Practical aspects of computational chemistry methods concepts and applications

Practical Aspects of Computational Chemistry


Jerzy Leszczynski

l

Manoj K. Shukla

Editors

Practical Aspects of
Computational Chemistry
Methods, Concepts and Applications


Editors
Prof. Jerzy Leszczynski
Jackson State University
Department of Chemistry
and Biochemistry
1325 J. R. Lynch St.
Jackson MS 39217
USA


Dr. Manoj K. Shukla
Jackson State University
Department of Chemistry
and Biochemistry

1325 J. R. Lynch St.
Jackson MS 39217
USA


ISBN 978-90-481-2686-6
e-ISBN 978-90-481-2687-3
DOI: 10.1007/978-90-481-2687-3
Springer Heidelberg Dordrecht London New York
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Preface

Very few areas of science enjoy such a fast progress as has been witnessed in the
last quarter of the 20th century for computational chemistry (CC). An access to
increasingly faster and more powerful computers in parallel with continuous
developments of more efficient computational programs and methods contributed
toward employment of the CC approaches in both basic science as well as commercial applications. As a result, the investigated molecules are larger than ever and can
be studied not only in vacuum but also in different solvent environments or in a
crystal. Such remarkable progress has not been unnoticed by scientific community.
In fact, the chemical and physical societies celebrated the great event of the 1998

Nobel prize in chemistry that was awarded to two leading theoretical chemists/
physicists: Walter Kohn and John A. Pople for their seminal contributions to the
development of efficient computational methods for quantum chemistry. Owing to
the meticulous and continuous efforts, the computational chemistry methods have
become complementary to the costly and time‐consuming experiments and in many
cases they provide the only reliable information when experiment is not possible or
investigated species exhibit a health hazard to the investigators.
The methods and applications of the CC are the topics of the current book
entitled ‘‘Practical Aspects of Computational Chemistry: Methods, Concepts, and
Applications. Special Issue of Annals–The European Academy of Sciences’’. It was
not our goal to collect specialized contributions aimed at a narrow group of experts.
Instead, we asked all authors to provide more general reviews, focusing toward
general interests of the affiliates of the academy and members of scientific society.
Though, it is not possible to cover all topics related to the CC in one volume, we hope
that the collected contributions adequately highlight this important scientific area.
This book encompasses 23 contributions on different aspects of CC applied to a
large arena of research field. The first contribution by Flores-Moreno and Ortiz
deals with the theoretical formulation of electron propagator methods developed to
compute accurate ionization potentials and electron affinity of system of different
sizes. This review describes recent implementations that can be used for more
challenging system without compromising the accuracy of the results. In the next

v


vi

Preface

contribution, Cammi et al. have reviewed the implementation of Polarizable

Continuum Model to describe the effect of different solvents on ground and excited
state structural properties of variety of systems. Alkorta and Elguero have reviewed
the chiral recognition from a theoretical perspective in the next contribution where
a meticulous theoretical and experimental analysis is presented. Multiscale modeling is key for more accurate simulations of solid materials. In the following
contribution, Horstemeyer has reviewed different aspects of computational muliscale modeling, its successes, limitations, current challenges, and possible ways for
improvement. The multiple minima problem is connected to all applications of
theory to structural chemistry. Protein folding as an example of multiple minima
problem is discussed by Piela in the next contribution.
s‐Hole bonding is defined as a highly directional noncovalent interaction between
a positive region on a covalently-bonded Group V – VII atom and a negative site on
another molecule, e.g. a lone pair of a Lewis base. Politzer and Murray have
discussed an overview of s‐hole bonding in variety of system in their contribution.
And this contribution is followed by the discussion of s‐ and p‐bonds in the main
group and transition metal complexes by Pathak et al. In this contribution, authors
have described possible mechanisms related to the phenomena where s-bonds
prevent p‐bonds from adopting their optimal shorter distances. We collected three
contributions discussing the structure-activity relationships. Two of them – one
written by Benfenati and other by Puzyn et al. are devoted to the description of the
REACH programs of the European Union for chemical regulatory purpose. The
possibility of application of this new regulation to nanomaterials is also discussed.
The third contribution by Vogt et al. discusses the structure-activity relationships in
nitroaromatic compounds to predict their physicochemical properties.
In the next contribution, Lipkowski and Suwin´ska have discussed the different
complications that may crop up in solving molecular structures using X-ray
crystallography. These authors have described how molecular modeling methods
can work as an auxiliary method in solving and refining such problems. Dihydrogen
bonds are considered as a special type of hydrogen bond and are formed when two
hydrogen atoms, one of them is negatively while other is positively charged, are
usually closer than the sum of their van der Waals radii. Grabowski and Leszczynski have reviewed the novelty of dihydrogen bonds in the next contribution.
And this is followed by a contribution from Michalkova and Leszczynski who have

summarized the results of theoretical and experimental studies on organophosphorus systems, which may be used to develop theoretical models in explaining and
predicting how clay minerals and metal oxides can affect the adsorption and
decomposition of selected organophosphorus compounds. Clean energy resources
is currently a major thrust area of fundamental and applied research. Dinadayalane
and Leszczynski have discussed the mechanism toward the hydrogen storage in
single-walled carbon nanotube via the chemisorption mechanism in the next
contribution.
There are four contributions based on Monte Carlo (MC) simulations of different
systems. These contributions include lucid discussion of the fundamentals of MC
methods used in electronic structure calculations by Lester, the MC simulation, and


