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CONFIGURATION INTERACTION WITH NON-ORTHOGONAL SLATER DETERMINANTS APPLIED TO THE HUBBARD MODEL, ATOMS, AND SMALL MOLECULES

DISSERTATION

Presented in Partied Fulfillment of the Requirement for

the Degree of Doctor of Philosophy in the Graduate

School of The Ohio State University

by SVEN PETER RUDIN, DIPL. EL. INC. ETH

The Ohio State University

1997

Dissertation Committee:

John VV. Wilkins Approved by

Charles A. Ebner

Arthur J. Epstein I/O. UD

Advisor

Department of Physics UMI Number: 9721160

UMI Microform 9721160 Copyright 1997, by UMI Company. All rights reserved.

This microform edition is protected against unauthorized copying under Title 17, United States Code.

UMI 300 North Zeeb Road Ann Arbor, MI 48103 © Sven Peter Rudîn 1997

ALL RIGHTS RESERVED ABSTRACT

The energy of a truncated configuration interaction (Cl) electronic wave func­ tion depends on the quality of the underlying basis set and the form of the expan­ sion, i.e., which determinants are included. Given an expansion with fixed basis set and form, a unitary transformation of the basis functions can significantly alter the result by changing the form of the orbitals.

Two examples are given for the advantage of one particular unitary transfor­ mation, natural orbitals, over the self-consistent-field orbitals: (i) For the nitrogen dimer the use of natural orbitals improves the energy of a 01 wave function by

1.3 millihartree over a benchmark study, in which the dissociation energy was un­ derestimated by approximately the same amount, (ii) A well-defined, orbital-based truncation of the Cl wave function in natural orbitals for first-row supermolecules

(dimers with large nuclear separation) accurately reproduces the energy of the two atoms treated independently.

The determinants in Cl expansions are traditionally required to be mutually orthogonal. This forces the entire expansion to be done with the same unitary transformation of the basis set. Without the restriction of orthogonality between determinants each determinant can have a different unitary transformation of the basis set. A computer code to calculate expansions in non-orthogonal determinants was implemented and applied to the Hubbaxd model, atoms and molecules. The

ii resulting wave functions have a lower energy than a wave function of equal length expanded in orthogonal determinants. Starting from a reference determinant with localized orbitals is found to further improve the energy. Application to the Hub­ bard model in one and two dimensions leads to a systematic expansion which dis­ plays a pronounced hierarchy observable only with non-orthogonal determinants.

For singlet states of atoms (Be, Ne) and small molecules (H 2, HeH"^, HgO, N2, C2,

C2H2, C2H4, C2H6) a few dozen non-orthogonal determinants account for roughly

90% of the correlation energy in the basis sets used. The definition of a degree of excitation for non-orthogonal determinants is used to illustrate the significant differences between expansions with orthogonal and non-orthogonal determinants.

Ill Dedicated to the memory of David Lera (1968 — 1993)

IV ACKNOWLEDGMENTS

It is amazing what you can accomplish if you do not care who gets the credit. - Harry S. Truman

John Wilkins and Matthew Steiner deserve a lot of credit for what this document represents: My understanding. So, John and Matthew: Thank you.

My thanks also goes to many other people, who contributed each in their own way: Marijan Adam, Edward Adelson, Mebarek Alouani, Todd Anderson, Wilfried Aulbur, Janna Auvinen, Sapna Batish, Mary Heather Bents, Ravi Bhagavatula, Sue Blaker, Jim Blatchford, Thomas Blcizek, Thomas Boltshauser, Verena Bolt- shauser, Richard Bomfreund, Richard Boyd, Judith Braendle, William Brinker- hoff, Jill Brotman, Francesca Brotman-Orner, Shoshana Brotman-Omer, Pamela Brown, Joe Broz, Joy Bums, Mary-Lou Capozzi, Dave Casdorf, Emily Cassani, Joe Cassani, Lou Cassani, Mary Ann Cassani, Ruggero Castagnetti, Jim Castiglione, Herve Castella, Leena Chandran, Ted Chase, Lauren Chase, Jian Chen, Joyce Chin, Oliver Chung, Sora Cho, Bunny Clark, Jeffrey Clayhold, Chet Cole, Pete Colo- vas. Bill Conable, Cheryl Conel, Erin Conel, Jim Conel, Dan Cooper, Bob Cope, Daniel Cox, Fleda Crawford, Todd Culman, A. Cristina Cunha, Mark Dauben- meier, Mike Degen, Brad Dielman, Heidi Dugger, Bryan Dunlap, Charles Ebner, Doris Eckstein, Carlos Egues, Keith Emptage, Arthur Epstein, Catrin Ericsson- Novak, Son Evans, Melinda Everman, Armin Ezekielian, Maria Faxias, Dres Fehr, Claudia Filippi, Miodrag Filipovic, Jenny Finnell, Linda Fox, Roger Fox, Karolyn Frasier, Sonia Frick, Richard Furastahl, Jorge Gal an. Pilar Galan, Darren Gebler, Avik Ghosh, Jason Gilmore, John Gratsias, Jeffrey Grossman, Peter Hansch, Ed­ ward Harris, Kathy Hart, HaxaJd Haughlin, Fernand Hayot, Saad Hebboul, Mary Heck, Rolf Held, John Heimaster, Steve Herbert, Ron Hinkle, J. B. Hoy, Hung- Chen Hsieh, Susanne Huber, Per Hyldgaard, Laurens Jansen, Mark Jarrell, Sashi Jasty, Ciriyam Jayaprakash, Scott Jessen, Darrell Jones, Lars Jonsson, Jinsoo Joo, Ben Kaczer, Seth Kantor, Brian Keller, Karen Keppler, Gregory Kilcup, Eimsik Kim, Eunsook Kim, Jeongnim Kim, Kihong Kim, Taesnk Kim, Karen Kitts, George Klinich, Randy Kohlman, Tracy Kotrly, Brian Kuban, Rahul Kulka- mi, Lisa Kurth, Alex Kuznetsov, Sanjay Khare, Lisa Kiner, Barbara King, Brian Kuban, Thomas Lemberger, Beth L’Esperance, Richard FumstaM, Lisa Lantz, Bea Latal, Zachary Levine, Jeff LePage, Washington Lima, Mark Mamrack, John Markus, Victor Majtisovits, Meg McConnell, Sylvia McDorman, Janis McKay, Ross McKenzie, Deanna Mears, Brenda Mellett, Ursina Metzger, Alfred Mieth, Robert Mills, Lubos Mitas, Chad Mitchell, Rene Monnier, Brian Morin, Dave Moser, Hydee Moser, Andreas Miinzner, Bernard Mulligan, Ralf Niehaus, William Novak, Pat O’Bannon, Lee Oesterling, Leigh Oesterling, Ursula Oetiker, Luiz Oliveira, Frederick Omer, Keith Omer, Phyllis Omer, Kathleen Paget, Shelley Palmer, William Palmer, Karen Papritan, Don Parsons, Bruce Patton, Steve Pat­ ton, Jack Pawlicki, Charles Pennington, Uri Perrin, Robert Perry, Derek Peter­ man, Cheri Petersen, Ed Petersen, Mary Lou Petersen, Doug Petkie, Lisa Petkie, Pia, Russell Pitzer, Shawn Prendergast, Geoff Prewett, Zack Protegeros, Punky, William Puttika, Eduardo Raposo, Christian Rappan, Jim Raynolds, Bill Reay, Jen­ nifer Recchia-Cox, Charlie Recchia, Lauren Reed, Mark Reed, Mike Reed, Michael Reyzer, Eric Roddick, Markus Roos, Shirley Royer, Andy Rudin, Ann Rudin, Annika Rudin, Edith Rudin, Erik Rudin, Harry Rudin, Jr., Harry Rudin, Sr., Heather Rudin, Karen Rudin, Katy Rudin, Kirsten Rudin, Linda Rudin, Mark Rudin, Jennifer Rufsvold, Seungoh Ryu, Shelly Sabo, David Sasik, Helena Sasik, Roman Sasik, Doug Scalapino, Beatrice Scheurer, Beat Scheurer, Patricia Scheurer, Rosli Scheurer, Thomas Scheurer, Sabina Schiesser, Avi Schiller, Tanya Schneider, Scooter, Jeff Seiple, Andrew Sergeev, Richard Seyler, Cookie Shand, Ernie Shand, Isaiah Shavitt, Vivek Shenoy, Diane Sherwood, Don Sherwood, Paulo Sigg, Amy Simon, Beth Smith, Jeff Smith, Robert Smith, Jenny Sokolski, R. SooryaJcumar, Melissa Spangler, Renee Speh, Rekha Srinivasan, Urs Staub, Ursula Staub, Philip Steden, Renate Steden, Teri Steiner, Andy Stenger, Deborah Storm, Suresh Subra- manian, Charles Thorne, Brad Trees, Dallas Trinkel, A run Tripathi, Brad Turpin, Cyrus Umrigar, Jon Vandegriff, Alan Van Heuvelen, Peter Vaterlaus-Rack, Kitty Wagner, Lukas Wagner, Betty Wallace, Wei Wang, Irene Warren, Chris Weait, John

VI Weax, Christoph Weder-Huber, Sandy Weder-Huber, Dorothea Wehlen, Wolfgang Wenzel, John Whitcomb, Philip Wigen, Ken Wilson, Les Wood, Shiwei Zhang, Paul Ziesche, the Zimering family, and Maja Zweifel.

I would like to thank the DOE and the Ohio Supercomputer Center for support.

I would also like to thank several institutions and their people: Caffé Fino, the Cajun Kitchen, Barley’s, Bemie’s, Brenen’s and the Yogurt Oasis, Columbus Camera Group, Davis Center, Graeter’s, Basting’s General Store, the Institute for Theoretical Physics, Kinko’s, Long’s, the Ohio State University Libraries, the Spot, and the Statsbiblioteket Aarhus.

VI1 CURRICULUM VITAE

September 10, 1964 ...... Born — Redbank, New Jersey, USA

1985-1990 ...... Undergraduate student, ETHZ, Switzerland

1990-1991 ...... Research assistant, ETHZ, Switzerland

1991-1992 ...... National Needs Fellow, Dept, of Physics, Ohio State University, Columbus, Ohio, USA

1993 ...... Teaching assistant. Dept, of Physics, Ohio State University, Columbus, Ohio, USA

1993-present ...... Research assistant. Dept, of Physics, Ohio State University, Columbus, Ohio, USA

PUBLICATIONS

S. Rudin and H. Baltes, “Treatment of Thermomagnetic Effects in Semiconductor Device Modeling,” (20th European Solid State Device Research Conference, Not­ tingham, UK, September 10-13, 1990), edited by: W. Eccleston and P. J. Rosser (Adam Hilger, Bristol, UK 1990), pp. 493-6.

S. Rudin, G. Wachutka, and H. Baltes, “Thermal effects in magnetic microsensor modeling” (Eurosensors IV ’90, Karlsruhe, West Germany, October 1-3, 1990), Sensors and Actuators A (Physical) A27, 731 (1991).

FIELD OF STUDY

Major Field: Physics

Vlll TABLE OF CONTENTS

Abstract ...... ii Dedication ...... iv Acknowledgements ...... v V i t a ...... viii Table of contents ...... ix List of T ables ...... xii List of Figures ...... xv

C hapter Page

1. Introduction ...... 1 1.1 The basic subject of inquiry: Electron—electron correlation ...... 2 1.2 The basic forms ...... 6 1.3 Size-extensivity and -consistency ...... 9 1.4 Single- and multi-reference systems ...... 10 1.5 Basis set orientation and basis set truncation ...... 11 1.6 Scope ...... 13 2. Methods and definitions ...... 15 2.1 General approximations ...... 16 2.2 Atomic units ...... 18 2.3 Operators to generate expansions and determinants ...... 19 2.4 One- and two-electron integrals ...... 21 2.5 Slater’s rules ...... 22 2.6 Contracted (Cartesian) Gaussian basis sets ...... 23 2.7 Density matrices ...... 27 2.8 SCF, virtual, and natural orbitals ...... 28 2.9 Pair theories ...... 33 2.10 Localized orbitals ...... 35

ix 2.11 Symmetries ...... 38 2.12 Excitation graphs ...... 41 3. Natural orbitals in traditional Cl ...... 43 3.1 Natural versus CASSCF orbitals ...... 45 3.2 Effectively size-consistent truncated C l ...... 48 4. Non-orthogonal Slater determinants and their configuration interaction: Method ...... 54 4.1 Basic math for non-orthogonal determinants ...... 60

4.2 Basic N o r t h C o in algorithms ...... 62 4.3 “Approximation” of restricted-spin excitations ...... 64 4.4 Degree of excitation ...... 66 4.5 “Slater’s rules” for non-orthogonal d e te rm in a n ts ...... 72 4.6 Taking advantage of sy m m e trie s ...... 77 4.7 Generating higher excitations ...... 79 4.8 Comments on re-optimization algorithms ...... 82 4.9 Other algorithms ...... 84 5. Non-orthogonal Slater determinants and their configuration interaction: Application to the Hubbard model ...... 86 5.1 Basics of the Hubbard model ...... 88 5.2 One-dimensional Hubbard model ...... 89 5.3 Higher excitations for the one-dimensional Hubbard ...... 103 5.4 Degenerate reference determinants: Multi reference cases ...... 112 5.5 Two-dimensional Hubbard model ...... 119 5.6 Hubbard model with mirror symmetry ...... 128 6. Non-orthogonal Slater determinants and their configuration interaction: Application to atoms and small molecules ...... 130 6.1 The helium atom H e ...... 131 6.2 The hydrogen dimer Hg and the helium hydrid cation HeH"^ . . . 135 6.3 The beryllium atom Be and the Beg supermolecule ...... 141 6.4 The beryllium dimer B e g ...... 148 6.5 The water molecule HgO ...... 152 6.6 Small hydrocarbons and the carbon dimer ...... 158 6.7 The nitrogen dimer N g ...... 166 6.8 Neon ...... 171 7. Conclusions ...... 174 Appendix Page

A. Quantum chemistry acronyms ...... 179 B. The SCF approximation and the Virial Theorem ...... 191 C. Mathematical formalism for Cl with non-orthogonal determinants . 192 D. The restricted-excitation “approximation” ...... 199 E. Recursion relations for Gaussian integrals ...... 203

Bibliography

Bibliography ...... 206

XI LIST OF TABLES

Table Page

1.1: The periodic table of the chemical elem ents ...... 2

2.1: Dimensions of first-row (Gaussian-based) basis sets ...... 25

2.2: Number of spherical atomic orbitals in Cartesian symmetry groups . . . 38

2.3: Number of molecular orbitals in Cartesian symmetries for a homonucleax dimer oriented along the z direction ...... 39

2.4: Number of molecular orbitals in the correlation-consistent basis sets obeying each cartesian symmetry for homonuclear dimers oriented along the z direction ...... 40

3.1: Dependence of the total frozen-core energy on types of orbitals used for Ng in the cc-pVQZ b a s is ...... 46

3.2: Energy per atom of supermolecule wave functions for comparison with uncorrelated atomic wave functions of first-row atoms in the cc-pVTZ basis s e t ...... 50

3.3: Energy per atom of supermolecule wave functions for comparison with correlated atomic wave functions of first-row atoms in the cc-pVTZ basis s e t ...... 51

4.1: Estimates for the number of non-orthogonal determinants in comparison with the number of configuration state functions determined by Weyl’s formula ...... 58

Xll 5.1: Energies of the SCF determinant and of the SCF determinant with first CSF wave function for the one-dimensional Hubbard model at (n) = 0 .7 5 ...... 93

5.2: Overlap between determinants with single peaks in the density .... 95

5.3: Energies resulting from the successive addition of CSFs for the one-dimensional Hubbaxd model with eight sites, (n) = 0.75, and £7/t = 4 104

5.4: Up-down correlation function for the 8 -site Hubbard model as a function of the number of CSF’s included in the expansion .... 106

5.5: Coefficients for the successive expansion in CSFs for the one-dimensional Hubbard model with eight sites ...... 108

5.6: Overlaps of the maximum coincidence orbitals of ID, 8 -site Hubbard model CSFs and their degrees of excitation ...... 110

5.7: Energies for the successive addition of CSFs for the one­ dimensional, anti-periodic Hubbard model with sixteen sites .... 115

5.8: Energies per site of the periodic 16-site Hubbard m o d e l ...... 118

5.9: Energies of SCF and SCF plus first CSF states for the 4-by-4 Hubbard model at (n) = 5/16 and with C//f =4 123

5.10: Energies of SCF and SCF plus first CSF states for the 6-by -6 Hubbaxd model at (n) = 13/36 and with U/t = 4 ...... 127

6.1: Orbital fillings of the FCI wave function for He in an aug-cc-pVDZ basis using HF and natural o rb ita ls ...... 131

6.2: Correlation energies (in millihartree) for Be in a (7s3p) basis set . . . 142

6.3: Coefficients of the determinants for the beryllium Cl expansion . . . 144

6.4: Energies of Be and Beg supermolecule wave functions ...... 146

X lll 6.5; Experimental equilibrium geometries for the carbon dimer, acetylene, ethylene, and ethane ...... 158

6.6: SCF and (orthogonal) Cl results for ethane, ethylene, acetylene, and the carbon dimer in a DZ basis set ...... 160

6.7: North C oin frozen-core correlation energies for ethane, ethylene, acetylene and the carbon dimer in a DZ basis s e t ...... 162-163

D.l: The ten linearly independent spin-adapted configurations constructed from double excitations from the orbitals i and into (f>rand (j>s .2 0 1

XIV LIST OF FIGURES

Figure Page

1.1: Qualitative types of correlation for atoms and dim ers ...... 4

1.2: The combined effect of basis set orientation and basis set truncation illustrated with the basis vectors of three-dimensionaJ space ...... 11

2.1: Dependence of (angular) correlation energy on the exponent of the virtual 2p orbital for the beryllium a t o m ...... 30

2.2: SCF and natural Is, 2s, and 2p orbitals for Be ...... 31

2.3: Extended and localized orbitals ...... 36

2.4: Energy-ordered (a) and symmetry-ordered {b, c) excitation graphs representing determinants ...... 42

3.1: Partitioning leading to effectively size-consistent dynamic correlation of the frozen-core N 2 supermolecule in the cc-pVTZ basis ...... 53

4.1: Natural 3s orbital for neon in the cc-pCVTZ basis resulting from correlating different o rb ita ls ...... 55

4.2: Estimated number of configuration state functions for orthogonal determinants and non-orthogonal determ inants ...... 59

4.3: Determinant overlap with the SCF determinant plotted against their degree of excitation for non-orthogonal determ inants ...... 68

4.4: Scaled determinant overlap with the reference determinant plotted against the degree of excitation for non-orthogonal determinants . . . 70

XV 4.5: Example of rotating the orbitals in two determinants into maximum coincidence ...... 73

4.6: Combining two excited determinants to maJce a new determinant of higher excitation ...... 80

5.1: One-particle orbitals for the one-dimensional Hubbard model (16 sites) for both periodic and anti-periodic boundary cond itions ...... 90

5.2: Electron densities of typical determinants from the first configuration state function of the one-dimensional Hubbaxd model ...... 91

5.3: Energies per site of the SCF determinant and of the SCF determinant plus first CSF wave function for the one-dimensional Hubbaxd model . 94

5.4: Overlap between different determinants of the first CSF for the one­ dimensional Hubbaxd model at (n) =0.75 for various values oiUft . .97

5.5: Orbitals of maximum coincidence of the SCF determinant and the determinants of the first CSF for the one-dimensional, eight-site Hubbaxd model ...... 98

5.6: Up-down correlation function for the one-dimensional Hubbard model, 16 sites and Ujt = 4, for free-electron (SCF) determinant alone and with first single-peaked density configuration state function .... 100

5.7: Up-down correlation functions for the one-dimensional 16-site Hubbard lattice for varying Uft ...... 101

5.8: Up-down correlation function at |s — s^| = 0 for the wave function consisting of the SCF determinant and the first CSF for the one-dimensional Hubbaxd model ...... 102

5.9: Electron density of representative CSF determinants for the one-dimensional, eight-site Hubbaxd m o d e l ...... 105

5.10: Scaled determinant overlap with the SCF determinant plotted against the degree of excitation for the 8 -site one-dimensional Hubbard model . . 107

x v i 5.11; Orbitals of majcimum coincidence of the SCF and typical correlating determinants for the one-dimensional, eight-site Hubbard model . . 109

5.12: Correlation energy for successively added determinants and coefficients in the expansion for the 16-site one-dimensional lattices with anti-periodic boundary conditions and (n) = 0.75 . . . Ill

5.13: Up-down correlation functions for the periodic 16-site Hubbard model 113

5.14: Electron density of representative CSF determinants for the one-dimensional, periodic, sixteen-site Hubbaxd m o d e l ...... 114

5.15: Orbitals of maximum coincidence of the two SCF and typical correlating determinant for the one-dimensional, periodic, sixteen-site Hubbard model ...... 116

5.16: Density of the SCF determinant and one of 16 determinants in the first CSF for the 4-by-4 Hubbard model at (n) = 5/16 and with U/t = A ...... 120

5.17: Correlation functions for the 4-by-4 Hubbard model at (n) = 5/16 and with Uft = A ...... 121

5.18: Densities of the basic determinants of the second CSF for the 4-by-4 Hubbard model at (n) = 5/16 and with U/f = 4 . . . 122

5.19: Densities of the SCF determinant and one determinant of the first CSF for the 6-by -6 Hubbard m odel ...... 124

5.20: Correlation functions for the two-dimensional, 6-by -6 Hubbard model 125

5.21: Scaled determinant overlap with the SCF determinant plotted against the degree of excitation for the two-dimensional Hubbard model . . 126

5.22: Orbitals and densities for the 8 -site Hubbard model with and without enforced mirror symmetry ...... 129

XVI1 6.1: Excitation graphs for He in SCF and natural orbitals in an aug-cc-pVDZ basis set ...... 133

6.2: Scaled determinant overlap with the SCF determinant plotted against the degree of excitation for helium in the aug-pVDZ basis ...... 134

6.3: H2 SCF and FCI energies in the aug-cc-pVDZ basis set for the singlet and triplet sta te s ...... 136

6.4: Correlation, SCF, and FCI energy for the helium hydride cation in the aug-cc-pVDZ basis ...... 137

6.5: Excitation graphs of the nine non-orthogonal determinants used for H 2 in the aug-cc-pVDZ basis at various bond lengths R . . 139

6.6: Comparison of results for nine optimized non-orthogonal determinants with the FCI results for H 2 and for HeH"*" in the aug-cc-pVDZ b a s is ...... 140

6.7: Excitation graphs for Be in SCF and natural orbitals in a (7s3p) basis s e t ...... 143

6.8 : Orbital coefficients for the first seven Be determinants ...... 147

6.9: Potential energy curves for Bc 2, SCF and FCI in comparison with 1-1-20 non-orthogonal determ inants ...... 149

6.10: Scaled determinant overlap with the SCF determinant plotted against the degree of excitation for Be 2 ...... 151

6.11: Effect of localizing the orbitals in the reference determinant for H2O in a DZ b a s i s ...... 153

6.12: ScaJed determinant overlap with the SCF determinant plotted against the degree of excitation for H 2O ...... 155

6.13: Correlation energies for expansions of water in a DZ b a s i s ...... 157

6.14: Structures of ethane, ethylene, acetylene, and the carbon dimer . . . 159

x v ii i 6.15: Scaled determinant overlap with the SCF determinant plotted against the degree of excitation for Cg, CgHg, C 2H4, ajid CgHg...... 164

6.16: Correlation energy for the nitrogen dimer in original expansion and with higher excitations ...... 167

6.17: Scaled determinant overlap with the SCF determinant plotted against the degree of excitation for the nitrogen dimer in a DZ basis .... 168

6.18: Correlation energy missing from expansion in non-orthogonal determinants for Ng in a DZ b a s is ...... 170

6.19: Correlation energy for the neon atom in the cc-pCVTZ basis set . . 172

6.20: Determinant weight and scaled overlap with the SCF determinant plotted against the degree of excitation for neon in the cc-pCVTZ b asis ...... 173

XIX CHAPTER 1 INTRODUCTION

He who wonders discovers that this in itself is wonder.

- M. C. Escher

This chapter discusses the basic problem, electron-electron correlation, with a minimal amount of formalism. Of the many possible approaches to solve the correlation problem the focus here is on methods that expand the wave function in

Slater determinants. The principle forms of such expansions are introduced. 1.1 The basic subject of inquiry:

Electron—electron correlation

The structure in the periodic table of the elements reflects the classical picture of electrons occupying orbitals. This orbital picture is very powerful in providing a qualitative understanding of many chemical processes.

H He

Li Be B C NOF Ne

Na Mg A1 Si P S Cl Ar

K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn G a Ge As Se Br Kr

Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe (1) Cs Ba La Hf Ta W Re Os Ir P t Au Hg TI Pb Bi Po At Rn (3) F t Ra Ac

(1) Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

(3) Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr

TABLE 1.1 The periodic table of the chemical elements. The table’s struc­ ture strongly supports the notion of electrons occupying orbitals. The table’s usefulness indicates that this orbital picture is a very good approximate de­ scription of atoms and molecules.

Picturing electrons in orbitals is an approximation, which in modern chemistry applications no longer suffices. In many cases the orbital picture is a good approx­ imation to describe where an electron can be found in relation to the nucleus. The

9 information about how the electrons interact with each other is contained only in an averaged way in the orbital picture — hence the modem name of the orbital pic­ ture, “the meam-field solution.” This solution is made up of one-particle functions, whereas what we really want is a many-body wave function: a mathematical ex­ pression that fully describes how every electron interacts with every other electron and the nucleus or the nuclei.

In the many-body wave function the electrons are approximately in the mean- held orbitals on the one-particle level, but they avoid each other as much as possible.

Figure 1.1 illustrates the qualitative ways in which electrons can increase the av­ erage distance to each other in atoms and dim ers:T w o electrons in an atomic orbital can stay away from each other by keeping the nucleus between them, or oc­ cupying different angular segments, this is known as angular correlation. Another possibility is for one electron to move closer to the nucleus while the other electron moves slightly further out, this is known as in-out correlation. In-out and angular correlation also exist for dimers, where these terms are used with respect to the axis connecting the two nuclei. Dimers also show a third type of correlation, left-right or alternant correlation, where one electron tends to be closer to one nucleus while the other electron tends to be closer to the other nucleus.

