaction energies computed with EFP have the same order of magnitude as for

theab initio methods. For most trimers studied here, theE3 correction is rather small

and miscellaneous errors may have a significant impact on its value. The exception is the

cyclic benzene trimer (see Fig. 1.18), which is a true three-body system demonstrating

substantial three-body interactions at all the considered levels of theory. For this trimer

E3 computed with EFP is within 0.01 kcal/mol from the CCSD(T) value. The fact that

46

these values almost coincide might be accidental, but their closeness implies a certain

level of accuracy in the treatment of three-bodyπ− π interactions by EFP. On the basis

of the above comparison, we conclude that EFP provides a good description of three-

body interactions in benzene trimers.

Another interesting conclusion can be drawn from the comparison of total non addi-

tive interaction energies (Enon−add). Firstly, Enon−add for EFP are greater than for

CCSD(T) and MP2 for all eight considered trimers. However, we concluded that this

effect is mostly due to the basis set superposition error (BSSE).Enon−add is computed

as a difference between trimer binding energy and energies of two dimers (which were

derived from the trimer by removing one of the edge benzenes). Thus, the basis sets

used for the trimer and the dimers energy computations are different. Since trimer has

a larger basis set, its nearest neighborπ − π interactions are stronger than in the corre-

sponding dimers. That is, the additional basis functions contribute to the wavefunction

relaxation, leading to binding energy lowering. In fact, table 1 of Ref. 78 shows the

energies of the dimers computed in the full basis set of the trimer. These energies are

0.05-0.10 kcal/mol lower than for the dimers in its own basis set. Thus, as much as 0.3

kcal/mol for the cyclic trimer and 0.2 kcal/mol for other trimers can be contributed to

the lowering ofEnon−add due to the basis set relaxation effect. Since EFP does not suffer

from the basis set relaxation, itsEnon−add for benzene trimers are systematically higher.

Enon−add computed byab initio methods can be corrected by using the binding ener-

gies of the dimers computed in the full basis set of the trimer, similarly to the counter-

poise correction often employed to rectify BSSE. Since these values are readily available

in table 1 of Ref. 78, we were able to compute corrected non-additive interaction energy

ECPnon−add, which is also provided in tables 1.3 and 1.4. EFP non-additive interaction

energies are within 0.03 kcal/mol fromECPnon−add for CCSD(T) and within 0.04 kcal/-

mol from MP2 values (except for the cyclic trimer). The share of the total non-additive

47

energy correction∆ does not exceed 4% for EFP and MP2. This share is larger for

CCSD(T) because the method significantly underestimates the total binding energy. It

also may attributed to the inclusion of higher-order three-body interaction energy terms

(e.g., three-body dispersion) in CCSD(T).

Based on the above comparisons of three-body and total non-additive interaction

energies, we conclude that EFP accurately describes the multipleπ − π interactions in

the considered benzene trimers.

Large benzene oligomers

The last part of our investigation is a study of the asymptotic behavior of the non-additive

energy contribution. Using the intermolecular separation parameters for dimers (table

1.1), we constructed the linear oligomers with 10, 20, 30 and 40 benzene molecules.

The oligomers S-N, T-N and PD-N, where N is the number of monomers, have the same

structural motives as trimers S, T1 and PD shown in Fig. 1.18. The geometries of the

oligomers are provided in Appendix B. The total binding energies (Etotal), the nearest

neighbor estimates (Etotal(dimer)), the non-additive energy (Enon−add) and its share (∆)

in the total energy are shown in table 1.5. The nearest neighbor estimate for the oligomer

with N benzene rings has been computed as:

Etotal(dimer) = (N − 1)Edimer (1.44)

From table 1.5, we can see that for all the oligomersEnon−add is positive and thus

destabilizing in nature, as it was correctly predicted in Ref. 78. It is also in good

agreement with the observation that nonadditive energy is positive in the bulk benzene

made in Ref. 84. Non-additive energy grows with the size of the system because of the

48

Table 1.5: Binding energies of benzene oligomers computed with EFP.

Etotal Etotal(dimer) Enon−add ∆

S-10 -17.14 -17.64 0.50 2.94%

S-20 -36.01 -37.24 1.23 3.41%

S-30 -54.89 -56.84 1.95 3.56%

S-40 -73.76 -76.44 2.68 3.63%

T-10 -20.89 -21.42 0.53 2.53%

T-20 -44.05 -45.22 1.17 2.65%

T-30 -67.21 -69.02 1.81 2.69%

T-40 -90.37 -92.82 2.45 2.71%

PD-10 -19.54 -19.62 0.08 0.41%

PD-20 -41.16 -41.42 0.26 0.63%

PD-30 -62.78 -63.22 0.44 0.70%

PD-40 -84.40 -85.02 0.62 0.73%

a S-N, T-N and PD-N are linear benzene oligomers with N benzene monomers, con-structed using sandwich, t-shaped and parallel-displaced structural motives, respec-tively.

large number of many-body and long-range two-body interactions possible in the larger

oligomers.

The dependence of the share of non-binding energy∆ on the system size is shown

in Fig. 1.19. For all 3 types of the oligomers,∆ asymptotically approaches a constant

value and does not exceed 4% of the total binding energy.

Based on the investigation of benzene trimers and higher-order oligomers at the EFP

level of theory, we conclude that the share of total non-additive energy is about 4%

for the systems considered. This result implies that total interaction energy in clustered

π−π systems can be estimated reasonably well as the sum of two-body nearest neighbor

interactions. However, it is important to note that neither EFP applied in our work

nor MP2 used in the studies of trimers in Ref. 78 account for many-body dispersion.

As demonstrated in the recent study79, many-body dispersion terms contribute about

49

1 0 2 0 3 0 4 00 . 5

1 . 0

1 . 5

2 . 0

2 . 5

3 . 0

3 . 5

Share

of th

e non

-addit

ive en

ergy,

%

T h e n u m e r o f b e n z e n e m o n o m e r s

S P D T

Figure 1.19: Dependence of the non-additive energy share on the length of benzeneoligomers.

50% to total three-body interaction energy of the cyclic benzene trimer. Additionally,

we expect that the share of the non-additive energy contributions will increase in the

three-dimensional clusters and bulk benzene due to the large number of many-body

interactions, which can occur between molecules located within a short distance from

each other.

1.8 Conclusions

This chapter presented a review of the theoretical approaches to modeling the envi-

ronmental effects and a detailed description of the EFP method. The details of our

EFP implementation in the Q-Chem package were presented. The developed code was

50

applied to investigate multipleπ − π interactions in benzene oligomers focusing on the

role of many-body and total non-additive energies in these systems.

We analyzed previously reported78 ab initio results for benzene trimers and con-

cluded that they underestimate non-additive energy due to the basis set superposition

error. Based on the data presented in Ref. 78, we were able to recomputeab initio

non-additive energy with the counterpoise correction. Using our EFP code, we com-

puted three-body and total non-additive energies for benzene trimers. These energies

for all eight timers included in our study appeared to be in excellent agreement with the

counterpoise-correctedab initio values (within 0.04 kcal/mol). The magnitude of non-

additive interaction energy was within 4% of total binding energy for all the considered

benzene clusters. Based on the detailed comparison of the binding energy components

(presented in tables 1.3 and 1.4), we concluded that EFP describes the intermolecular

π − π interactions with the accuracy comparable to that ofab initio methods.

In the final part, our EFP code was applied to study linear benzene oligomers. Based

on the results (presented in table 1.5), we concluded that the share of non-additive inter-

action energy in linear benzene clusters asymptotically approaches a constant value with

the increase in cluster size, and does not exceed 4% of total binding energy. Thus, it is

approximately of the same magnitude as for the benzene trimers. This result implies that

there are no significant non-local effects in the binding mechanism of the linear benzene

clusters.

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56

Chapter 2

Structure, vibrational frequencies,

ionization energies, and photoelectron

spectrum of the para-benzyne radical

anion

2.1 Introduction

Para-benzyne1, an intermediate in Bergman cyclization2, is believed to be a warhead in

antitumor enediyne antibiotics3–5 due to its ability to cause double-strand DNA cleavage

and self-programmed cell death or apoptosis6,7, and extensive studies have been carried

out in the past 20 years to understand the factors that control the reaction8–10. Originally,

it was interest in para-benzyne electronic structure that motivated the development of

procedures to generate the corresponding anion for use in photoelectron spectroscopy

experiments11,12. Recent studies suggested that anionic cycloaromatization reactions

are also feasible13,14 including the anionic version of the Bergman cyclization15. Larger

sensitivity to substituents15 suggests that the reactions of the anions might be designed

to be more selective than the prototypic non-anionic Bergman cyclization, important for

their applications in cancer-DNA damaging drugs.

Previous theoretical studies of the para-benzyne anion16 predicted strong vibronic

interactions (often referred to as pseudo or second-order Jahn-Teller effect) that produce

57

a lower symmetryC2v structure in addition to theD2h minimum. The energy difference

between the two structures was found to be highly sensitive to the correlation method,

with higher-level models favoringD2h. The analysis of the photoelectron spectrum of

the anion12 supported the assignment of theD2h symmetry, and it was concluded that

the lower symmetry structures are artifacts of incomplete correlation treatment, a well

known phenomenon in electronic structure17. Interestingly, density functional calcula-

tions produced onlyD2h structures, which was regarded as success of DFT methodol-

ogy and the authors concluded that “this level of theory is particularly well-suited for

computational studies of distonic radical anions derived from diradicals”16. The robust

behavior of DFT in cases of symmetry breaking has been noted by other researches as

well18.

The goal of this work is to investigate the performance of coupled-cluster (CC) and

equation-of-motion (EOM) methods in this challenging case of strong vibronic interac-

tions and to assess the quality of theab initio and DFT results by comparing the cal-

culated photoelectron spectrum with the experimental one. We quantify the symmetry

lowering by the selected geometrical parameters, charge distributions, and relative ener-

gies. In addition, the shape of the potential energy surface along the symmetry breaking

coordinate is analyzed. We also present an efficient scheme for calculating ionization

energies (IEs) of the anion in the spirit of isodesmic reactions.

Several recent studies19–21discussed different aspects of symmetry breaking in elec-

tronic structure calculations distinguishing between: (i) purely artefactual spatial and

spin symmetry breaking of approximate wave functions, i.e., Lowdin dilemma22; (ii)

real interactions between closely-lying electronic states that result in lower-symmetry

structures, significant changes in vibrational frequencies (see Fig. 1 from Ref. 21), and

even singularities (first order poles) in force constants (e.g., Fig. 3 from Ref. 20). While

the latter is a real physical phenomenon, it may or may not be accurately described by an

58

approximate method. If vibronic interactions are overestimated, calculations may yield

incorrect lower-symmetry structure, and vice verse.

Formally, the effect of the interactions between electronic states on the shape of

potential energy surface can be described20 by the Herzberg-Teller expansion of the

potential energy of electronic statei:

Vi = V0 +∑

α

< Ψi|∂H

∂Qα

|Ψi > Qα+

1

2

∑α

< Ψi|∂2H

∂2Qα

|Ψi > Q2α−

∑α

∑j 6=i

(< Ψi| ∂H

∂Qα|Ψj >

)2

Ej − Ei

Q2α,

(2.1)

whereQα denotes normal vibrational modes, wave functionsΨk and energiesEk are

adiabatic wave functions and electronic energies atQα = 0. The last term, which is

quadratic in nuclear displacement — hencesecond-order Jahn-Teller, describes vibronic

effects. For the ground state, i.e., whenEj > Ei, it causes softening of the force constant

alongQα, which may ultimately result in a lower symmetry structure, e.g., see Fig. 1

from Ref. 21. The above mentioned poles in force constants originate in the energy

denominator.

The formal analysis of the CC and EOM-CC second derivatives by Stanton20

demonstrated that the quadratic force constant of standard coupled-cluster methods,

e.g., coupled-cluster with singles and doubles (CCSD), necessarily contains unphysi-

cal terms, which may become significant and spoil the CC force constant in the cases

of strongly interacting states. Nevertheless, the standard CC methods are quite reliable

for systems exhibiting pseudo Jahn-Teller interactions, as follows from Refs. 20, 21, as

well as a host of numerical evidence. EOM-CC methods, on the other hand, provide

most satisfactory description of the interacting states20, not at all surprising in view of a

59

multistage nature of EOM. It should be noted that both CC and EOM-CC may exhibit

spurious frequencies if the orbital response poles, i.e., coupled-perturbed Hartree-Fock

(CPHF) poles, are present due to near-instabilities in the reference, a condition closely

related to (i). Perturbative corrections, e.g., as in CCSD(T), do not provide systematic

improvement, and often results in wider instability volcanoes19.