Preface

vii

quantum mechanical calculations to compute the static dipole polarizability and the
related dielectric constant of atomic argon in the liquid phase by Coutinho and
Canuto and the application of free energy perturbation/MC simulations in molecular mechanics parameterization of CO2(aq) for use in CO2 sequestration modeling
studies and that of similar investigations of liquid and solid phases of water to
determine the melting temperature of several popular 3‐ and 4-site water models
by Dick et al. In the next contribution, Latajka and Sobczyk have reviewed the lowbarrier hydrogen bond problem in protonated naphthalene proton sponges. Experimental data related to the infra-red and NMR spectra and contemporary theoretical
approaches to the barrier height for the proton transfer are also discussed.
The last four contributions are devoted to the structures and properties of nucleic
acid fragments. Czyz˙nikowska et al. have discussed the most accurate and reliable
framework for the analysis of intermolecular interactions in nucleic acid bases by
the quantum chemical method. Shishkin et al. have reviewed the recent results of
the conformational flexibility of nucleic acid bases and model systems. Such
conformational flexibility arises from the high deformability of the pyrimidine
ring where transition from a planar equilibrium conformation to a sofa configuration results in an increase of energy by less than 1.5 kcal/mol. DNA is constantly

attacked by a large number of endogenous and exogenous reactive oxygen species
(ROS), reactive nitrogen oxide species (RNOS), and alkylating agents. As a result
of these interactions several lesions are produced and some of them are implicated
in several lethal diseases. In the next contribution, Shukla and Mishra have
reviewed recent results of interaction of ROS and RNOS with guanine. Nucleic
acids can form complex structures that consist of more than two strands. Recent
investigations of the polyads of the nucleic acid bases strongly suggest that all of the
NABs can form stable tetrad structure in cyclic form through the H-bonding
between the neighboring bases. The last contribution of this special issue is
provided by Gu et al. where authors have reviewed the results of recent studies
on structural properties of nucleic acid tetrads and role of metal ions in such
formation.
With great pleasure, we take this opportunity to thank all the authors for
devoting their time and hard work in enabling us to complete this book. We are
grateful to the excellent support from the President of the EAS, Editor in Chief of
the Annals, as well as the editors at Springer. Many thanks go to our families and
friends without whom the realization of this book is not possible.
MS, USA

Jerzy Leszczynski and Manoj K. Shukla


Contents

1

Efficient and Accurate Electron Propagator Methods
and Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Roberto Flores-Moreno and J.V. Ortiz

2

Properties of Excited States of Molecules in Solution Described
with Continuum Solvation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
R. Cammi, C. Cappelli, B. Mennucci, and J. Tomasi

3

Chirality and Chiral Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Ibon Alkorta and Jose´ Elguero

4

Multiscale Modeling: A Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
M.F. Horstemeyer

5

Challenging the Multiple Minima Problem: Example
of Protein Folding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Lucjan Piela

6

An Overview of s-Hole Bonding, an Important
and Widely-Occurring Noncovalent Interaction . . . . . . . . . . . . . . . . . . . . . . . 149
Peter Politzer and Jane S. Murray

7

s‐Bond Prevents Short p-Bonds: A Detailed Theoretical
Study on the Compounds of Main Group and Transition
Metal Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Biswarup Pathak, Muthaiah Umayal, and Eluvathingal D. Jemmis

8

QSAR Models for Regulatory Purposes: Experiences
and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Emilio Benfenati

ix


x

Contents

9

Quantitative Structure–Activity Relationships (QSARs)
in the European REACH System: Could These Approaches
be Applied to Nanomaterials? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Tomasz Puzyn, Danuta Leszczynska, and Jerzy Leszczynski

10

Structure–Activity Relationships in Nitro-Aromatic Compounds . . . . 217
R.A. Vogt, S. Rahman, and C.E. Crespo-Herna´ndez

11

Molecular Modeling as an Auxiliary Method in Solving
Crystal Structures Based on Diffraction Techniques . . . . . . . . . . . . . . . . . 241
Janusz Lipkowski and Kinga Suwin´ska

12

Dihydrogen Bonds: Novel Feature of Hydrogen
Bond Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
Sławomir J. Grabowski and Jerzy Leszczynski

13

Catalytic Decomposition of Organophosphorus Compounds . . . . . . . . 277
A. Michalkova and J. Leszczynski

14

Toward Understanding of Hydrogen Storage in Single-Walled
Carbon Nanotubes by Investigations of Chemisorption
Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
T.C. Dinadayalane and Jerzy Leszczynski

15

Quantum Monte Carlo for Electronic Structure . . . . . . . . . . . . . . . . . . . . . . 315
William A. Lester Jr.