Since electrons are fermions, the exchange of two electrons must change the sign of the wave function. This antisymmetry requirement is elegantly incorporated by writing the wave function in terms of Slater determinants. Electrons with the same spin are now forced to be in different orbitals and so are to some degree correlated, or “Fermi correlated”. •X e x

æ

FIGURE 1.1 Qualitative types of correlation for atoms and dimers. The orbital picture permits two electrons with opposite spin to occupy the same orbital, but says nothing about the position of one electron in relation to the other. Qualitative ways of thinking about how the electrons in an orbital are kept apart, i.e., correlated, are angular (a—>a’) and in-out {b—*b’) correlation for an atom, while for a dimer one distinguishes left-right or alternant (c—+c ’), angular or axial-plane (

a single determinant consisting of orbitals that are optimized in each other’s av­

erage field. Also called the Hartree-Fock (HF) solution, this is essentially solving

Schrodinger’s equation with a potential set up according to the Coulomb inter­

action (Hartree potential) and with the correct quantum statistics (the exchange

interaction of Fock term).

The energy difference between the truth (the exact eigenvalue of the Hamil­

tonian) and the mean-field solution (the expectation value of the Hamiltonian in

the HF approximation) is the correlation energy. In practice, calculations are done

using a set of basis functions. HF limited to the space spanned by the basis set

is one of many “self-consistent field” (SCF) methods.^ Throughout this document

SCF will always refer to HF in a basis set. The exact solution within a basis set

is the full configuration interaction (FCI) wave function. The term “correlation

energy” is often used for the energy difference within a basis set, i.e., the difference

between the FCI energy and the SCF energy.

^ HF and SCF are often used interchangeably in the literature.

5 1.2 The basic forms

Once we have the self-consistent-field (SCF) solution in a given set of one-

electron basis functions there are several ways to proceed with finding the correlated

wave function and energy. The methods discussed in this document rely on writing

the wave function as a sum of many determinants, i.e., adding other determinants to

the one we found by the SCF method. The purpose of these additional determinants

is to take the SCF wave function and put in the missing many-body character. Or,

from another point of view, the fact that electrons of opposite spin can sit right on

top of each other in the SCF wave function is an “imperfect many-body character,”

which is corrected with additional determinants.^

The best wave function allowed by the chosen basis functions is the full con­

figuration interaction (FCI) wave function. The FCI wave function is a linear

combination of all possible determinants (for a system with N electrons every pos­

sible choice of M basis functions combined as a determinant). The number of

symmetry-adapted Cl configurations is approximately

where i is the level of excitation and 5 is a symmetry factor between 2 and 10.^

Because of this scaling, FCI can only be done for small systems with a small number

of basis functions.

Calculations on systems where FCI in a reasonable basis set is not possible are

done with a subset of all possible determinants. Choosing which determinants to

^ The idea that the electrons spend too much time too close to each other in the mean- field solution is further supported by applying the virial theorem, see Appendix B. include — and how to include them — is the focus of intense research. One trun­ cated version of FCI — “Cl” — is obtained by variationaily minimizing the energy within the space defined by a subset of determinants. The choice of determinants is specified with extended acronyms, e.g., if all determinants that differ from the

SCF determinant by one or two orbitals are included, the correct term is “Cl with single and double substitutions” or “CISD.”

As the name “configuration interaction” implies, the correlation is described by the interaction of configurations, i.e., sums of determinants that have the correct symmetry properties. These symmetry properties include both the spatial and the spin functions. The SCF determinant contains only one possible coupling of the spins, with which the orbitals are optimized. A more general self-consistent solution can be found by including many different spin couplings of the orbitals in the op­ timization procedure, this leads to the spin-coupled valence bond theory (SCVB).'*

It is also possible to optimize a collection of determinants in which the electrons occupy different orbitals, this is the multi-configuration SCF method (MCSCF).

One special case is the inclusion of all chemically active orbitals in the complete active space SCF method (CASSCF). All of these SCF-type solutions can be used as starting points for more refined calculations.

A powerful, non-variational approach is the coupled-cluster (CC) method. The determinants that are included here are again the result of substituting orbitals in the SCF determinant, e.g., all doubles. CC with doubles (CCD), includes more than just double substitutions: Quadruple, sextuple, etc. substitutions are also part of the wave function, but only those that can be constructed as products of two, three, etc. successive doubles. The CC formalism considers only this type 7 of higher substitutions (or excitations), and does not treats them as independent excitations, hence the non-vaxiational nature of the method: Higher excitations have a coefficient in the expansion that is equal to the product of the coeflScients of those determinants used to form the higher excitation. This is put a bit simplistically, the higher excitations can be formed by combining many different lower-excitation determinants, and so many coefficients must be multiplied and added accordingly. 1.3 Size-extensivity and -consistency

In quantum chemistry the calculation of an observable quantity often involves looking at systems with different numbers of particles. To calculate the dissociation energy of a homonuclear dimer, e.g., the energy of an atom is compared to the energy of the dimer, which involves twice as many electrons as the atomic system.

A correct comparison of the energies requires that both systems be treated equally.

An increase of the CPU time with system size is unavoidable, but the quality of the result should be independent of the number of particles in the system, i.e., the system size. Any method that fulfills this requirement is “size-extensive”.^

“Size-consistency” is a pragmatic, special case of size-extensivity and refers to the energy of independent molecular fragments.® A method can be applied sepa­ rately to each fragment or all fragments at once. With a size-consistent method, the energy of the entire (fragmented) system calculated at once is equal to the sum of the energies of each subsystem calculated separately, i.e., E[A + B] = E[A] -f E[B],

In quantum mechanics the requirement of size-extensivity is not easy to fulfill.

Energy is an additively separable quantity: The energy of two independent systems is the sum of their individual energies. The quantum-mechanical wave function, however, is multiplicatively separable: For two independent systems treated to­ gether the overall wave function is the product of the two partial wave functions, but the total wave function must be antisymmetrized.

Truncated Cl, i.e.. Cl that includes only excitations up to a given degree, has a major drawback: It is neither size-extensive nor size-consistent. Coupled-cluster methods, on the other hand, are size-extensive. 1.4 Single- and multi-reference systems

The Hartree-Fock (HF) solution is often required to be more than one determi­ nant to preserve the symmetry of the system. A possible conhguration of the oxygen atom, e.g., consists of three determinants, 1 2py2p%, Is^2s^2pr2py2pr, and

Is^2s^2px2py2pr. These three determinants axe degenerate and hence must enter the wave function with equal coeflScients. Since they maice up a single configuration, this is a “single-reference” case.

Some systems require a reference that consists of several determinants that are nearly degenerate, e.g., the beryllium atom. The HF solution is only one deter­ minant, ls^2s^, but because of the near degeneracy of the 2s and 2p orbitals, the correct reference should aJso include the ls^2p|, ls^2py, and ls^2pj determinants.

The beryllium atom is a “multi-reference” system.

Including the (ls)^(2p)^ configuration correlates the electrons in the 2s orbital.

This angular correlation contributes to the correlation energy, but because of the near degeneracy the correlation is referred to as “non-dynamicaJ” or “internal”.^

When describing dissociation processes, different states become important as the nuclear separation is varied. States that differ significantly in energy for a molecule can become degenerate as the molecule dissociates into fragments. In the dissociated limit the reference must contain the degenerate states. A smooth description of the potential energy surface requires that the reference includes the same states at all bond lengths. Hence a molecule may be well described with a single reference at the equilibrium geometry, but to describe its potential energy surface a multi-reference treatment is required.

10 1.5 Basis set orientation and basis set truncation

A set of basis functions (or vectors) can be linearly combined to form a new basis, as long as they they remain linearly independent.

F IG U R E 1.2 The combined effect of basis set orientation and basis set truncation illustrated with the basis vectors of three-dimensional space. The basis vectors 1 and 2 form a truncated basis with which the bold vector can only be approximated. With a suitable re-orientation of the basis vectors, the truncated basis (now formed by 1 'and 2 ’) spans a plane that the bold vector lies in. The orientation of the basis set determines how well a truncated basis set can describe a vector.

Figure 1.2 illustrates a re-orientation of the basis vectors of three-dimensional space. A vector is only approximated with a truncated bcisis (two out of three vectors), unless the vector lies in the plane spanned by the truncated basis. A very

11 poor approximation would result from using a truncated basis that spans the plane orthogonal to the vector in question.

Quantum chemistry methods based on expansions in Slater determinants of one- particle orbitals involve two spaces. The basis sets that axe available in the literature

(or online®) are for the one-particle orbitals, which make up the determinants. The

Slater determinants axe the “basis vectors” for a wave function. This leads to a two-step process,

basis set of one-particle orbitals —» determinants —+ wave function.

Basis set orientation refers to the orientation of the one-particle orbitals, which in turn affects the orientation of the basis vectors (the Slater determinants) in the space the wave function is expanded in. How the basis set is oriented deals with the first step,

basis set of one-particle orbitals determinants —+ ______of the above process. “Truncated Cl” refers to a Cl expansion that does not include all possible determinants; which determinants are included determines the form of the wave function. How the expansion is truncated deals with the second half,

—» determinants —*■ wave function. of the above two-step process. Both steps axe important, and though they axe often discussed separately, their effect on each other cannot be neglected.

12 1.6 Scope

In the following chapters effects of basis set orientation for Cl calculations of the correlation energy are demonstrated. For Cl with orthogonal determinants the ba­ sis set orientation is chosen for the entire expansion, in this case “natural orbitals” define the most meaningful basis set orientation. The use of non-orthogonai deter­ minants allows a different basis set orientation for each determinant, leading to a significantly shorter expansion. A computer code, North C oin , was implemented to handle expansions in non-orthogonaJ determinants.

Chapter 2 introduces many of the more formal aspects of quantum chemistry methods. Cl in particular. Much of the traditional nomenclature is presented, some of which is adapted in later chapters for the purposes of discussing North C o in .

In Chapter 3 the importance of basis set orientation in traditional Cl methods is illustrated. The rotation of the basis set into natural orbitals allows the definition of a truncated Cl method that is effectively size-consistent for small molecules.

Chapter 4 introduces the method of Cl with non-orthogonal Slater determi­ nants, as used in the N o RTHC o IN code. The use of non-orthogonal determinants involves a certain amount of mathematics, most of which, however, has been banned

to Appendix C. In Chapter 4 ideas from orthogonal Cl are adapted for a more in­

tuitive understanding of Cl with non-orthogonal determinants.

Chapter 5 deals with the application of Nor ThC oin to the one- and two- dimensional Hubbard model. The Hubbard Hamiltonian contains two parts; each one taken alone can be solved exactly. The two solutions define two very different

basis set orientations. The combined Hamiltonian is a complex model system, where 13 the ability to have multiple basis set orientations present in the sajne expansion is studied for the first time here.

Chapter 6 presents the application of No RTHCoIN to a collection of atoms and small molecules. The focus is not so much on the results, but more on understanding the possibilities permitted by the use of non-orthogonal determinants, and on how the shorter expansions describe electron correlation effects.

Chapter 7 concludes this document with a more general discussion of the ad­ vantages and disadvantages of using non-orthogonal Slater determinants.

14 CHAPTER 2 METHODS AND DEFINITIONS

Computers are useless. They can only give you answers.

- Pablo Picasso

Since the full configuration interaction (FCI) wave function is computationally too expensive even for modest-sized molecules, truncated Cl methods are necessary.

Depending on

(i) which determinants are allowed,

(ii) what restrictions are placed on their coefficients in the expansion, and

(iii) in what basis set orientation the determinants are written, the results can be quite different.

In this chapter the general formalism of expansions in determinants is intro­ duced. The discussion is far from complete, the main focus being on essential ideas necessary for later chapters.

15 2.1 General approximations

The energy of a molecule is the expectation value of the Hamiltonian

where the sums are over electrons (i) and nuclei (a). Za is the nuclear charge and Ma is the mass of the nucleus a, is the distance between the two nuclei a and /d, and r,-y is the distance between two electrons i and j. The last term in (2.1) is the nuclear-nuclear repulsion and is constant in the first approximation that is generally made, the Bom-Oppenheimer approximation.® Since the difference between the electrons and the nuclei masses is a factor of 10^ to 10^, one assumes that the electrons move on a faster time scale than the nuclei and so the position of the nuclei are held fixed while the electron problem is solved. Holding the nuclei fixed also removes the second-to-last term in (2.1). Even at absolute zero the nuclei are not at rest, due to Heisenberg’s uncertainty relation, but the zero-point energy is small in comparison to the electronic energy, and can be added later as a correction.

Schrodinger’s equation,

H\^) = E |^ ), (2.2) can in general not be solved exactly. A representation of the wave function I'P) is required, the form used throughout this document is an expansion in Slater determinants jV’A')- The Slater determinants are made up of one-particle orbitals

éi- The exact solution of (2.1) requires a complete set of orbitals, i.e., an infinite set. In practice the use of an infinite set is neither possible nor necessary.

The basis sets used in our calculations are discussed in detail in Section 2.6.

All basis sets used are linear combinations of atom-centered functions, though basis 16 functions centered on bonds can also be used.^® All our functions are one-particle orbitals, though the electron-electron distance can be explicitly included in deter- minantal expansions.^

Most expansion-based calculations begin with a single determinant, the Hartree-

Fock determinant. The advantage of using Hartree-Fock as a starting point is that the long-range electron interaction is well approximated and the focus can be on the local correlation effects. This is achieved by using the Mpller-Plesset partitioning of the Hamiltonian,^^

H = Hq + V

= ( e N') + ""(')]) + f i ; - E where Hq includes not only the “true” one-electron terms (kinetic and nuclear attraction energy), but also the average Coulomb repulsion of the other elec­ trons on electron i. The (orthogonal) one-particle orbitals that diagonalize Hq are the canonical molecular orbitals (CMO’s).

The current implementation of North C oin is for closed-shell singlet states only, in which case the SCF determinant is made up of doubly-occupied spatial orbitals. Our calculations will thus begin with restricted HF (RHF).

17 2.2 Atomic units

In equation (2.1) there was a suspicious absence of physical constants. It should

have been

2me 4 ^ 47T£o \ 2 rj

where only the electronic parts axe kept. Since V? has the dimensions of

and both l/r^ and l/raf have the dimensions of , a rescaling of the unit for

length in terms of the bohr, defined as

iTTEQh^ eg — 2~-

leads to the equation in “atomic units”, as written in (2.1).

Conversion factors to other units are:

(1) Energy. Atomic unit: hartree

1 hartree = 4.3598 • 10~^® J = 27.211 eV = 627.51 kcal/mol = 219475 cm~^

(2) Length. Atomic unit: bohr

1 bohr = 5.2918 • 10“ ^^ m = 0.52918 Â

To compare the results of a calculation with those of a measurement, theory

and experiment should have a similar accuracy. This is the “chemical accuracy”,

and is about 1 kcal/mol, which is about 1.6 millihartree, 43 meV, or 350 cm~^,

slightly larger than room temperature.

18 2.3 Operators to generate expansions and determinants

In a given basis set, the many-electron wave function is expanded in terms of

Slater determinants, in the sense that one particular determinant is unsubstituted and ail other are viewed as excitations of it. The unsubstituted determinant, |0o); is most often the SCF solution, but does not have to be. The formally complete solution is the full configuration interaction (FCI) wave function,

l ^ F C l ) = Cq I ^ o ) + ^ ^ ^ ) + .... i,a i

= Uo + X] cfaUi + X E (^ÿ^aaiolaj + .... jV’o), \ i.a i

The energy of the FCI expansion is invariant to any rotation of the basis set and is size-extensive. Unfortunately, it is also not practical except for small systems in medium-sized basis sets since the number of configurations increases nonlinearly with the level of excitation. In practical applications one is thus forced to use a partial expansion.

With the operators Tn that generate excitations of a specific degree n, e.g.,

f i = ^ tfaiai ,

T2 — ^ij 1 i>j,a>b the operator that generates ail possible excitations is T = Ti-t-T 2-f-T’3....-hT’n. These can be reordered such that excitations which axe actually the product of lower- degree excitations also enter the formalism as such products. Such a reordering is 19 done in the coupled-cluster expansion. Starting from the Hartree-Fock determinant

|$o), the wave function is

l^cc) =exp(f)|^o> = f 1 4- 4- %2 + -f- —Ti 4- —%2 4-.... 4- FiTg + •••• J l^o)-

The truncation of the wave function is determined by which Tn are included. CCSD, for example, keeps T\ and J 2. Excitations of higher order than two (in this example) are present in the wave function, but they are restricted to those that can be viewed as two (or more) separate excitations created by the Tn included. The coefiBcients of these determinants are forced to be the linear combinations of the coefficients of the determinants that can be used to form the higher excitations. This guarantees size-extensivity.

For the configuration interaction method we have a linearized/truncated version of ( 2.6), |'5) = (l4-Ci-hC2 + ....)|^0), (2.7) where Cl = n ,

C2 = f2 + j f f , (2.8 ) C^ = Tz + TiT2 + ^ T f ,

C4 = T4 + j r | + T1T3 + , etc.

The coefficients of the determinants are not forced to have the type of relation as in the CC method. In Chapter 3, partitioning is introduced, wherein the level of

Cn, at which the wave function is truncated, is orbital-dependent.

2 0 2.4 One- and two-electron integrals

Throughout this document we will follow the notation used by Szabo and

Ostlund.^^ The SCF step, with which most quantum chemistry calculations be­ gin, leaves us with an orthogonalized basis set of one-electron functions, either as spin orbitals Xi or spatial orbitals 0,-. The notation for the integrals is:

0 One-electron integrals over spin orbitals:

( # |j ) = J «/x:ix^(xi)/i(xi)xy(xi) (2.9)

0 Two-electron integrals over spin orbitals (physicist’s notation):

(ij\kl) = ixiXjlXkXl) = J <^xidx 2X f(xi)xj(x 2)^Xifc(xi)x/(x 2) (2.10)

0 Antisymmetrized two-electron integrals over spin orbitals:

{ij\\kl) = {ij\kl) - (ij\lk) (2.11)

0 One-electron integrals over spatial orbitals:

(i|A|j) = J *i^Kri)*(ri)«‘i(ri) (2.12)

0 Two-electron integrals over spatial orbitals:

{ij\kl) = {(l>ij\ékl) = [ dridT24>i{ri)l{^2) (2.13) J ^ n 2

0 Coulomb integrals:

Jij = {ij\ij) (2.14)

0 Exchange integrals:

Kij = {ii\jj) (2.15)

21 2.5 Slater’s rules

In traditional Cl the computation of the Hamiltonian matrix element between two iV-electron Slater determinants can be done using “Slater’s rules”. In using the rules one assumes that the two determinants are in maximum coincidence, i.e.,that any orbitals common to both determinants are also in the same place. For example, the two determinants

\asàcd4>t) are in maximum coincidence, while

\aéb(i>c(t)de) and \4>ca4>s(i>d^) are not in maximum coincidence.

Which rule applies depends on how many spin orbitals cire different in the two determinants.

(0) Identical determinants: M M {il)\H\'ib) = ^ {m\h\m) + ^ (mn||mn) (2.16) m=l m>n (1) Determinants that differ by one orbital: N (• •-mn ■ • ■ \H\ • ■ ■ pn • ■•) = {m\h\p) + ^ ( m n ||p n ) (2.17) n=l (2) Determinants that differ by two orbitals:

(• • -mn ■ ■ ■ \H\ ■ ■ ■ pq ■ ■•) = {mn\\pq) (2.18)

(3) Determinants that differ by more than two orbitals:

(• • -mno ■ ■ ■ \H\ ■ ■ ■ pqr • ••) = 0 (2.19)

22 2.6 Contracted (Cartesian) Gaussian basis sets

The use of Gaussian functions as AO basis sets was proposedbecause the

two-electron integrals are more easily evaluated for Gaussian type orbitals (GTO’s)

than for Slater-type orbitals (STO’s). This is one of many reasons why GTO’s are

a popular choice today:

(i) Multi-center two-electron integrals can be evaluated easily and accurately.^

(ii) Since GTO’s do not introduce a cusp, they need not be centered on atoms.

(iii) If the point nucleus is replaced by a more realistic, finite nucleus, then GTO’s

become the “natural” choice.

(iv) For large basis sets the near-linear dependence is less severe for GTO’s than

for STO’s.

(v) GTO’s are smooth.

(vi) GTO’s are suited to describe both core and valence regions.

Not all of these are advantages over STO’s, furthermore, in order to approximate

a STO with high quality, many GTO’s must be used. But even taking this into

account it is computationally significantly more favorable to use GTO’s.

A contracted Cartesian Gaussian orbital is the sum of G Gaussians, centered

around a point R (such that r = r — R, î = x — Rx-, y = y — Ryi z = z — Rz),

G H r - R ) = ^ 9i N{fii,Y,nd) , (2.20) 1=1 d

^ In Appendix E three examples of the recursion relations are given. 23 where the contraction coeflScients gi and the Gaussian exponents are tabulated

for each orbital, and the normalization constants are given by

= ------J/2 - (2-21) j ((2nx - l)!!(2nj, - l)!!(2n^ - 1)!!)^'^

Basis sets come in many variations and sizes. Some of the most important types

and characteristics are:

(i) A minimal basis set is one that has a single basis function for each of the AO’s

occupied in the atom.

(ii) A double-zeta basis set consists of two basis functions per occupied atomic or­

bital — the name stems from the tradition of using a zeta ((") for the exponential

factor in Slater type orbitals.

(iii) A split-valeace basis is of double-zeta quality for the valence atomic orbitals,

with a minimal basis for all other atomic orbitals.

(iv) Basis sets can include polarization functions, which axe basis functions of higher

angular momentum to allow mainly angular correlation.

(v) Diffuse functions have exponents that are smaller than those normally used,

these allow for a more accurate account of the outer charge density cloud — an

atom’s polarizability, etc.

All calculations presented in this document were done with GTO’s,® mainly the

so-called correlation-consistent basis The correlation consistent basis sets

are built up of contracted GTO’s (for the SCF solution) and shells of correlating

functions, where each function (s, p, d, etc.) in a shell contributes approximately

the same amount of correlation energy. These basis sets have been optimized for 24 aumber of orbitals of symmetry name s P d / 9 h

DZ 4 2 0 0 0 0 DZP 4 2 1 0 0 0 cc-pVDZ 3 2 I 0 0 0 cc-pVTZ 4 3 2 1 0 0 cc-pVQZ 5 4 3 2 I 0 cc-pV5Z 6 5 4 3 2 1

cc-pCVDZ 4 3 1 0 0 0 cc-pCVTZ 6 5 3 1 0 0 cc-pCVQZ 8 7 5 3 I 0 cc-pCV5Z 10 9 7 5 3 1

TABLE 2.1 Dimensions of first-row (Gaussian-based) basis sets. “DZ” stands for “double zeta”, indicating that for each occupied orbital there are two basis functions. The addition of a “P”, for “polarizing”, indicates the inclusion of a polarization function, e.g., the d function in the DZP basis set. The basis sets of Dunning et al. are “correlation-consistent”; They are built up in terms of shells — much like atomic shells — and within a given shell each basis function contributes approximately the same amount of correlation energy. The correlation-consisten basis sets include polarization functions, and are meant for valence-only (cc-pVXZ) or all-electron calculations (cc-pCVXZ).

25 hydrogen and the first row elements. For hydrogen the polarization exponents were determined by optimizing them at the CISD level for molecular hydrogen in its ground state. The (s, p) exponents for B through Ne were optimized in atomic

SCF calculations on the ground state. The polarization exponents were optimized at the CISD level.

One element behind the systematic construction of the correlation-consistent basis sets is the hope of being able to define a complete basis set (CBS) limit.^^

Results for a spectroscopic constant A calculated in the various cc-pVXZ basis sets are fitted to a function of the “cardinal number of the basis” (%),

A{X) = A(oo) -k (2.22) to extrapolate to the CBS limit, X —* oo.

The notation for the number of primitives and contracted GTO’s is: The prim­ itives are listed in parentheses, the number of GTO’s are listed in square brackets.

For example, a basis that consists of four s- and two p-type primitives that are contracted into two s- and one p-type GTO’s is written as (4s2p)-+[2slpj.

In Section 2.8 natural orbitals (NO’s) are introduced. One school of thought uses atomic natural orbitals (ANO’s) as basis sets. Here large sets of primitive

Gaussian orbitals were used in atomic calculations to optimize the total energy.

Based on these calculations the NO’s were formed, these provide the ANO basis sets for molecular calculations. ANO’s have advantages (e.g., they minimize the so- called basis set superposition errors, BSSE), but require very large primitive sets.^^

Correlation consistent basis sets have fewer primitives and can result in significant computer time savings.

26 2.7 Density matrices

Following Lowdin,^^ the density matrices can be constructed for a given wave function. For the Hamiltonian we axe concerned with, the density matrices of interest are the first-order density matrix,

7(fl,|xi) = AT j 'iH*{l'2Z....N)^{123....N)dx2dxz....dxN, (2.23) and the second-order density matrix

F(zi,x2,|xiz2) = J ^*il'2'3....N)^(l23....N)dxsdx4....dx,y. (2.24)

The expectation value of an operator can be expanded in terms of density matrices. e.g., our Hamiltonian requires only the first two density matrices,

(H) = - ^ j (vf7(x,v|z,-)]| dxi f ^-f(xi\xi)dxi a (2.25)

- t - ^ f — T(xiXj\xiXj)dxidxj. I J Tij For the special case that the wave function is a single determinant (e.g., the SCF solution), all higher-order density matrices can be generated from the first-order density matrix.The second-order density matrix in this case is

1 7(zp 7(xp f l ) ^2) (2.26) 2 7(^2' z i) 7(^2' ^2) The density matrices can be transformed from the real and spin space (used here) into the space of the one-particle basis functions. The first-order density matrix then gives the average occupations of the orbitals.