By analyzing the properties of linear response of multi-configurational SCF

(MCSCF) and the relationship between response equations and poles’ structure, Stanton

dispelled a common misconception of relying on multi-reference treatment of systems

with strong vibronic interactions20. Using the same arguments, he also concluded that

DFT describes pseudo Jahn-Teller effects reasonably well. While some numerical evi-

dence18 seems to support this conclusion, other examples21 indicated that DFT (B3LYP)

tend tounderestimatethe magnitude of vibronic interactions. Our results are in favor of

the latter, and we attribute this behavior to self-interaction error.

The structure of this chapter is as follows. The next section describes molecular

orbitals (MOs) and relevant electronic states of the para-benzyne anion, as well as the

nature of vibronic interactions in this system. Section 2.3 outlines computational details.

Section 2.4 discusses the competition between lower and higher symmetry structures at

different levels of theory. Photodetachment spectra are presented in section 2.5. Differ-

ent computational strategies for calculating IEs are discussed in section 2.7. Our final

remarks are given in the last section.

2.2 Molecular orbital picture and the origin of vibronic

interactions in the para-benzyne anion

The frontier MOs of the para-benzyne anion and the diradical are shown in Fig. 2.1. At

D2h, the two orbitals belong to different irreps,b1u andag, and their density is equally

60

distributed between the two radical centers (C1 and C4). At C2v, both orbitals become

a1, and their densities are localized at different carbons. As in the benzyne diradical,

the lowest MO, which hosts two electrons in the anion, is of anti-bonding character with

respect to C1-C4, whereas the singly occupied MO is bonding. In the case of the dirad-

ical, three singlet states and one triplet state are derived from different distributions of

two electrons on these two orbitals: X1Ag, 21Ag, 11B1u, 23B1u. The ground singlet state

belongs to the same irrep as the doubly excited singlet, and both have significant multi-

configurational character. Likewise, distributing three electrons on these two orbitals in

the anion gives rise to two electronic configurations shown in the lower panel of Fig. 2.1.

At D2h, these two determinants are of different symmetry and correspond to two distinct

electronic states, X2Ag and2B1u, separated by the energy gap of about 0.95 eV, as cal-

culated by EOM-CC for electron-attached states (see below). AtC2v, both determinants

are ofA1 symmetry and, therefore, can interact. This interaction increases the energy

separation between the two electronic states that acquire multi-configurational character

and, therefore, “softens” the corresponding normal mode in the lower state. If the inter-

action is sufficiently strong, that is, if the coupling matrix element is large relative to the

energy gap between the interacting states, the lower potential energy surface (PES) may

develop a lower symmetry minimum. In other words, lowering symmetry stabilizes the

lowest electronic state through configurational interaction, and this is the driving force

for symmetry breaking.

This analysis of the wave functions of interacting states complements the lessons

learned from Eq. (2.1) and poles’ structure of quartic force constants20,21. For example,

it shows that two factors are important for accurate description of systems with consid-

erable vibronic interactions: (i) balanced description of possibly multi-configurational

wave function (non-dynamical correlation); and (ii) quantitative accuracy in describing

61

C2v2hD

a1

a1

ag

b1u

C1

C1 C1

C1

C4 C4

C4 C4

X2Ag

A2B1u

X2A1

A2A1

6ag

5b1u

6

5

6ag

5b1u

g

5

11a1

10a1

11

10

11a1

10a1

11

10

Figure 2.1: Frontier MOs and leading electronic configurations of the two lowest elec-tronic states atD2h andC2v structures.

62

excited state ordering (dynamical correlation). Unbalanced treatment may overempha-

size the magnitude of vibronic interaction and, therefore, may lead to so-called arte-

factual symmetry breaking17. While (i) can be achieved by multi-configurational self-

consistent field (MCSCF), (ii) is much more difficult to satisfy. For example, MCSCF

exhibits artificial symmetry breaking in NO3, which is removed when dynamical cor-

relation is included23. Ground-state coupled-cluster methods, e.g., CCSD or CCSD(T),

may fail due to the insufficient description of non-dynamical correlation, as it happens

in NO324,25, although they often tackle such challenging systems successfully26.

The analysis of the MOs and the electronic states of the para-benzyne anion

(Fig. 2.1) suggests a moderate multi-configurational character of the ground state wave

function, and this is confirmed by EOM-EA amplitudes (as calculated at EOM-EA-

CCSD/6-311+G**) For theD2h structure, the EOM coefficient of the leading config-

uration (theAg deteminant from Fig. 2.1 and 2.2) is 0.86. The three next important

configurations have weights of 0.28, 0.26 and 0.07. Likewise, for theC2v structure the

weights of the four leading configurations are 0.87, 0.27, 0.16 and 0.15. The EOM-

SF-CCSD/6-311+G** amplitudes are similar: the weights of the leading configurations

are 0.81 and 0.80 for theD2h andC2v structures, respectevily. This confirms minor

multi-configurational character of the para-benzyne anion. For comparison, the weights

of the four leading configurations of the singlet para-benzyne diradical, which is a truly

multiconfigurational system, are 0.57, 0.38, 0.32 and 0.31 and the two leaing configu-

rations are double excitations with respect to each other. Thus, we expect ground-state

CC methods to be capable of treating C6H−4 reasonably well, unlike heavily multicon-

figurational wavefunctions, as those of the singlet diradicals24.

The EOM methods that simultaneously include both dynamical and non-dynamical

correlation are particularly attractive for the systems with vibronic interactions. More-

over, as formal analysis of Stanton shows, the pole structure of the exact force constant

63

6ag

5b1u

6

5

6ag

5b1u

g

5

6ag

5b1u

g

5

6ag

5b1u

g

5

Reference:

7agg

p-C H 62-4

Quartet

Tripletp-C H 6 4

p-C H 6-4

Singlet

+

Doublet p-C H 6-4

Ψtarget = R IP Ψ

ref

Ψtarget = REA Ψ ref

Ψ target = RSF Ψ ref

Figure 2.2: The two lowest electronic states of C6H−4 can be described by: (i) EOM-IP

with the dianion reference; (ii) EOM-EA with the triplet neutral p-benzyne reference;(iii) EOM-SF with the quartet p-benzyne anion reference.

of Eq. (2.1) is correctly reproduced within EOM-CC formalism. From the configura-

tion interaction point of view, the balanced description of two important determinants

from Fig. 2.1 is easily provided by several EOM-CC methods, i.e., ionization poten-

tial, electron-attachment, and spin-flip EOM-CC (EOM-EA, EOM-IP, and EOM-SF,

respectively), as explained in Fig. 2.2. All three EOM models include dynamical corre-

lation as well. Their reliability depends on how well the corresponding reference state is

described by the single-reference CCSD method and possible (near)-instabilities of the

Hartree-Fock references.

64

2.3 Computational details

Two basis sets were employed in this work. All the geometry optimizations, fre-

quencies’ calculations and most single point calculations were performed with the 6-

311+G**27,28basis. Additional single point calculations of IEs used the aug-cc-pVTZ29

basis.

The choice of reference in our calculations requires additional comments. The

ground-state methods [i.e., MP2, CCSD, CCSD(T)], which employ2Ag reference, were

conducted using pure-symmetry ROHF and UHF references. The< S2 > values for

UHF were 0.869 and 1.226 atD2h andC2v, respectively. The stability analysis revealed

several symmetry-breaking triplet instabilities atD2h. The lowest-energy UHF solution

is heavily spin-contaminated,< S2 >=1.248, and is also of broken symmetry. When

employed in the CCSD calculations, the lower-energy broken-symmetry UHF reference

yielded higher correlated energies, in agreement with our experience (see also foot-

note 17 in Ref. 20). Thus, we consider symmetry-pure UHF references to be more

appropriate in CC calculations, even though this results in cusps in the PES scans along

theD2h→ C2v distortion. Using symmetry-broken reference atD2h yields continuous

curves. In the cases of moderate spin-contamination (as for pure-symmetry UHF refer-

ences), CCSD is orbital insensitive, however, the (T) correction is often more reliable

for ROHF (see, for example, results from Ref. 30). The optimized-orbital CC models

avoid ambiguity of the reference choice for non-variational CCSD, and, in view of UHF

instabilities, we consider optimized orbitals coupled-cluster with doubles (OO-CCD)

reference to be most appropriate (among the ground-state CC methods) for this system.

References for the EOM calculations (see Fig. 2.2) were not well behaved either.

The closed-shell1A1 reference for EOM-IP corresponds to the C6H2−4 dianion, which is

unstable. This caused severe convergence problems and we abandoned EOM-IP treat-

ment. The EOM-EA calculations employed the3B1u state (3A1 atC2v) of the diradical.

65

Unexpectedly, the corresponding UHF reference develops strong spin contamination

at relatively small displacements fromD2h: for example,< S2 >=2.026 and 2.468

andD2h andC2v, respectively. This compromises the reliability of EOM-EA-CCSD at

C2v. The problem can be resolved by performing the EOM-EA calculations using the

OO-CCD triplet reference (EOM-EA-OD). The EOM-SF calculations employ the4B1u

reference (4A1 atC2v), which can be derived from the3B1u configuration by placing an

additional electron on aσ-like ag orbital. Despite the similarity to the parent triplet, the

quartet reference has been found to be well behaved throughout the whole range of the

D2h→C2v distortions, e.g.,< S2 >=3.776 and 3.779 andD2h andC2v, respectively.

Optimized geometries were obtained at the UHF, ROHF, UHF-CCSD31, UHF-

CCSD(T)32, OO-CCD31,33, EOM-EA-CCSD34, EOM-SF-CCSD35,36, and B3LYP37 lev-

els of theory. The latter two methods have produced onlyD2h structures. For all other

methods, two minima were found, and theC2v-D2h energy differences∆E’s were cal-

culated from the corresponding total energies. In addition,∆E’s were calculated by

a variety of methods at the UHF-CCSD optimized geometries. The lowest electron-

ically excited states forC2v andD2h structures where computed by EOM-EA-CCSD

at the ground state geometries optimized at the same method. We also performed

the UHF-CCSD, UHF-CCSD(T), OO-CCD, OO-CCD(T), EOM-EA-CCSD, EOM-SF-

CCSD and EOM-EA-OD PES scans along theD2h-C2v displacement coordinateR(x)

defined as:

R(x) = (1− x) ∗ r(D2h) + x ∗ r(C2v), (2.2)

wherex is a fraction ofC2v structure and vectorr denotes Cartesian coordinates of the

D2h andC2v structures.

The photodetachment spectra were calculated within double harmonic and parallel

normal modes approximations (using normal modes of anion) by programPES 38. In all

spectra calculations, we employed the SF-DFT geometry and frequencies of the singlet

66

state of the diradical39, while the triplet state was described by CCSD. For the anion, we

considered UHF-CCSD and B3LYP geometries and harmonic vibrational frequencies

of the anion.

Charge distributions were obtained from the natural bond orbital (NBO) analysis40.

Dipole moments were computed relative to the molecular center of mass. The Mulliken

conventions41 for symmetry labels were used: for bothC2v andD2h, the plane of the

molecule is YZ and the Z-axis passes through the two carbons that host the unpaired

electrons. Electronic structure calculations were performed usingQ-Chem42 andACES

II 43.

2.4 Equilibrium structures, charge distributions, and

the D2h→ C2v potential energy profiles at different

levels of theory

This section discusses theD2h andC2v optimized structures, respective energy differ-

ences, PES scans, as well as changes in charge distributions upon symmetry lowering.

As pointed out by Nash and Squires16, most theoretical methods predict two minima on

the PES, withD2h andC2v symmetries. As follows from the MOs (see Fig. 2.1), the

latter structures are characterized by a localized charge, i.e., are of a distonic type.