16

Sequential Monte Carlo and Quantum Mechanics Calculation
of the Static Dielectric Constant of Liquid Argon . . . . . . . . . . . . . . . . . . . . 327
Kaline Coutinho and Sylvio Canuto

17

CO2(aq) Parameterization Through Free Energy Perturbation/
Monte Carlo Simulations for Use in CO2 Sequestration . . . . . . . . . . . . . 337
Thomas J. Dick, Andrzej Wierzbicki, and Jeffry D. Madura

18

Free Energy Perturbation Monte Carlo Simulations of Salt
Influences on Aqueous Freezing Point Depression . . . . . . . . . . . . . . . . . . . . 359
Thomas J. Dick, Andrzej Wierzbicki, and Jeffry D. Madura

19

The Potential Energy Shape for the Proton Motion in Protonated
Naphthalene Proton Sponges (DMAN-s) and its Manifestations . . . . 371
Z. Latajka and L. Sobczyk

20

Nucleic Acid Base Complexes: Elucidation of the Physical
Origins of Their Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
Z˙aneta Czyz˙nikowska, Robert Zales´ny, and Manthos G. Papadopoulos

Contents

xi

21

Conformational Flexibility of Pyrimidine Ring in Nucleic
Acid Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
Oleg V. Shishkin, Leonid Gorb, and Jerzy Leszczynski

22

DNA Lesions Caused by ROS and RNOS: A Review
of Interactions and Reactions Involving Guanine . . . . . . . . . . . . . . . . . . . . . 415
P.K. Shukla and P.C. Mishra

23

Stability and Structures of the DNA Base Tetrads: A Role
of Metal Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
Jiande Gu, Jing Wang, and Jerzy Leszczynski

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455


Contributors

Ibon Alkorta
Instituto de Quı´mica Me´dica (CSIC), Juan de la Cierva 3, E-28006 Madrid, Spain

Emilio Benfenati
Istituto di Ricerche Farmacologiche “Mario Negri,” Via Giuseppe La Masa 19,
20156 Milano, Italy,
R. Cammi
Dipartimento di Chimica G.I.A.F., Universita` di Parma, Parco Area delle Scienze,
I-43100 Parma, Italy,
Sylvio Canuto
Instituto de Fı´sica, Universidade de Sa˜o Paulo, CP 66318, 05315-970 Sa˜o Paulo,
SP, Brazil,
C. Cappelli
Dipartimento di Chimica e Chimica Industriale, Universita` di Pisa, Via Risorgimento 35, I-56126 Pisa, Italy
Kaline Coutinho
Instituto de Fı´sica, Universidade de Sa˜o Paulo, CP 66318, 05315-970 Sa˜o Paulo,
SP, Brazil,
C.E. Crespo-Herna´ndez
Department of Chemistry, Case Western Reserve University, 10900 Euclid
Avenue, Cleveland, OH 44106, USA,
Z˙aneta Czyz˙nikowska
Institute of Organic and Pharmaceutical Chemistry, The National Hellenic
Research Foundation, 48 Vas. Constantinou Avenue, 11635 Athens, Greece
xiii


xiv

Contributors

Thomas J. Dick
Department of Chemistry and Physics, Carlow University, 3333 Fifth Ave., Pittsburgh, PA 15213, USA
T.C. Dinadayalane

NSF CREST Interdisciplinary Nanotoxicity Center, Department of Chemistry and
Biochemistry, Jackson State University, Jackson, Mississippi 39217, USA
Jose´ Elguero
Instituto de Quı´mica Me´dica (CSIC), Juan de la Cierva 3, E-28006 Madrid, Spain
Roberto Flores-Moreno
Facultad de Quı´mica, Universidad de Guanajuato, Noria Alta s/n, Guanajuato, Gto.
36050, Me´xico,
Leonid Gorb
NSF CREST Interdisciplinary Nanotoxicity Center, Department of Chemistry and
Biochemistry Jackson State University, P.O. Box 17910, Jackson, MS 39217, USA
Department of Molecular Biophysics, Institute of Molecular Biology and Genetics,
National Academy of Science of Ukraine, 150 Zabolotnogo St., Kyiv 03143,
Ukraine
Sławomir J. Grabowski
NSF CREST Interdisciplinary Nanotoxicity Center, Department of Chemistry and
Biochemistry Jackson State University, Jackson, MS 39217, USA
Department of Chemistry, University of Ło´dz´, 90-236 Ło´dz´, ul.Pomorska 149/153,
Poland
Jiande Gu
NSF CREST Interdisciplinary Nanotoxicity Center, Department of Chemistry and
Biochemistry, Jackson State University, Jackson, MS 39217, USA
M.F. Horstemeyer
Department of Mechanical Engineering, Mississippi State University, MS 39760,
USA,
Eluvathingal D. Jemmis
Department of Inorganic and Physical Chemistry, Indian Institute of Science,
Bangalore 560-012, India,
Z. Latajka
Faculty of Chemistry, University of Wrocław, Joliot-Curie 14, 50-383 Wrocław,
Poland

Contributors

xv

William A. Lester, Jr.
Kenneth S. Pitzer Center for Theoretical Chemistry, Department of Chemistry,
University of California, Berkeley, CA 94720-1460, USA
Chemical Sciences Division, Lawrence Berkeley National Laboratory, Berkeley,
CA 94720, USA,
Danuta Leszczynska
NSF CREST Interdisciplinary Nanotoxicity Center, Department of Chemistry and
Biochemistry, Jackson State University, 1325 Lynch St, Jackson, MS 39217-0510,
USA
Department of Civil and Environmental Engineering, Jackson State University,
1325 Lynch St, Jackson, MS 39217-0510, USA,
Jerzy Leszczynski
NSF CREST Interdisciplinary Nanotoxicity Center, Department of Chemistry and
Biochemistry, Jackson State University, 1325 Lynch St, Jackson, MS 39217-0510,
USA,
Janusz Lipkowski
Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52,
Warszawa 01 224, Poland,
Jeffry D. Madura
Center for Computational Sciences, Department of Chemistry and Biochemistry,
Duquesne University, 600 Forbes Ave., Pittsburgh, PA 15282, USA
B. Mennucci
Dipartimento di Chimica e Chimica Industriale, Universita` di Pisa, Via Risorgimento 35, I-56126 Pisa, Italy
A. Michalkova