The concept of density matrices can be generalized to describe the interaction of two states in terms of transition matrices, e.g., the first-order transition matrix between two states and

7/_//(x'i|xi) = iV j ^*j(l'23....N)<Î!iiil23....N)dx2dxi....dx^. (2.27) 2.8 SCF, virtual, and natural orbitals

A full Cl expansion of the wave function can be done in any orientation of a given basis set, i.e. a unitary transformation of the orbitals does not change the available Hilbert space, and the resulting energy will be the same. This is only true for full Cl: For truncated Cl methods the orientation of the basis set is crucial.

The simplest example is a wave function with only one determinant consisting of a set of orbitals {^^,^2,^3, the basis set (which contains more than N orbitals) is oriented so that are the SCF orbitals, then this wave function has the SCF energy. If the basis set is oriented so that the SCF orbitals are not part of the space spanned by {<^ 1, <^2,9^3, then the energy of this determinant will be higher than the SCF energy.

The orbitals in the SCF determinant have a physical meaning: They make up the best single-determinant wave function within the particular basis set. Besides these SCF orbitals, the result of the SCF step is a set of orthogonal orbitals that make up the remaining one-particle space. These orbitals are called virtual orbitals.

Though an energy can be assigned to them, the energies strictly have only a math­ ematical meaning. Virtual orbitals result from test particles in the field of the real particles — without influencing the real ones. One possible way to give the virtual orbitals more physical meaning is by using potentials that are better suited as is done in 0 theories.^®’^®’^^

For a given form of Cl expansion such as CISD, there exists one orientation of the basis set that leads to the best possible energy. These are the natural orbitals.

Natural orbitals are found by diagonalizing the first-order density matrix of the FCI

2 8 wave function. In the SCF picture each electron occupies an orbital and the diagonal elements of the first-order density matrix projected into the orbital basis, also called the one-particle density matrix, are either zero or one. The diagonal elements of the density matrix of a correlated wave function are no longer necessarily integer values; their values lie between zero and one.

For two-electron singlet systems the transformation into NO s reduces the length of the Cl expansion from M{M -f-l)/2 to M configurations: The Cl expansion itself is diagonalized,^®’^® i.e., the expansion consists of a sum of doubly-occupied spatial orbitals with singlet-coupled electrons.

In the general czise with more electrons, there is no strict reduction in the number of terms.^® However, if one wishes to limit the number of basis functions in (and hence the length of) a truncated expansion, the use of NO’s is powerful.

To diagonalize the first-order density matrix of the FCI wave function the FCI calculation has to be done, but avoiding the FCI calculation was the reason to find the NO s. However, approximate NO s from, e.g., CISD, used as one-particle orbitals often lead to a considerable improvement of the energy.

Figures 2.1 and 2.2 show a physical reason why NO’s can improve the energy of a truncated Cl wave function: Rotation into NO’s brings the amplitude of the correlating orbitals into the same region of space as that of the occupied orbitals.

The lowest-lying canonical SCF virtual orbitals are poorly suited for describ­ ing the most important non-dynamical correlation effects.Furthermore, if an expansion is to be truncated by choosing how different orbitals are to be used (par­ titioning, see Chapter 3), the orbitals should have a more physical meaning. NO’s are inherently better suited for these purposes. 29 — If AO —■ 2s AO

Gaussian orbitals

30

20

10

0 0 2 3 4 r (bohr)

FIGURE 2.1 Dependence of (angular) correlation energy on the exponent of the virtual 2p orbital for the beryllium atom. To correlate the electrons in the Is and 2s orbitals (top) several Gaussian 2p orbitals (middle) are used. The largest correlation energies (bottom) result from excitation into a 2p orbital where the first moment equals that of an occupied s orbital. This correlates angularly the two electrons in an s-type orbital without changing their average distance to the nucleus. 30 SCF orbitals

-e- — Is

Natural orbitals

— 2s

0 3 4 r (bohr)

FIGURE 2.2 SCF and natural Is, 2s, and 2p orbitals for Be. The occupied Is and 2s orbitals are negligibly affected by the rotation into natural orbitals (NO’s), while the virtual orbitals now take on a more physical meaning: Here the 2p orbital takes on a shape to allow an angular correlation of the 2s electrons that does not change their average distance to the nucleus.

31 Finally, we mention Brueckner orbitals, which are the basis functions in the basis set orientation in which all single excitations vanish.^^ This is also the condition for the reference determinant to have the maximum overlap with the exact wave function,^^ thus Brueckner orbitals are sometime referred to as the orbitals of an exact SCF theory.^^ For closed-shell states, Brueckner orbitals are very similar to the SCF orbitals.

32 2.9 Pair theories

As a compromise between the one-particle, SCF determinant and the fully- correlated, many-body wave function one can consider wave functions made up of two-electron orbitals (“geminais”). These electron-pair theories are a natural step to take beyond SCF, since (i) the Hamiltonian contains only I- and 2-particle oper­ ators, and (ii) the Pauli principle prevents three electrons from occupying the same point in space. Still, though a Slater determinant would be exact for a Hamilto­ nian with only one-electron operators, an electron-pair theory is not exact for a

Hamiltonian that contains only 2-electron operators.

If we go back to our SCF solution, each orbital will have two electrons in it

(for restricted Hartree-Fock treatment of singlet states). Pair theories attempt to correlate the electrons in groups of two. If each electron is correlated with only one other electron one has the independent-pair approximation (IPA) or the separated- pair approximation.^^

The separated-pair approximation splits the N electrons into N/2 pairs and optimizes each pair in the mean field of the others. A better electron pair theory for a closed-shell state is characterized by as many two-electron functions as there are electron pairs, i.e., N{N — l)/2 for an iV-electron system. This leads to the coupled-pair approximation,

I'^CPMEt ) = exp(!^)|#o) / - 1 -o \ (2.28) = U + r2 + + ....) |*o>, where the form of the wave function is identical to that of a CCD expansion.

33 A pair theory of paxticulax interest in terms of basis set orientation is Cl with pair natural orbitals (PNO-CI).^® Like Cl, it is an expansion in determinants that is optimized vaxiationally with the degree of excitation two for each determinant.

The expansion is divided into parts associated with different electron pairs. The determinants in each pair’s part of the expansion have two difFerent orbitals replaced in the SCF determinant. The advantage of this is that the matrix element for the energy is only non-zero within the partial expansion, as can be seen by applying

Slater’s rules (equation (2.19)). The different parts of the expansion can thus have a different basis set orientation without increasing the computational effort. Though the virtual orbitals of different parts of the expansion are no longer necessarily orthogonal, one never needs to calculate the matrix elements of the energy between such non-orthogonal determinants. The choice for the basis set orientation of each pair’s part of the expansion is an approximation to that pair’s natural orbitals.

34 2.10 Localized orbitals

At the end of the SCF step, one is usually left with canonical molecular orbitals

(CMO’s). Formally, canonical molecular orbitals are those that diagonalize the Fock operator. For extended systems the CMO’s will tend to be extended over the entire system, which is not necessarily the best starting point if one wants to correlate electrons in a given orbital, pair theory or not: With extended MG’s, the electrons within an orbital already tend to be less likely to sit right on top of each other, so one really wants to correlate electrons in different orbitals. A convenient way to do this is by a unitary transformation of the SCF orbitals into more localized orbitals.

One possibility of performing this localization is to transform the MO’s so that the intra-orbital Coulomb repulsion is maximized (while the inter-orbital repulsion is minimized). This method has the advantage that it uses the two-electron in­ tegrals that are readily available. Other, computationally more efïïcient methods exist, however, these require integrals that are not necessarily already included or needed for the Cl calculation. Methods to localize the orbitals by a more general nonsingular, i.e., not necessarily unitary, transformations of the orbitals have also been examined.Since the orthogonality requirement between orbitals is lifted, these localized orbitals are better suited for describing purely physical effects.

For the purposes of localizing orbitals for N o RTHC o in the method based on the two-electron integrals is used. The method of C. Edmiston and K. Ruedenberg^®’^^ requires a minimization of the off-diagonal integrals for the exchange,

X = Y i (2.29)

35 a

GG

jÆ lllw ItlW.

.nWllI G GG

FIGURE 2.3 Extended eind localized orbitals. The extended orbitals (a) are eigenfunctions of the symmetry operations that can be applied to the molecule. Localized orbitals (b) tend not to obey these symmetry operations, but usually allow more traditional chemical interpretations such as the grouping into core, bond, and lone-pair orbitals.

36 and for the Coulomb interaction,

C = ^ Jij. (2.30)

This is identical to maximizing

the sum of each orbital’s “Coulomb repulsion of itself”: A localized orbital will concentrate a lot of charge in a small region, and the Coulomb self-interaction of such an orbital can be used as a measure of how localized an orbital is.

The virtual SCF orbitals tend not be localized in the literature, since they cannot be uniquely ascribed to one particular region of localization of the occupied orbitals. There are exceptions,'^® and while NORTHCoiN applications presented here have only the occupied orbitals localized, the possibility to localize virtual orbitals is implemented. Another possibility if to leave the virtual orbitals as close to their original atomic orbital representation as is allowed by the SCF optimization of the occupied orbitals.

37 2.11 Symmetries

The symmetry operations allowed by the codes^^ used here axe based on the

Cartesian coordinate system. For atoms this means that orbitals of the same shell appear in different symmetry groups, e.g., the fivÇ d orbitals are found in four symmetry groups as determined by which symmetry operations they follow.

L ______Cartesian symmetry operations ______X—^-x + + — — + + — y -» -y + + + + z—^-z + + + — + — 5 1 0 0 0 0 0 0 0 P 0 1 I 0 I 0 0 0 d 2 0 0 1 0 I I 0 I 0 2 2 0 2 0 0 I 9 3 0 0 2 0 2 2 0

TABLE 2.2 Number of spherical atomic orbitals in Cartesian symmetry groups. All calculations presented here are done with spherical atomic orbitals grouped according to the Cartesian symmetry operations of which they are eigenfunctions.

For molecules the orientation of the molecule along with the symmetry of the

AO’s determines the number of MO’s of a given symmetry. Table 2.3 shows the number of MO’s for a homonuclear dimer oriented along the z axis. A heteronuclear dimer with the same orientation would keep only four of the eight symmetry groups by combining pairs of groups that differ by the z —»■ —z operation.

38 Table 2.4 shows the number of molecular orbitals in the correlation-consistent basis sets for dimers.®’ The total number of basis functions grows rapidly with system size and cardinality of the basis set. For a first-row dimer in the cc-pV5Z bcisis, the total number of basis functions is 160, approximately the limit of current computational resources.

______Cartesian symmetry operations X—^-x -|- -|- — — -f" -f- y -^-y -h + + -f- z—»-z -b -t- -t- — + — s ± s I I 0 0 0 0 0 0 p ± p I I I 1 I I 0 0 d àzd 2 2 1 1 I I 1 I f ± f 2 2 2 2 2 2 1 I 9±g 3 3 2 2 2 2 2 2

TABLE 2.3 Number of molecular orbitals in Cartesian symmetries for a homonuclear dimer oriented along the z direction. The symmetry of the AO’s and their symmetric or antisymmetric combination determines the number of MO’s found obeying a given Cartesian symmetry.

39 basis set ______Cartesian symmetry operations X—>-x + + — ~ + y -^-y + + + + — — — — z—y-z — + — + — + — cc-pVDZ 7 7 3 3 3 3 1 1 cc-pVTZ 13 13 7 7 7 7 3 3 cc-pVQZ 22 22 13 13 13 13 7 7 cc-pV5Z 31 31 19 19 19 19 11 11 cc-pCVDZ 9 9 4 4 4 4 1 1 cc-pCVTZ 19 19 10 10 10 10 4 4 cc-pCVQZ 34 34 20 20 20 20 10 10 cc-pCV5Z 52 52 32 32 32 32 18 18

TABLE 2.4 Number of molecular orbitals in the correlation-consistent basis sets obeying each Cartesian symmetry for homonuclear dimers oriented along the z direction. For heteronuclear dimers the z —» —z symmetry operation is no longer allowed, and so each pair of columns is combined in such cases.

40 2.12 Excitation graphs

One of the main advantages of Cl with non-orthogonal determinants is that, compared to traditional Cl, orders of magnitude fewer determinants suflBce to ac­ count for the same amount of correlation. This allows an analysis of the wave function on a more intuitive level. A visual aid which will be used in this document to represent determinants is the “excitation graph”.

In Figure 2.4 three versions of the excitation graph for a determinant are shown.

The determinant represented here is a double excitation of the SCF determinant, a quadruple excitation would have two roads leading away from the SCF orbitals, etc. These excitation graphs are only guides to how a determinant differs from the

SCF determinant. For non-orthogonal determinants in particular, when we have fractional excitations out of many orbitals simultaneously, they can be somewhat misleading in terms of which orbital is excited into which. For such purposes a more refined method is required, which describes the excitation on an orbital level rather than in terms of the total occupations.

41 a HZ 1 2 3 4 5 6 7 8 9 10 11

i m i y 12 3 4 6 7 8 9 10 11

12 3 4 5 6 7 8 9 10 11

FIGURE 2.4 Energy-ordered (a) and symmetry-ordered (6, c) excitation graphs representing determinants. The orbitals of a given determinant are represented with respect to the SCF determinant: Each “road” staxts in an orbital that is occupied in the SCF determinant and ends in an orbital occupied in the determinant it represents. The energy-ordered version has the advantage of clearly separating SCF orbitals {1 through 3 here) from virtual orbitals {4 through 11 here). It is often easier to see the excitation when the orbitals are ordered into symmetry groups, though the occasional road leading to the left (e.g. 9—^8 shown in b) can be misread as an excitation into an orbital of lower energy. The third excitation graph (c) is also symmetry ordered, and illustrates an excitation from 9 into an orbital that is a linear combination of the SCF orbitals 8 and 9 — this type of excitation appears for non-orthogonal Slater determinants.

42 CHAPTER 3 NATURAL ORBITALS IN TRADITIONAL Cl

A physicist is an atom’s way of knowing about atoms.

- George Wald

In this chapter two examples on first row dimers illustrate the advantages of natural orbitals (NO’s) over SCF orbitals for correlated wave functions. The first example deals with SCF orbitals that result from a procedure which is a gen­ eralization of Hartree-Fock theory, the complete-active-space self-consistent-field

(CASSCF) method. Natural orbitals (NO’s) obtained from a Cl wave function with single and double excitations (CISD) have advantages over the orbitals ob­ tained from a CASSCF wave function.^^ For an orbital-based truncation of the many-body space the NO virtual orbitals (those outside the CAS) are inherently better suited. Upon rotation of the basis into NO’s we often find a shell-like struc­ ture in the orbital filling that is not present in (virtual) orbitals determined by a self-consistent method that optimizes the chemically active orbitals. This shell structure suggests a partitioning of the orbitals into groups with different allowed degrees of excitation.'^^

43 The form of the wave functions is determined by a partitioning of the molecular orbitals (MO’s). A partitioning of the one-electron basis set yields MO’s grouped according to their filling: MO’s with a larger filling appear in more determinants.

For this procedure to be effective, it is crucial that the MO’s with a larger filling account for the most important correlation effects.

Partitioning of the MO’s begins with a division into a reference and a virtual space. The reference-space orbitals for first-row dimers are

The crls and o’*ls MO’s are kept inactive. The a2s and a* 2s MO’s axe kept semi-active: At least two of these four spin-orbitaJs are required to be occupied. The remaining six spatial MO’s axe active. Out of this reference space we allow all single and double excitations (CAS-CISD). In the natural orbitals of this wave function we calculate the new CAS-CISD wave function and include the most important triples and quadruples (CAS-CISD(TQ)).

44 3.1 Natural versus CASSCF orbitals

The nitrogen atom has a half-fiUed 2p shell, hence of the molecular orbitals orig­

inating from the 2p atomic orbitals all bonding orbitals are occupied and all anti­

bonding orbitals remain empty. This results in a short bond distance and large bind­

ing energy. The small intemuclear distance and purely bonding nature of the occu­ pied orbitals lead to a high electron density, i.e.,the electron-electron correlation is expected to be large, maJdng nitrogen a “standard” test caseh), 12,31,44,45,46,47,48,49 for electronic structure methods.

Many previous Cl studies of the nitrogen system have investigated the com­ putation of the dissociation energy^^’^^’'^^’"*®’'^^ of N 2. The largest reference space in these studies was the complete active space (CAS) of the 2s-2p manifold, out of which single and double (SD) excitations were allowed. Any contribution from higher excitations was estimated with the Davidson correction.'^^ Here we explic­ itly include the most important higher excitations, triples and quadruples (TQ), in the Cl expansion. Partitioning of the one-electron orbitals makes it possible to systematically include higher excitations in large basis sets without the number of determinants becoming prohibitively large.

45 Method Orbitals R (bohr) E (hartrees)

P eterso n et al. SCF SCF 2.0137 -108.9945

SCF-hl-f-2 SCF 2.0511 -109.3573

CASSCF-t-l-h2 CASSCF 2.0796 -109.3899

th is w ork SCF SCF 2.0137 -108.9945

CAS-CISD SCF 2.0700 -109.3770

CAS-CISD natural* 2.0701 -109.3914

CAS-CISD(TQ) natural* 2.0701 -109.3998

Ref. 21, with R converted into bohr to compare with our results *NO’s resulting from the CAS-CISD calculation in SCF orbitals.

TABLE 3.1 Dependence of the total frozen-core energy on types of orbitals used — Hartree-Fock (SCF), CASSCF, or natural orbitals (NO’s) — for Ng in the cc-pVQZ basis. In SCF orbitals the lowering from the SCF-t-I-t-2 to the CAS-CISD energy results from the increased number of reference states. The CASSCF-fl-f2 wave function of Peterson et al. has the same form as our CAS-CISD, but differs in energy because of how the orbitals were ob­ tained: CASSCF-t-l-h2 lies lower than the CAS-CISD in SCF orbitals since the CASSCF step optimizes all orbitals in the CAS (not just those of the SCF determinant). The CAS-CISD result is further improved by using NO’s; this leads to a 1.3 millihartrees lower energy than the CASSCF-fl-|-2 result. Inclu­ sion of the most important triple and quadruple excitations lowers the energy another 8.6 millihartree. A 1.3 millihartrees improvement of the energy by rotation into NO’s, i.e., without changing the form of the wave function, is too large to be neglected in dissociation energy calculations.

46 A common qualitative ordering of correlation effects is into static and dynaxnic.^®

Static correlation effects, due to near degeneracies of the chemically active orbitals,

can be accounted for by optimizing all valence orbitals in a CASSCF calculation

or by using natural orbitals from a CISD wave function (CINOs). CINO’s can

be determined in cases where the CAS becomes prohibitively large for a CASSCF

calculation, and the CINOs outside the CAS are better suited than CASSCF vir­

tual orbitals to describe the dynamic correlation effects in a truncated Cl wave

function.^^’^^

Table 3.1 shows the effect of using NO’s from a CAS-CISD wave function for

the nitrogen dimer in a cc-pVQZ basis with a frozen core. The energy we obtain

with all singles and doubles lies about I mH below that of the CASSCF-M4-2 of

Peterson et al. This energy difference is due to the choice of orbitals, since the

form of the wave function is identical with the exception of the treatment of the (t 2s

and a*2s MO’s (Peterson et al. have them active while we keep them semi-active).

The difference between the CASSCF-M-f2 and the CAS-CISD results is of the

same magnitude as the difference between the dissociation energy of Peterson et

al. in the complete bcisis limit^ (363.0 millihartree) amd the experimental value

(364.0 millihartree).

^ Using equation (2.22) defined in Section 2.6. 47 3.2 Effectively size-consistent truncated Cl

A disadvantage of truncated Cl methods is the lack of size consistency — defined by the requirement that, with a single method, the energy of two well-separated fragments A and B treated as one system equal the sum of the energies of each fragment treated separately,^

E[A + B] = E[A] + E[B]. (3.1)

To derive dissociation energies with a theoretical method that is not size consistent the dissociation fragments are treated as a single system, a “supermolecule”.

For the supermolecules of first-row dimers (two essentially independent atoms treated as one system) we introduce an effective size-consistency, where one aspect of the method (partitioning) is varied with system size in an a priori way. Only valence orbitals are considered; calculations are done in the cc-pVTZ basis sets.

As defined by equation (3.1), size consistency requires that the energy for a dimer with very large nuclear separation be twice the energy of a single atom. Two values of the atomic energy are considered in this section, the SCF energy and the CISD energy. To achieve effective size consistency in both cases requires the correct form of the wave function and the correct form of the orbitals.

(i) Effectively size consistent static correlation.

In supermolecules of first-row elements, the 2p AO’s combine to form degenerate a- and 7r-type MO’s. A SCF wave function that consists of only one determinant does not treat these orbitals equally. However, by going to the complete active space (CAS), all of the MO’s resulting from the 2p AO’s are included in the wave 48 function. But since the orbitals themselves resulted from an unequal treatment, the corresponding energies for N, 0, and F are well above twice the atomic SCF energy

(see Table 3.2). This can be corrected by using natural orbitals (NO’s) based on a

CAS-CISD expansion.

For boron and carbon the CAS calculations in NO’s lead to energies that lie below those of the corresponding two uncorrelated atoms, for boron this happens even in SCF orbitals. Because the subshell resulting from the 2p AO’s is less than half full, the electrons in the 2s AO’s can be correlated by excitation into unoccupied 2p AO’s. This can be prevented by either freezing the MO’s formed by the 2s AO’s or by using a smaller space, denoted here by CAS’. CAS’ is defined by including only those anti-bonding orbitals where the corresponding bonding orbital is occupied in the SCF determinant. For B, CAS’ includes the ag2s, au 2s, crg2p, and au'2p orbitals; for C, CAS’ includes the cr^2s, (Tu2s, ^{z,y}g2p, and tt^^. yj„2p orbitals. Table 3.2 shows the effective static-correlation size consistency that results from the appropriate partitioning and by using natural orbitals.

(ii) Effectively size consistent dynamic correlation.

To describe two atoms treated at the CISD level, the supermolecule wave func­ tion must include some triple and quadruple excitations. In an AO basis these would be combinations of single and double excitations on each of the atoms. Here a MO basis is used, and partitioning is used to achieve effective dynamic-correlation size consistency.

Table 3.3 shows for nitrogen how by partitioning the one-electron MO’s in a well-defined way the resulting energy differs by only 0.4 mH from that of two atoms treated at the CISD level. The partitioning is based on the filling of the 49 atom® supennolecule (two dissociated atoms) ______SCF orbitals ______NO’s ____ SCF CAS’ CAS CAS’ CAS B -24.5281 -24.5203 -24.5363 -24.5274 -24.5620 C -37.6867 -37.6701 -37.6810 -37.6860 -37.7056 N -54.3974 -54.3706 -54.3970 0 -74.8031 — -74.7588 — -74.8022

F -99.3992 — -99.3886 ------99.3998 “ T. H. Dimning, Jr., J. Chem. Phys. 90, 1007 (1989).

TABLE 3.2 Energy per atom of supermolecule wave functions for compari­ son with uncorrelated atomic wave functions of first-row atoms in the cc-pVTZ basis set. Effectively size-consistent static correlation is achieved when the su­ permolecule wave function reproduces (within chemical accuracy) twice the energy of an atom treated at the mean-held level. The correct description of two independent atoms at the mean-held level requires more than one determi­ nant when the dissociated atoms are treated as one system (“supermolecule”). The corresponding energies are in boldface. For N, 0, and F, all chemically active orbitals must participate in the wave function. The MO’s formed by the 2p AO’s must all be treated equivalently not only in their participation in the wave function, but also in their form: The single-determinant SCF treats occupied and virtual orbitals differently, but a rotation into NO’s based on a CAS-CISD wave function leads to the correct form. For the atoms with less than half full shells, boron and carbon, the use of the CAS allows for some correlation of the 2s AO’s (excitations into empty orbitals resulting from the 2p AO’s), and the supermolecule energy lies below the atomic SCF energy. A smaller space, CAS’, is defined by including only those anti-bonding orbitals where the corresponding bonding orbital is occupied in the SCF determinant. For B, CAS’ includes the cr^2s,

TABLE 3.3 Energy per atom of supermolecule wave functions for compari­ son with correlated atomic wave functions of first-row atoms in the cc-pVTZ basis set. Effectively size-consistent dynamic correlation is achieved when the supennolecule wave function reproduces (within chemical accuracy) twice the energy of an atom treated at the CISD level. Rigorous size consistency would require single and double excitations on both atoms to be treated indepen­ dently, which involves a distinct set of triple and quadruple excitations when the two atoms are treated as one system (the supermolecule). Here the re­ sulting energy is approximated by partitioning the one-electron orbitals as suggested by their occupation numbers shown in Figure 3.1. This partitioning is equivalent to (i) forming a cc-pVDZ basis set from the cc-pVTZ basis, (ii) al­ lowing the orbitals in the double-zeta basis to be treated with up to quadruple excitations, and (iii) allowing up to double excitations into the triple-zeta basis orbitals that are not included in the double-zeta set. As with the calculations in Table 3.2, the use of NO’s from a CAS-CISD wave function is required. As the 2p shell becomes more than half full, this approximation works less well: Triple and quadruple excitations on the individual 0 and F atoms contribute significantly to the correlation energy.

51 supennolecule’s MO’s in the CAS-CISD expansion and is shown in Figure 3.1. In the CAS we have the MO’s corresponding to combinations of the atomic valence orbitals. The remaining MO’s are partitioned into two groups: The MO’s that correspond to the first shell of correlating AO’s are allowed to contribute in up to quadruple excitations; all other MO’s participate only at the SD level.

The structure of this partitioning can also be described in terms of correlation- consistent basis sets: Using atomic cc-pVTZ basis sets centered at each atom we calculate the natural orbitals of a CAS-CISD wave function. We partition these

NO’s into (i) a molecular cc-pVDZ basis set and (ii) the remaining MO’s. Within

(i) we calculate the CAS-CISDTQ wave function, this we augment by allowing single and double excitations firom the CAS into the MO’s in (ii).