The optimizedD2h structures are presented in Fig. 2.3. Fig. 2.4 and Fig. 2.5 sum-

marizes differences between theD2h andC2v structures calculated at different levels of

theory. For coupled-cluster methods, the changes in geometries between different levels

of theory (Fig. 2.4, 2.5) are not significant and are smaller for theD2h structure than

for C2v. For example, the variation in bond lengths and angles forD2h structures are

67

H

H

H

H

1.3911.3981.3841.3901.3931.393

1.0931.0961.0921.0931.0931.092

121.9121.8122.0121.9121.6121.5

1.4351.4401.4331.4351.4331.433

116.5116.5117.2116.8116.6116.6

CCSDCCSD(T)B3LYPOO-CCDEA-CCSDSF-CCSD

Figure 2.3:D2h structures optimized at different levels of theory.

below 0.02A and 2, respectively. The differences are larger for theC2v structures, pos-

sibly because of larger spin-contamination of the doublet and triplet UHF references.

TheD2h structures calculated by the CC methods with double substitutions, i.e., CCSD,

OO-CCD, EOM-EA-CCSD, and EOM-SF-CCSD, are very close to each other, which

is reassuring in view of the instability of the doublet UHF reference. The changes in the

bond lengths and angles between CCSD(T) and CCSD are within 0.01A and 1, which

is consistent with typical differences for well behaved molecules (Ref. 44). The largest

change due to (T) is observed for the distance between the radical centers. The B3LYP

structure is considerably different, e.g., the corresponding C1-C4 distance is 0.035A

shorter than the CCSD(T) value. The above trends suggest that the accuracy of the CC

structures is not affected by the HF instabilities and vibronic interactions, and question

the reliability of B3LYP results.

68

B 3 L Y P C C S D O O - C C D C C S D ( T ) E A - C C S D S F - C C S D1 1 2

1 1 4

1 1 6

1 1 8

1 2 0

1 2 2

Angle

, deg

M e t h o d

C 2 v : C 4 D 2 h : C 1 C 2 v : C 1

B 3 L Y P C C S D O O - C C D C C S D ( T ) E A - C C S D S F - C C S D1 . 3 8

1 . 3 9

1 . 4 0

1 . 4 1

1 . 4 2

1 . 4 3

1 . 4 4

1 . 4 5

Bond

length

, A

M e t h o d

D 2 h : C 2 - C 3 C 2 v : C 2 - C 3 C 2 v : C 1 - C 2 D 2 h : C 1 - C 2 C 2 v : C 3 - C 4

Figure 2.4: CCC angles, bonds lengths calculated by different coupled-cluster methodsand B3LYP.

The comparisons between theD2h and C2v structures allows one to quantify the

degree of symmetry lowering at each level of theory. For example, the magnitude of

69

B 3 L Y P C C S D O O - C C D S F - C C S D E A - C C S D C C S D ( T )2 . 8 7

2 . 8 8

2 . 8 9

2 . 9 0

2 . 9 1

Distan

ce, A

M e t h o d

D 2 h : C 1 - C 4 C 2 v : C 1 - C 4

Figure 2.5: The distance between the radical-anion centers calculated by differentcoupled-cluster methods and B3LYP.

C2v distortions can be characterized by the difference between C1-C2 and C3-C4, which

are identical inD2h. As shown in the lower panel of Fig. 2.4, the difference in C1-C2

between theD2h andC2v structures is largest for CCSD and OO-CCSD and is reduced

by the inclusion of triples. At the EOM-EA-CCSD level, it shrinks even further. The C1-

C2 — C2-C3 difference, as well as difference in the C1 and C4 angles, follows the same

trend. Thus, the magnitude of symmetry breaking is smaller for higher-level methods,

or methods that use better references. This, along with the absence of aC2v minimum

at the EOM-SF-CCSD level, suggests that its occurrence is artefactual.

The degree of symmetry breaking can also be quantified by the charge localization,

which gives rise to a non-zero dipole moment shown in Fig. 2.6. The correspond-

ing atomic charges calculated using the NBO procedure are consistent with the dipole

moment changes, except for ROHF, which may be an artifact of the NBO analysis. In

agreement with the trends in structural changes, the charge localization is smaller for

70

- 0 . 4 8

- 0 . 4 4

- 0 . 4 0

- 0 . 3 6

- 0 . 3 2

R O H F - E A - C C S D

q(C1), a

.u

U H F

R O H F

U H F - C C S D

R O H F - C C S D U H F - E A - C C S D

2 . 0

2 . 5

3 . 0

3 . 5

4 . 0

4 . 5

5 . 0

5 . 5

Dipole

, deb

ye

U H FR O H F

U H F - C C S D

R O H F - C C S D

U H F - E A - C C S D

R O H F - E A - C C S D

Figure 2.6: The NBO charge at C1 (top panel) and dipole moments (bottom panel)at the optimizedC2v geometries. The ROHF-CCSD and ROHF-EA-CCSD values arecalculated at the geometries optimized by the corresponding UHF-based method.

71

more correlated wave functions. Interestingly, the overall dipole moment decreases in

spite of increased C1-C4 distance. Thus, the charge localization patterns also suggest

that the symmetry lowering is an artifact of an incomplete treatment of electron correla-

tion.

Finally, symmetry lowering can be characterized by energy differences (∆E’s)

between theD2h andC2v structures summarized in Table 2.1 and Fig. 2.7. In addition,

several PES scans are shown in Fig. 2.8. The first part of the table and the upper panel of

Fig. 2.7 summarize single point calculations at the CCSD optimized structures (MCSCF

and CASPT2N employ the MCSCF geometries and are from Ref. 16), while the second

part and the lower panel of Fig. 2.7 present energy differences calculated using the

respective optimized structures. All∆E’s from Table 2.1 and Fig. 2.7 are calculated

using symmetry-pure UHF references atD2h (see section 2.3). Fig. 2.8 shows both the

symmetry-pure and symmetry-broken CCSD/CCSD(T) values atD2h. For these meth-

ods, the correct-symmetry reference yields lower correlated energies, which results in

cusps on the PESs (see section 2.3). OO-CCD has only one solution atD2h minimum,

and the respective curves are smooth.

Similarly to the trends in structures and charge localization, the energy difference

between theD2h andC2v structures is also very sensitive to correlation treatment, in

agreement with earlier studies16. UHF and ROHF place theC2v structure significantly

lower (by about 1 eV), than theD2h one, while MP2 gives the almost exactly oppo-

site result. At higher level methods, the energy difference becomes much smaller. At

the CCSD and OO-CCD levels, theC2v structure is just a little (<0.1 eV) more stable

than theD2h one. Finally, (T) correlation at the CCSD or OO-CCD levels changes the

balance and theD2h structure becomes the global minimum. Similarly, at the EOM-EA-

CCSD level,D2h is also the lowest minimum. The B3LYP and EOM-SF PES do not

have aC2v minimum. Overall, the PES along this distortion is rather flat (due to vibronic

72

- 1 . 2

- 0 . 8

- 0 . 4

0 . 0

0 . 4

0 . 8

1 . 2

UHF-E

A-OD

OO-C

CDOO

-CCD

(T)

ROHF

-CCS

D(T)

E(C2v

) - E(D

2h), e

V

UHF

ROHF

UHF-M

P2RO

HF-M

P2UH

F-CCS

DRO

HF-C

CSD

DFT/B

3LYP

UHF-C

CSD(

T)

UHF-E

A-CCS

DUH

F-SF-C

CSD

MCSC

FCA

SPT2

N

- 1 . 2

- 0 . 8

- 0 . 4

0 . 0

0 . 4

0 . 8

1 . 2

OO-C

CD

E(C2v

) - E(D

2h), e

V

UHF

ROHF

UHF-C

CSD

UHF-C

CSD(

T)

UHF-E

A-CCS

D

Figure 2.7: Energy differences between theC2v andD2h structures (6-311+G** basis).Upper panel: calculated using the UHF-CCSD optimized geometries. MCSCF andCASPT2N employ the MCSCF geometries. Lower panel: calculated at the respectiveoptimized geometries.

73

Table 2.1: Energy differences between theC2v andD2h minima (eV) in the 6-311+G**basis set. Negative values correspond to theC2v minimum being lower. ZPEs are notincluded.

Method UHF ROHF

at UHF-CCSD optimized geometries

HF -0.999 -1.049

MP2 1.191 0.887

CCSDa -0.046 (-0.116) -0.130

CCSD(T)a 0.140 (-0.052) 0.077

B3LYP 0.070

5050b -0.149

OO-CCD -0.096

OO-CCD(T) 0.081

EOM-EA-CCSD 0.086

EOM-EA-OD 0.050

EOM-SF-CCSD 0.052

at the geometries optimized by the corresponding method

HF -0.971 -1.039

5050 -0.136

CCSD -0.046

CCSD(T) 0.140

OO-CCD -0.097

EOM-EA-CCSD 0.071

a Calculated using symmetry-pure and symmetry-broken (values in parenthesis) doubletUHF references atD2h.b DFT functional with 50% of the HF exchange.

interactions): the most reliable estimates of∆Es between the two CCSD optimized min-

ima range between 0.05 eV (EOM-SF-CCSD) to 0.08 eV [CCSD(T) and OO-CCD(T)].

Optimized-orbital and regular CC methods give similar energy orderings, although OO-

CCD and OO-CCD(T) place theD2h structure a little higher than the corresponding CC

methods based on the Hartree-Fock reference. EOM-SF behave similarly to EOM-EA.

74

0 2 0 4 0 6 0 8 0 1 0 0- 0 . 1 6

- 0 . 1 2

- 0 . 0 8

- 0 . 0 4

0 . 0 0

0 . 0 4

0 . 0 8

0 . 1 2

E-E

(C2v

), eV

% o f C 2 v

C C S D C C S D ( T ) O O - C C D O O - C C D ( T )

E A - C C S D S F - C C S D E A - O D B 3 L Y P

Figure 2.8: The potential energy profiles along theD2h-C2v scans. At theD2h struc-ture, CCSD and CCSD(T) energies are calculated using symmetry-broken and pure-symmetry UHF references (see section 2.3).

Similar trends, although with larger variations, were observed for multi-reference

methods, e.g. MCSCF and CASPT2. The previous study16 reported the energy dif-

ferences between theC2v andD2h structures calculated by multi-reference methods as

-0.286 eV for MCSCF(9,8)/cc-pVDZ and 0.388 for CASPT2N/cc-pVDZ. Thus, only

when dynamical correlation is included in a multi-reference method, theD2h structure

becomes the lowest in energy.

As follows from Table 2.1, using different optimized geometries has only a minor

effect on∆E’s. For example, the energy differences calculated at the CCSD optimized

geometries are within 0.03 eV from those calculated using the structures optimized at

the same level of theory, and the observed trends are the same.

75

Thus, the presented∆E’s exhibit the same trend as the optimized geometries and

charge localization patterns — the more correlation we add the more stable theD2h

structure becomes.

Energy profiles from Fig. 2.8 give more detailed information on the shape of the

PES. Due to the HF instabilities, the CCSD and CCSD(T) curves are discontinuous,

unless the symmetry-broken UHF reference is used atD2h. Apart from the cusp, the

shapes of the OO-CCD and CCSD curves are surprisingly similar. The difference

becomes more profound when triples are included: the OO-CCD(T) curve smoothly

descends towards theD2h minimum, while the CCSD(T) curve is gradually climbing up

all the way till the final drop atD2h. Even though the correspondingD2h→ C2v ∆E’s

are close, the shape of the EOM-SF-CCSD and OO-CCD(T) curves are quite different:

the former is much flatter atD2h than the latter. The EOM-EA-CCSD curve parallels

EOM-SF-CCSD aroundD2h, but soon becomes spoiled by the instability in the triplet

reference. Using optimized orbitals triplet reference solves the problem of HF instabil-

ity and the EOM-EA-OD curve is very similar to the EOM-SF-CCSD one. The B3LYP

curve is steeper than the EOM-SF one, consistent with large value of the corresponding

frequency (see next section). Overall, Fig. 2.8 demonstrates that describing the shape of

PES is more challenging than calculating energy difference between two minima. One

should expect considerable differences in the corresponding vibrational frequency. Note

that the analytic CCSD or CCSD(T) frequencies would differ dramatically from those

calculated by finite differences: the analytic frequencies will characterize the curvature

of the PES corresponding to the symmetry-pure reference, whereas finite difference cal-

culations will sample the cusp.