NSF CREST Interdisciplinary Nanotoxicity Center, Department of Chemistry and
Biochemistry, Jackson State University, Jackson, MS 39217, USA
P.C. Mishra
Department of Physics, Banaras Hindu University, Varanasi 221 005, India

Jane S. Murray
Department of Chemistry, University of New Orleans, New Orleans, LA 70148,
USA
Department of Chemistry, Cleveland State University, Cleveland, OH 44115, USA


xvi

Contributors

J.V. Ortiz
Department of Chemistry and Biochemistry, Auburn University, Auburn, AL
36849, USA,
Manthos G. Papadopoulos
Institute of Organic and Pharmaceutical Chemistry, The National Hellenic Research Foundation, 48 Vas. Constantinou Avenue, 11635 Athens, Greece

Biswarup Pathak
Department of Inorganic and Physical Chemistry, Indian Institute of Science,
Bangalore 560-012, India
Lucjan Piela
Department of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw,
Poland,
Peter Politzer
Department of Chemistry, University of New Orleans, New Orleans, LA 70148,
USA

Department of Chemistry, Cleveland State University, Cleveland, OH 44115, USA,

Tomasz Puzyn
NSF CREST Interdisciplinary Nanotoxicity Center, Department of Chemistry and
Biochemistry, Jackson State University, 1325 Lynch St, Jackson, MS 39217-0510,
USA
Laboratory of Environmental Chemometrics, Faculty of Chemistry, University of
Gdan´sk, Sobieskiego 18, 80-952 Gdan´sk, Poland
S. Rahman
Department of Chemistry, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA
Oleg V. Shishkin
STC ‘‘Institute for Single Crystals,’’ National Academy of Science of Ukraine, 60
Lenina Ave., Kharkiv 61001, Ukraine
NSF CREST Interdisciplinary Nanotoxicity Center, Department of Chemistry and
Biochemistry, Jackson State University, P.O. Box 17910, Jackson, MS 39217, USA
P.K. Shukla
Department of Physics, Banaras Hindu University, Varanasi 221 005, India


Contributors

xvii

L. Sobczyk
Faculty of Chemistry, University of Wrocław, Joliot-Curie 14, 50-383 Wrocław,
Poland,
Kinga Suwin´ska
Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01
224 Warszawa, Poland
J. Tomasi

Dipartimento di Chimica e Chimica Industriale, Universita` di Pisa, Via Risorgimento 35, I-56126 Pisa, Italy
Muthaiah Umayal
Department of Inorganic and Physical Chemistry, Indian Institute of Science,
Bangalore 560-012, India
R.A. Vogt
Department of Chemistry, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA
Jing Wang
NSF CREST Interdisciplinary Nanotoxicity Center, Department of Chemistry and
Biochemistry, Jackson State University, Jackson, MS 39217, USA
Andrzej Wierzbicki
Department of Chemistry, University of South Alabama, Mobile, AL 36688, USA
Robert Zales´ny
Institute of Organic and Pharmaceutical Chemistry, The National Hellenic
Research Foundation, 48 Vas. Constantinou Avenue, 11635 Athens, Greece


Chapter 1

Efficient and Accurate Electron Propagator
Methods and Algorithms
Roberto Flores-Moreno and J.V. Ortiz

Abstract Recent developments in electron propagator methods that employ the
quasiparticle approximation can facilitate calculations on molecules of unprecedented size. Reductions of arithmetic and storage requirements are considered. New
and reliable approximations that offer a better compromise of accuracy and feasibility are proposed. Transition operator orbitals, in combination with the second-order
self-energy, provide reliable predictions for valence and core electron binding energies with algorithms that are comparable in efficiency to their counterparts that
employ ordinary Hartree–Fock orbitals. Quasiparticle virtual orbitals enable accurate evaluation of third-order self-energy contributions, while significantly reducing
storage and arithmetic requirements. Algorithms that employ the resolution-ofthe-identity approach to the evaluation of electron repulsion integrals require less
memory but retain the accuracy of ordinary calculations. Numerical tests confirm
the promise of these new approaches.

1.1 Introduction
Several methods of electron propagator theory (EPT) [1–4], or the one-electron
Green’s function [5, 6] approach, are now well established techniques [4, 7–16] for
the theoretical description of molecular photoelectron spectra. Quasiparticle approximations in EPT can be viewed as correlated corrections to Koopmans’s theorem
(KT) results for electron attachment or detachment energies (EADEs). Systematic
improvements in electron propagator approximations produce better results for ionization energies and electron affinities. However, these improvements may require
lengthy calculations or large quantities of memory [11]. The diagonal, third order
approximation (and therefore, the outer valence Green’s function (OVGF) methods [6] as well) requires a step with ov4 arithmetic scaling (where o and v are the
R. Flores-Moreno1 and J.V. Ortiz2 (B)
1
Facultad de Química, Universidad de Guanajuato, Noria Alta s/n, Guanajuato, Gto. 36050,
México; 2 Department of Chemistry and Biochemistry, Auburn University, Auburn, AL 36849,
USA
e-mail: ;