Also shown in Table 3.3 are the results of applying the same procedure described above for nitrogen to other first-row supermolecules. In all cases the supermolecule result lies below that of the atom. For boron and carbon the respective energy differences (0.1 mH and 0.6 mH) are within chemical accuracy, while for oxygen and fluorine the results (0.7 mH and 1.8 mH) show a tendency for being too low as the 2p shell becomes increasingly full. The electron density is high, and the electrons are more strongly correlated, hence triple and quadruple excitations on a single atom contribute significantly to the correlation energy. The results for the supermolecule calculations, which allow these higher excitations, are thus increasingly lower than the atomic energies, where only singles and doubles are included.

52 .0 10

CAS

10 u Q 00 00 < U 10■2 SDTQ 1 s o

10 •3 SD

,-4 10

s p d f AO symmetry

FIG U RE 3.1 Partitioning for effectively size-consistent dynamic correlation of the frozen-core N 2 supermolecule in the cc-pVTZ basis (see Table 3.3). The orbital fillings of the MG’s are plotted against the symmetry of the AO’s from which the MO’s stem. The shell structure exhibited by the filling of the one- electron orbitals in the CAS-CISD expansion for the supermolecule suggests the partitioning. In the CAS we have the eight orbitals required for effectively size-consistent static correlation (

There is no safety in numbers, or in anything else.

- James Thurber

Natural orbitals, as defined for a truncated Cl wave function, lead to better energies. They are basically a re-orientation of the basis set orbitals resulting from the SCF step. For two-electron systems this re-orientation leads to a diagonal expansion of the wave function. To get a crude idea as to what the NO’s of a many- electron system are, we assume that the electrons can be grouped into distinct, non-interacting pairs. The correlation of one such pair could then, in principle, be done with a diagonal expansion in the pair’s natural orbitals (see the discussion of PNO-CI Section 2.9). Each electron pair would have a set of “pair-specific

NO’s”, i.e., each electron pair would require a different basis set orientation. In a traditional Cl expansion, only one basis set orientation is allowed, NO’s defined for the many-pair system would then presumably represent a basis set orientation which is a weighted average of the pair-specific basis set orientations. Figure 4.1 54 1 correlated - - 2p correlated — Is and 2p correlated

0 0 1 2 3 r(bohr)

F IG U R E 4.1 Natural 3s orbital for neon in the cc-pCVTZ basis resulting from correlating different orbitals. Correlating the electrons in the 2s orbital results in a different 3s natural orbital than correlating the electrons in the 2p orbital. When all electrons in the 2s and the 2p orbitals are correlated simultaneously, the resulting natural 3s orbital lies somewhere inbetween the two pairs’ separated result.

55 shows the 3s orbital for neon that results from the trajisformation into NO’s of

three different CISD expansions:

(i) electrons in 2s orbital correlated, keeping 2p frozen,

(ii) electrons in 2p orbital correlated, keeping 2s frozen, and

(iii) electrons in 2s and 2p orbitals correlated.

The electrons in the 2s and the 2p orbitals of neon cannot be assumed to be in­

dependent. However, the 3s NO resulting from the expansion where all valence electrons are correlated resembles the average of the 3s NO’s resulting from the 2s-

and the 2p-correlated wave functions.

A natural question one might ask is, what if instead of rotating the basis set

for the entire expansion one were to rotate it separately for different parts of the expansion? Or in the extreme case, what if each determinant had its own rotated

version of the basis set? A computer code that allows complete freedom of the

basis set orientation for each determinant, Nor THC oin , is introduced in this chap­ ter. Nor ThC oin is the acronym for Non-ORTHogonal determinants and their configuration INteraction. Some of the more involved mathematical details for Cl with non-orthogonal determinants can be found in Appendix B, here the method is discussed in a more qualitative way.

The number of configuration state functions in a full Cl expansion is given, neglecting symmetry arguments, by Weyl’s formula,®

1 5) 1)' where S' is total spin, N is number of electrons, M is number of basis functions.

NorthCoin in its current implementation is restricted to singlet states {S = 0), 56 where Weyl’s formula reduces to

We can estimate the number of determinants in an expansion of non-orthogonal determinants by assuming individual pairs of electrons have their own “pair-specific

NO’s”. In such a scenario, each pair would have its own expansion in determinants that differ from the reference determinant by an excitation of the electron pair in question. Of the M orbitals, {N/2) — l are held occupied by the electron pairs which are not being correlated in this partial expansion, leaving M — (iV/2) -|- 1 orbitals with which to form the pair-specific NO’s. This includes a re-optimization of the orbital occupied by the pair in the SCF determinant — unlike, e.g., in the PNO-

CI method. The PCI expansion of a two-electron system in NO’s is diagonalized, i.e., the number of determinants is equal to the number of orbitals. For our case this means that by choosing optimized orbitals, an electron pair is ideally correlated by an expansion in M — (A^/2) -f- 1 determinants. If we assume a system in which the electrons are well described as distinct pairs, the number of non-orthogonal determinants is estimated to be

y ( m - y + l ) . (4.3)

If instead all possible electron pairs require a separate expansion, the number of determinants is estimated to be

0 (»-?•')■

These are crude estimates; inter-pair correlation effects are considered to be entirely absent, though in practice these effects should not be neglected. However, since •57 system basis N M estimated number of _ Weyl’s non-orthogonal determinants formula y pairs (^ 2 ) pairs all-electron calculations H2 0 DZ 10 14 1 • 10® 55 495

Ne cc-pCVTZ 10 43 174 • 10^ 195 1755 frozen- core calculations Ng cc-pVDZ 10 26 885 • 10® 110 990

C2 cc-pVDZ 8 26 52 • 10® 92 644 C2H2 cc-pVDZ 10 36 27 • 10^ 160 1440

C2H4 cc-pVDZ 12 41 3•IOI2 216 2442

C2H6 cc-pVDZ 14 46 421 • IOI2 287 3731

TABLE 4.1 Estimates for the number of non-orthogonal determinants in comparison with the number of configuration state functions determined by Weyl’s formula. In all cases 5 = 0.

58 log(CSFs) 5 I’s formula

possible pairs

distinct pairs

2 4 6j^8 10

FIGURE 4.2 Estimated number of configuration state functions for orthog­ onal determinants (Weyl’s formula) and non-orthogonal determinants. N is the number of electrons; M is the number of basis functions.

North C oin is not implemented to form expansions for electron pairs as described above, the expansions that would appear are inherently relaxed to allow for such inter-pair correlation effects.

To get an idea of the numbers involved, the number of determinants for several systems that axe considered in Chapter 6 are given in Table 4.1, as estimated by equations (4.3) and (4.4). The number of determinants are many orders of magnitude fewer than the number of CSF’s estimated by Weyl’s formula. These savings in the length of the expansion are not without their cost somewhere else:

There is no guarantee that the orbitals in one determinant are orthogonal to those in another determinant. Slater’s rules for calculating the matrix element between two determinants can no longer be applied; they have to be modified.

59 4.1 Basic math for non-orthogonal determinants

As in traditional Cl, a wave function in North C oin is expanded in Slater determinants,

1^) = (4-5) K the difference being that now two determinants and ) of our expansion can be non-orthogonal. The overlaps between the determinants must be included in the Schrddinger equation,

H|^)=S|^), (4.6) where S is the matrix of overlaps The normalization condition for the total wave function is now

= = (4 7) K L

Based on the normalization (4.7), the occupation of a determinant IV’a Oi z-c., its weight in the expansion, can be defined as^^

= c/v ^lSkl- (4.8) L

The form of a given determinant can be represented in several ways. In Ap­

pendix C the formulation used by Koch and D algaard^4 jg discussed in detail. Here

the representation is discussed in terms of matrices that act on the orbitals in the

basis for which the integrals were calculated. A determinant |V’a ) is then a set of occupied and the associated unoccupied orbitals,

H’k ) Wk \Qk]^ (4.9) 60 where represents the occupied orbitals and represents the unoccupied or­

bitals of determinant IV’K’)- In NorTHCoin, every determinant is represented by a

matrix [Df(\Qf{] where the columns of Dj( represent the n = NJ2 occupied orbitals

and the columns of Q[^ represent the m = M — N/2 unoccupied orbitals.

The reference determinant |V’o) is represented by the unit matrix,!

‘1 0 0 ... 0 o' - - 0 1 0 ••• 0 0 Inxn Onxm 0 0 1 ••• 0 0 [^olQo] = — (4.10) ÔÔ Ô ••• 1 Ô Omxn Imxm

0 0 0 ... 0 1 -- and the orbitals of any determinant in the expansion can be viewed as resulting

from either a matrix multiplication of the reference determinant’s orbitals or of the

basis functions. A double excitation is represented as, e.g.. '1 0 0 ... 0 o' 0 0 0 ... 1 0 0 0 1 ... 0 0 (4.11)

Ô 1 Ô ... Ô Ô _0 0 0 ... 0 1 where the second occupied orbital (d^) was replaced by the second-last unoccupied

orbital (<^n_[).

The overlap matrix S used in equation (4.6) is calculated from the determinant

overlap matrices

^KL = \^K l (^ = \^ K ^ l \^- (4.12)

The square of the A/^-j[,’s is taken because all orbitals are doubly occupied. More

detail on the calculation of the Sfd's is in Section 4.4 and Appendix C. ! In general the reference determinant is the SCF determinant and the integrals are stored in the SCF basis. In the Hubbard model applications, the integrals are stored in the site basis, but the SCF determinant is not the unit matrix. Another exception is the use of localized orbitals in the reference determinant, in this case a unitary transformation (applied to the Inxn matrix) represents the localized orbitals. 61 4.2 Basic North C oin algorithms

A new detenninajit is initiated by an excitation of a previous “source” deter­ minant in the expansion. This is always a double excitation, i.e., the replacement of an occupied orbital with an unoccupied orbital of the source determinant. For

M orbitals and N electrons, a given source determinant allows (N/2) x M — {N/2) possible excitations, the best new determinant is chosen by how much its addition to the expansion improves the energy. N orThCoin is implemented to initiate a new determinant from the following source determinants:

(i) The SCF determinant in canonical molecular orbitals (CMO’s)

(ii) The SCF determinant in localized molecular orbitals (LMO’s)

(iii) The SCF determinant in CMO’s and LMO’s

(iv) The last determinant in the expansion

(v) The first determinant in the expansion

(vi) All determinants in the expansion

(vii) All determinants in the expansion and the SCF determinant in CMO’s

Following the initiation of a new determinant, it needs to be optimized. A determinant is optimized by multiplication with a matrix that describes a unitary transformation between the occupied and the unoccupied space. The transforma­ tion can mix one or more pairs of (occupied unoccupied) orbitals. The decision which orbital(s) to mix is based on the gradient, represented by a {N/2) x M matrix with elements V,> given by the first derivative of the total energy with respect to 62 a mixing of orbitaJs (j>i {i = 1,2,..., iV/2) and 4>r (r = 1,2,..., M ),

= (4.13)

where is the angle describing the mixing,

i - + cos{0ir) i + sin(0,>) r -*■ cos(0,v) r - sin(0,>) ,

of orbitals in the last determinant of the expansion. The gradient matrix is calcu­

lated by explicit mixing of the orbitals by a small angle 0test specified in the input

file. The finite difference of the resulting energies specifies V.

North C oin is implemented to allow three basic algorithms to use the gradient

matrix V in determining which orbitals to mix:

(i) Use only the largest V,> to mix one pair of orbitals.

(ii) Use a specified number of largest Viv’s to mix pairs of orbitals.

(iii) Use those V,>’s which axe larger than a given threshold to mix pairs of orbitals.

For cases (i) and (ii) a ‘memory’ version also exists, where NORTH COIN optimizes

the orbital mixing in the direction of one V,> at a time, stepping through the

specified possible V,> repeatedly. This makes optimal use of a gradient matrix.

63 4.3 “Approximation” of restricted-spin excitations

In the current implementation of N o r t h C o in , the spatial orbitals in every

determinant are doubly occupied. For two-electron systems in the singlet ground

state, natural orbitaJs eliminate the need for any other type of determinant. With

the freedom to change the basis set orientation for each determinant, N o r t h C o in

can, in principle, “find” the NO’s needed for the full Cl wave function. For sys­

tems with approximately independent electron pairs, such as Be or a collection of

separated He atoms, this approximation is expected to have little influence on the

results since NORTH CoiN can find the NO’s for the individual pairs.

Systems where electrons can be grouped into approximately independent pairs

are not the norm. The general case must be considered, where, e.g., no basis set

orientation can eliminate the need double excitations into different orbitals.

For systems with four electrons, the following can be shown:^ If, in a determi­

nant, one of the doubly-occupied orbitals is a linear combination of two orthogonal

orbitaJs, (f>a = àc-\- oi

\a^abh) = \àc(i>c4>bh) + û: [\4>c^dhh) + \dMbh)) + Ot^\d^d4>b^b)^ (4.15)

where a represents the spatial

orbital with down spin. Hence, even though the orbitals in every determinant

are doubly occupied, due to the non-orthogonality between determinants we do

get symmetric combinations of single excitations and double excitations into two

different orbitals. By generalizing this example, in principle all spin couplings

^ see Appendix D for more details 64 are possible with the current implementation of North C oin . However, whether the code’s optimization procedures can actually find the correct determinants to describe all the necessary couplings is not clear from the outset. In particular, cases where several different spin couplings need to be considered in the same wave function wiU be challenging.

65 4.4 Degree of excitation

In traditional configuration interaction, a given determinant will differ from

the reference determinant by an integer number of spin orbitals — this number is

referred to as the “degree of excitation”, Uex- A Cl wave function expansion is then

truncated at a chosen degree of excitation, such as:

CISD: all single and double excitations (max nex=2)

CISDTQ: up to quadruple excitations (max nex=4)

For non-orthogonal determinants, the difference between a determinant and the

reference determinant is no longer required to be an integer number of spin or­

bitals. To describe such determinants the definition of the degree of excitation is

generalized in this section.

Unitary transformations^ can be found that bring the two sets of n (occupied)

spin orbitals {i} and {y?,} of two determinants into “maximum coincidence”, where

(i) 0 < {4>i\tpi) < 1, and (ii) {(l>i\^j) = 0 for i ^ j.

While the transformations are not unique, the sum of is. We define the

degree of excitation for a determinant to be

N n,,{K) = N-Y,{4>iW i), (4.16) :=1

where are the spin orbitals of the reference determinant |^o), and {y,} are

the spin orbitals of the excited determinant For all systems studied here, the

^ A unitary transformation of the occupied orbitals within a determinant leave the energy unchanged.

6 6 spatial orbitals in every determinajit are doubly occupied. Written in terms of the n = N /2 spatial orbitals, the degree of excitation is

n^y,{K) = 2 X . (4.17)

The degree of excitation n^xiK) gives a measure of how a determinant differs from the reference determinant |^o). Another measure for this difference is the overlap = (^oIV’a:)- With the orbitals of {iPq) and IV’A') maximum coincidence, Tiex{K) is essentially the sum of the orbital overlaps, whereas S qj( is the product of the orbital overlaps. By considering both measures together as points

{nex(A'), •S'oa} & graph, different “types” of determinants can be distinguished.

To acquire some idea of where in such a graph a certain type of determinant will appear, several types and their associated points are considered below.

Type 1: orthogonal determinants.

In a traditional Cl expansion, a determinant |^ a ) will differ from |^o) by an integer number of orbitals. Orthogonal determinants have points {26,0}, where 26 is the number of spatial orbitals replaced in the reference determinant.

Type 2: determinants differing in two orbitals.

For non-orthogonal determinants, a determinant I^a) a^so differs from [t/’o) by replacing a number of orbitals, but now a new orbital is not required to have a non-zero overlap S with the orbital it replaces. If, e.g., two orbitals (/){, (f>j of iV'o) have been replaced by ipj, yj in |0a)i tben the determinant overlap is

S qk = (4.18) and the degree of excitation is

nex(AT) = 2(2 - 5,-- (5y), (4.19) 67 1.0

0.8

a = 0 a= 1 CL — 2

a, Ui Branching points 0.4

0.2

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

FIGURE 4.3 Determinant overlap with the reference determinant plotted against the degree of excitation for non-orthogonaJ determinants. Cases where only one (spatial) orbital differs from the reference determinant correspond to a = 0. Branching off the a = 0 line are curves describing determinants which differ from the reference determinant by two or three orbitals. When the difference is in two orbitals, q is the ratio between the overlaps of the two pairs of orbitals. The “3(5’s” line describes determinants that differ from the reference determinant in three orbital pairs, provided all three pairs have the same (orbital) overlap.

6 8 where Si = {4>i\'fi) and Sj = {j\'Pj). Talcing one orbital overlap to be a factor a

larger than the other,

Si = ocSj, (4.20)

we can write the determinant overlap as a function of the degree of excitation,

SoA-(cr,n„(A:)) = ^ 4' (4 2D

where the lower limit for riex{K) is determined by Si = 1 in equation (4.19).

Figure 4.3 shows the plots for selected values of a. The limiting case a = 0

(equivalent to a = oo) describes a determinant that differs from the reference

determinant by only one orbital, in this case the relation between the overlap and

the degree of excitation is

5oA(0,n..(A:))= . (4.22)

There can be no points to the left of the q = 0 line (for determinants in which all occupied spatial orbitals are doubly occupied).

The curves for determinants that differ from the reference determinant by two orbitals begin at the “branching points” on the a = 0 line. The a = 2 line, e.g.,

begins at n«x = 1,5 = 0.25, where one orbital pair has an overlap of 1/2 — the other orbital pair has an overlap equal to unity.

Type 3: determinants differing in three orbiteds.

If three orbitals of |^o) have been replaced, then the determinant overlap is

SoK = S}6jsl (4.23) and the degree of excitation is

"«x(^) — 2(3 — Si — Sj — Sf.). (4.24)

69 1.0

a = 1 0.8 a = 2

0.6 §■

I0

1u a: 0.4 Branching points

0.2 0= 0 line

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 n

FIGURE 4.4 Scaled determinant overlap with the reference determinant plotted against the degree of excitation for non-orthogonal determinants. This figure is the scaled version of Figure 4.3; the same symbols are used in both figures. For intermediate values of the ordinate is a rough measure of the number of orbitals in which the determinant differs from the reference determinant. The abscissa (the “a = 0 line”) described determinants that differ from the reference determinant in one (spatial) orbital.

70 The most distinct case with three replaced orbitals occurs when all the orbital overlaps are approximately equal. For 5,- = Sj = Sf., the determinant overlap can again be expressed as a function of the degree of excitation,

SokMK)) = A - • (4.25)

This curve is also displayed in Figure 4.3 and is labeled “35’s”.

The lines plotted in Figure 4.3 distinguish different types of determinants, but the lines axe not very different. To graph the determinants in such a way that makes their types better distinguishable, the determinant overlap can be scaled.

The result of the scaling chosen here is shown in Figure 4.4. The scaling used is:

(1 + K x - 4)2) , Hex < 2 scaled overlap = \ ^ ^ , (4.26) i^OK ~ 0) (l + (Uex — 4) ) , Uex ^ 2

This choice is somewhat arbitrary, however, the resulting graph makes the different types of determinants more readily distinguishable. As shown in Figure 4.4, the

Û = 0 line has become the abscissa, hence any determinant differing from the reference determinant in only one orbital has a scaled overlap of zero. The ordinate now gives a rough measure of the number of orbitals in which a determinant differs from the reference determinant, at least for intermediate values of Hex-

Throughout this document Figure 4.4 is used to plot the points corresponding to determinants of the calculated expansions. The three curves introduced here are kept as reference lines.

71 4.5 “Slater’s rules” for non-orthogonal determinants

As with orthogonal determinants, the calculation of matrix elements between

non-orthogonal determinants can be described in terms of rules. These are de­

veloped here to provide a more intuitive understanding of how matrix elements

between non-orthogonal determinants axe calculated.

In Section 2.5, where Slater’s rules for orthogonal determinants were presented,

the orbitals were assumed to be in maximum coincidence. Two non-orthogonal

determinants can also be brought into a generalized maximum coincidence. This

involves establishing a one-to-one correspondence between the orbitals of one deter­

minant with those in the other determinant. Diagonalizing (with a singular value

decomposition) the overlap matrix between the two sets of orbitals brings any or­

bitals that are common to both determinants to the same place in determinants,

as in the orthogonal case. The difference between the two determinants then lies in

the orbitals that partially overlap each other. Because the overlap matrix is now

diagonal, each orbital of one determinant has a non-zero overlap with at most one

orbital from the other determinant.

As an example. Figure 4.5 shows the orbitals from two non-orthogonal de­

terminants (the determinant overlap is approximately 0.3) rotated into maximum

coincidence.^ Two of the three orbitals in maximum coincidence are equivalent,

({^1^^ = for i = 1,2) while the third orbital pair has an overlap of approxi­

mately 0.56 (hence the determinant overlap 0.3 % 0.56^). All orbitals from different

pairs are orthogonal, i.e., = 0 for all i ^ j.

^ This example is from calculations on the anti-periodic 8-site Hubbard model, see Table 5.7. 72 SCF determinant excited determinant

(I)

i l f

FIGURE 4.5 Example of rotating the orbitals in two determinants into maximum coincidence. On the left are the orbitals of the SCF determinant, on the right those of an excited determinant. In the upper half the orbitals are given in their original form (sines and cosines for the SCF determinant). The lower half shows their maximum coincidence form, where each orbital of the SCF determinant has a non-zero overlap with at most one orbital of the excited determinant. These two determinants differ only in a single orbital, as can be seen once their orbitals have been rotated into maximum coincidence by the unitary transformations and U^. 73 The two sets of majdmura coincidence orbitals do not correspond to the or­ bitals for which the integrals are given. In traditional Cl the one- and two-electron integrals are in the basis set orientation where Slater’s rules are applied; if neces­ sary (e.g. for NO’s) one transforms the integrals to a new basis set orientation.

In an expansion of non-orthogonal determinants it is possible that some pairs of determinants are orthogonal to each other, in which case Slater’s rules can be ap­ plied after an appropriate basis set transformation. The real problem is between non-orthogonal determinants that have a finite overlap, since each determinant will have its own baisis set orientation.

One-electron integrals. In Slater’s rules, the one-electron integrals enter in two ways: For identical determinants the one-electron integrals of all orbitals are summed; for determinants that differ in one orbital the one-electron integral of the two differing orbitals is taken. This can be formulated more generally as, “for each orbital pair, talce the one-electron integral and multiply by the overlap of the remaining orbital pairs’ overlaps.” A graph that reflects this thus contains “overlap operators” A for each orbital pair, in the case where the two determinants differ by one orbital pair we have the following (non-orthodox) graph:

(4.27)

For two orthogonal determinants that differ by the orbitals and <^s, this is the only one-electron graph that gives a non-zero contribution, in agreement with 74 Slater’s rules: While it is possible to have, e.g., a non-zero (^rl^l^f) in the corre­ sponding graph.

(4.28) the overlap operator A between orbitals (f>i and 4>s, which are orthogonal, makes the contribution of this graph zero.

These graphs can be applied to two non-orthogonal determinants, the difference to orthogonal determinants is that the overlap operators now can result in any number between zero and one. For the example of two determinants that differ in only two orbitals, (pr and s, this results in the sum of two terms,

(i) the sum of one-electron integrals of all the orbitals that are the same, ^i{i),

multiplied by (<^r|A|(6g)^ and a factor of two for spin, and

(ii) the one-electron integral multiplied by 2{4>r\^\s) to account for the

two spins (since we require all orbitals to be doubly occupied).

Taken together these terms result in

-2^<4&,|AK) -b2(^r|A|^,>(,^r|/i|<)s>, (4.29) i where the first factor of 2 comes from the sum over the doubly-occupied orbitals i, while the second factor of 2 results from orbitals r and 4>s being doubly occupied.

The one-electron integral can be formed by projecting the one-electron in­ tegrals into the basis set orientation of the determinant containing d»r, and summing

75 them with weights given by the projection of (f>s onto the corresponding orbitals in this basis set orientation.

The two-electron integrals can be illustrated by a graph where overlap operators are coupled to the two-paxticle operator:

(4.30)

Again, this graph results in Slater’s rules for orthogonal determinants and gives an idea of how the matrix elements between non-orthogonaJ determinants involve the two-body operator and orbital overlaps.

76 4.6 Taking advantage of symmetries

The symmetries found in the systems are often taken advantage of when using

No r t h C o in . In the Hubbard model, for instance, we write all orbitaJs in terms

of the site orbitals, which are all degenerate with each other.^ Ideally, one would

formulate an algorithm that simultaneously optimizes an entire configuration state

function, i.e., the sum of determinants required to preserve the symmetry of the

system. In the absence of such a formulation, we optimize a single determinant

of the CSF in the field of the entire expansion, i.e., the reference dcterminant(s),

any previous CSF’s, and the remaining, symmetrically translated determinants of

the current CSF. Upon optimization of one of the determinants, the symmetry

operation (translation, rotation) is applied to it to form an improved CSF. The

procedure (optimize, symmetrize) is repeated until convergence is reached.

In some cases, this procedure does not converge and leads to oscillations in the

energy. For these cases, more than one of the determinants making up the CSF

were optimized. These were then used as seeds to form “candidate CSF’s” — the

energetically best was then chosen to proceed.

For the Hubbard model inversion symmetry was initially not enforced. For sin­

gle determinants this is evident in plots of the densities. While these may not be

the best possible determinants, taken together in CSF’s they do preserve this sym­

metry as is seen in the correlation functions. Section 5.6 presents some preliminary

results with enforced mirror symmetry.

^ for periodic lattices

( I For atoms and molecules, symmetries were not enforced. In particular, for local­ ized orbitals symmetries are often broken purposefully. Comparisons of optimized determinants with their symmetric counterparts shows that while optimizing the energy symmetry is kept or re-established in most cases. Provided one allows for all symmetrically-equivalent determinants, this may be potentially used as a criterion of convergence.

78 4.7 Generating higher excitations

In coupled-cluster (CC) methods, higher-order excitations are generated from

products of lower-order excitation operators with an operator,

l + fi + f2 + if? + i f | + .... (4.31)

and their coeflScients in the expansion are treated accordingly as products of the

lower-excitation coefficients.

For non-orthogonal determinants, a similar procedure can be used — with some

restrictions. The two excitation operators must, in essence, excite two different

orbitals occupied in the SCF determinant into two different virtual orbitals. Note

that these do not have to be canonical SCF orbitals, the spaces excited out of must

be orthogonal (or nearly so) to each other, as must the spaces excited into. If these

restrictions hold for two determinants,

and rpe =

where e^'^xo and e^^xo generate the determinants from the SCF determinant, then

the new determinant is

i’A®B = e^^e^^xo- (4.32)

Such a generation is illustrated with excitation graphs in Figure 4.6.