To conclude, the analysis of the equilibrium structures, charge distributions, and

energy differences calculated at different levels of theory suggests that theC2v minimum

is an artifact of incomplete treatment of electron correlation and HF instabilities in the

76

doublet or triplet references. The strong vibronic interactions result in rather flat PES

along theD2h→ C2v distortion. In the next section, we present our calculations of the

photoelectron spectrum, which allows us to assess the quality of calculated geometries

and to unambiguously determine the symmetry of the anion.

2.5 Vibrational frequencies and the photoelectron spec-

trum of para-C 6H−4

Photoelectron spectroscopy is a very sensitive structural tool. The spectra depend on

equilibrium geometries, frequencies and normal modes of the initial (radical anion) and

target (diradical) states. The equilibrium geometries were discussed in the previous sec-

tion. The vibrational frequencies of the anion (atD2h) and the diradical are given in

Table 2.2. The CCSD and B3LYP frequencies of the anion appear to be similar, except

for the lowestb1u mode that describesD2h→C2v displacements and is most affected by

vibronic interactions. This mode also exhibits the largest change relative to the neu-

tral, i.e., it is considerably softer in C6H−4 . The B3LYP frequency is 40% higher than

the CCSD value, in favor of our assumption that B3LYP underestimates the magni-

tude of vibronic interaction. Unfortunately, the only significant vibrational progressions

present the Franck-Condon factors (see section 2.3) are those for fully-symmetric nor-

mal modes that exhibit displacements relative to the anion. Modes with significant fre-

quency change, such as the aboveb1u mode, yield much smaller features that are likely

to be buried under major progressions, as it happens for the para-benzyne anion.

The electron photodetachment spectrum of the para-benzyne radical anion reported

by Wenthold and coworkers12 consists of the two overlapping bands corresponding to

the singlet (X1Ag) and triplet (3Bu) states of the diradical with the origins, i.e., electron

77

Table 2.2: Vibrational frequencies of the para-benzyne radical anion and the diradical.

Symmetry X2A1a X2A1

b a3B1ua X1A1

c

ag 3137 3085 3221 3317

ag 1471 1434 1565 1440

ag 1189 1175 1166 1202

ag 992 981 1028 1065

ag 624 631 615 660

au 879 917 920 994

au 380 398 368 450

b1g 691 721 810 722

b2g 1120 890 871 961

b2g 647 601 437 616

b3g 3114 3061 3207 3302

b3g 1579 1549 1652 1738

b3g 1297 1284 1300 1327

b3g 608 612 585 597

b1u 3111 3056 3206 3300

b1u 1445 1425 1471 1511

b1u 1071 1060 1046 1101

b1u 452 639 947 973

b2u 3136 3082 3220 3316

b2u 1331 1332 1375 1448

b2u 1189 1203 1267 1254

b2u 1027 1020 1081 1076

b3u 715 718 753 796

b3u 422 447 400 477

a Calculated at UHF-CCSD/6-311+G**b Calculated at B3LYP/6-311+G**c From Ref. 39, SF-5050/6-31G**

affinities (EAs), at 1.265± 0.008 eV and 1.430± 0.015 eV, respectively. Long vibra-

tional progressions for frequencies at 635± 20 cm−1 and 990± 20 cm−1 (assigned to

the singlet band), at 610± 15 cm−1 and 995± 20 cm−1 (assigned to the triplet band),

78

as well as a hotband at 615± 30 cm−1 were reported. For both the singlet and the

triplet bands, the lower (about 600cm−1) and the higher (about 1000cm−1) energy pro-

gressions were assigned to symmetric ring deformation and ring breathing, respectively.

These bands, four lowestag vibrations, are also present in all the computed spectra.

Including the lowestb1u mode resulted in very small features, completely obscured by

the dense lines of major progressions.

In the original paper12, the normal displacements were derived from the fitting pro-

cedure. Table 2.3 and Fig. 2.9 compare these values with the shifts computed in this

work from the correspondingD2h optimized geometries and normal modes of the anion.

The triplet and the singlet states of the diradical are described by the CCSD and spin-flip

DFT methods, respectively (see section 2.3). The computed CCSD shifts are very close

to those derived from the experiment. Values obtained with the CCSD normal modes

and different equilibrium structures [e.g., CCSD(T), EOM-EA-CCSD, OO-CCD, and

B3LYP] are quite similar to each other. However, the full B3LYP shifts (the last line

in Table 2.3), i.e., those computed using the B3LYP normal modes and B3LYP equi-

librium structures, are much larger for the ring breathing mode, which will result in a

longer progression.

Finally, Fig. 2.10 compares the calculated and experimental photodetachment spec-

tra. The upper panel of Fig. 2.10 shows the singlet and triplet bands of the spectrum

for theD2h structures calculated using the CCSD description of the anion. To compare

with the experimental spectrum, the scaled intensities of the singlet and triplet bands

were added together. Two scaling coefficients were determined from the following con-

ditions: (i) the combined spectrum is normalized to 1; and (ii) the intensity of peak

number 3 in the experimental and the calculated spectra coincide. The experimental

spectrum was normalized to 1 as well. The bottom part of Fig. 2.10 compares the

computedD2h and the experimental spectra. The agreement between the two spectra

79

Table 2.3: Displacements in normal modes upon ionization,A ∗√

amu.

Method ν5a ν4

b ν5a ν4

b

Singlet Triplet

Exp. fitc] -0.44 -0.14 0.76 -0.12

CCSDd -0.49 -0.21 -0.69 -0.09

CCSD(T)d -0.50 -0.27 -0.71 -0.14

EOM-EA-CCSDd -0.49 -0.22 -0.70 -0.10

OO-CCDd -0.49 -0.21 -0.69 -0.09

DFT/B3LYPd -0.47 -0.17 -0.67 -0.05

DFT/B3LYPe -0.40 -0.29 -0.60 -0.23

aRing deformationbRing breathingcRef. 12dusing the CCSD normal modeseusing the DFT/B3LYP normal modes

S - Q 6 3 5 S - Q 9 9 0 T - Q 6 1 0 T - Q 9 9 5- 0 . 8

- 0 . 6

- 0 . 4

- 0 . 2

0 . 0 e x p . f i t C C S D C C S D ( T ) E O M - E A - C C S D O O - C C D D F T D F T - f u l l

Displa

ceme

nts, A

∗amu 1/2

N o r m a l m o d e s

Figure 2.9: Displacements along the normal modes upon ionization (D2h structure).

80

1 . 2 1 . 4 1 . 6 1 . 8 2 . 0 2 . 2 2 . 4

Inten

sity

E l e c t r o n b i n d i n g e n e r g y , e V

1 . 2 1 . 4 1 . 6 1 . 8 2 . 0 2 . 2 2 . 4

Inten

sity

E l e c t r o n b i n d i n g e n e r g y , e V

Figure 2.10: Top: Singlet (solid line) and triplet (dotted line) bands of the electron pho-todetachment spectrum forD2h using the CCSD equilibrium geometries and frequenciesof the anion. Bottom: the experimental (solid line) and the calculated (dotted line) spec-tra forD2h using the CCSD structure and normal modes.

81

is excellent: the computed spectrum has the same features as the experimental one and

differs only slightly in the peaks’ intensities. Fig. 2.11 shows the spectrum calculated

for theC2v CCSD structure, which is markedly different — it features additional pro-

gressions that are not present in the experimental spectrum. They correspond to the

ring deformation band and the combination bands involving this mode. To conclude,

an excellent agreement between the spectrum calculated using theD2h structure and

the experimental one, along with significant differences betweenD2h andC2v, unam-

biguously proves theD2h structure of the para-benzyne radical anion. Moreover, it also

confirms high quality of the CCSD equilibrium structure and frequencies for the anion,

as well as the accuracy of the spin-flip description of the diradical. We also calculated

the spectrum using the B3LYP optimized geometries and frequencies for p-C6H−4 (and

the same structures and frequencies for the diradical, as above). The resulting spectrum

(Fig. 2.12) has significantly different ratio of intensities for the two main bands and

much longer progressions. Thus, even though DFT correctly predicts symmetry of the

para-benzyne radical anion, the overall structure is of poor quality. The reasons for this

are discussed below.

2.6 DFT self-interaction error and vibronic interactions

The analysis of equilibrium structures, charge localization, and energy differences

betweenD2h andC2v structures presented in section 2.4 demonstrated that the magni-

tude of symmetry breaking decreases when higher level methods are employed. More-

over, the PES scans and the analysis of UHF references strongly suggest that the exis-

tence of aC2v minimum is an artifact. Indeed, EOM-SF-CCSD, the only CC method

82

1 . 2 1 . 4 1 . 6 1 . 8 2 . 0 2 . 2 2 . 4

Inten

sity

E l e c t r o n b i n d i n g e n e r g y , e V

Figure 2.11: The experimental (solid line) and the calculated (dotted line) spectra forC2v using the CCSD structure and normal modes.

that employs a well-behaved UHF reference (that is, symmetry-pure, not strongly spin-

contaminated and stable), yields only aD2h minimum and a smooth PES. The EOM-

EA-OD potential energy profile parallels the EOM-SF one. Finally, modeling of the

photoelectron spectrum ruled out theC2v structure. Thus, we conclude that symmetry

breaking in p-C6H−4 is purely artefactual.

DFT/B3LYP appears to be more robust with respect to this artefactual symmetry

breaking as it yields only aD2h structure. The important question iswhy, and whether

or not one should consider DFT a reliable tool for modeling systems with strong vibronic

interactions. Detailed benchmark studies (e.g., Ref. 18) provide many examples of DFT

giving “the right answer” for difficult symmetry breaking systems. Stanton explained

these observations by making connections between DFT response theory, TD-DFT exci-

tation energies and their Tamm-Dancoff approximation (footnote 23 in Ref. 20), and

83

1 . 2 1 . 4 1 . 6 1 . 8 2 . 0 2 . 2 2 . 4

Inten

sity

E l e c t r o n b i n d i n g e n e r g y , e V

Figure 2.12: The experimental (solid line) and the calculated (dotted line) spectra forD2h using the B3LYP structure and normal modes.

concluded that DFT describes pseudo Jahn-Teller effects reasonably well “for the right

reason”, that is, because the poles appear in approximately the right places. Other

examples, however, suggested that DFT might actually underestimate the magnitude

of vibronic interactions21. To complicate matters even further, we would like to mention

the dehydro-meta-xylylene anion (DMX−) for which DFT yields low symmetry (Cs or

C2) equilibrium geometries26, while wave function based methods predict planar C2v

structures45.

To gain insight, we begin by analyzing the character of interacting electronic states

in p-C6H−4 and DMX− at symmetric and symmetry-broken geometries. As described in

sections 2.2 and 2.4, the two interacting states of p-C6H−4 have the excess charge equally

delocalized between two radical centers (C1 and C4) at D2h, whereasC2v distortions

84

result in the charge localization at one of the carbons. DMX− shows exactly the oppo-

site behavior26,45: at the planarC2v geometries, most of the excess charge resides on the

σ-radical center, whereas C2 or Cs displacements facilitate the flow of the charge into

theπ-system. Thus, in both cases B3LYP favors the structures with more delocalized

excess charge, which immediately brings about the infamous H+2 example46. While HF

dissociation curve for a one-electron H+2 is exact, many DFT methods result in quasi-

bound potential energy curves and with errors as large as 50 kcal/mol! This happens

because DFT over-stabilizes the solutions with an electron being equally split between

the two hydrogens, which is an artifact of self-interaction error (SIE) present in many

functionals47–49. SIE is considerably larger in systems with odd number of electrons.

Self-interaction energy, Coulomb interaction of an electron with itself, is always posi-

tive, and electron delocalization lowers its magnitude thus causing artificial stabilization

of delocalized configurations. Other well understood examples of artificial stabilization

of delocalized states due to SIE include underestimated barriers for radical reactions,

dissociation of cationic radicals, and mixed-valence transition metal dimers49.