J. Leszczynski and M.K. Shukla (eds.), Practical Aspects of Computational Chemistry,
DOI 10.1007/978-90-481-2687-3_1,
c Springer ScienceCBusiness Media B.V. 2009


1


2

R. Flores-Moreno and J.V. Ortiz

number of occupied and virtual spin-orbitals, respectively), and its self-energy formulae depend on the full set of transformed two-electron repulsion integrals [11].
Even the diagonal, partial third order method (P3) for ionization energies has an

o2 v3 step and calls for transformed integrals with one occupied and three virtual
indices in its rate-limiting contraction [9]. The second order approximation (EP2)
has a much better scaling behavior: for a single EADE calculation, the rate limiting
contraction scales as ov2 . This step is even faster than a conventional, self-consistent
field [17,18] iteration in the calculation of the reference Hartree–Fock [19,20] state.
Unfortunately, the reliability of results from EP2 calculations is very poor compared
to P3 or OVGF. The following order of reliability has been found:
P3, OVGF > EP2 > KT:
This hierarchy of approximations constitutes a guide for judging the quality of
the methods discussed below that aim to improve the efficiency of quasiparticle
electron propagator calculations.
Even with efficient, modern implementations [21,22] of the P3 and OVGF quasiparticle approximations, only medium size systems [12–15] can be treated with
average computer resources. A new set of approximations that may enable treatment of larger molecular systems using quasiparticle electron propagator methods
has been introduced. This set includes the use of transition operator orbitals [23–25]
in the second-order self-energy [26], reduction of the virtual orbital space for
higher order quasiparticle electron propagator calculations [27] and use of the
resolution-of-the-identity (RI) technique [28].
This paper is organized as follows. In Sect. 1.2, the superoperator formulation of
EPT is outlined to introduce nonspecialist readers to the terminology that is used
to discuss various approximations. In Sect. 1.3, approximations proposed for the
treatment of large molecules are described. The benefits of these techniques and the
reliability of their results are discussed in Sect. 1.4. Concluding remarks are made
in Sect. 1.5.

1.2 Superoperator Formulation
The superoperator formalism that has been used in previous publications is outlined here [2, 9, 29]. The alternative diagrammatic and algebraic-diagrammatic
representations can be found in other works [6].
After Fourier transformation, the time domain of the Green’s function is translated to frequency dependency. We start with the resulting spectral representation of
the one-electron propagator

1

Efficient and Accurate Electron Propagator Methods and Algorithms

3

X hN jap jN C 1; nihN C 1; njaqŽ jN i
!0
! En .N C 1/ C E0 .N / C i
n

Gpq .!/ D lim

C lim

!0

X hN jaqŽ jN
n

! C En .N

1; nihN 1; njap jN i
;
1/ E0 .N / i

(1.1)

where E0 .N / is the energy of the reference state with N electrons, jN i, and

En .N ˙ 1/ is the energy of the nth state of the system with N ˙ 1 electrons,
jN ˙ 1; ni. Creation and annihilation operators for the pth canonical molecular
Ž
orbital (MO) of the reference system are symbolized, respectively, by ap and ap .
From this expression, it is obvious that poles of the electron propagator correspond
to EADEs. When Hartree–Fock reference states are used, the following expansion
is employed:
8
<
X
X
Ž
jN i D W 1 C
ia aaŽ ai C
ijab aaŽ ab ai aj C
:
ia

i >j;a>b

9
=

jHF i

(1.2)

;

where W is a normalization constant and jHF i is the Hartree–Fock approximation

to the wavefunction. The correlation coefficients, , are obtained from Rayleigh–
Schrödinger perturbation theory [30]. Occupied MOs are labeled with i or j, and
virtual MOs with a or b.
In the superoperator approach, an abstract linear space is introduced [2]. The elements of this space are fermion operators generally expressed as linear combinations
of products of creation or annihilation operators,
fas ; aqŽ as at .s < t/; apŽ aqŽ as at au .p < q; s < t < u/; : : :g;

(1.3)

where the limits of the indices avoid double counting. Here, p, q, s, t, and u are
general MO indices. If Y and Z are two arbitrary operators, i.e., linear combinations
of products of creation or annihilation operators, the identity superoperator, IO, is
defined by
IOY D Y

(1.4)

and the Hamiltonian superoperator, HO , by
HO Y D ŒY; H  D YH

HY:

(1.5)

H is the Hamiltonian operator expressed in the language of second quantization,
H D

X
p;q

hpq apŽ aq C

1 X
hpqjjstiapŽ aqŽ at as ;
4 p;q;s;t

(1.6)


4

R. Flores-Moreno and J.V. Ortiz

where hpq is a matrix element of the one-electron contributions including kinetic
energy and external potentials acting on the electrons, such as the electrostatic
nuclear attraction. hpqjjsti is an antisymmetrized electron repulsion integral in
Dirac notation. The superoperator linear space is defined in terms of the following
rule for the inner product
.Y jZ/ D hN jŒY Ž ; ZC jN i D hN jY Ž Z C ZY Ž jN i:

(1.7)

This rule associates a complex number to each pair of operators. The value of this
number depends on the reference state used and the truncation of the perturbation
series,
.Y jZ/ D .Y jZ/.0/ C .Y jZ/.1/ C .Y jZ/.2/ C