For cases like the Hubbaxd model, or for multi-reference cases, a difficulty arises:

The determinants are not necessarily written in terms of the SCF orbitals. In the

Hubbard model, e.g., the basis orbitals are the site orbitals — not the SCF or­ bitals. However, the SCF determinant’s space V’Oi f-e., its occupied and unoccupied

79 II11/ 1II1III III / 1 2 3 4 5 6 7 8 9 10 11 1 2 3 4 5 6 7 8 9 10 11

a I. / f\ 1----- IL / z 3 4 5 6 7 8 9 10 11

11/ 1 1 1 1 2 3 4 5 6 V-i\9 n) 11 1 2 3 4 5 6 7 8 9 10 11 / \ / b ... 1 1/ V 11\J\ / : 3 4 5 6 7 8 9 10 11

FIGURE 4.6 Combining two excited determinants to make a new deter­ minant of higher excitation, (a) For two orthogonal-CI double excitations the combination leads to a quadruple excitation, (b) Two non-orthogonal-Cl excitations can aiso be combined. In both cases the excitations must affect different orbitals, i.e., excite from different orbitals and into different orbitals.

80 orbitals, are given as a unitaxy trajisformation of the basis orbital space %o,

^0 = r%o, and so two determinants

and

V’S = e^®XO = are combined according to

4’A®B = e^^r-le-'^«r~Vo = e^'^r'^e^^xo- (4.33)

With the exception of one case to demonstrate the principle of combining lower excitations to form higher excitations, the application will be done auto­ matically. Starting from an expansion in the SCF determinant |V>o) and optimized, non-orthogonal determinants \iPk )i

1^) = colt^o) + ^ (4.34) K all possible determinants using (4.33) are formed, symbolically indicated as

E (4M ) K No restrictions are placed on the coefficients, i.e., the system is still treated varia- tionally. This procedure results in many determinants that do not obey the require­ ments of orthogonality on the excitation spaces, i.e., they will contribute very little to the energy. In practice, many determinants created this way are linear combi­ nations of previous determinants in the expansion and are automatically cancelled from the expansion. 81 4.8 Comments on re-optimization algorithms

Instead of viewing the problem as, “How many determinants are required to account for a certain percentage of the correlation energy?” one can ask, “Given a number of determinants d, what is the most correlation energy that can be ac­ counted for?” To address this question we have experimented with three different ways of re-optimizing the determinants:

(i) After each determinant, re-optimize all determinants.

(ii) Optimize the last one or two determinants, but leave the re-optimization

of all determinants until the desired number d has been acquired.

(iii) Find the best d orthogonal determinants, then go back and re-optimize

them allowing non-orthogonality.

Experience shows that (ii) leads to better final energies. The re-optimization of all determinants at every step (i) is not only slower, but also often leads to poor final energies. This is interpreted as being due to determinants with a basis set orientation that is an “average” of two optimal orientations, i.e., correlation effects that would be best described by two determinants are approximated with a single, well-optimized determinant. With such a determinant in place, the initial excitation for a second determinant to describe the correlation effects in question is no longer energetically favorable. Using d orthogonal determinants to start with

(iii) is quick, but here it is possible that several determinants are used to describe a correlation effect which would require only one well-optimized determinant: Since more orthogonal determinants than needed for such an effect have been established, the number of determinants available for other correlation effects has been reduced.

82 One case where re-optimization of the entire expansion has been found to be quite useful is the calculation of potential energy surfaces. Here, instead of creating an expansion from the start for each bond length, the expansion from a neighboring bond length can be re-optimized. Without the re-optimization the wave function is often energetically very poor, especially across regions where bonds are being broken and spins re-coupled. With the re-optimization, the energy converges quite quickly

— sometimes even below that obtained by starting the expansion from scratch. For the Hubbard model calculations a similar procedure was used for different ratios of

Ujt: The wave function for a given lattice at one value o( U ft was re-optimized at other U/t ratios.

83 4.9 Other algorithms

We mention here briefly algorithms that were experimented with but found

to be of little use. They are aimed at initiating a new determinant closer to an

optimal determinant than those obtained from the method outlined in Section 4.2.

The optimization of a new determinant by mixing occupied and unoccupied orbitals

is both slow and not capable of talcing large steps in the multi-dimensional space.

Partitioning of the rotation. In the formulation of Koch and Dalgaard,^"^ one

way of representing a determinant can be understood as two separate unitary trans­

formations of the occupied and unoccupied orbitals in the reference determinant,

followed by a mixing of pairs in the two spaces. NORTH COIN has an implemen­

tation that partitions the unitary transformation of the reference determinant into

an excited determinant along these lines: Rather than optimizing the full M x M

matrix that represents the transformation, three smaller transformation matrices

are optimized separately:

(i) A n X n matrix that describes the transformation of the occupied orbitals.

(ii) A [M — n) X {M — n) matrix that describes the transformation of the unoccupied

orbitals.

(iii) An angle of mixing to describe each pair of (occupied, unoccupied) orbitals that

are mixed.

The optimization is thus constrained to fewer variables, and optimization is faster.

Our experience has been that this constraint works against finding the optimal

determinants. 84 Genetic algorithms. The optimization of non-orthogonal determinants suffers from being done in a phase space with a large number of local minima. Thus algorithms that make large intelligent ‘jumps’ in the available phase space would be extremely useful. One such possibility is offered by genetic a lg o rith m s.H ere one uses a “mating operator” that takes two “parent” determinants to produce a “child” determinant. North C oin was implemented to take two determinants A and B. and calculate the unitary transformation which takes one determinant into the other. Starting from determinant A and performing the transformation half way produces a “child” determinant. The use of this mating procedure in various systems did not prove useful, i.e., the child determinants were rarely better suited for the expansion than their parent determinants.

Scouting. The initiation of a new determinant is based on an excitation of a previous determinant. The contribution to the total energy of such an initial determinant can be quite different than the contribution once the determinant has been optimized. Ideally, one would want to compare the energy contribution of all possible initial determinants in their optimized form, but this is prohibitively expensive. However, a partial optimization may give a better starting point than the initial determinant.^® Our experience with implemented algorithms to use such partial optimization has been that it is useful for very short expansions (less than ten determinants). The improvement in the energy for moderate-length expansions is rather modest in view of the increased CPU time required.

85 CHAPTER 5 NON-ORTHOGONAL SLATER DETERMINANTS AND THEIR CONFIGURATION INTERACTION: APPLICATION TO THE HUBBARD MODEL

In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in poetry, it ’s the exact opposite.

- Paul Dirac

In this chapter No RTHCoiN is applied to the Hubbaxd model, a highly over­ simplified, yet intensively studied model for interacting electrons on a lattice.^^’^®

The Haxniltonian of the Hubbaxd model consists of two parts: Hhop which describes quantum mechanical hopping of electrons from one lattice site to another, and which describes an on-site Coulomb repulsion. Either part of the Hamiltonian taJcen alone can be solved exactly. The combination of the two terms, H = Hhop + Hmt, is known (in some cases believed) to exhibit various complex properties, e.g., ferro­ magnetism and superconductivity.

In terms of basis set orientation, the two parts of the Hamiltonian require very different descriptions: The hopping term describes free electrons, for which a basis set of sines and cosines is best suited. For the interaction term a basis set of single­

8 6 site orbitals is preferable. A unitary transformation taJces one basis set orientation into the other. The Hamiltonian consisting of both parts lends itself naturally to be studied with a method that allows for complete freedom of the basis set orientation.

As an application of Nor THCoiN, the Hubbard model has an additional advan­ tage: Since the two-electron part of the Hamiltonian is entirely local, both direct and exchange Coulomb effects are grouped into one term: Exchange effects enter the formalism only implicitly. In terms of integrals and calculating matrix elements this simplifies matters considerably compared to a system with the same number of orbitals that includes exchange explicitly.

87 5.1 Basics of the Hubbard model

The Hamiltoniaji of the Hubbaxd model,

* (5.1) = —i ^2 ^ O'"*,-0-1 (i,j),o- hO- contains two distinct parts: A term describing the hopping between neighboring sites (favored by —t) and a term describing the double occupation of a site (dis­ couraged by a positive U). Each part taken alone can be easily solved, but taken together they provide a model with many facets.

The parameters in the Hamiltonian axe usually given by their ratio U /t, typical values are between 4 and 8. Furthermore, the number of electrons,

iV = ^ n,- 0- (5.2, ) i,a is important. The maximum value N can take on is 2A, where A is the number of sites. The most interesting physics occurs around N = A (or (n) = N/A = I), at

“half filling”. We only consider the ground state.

For U /t = 0 we have only hopping in the model, and the Schrodinger equa­ tion can be solved — this is often called the free-electron result. The one-particle eigenstates are of the form exp(ifc(n) • r), with wave number kx{nx) = 2irnx/Lx, where Lx is the length of the lattice in dimension i, and rix = 0,±1, ±2,..... The eigenvalues are e{k) = —2tcosk. This is also the SCF (RHF, in particular) solu­ tion. Here the one-particle states with Ux = dbn are combined to form real orbitals

(sines and cosines).

8 8 5.2 One-dimensional Hubbard model

The filling is (n) = 0.75 for aU systems in this section; the lattices presented vary in length, A = S, 16,24,32, and 40. In Figure 5.1 the one-paxticle orbitals are shown for the 16-site, one-dimensional Hubbard model. If the boundary conditions are chosen to be periodic, any filling with an even number of electron pairs (e.g.,

(n) = 12/16 = 0.75) will have two degenerate free-electron determinants. To alleviate this problem, amti-periodic boundary conditions are introduced for the 16- site (as well as the 32-site) Hubbard lattice in this section, making it a closed-shell system. This affects the total energy of the system, but by a vanishing amount as the system size grows.^® Section 5.4 deals with the 16-site model with periodic boundary conditions.

The one-particle orbitals shown in Figure 5.1 are one possible choice for the orientation of the basis set: Rather than using sines and cosines, we could have stayed with the imaginary exponentials. Another natural choice is the orientation of the basis in which the Hamiltonian is written, where the site orbitals are the basis functions.

The expansions presented here were obtained by starting with the site-orbital basis set orientation for a preliminary set of determinants. The first determinant consists of N /2 occupied sites, which has a very poor energy and also does not have the correct symmetry. It does have the advantage of being very localized.

A new determinant was initiated from this first determinant and then (partially) optimized. Based on this second determinant, a new determinant was initiated and optimized. After several steps of this form, a reasonable set of determinants was 89 anti-periodic lattice

periodic lattice

FIG U RE 5.1 One-particle orbitals for the one-dimensional Hubbard model (16 sites) for both periodic and anti-periodic boundary conditions. With the exception of the lowest-energy orbital in the periodic case, every energy level is degenerate (for each level one orbital is shown with a full circle and the degenerate orbital with an empty circle). An odd number of electron pairs has a unique free-electron determinant for the periodic lattice and two degenerate free-electron determinants for the anti-periodic lattice. For an even number of electron pairs the situation is reversed.

90 A = 32

A = 24 A = 16

A = 40

FIG U RE 5.2 Electron densities of typical determinants from the first config­ uration state function of the one-dimensional Hubbard model. For all lattices — A = 8, 16, 24, 32, 40 — the filling is (n) = 0.75 and U ft = A. The 16- and 32-site lattices were taken with anti-periodic boundary conditions, all other lattices have periodic boundaries. The number of determinants in the first CSF is equal to the number of sites A in each case.

91 obtained to “seed” a wave function with the correct symmetry. A set of A-fold translated versions of one of the determinants from the preliminary set was added to the SCF determinant to form a correlated wave function. This expansion was then optimized as outlined in Section 4.6.

The determinants that make up the expansion of a No RTHCoIN wave function for the Hubbard model are found to appear in groups. Since it is only the complete group of symmetry-equivalent determinants that has the correct symmetry of the problem, these groups are correctly called configuration state functions (CSF’s).

The expansion then takes on the form

A 1^) = c o l^ o )+ ) (5-3)

5 = 1 with the reference determinant |^o) ^.nd one CSF, or

A 1^) = cqH'o) + 1% ) (5.4) 0 = 1 s=l for several CSF’s of order o.

In Figure 5.2 the electron densities for determinants of the first CSF are shown for the lattices considered here. All of them show a density peaked at one site. For one such determinant, this indicates an increased probability of finding two electrons on the site with the peak, which in turn increases the energy of such a determinant relative to |^o) due to the electron repulsion U. The effect of correlating the electrons, however, is expected to reduce the probability of finding two of them simultaneously on the same site. The determinants of the first CSF describe what the system is trying to avoid, and when they are added to the SCF determinant, they enter the expansion with a negative coefficient. This can be

92 Sites Ult = 2 U lt = i U lt= 8 UH = 12 U lt = 16 SCF energies 8 -0.9259 -0.6446 -0.0821 +0.4804 +1.0429 16* -0.9027 -0.6214 -0.0589 +0.5036 +1.0661 24 -0.8984 -0.6172 -0.0547 +0.5078 +1.0703 32* -0.8970 -0.6157 -0.0532 +0.5093 +1.0718 40 -0.8963 -0.6150 -0.0525 +0.5100 +1.0725 SCF-t-CSF(l) energies 8 -0.9755 -0.8085 -0.5457 -0.3148 -0.0950 16* -0.9493 -0.7644 -0.4327 -0.1151 +0.1982 24 -0.9427 -0.7463 -0.3788 -0.0198 +0.3435 32* -0.9392 -0.7348 -0.3444 +0.0497 +0.4383 40 -0.9368 -0.7263 -0.3197 +0.0828 +0.4843 using anti-periodic boundary conditions

TABLE 5.1 Energies per site of the SCF determinajit and of the SCF deter­ minant plus first CSF wave function for the one-dimensional Hubbard model at (n) = 0.75. For a A-site lattice the first CSF consists of A symmetrically- equivalent determinants, each with an electron density peaked at one of the A sites. In all cases the energy is lower for the correlated wave function than for the free-electron result.

93 G--0 A = 8: \|/g l i l t G - a A=16: ij/g 1.0 . ■ « A=16: Yo + 0- 0 A=24: % ♦ A=24: \|fj, + 9 V A=32: ijfo ▼ - ▼ A=32: 0.5 X X A=40: v|f(, A=40: v|fn 4- (U /: (U d. 0.0 î

-0.5

- 1.0

! 1 r ! . J. 1 1 1— 8 12 16 U/t

FIGURE 5.3 Energies per site of the SCF determinant and of the SCF determinant plus first CSF wave function for the one-dimensional Hubbard model at (n) = 0.75. The difference between the results for successive lattice sizes A becomes smaller as A grows. This edge effect is not zero for the largest lattice (A = 40).

94 8 sites

U/t 1 2 3 4 5 6 7 2 .0177 .0595 .0905 .0481 .0905 .0595 .0177 4 .0152 .0644 .0917 .0563 .0917 .0644 .0152 8 .0104 .0686 .0916 .0637 .0916 .0686 .0104 12 .0067 .0681 .0884 .0646 .0884 .0681 .0067 16 .0053 .0694 .0882 .0663 .0882 .0694 .0053 16 sites Is1V — s^l ^ 1 U/t 2 3 4 5 6 7 8 2 .0460 .0713 .0491 .0712 .0663 .0618 .0746 4 .0268 .0455 .0323 .0474 .0441 .0414 .0497 8 .0357 .0528 .0427 .0557 .0531 .0511 .0576 12 .0395 .0553 .0469 .0586 .0564 .0547 .0602 16 .0382 .0525 .0453 .0557 .0537 .0522 .0573

TABLE 5.2 Overlap between déterminants and with single peaks in the density. |s — s^| is the distance between the two density peaks. Nearest-neighbor overlaps are not shown for the 16-site case, they are 0.008, 7-10“ ~, I -10"^, 5T 0“ ®, and 0.0003 for U/t = 2, 4, 8 , 12, and 16, respectively.

95 interpreted as “subtracting out” part of the energetically-unfavorable many-body character present in the SCF determinant.

Table 5.1 shows the energy per site for the SCF determinant |V’o) aJid for the wave function cqIV’o) +^1 I^s=l energies are also plotted in Figure 5.3 as a function oiUjt. For a lattice size A the wave function was obtained for [//t = 4, and then re-optimized for other values of Ult. For all lattices and all values oiU/t studied, the inclusion of the first CSF lowers the total energy. Edge effects, due to the finite length of the lattices, axe not negligible at A = 40, nonetheless the total energy for a given Uft does show a converging behavior as A is increased.

In Table 5.2 the overlap between determinants of the first CSF are given for the A = 8 and A = 16 lattices. There is a significant overlap between almost all the determinants, indicating a non-negligible change in the basis set orientation. The overlap between determinants of the first CSF for the A = 24,32, and 40 lattices, are plotted in Figure 5.4. For all three lattices the largest overlap values are found for the highest ratio Uft for which the correlated wave function (SCF plus first

CSF) has a negative total energy (see Figure 5.3 or Table 5.1).

The degrees of excitation for determinants of the first CSF are 0.889132

(for A = 8 ), 1.051460 (A = 16), 1.358648 (A = 24), 1.597718 (A = 32), and 1.661862

(for A = 40). In all lattices, the difference between a determinant of the first CSF and the SCF determinant can be described by the excitation from a single orbital.

Figure 5.5 shows the orbitals of maximum coincidence for the determinants of the first CSF and the SCF determinant in the 8-site lattice. To identify the SCF orbital that is replaced requires a different unitary transformation of all SCF orbitals for each lattice site. 96 24 sites 32 sites 40 sites 10

U/t=I2

I 10'-

22 U/t=8 U/t=8 ca

•3 TJ1 10 g

I O ,-4 10

121 1 1 16 1 1 20 1 U - ^ ’l

FIGURE 5.4 Overlap between different determinants of the first CSF for the one-dimensional Hubbard model at (n) = 0.75 for various values of C//i. The highest-lying curve corresponds to Ult = 12 for the 24-site lattice and to U/t = S (ov the 32- and 40-site lattices. These correspond to the largest U/t ratios for which the correlated wave functions have a negative total energy. The overlaps for the 16-site lattice (see Table 5.2) also reach a maximum at the largest U/t ratio that results in a negative total energy.

97 third maximum coincidence orbitals are different for the SCF determinant and, e.g., t|f

3 I

o

. first and second maximum coincidence orbitals are the same for the SCF determinant and y

3 o

FIGURE 5.5 Orbitals of maximum coincidence of the SCF determinant (dashed lines) and the determinants of the first CSF (solid lines) for the one­ dimensional, eight-site Hubbaxd model {U/t = 4). The excitation can always be presented as from one orbital in the SCF determinant, but each CSF de­ terminant requires a different basis set orientation.

98 The excitations described by the detenninajits of the first CSF have a dramatic effect on the probability of finding two electrons on the same site simultaneously.

Since the exclusion principle only allows electrons of opposite spin on the same site, the probability of finding two electrons with the same spin on a site is zero even in the single-particle description. For electrons with opposite spin the SCF determinant does not differentiate between orbitals, i.e.,the probability of finding an up-spin on one site and a down-spin on another site does not depend on which sites are considered. The up-down correlation function,

(cScscJ,Cs/), (5.5) is independent of js — for the mean-field result, see Figure 5.6.

Figure 5.6 shows that the up-down correlation function for electrons on the same site is significantly reduced by including the first CSF. With the on-site re­ pulsion U larger than zero, the system can reduce its energy by reducing the prob­ ability of having two electrons on the same site. The effect of changing the ratio

U/t on the correlation function is shown in Figure 5.7. The long-range correlation changes very little, but the short-range correlation is strongly affected hy U/t: A larger value oi Ult further suppresses the probability of finding two electrons on the same site.

Figure 5.8 shows the on-site correlation function’s dependence on Uji and the number of lattice sites. The edge effect, i.e., the change in the on-site double­ occupancy probability as A is varied, tends to zero in two limits: As the number of sites increases and as Uji goes to zero.

99 0.20

0.15

> 0.10

+ Z

0.05

0.00 0 1 2 3 5 6 7 84 1^-5’I

FIGURE 5.6 Up-down correlation function for the one-dimensional Hub­ bard model, 16 sites and l//t = 4, for free-electron (SCF) determinant alone and with the first, single-peaked density configuration state function. The first CSF enters the wave function with a negative coefficient, reducing the proba­ bility of finding two electrons simultaneously on the same site and so lowering the energy.

100 0.20 o U/t = 2 s U/t = 4 -o U/t= 8 -A U/t = 12 ^ U/t = 16 0.15

0.18 A U + 0.10 0.10

+ 0.17 o V 0.09 0.16 0.05 0.08

0.00 0 1 2 3 4 5 6 7 8 \s-s'\

FIGURE 5.7 Up-down correlation functions for the one-dimensional 16-site Hubbard lattice for varying U/t. Insets show more detail at |s — s^| = 0 and |s — s'l = 1. With increasing U the probability of finding two electrons on the same site decreases.

101 0 . 10--

0.09-

0.08-

0.07-

0.06-

32

number of sites

16 8

FIGURE 5.8 Up-down correlation function at |g — g'] = 0 for the wave function consisting of the SCF determinant and the first CSF for the one­ dimensional Hubbard model. As the on-site Coulomb repulsion U increeises, the probability of finding two electrons on the same site decreases. The effect of having a finite lattice is strongest for a large U, as U tends towards zero the SCF solution becomes a better approximation to the exact solution. At U = 0 the SCF solution, the free-electron result, is exact.

102 5.3 Higher excitations for the

one-dimensional Hubbard model

The wave function including the first configuration state function of the eight- site lattice Hubbard model,

8 ^ 1-H = cqV’O + Cl rl’i , (5.6) !=1 can be generalized to include higher excitations by including more CSF’s. For a

total of Ncsf CSF’s we have a wave function

■iVcsF—1 8 ^iVcsF=^O0O+ ■ (5.7) 0= 1 i= 1

The notation for the determinants of a CSF are chosen to reflect the order (o) of

the CSF, i.e., the order in which they enter the expansion, and the site (s) around which the determinant shows the largest difference to the SCF determinant. For the single-peaked-density CSF determinants the definition of s is the site with the

highest peak in the density. For higher-order CSF determinants the distinction of

which site shows the largest difference is sometimes less obvious, but does not affect our discussion here.

The first CSF contributes 87% of the exact correlation energy (see Table 5.3).

With all five CSF’s in the expansion 99% of the correlation energy is accounted

for. Unlike the atomic and molecular systems studied in the next chapter, the one-electron basis set used for the Hubbard model is complete, hence this is one

case where we are in fact comparing to the true correlation energy (from an exact

diagonalization calculation^®). The electron densities for determinants of the second

103 CSF (Figure 5.9) show a dip at one site, and this CSF enters the expansion with a

positive coeflBcient, leading to a further reduction in the on-site correlation function

(see Table 5.4).

N csf wave function energy ■^correlation

1+0 V’O -5.1569 0.0% 1 + 1 CQ^O + Cl Et -6.4683 86.5% 1+2 CO^O + ELl Co Et -6.5794 93.9%

1 + 3 CQ^O + ELi Co Et ^ 0 ^ -6.6449 98.2% 1 + 4 CO^O + Eo=l Co Et -6.6532 98.7% 1 + 5 CO’/’O + ELl Co Et -6.6637 99.4% exact®° -6.6722 100%

TABLE 5.3 Energies resulting from the successive addition of CSF’s for the one-dimensional Hubbard model with eight sites, (n) = 0.75, and 17/i = 4. The first CSF accounts for 87% of the total correlation energy, while the wave function with five CSF’s misses less than 1%.

The density of the determinants of the laat three CSF’s all show a similar structure with two peaks in the density. In terms of other variables these three

CSF’s differ:

(i) The degree of excitation for the third CSF is considerably smaller than those

of the fourth and fifth CSF’s (see Figure 5.10). 104 FIGURE 5.9 Electron density of representative CSF determinants for the one-dimensional, eight-site Hubbard model. The first CSF, representing an increased probability for two electrons on the same site, enters the expansion with a negative coefficient. The second CSF, representing a decreased proba­ bility for two electrons on the same site, enters the expansion with a positive coefficient. The remaining three CSF’s have similar structures in the electron density, but differ considerably in their orbitals. 105 (ii) The coefficients, shown in Table 5.5, are positive for the third and the fifth CSF

and negative for the fourth CSF.

(iii) Based on the number of orbitals in which the CSF’s differ from the reference

determinant (see Figure 5.11 and Table 5.6) CSF’s 3 and 4 are more similar to

each other than with CSF 5.

Thus, though the structure of the electron densities for determinants of the last three CSF’s appear similar, these CSF’s are otherwise very different.

SF’s 0 1 2 3 4 0 0.1406 0.1406 0.1406 0.1406 0.1406 1 0.0743 0.1741 0.1414 0.1353 0.1491 2 0.0647 0.1785 0.1422 0.1336 0.1518 3 0.0591 0.1811 0.1421 0.1332 0.1530 4 0.0580 0.1814 0.1423 0.1330 0.1536 5 0.0571 0.1819 0.1425 0.1329 0.1533

TABLE 5.4 Up-down correlation function for the 8-site Hubbard model as a function of the number of CSF’s included in the expansion. The probability for two electrons to be found on the same site decreases as more of the correlation is put into the wave function.

Table 5.5 show the coefficients for the various CSF’s with their dependence on the number of CSF’s included. All coefficients but for the SCF one ( c q ) increase

106 1.0

— oc = I 0.8 -- a = 2 - 3 ô ’s

& 0.6

0.4

0.2

0.0I I/. ■ i #_i ------1------1 . ■ I------— "r . .. #1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 n_

FIGURE 5.10 Scaled determinant overlap with the SCF determinant plotted against the degree of excitation for the 8-site one-dimensional Hubbard model. The first CSF’s determinants are the only ones that have a significant overlap (approximately 0.3) with the SCF determinant. The degree of excitation is very similar for the second and third CSF’s as well as for the last two CSF’s, though they differ considerably in other respects.