As shown in Ref. 48, the magnitude of SIE in H+2 strongly depends on the distance

between the atoms, varying from a negligibly small value around the equilibrium to 55

kcal/mol (B3LYP) at the dissociation limit. The charge-bearing radical centers in the

para-benzyne radical anion are 2.9A apart. At this distance, the H+2 curve exhibits as

much as 60% of its maximum SIE. This suggests substantial SIE in the para-benzyne

radical anion.

To test this assumption, we computed energy differences between theD2h andC2v

minima (using the CCSD optimized geometries) employing functionals with a varying

fraction of HF exchange. The results are shown in Fig. 2.13. The trend is clear: the

relative stabilization of theD2h structure is inversely proportional to the fraction of the

HF exchange in the functional, and, consequently, is proportional to SIE. Combining

85

different correlation functionals with the fixed fraction of the HF exchange has little

effect on the∆E’s, for example, all functionals with 0% HF exchange yield approxi-

matly the sameD2h-C2v enegry separition. Therefore, we conclude that for DFT, the

strengths of vibronic interactions and, consequently, a relative ordering of theD2h and

C2v structures, are governed by SIE. Thus, B3LYP yields the right structure for the

“wrong reason”. Therefore, it is not at all surprising that the quality of the B3LYP

structure and the shape of PES is relatively poor, as follows from the comparison of

the computed photo-electron spectrum with the experimental one. In DMX−, B3LYP

yields the wrong structure, but for the same “right reason” (SIE). Therefore, B3LYP (or

any functional with SIE) is not a reliable method for the systems with strong vibronic

interactions that affect charge localization patters, or when the interacting states are

characterized by considerably different charge distributions.

0 2 0 4 0 6 0 8 0 1 0 0- 1 . 0

- 0 . 8

- 0 . 6

- 0 . 4

- 0 . 2

0 . 0

0 . 2

H FV W NL Y P

7 5 - 2 5

5 0 - 5 0B 3 L Y PB 3 L Y P 5B 3 P W 9 1

E D F 2

E D F 1G i l lB L Y PS V W NB V W NG G 9 9

E(C2v

) - E(D

2h), e

V

H F e x c h a n g e , %

Figure 2.13: Energy differences between theD2h andC2v minima (using the CCSDstructures) calculated using different density functionals.

86

2.7 Calculation of electron affinities of p-benzyne

Different strategies for calculating electron affinities in problematic open-shell systems

were discussed in Ref. 26. The simplest ”brute-force” approach, i.e., to take the differ-

ence between the total energies of the anion and the neutral calculated at the same level

of theory, can give accurate values only if the errors in the corresponding total energies

are very small (which can be achieved only at very high levels of theory), or if both

species are described on the equal footing by the method and errors in the total ener-

gies cancel out. The latter is difficult to achieve in the case when both the neutral and

anion wave functions have a complicated open-shell character. For example, the singlet

state of the diradical is significantly multi-configurational and it would be described less

accurately than the anion by single reference methods. The triplet state of the diradical,

however, is single-configurational and can be described by single-reference methods

with approximately the same accuracy as the doublet. Thus, triplet EAs calculated by

energy differences employing correlated single-reference methods are more reliable than

singlet EAs. The latter can be accurately computed by subtracting the experimental (or

a calculated by an appropriate method) singlet—triplet gap from the calculated triplet

electron affinity. This approach was used for calculating EAs of triradicals26,45 and

diradicals50. Below we demonstrate that this scheme in the spirit of isodesmic reactions

indeed yields accurate singlet EAs.

EAs calculated at different levels of theory for the singlet and the triplet states of the

diradical are presented in Table 2.4 and Fig. 2.14. The first two columns present triplet

and singlet EAs computed by energy differences. The last column presents singlet EA

calculated from the corresponding triplet EA and the best theoretical estimate of the

singlet-triplet gap. For the triplet state, which is predominantly single-configurational,

even relatively low level methods using a moderate basis, e.g. B3LYP and MP2 in 6-

311+G**, yield reasonable EAs, e.g., within 0.11 eV and 0.07 eV from the experiment,

87

respectively. For the singlet state, however, EAs calculated by DFT and MP2 are 1.03 eV

and 0.67 eV off, because these methods are not appropriate for the multi-configurational

wave function of the singlet. Note that such brute-force approach reverses the states

ordering in the diradical even at the CCSD level. For both states coupled-cluster meth-

ods, especially with triples’ corrections, are in a reasonable agreement with the exper-

imental values. For example, ROHF-CCSD(T) is within 0.16 eV for triplet and within

0.05 eV for singlet.

Table 2.4: Electron affinities (eV) of the a3B1u and X1Ag states of the para-benzynediradical.

Method Triplet EA Singlet EA Singlet EAa

Experiment 1.430 1.265

Experiment-ZPE 1.375 1.140

6-311+G** basis

B3LYP 1.481 2.172 1.310

UHF -0.994 2.679 -1.165

UHF-MP2 1.547 0.467 1.376

UHF-CCSD 0.893 1.659 0.722

UHF-CCSD(T) 1.207 1.085 1.036

ROHF -1.148 2.356 -1.319

ROHF-MP2 2.053 0.914 1.882

ROHF-CCSD 0.872 1.629 0.701

ROHF-CCSD(T) 1.216 1.094 1.045

OO-CCD 0.875 1.644 0.704

OO-CCD(T) 1.210 1.080 1.039

aug-cc-pVTZ basis

ROHF -1.199 2.297 -1.370

ROHF-MP2 2.266 1.010 2.095

ROHF-CCSD 0.998 1.766 0.827

ROHF-CCSD(T) 1.395 1.258 1.224

aCalculated by using triplet EA and the singlet-triplet gap from Ref. 51.

88

0 . 5

1 . 0

1 . 5

2 . 0

2 . 5 6 - 3 1 1 + G * * a u g - c c - p V T Z

ROHF

-CCS

DRO

HF-C

CSD(

T)

EA, e

V

UHF

DFT/B

3LYP

UHF-M

P2

UHF-C

CSD(

T)UH

F-CCS

D

OO-C

CD

OO-C

CD(T)

E x p .

- 1 . 0

- 0 . 5

0 . 0

0 . 5

1 . 0

1 . 5

2 . 0 6 - 3 1 1 + G * * a u g - c c - p V T Z

ROHF

-CCS

DRO

HF-C

CSD(

T)

EA, e

V

UHF

DFT/B

3LYP

UHF-M

P2

UHF-C

CSD(

T)UH

F-CCS

D

OO-C

CD

OO-C

CD(T)

E x p .

Figure 2.14: Adiabatic electron affinities calculated by energy differences for the singlet(top panel) and triplet (bottom panel) state of the para-benzyne diradical.

89

For quantitative thermochemical results, larger basis sets are required. We performed

additional coupled-cluster calculations with the aug-cc-pVTZ basis (see Table 2.4). EA

for the triplet calculated by CCSD(T) is within 0.02 eV from the experimental one, as

expected for this method. However, the singlet CCSD(T) value is 0.12 eV off, because

of the poor description of the multi-configurational singlet state by ground-state single-

reference methods.

By employing the best theoretical estimate for the singlet-triplet gap in para-

benzyne, i.e., 0.171 eV from Ref. 51, singlet EA calculated from the CCSD(T) triplet

value is within 0.08 eV from the experiment, as compared to 0.12 eV for the brute-force

approach. As follows from the excellent agreement of triplet EA, the accuracy of singlet

EA can be improved by refining the value of the singlet-triplet gap.

2.8 Conclusions

The shape of ground-state PES of C6H−4 is strongly affected by the vibronic interactions

with a low-lying excited state. The magnitude of vibronic interactions depends cru-

cially on the the energy gap and couplings between the interacting states. Complicated

open-shell character of the wave functions of the interacting states and HF instabilities

challengeab initio methodology: the shape of PES along the vibronic coupling coordi-

nate differs dramatically for different methods. For example, many wave function based

methods predict vibronic interactions to be strong enough to result in a lower-symmetry

C2v minimum, in addition to theD2h structure. EOM-SF-CCSD and B3LYP predict

only a single minimum, however, the curvature of the surface is quite different.

The degree of this symmetry lowering can be quantified by structural parameters

(e.g., the magnitude of nuclear displacements), charge localization and the resulting

dipole moment, as well as energy difference between the two minima. We found that

90

the magnitude of symmetry breaking (as characterized by the above metrics) decreases

when higher-level methods are employed. For example, the dipole moment of theC2v

structure decreases in the HF→CCSD→EOM-EA-CCSD series. Likewise, coupled-

cluster methods with triples corrections as well as EOM-EA/SF-CCSD predict thatD2h

is a lower energy structure. Moreover, EOM-SF predicted onlyD2h minima. Calculation

of the potential energy profiles along theD2h→ C2v distortion along with the stability

analysis of UHF references allowed us to attribute the lower-symmetry minimum to the

incomplete treatment of electron correlation and UHF instabilities.

The quality ofD2h structure was assessed by comparing the computed photoelectron

spectrum against the experiment. The spectrum calculated using the CCSD description

of the anion and the SF-DFT results for the diradical is in an excellent agreement with

the experimental one, which is particularly encouraging in view of its dense and com-

plicated nature. The spectrum calculated using theC2v geometry and frequencies is

markedly different, which allows us to rule out theC2v structure.

As pointed out in earlier studies16, B3LYP appears to be robust w.r.t. this artefactual

symmetry breaking, and yields onlyD2h structure. The analysis of the structure and the

potential energy profile along theD2h→ C2v distortion reveals that the structure and the

curvature are considerably different from those calculated by reliable wave functions

methods. Moreover, the computed spectrum differs significantly from the experimental

one. The shape of the surface, as well as the corresponding harmonic frequency reveals

that B3LYP underestimates the magnitude of vibronic interactions, as pointed out ear-

lier by Crawford and coworkers21. Further analysis allowed us to attribute the failure

of B3LYP to the self-interaction error. A similar behavior (that yielded seemingly dif-

ferent result) was observed in DMX−, where SIE resulted in B3LYP overestimation

of vibronic interactions and, consequently, a lower symmetry structure. In both cases,

91

large SIE originates in significant changes in charge localization patterns along sym-

metry breaking coordinate. Thus, B3LYP is not a reliable method for systems where

vibronic interactions are coupled to different charge localization patterns, as happens in

distonic radical and diradical anions.

The work presented in this chapter was published in Ref. 52.

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95

Chapter 3

Future work

The effective fragment potential method, which emerged in the last decade, is currently

an active area of research in both the methodology and applications. The implementation

developed in this work increases the availability of the method and opens the possibil-

ity to combine it with variousab initio theories available in Q-Chem1 (e.g., RI-CCSD,

EOM-SF-CCSD). Thus, it is likely to increase the interest to this method and its poten-

tial even further. Since EFP is a relatively new method, the opportunities for the future

developments are abound. Below the most interesting directions of the future work are

presented in the separate sections for the method, implementation and applications.

3.1 Methodology

Three-body dispersion

A recent work by Podeszwa ar al.2 demonstrated that three-body dispersion can con-

tribute up to 50% to total three-body interaction energy for non-polar molecular clus-

ters, such as cyclic benzene timer. Therefore, we expect that three body dispersion will

become increasingly important in the three-dimensional aromatic clusters and in the

bulk. To model intermolecular interactions in such systems accurately, the three-body

dispersion term needs to be included in the EFP model. The approach, which must not

compromise the computational efficiency of EFP, is yet to be developed. One of the

96

possible directions is to apply the approximations made in the derivation of the two-

body EFP dispersion3 to the three-body dispersion expression in terms of frequency-

dependent electric polarizabilities derived in Ref. 4.

Impoved electrostatics expansion

In the present EFP formulation5, the exchange-repulsion, polarization and dispersion

points are the LMO centroids, whereas the electrostatics expansion points are the nuclei

and bond midpoints. The LMOs centroids account for both bond midpoints and lone

pairs. Thus, the lone pairs, which also represent concentration of the electronic density,

are not included in the electrostatic expansion. An interesting question is if using the

LMO centroids and the nuclei (instead of the bond midpoints and the nuclei) as the

electrostatics expansion points will make the EFP description more balanced and thus

more accurate. Additionally, the multipoles at the LMO centroids might better fit in the

unified EFP damping scheme proposed in Ref. 29.