;

(1.8)

where orders are defined with respect to the fluctuation potential operator, i.e., the
partitioning of Møller and Plesset [31] applies.
Using superoperators in combination with (1.1), we get the electron propagator
matrix
G.!/ D .aj.! IO

HO / 1 ja/;

(1.9)

where a is the set of simple field operators, fap g. By applying Löwdin’s inner projection technique [32], the inversion of an operator is avoided. Therefore, only a
matrix must be inverted according to
G.!/ D .ajh/.hj! IO

HO jh/ 1 .hja/;

(1.10)

where the projection space, h, is composed of elements of the superoperator linear
space of (1.3). When the set h contains the complete space, there is no approximation.
From (1.10), it follows that eigenvalues of the Hamiltonian superoperator correspond to poles of the Green’s function, and therefore, to EADEs. Thus, we are faced
with an eigenvalue problem [33, 34],
O D .hjHO jh/ C D ! C:
HC

(1.11)

The projection space can be decomposed for convenience into a primary space,
a, and a complementary space, f. The latter space contains operators associated

with ionizations coupled to excitations: triple products (two-hole particle, 2hp, and
two-particle hole, 2ph, subspaces), quintuple products, heptuple products and so
on. With this partition of the projection space, the eigenvalue problem can be rewritten as





Ca
.ajHO ja/ .ajHO jf/
Ca
(1.12)
D
!
:
Cf
Cf
.fjHO ja/ .fjHO jf/


1

Efficient and Accurate Electron Propagator Methods and Algorithms

5

The principal part of the eigenvectors, Ca , contains the combination of coefficients that is required to build Feynman–Dyson amplitudes (Dyson orbitals) from
the reference Hartree–Fock orbitals. Dyson orbitals as they result from (1.12) are
not normalized, i.e., the sum
X

(1.13)
Pp D
jCqp j2
q

is, in general, different from 1. Pp is known as the pole strength for the pth Dyson
orbital. A larger pole strength reflects an attachment or detachment process that
is well represented by an orbital. Small pole strengths correspond to correlation
final states where orbital descriptions are invalid. If we are interested only in the
solutions that are described chiefly by the principal space, the partitioning technique
introduced by Löwdin [32] allows us to reduce considerably the size of the system
of equations. The resulting system has an implicit dependence on the poles of the
electron propagator matrix,
n
.ajHO ja/


.ajHO jf/ !I

.fjHO jf/



1

o
.fjHO ja/ Ca D !Ca :

(1.14)

Equation (1.14) is generally presented in the following, alternative form:
Œ"

†.!/Ca D !Ca ;

(1.15)

where " is a diagonal matrix containing Hartree–Fock canonical orbital energies
as its nonzero entries and, †.!/ is known as the self-energy matrix. In actual
calculations, the self-energy matrix is approximated to a certain order in the perturbation series. The order of this matrix defines the order of the electron propagator
calculation. The first nonzero contribution occurs at second order, where
†.2/
pq .!/ D

X hpajjij ihj i jjaqi
X hpi jjabihbajji qi
C
:
! C "a "i "j
! C "i "a "b
a;i
(1.16)

i;a
Because second order is the first nonvanishing contribution, zero and first order
electron propagator calculations correspond to KT results. In the diagonal, P3 selfenergy, terms with three virtual indices, such as
X hbcjjqaihij jjbci
hpajjij i

1X
C
4 a;i;j ! C "a "i "j
"i C "j "b "c

(1.17)

b;c

occur. In OVGF calculations, a term with four virtual indices must be evaluated:
X habjjcd ihcd jjpi i
1X
hpi jjabi
:
4
E C "i "a "b
E C "i "c "d
i;a;b

c;d

(1.18)


6

R. Flores-Moreno and J.V. Ortiz

The evaluation of such terms imposes a greater arithmetic burden than that
encountered in second order calculations.

1.3 Quasiparticle Methods
Neglecting off-diagonal elements of the self-energy matrix in the canonical Hartree–
Fock basis in (1.15) constitutes the quasiparticle approximation. With this approximation, the calculation of EADEs is simplified, for each KT result may be improved
with many-body corrections that reside in a diagonal element of the self-energy
matrix.
The quasiparticle approximation has succeeded in the description of valence ionization spectra of many systems. Recently, it has been shown that reliable results
also can be obtained for core electron binding energies [26]. In this section, we will
describe some recent developments that have been realized with the quasiparticle
approximation.
The proposals found here can be seen as the result of a two-way strategy for the
treatment of large molecules. First, we improve on the accuracy of the very efficient second order approximation. In addition, we introduce approximations that
lower considerably the required computer resources for the use of higher-order
approximations to the electron propagator within the quasiparticle approach.