107 in magnitude at every step of the procedure — inclusion of more CSF’s and re­

optimization. The exception is in the inclusion of the fifth CSF, which upon re­

optimization has a larger coefficient in the wave function than the fourth CSF.

A cs F CO Cl C2 C3 C4 C5 1-hO 1.0000 1-t-l 0.9342 -0 .1 2 6 2 1 + 2 0.9258 -0 .1 3 1 8 0.0223 (reopt) 0.9250 -0.1323 0.0233 1 + 3 0.9199 -0.1358 0.0218 0.0174 (reopt) 0.9174 -0 .1 3 7 5 0.0225 0.0196 1 + 4 0.9167 -0 .1 3 7 7 0.0237 0.0201 -0.0052 (reopt) 0.9161 -0 .1 3 8 1 0.0241 0.0204 -0.0054 1 + 5 0.9161 -0 .1 3 8 3 0.0223 0.0202 -0.0055 0.0038 (reopt) 0.9151 -0 .1 3 9 0 0.0229 0.0200 -0.0058 0.0060

TA B LE 5.5 Coefficients for the successive expansion in CSF’s for the one­ dimensional Hubbard model with eight sites, (n) = 0.75, and Uft = 4 . The coefficient of the reference determinant decreases as more CSF’s are added and optimized. The coefficients of the CSF’s all increase as the correlation of the wave function is improved, with one minor exception (eg).

Figure 5.11 shows the orbitals of four typical determinants from the five CSF’s. rotated into maximum coincidence with the SCF determinant’s orbitals. Only the first CSF has determinants that differ from the SCF determinant by essentially one orbital — see also Table 5.6. A comparison of the orbitals in Figure 5.11 shows that 108 5 •S o

(N B 'ë o

B •e o

FIGURE 5.11 Orbitals of maximum coincidence of the SCF and typical correlating determinants for the one-dimensional, eight-site Hubbard model. The determinants shown here are the same as in Figure 5.9. The determinants in differ from the SCF determinant by only one orbital, the determinants in and differ by two orbitals, while the determinants in differ in all three orbitals from the SCF determinant.

109 none of the higher-excitation determinants can be constructed by combining lower- order determinants as outlined in Section 4.7: The maximum-coincidence basis set orientation of the reference determinant appears similar for CSF’s 1 and 3, but the orbitals that are excited into are very different. The application of (4.33) indeed does not produce determinants that lower the energy.

maximum coincidence overlaps CSF All A 22 A 33 ^ex 1 0.999907 0.999404 0.556123 0.889132 2 0.996027 0.448692 0.364805 2.380952 3 0.999658 0.482957 0.397052 2.240666 4 0.999999 0.019712 0.004042 3.952494 5 0.626397 0.519801 0.025979 3.655646

TABLE 5.6 Overlaps of the maximum coincidence orbitals of ID, 8-site Hubbard model CSF’s and their degrees of excitation, for C//< = 4 and (n) = 0.75. None of the overlaps are zero, indicating that these types of excitation cannot be seen in orthogonal-determinant methods.

Figure 5.10 shows that only the determinants of the first CSF have a large overlap with the reference determinant. However, the non-orthogonality between determinants is also important for the other CSF’s. The overlap between determi­ nants of the same CSF, e.g., is often not very large, but it is clearly not negligible.

Figure 5.12 summerizes preliminary results for the 16-site lattice. The second set of determinants shows a dip in the density similar to the ^ 2*^ of the eight-site lattice (Figure 5.9). 110 1 0 0 %

bû 80%

(16-site lattice)

c.x(-l)

0 16 32 48 64 80 number of determinants in expansion

FIG U RE 5.12 (a) Correlation energy for successively added determinants and (b) coefficients in the expansion for the 16-site one-dimensional lattice with anti-periodic boundary conditions and (n) = 0.75. The first set, 53s=l accounts for 75% of the correlation energy.®^ Further optimized sets each add approximately 4% of the correlation energy. The electron density is shown for one determinzmt in each set, the other 15 determinants of each set are obtained by translation.

Ill 5.4 Degenerate reference

determinants: Multi-reference cases

For lattices with 16 and 32 sites the periodic boundary conditions were replaced with anti-periodic boundary conditions in Section 5.2. This was to avoid having to deal with the two degenerate reference determinants that result for an even number of electron pairs. Here we return to the 16-site lattice and use periodic boundary conditions for six electron pairs ((n) = 0.75).

The reference state for an open-shell system requires more than a single de­ terminant, in the one-dimensional Hubbard model two determinants and are necessary. The two determinants differ in the orbital with the highest one- particle energy, each determinant contains one of the degenerate orbitals shown at the top of the left column (“periodic”) in Figure 5.1. The electron density of the two-determinant reference state.

(5.8) is uniform in space, and the up-down correlation function (Figure 5.13) is without structure. Figure 5.13 also shows the up-down correlation function for one of the degenerate determinants alone: The long-range correlation shows the correct behavior, while the short-range correlation is worse than the mean-field solution

(with both determinants).

The electron density for either of the two reference determinants, displayed in

Figure 5.14, is clearly not uniform. Also displayed are the electron densities for determinants of the first four CSF’s. As in the single-determinant reference cases each CSF consists of sixteen determinants that differ from each other by a linear 112 0.16

0.14

+

+ 0.12

0.10

0.08

\s-s’\

FIGURE 5.13 Up-down correlation functions for the periodic 16-site Hub­ bard model. The “SCF” solution requires two determinants, as can be seen from the fact that only the two-determinant reference state shows a flat cor­ relation function. Compared to the correct reference state, has a better long-range and a worse short-range character. The inclusion of the first CSF reduces the probability of finding two electrons on the same site.

113 FIGURE 5.14 Electron density of representative CSF determinants for the one-dimensional, periodic, sixteen-site Hubbard model. The density of and V’o combined is translationally invariant, as that of the correct reference state must be.

114 translation. The density is no longer peaked at a single site in the first CSF, as it was for the single-reference case: The density now shows two peaks.

As in the single-determinant reference case, the difference between a determi­ nant of the first CSF can be attributed to mainly a single orbital. Whereas in the single-reference case the remaining maocimum-coincidence orbitals of the SCF and the CSF determinants are nearly identical, in the two-reference case a second orbital also is changed. Figure 5.15 shows the orbitals of maximum coincidence for a determinant of the first CSF with the two reference determinants and

wave function energy energy(reoptimized)

( 4 '* + 4 ^ ’) - 9.5786 —

( 4 ’* + 4^*) + '= 1Z. -11.4997 —

co(4"+4’')+zLiA ,Zi4" -11.7462 -11.7731

co(4‘>+4'>)+ELi«^Ei4‘' -11.8930 —

co(4'’+ 4”)+ELi<^Zi4*’ -11.9782 —

TA B LE 5.7 Energies for the successive addition of CSF’s for the one­ dimensional, periodic Hubbard model with sixteen sites and U/t = 4. Only for two CSF’s was the entire expansion re-optimized.

In the single-reference case, a determinant of the first CSF can be described as a somewhat localized orbital in the reference determinant being replaced by 115 (1) (2) mapped onto ij/g mapped onto Vjfg

FIGURE 5.15 Orbitals of maximum coincidence of the two SCF (dotted lines) and typical correlating determinant (solid lines) for the one-dimensional, periodic, sixteen-site Hubbard model. The determinant here is the same (8) aa in Figure 5.13. The orbitals overlap somewhat better with those of (2) than with those of

1 1 6 an orbital ^3 that is more strongly peaked at one site in the localized region of

<^3. The multi-reference case presented here can be understood by generalizing this understanding of the single-reference case. The reference is now two determinants that differ in a very specific way: One determinant contains the cosine of wavelength

A = 16/3 as a single-particle orbital, the other contains the sine with the same wavelength (see Figure 5.1). Another way to put this is that the two determinants differ from each other by a translation of A/4 = 1/4 x 16/3 = 4/3 lattice spacings, i.e., somewhat more than one lattice spacing. Thus, if a similar excitation as in the single-reference case is to appear here, we must consider two somewhat-localized orbitals and replace them by one orbital that is peaked in the localization region.

However, we now have two localized regions (approximately one lattice spacing apart), and so it is not surprising that determinants of the first CSF show two density peaks one lattice spacing apart.

In Table 5.8, the results from the periodic 16-site lattice are compared with the results from the anti-periodic lattice in Section 5.2. The reference-state energy of the periodic lattice lies 0.023t above that of the anti-periodic lattice. This energy ordering agrees with the structure of the two reference states: In both cases the up-down correlation function is without structure, i.e., the potential energy of both reference states is the same. For the kinetic energy Figure 5.1 shows the ordering of the one-particle orbitals, and the kinetic energy of the highest-lying one-particle orbital is higher in the periodic case. For the correlated wave functions the difference in the energy is found to increase approximately linearly with U/t; the small nonlinearity is due to the finite size of the lattices.

117 rpfprpncp

U/t e(4"+4^') 2 -0.8760 -0.8799 -0.9163 4 -0.5909 -0.5987 -0.7187 8 -0.0205 -0.0362 -0.3604 12 +0.5498 +0.5263 -0.0158 16 + 1.1201 +1.0888 +0.3246 anti-periodic results from Table 5.1 2 -0.9027 -0.9493 4 -0.6214 -0.7644 8 -0.0589 -0.4327 12 +0.5036 -0.1151 16 +1.0661 +0.1982

TA BLE 5.8 Energies per site of the periodic 16-site Hubbard model with (n) = 0.75. The energy per site of the anti-periodic reference state lies 0.023 t lower than the energy per site of the periodic reference state for all values of U/t. For the correlated wave function, the difference between periodic and anti-periodic energy per site increases approximately linearly with U/t.

118 5.5 Two-dimensional Hubbard model

In the two-dimensional Hubbard model we find results similar to those in the one-dimensional case. Nor THC oin was applied to lattices of 16 (4-by-4) and 36

(6-by- 6) sites with five and thirteen electron pairs, respectively — fillings at which there is a unique reference determinant. Again, the first CSF that enters to correlate the electrons in the SCF determinant is made up by determinants with densities peaJced at a given site (see Figure 5.16).

While for the one-dimensional Hubbard model these CSF’s were optimized, for the 4-by-4 case presented here they are those resulting from letting North C oin find and optimize a number of determinants. The resulting expansion showed a hierarchy in the coefficients of the determinants, and inspection of the electron density provided insight as to what the hierarchy is due. By using the determi­ nants from a given level in the hierarchy and symmetrically translating them the energetically best one was chosen. No optimization was performed. For the 6-by -6 case the determinants of the first CSF were optimized.

The effect of the first CSF on the correlation function is similar to the ef­ fect found for the one-dimensional lattices. Figure 5.17 shows how in the two- dimensional case the on-site repulsion results in a smaller probability of finding the two electrons on the same site. The two electrons avoid sitting on the same site due to the on-site repulsive interaction U. This increases the correlation function for the neighboring sites. Figure 5.17 shows an additional increase at [2,2] for the

4-by-4 lattice. This increase is an effect of the lattice size, for the 6-by -6 lattice the increase is less severe (point [3,3] in Figure 5.20). 119 FIG U RE 5.16 Density of the SCF determinant (top) and one of 16 determi­ nants in the first CSF (bottom) for the 4-by-4 Hubbard model at (n) =5/16 and with = 4. As in the one-dimensional case, the most important set of correlating determinaiits has a peak in the density at one site.

120 0.15

0.10

A o + O CJ +

0 . 0 5

0.00 0 [1,0] [1,1] [2,0] [2,1] [2,2] ‘Is-s’l”

FIGURE 5.17 Correlation functions for the 4-by-4 Hubbard model at (n) = 5/16 and with U/t = A. The effect of adding CSF’s is to decrease the probabil­ ity of finding two electrons on the same site. The increased correlation at [2,2] is an artificial effect due to the small lattice size (the sum of the correlation function must add up to the same value for correlated and uncorrelated wave functions because of particle number conservation).

121 FIG U RE 5.18 Densities of the basic determinants of the second CSF for the 4-by-4 Hubbard model at (n) = 5/16 and with U/t = 4. These determinants were chosen from an initial expansion and used to form CSF’s by translation and rotation. They were optimized without enforcing symmetry.

122 wave function energy per site %^corr 00 -1.1094 CO0O + Cl '£ i -1.2053 83.8

CQ0O + Z L l Co E i -1.2185 95.4

exact^2 -1.2238 100.0

TABLE 5.9 Energies of SCF and SCF plus first CSF states for the 4-by-4 Hubbard model at (n) = 5/16 and with U /t =4. The number of determinants in a CSF is 16, so the final expansion contains 155 determinants. These deter­ minants were chosen from an optimized expansion where symmetry was not enforced, i.e., re-optimization with enforced symmetry would lead to some­ what improved energies.

Figure 5.18 shows the electron densities of the four basic determinants found

for the second CSF. All densities show two peaks, and can be translated in the two

spatial directions (each leading to a total of 16 determinants) and rotated by tt/ 2

and translated (another 16 determinants). Of the possible ways in which two peaks

can be arranged on a square lattice, the diagonally most-distant arrangement is

not present. While this may be due to the lack of a systematic optimization with all symmetrically equivalent determinants present, an inspection of the up-down

correlation function (Figure 5.17) indicates otherwise: The increased correlation

function at [2,2] is due to the size of the lattice, not the expansion. Including determinants analogous to those in Figure 5.18 but with density peaks halfway across the diagonal would reduce the correlation peak at [2,2] (if they enter the wave function with a negative sign). For larger lattices, where the point [2,2] is no

longer halfway across the diagonal, such determinants will be important. 123 0.6

0.4 -

0.2 -

0.6

0.4

0. 2 -

FIGURE 5.19 Densities of the SCF determinant (top) and one determinant of the first CSF (bottom) for the 6-by-6 Hubbard model with (n) = 13/36 and at Ujt = 4.

124 0 [1,1] [2,1] [2,2][3,1][3,2] [3,3] 0.15

A u + u 0.10 u o V

0.05 0 [1,0][1,1] [2,0] [3,0] [3,3] “ Is-s’l”

F IG U R E 5.20 Correlation functions for the two-dimensional, 6-by-6 Hub­ bard model. The on-site correlation is reduced for the correlated wave func­ tion. The correlation function peak that appears halfway across the diagonal (at [3,3]) is due to the finite lattice size, but is less pronounced than the analogous peak for the 4-by-4 lattice (at [2,2] in Figure 5.17).

125 1.0

—...... OC = 1

0.8 - oc = 2 3 ô’s • 4-by-4 lattice 06-by-6 lattice

CQCL 0 . 6 5 > o \ T3 \ (U \ "a , \ \ \ ^ 0.4 \ \ \ \ \ \ ;. ; / ;/ 0.2 \\\

0.0 ^ ' ' 0 ' --- 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 n _

FIGURE 5.21 Scaled determinant overlap with the SCF determinant plotted against the degree of excitation for the two-dimensional Hubbard model. The determinants in the first CSF have an overlap of nearly zero with the reference state — but have non-zero overlaps with each other.

126 The 4-by-4 lattice is rather small. On the other hand the 6-by-6 is already at

the basis set size limits of our current code’s capabilities. Nonetheless, the first CSF

in this system was obtained. The filling in this case was 13/36. The density is again

peaked at a single site (Figure 5.19 shows the SCF density as well as the density of one of the 36 single-peaked-density determinants). The effect of including the first CSF on the correlation function is similar to that of the 4-by-4 system, and is shown in Figure 5.20.

wave function energy per site %F'corr —1.0340 COV'D + Cl -1.1272 65.6

CPQMC®^ -1.1761 ±0.0006 100.0

TA B LE 5.10 Energies of SCF and SCF plus first CSF states for the 6-by-6 Hubbard model at (n) = 13/36 and with Ujt = 4. Compared ot the 4-by- 4 lattice result, the first CSF in this lattice accounts for significantly less of the correlation energy. As the “FCI” energy we take the constrained-path quantum Monte Carlo (CPQMC) result of Zhang et al.^^

127 5.6 Hubbard model with mirror symmetry

The results for the Hubbard model in the previous sections were all calculated

without enforcing symmetry. In this section preliminary results with enforced sym­

metry are presented.

The mirror symmetry is enforced at the orbital level by forcing all orbitals of each determinant to be either symmetric or antisymmetric around a reference site.

The SCF determinant for the 8-site lattice with three electron pairs, e.g., has

(i) in the occupied space two cosines and one sine, and

(ii) in the virtual space three cosines and two sine.

Excited determinants are then formed by allowing only cosines to mix among them­ selves, and sines to mix with other sines.

Figure 5.22 shows the resulting maximum coincidence orbitals of the SCF deter­

minant with a determinant of the first set. The energy does not change significantly

with enforced symmetry, however, the mirror symmetry does maJce the calculations cleaner and thus more reliable for physical interpretation. Allowing the sines to mix, e.g., is surprisingly important for the total electron density, though the amount of

mixing among the sines is small (see Figure 5.22).

128 no symmtery symmetry enforced any mixing only cosines sines, cosines

(maximum coincidence) orbital in SCF determinant orbitals of determinant in first set resulting density

FIGURE 5.22 Orbitals and densities for the 8-site Hubbard model with and without enforced mirror symmetry. The fourth site from the left is the reference around which symmetry are defined. On the left are the orbitals and density as optimized without enforcing mirror symmetry. In the middle and on the right are the symmetry-enforced cases. When only the symmetric orbitals (cosines) are allowed to mix, the peak in the density is not very pronounced. Once the antisymmetric orbitals (sines) are allowed to mix, the occupied sine orbital changes very little, but this small change has a dramatic effect on the optimization of the symmetric orbital and the resulting density.

129 CHAPTER 6 NON-ORTHOGONAL SLATER DETERMINANTS AND THEIR CONFIGURATION INTERACTION: APPLICATION TO SMALL MOLECULES AND ATOMS

The cure for boredom is curiosity. There is no cure for curiosity.

- Ellen Parr

The application of N o r t h C o iN to atoms and small molecules presented here

illustrates the concepts discussed in chapters 2 and 4. For two-electron systems

non-orthogonal determinants are shown to form an expansion that is equivalent

to a traditional-CI expansion using natural orbitals. For systems with more then

two electrons, non-orthogonality between the determinants is shown to significantly

reduce the length of the expansion.

130 6.1 The helium atom He

With only two electrons, the helium atom is a simple example of correlation.

In an aug-cc-pVDZ basis (3s2p) there are 21 symmetry-allowed determinants. FCI of these 21 determinants results in a correlation energy of 33.844 millihaxtree. As mentioned in section 1.4, Shull and Lowdin showed for Helium that by transforming a set of Laguerre functions into natural orbitals, the expansion of the wave function can be written in terms of doubly-occupied spatial orbitals.T he aug-cc-pVDZ basis for He consists of nine orthogonal orbitals. In accordance with Shull and

Lowdin’s result, the FCI expansion in NO’s would require nine determinants.

symmetry fillings FCI expansion in SCF orbitals

s 1.9834549 0.0020763 0.0065739 p 0.0033504 0.0045444 FCI expansion in natural orbitals

s 1.9834867 0.0085969 0.0000216 p 0.0077328 0.0001620

TA BLE 6.1 Orbital fillings of the FCI wave function for He in a aug-cc- pVDZ basis using SCF and natural orbitals. Weight in excitation level 1 in SCF MG’s is 3.38499 • 10“ ^, while in NO s it is zero.

In Table 6.1 the fillings of the orbitals for the FCI expansion are shown. The filling for the Is orbital does not change significantly with the rotation into NO’s.

131 For the other s and the p orbitals the change is large, the second shell of correlating

NO’s is almost negligible.

In Figure 6.1 two expansions in non-orthogonal determinants are shown. With only nine determinants, North C oin optimizes the orbitals in each determinant with the result that the linear combinations reflect the fillings in the FCI wave function shown in Table 6.1. North C oin optimizes the one-particle orbitals to find the NO’s necessary to diagonalize the expansion, and converges with nine determinants to a correlation energy of 33.844 millihartree, the exact difference to

FCI is 0.92 • IQ-^.

Figure 6.2 shows the overlap of the correlating determinants with the SCF determinant plotted against their degree of excitation. In NO’s the eight correlating determinants would all have zero overlap and a degree of excitation of 2. In our expansion two determinants have a degree of excitation smaller than 2, in particular one at Ug* % 0.8 (determinant “1” in Figure 6.1). The reason for this “imperfection” is due to the optimization of one determinant at a time, rather than a simultaneous optimization of the orbitals in all determinants. The rotation into NO’s affects all orbitals, including those occupied in the SCF determinant. To account for this in non-orthogonal determinants, optimization of the first correlating determinant mixes the Is orbital with the other s-type orbitals, which leads to the ng* ~ 0.8.

132 C3c C A 2 s T3

0

Px fy Px HFMOs

F IG U R E 6.1 Excitation graphs for He in SCF and natural orbitals in an aug- cc-pVDZ basis set. In both cases the four correlating determinants account for 99% of the correlation energy. On the left the wave function is optimized by using linear combinations of SCF orbitals to form MO s that best correlate the SCF determinant. This “simulates” the construction of NO’s.

133 1.0

a= I 0.8 a = 2 3 5’s

D. 0.6

0.4

SCF determinant \ Double excitations \ (with respect to yg) 0.2

0.0 ^ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 n

FIGURE 6.2 Scaled determinant overlap with the SCF determinant plotted against the degree of excitation for helium in the aug-pVDZ basis. The SCF determinant has unit overlap with itself, and 7%,% = 0. A expansion in orthog­ onal determinants with the basis set in natural orbitals would have all but the SCF determinant at = 2, = 0 in the NO’s basis. Here we are dealing with non-orthogonal determinants, where the expansion can have more than one basis set orientation, and so the ‘pure’ solution is not reproduced exactly.

134 6.2 The hydrogen dimer Hg and

the helium hydride cation HeH^

H2 and HeH+ are two-electron systems; their ground states near their equi­ librium bond lengths are singlets. As the two molecules dissociate, however, they show very different behavior: The hydrogen dimer dissociates into two hydrogen atoms. I.e., two doublets, while the helium hydride cation’s separated fragments are a helium atom (a singlet state) and a cation. Increasing the nuclear separation thus has very different consequences for the two dimers and the correlation of the electrons: Left-right correlation, e.g., becomes increasingly important for H 2 but negligible for HeH"*". The HeH+ molecule is a heteronuclear diatomic molecule and, since the ionization energy for He (24.6 eV) is considerably larger than that for H

(13.6 eV), as the two nuclei are pulled apart both electrons follow the He nucleus, and the ground state remains a singlet even in the supermolecule system.

In Figure 6.3 the potential energy surfaces for the singlet and the triplet states of the H 2 are shown. The curves are shown for the uncorrelated (SCF) determinant and for the correlated, FCI wave function in an aug-cc-pVDZ basis set,®"^’® which is a cc-pVDZ basis set ((4slp)—>-[2slp]) augmented with one s- and one p-type diffuse primitive. For the dimer this translates into five bonding and five anti-bonding c-type MO’s, and two bonding and two anti-bonding MO’s for each tt symmetry, resulting in a total of 18 MO’s. Figure 6.4 shows how with increasing bond length the SCF, the FCI, and the correlation energy for HeH"*” all tend towards their respective values for a helium atom in the same basis (aug-cc-pVDZ). The total number of MO’s is the same as for H 2, but we can no longer use symmetry to distinguish between bonding and anti-bonding orbitals. 135 -0.5 G----- 0 singlet

- 0.6 V----- 9 triplet w "0 7

I - 0.8 = -0.9

- 1.0

-0.5 ♦ singlet - 0.6 -V triplet -0.7 c S -0.8 u Ü- -0.9

- 1.0

1.0 2.0 3.0 bond length (bohr)

FIGURE 6.3 Hg SCF and FCI energies in the aug-cc-pVDZ basis set for the singlet and triplet states. All energies are in hartrees. In its current implementation, North C oin is restricted to singlets.

136 38

JZI E ^ o fH eH I 36 u 34 I ^ 0 0 . of He 32 1u

-2.7 s e(3 £crTT of He I - 2.8 of HeH g of He u. u

1 -2.9 g £ ~ , of HeH

1.0 2.0 3.0 4.0 bond length (bohr)

FIGURE 6.4 Correlation, SCF, and FCI energy for the helium hydride cation in the aug-cc-pVDZ basis. He"^ has a larger electron affinity than The molecule dissociates into a helium atom and a ion, hence all energies tend towards their respective values for He.

137 In Figure 6.5 the excitation graphs are shown for nine non-orthogonal determi­

nants at selected nuclear separations of H 2. The determinants are ordered according

to their coefficients in the expansion, the lowest (the SCF determinant) being the

most important. With increasing bond length, the excitations into c-type orbitals

(bonding and anti-bonding) become increasingly more important than excitations

into TT-type orbitals. In terms of the qualitative correlation types (see Section 1.1),

left-right correlation becomes increasingly important as the dimer dissociates.

Figure 6.6 shows the results for nine non-orthogonal determinants for and

HeH+, compared with the FCI (66 and 132 determinants, respectively) values. The

difference is well within chemical accuracy (1 kcal/mol, or 1.6 millihaxtree). For

HeH"*" North C oin does better.

The shape of the difference between the non-orthogonal result and the FCI en­

ergies is similar to an observation of Murphy and Messmer,"^" who compared GVB-

RCI energies with CASSCF energies for the nitrogen dimer.^ While in our example

there are only two electrons forced to couple as a singlet, we are dealing with a

similar problem of not having enough degrees of freedom in the bond-breaking re­

gion. On either side of the bond-breaking region, nine non-orthogonal determinants

suffice to optimize the orbitals. In the bond-breaking regions, nine determinants

are not sufficient (compared to 18 determinants of a FCI wave function in NO’s).

Nine non-orthogonal determinants are closer to the FCI result for HeH'*’, where the

bond breaks without spatially separating electrons.