Linear scaling EFP

The computational complexity of all EFP components in their straightforward formula-

tion is O(N2), whereN is the number of fragments. Application of the fast multipole

method6 to the electrostatics and polarization components along with a cutoff procedure

to fast decaying exchange-repulsion and dispersion, will bring the computational com-

plexity down toO(Nlog(N)). This will open the possibility for modeling much larger

molecular systems and the properties of the bulk condensed phase at the EFP level.

Correlated treatment of the active region

In the present EFP formulation5 the active region is treated at the HF level of theory,

which does not account for electron correlation. However, it is well established7–18 that

97

accurate description of many molecular properties requires correlated methods. Elec-

tron correlation is especially important for the open-shell species, such as multiconfigu-

rational excited states, which require inclusion of at least static correlation.

The implementation of EFP developed in this work is a part of Q-Chem. This state

of the artab initio electronic structure package implements a wide variety of correlated

methods1. Therefore, a promising direction of the future work is to develop a correlated

EFP theory and to extend our implementation so that the active region can be treated

with any of correlation methods available in Q-Chem. This theory will allow one to

account for the environmental effects in the electronic structure studies of radicals and

electronically excited species, as well as to model chemical reactions in the condensed

phase. Relevant to the research conducted in this work (chapter 2), correlated EFP will

allow to model the reaction of Bergman cyclization in solutions.

Below a possible framework for such correlated EFP theory is outlined. In the

present EFP formulation, the effective fragments affect the active region via one-electron

terms in the Hamiltonian. These terms originate from the electrostatics and polariza-

tion components. Correlated methods can similarly use this modified Hamiltonian to

account for the field of the effective fragments. However, a complication arises from

the polarization component. Its contribution to the Hamiltonian depends on the values

of the induced dipoles. The induced dipoles depend on the wavefunction of the active

region, which in turn depends on the Hamiltonian. Thus the correlated wavefunction

and the induced dipoles should be consistent with each other. To satisfy the consistency

requirement, the induced dipoles and the correlated wavefunctions should be computed

self-consistently with the following iterative procedure:

1. Compute the correlated wavefunction for a given values of the induced dipoles.

2. Update the induced dipoles to account for the new wavefunction.

98

3. If the induced dipoles and the wavefunction are converged exit, otherwise return

to step 1.

However, such strict self-consistency requires to recompute the correlated wavefunc-

tion multiple times and, therefore, is computationally expensive. Alternatively, one may

use the induced dipoles consistent with the HF wavefunction, which is computed as a

prerequisite for all single reference correlated methods. This way the consistency is

compromised to reduce the computational cost. Yet another approach is to perform just

one step toward the consistency, i.e., to compute the correlated wavefunction based on

the HF induced dipoles, update the induced dipoles and recompute the correlated wave-

function again. The accuracy of the last two approximations is yet to be investigated.

For the methods computing the correlated wavefunction iteratively (such as CCSD), it

is possible to design a more efficient integrated approach, similar to that for the Hartree-

Fock SCF procedure (section 1.3). In this approach, at every iteration both the correlated

wavefunction and the induced dipoles are updated and thus consistence of both will be

achieved once they converge. The drawback is the increased complexity in the imple-

mentation.

3.2 Implementation

Gradients

As discussed in section 1.7, the energy minima for the benzene dimer structures were

obtained using scans of the potential energy surface. For large systems, a better approach

would be to use a gradient-based optimization procedure. The gradient of the electro-

static energy was formulated in the original EFP paper19. Gradient for the dispersion

energy is simply a derivative of1R6 . More intricate and difficult to implement are recently

99

published gradients for polarization20,21and exchange-repulsion energies22. Implement-

ing the gradient of EFP energy will allow one to perform geometry optimizations and

molecular dynamics simulations for large molecular systems.

Fragment paramters

In the developed implementation, the EFP fragment parameters are defined in the Q-

Chem standard input, as described in section 1.6. The parameters of an effective frag-

ment are computed from an individual isolated molecules, as discussed in sections 1.2,

1.3 1.4 and 1.5. Thus, for a given molecule the same set of EFP parameters can be used

in all computations. This immediately prompts creating a library of the pre-computed

EFP parameters for molecules that frequently occur in the systems studied with EFP.

Such library can, for instance, include the parameters for common organic solvents

(e.g., water, alcohols, acetone, DMSO, acetonitrile, formic acid, dichloromethane, car-

bon tetrachloride, chloroform, tetrahydrofuran, benzene, phenol, toluene, etc.). The EFP

parameters included in the library can be carefully selected and benchmarked against

high levelab initio theories. Therefore, the benefits of such a library are not just the

convenience it provides, but also a high-quality assurance for the EFP parameters.

QM-EF dispersion and exchange-repulsion interactions

In the current state of EFP theory, the interaction between the effective fragments and the

ab initio region is due to the electrostatics and polarization components only. Thus, there

is no account for dispersion and exchange-repulsion interactions between them. The

proper theories for such interactions are presently developed in M.S. Gordon’s group.

Meanwhile, it is possible to account for such interactions by treating theab initio system

as an effective fragment for these two types of interactions. Since the wavefunction of

theab initio part is readily available, the dispersion and exchange-repulsion parameters

100

can be computed as discussed in sections 1.4 and 1.5, respectively. From these param-

eters, dispersion and exchange-repulsion interaction energies between the active region

and the effective fragments can be evaluated. Implementing such scheme will allow one

to properly compute the binding energy in the systems with anab initio region. In the

present EFP theory, binding energy is properly computed for systems composed of the

effective fragments only, whereas molecular systems with anab initio region are used

to model the solvation effects. Implementing the suggested technique will allow one to

combine accurate description of the solvation effects and binding energies.

3.3 Applications

Numerous applications of EFP for modeling the solvation effects and the binding ener-

gies in the clusters and condensed phase can be envisioned. Here, the future direction of

the research of multipleπ − π interactions conducted in this work (section 1.7) is pre-

sented. As demonstrated in Ref. 2 using the cyclic benzene trimer example, three-body

dispersion has a substantial contribution to many-body interaction energy. Introduction

of the three-body dispersion term in EFP is proposed in section 3.1. Once this term is

available, the extended EFP method can be applied to:

• study the role of many-bodyπ− π interactions in the multi dimensional aromatic

clusters and the bulk;

• model the mechanical properties of carbon-based materials, such as graphite and

fullerene powders;

• investigate the role ofπ − π stacking in the rigidity of DNA.

101

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116

Appendix A

GAMESS input for generating EFP

parameters of benzene

The GAMESS input file for generation the EFP parameters at the HF/6-311G++(3df,2p)

level of theory for the benzene monomer used in this work is given below.

$CONTRL RUNTYP=MAKEFP ISPHER=1 $END$SYSTEM TIMLIM=99999 MWORDS=150 $END$BASIS GBASIS=N311 NGAUSS=6 NFFUNC=1 NDFUNC=3 NPFUNC=2

DIFFSP=.T. DIFFS=.T. $END$SCF SOSCF=.F. DIIS=.T. $END$STONE

BIGEXP=4.0$END$DAMP IFTTYP(1)=3,2 IFTFIX(1)=1,1 THRSH=500 $END$DAMPGSC3=C1H4=H2C5=C1H6=H2C7=C1H8=H2C9=C1H10=H2C11=C1H12=H2BO43=BO21BO53=BO31BO65=BO21BO75=BO31BO87=BO21BO97=BO31BO109=BO21BO111=BO31BO119=BO31BO1211=BO21$END$DATA

benzene

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C1C1 6.0 -1.3942300 0.00000000 0.0000000H2 1.0 -2.4764870 0.00000000 0.0000000C3 6.0 -0.6971150 0.00000000 1.2074388H4 1.0 -1.2382435 0.00000000 2.1447002C5 6.0 0.6971150 0.00000000 1.2074388H6 1.0 1.2382435 0.00000000 2.1447002C7 6.0 1.3942300 0.00000000 -0.0000000H8 1.0 2.4764870 0.00000000 0.0000000C9 6.0 0.6971150 0.00000000 -1.2074388H10 1.0 1.2382435 0.00000000 -2.1447002C11 6.0 -0.6971150 0.00000000 -1.2074388H12 1.0 -1.2382435 0.00000000 -2.1447002

$END

118

Appendix B

Geometries of benezene oligomers

Geometries of all the benzene oligomers studied in this work are provided below in the

format of the Q-Chem$efp fragmentssection. Fig. 1.16 provides the definition of the

format.

Sandwich (S) structure of benzene dimer:

benzene 0.0 0.0 0.0benzene 0.0 4.0 0.0

T-shaped (T) structure of benzene dimer:

benzene 0.0 0.0 0.0 1.57079633 1.57079633 1.57079633benzene 0.0 5.2 0.0 1.57079633 1.57079633 0.00000000

Parallel-displaced (PD) structure of benzene dimer:

benzene 0.0 0.0 0.0benzene 1.2 3.9 0.0

S benzene trimer:

benzene 0.0 0.0 0.0benzene 0.0 4.0 0.0benzene 0.0 8.0 0.0

T1 benzene trimer:

benzene 0.0 0.0 0.0 1.57079633 1.57079633 1.57079633benzene 0.0 5.2 0.0 1.57079633 1.57079633 0.00000000benzene 0.0 10.4 0.0 1.57079633 1.57079633 1.57079633

T2 benzene trimer:

benzene 0.0 0.0 0.0 1.57079633 1.57079633 1.57079633benzene 0.0 5.2 0.0 1.57079633 1.57079633 0.00000000benzene 0.0 -5.2 0.0 1.57079633 1.57079633 0.00000000

PD benzene trimer:

119

benzene 0.0 0.0 0.0benzene 1.2 3.9 0.0benzene 0.0 7.8 0.0

C benzene trimer:

benzene 2.6 0.0 0.0 -0.25943951 0.00000000 0.00000000benzene -2.6 0.0 0.0 1.83495559 0.00000000 0.00000000benzene 0.0 -4.503332 0.0 3.92935069 0.00000000 0.00000000

PD-T benzene trimer:

benzene 0.0 0.0 0.0 1.57079633 1.57079633 1.57079633benzene 0.0 5.2 0.0 1.57079633 1.57079633 0.00000000benzene 0.0 -3.9 1.2 1.57079633 1.57079633 1.57079633

PD-S benzene trimer:

benzene 0.0 -4.0 0.0benzene 0.0 0.0 0.0benzene 1.2 3.9 0.0

S-T benzene trimer:

benzene 0.0 0.0 0.0 1.57079633 1.57079633 1.57079633benzene 0.0 5.2 0.0 1.57079633 1.57079633 0.00000000benzene 0.0 -4.0 0.0 1.57079633 1.57079633 1.57079633

S-10 benzene oligomer (larger S-oligomers are constructed similarly):

benzene 0.0 0.0 0.0benzene 0.0 4.0 0.0benzene 0.0 8.0 0.0benzene 0.0 12.0 0.0benzene 0.0 16.0 0.0benzene 0.0 20.0 0.0benzene 0.0 24.0 0.0benzene 0.0 28.0 0.0benzene 0.0 32.0 0.0benzene 0.0 36.0 0.0

T-10 benzene oligomer (larger T-oligomers are constructed similarly):

benzene 0.0 0.0 0.0 1.57079633 1.57079633 1.57079633benzene 0.0 5.2 0.0 1.57079633 1.57079633 0.00000000benzene 0.0 10.4 0.0 1.57079633 1.57079633 1.57079633benzene 0.0 15.6 0.0 1.57079633 1.57079633 0.00000000benzene 0.0 20.8 0.0 1.57079633 1.57079633 1.57079633benzene 0.0 26.0 0.0 1.57079633 1.57079633 0.00000000benzene 0.0 31.2 0.0 1.57079633 1.57079633 1.57079633benzene 0.0 36.4 0.0 1.57079633 1.57079633 0.00000000benzene 0.0 41.6 0.0 1.57079633 1.57079633 1.57079633benzene 0.0 46.8 0.0 1.57079633 1.57079633 0.00000000

120

PD-10 benzene oligomer (larger PD-oligomers are constructed similarly):

benzene 0.0 0.0 0.0benzene 1.2 3.9 0.0benzene 0.0 7.8 0.0benzene 1.2 11.7 0.0benzene 0.0 15.6 0.0benzene 1.2 19.5 0.0benzene 0.0 23.4 0.0benzene 1.2 27.3 0.0benzene 0.0 31.2 0.0benzene 1.2 35.1 0.0

121

Appendix C

Converter of the EFP parameters

A script, which converts the GAMESS input parameters to the Q-Chem format is given

below:

#!/usr/bin/env python"""EFP parameters converter.Converts from GAMESS to Q-Chem input format.