1.3.1 Transition Operator Method
The transition operator method combined with the second-order quasiparticle electron propagator (TOEP2) may be used to calculate valence and core electron
binding energies [26]. Because this approach adds relaxation corrections to second
order electron propagator calculations, the accuracy of the results is consistently
improved. For valence ionization energies, well known methods that include thirdorder terms achieve higher accuracy, but only with much more difficult computations. TOEP2 is proposed for the calculation of valence electron binding energies in
large molecules where third-order methods are infeasible. For core-electron binding
energies, TOEP2 results are more accurate than those obtained with the perturbative methods that have been applied extensively to valence ionization energies and
electron affinities, such as P3 or OVGF [26].
Instead of the standard Hartree–Fock reference calculation, a grand-canonical
Hartree–Fock calculation [35] is used with the occupation number of a single spinorbital (i.e., the transition spin-orbital) set to 0.5. Upon convergence, appreciable
corrections to the relaxation energy are included in the transition spin-orbital’s
energy [23, 24]. Usually a very close agreement with the SCF method [36] is
obtained [26]. The second order electron propagator is applied to the ensemble

1

Efficient and Accurate Electron Propagator Methods and Algorithms

7

reference state. After taking fractional occupation numbers into account [26,37–40],
the Dyson equation in the quasiparticle approximation reads
!p D "p C

X

a;i
C

X
jhpajjij ij2 ni nj
jhpi jjabij2
C
!p C "a "i "j
!p C "i "a "b
i;a
X jhpi jjpaij2 .1
a;i

!p C "i

"p

np /
;
"a

(1.19)

where p is the transition spin-orbital with an occupation number of 0.5. Occupation
numbers are designated by n. Such calculations [26] have been performed recently
with a modified version of the Gaussian 03 [36] suite of programs.

1.3.2 Reduction of Virtual Space
Electron density difference matrices that correspond to the transition energies in the
EP2 approximation may be used to obtain a virtual orbital space of reduced rank [27]
that introduces only minor deviations with respect to results produced with the full,
original set of virtual orbitals. This quasiparticle virtual orbital selection (QVOS)
process provides an improved choice of a reduced virtual space for a given EADE
and can be used to speed up computations with higher order approximations, such
as P3 or OVGF. Numerical tests show the superior accuracy and efficiency of this
approach compared to the usual practice of omission of virtual orbitals with the
highest energies [27].
For the pth EADE, the first-order, density-difference matrix in the virtual–virtual
subspace [27, 29, 42], where
Dab D ıap ıbp

X
i
C

X
i;c

hpajjij i
.!p C "a "i

hpi jjaci
.!p C "i "a

hpbjjij i
"j / .!p C "b "i

hpi jjbci
"c / .!p C "i "b

"c /

;

"j /
(1.20)

is used to select an EADE-specific, reduced virtual space [27]. The computational
procedure has three stages. First, an EP2 calculation is performed. In the second step, the density difference matrix of (1.20) is constructed and diagonalized.
Eigenvectors that correspond to eigenvalues with the lowest absolute values are discarded and the Fock matrix is reconstructed and diagonalized in the reduced virtual
orbital space. Finally, higher order calculations are performed with a new, smaller
set of canonical virtual orbitals. Higher-order calculations proceed with the same
algorithm as in ordinary, all-virtual calculations.

8

R. Flores-Moreno and J.V. Ortiz

1.3.3 Resolution of Identity
RI methods also can be applied to electron propagator calculations in the quasiparticle approximation [28]. Savings in storage are dramatic. When compared to
semi-direct algorithms [21, 22], the pre-factor for RI results in a considerable
speed-up. The implementation is also much simpler. Test calculations with different
approximations and basis sets show the reliability of this approach.
The RI approximation is based on the evaluation of individual electron repulsion
integrals according to
X
.psjK/GKL1 .Ljqt/

.psjqt/ D

(1.21)

KL

with
.psjqt/ D

'p .r/'q .r0 /'s .r/'t .r0 /

jr r0 j
Z Z 
'p .r/'s .r/K.r0 /

dr dr0 ;

jr r0 j
Z Z
K.r0 /L.r/
D
dr dr0 ;
jr r0 j

.psjK/ D
GKL

Z Z

dr dr0 ;

(1.22)
(1.23)
(1.24)

where 'p .r/ is any canonical MO, and K and L are auxiliary basis functions. We
have employed Cartesian Hermite Gaussian functions [43] for the auxiliary basis.
The use of RI [44–48] can be seen as expansion of orbital products in terms of linear
combinations of auxiliary functions.
The formulation used here corresponds to the use of RI with the Coulomb
norm [47]. Although there are other formulations of the RI [48, 49], we will not
use them here. Because we need the matrix G 1 only at this level of the calculation,
its absorption into three-index quantities can be exploited [50] as follows:
q

TKt 

X

1=2

GKL .Ljqt/:

(1.25)

L

Thus, four center integrals may be obtained as simple matrix multiplications of
the three-index fields. This allows one to combine RI with most approximations
without needing major modifications to existing algorithms and codes.
Considerable savings in storage can be made. Using RI as described here, the
conventional implementation can be used, but with much less demand for disk
storage. Furthermore, the transformation from atomic to MO bases can be realized
for the three-index matrices with a formal gain of one order in the scaling of this
task [28].
Note that the method described here does not use RI for the SCF solution
of the reference system. Such an approximation would result in an approximate


1

Efficient and Accurate Electron Propagator Methods and Algorithms

9

description of the pole structure because the orbital energies would be affected [51].
We have not yet tested the combination of such an approximation with the approach

proposed here.
If the Kohn–Sham orbitals [52] of density functional theory (DFT) [53] are used
instead of Hartree–Fock orbitals in the reference state [54], the RI can become
essential for the realization of electron propagator calculations. Modern implementations of Kohn–Sham DFT [55] use the variational approximation of the Coulomb
potential [45,46] (which is mathematically equivalent to the RI as presented above),
and four-index integrals are not used at all. A very interesting example of this
combination is the use of the GW approximation [56] for molecular systems [54].
The RI can be combined with any of the methods presented here, including
TOEP2 and QVOS. We do not recommend combining it with second-order approximations, such as EP2 and TOEP2, because they can be equally efficient without the
RI if properly implemented.