^ GVB-RCI (restricted Cl from a generalized valence bond reference) is an alternative to CASSCF for generating reference states for perturbation theory calculations. A CASSCF wave function is the FCI wave function in the CAS, and GVB-RCI is an approximation to this FCI. Consequently, CASSCF has more degrees of freedom in optimizing the orbitals, which is especially important in regions where there is a significant re-coupling of the spins (bond-breaking regions). 138 /? =1.0 /?= 1.4 R = l .l

R = 2.6 R = 3.2 R = 3.6

m . I 7U^ 7C O 7C, 7ly CT 7I_ K molecular orbitals (bonding I antibonding)

FIGURE 6.5 Excitation graphs of the nine non-orthogonal determinants used for H 2 in the aug-cc-pVDZ basis at various bond lengths R. The deter­ minants are ordered according to their coefficient in the wave function, “0” is the HF determinant and always haa the largest coefficient. The first two sets of 5 orbitals are cr bonding and antibonding MO’s, the remaining four pairs are 7r-type MO’s. With increasing bond length, determinants involving excitations into the r MO’s become less important. 139 0.25 I« 0.20 i 0.15 1 I 0.10 g" 0.05

99.9 99.8

99.7

J 99.6 99.5 99.4

99.3

1.0 1.5 2.0 2.5 3.0 3.5 bond length (bohr)

FIGURE 6.6 Comparison of results for nine optimized non-orthogonal deter­ minants with the FCI results (132 determinants) for HeH'*’ in the aug-cc-pVDZ basis. For comparison the results of Hg in the same basis are also shown as empty diamonds. Though the differences with the FCI results are well within chemical accuracy, the NORTH COIN results appear to do systematically worse in the bond-breaking region.

140 6.3 The beryllium atom Be and the Beg supermolecule

The beryllium atom’s four electrons occupy the Is and 2s orbitals, and axe spatially well separated. This makes correlating the two pairs close to independent.

Methods that are based on separated pairs perform well in this system. The 2s orbital is nearly degenerate with the 2p orbitals, malcing a multi-reference treatment preferred for Be.

The basis used in this section is that of Dykstra et al. without the d function.

This rather small basis is used here only to illustrate the construction of higher excitations from lower-excitation determinants as outlined in Section 4.7. The SCF energy, i.e., the energy of the determinant 'Pq = |(ls)^(2s)^| is —14.573 hartree in the (7s3p) basis.

Table 6.2 contrasts the energy of an expansion in 30 optimized non-orthogonal determinants with that of the FCI wave function and the CISD result. If the electrons in the Is orbital could be correlated independently of the electrons in the

2s orbital, then according to equation (4.3) the FCI result would be obtained with

30 determinants. The resulting energy lies approximately 0.4 millihartree above

the FCI energy, indicating that the electron pairs are not completely independent.

Compared to CISD, non-orthogonal determinants can account for more correlation energy with nearly an order of magnitude fewer determinants.

In Figure 6.7 the excitation graphs for the first nine (non-orthogonal) determi­

nants of a wave function are shown — both in the canonical atomic orbitals (AO’s)

and in natural orbitals (NO’s). The energy is the same (within chemical accuracy)

for both expansions. In the canonical-AO expansion, Nor THCoin had to first ‘cre- 141 method E^orr % ^con- (FCI) CISD in N O V 59.969 97.5 30 non-orthogonal determinants 61.125 99.4 ______FCI______61.501______100.0

* 267 orthogonal determinants

TABLE 6.2 Correlation energies (in millihartree) for Be in a (7s3p) basis set. According to equation (4.3), 30 optimized non-orthogonal determinants would reproduce the FCI result — if the electrons in Be could be separated into two independent pairs. While the 30 determinants do fall short of the FCI correlation energy by ~ 0.4 millihartree, the result is better than the CISD energy by 1 millihartree. The improvement over CISD is attributed to (i) some of the non-orthogonal determinants having degrees of excitation larger than 2, and (ii) the freedom to re-orient the basis set for each non-orthogonal determinant. While some of the non-orthogonal determinants are excitations out of either the Is or the 2s orbital, excitations out of orbitals that are linear combinations of the Is and 2s orbitals are also present.

142 Px Py Pz SCF AO’s NO’s

FIGURE 6.7 Excitation graphs for Be in SCF and natural orbitals in a (7s3p) basis set. On the left the wave function is optimized by using linear combinations of SCF orbitals to form AO’s that best correlate the SCF deter­ minant. This “simulates” the construction of NO’s. Determinants 6, 7, and 8 are higher excitations, i.e., appear to be a superposition of determinant 5 with 1, 2, and 3.

143 ate’ the best orbitals from the AO’s. These new orbitals axe very close to NO’s,

as can be seen by comparing the two sets of excitation graphs, especially the first

five: What required a linear combination of several AO’s in the canonical case looks

much more like a traditional double excitation in NO’s.

determinant(s) 0 1,2,3 4 5 6,7,8 coefficient(s) 0.9510 -0.1763 -0.0417 -0.0243 0.0045

TABLE 6.3 CoefBcients of the determinants for the beryllium Cl expansion shown in Figure 6.7. The determinants 6, 7, and 8 can be viewed as the excitations of determinants 1, 2, and 3 combined with the excitation in 5. The product of the coefficients for determinants 1, 2, and 3 with that of determinant 5 is approximately equal to the coefficients of determinants 6, 7, and 8.

Closer inspection of the determinants 6, 7, and 8 in Figure 6.7 reveals that the

Is electron is excited in the same way as in determinant 5, while the 2s electron is

excited into one of the 2p orbitals in the same way in either determinant 1,2, or 3.

In Table 6.3 the coefficients for these determinants are shown. The product of the

coefficients for two of the determinants that appear to be combined is approximately equal to the coefficients of the determinants 6, 7, and 8,

-0.1763 X (-0.0243) = 0.0043 % 0.0045. (6.1)

This observation, that the coefficients of a quadruple excitation can be estimated by

the product of the coefficients of the two corresponding doubly excited configura­

tions, was made by Sinanoglu for an expansion in orthogonal Slater determinants.®®

144 Instead of finding the apparent superposition of lower excitations we can at­ tempt to construct them as outlined in Section 4.7. Rather than using the auto­ matic generation of determinants by combining all possible pairs of determinants in the expansion, in this example the concept is illustrated by specifically choosing pairs. This example is done for the berylUum supermolecule system. The goal is to achieve effective size-consistency by combining pairs of determinants that describe an excitation on one atom each.

Table 6.4 shows the energy of various Nor THC oin expansions for the beryllium atom and for the beryllium supermolecule. The notation []® indicates the formation of aU determinants that combine at most one excitation on each atom, i.e., the use of equation (4.33). These higher excitations were generated using the reference determinant with localized orbitals. The generation with respect to the localized reference determinant is required — attempts to use the reference determinant in canonical form lead neither to higher excitations nor to improvements in the energy.

In all cases effective size consistency is achieved within 0.1 millihartree per atom.

The orbital coeflBcients of the atomic determinants are shown in Figure 6.8. The two excitations 2s —* 3s and Is —*• 3s' result in two non-orthogonal determinants:

The two orbitals 3s and 3s' are two different, non-orthogonal linear combinations of s-type AO’s, i.e., each excitation requires a different basis set orientation.

145 system wave function energy

ISe 4%, + I : : 4%%' - 43.54

Be2 00 + T,A E r - 79.85

^Q+ Eyl E z 0 % - 86.88 per atom: - 43.44

Be - 56.84

Be2 ’t’i + Ea (Ex -104.91

00 + E „ ( E x -113.57 per atom: — 56.79 3s I j.Zs' f,3s" Be (»ô+ExV'lr+4:+v.!:'+V'g - 58.43

Be2 v-i+ e a ( E x v - % + -107.98

■116.77 00 + E a ( e x 4 $ + '/'If+ ;j per atom: - 58.39

TABLE 6.4 Energies of Be and Be 2 supermolecule wave functions. All ener­ gies are given in millihartrees; coefficients are omitted. To obtain twice the en­ ergy of a given atomic wave function, the wave function for the supermolecule must include more than just those determinants with an excitation on one of the supermolecule’s atoms: The inclusion of ail determinants that describe the product of one double excitation on each atom, symbolically indicated by [ ]®, results in the desired energy per atom.

146 I s ^ S s ' If—>4f

2 j — > 2 / 7 , 2j—>2/7y

à

' ' ' ' ' ' I ' ' ' ' ' I ' '

SCF’ determinant

2s’ orbital

I5 ’ orbital

s Px Py Pz

FIGURE 6 . 8 Orbital coefficients for the first seven Be determinants. The reference determinant is close to the SCF determinant, but the orbitals have been combined linearly to some degree, hence the prime. The first three cor­ relating determinants are those where the 2s orbital has been excited into one of the 2p orbitals. In the top row are excitations into s-type orbitals, these three determinants all have non-zero overlaps with each other.

147 6.4 The beryllium dimer Beg

The beryllium dimer should not be bound — according to qualitative molecular orbital theory: The Is and the 2s atomic orbitals (AO’s) combine to form bonding and anti-bonding <7-type molecular orbitals (MO’s). Beg has eight electrons with which to occupy these MO’s, and so ail four MO’s are filled. An equal number of bonding and anti-bonding orbitals leads to an unbound state in qualitative MO theory, and the SCF solution is indeed unbound. The inclusion of electron-electron correlation, however, leads to a weak bond with a well depth of approximately

2 kcal/mol near 4.8 bohr, and a shallower van der Waals minimum around 9.4 bohr..

There has been one experimental study of Beg,®" as well as a plethora of theoretical studies.®®'®^'®^,70,71,72.73,74

The basis used in this section is that of Dykstra et al. ®® with the d functions; in this basis FCI results are available.®® The core is kept frozen, leaving 40 molecular orbitals. This is at the limit of our code’s current capability, hence the results pre­ sented here are not fully converged. Figure 6.9 shows the potential energy curves for the SCF determinant, the FCI wave function,®® and the result of expansions in

21 non-orthogonal determinants. Also shown are the results for the supermolecule.

The North C oin results do show a minimum at the correct bond length by account­ ing for ~ 90% of the correlation energy. However, the result for the supermolecule is considerable better (97% of the correlation energy), i.e., the overall shape of the potential energy curve calculated with non-orthogonal determinants is wrong.

Though a coupled-cluster treatment with up to doubles gives the exact FCI energy for the supermolecule, the same treatment near equilibrium only gives the 148 - — .j - - ! -- — 1 -29.13

.. . ..O...... o -29.14 o —..o SCF

1

I # 1+20 determinants e -29.22 I ^ F C I 3 Q -29.23

o

-29.24

. 0.04.75 5.004.50 5.25 100 nuclear separation (bohr)

FIGURE 6.9 Potential energy curves for Beg, SCF and FCI in comparison with 1+20 non-orthogonal determinants. North C oiN does significantly bet­ ter for the supermolecule, resulting in a non-bonding potential energy curve, though there is a local minimum at the correct nuclear separation.

149 shallow van der Waals minimumThe bond requires higher excitations that are not simple products of single and double excitations: The inclusion of triple ex­ citations approximated by their linear contribution alone, i.e., no terms with n > 1, does not produce the minimum at short distances.®^ To correctly describe the bond the connected quadruple excitations are required.Our results support the necessity of linked higher excitations: Attempts to generate higher excitations for Beg by combining lower-excitation determinants improves the energy on the order of tens of microhartrees — much smaller thaxi in other systems. The degrees of excitation of many non-orthogonal determinants are considerably larger than 2 at all bond lengths (see Figure 6.10), i.e., higher excitations are clearly present.

The inclusion of the d-type orbitals is necessary: The Beg bond cannot be de­ scribed using only s and p-type AO’s."^ While the d-type AO’s do contribute to the determinants optimized by NORTH COIN, we find that the MO’s which result from the dxy and dj.2_y2 atomic orbitals do not participate in any of the determinants.

Thus it would appear that the contribution from the d-type AO’s is not for angular, but rather left-right or in-out correlation.

150 1.0 "T" -T" "T“ /?=4.50bcAr...... a = 1 a = 2 0.5 3 8’s 0.0 a ! m ^ . . ■— .—m 1.0 ...... ^ R=4-J5 bohr. x : ' .... a = 2 0.5 / . X 3 8’s Xi_ 0.0 g- 1-0 R=5.00 bohr a = 1 I xC X '* o l = 2 o 0.5 "u %. 3 8’s g 0.0

R=5.25y bohr / / / . /.

superaïolecule..

FIG U RE 6.10 Scaled determinant overlap with the SCF determinaxit plotted against the degree of excitation for Beg. The degree of excitation appears quite different for the separated atoms than for the molecule in the bonding region. The determinants of the supermolecule system are 12 doubles (or nearly such) on one of the two atoms. The remaining 8 determinants are quadruples with one excitation on each atom, i.e., the most important of the first twelve determinants.

151 6.5 The water molecule H 2O

Calculations on the water molecule are done in a double zeta basis ((9s, 5p) —^

[4s, 2p] on oxygen, (4s) [2s] on hydrogen)."® Correlating all ten electrons in this

rather small basis (14 molecular orbitals) allows a more systematic look at the effect of localization. We consider three different ways of letting North C oin choose the energetically most favorable initial excitation of each new determinant,

(i) from the SCF determinant incanonical molecular orbitals (“CMC”),

(ii) from the SCF determinant inlocalized molecular orbitals (“LMO”), and

(iii) from the SCF determinant in either type of MO (“CMO/LMO”).

Once the initicd excitation is chosen, the determinant is optimized. With the new

determinant in the expansion, it can be energetically favorable to re-optimize pre­

vious determinants. We consider two possibilities,

(i) re-optimizing all determinants in the expansion (“reopt-all”), and

(ii) only re-optimizing the last two determinants (“reopt- 2”).

In Figure 6.11 the correlation energy is shown as a function of the number of

determinants in the expansion. Choosing which initial-excitation determinant to

keep and optimize is clearly not only a question of how much energy is gained by

the initial excitation. The excitation from a localized orbital is only energetically

favorable once the initialized determinant is optimized; forcing North C oin to take

this route is clearly favorable when the determinants are all optimized in every step.

When previous determinants are not all re-optimized, the effect is somewhat

different: For the two cases where the initial excitation is chosen from either the 152 100%

80% -

I 60% c I v -v L M O - reopt-2 I 40% ■ 0—0 LMO/CMO - reopt-2 8 0 - 0 CMO - reopt-2 LMO - reopt-all 20% CMO/LMO - reopt-all CMO - reopt-all

10 15 20 30 determinants

F IG U R E 6.11 Effect of localizing the orbitals in the reference determinant for HgO in a DZ basis. New determinants were initiated by choosing the energetically most favorable excitation of the SCF determinant in canonical (CMO) or localized (LMO) molecular orbitals, or in the energetically best ex­ citation of either (CMO/LMO). The initiated determinant is then optimized. After every two determinants the last two were re-optimized for the hollow symbols, while the filled symbols indicate expansions that were re-optimized at every step. Re-optimizing all determinants with every new determinant for the LMO/CMO caae leads to an expansion that is energetically worse than re-optimization of only two determinants after every two steps. (The “LMO/CMO reopt-air curve follows the “CMO reopt-all” curve very closely and is hence barely visible.) For the LMO case, the difference between the two ways of re-optimization is negligible after 18 determinants, while for the CMO case the difference disappears after 18 determinants. The two determi­ nants (l?/> 2o)and | 02l)i circled) that give a large contribution after apparent convergence of the LMO expansion are also noted in Figure 6.13. 153 CMO or the LMO representations of the SCF determinant, the difference between

re-optimizing all or the last two is small after 20 determinants. For the case where

NORTH COIN compares the energies of initial excitations out of both representations

(CMO/LMO), the re-optimization of only two determinants is clearly better. This

indicates that the re-optimization of the determinants already in the expansion,

partially describes the correlation effects which could be taken care of more thor­ oughly by a new determinant

l^d+I>-

With the particular effects partially described, the initial excitation for the new determinant

\^d+l)

becomes energetically less favorable compared to other excitations which, after op­

timization, do not improve the energy as much.

Forcing the initial excitation to be from the SCF determinant in localized or­

bitals is at first less favorable than talcing the initial excitation from the canonical

representation. However, as the new determinant is optimized, the resulting energj"^

is improved — in this particular system for all expansion lengths.

In Figure 6.12, the scaled determinant overlap with the reference determinant

is plotted against their degree of excitation. The expansion where North C oin

is forced to use LMO’s leads to determinants with higher degrees of excitation.

Several determinants differ from rpQ in more than one orbital. In particular, two determinants have a scaled overlap much larger than zero, i.e., differ in at least 154 1.0

0.8 - - S=S5 5 0.6 — s = s V 8=5^26) 0.4 ▼ LMO 0.2

a 0.8 cd - - 8= 8 6-8 I 0.6 — 8= 8V 0 1 0.4 C3 • CMO O “ 0.2

0.8 - - 8=8 8 8 0.6 — 8 = 8 V S = b\2S f 0.4 ♦ LMO/CMO 0.2 0.0 0 1 2 3 4 n

F IG U R E 6 .1 2 Scaled determinant overlap with the SCF determinant plot­ ted against the degree of excitation for HgO. Forcing the code to start from localized orbitals leads to determinants that differ by more than a single or­ bital from the reference determinant and some with a degree of excitation above 2. In particular, the two circled determinants (here and in Figure 6.12) differ from the SCF determinant in four of the five orbitals.

155 three orbitals from the SCF determinant. This type of determinant is absent in the expansion where the choice between LMO and CMO was allowed.

These two determinants |^2o) &nd |V’2l) based on localized orbitals are circled in Figures 6.11 and 6.12. In Figure 6.11, these two determinants are shown to contribute significantly to the energy after the expansion appears to be converging.

Determinants that differ from the SCF determinant in more than one orbital are not found by the initial excitation of the SCF determinant alone. Unless their initial excitation improves the energy more than other initial excitations, they will not be found until the other initial excitations have been used.

Application of the formalism to create higher excitations from the determi­ nants optimized by North C oIN is shown in Figure 6.13 to improve the energies only marginally. The number of determinants involved is quite large, creating all products from an expansion of 30 determinants (including the SCF determinant) results in a total of 436 determinants. Only a small fraction of these are kept — in this case only 66 (CMO) and 34 (LMO) — as many resulting determinants are linearly dependent on the determinants already present in the expansion and are hence thrown away.

156 100%

80%

M 60% I § ca 0 -0 CMO: l4-^ expansion 1o 40% # -# CMO: expansion u v - 9 LMO: \+K expansion v -v LMO: expansion

20%

0% 0 10 20 30 determinants K

FIGURE 6.13 Correlation energies for expansions of water in a DZ basis. The construction of determinants with higher excitations from products of lower excitations is ineffective in this system, both for initial excitations based on the canonical MO s and the localized MG’s of the SCF determinant.

157 6.6 Small hydrocarbons and the carbon dimer

The four molecules considered here — ethane (CgHg), ethylene (C 2H4), acety­

lene (C 2H2), and the carbon dimer (C 2)— are similar in that they consist of two

bonded carbon atoms. The character (and strength) of the carbon-carbon bond varies with the number of hydrogen atoms that participate in the molecules.

molecule re(C-C) re(C-H) ZHCH

carbon dimer' ' 2.35 bohr —— acetylene^® 2.27 bohr 2.00 bohr — ethylene 2.53 bohr 2.03 bohr 117° ethane^® 2.89 bohr 2.07 bohr 108°

TABLE 6.5 Experimental equilibrium geometries for the carbon dimer, acetylene, ethylene, and ethane. Beginning with C 2, the addition of hydro­ gen atoms reduces the number of electrons participating in the C-C bond. In going from C 2 to acetylene this shortens the bond length, because fewer elec­ trons need to “fit between the carbon nuclei.” As more electrons are pulled away from the C-C bond, the bond weakens, the shielding of the carbon nuclei repulsion is reduced, and the bond lengthens.

The more hydrogen atoms that participate in the molecule, the more electrons are ‘pulled’ away from the carbon-carbon bond, reducing the electron density be­ tween the carbon nuclei. At first, this reduces the length of the bond in going from

C2 to acetylene, see Table 6.5. As fewer electrons participate in the bond, the bond

158 weakens, the carbon nuclei axe less shielded from each other, and the bond length­ ens. In going from C 2 to C 2H6 we proceed from a system with strongly-correlated electrons in one region (C 2) to a system with weaJdy-correlated electrons spread out over several bonds.

ethane

H H

H A / ethylene C C H H H / \ / HH C = C / \ H H acetylene

H - C = C - H carbon dimer :c = c:

FIGURE 6.14 Structures of ethane, ethylene, acetylene, and the carbon dimer. As the number of hydrogen nuclei is reduced, more electrons participate in the C-C bond. The bond in the carbon dimer is drawn as a double bond, though in actuality it cannot be uniquely described as a double or triple bond.

A transformation of the canonical molecular orbitals (CMO’s) into localized molecular orbitals (LMO’s) leads to two core orbitals, one for each carbon nucleus, in all four molecules. The number of C-H bonds equals the number of hydrogen 159 nuclei, and the remaining electrons form the C-C bond. The carbon dimer has a

C-C bond that cannot be uniquely described as a double or a triple bond,®^’®^ and in fact is a difficult multi-reference problem.^^

molecule DZ cc-pVDZ* SCF CISD SCF®"* MP4®"* ethane -78.549 -78.819 -79.241 -79.589 Ecott 270 348 ethylene -78.011 -78.193 -78.045 -78.357 -Fcorr 182 312 acetylene -76.799 -76.976 -76.828 -77.109 Ecorr 177 280 carbon dimer -75.356 -75.586 Ecorr 230

* Dunning’s cc-pVDZ basis set^® with s exponents scaled by 1.44

TA BLE 6.6 SCF and (orthogonal) Cl results for ethane, ethylene, and acety­ lene in a DZ basis set.®^ Total energies are in hartree, correlation energies are in millihartree. For comparison, the results of Del Bene et in a cc-pVDZ basis set are given. The lack of polarization functions in the DZ basis set affects the SCF energies very strongly.

In Table 6.6 the SCF and CISD energies for the four molecules are given in the

DZ basis set used here.^^ For comparison the results of Del Bene et al. in a cc-pVDZ basis are shown,®"* where the correlated energies were calculated with fourth-order

Mpller-Plesset perturbation theory (MP4). In all molecules the core electrons are 160 frozen. Comparison of the SCF energies in the two basis sets reveaJs that the DZ basis used here is too small for quantitatively reliable calculations, hence the focus will be on qualitative trends.

Table 6.7 shows the correlation energies of the four molecules for selected ex­ pansions in non-orthogonal determinants in comparison with CISD results. All expansions start with a SCF determinant whose orbitals have been localized. Non- orthogonal determinants are initiated by excitations of the SCF determinant. Each initiated determinant is partially optimized; every time two determinants have been added to the expansion these two are re-optimized. The percentage of correlation energy accounted for depends strongly on the size of the system and the basis set:

For the carbon dimer, with 8 electrons and 18 basis functions, two dozen non- orthogonal determinants result in an energy below the CISD energy, while ethane

(14 electrons, 30 basis functions) clearly requires more determinants.

The construction of higher-excitation determinants from pairs of lower-excitation determinants reflects the structure of these systems, i.e., the spatial distribution of electrons. For ethéine the determinants constructed from the 14-32 expansion add another 8 mH to the correlation energy in contrast to C 2, where the gain is only

1 mH. Ethylene and acetylene lie inbetween with 5 mH and 2 mH, respectively.

This trend is not surprising, since the construction of higher excitations works best when the lower excitations can be spatially well separated.

The number of constructed higher-excitation determinants that are actually kept, i.e., that are linearly independent of all determinants in the expansion, de­ creases as the number of optimized determinants increases beyond 16 — the only exception being ethane. The constructed determinants tend to have larger degrees 161 molecule N d •^corr %Ecorr(CISD) ethane 1 + 8 139 51.5 1 -(- [8 ]® => 1 + 36 152 56.3 1 + 16 214 79.3 1 + [16]® =>1 + 88 231 85.6 1+ 24 235 87.0 1 + [24]® = > 1+ 62 244 90.4 1+ 32 244 90.4 1 + [32]® =>1 + 73 252 93.3 ethylene 1 + 8 124 68.1 1 + [8 ]® = > 1+ 36 129 70.9 1 + 16 151 83.0 1 + [16]® => 1 + 51 156 85.7 1+ 24 161 88.5 1 + [24]® =>1 + 56 165 90.7 1+ 32 165 90.7 1 + [32]® = > 1+ 49 170 93.4 acetylene 1 + 8 135 76.3 1 -f [8 ]® = > 1+ 29 141 79.7 1 + 16 153 86.4 1 + [16]® =>1 + 51 160 90.4 1 + 24 169 95.5 1 + [24]® = > 1+ 53 172 97.2 1+ 32 173 97.7 1 + [32]® = > 1+ 50 175 98.9

C2 1 + 8 197 85.7 1 + [8 ]® = > 1+ 36 217 94.3 1 + 16 226 98.3 1 + [16]® =>1 + 53 236 102.6 1 + 2 4 243 105.7 1 + [24]® = > 1+ 52 246 107.0 1+ 32 252 109.6 1 + [32]® =>1 + 51 253 110.0

TA BLE 6.7 North C oin frozen-core correlation energies for ethane, ethy­ lene, acetylene and the carbon dimer in a DZ basis set.®^ The energies axe 162 continued #- TA B LE 6.7, continued given in millihaxtrees. Nf) is the number of deter­ minants. The correlation energy of expansions in non-orthogonal determinants is compared with the CISD correlation energies obtained from an expansion in orthogonal determinants. The number of correlated electrons (14 for ethane, 8 for the carbon dimer) and the size of the basis sets (30 for ethane, 18 for carbon) strongly affect how much correlation energy a specific number of determinants can account for. The column for the number of determinants, N p, also shows the number of determinants kept in the expansion with explicitely-constructed higher excitations. If all determinants were kept {i.e., if they were all lin­ early independent), the number of determinants would be 1 -|- [ 8 ]® => 1 -t- 36, 1 -t- [16]® 1 + 136, 1 -I- [24]® 1 -f- 300, and 1 -f- [32]® 1 -t- 528. With the exception of acetylene, the entire 1 -f- [ 8 ]® =>■ 1 -f 36 expansions are kept. Of the determinants constructed from the longer expansions, many are not kept. In particular, the total number of determinants does not change signif­ icantly between the 1 + [16]®, 1 -f- [24]®, and 1 + [32]® expansions, i.e., the number of constructed determinants is reduced in going from the 1 -f- [16]® to the 1 -I- [24]® and from the 1 + [24]® to the 1 -|- [32]® expansion (with the exception of ethane).