Usage: python converter.py <GAMESS INPUT>"""

import sysimport re

# Bohrs to Angstroms convertion factorB2A = 0.529177249

__version__ = "1.15"__author__ = "Vitalii Vanovschi"

if "set" not in globals():from sets import Set as set

(FRAG, COORDS, MONOPOLES, DIPOLES, QUADRUPOLES, OCTUPOLES, \BASIS, WAVEFUNCTION, FOCK, LMO, MULTIPLICITY, SCREEN2,\POLARIZABLE, DYNAMIC, CANONVEC, CANONFOK, SCREEN3) = xrange(0, 17)

def dump_qchem_frag(frag_name, labels, charges, dipoles, quadrupoles, \octupoles, wavefunction, fock, centroids, screen2, \ordered_labels, nuc_charges, pols, disps, basis,coordinates):

"""Outputs EFP fragment parameters in the Q-Chem format"""

print "fragment", frag_name

# output atomsfor coordinate in coordinates:

if coordinate[0][0:2] != "bo":print coordinate[0].rstrip("0123456789") + " " \

+ " ".join(coordinate[1:])

# output atomic charges (as multipoles)for label in ordered_labels:

coord = labels[label]if label in nuc_charges and float(nuc_charges[label]) > 0.1:

print "mult ", " ".join(coord)print nuc_charges[label]

# output multipolesfor label in ordered_labels:

122

coord = labels[label]print "mult ", " ".join(coord)if label in screen2:

print "cdamp " + screen2[label][1]if label in charges:

print charges[label]if label in dipoles:

print " ".join(dipoles[label])if label in quadrupoles:

print " ".join(quadrupoles[label])if label in octupoles:

print " ".join(octupoles[label])

# output polarization pointsfor pol in pols:

print "pol ", pol[1], " ", pol[2], " ", pol[3]print " ".join(pol[4:])

# output dispersion pointsdispindexes = list(set([x[0] for x in disps]))dispindexes.sort()for i in dispindexes:

isfirst = Truefor disp in disps:

if disp[0] == i:if isfirst:

print "disp ", disp[1], " ", disp[2], " ", disp[3]isfirst = False

print disp[4],print

# output exchange-repulsionprint "er_basis"print "er_wavefunction"wfindexes = wavefunction.keys()wfindexes.sort()

if len(wfindexes) * (len(wfindexes) + 1) != 2 * len(fock):raise AssertionError("Inconsistent input.\n"

"The following assertion has failed:\n""len(wfindexes) * (len(wfindexes) + 1) == 2 * len(fock)\n""Values: len(wfindexes) = " + str(len(wfindexes)) + ", ""len(fock) = " + str(len(fock)) + ".\n""Did you remove ’>’ from the last line of the FOCK MATRIX \n""section in GAMESS input? (an old GAMESS bug)")

for ind in wfindexes:orb = wavefunction[ind]i = 0for shell in basis:

if shell == ’s’:i += 1

if shell == ’p’:i += 3

if shell == ’sp’ or shell == ’l’:i += 4

if shell == ’d’:tmp = orb[i:i+6]

#10 -> 10#11 -> 12#12 -> 15#13 -> 11#14 -> 13

123

#15 -> 14orb[i] = tmp[0]orb[i+1] = tmp[3]orb[i+2] = tmp[1]orb[i+3] = tmp[4]orb[i+4] = tmp[5]orb[i+5] = tmp[2]i += 6

if shell == ’f’:tmp = orb[i:i+10]orb[i] = tmp[0] #fxxxorb[i+1] = tmp[3] #fxxyorb[i+2] = tmp[5] #fxyyorb[i+3] = tmp[1] #fyyyorb[i+4] = tmp[4] #fxxzorb[i+5] = tmp[9] #fxyzorb[i+6] = tmp[6] #fyyzorb[i+7] = tmp[7] #fxzzorb[i+8] = tmp[8] #fyzzorb[i+9] = tmp[2] #fzzzi += 10

for i in range(len(orb)):if i % 5 == 0:

print ind,if orb[i][0] == "-":

print orb[i],else:

print " " + orb[i],if i % 5 == 4:

printprint

print "er_fock_matrix"for i in range(len(fock)):

print fock[i],if i % 5 == 4:

printprintprint "er_lmos"for lmo in centroids:

print " ".join(lmo)

def parse_gamess(gamess_file):"""Parses GAMESS input file"""

str2int = "frag" : FRAG,"coordinates" : COORDS,"monopoles" : MONOPOLES,"dipoles" : DIPOLES,"quadrupoles" : QUADRUPOLES,"octupoles" : OCTUPOLES,"basis" : BASIS,"wavefunction" : WAVEFUNCTION,"fock" : FOCK,"lmo" : LMO,"multiplicity" : MULTIPLICITY,"screen2": SCREEN2,"polarizable": POLARIZABLE,"dynamic": DYNAMIC,"canonvec": CANONVEC,"canonfok": CANONFOK,

124

"screen3": SCREEN3

frag_name = Noneordered_labels = []labels = coordinates = []charges = nuc_charges = dipoles = quadrupoles = octupoles = screen2 = centroids = []wavefunction = fock = []expect = FRAGprefix = ""pols = []disps = []basis = []

print "$efp_params"for line1 in gamess_file:

line_orig = line1line1 = line1.strip()

if not line1 or line1.lower() == ’STOP’:continue

if line1[-1] == ">":prefix += " " + line1continue

else:line = prefix + " " + line1prefix = ""

fragment_start = re.search("\$([A-Za-z0-9]+)", line)if fragment_start:

text = fragment_start.groups()[0].lower()if text in ("efrag", "contrl"):

continue

if text == "end" and frag_name:# print "frag_name:", frag_name# print "labels:", labels# print "dipoles:", dipoles# print "quadrupoles:", quadrupoles# print "octupoles:", octupoles# print "wavefunction:", wavefunction# print "fock:", fock# print "centroids:", centroids

dump_qchem_frag(frag_name, labels, charges, dipoles,quadrupoles, octupoles, wavefunction, fock, centroids,screen2, ordered_labels, nuc_charges, pols, disps,basis, coordinates)

frag_name = Nonelabels = charges = nuc_charges = dipoles =

125

quadrupoles = octupoles = screen2 = centroids = []coordinates = []ordered_labels = []wavefunction = fock = []expect = FRAGprefix = ""pols = []disps = []basis = []

if text != "end":frag_name = textwhile 1:

line = gamess_file.next()words = re.findall("([A-Za-z0-9\.+-]+)", line)if words[0].lower() == "coordinates":

breakexpect = COORDS

continue

words = re.findall("([A-Za-z0-9\.+-]+)", line)if words:

firstword = words[0].lower()else:

firstword = None

if firstword in str2int:expect = str2int[firstword]continue

if firstword == ’projection’ and len(words) > 1:expect = str2int[words[1].lower()]continue

if expect == COORDS and len(words) > 3:atom_x = str(float(words[1])*B2A)atom_y = str(float(words[2])*B2A)atom_z = str(float(words[3])*B2A)labels[firstword] = (atom_x, atom_y, atom_z)coordinates.append((firstword, atom_x, atom_y, atom_z))ordered_labels.append(firstword)

if expect == MONOPOLES and len(words) > 1:charges[firstword] = words[1]nuc_charges[firstword] = words[2]

if expect == DIPOLES and len(words) == 4:dipoles[firstword] = words[1:]

if expect == QUADRUPOLES and len(words) == 7:quadrupoles[firstword] = words[1:]

if expect == OCTUPOLES and len(words) == 11:octupoles[firstword] = words[1:]

if expect == WAVEFUNCTION:# 3 1-1.40747045E-01 2.10323654E-01 2.42989533E-01-4.68907985E-01-1.03447318E-01

key = int(line_orig[:3].strip())

126

if key not in wavefunction:wavefunction[key] = []

values = line_orig[5:].rstrip()while values:

wavefunction[key].append(values[:15].strip())values = values[15:]

if expect == FOCK:fock += words

if expect == LMO and len(words) == 4:centroids.append([str(float(x)*B2A) for x in words[1:]])

if expect == SCREEN2:screen2[firstword] = words[1:]

if expect == POLARIZABLE:if len(words) == 4:

pol_x = str(float(words[1])*B2A)pol_y = str(float(words[2])*B2A)pol_z = str(float(words[3])*B2A)pols.append([words[0], pol_x, pol_y, pol_z])

elif len(words) == 9:pols[-1].extend(words)

if expect == DYNAMIC:if len(words) == 5 or len(words) > 5 and words[5] == "--":

index = int(words[1])disp_x = str(float(words[2])*B2A)disp_y = str(float(words[3])*B2A)disp_z = str(float(words[4])*B2A)disps.append([index, disp_x, disp_y, disp_z])

elif len(words) == 9:disps[-1].append(sum([float(x) for x in words[:3]]) / 3)

if expect == BASIS:if firstword in (’s’, ’sp’, ’p’, ’d’, ’l’, ’f’):

basis.append(firstword)

print "$end"

if __name__ == "__main__":if len(sys.argv) != 2:

print "Usage: python converter.py <GAMESS INPUT>"sys.exit()

INPUT = open(sys.argv[1])parse_gamess(INPUT)INPUT.close()

127

Appendix D

Fragment-fragment electrostatic

interaction

The equations for electrostatics inter-fragment interaction energy are presented below.

k and l are the multipole expansion points.q is the charge,µ is the dipole,Θ is the

quadrupole,Ω is the octupole,R is the distance between the expansion pointsk andl,

a, b andc are the components of the distance.

Charge-charge interaction energy:

Ech−chkl =

qkql

Rkl

(D.1)

Charge-dipole interaction energy:

Ech−dipkl =

qk∑x,y,z

a µlaa

R3kl

(D.2)

Charge-quadrupole interaction energy:

Ech−quadkl =

qk∑x,y,z

a

∑x,y,zb Θl

abab

R5kl

(D.3)

Charge-octupole interaction energy:

Ech−octkl =

qk∑x,y,z

a

∑x,y,zb

∑x,y,zc Ωl

abcabc

R7kl

(D.4)

128

Dipole-dipole interaction energy:

Edip−dipkl =

∑x,y,za µk

aµlb

R3kl

− 3

∑x,y,za

∑x,y,zb µk

aµlbab

R5kl

(D.5)

Dipole-quadrupole interaction energy:

Edip−quadkl = −2

∑x,y,za

∑x,y,zb Θk

abµlab

R5kl

+ 5

∑x,y,za

∑x,y,zb Θk

abab∑x,y,z

c µlcc

R7kl

(D.6)

Quadrupole-quadrupole interaction energy:

Equad−quadkl = 2

∑x,y,za

∑x,y,zb Θk

abΘlab

3R5kl

−20

∑x,y,za

∑x,y,zb Θk

abb∑x,y,z

c Θlacc

3R7kl

+ 35

∑x,y,za

∑x,y,zb Θk

abab

3R9kl

(D.7)

Nuclei-charge interaction energy:

Enuc−chIl =

ZIql

rkl

(D.8)

Nuclei-dipole interaction energy:

Enuc−dipIl =

ZI∑x,y,z

a µlaa

R3kl

(D.9)

Nuclei-quadrupole interaction energy:

Enuc−quadIl =

ZI∑x,y,z

a

∑x,y,zb Θl

abab

R5kl

(D.10)

Nuclei-octupole interaction energy:

Enuc−octIl =

ZI∑x,y,z

a

∑x,y,zb

∑x,y,zc Ωl

abcabc

R7kl

(D.11)

129

Appendix E

Buckingham multipoles

E.1 The conversion from the sperical to the Bucking-

ham multipoles

The Buckingham chargeq in terms of the spherical multipoleQ with l = 0 angular

momentum:

q = Q0,0 (E.1)

The Buckingham dipoleµ in terms of the spherical multipoleQ with l = 1 angular

momentum:

µx = −2Q11

µy = 2Q11

µz = Q10

(E.2)

130

The Buckingham quadrupoleΘ in terms of the spherical multipoleQ with l = 2

angular momentum:

Θxx = 6Q22 −Q20

Θxy = −6Q22

Θyy = −6Q22 −Q20

Θxz = −3Q21

Θyz = 3Q21

Θzz = 2Q20

(E.3)

The Buckingham octupoleΩ in terms of the spherical multipoleQ with l = 3 angular

momentum:

Ωxxx = −30Q33 + 6Q31

Ωxxy = 30Q33 − 2Q31

Ωxxz = −3Q30 + 10Q32

Ωxyy = 30Q33 + 6Q31

Ωxyz = −10Q32

Ωxzz = −8Q31

Ωyyy = −30Q33 − 6Q31

Ωyyz = −3Q30 − 10Q32

Ωyzz = 8Q31

Ωzzz = 6Q30

(E.4)

131

E.2 The conversion from the Cartesian to the Bucking-

ham multipoles

.