1.4 Performance
The approximations discussed here are all ab initio. The performance of these
approximations is analyzed in terms of mean absolute deviations obtained from
calculations on many small molecules.

1.4.1 Transition Operator Method
Tables 1.1–1.6 show ionization energies of atoms of the second and third row of
the periodic table, some valence ionization energies of molecules and a number
of core electron-binding energies (CEBEs) of molecules for the 1s core orbital of
C, N, O, and F, respectively. Comparison to experiment [57, 58] is provided in all
these tables and is quantified in terms of mean absolute deviations in the last row.
TOEP2 results are compared with other quasi-particle methods. In all these calculations, the transition spin-orbital occupation number was set to 0.5 electrons. The
cc-pVTZ [59] basis set was used. From these tables, one may conclude that TOEP2
always improves over EP2. In Table 1.2, it is observed that for valence ionization
energies of molecules, the more computationally demanding P3 and OVGF methods are clearly more accurate than TOEP2. For CEBEs, TOEP2 is the best method,
being the most accurate and almost as fast as EP2.
Unlike EP2, TOEP2 is reliable enough in the calculation of valence EADEs
and can be used as a very efficient alternative for the treatment of large molecular
systems. For valence EADEs, the following reliability ordering is obtained:

P3, OVGF > TOEP2 > EP2 > KT:


10

R. Flores-Moreno and J.V. Ortiz

Table 1.1 Ionization energies of atoms (eV)
Atom

Ionization

Li
Be
B
C
N
O
F
Ne
Na
Mg
Al
Si
P
S
Cl
Ar
Av. jj

S! S
S!2 S
2
P!1 S
3
P!2 P
4
S!3 P
3
P!4 S
2
P!3 P
1
S!2 P
2
S!1 S
1
S!2 S
2
P!1 S
3
P!2 P
4
S!3 P
3
P!4 S
2
P!3 P
1
S!2 P

a

2

1

1

KT

EP2

TOEP2

P3

OVGF

Expt.a

5:34
8:42
8:65
11:91
15:48
14:15
18:40
23:00
4:96

6:89
5:93
8:18
10:65
11:00
13:05
16:06
0:50

5:35
8:89
8:40
11:30
14:44
12:93
16:37
20:12
4:98
7:34
5:91
8:10
10:49
10:83
12:58
15:39
0:36

5:35
8:89
8:36

11:25
14:48
13:04
16:85
21:03
4:98
7:34
5:90
8:08
10:49
10:03
12:70
15:61
0:23

5:35
8:84
8:22
11:10
14:31
13:07
16:92
21:21
4:98
7:30
5:81
7:98
10:35
10:77
12:58

15:49
0:28

5:35
9:23
8:50
11:35
14:58
13:39
17:14
21:44
5:00
7:55
5:93
8:11
10:48
10:95
12:74
15:60
0:15

5:39
9:32
8:30
11:26
14:53
13:62
17:42
21:56
5:14

7:65
5:98
8:15
10:49
10:36
12:97
15:76

P3
12:14
14:22
10:55
12:98
14:89
16:11
19:44
13:96
14:11
10:73
17:18
15:93
19:30
14:27
17:04
10:90
12:49
14:77
18:74
15:94
19:84

15:62
21:04
18:89
0:25

OVGF
12:36
14:32
10:48
13:06
14:77
16:17
19:61
13:61
14:02
10:74
16:63
15:35
19:08
13:85
17:07
11:04
12:61
14:91
18:87
16:02
19:84
15:58
21:00
19:01

0:25

Expt.a
11:9
14:40
10:51
12:85
14:66
15:87
19:23
13:61
12:55
10:8
16:98
15:60
18:78
14:01
16:91
10:9
12:78
14:74
18:51
16:19
20:00
15:83
21:1
18:8

See [57]

Table 1.2 Valence ionization energies of molecules (eV)
Molecule
B2 H6
CH4
C2 H4

HCN
HNC
NH3
N2
CO
H2 CO
H2 O
HF
F2
Av. jj
a

Orbital
1b3g
1t2
1b3u
1b3g
3ag
1b2u
2b1u
1
1
3a1
1u

3g
2u
5
1
2b2
1b1
3a1
1b2
1
3
1g
3g
1u

KT
12:85
14:80
10:24
13:77
15:94
17:48
21:52
13:49
14:13
11:60
16:47
17:17
21:30
15:09
17:28

11:99
13:73
15:76
19:21
17:50
20:68
18:05
20:46
22:05
1:17

EP2
12:21
14:07
10:33
12:75
14:48
15:89
19:34
13:68
13:74
10:17
17:05
15:02
18:20
14:06
16:37
9:94
11:50
13:86

18:08
14:70
18:94
14:20
20:46
17:35
0:62

TOEP2
12:30
14:34
10:65
13:00
14:94
16:25
19:69
14:00
14:11
10:65
17:38
15:47
18:59
14:13
16:88
10:38
12:29
14:56
18:68
15:66
19:64

14:89
21:06
18:10
0:36

See [9, 60] for geometry details and experimental values