The Buckingham chargeq in terms of the Cartesian chargeq′:

q = q′ (E.5)

The Buckingham dipoleµ in terms of the Cartesian dipoleµ′:

µx = µ′x

µy = µ′y

µz = µ′z

(E.6)

The Buckingham quadrupoleΘ in terms of the Cartesian quadrupoleΘ′:

halftrace =Θ′

xx + Θ′yy + Θ′

zz

2

Θxx =3

2Θ′

xx − halftrace

Θxy =3

2Θ′

xy

Θyy =3

2Θ′

yy − halftrace

Θxz =3

2Θ′

xz

Θyz =3

2Θ′

yz

Θzz =3

2Θ′

zz − halftrace

(E.7)

132

The Buckingham octupoleΩ in terms of the Cartesian octupoleΩ′:

traceX = Ω′xxx + Ω′

xyy + Ω′xzz

traceY = Ω′xxy + Ω′

yyy + Ω′yzz

traceZ = Ω′xxz + Ω′

yyz + Ω′zzz

Ωxxx =5

2Ω′

xxx −3

2traceX

Ωxxy =5

2Ω′

xxy −3

2traceY

Ωxxz =5

2Ω′

xxz −3

2traceZ

Ωxyy =5

2Ω′

xyy −1

2traceY

Ωxyz =5

2Ω′

xyz −1

2traceZ

Ωxzz =5

2Ω′

xzz −1

2traceX

Ωyyy =5

2Ω′

yyy −1

2traceX

Ωyyz =5

2Ω′

yyz −1

2traceZ

Ωyzz =5

2Ω′

yzz −1

2traceY

Ωzzz =5

2Ω′

zzz

(E.8)

133

Appendix F

The Q-Chem input for EFP water

cluster

The Q-Chem input filea for a molecular system with two effective fragment water

molecules and another water moleucle in the the active region is presented below:

$molecule0 1O 2.98679899864 0.000000000000 0.000000000000H 3.37280799847 0.408736299796 0.759199899653H 3.37280799847 0.408736299796 -0.759199899653$end

$rembasis 6-31G*exchange hf$end

$efp_fragmentswater 2.5 0.0 0.0water 0.0 2.5 0.0$end

$efp_paramsfragment watero 2.98679899864 0.000000000000 0.000000000000h 3.37280799847 0.408736299796 0.759199899653h 3.37280799847 0.408736299796 -0.759199899653

mult 2.98679899864 0.000000000000 0.0000000000008.00000

mult 3.37280799847 0.408736299796 0.7591998996531.00000

mult 3.37280799847 0.408736299796 -0.759199899653

aLines ending at a backslash symbol should be joined with the next line in the actual Q-Chem input.

134

1.00000

mult 2.98679899864 0.000000000000 0.000000000000cdamp 2.432093890-5.9880774156

0.9069997308 0.960401737 0.000000000-2.0776300674 -2.055053110 -1.094067772 0.1972105926\

0.0000000000 0.0000000001.2431711201 1.307173512 0.0000000000 0.3882335020\0.0000000000 0.357964785 0.0000000000 0.3997408222\0.4232766191 0.000000000

mult 3.37280799847 0.408736299796 0.759199899653cdamp 1.856169014-0.6161272254-0.0607832521 -0.0643620268 -0.1385305992-0.2984353445 -0.2970178207 -0.2629326306 0.0123821246\

0.0186582649 0.0197568195-0.1419121249 -0.1496397602 -0.2838114377 -0.0466364802\-0.1010324765 -0.0434504063 -0.0996406903 -0.0238456713\-0.0252496482 0.0121573058

mult 3.37280799847 0.408736299796 -0.759199899653cdamp 1.856168614-0.6161272254-0.0607832521 -0.0643620268 0.1385305992-0.2984353445 -0.2970178207 -0.2629326306 0.0123821246\-0.0186582649 -0.0197568195-0.1419121249 -0.1496397602 0.2838114377 -0.0466364802\

0.1010324765 -0.0434504063 0.0996406903 -0.0238456713\-0.0252496482 -0.0121573058

mult 3.17980349856 0.204368149898 0.379599949853cdamp 10.000000000-1.3898340671

0.0671975530 0.0711539868 -0.0800483392-0.7200126389 -0.7080787920 -0.3459299814 0.1042426037\

0.1265126185 0.13396138360.0823824078 0.0860368352 -0.1633636547 0.0224998491\

-0.0684206741 0.0201192189 -0.0686914603 0.0517621518\0.0548097854 -0.0023653270

mult 3.17980349856 0.204368149898 -0.379599949853cdamp 10.000000000-1.3898340671

0.0671975530 0.0711539868 0.0800483392-0.7200126389 -0.7080787926 -0.3459299814 0.1042426037\-0.1265126185 -0.1339613836

0.0823824078 0.0860368352 0.1633636547 0.0224998491\0.0684206741 0.0201192189 0.0686914603 0.0517621518\

135

0.0548097854 0.0023653270

pol 3.225043346880 0.252271132154 -0.4488098121240.3364082841 0.3771885556 2.0857377185 0.3562152772\

-0.6146554330 -0.6508454378 0.3562153645 -0.9006532748\-0.9536819287

pol 3.22504358327 0.252271266354 0.4488097842360.3364078945 0.3771881750 2.0857352855 0.3562148807\0.6146555083 0.6508456481 0.3562149890 0.9006533941\0.9536821303

pol 2.68786044569 0.0680155240902 0.0000000000000.0546318425 0.2156829505 0.4678560525 0.0987167685\

-0.0000001586 -0.0000002666 0.1193632771 -0.0000002037\-0.0000002725

pol 3.07178559579 -0.294563387213 0.0000000000000.2027164107 0.0676015168 0.4678627585 0.1278414717\0.0000000833 0.0000000562 0.1071947676 0.0000000844\0.0000000709

disp 3.22504334688 0.252271132154 -0.4488098121240.9330994836 0.9328329368 0.9312726852 0.9258111592\0.9104538267 0.8718871966 0.7834519894 0.6086537072\0.3529710151 0.1263476070 0.0217816395 0.0007641188

disp 3.22504358327 0.252271266354 0.4488097842360.9330984158 0.9328318692 0.931271618733 0.92581009\0.9104527744 0.8718861726 0.7834510398 0.6086529349\0.3529705524 0.1263474433 0.0217816120 0.0007641179

disp 2.68786044569 0.068015524090 0.0000000000000.2460531659 0.2459405339 0.2452820477 0.2429879390\0.2366256365 0.2211913563 0.1883852733 0.1320296167\0.0651263370 0.0192960190 0.0029443470 0.0001007762

disp 3.07178559579 -0.294563387213 0.0000000000000.2460564460 0.2459438126 0.2452853183 0.2429911814\0.2366288005 0.2211943284 0.1883878314 0.1320314467\0.0651272751 0.0192963144 0.0029443942 0.0001007779

er_basis sto-3g

er_wavefunction1 9.12366435E-02 -2.55851988E-01 -2.74993695E-01 -2.91183082E-011 4.26891343E-01 1.09661524E-01 -5.16313349E-012 9.12367358E-02 -2.55852205E-01 -2.74994306E-01 -2.91183344E-012 -4.26890773E-01 -5.16313220E-01 1.09660818E-01

136

3 1.54680314E-01 -6.48520365E-01 7.87066922E-01 -1.96453849E-013 -1.20289533E-07 8.88260577E-02 8.88262341E-024 1.54680523E-01 -6.48521682E-01 -2.41107828E-01 7.74554226E-014 3.89668406E-08 8.88271814E-02 8.88271243E-02

er_fock_matrix-0.8097112963 -0.1811928224 -0.8097117054 -0.1951745608-0.1951748560 -0.5598738582 -0.1951742273 -0.1951744336-0.1696642207 -0.5598736581

er_lmos3.22504334688 0.252271132154 -0.4488098121243.22504358327 0.252271266354 0.4488097842362.68786044569 0.068015524090 0.0000000000003.07178559579 -0.294563387213 0.000000000000$end

137

Appendix G

Q-Chem’s keywords affecting the EFP

parameters computation

Q-Chem’s keywords affecting the computation of the electrostatics and polarization

parameters are presented below in the format compartiblie with the Q-Chem manual:

POL GENWhether distributed polarizabilities should be computed for makeefp jobtype.

TYPE:LOGICAL

DEFAULT:TRUE

OPTIONS:TRUE/FALSE

RECOMMENDATION:none

POL PRINT BOHRPrint polarizabilities in Bohr instead of Angstroms.

TYPE:LOGICAL

DEFAULT:FALSE

OPTIONS:TRUE/FALSE

RECOMMENDATION:none

138

POL DIFF TYPEFinite differentitation type

TYPE:INTEGER

DEFAULT:2

OPTIONS:2 two steps finite difference (for higher accuracy)

1 one step finite difference (faster)RECOMMENDATION:

none

POL FIELD DELTAElectric filed (step size for numerical differentiation of polarizabilities)

TYPE:INTEGER

DEFAULT:100

OPTIONS:n n ∗ 10−7 step size.

RECOMMENDATION:none

POL LMO FREEZEHow many core orbitals to freeze for LMO computation

TYPE:INTEGER

DEFAULT:-1

OPTIONS:-1 freeze all core orbitals

n freezen core orbitalsRECOMMENDATION:

none

DMA GENWhether to perform distributed multipole analysis (DMA) for jobtype makeefp.

TYPE:LOGICAL

DEFAULT:TRUE

OPTIONS:TRUE/FALSE

RECOMMENDATION:none

139

DMA MIDPOINTSAdd bond midpoint to DMA expansion points on bond centers (in addtion to expansionpoints on nucleai)

TYPE:LOGICAL

DEFAULT:TRUE

OPTIONS:TRUE/FALSE

RECOMMENDATION:none

DAMP STO1G GUESSWhether to use STO1G guess procedure

TYPE:LOGICAL

DEFAULT:TRUE

OPTIONS:TRUE/FALSE

RECOMMENDATION:none

DAMP GEN EXP FITWhether to compute exponential damping

TYPE:LOGICAL

DEFAULT:FALSE

OPTIONS:TRUE/FALSE

RECOMMENDATION:none

DAMP GEN GAUSS FITWhether to compute gaussian damping

TYPE:LOGICAL

DEFAULT:FALSE

OPTIONS:TRUE/FALSE

RECOMMENDATION:none

140

DAMP EXP TOLTolerance for exponential fitting

TYPE:INTEGER

DEFAULT:10

OPTIONS:n n ∗ 10−7 tolerance

RECOMMENDATION:none

DAMP GAUSS TOLTolerance for gaussian fitting

TYPE:INTEGER

DEFAULT:1000

OPTIONS:n n ∗ 10−7 tolerance

RECOMMENDATION:none

DAMP GRID STEPGrid step for damping of electrostatic potential

TYPE:INTEGER

DEFAULT:50

OPTIONS:n n/100 Angstroms

RECOMMENDATION:none

DAMP RMINMinimum distance from a nucleai to a grid point

TYPE:INTEGER

DEFAULT:67

OPTIONS:n n/100 Angstroms

RECOMMENDATION:none

141

DAMP RMAXMaximum distance from a nucleai to a grid point

TYPE:INTEGER

DEFAULT:300

OPTIONS:n n/100 Angstroms

RECOMMENDATION:none

142