163 1.0

0.8

T? 0.6

0.4

0.2

■ I

0.8 C,H C,H.

0.6 •a 0.4

0.2

0.0 0.0 1.0 2.0 3.0 0.0 1.0 2.0 3.0 4.0 n _

FIGU RE 6.15 Scaled determinant overlap with the SCF determinant plot­ ted against the degree of excitation for Cg, C 2H2, C2H4, and C 2Hg. All plots include 32 points representing the determinants beyond the SCF determinant. Only the carbon dimer, which is known to require a multireference approach, has determinants with > 2. All expansions contain determinants with riex < I with a scaled overlap well above zero, i.e., they differ from the SCF determinant in two or more orbitals. Determinants with n,* > 1 in the ex­ pansions for C 2H4 and C 2H6 all have a scaled overlap near zero, with the exception of one C 2H6 determinant (circled).

164 of excitation théin the optimized determinants. For all but the caxbon dimer the

optimized determinants all have n,* < 2 (see Figure 6.15), while most of the con­

structed higher excitations have > 2. This reflects one of the main differences

between expansions in (non-orthogonal) determinants with optimized orbitals and expansions where the orbitals are not optimized: Many correlation effects that re­

quire higher-order orthogonal excitations can be described by fewer non-orthogonal

determinants that differ very little from the SCF solution.

An exception to this is the carbon dimer, where the construction of higher exci­

tations from the 1 -1-8 expansion gives the largest increase in the correlation energy for the four molecules considered here. C 2 is known to require a multi-reference description in orthogonal expansions. Figure 6.15 shows that many optimized non- orthogonal determinants have n^x > 2 for C 2, as was the case for Be 2, which also requires a multi-reference treatment (see Section 6.4).

Also apparent in Figure 6.15 is how different types of determinants appear in the expansions for the different hydrocarbons. Determinants that differ in several orbitals appear in all expansions. The degrees of excitation for these determinants are (almost) all below one for C 2H4 and C 2H6, while for C 2H2 several of these determinants have n^x % 2. This difference in the types of determinants does not change gradually with the number of hydrogens participating in the molecules decreases, i.e., as one proceeds from C 2H0 through C 2H4 to C 2H2. The difference occurs between C 2H4 and C 2H2, and so one might speculate that the difference is due to the change from non-linear to linear molecules.

165 6.7 The nitrogen dimer N 2

The nitrogen dimer has a triple bond that is very short. Thus many electrons

are densely packed and strongly correlated. In this system, electron pairs cannot be

identified separately, with the exception of the core orbitals, which are kept frozen in

our calculations. The lack of independent electron pairs is reflected in Figure 6.16,

where the explicit construction of higher excitations in LMO’s is ineffective once

approximately 75% of the correlation energy has been accounted for.

Figure 6.17 shows the scaled determinant overlap with the SCF determinant and

their degree of excitation. Our results, in a DZ basis set,^^ are for the experimental

bond length and at 10% stretched and 10% shortened geometries. At all bond

lengths determinants axe present that differ from the SCF determinant in more

than 2 orbitals. In particular, two determinants with n^x between 1 and 2 appear

at all bond lengths along the “3^’s” line (representing determinants that differ from

the SCF determinant in three orbitals with similar orbital overlaps).

Figure 6.18 shows the missing correlation energy for the nitrogen dimer at

the different nuclear separations. The reference energy is the results of calculations

including up to quadruple excitations out of the CAS. Also shown are results for the

supermolecule, for which a few dozen determinants are not sufficient. In principle, a

fully correlated singlet supermolecule wave function could describe the two nitrogen

atoms in their "^S state.^ Such a description requires not only optimized orbitals,

but also enough determinants to reproduce all the necessary spin couplings. Spin

coupling is the reason why the expansion for the N 2 supermolecule here is so poor in

^ see Section 3.2 166 100%

80%

I 60% g I

1u 40%

0 - 0 CMO: I+2\j/^ * -# CMO: 1+[E\|/^] 20% 0 - 0 LMO: LMO: I+[S\l/j'

0% 1+0 1+10 1+20 determinants

FIGURE 6.16 Correlation energy for the nitrogen dimer in the original ex­ pansion and with higher excitations. All expansions are in a DZ basis set,^^ with a frozen core, at the experimental equilibrium bond length. The expan­ sion based on the SCF determinant in localized orbitals (LMO) is significantly superior to the expansion based in canonical molecular orbitals (CMO). The explicit construction of higher excitations is ineffective once the expansion has accounted for a significant amount of correlation energy. 167 1.0 0.8 R = 0.90 R,■exp 0.6

0.4

0.2

g.0.8 R=1.00R.‘exp (U > 0.6 O 73(U 0.4 s 0.2 ##

0.8 ■exp 0.6

0.4 36’s line 0.2 0.0 0.0 0.5 1.0 1.5 2.0 n

FIG U RE 6.17 Scaled determinéint overlap with the SCF determinant plotted against their degree of excitation for the nitrogen dimer in a DZ basis. Ail determinants have a <2. Two determinants with a scaled overlap that is significantly larger than zero (along the “3^’s” line) appear at all bond lengths. The remaining determinants lie close to the abscissa, though at the experimental bond length they are distributed somewhat further from a scaled overlap of zero. 168 comparison to the expansion for the beryllium supermolecule in Section 6.3, where the two independent atoms are both singlets.

For all three expansions shown in Figure 6.18, the energy appeaxs to suddenly drop after behaving rather smoothly as a function of the number of determinants.

This is similax to an observation for water in Section 6.5, where two determinants contribute significantly to the energy after apparent convergence (Figure 6.11).

These two determinants were identified to be those in Figure 6.12 that have large scaled overlaps. In the case considered in this section, N 2, the two determinants that have large scaled overlaps axe \rp2i) and |^ 22) = 0.90i2exp), l^ie) and \ipij)

{R = 1.00i?exp), and IV’ 22) and \1p23) = l-10i?exp)- These axe consistently the determinants that cause the sudden drops in the energy curves in Figure 6.18.

169 300 —----- ' ' f-iT'- supermolecule *, % a t 1162

J=I

‘2

c/3I U C/5t < u

tjq 100 I

0 0 10 20 determinants

FIG U RE 6.18 Correlation energy missing from expansion in non-orthogonal determinants for Ng in a DZ basis. The correlation energy is compared to that of CAS-CISDTQ in NO’s. The core orbitals are frozen.

170 6.8 Neon

As our last example, we tackle the neon atom in the cc-pCVTZ basis (43 AO’s), with all 10 electrons correlated. Figure 6.19 shows the results for the initiai expan­ sion, which is based on the SCF determinant in localized orbitals. Also shown is the energy for expansions where higher-excitation determinants are explicitly con­ structed. These additional determinants do not affect the energy significantly. In principle it should be possible to have excitations out of the Is orbital and separate excitations out of the second shell — these could then be combined to construct a higher-excitation determinant. However, the only determinant that contains an excitation out of the Is orbital that carries significant weight in the expansion

(see Figure 6.20) differs from the SCF determinant in several orbitals, i.e., some excitation out of the second shell is already present.

That the degrees of excitation shown in Figure 6.20 all have < 2 is not surprising, since doubles account for approximately 96% of the correlation energy of neon in FCI wave functions.T he 30 non-orthogonal determinants calculated here accounts for approximately 80% of the CISD correlation energy. Furthermore, since neon is a closed-shell system a single determinant is adequate as a reference state, and only systems requiring multi-reference treatment have so far shown optimized non-orthogonal determinants with Rg* > 2.

171 300

250

ë 150 u c o

& - 0 1+AT expansion #-# l+[^® expansion

determinants K

FIGURE 6.19 Correlation energy for the neon atom in the cc-pCVTZ basis set. The CISD correlation energy of Woon et al. is 323.30 millihartree.^^ The expansion in non-orthogonal determinants is based on the SCF determinant in localized orbitals. The inclusion of explicitly-constructed higher excitations does not affect the energy significantly.

172 10' «♦ M 10' K . 0 "

♦ excitations out of 2 s and 2p orbitals O excitations out of Is orbital

0.4

Èu. o T3 W U3S 0.2

0.0 œ -

n.ex

FIGURE 6.20 Determinant weight and scaled overlap with the SCF de­ terminant plotted against the degree of excitation for neon in the cc-pCVTZ basis. The eight excitations that are mostly out of the Is orbital are all nearly doubles, the only exception (near the origin) is also the only determinant of these eight that has a significant weight in the expansion. Its scaled overlap is well above zero, indicating excitation out of orbitals besides the Is.

173 CHAPTER 7 CONCLUSIONS

Quotation, n: The act of repeating erroneously the words of another.

- Ambrose Bierce: The Devil’s Dictionary

It is a good thing for an uneducated man to read books of quotations.

- Winston Churchill

I hate quotations. Tell me what you know.

- Ralph Waldo Emerson

The problem addressed in this document is the effect of basis set orientation on truncated configuration interaction wave functions. A configuration interaction

(Cl) wave function is an expansion in déterminants, which in turn are made up of one-particle orbitals. High-quality basis sets of one-particle orbitals are available in the literature, and the expansion in all possible determinants of a basis set — the full configuration interaction (FCI) wave function — would lead to very precise energies.

FCI is, however, computationally too expensive for even modest-size molecules, and the wave function must be truncated. Any unitary transformation of the basis set does not change the space spanned by the basis set, but the basis set does take on a different orientation. While FCI is not affected by a unitary transformation of the 174 basis set, the energy of a truncated Cl wave function depends on the orientation of the basis set.

In the previous chapters the effect of basis set orientation on truncated Cl wave functions is discussed in two contexts: (I) In traditional Cl, where the determinants are required to be orthogonal, the entire expansion must be done in one basis set orientation. The two most prevalent orientations are those given by the self- consistent-field (SCF) orbitals and the natural orbitals (NO’s). Two examples are presented to show the importance of using NO’s. (II) The main focus of this document is on expansions where each determinant can have its own basis set orientation. A computer code, North C oin , was developed and applied to atoms, molecules, and the Hubbard model in one and two dimensions.

(I) — Cl with orthogonal Slater determinants

Natural orbitals (NO’s) are formally defined as those orbitals for which the one-particle density matrix of the FCI wave function is diagonal. In practice, NO’s are approximated by diagonalizing the one-particle density matrix of a truncated

Cl wave function. The energy of the wave function with NO’s is often improved.

(I.a) In the first example, the energy of the nitrogen dimer is calculated with the truncated wave function and basis set of a benchmark study in the literature.

By rotating the basis into NO’s, the energy improves by 1.3 millihartree, which is approximately equal to the error in the dissociation energy calculated in the benchmark study.

(I.b) The second example concerns the energy calculated for systems of two independent atoms — B 2, Cg, Ng, O2, and F 2 all with large nuclear separation.

175 Since the individual atoms are independent, the energy of such a system (a “super- molecule”) should ideally be equal to the energy of the two atoms treated separately.

In Cl truncated at one particular level of excitation this is not the case. However, we show that the correct energy is approximately reproduced by truncating the wave function of the supermolecule system in a well-defined way with an orbital- dependent truncated level of excitation.

(II) — Cl with non-orthogonal Slater determinants

(Il.a) For two-electron systems (e.g.. He) North C oin reproduces the NO’s of an expansion in orthogonal determinants.

(II.b) For many-electron systems with spatially well-separated electron orbitals

(e.g. Be), North C oin approximates the NO’s of an expansion in orthogonal de­ terminants. This is interpreted as “pair-specific NO’s”: A pair of electrons can, in principle, be correlated by freezing all but the two electrons in question. The expansion for the correlation of one such pair will be shortest in the NO’s of that pair. The orientation of the basis set for the NO’s of different pairs will in general be different. An expansion that contains determinants in the NO’s of all pairs — by varying the basis set orientation — requires non-orthogonal determinants.

(II.c) The energy of the reference determinant (the SCF determinant) is not affected by a unitary transformation of the occupied orbitals. By rotating these orbitals into localized orbitals we thus get an equivalent determinant to start the expansion with. In terms of how much correlation energy such an expansion picks up, we find the localized orbitals to be superior. This observation is consistent with the pair-specific NO’s interpretation of point (Il.b): The localized electron pairs are correlated by finding their own natural orbitals within the basis set. 176 (Il.d) Certain higher excitations can be viewed as the product of two (or more) independent excitations of a lower degree. We show that these higher excitation can be generated by forming the products of determinants, provided

(i) the determinants excite from different orbitals and into different orbitals, and

(ii) the product is formed with respect to the correct reference determinant.

The generation of higher excitations from such products is successful in cases where the electron pairs are spatially separated (e.g., C 2Hg), while in systems where the electrons cannot be grouped into spatially separated pairs (e.g.. Ne, Ng) the proce­ dure is ineffective in accordance with (i) above.

(li.e) The concept of single, double, etc. excitations is generalized to the degree of excitation n^x of a non-orthogonal determinant with respect to the reference determinant. For orthogonal determinants, rigx is required to be an integer; for non-orthogonal determinants, n^x can take on fractional values. The freedom to have a different basis set orientation for each determinant significantly changes the values of n„- Determinants with n^x > 2 are nearly absent from the expansions. The only exceptions are molecules that require a multi-reference treatment in traditional

Cl (Be2, C2). In traditional Cl expansions determinants with Ugx = 1 carry very little weight (in Brueckner orbitals they vanish), while North C oIN expansions consistently contain many determinants with n^x < 1. A systematic study of the distribution of n,* for various molecules is necessary to fully understand patterns in the distribution. We note here a similarity between the distributions of n,* of N 2 and C 2H2 (Figures 6.16 and 6.18): Both of these molecules contain triple bonds.

(Il.f) North C oin is applied to the Hubbard model in one and two dimensions, resulting in systematic expansions in configuration state functions (CSF's). The 177 first CSF — most important in terms of its weight in the expansion — consists of A determinants, where A is the number of sites. Each of these determinants displays an electron density peaked at one site. This CSF enters the wave function with a sign opposite to that of the reference determinant, effectively removing energetically unfavorable parts of the many-body wave function from the reference wave function.

The lowering in the total energy is accompanied by a reduced probability of two electrons sitting on the same site. The overlaps between determinants of the first

CSF differ significantly from zero, indicating the re-orientation of the basis set for the correlation of each on-site electron pair.

Cl with non-orthogonal determinants is not as powerful as traditional quan­ tum chemistry methods, and certainly lacks these method’s high computational efficiency. However, the freedom to re-orient the basis set for each determinant produces much shorter expansions, which, as demonstrated for the systems above, are more accessible to the human mind.

NnttGoft

178 APPENDIX A QUANTUM CHEMISTRY ACRONYMS

Publications in quantum chemistry tend to have many acronyms. The following

list is far from complete, but contains the most important ones in use today.

ACPF Averaged coupled-pair functional

AE All electron

AE Atomization energy

AEL Approximate excitation level

AO Atomic orbital

AND Atomic natural orbital

APSG Antisymmetrized product of strongly orthogonal geminaJs

B-CC CC with Brueckner determinants

BE Bond functions

BLB Brillouin-Levy-Berthier

BSR Basis set reduction

BSR-MR-AQCC MR-AQCC-BSR

179 BSSE Basis set superposition error

BW Brillouin-Wigner

CAS Complete active space

CAS+1+2 CAS multi-reference single and double excitation Cl

CASPT2 Second-order perturbation theory with CASSCF wave function as

reference function

CASSCF Complete active space self consistent field

CBS Complete basis set

CC Excitations from core orbitals (also called KK terms)

CCI Contracted Cl

CCD Coupled cluster with double excitations

CCSD Coupled cluster with single and double excitations

CCSDPPA CCSD polarization propagator approach

CCSD(T) CCSD with perturbative treatment of triple excitations

CCSD-T CCSD with linearized triple excitations

CEPA Coupled electron pair approximation

CGTO Contracted GTO

CHELP Charges from electrostatic potential

Cl Configuration interaction

CI-D Doubles Cl 180 CINQ NOfromCISD

CISD Singles and doubles CI (also called CI-SD)

CISDfT] MRCI with references limited to those generated by single excitations

of the dominant reference

CISD[TQ] MRCI with references limited to those generated by single and double

excitations of the dominant reference

CMRCI Internally contracted MRCI

CMO Canonical molecular orbital

CNDO Complete neglect of differential overlap

CNO Constrained natural orbital

CNO Correlating natural orbital

CPA Coupled pair approximation

CPF Coupled-pair functional

CPHF Coupled-perturbed HF

CSF Configuration state function

CSOV Constrained space orbital variation

CT Charge transfer

CV Excitations from core and valence orbitals (also called KL terms) cc-pCVDZ Correlation-consistent polarized core valence double-zeta cc-pCVTZ Correlation-consistent polarized core valence triple-zeta

cc-pVDZ Correlation-consistent polarized valence double-zeta 181 cc-pVTZ Correlation-consistent polarized valence triple-zeta

DBS Dipole-bound state

DC Davidson correction or double configuration

DE Doubly excited

DBA Double-electron attachment

DFT Density functional theory

DIIS Direct inversion in the iterative space

DIP Double-ionization potential

D-MBPT(oo) Sum of doubles in MBPT to infinite order

DMF Dipole moment function

DM Dipole moment

DODS Different orbitals for different spins

DRT Distinct row table

DZP Double zeta polarization

EA Electron attachment

EA-EOMCC Electron attachment EOM-CC

ECC Extended CC

EHT Extended Hûckel theory

EOM-CC Equation-of-motion CC

E0M-MBPT(2) EOM-CC with similarity-transformed Hamiltonian truncated at sec­

ond order

182 EOM-QCISD Equation-of-motion quadratic Cl

EN Epstein-Nesbet

EPV Exclusion principle violating

ETGC Even-tempered Gaussian contraction

FCI Full Cl

FMO Frontier molecular orbital theory

FNC First natural configuration

FOCI First-order configuration interaction

FP-CC Finite perturbation CC

FP-CI Finite perturbation Cl

FORS Fully optimized reaction space

FSCC Fock space CC

FSGO Floating spherical Gaussian orbital

FSMRCC Fock space MR-CC

FVIP First vertical ionization potential

GAIO Gauge including atomic orbitals

GAPT Generalized atomic polar tensor

GDC Generalized Davidson correction

GEN Generalized Epstein-Nesbet

GHF General HF

183 GTO Gaussian type orbital

GUO A Graphical unitary group approach

GVB Generalized valence bond

GVB-RCI Restricted Cl from GVB reference

GVB+1+2 GVB multi-reference single and double excitation Cl

Gl Gaussian-1

HF Hartree-Fock

HFDFT Hajtree-Fock density functional theory

HF+1+2 HF single and double excitation Cl

HFR Hartree-Fock-Roothaan

HMD Hûckel molecular orbital theory

HOMO Highest occupied molecular orbital

ICACPF Internally contracted ACPF

IGF Interacting correlated fragment

ICMRCI Internally contracted MRCI

ICMRCI+Q Internally contracted MRCI with Davidson correction

ICSCF Internally consistent SCF

lEPA Independent electron pair approximation

IGLO Individual gauge for localized orbitals

INDO Intermediate neglect of differential overlap 184 INO Iterative NO

IP Ionization potential

IPA Independent pair approximation

IPEA Independent pair excitation approximation

IPNO Independent pair natural orbital

IPNSO Independent pair natural spin orbital

IPPA Independent-pair potential approximation

rVO Improved virtual orbitals

KK Excitations form core orbitals (also called CC terms)

KL Excitations from core and valence orbitals (also called CV terms)

KSCED Kohn-Sham equations with constrained electron density

KT Koopmans’ theorem

LHP Localized Hartree product

LL Excitations from valence orbitals (also called VV terms)

LCAO Linear combination of atomic orbitals

LCCD Linearized CC doubles

L-CPMET Linearized coupled-pair many-electron theory

LDGUGA Loop-driven GUGA

LORG Localized orbitals, local origin

LRCI Low-range Cl or low-rank Cl 185 LUMO Lowest unoccupied molecular orbital

MBOHO Maximum bond order hybrid orbital

MBPT Many-body perturbation theory

MC Multi-configurationaJ

MCI First-order multi-configurational

MCCEPA MulticonfigurationaJ CEPA

MCCHF Multiconfiguration coupled HF

MCLR MulticonfigurationaJ linear response

MCLRT Multiconfiguration linear response theory

MCSCF Multi-configurational self-consistent field

MINDO Modified intermediate neglect of differential overlap

MNDO Modified neglect of diatomic overlap

MNDOC MNDO correlated

MO Molecular orbital

MOR Method of optimal relaxation

MR-ACPF Multi-reference ACPF

MR-AQCC Multi-reference averaged quadratic CC

MR-AQCC-BSR Multi-reference averaged quadratic CC with BSR

(same as BSR-MR-AQCC)

MR-CC Multi-reference coupled cluster 186 MR-CEPA(O) Multi-reference CEPA (to zeroth order?)

MRCCI Multi-reference contracted configuration interaction

MRCI Multi-reference configuration interaction (also called MR-CI)

MRCI(n act) MRCI with n active electrons

MR-CISD Multi-reference CISD

MR-LCCM Multi-configurational reference, linearized CC method

MP M0ller-Plesset

MP2 M0ller-Plesset, second order

MP2-R12 MP2 with explicit inter-electronic coordinate dependence

MP3 Mpller-Plesset, third order

MP4 Mpller-Plesset, fourth order

MSO Molecular spin orbital

NDDO Neglect of differential diatomic overlap

NBO Natural bond orbital

NCCM Normal CC method

NHO Natural hybrid orbital

NO Natural orbital

NPA Natural population analysis

NSC Natural spin orbital

GDC Optimal double configuration

187 OHP Orthogonalized Hartree product

OR Orbital reference

PEF Potential energy function

P-EOM-CC Partitioned EOM-CC

PES Potential energy surface

PMC Perturbation molecular orbital theory

PNG Pair, principle, or pseudo natural orbital

PNSO Principle natural spin orbital

PP Perfect pairing

PP Polarization propagator

PPP Pariser-Parr-Pople

PSNO Pseudo-natural orbital

QCISD(T) Quadratic Cl with triples correction

QCPE Quantum chemistry program exchange

QD-VPT Quasidegenerate variational perturbation theory

QDVPT-APC QDVPT with averaged pair correction

QE Quadruply excited

QRHF Quasi-restricted Hartree-Fock

RAS Restricted active space

RDC Renormalized Davidson correction

1 8 8 RECP Relativistic effective core potential

RHF Restricted Hartree-Fock

ROHF Restricted open-shell Hartree-Fock

RS Rayleigh-Schrôdinger

SA-CC Spin-adapted CC

SAC-CI Symmetry-adapted cluster Cl

SAF Symmetry adapted function

SCEP Self-consistent electron pairs

SCF Self-consistent field

(SC pSC I Size-consistent, self-consistent, selected Cl

SC-XCC Strongly connected XCC

SDGUGA Shape-driven GUGA

SOCI Second-order configuration interaction

SOPP Strongly orthogonal perfect pairing

SOPPA Second-order polarization propagator approach

SOS Sum-over-state

SPNO Separated pair natural orbital

SSMRCC State-selective multi-reference CC

STEOM-CC Similarity transformed EOM-CC

STD Slater type orbital 189 sxcc Symmetric expectation value CC

TAE Total atomization energy

(TD)-CCSD Two-determinant CCSD

TDHF Time-dependent HF

UCC Unitary CC

UCEPA Unitary CEPA (same as MR-CEPA(O))

UGA Unitary group approach

UHF Unrestricted Hartree-Fock

UMP2 Second-order Moller-Plesset theory

UMP4 Fourth-order MoIler-PIesset theory

VB Valence bond

VBS Valence-bound state

VE Valence electron calculation

VPT Variational perturbation theory (same as MR-CEPA(O))

W Excitations from valence orbitals (also called LL terms)

XCC Expectation value CC

ZPE Zero-point vibrational energy

190 APPENDIX B The SCF approximation and the Virial Theorem

Since we axe using the variational principle to calculate the SCF energy, the correlation energy is negative. The virial theorem,

T ^ = - ^ , ( B . l ) can be applied, and hence

Ecorr = Tcorr + Korr = T:orr ~ “^T cott = ~ T cott < 0 , (B .2 ) which means that Hartree-Fock (or the SCF approximation) underestimates the kinetic energy of the electrons. The potential energy is overestimated, which can be interpreted as the SCF solution zdlowing electrons to get closer to each other than is optimal. Correlating the electrons increases the average distance between them, thus reducing the potential energy. The increased kinetic energy can be interpreted as the electrons having to undergo more complex motion in order to avoid each other.

191 APPENDIX C MATHEMATICAL FORMALISM FOR Cl WITH NON-ORTHOGONAL DETERMINANTS

In this appendix the formula and derivations used in Koch and Dalgaaxd’s paper^"^ axe given with a more explicit description of the relevant algebra.

Beginning with the reference determinant, |ç>o), a determinant is written in terms of a unitary transformation,

|<^m) = exp(zA("'))|

The exponent is written in terms of the SCF creation and annihilation operators:

^ ^ ~ (C.2) k j 0-

The are taken to be independent of spin a, this restriction is discussed in

Appendix D.

A unitary operator of the form U = exp(f -^rsOras), with Xrs = A*g, gen­ erates transformations of the creation operators.

U alu^ = Or + z ^ alAgr - + •••• = (C.3) st 192 What do the orbitals in the new determinant, look like? To answer this, assume they are created (out of the vacuum state |uac)) by a new set of creation operators, such that \m) — ...6" \va(^. Using the fact that the transfor­ mation is unitary, exp(— exp(zA^"‘^) = 1, and that the transformation does not create something out of nothing, exp(— |uac) = |uac), the transformed determinant can be rewritten as

exp(zA^"*^)|^o) = exp(zA('")) aja^-.-.an |uac)

= exp(iA^"*^) a | exp(— exp(zA("*^) exp(—

exp(zA^"’^) ajjexp(—zA^*^^) |uac)

= Y^oIDs,! Y^a\Dt ,2 •••• '^ a lD u ,n |uac). s t u (C.4)

Hence, for all z, the new creation operators are linear combinations of the SCF creation operators,

s A given orbital Vj{f) in \(j>m) is written as

Vj{f) = (r|uy) = (r|6t|uac) = ^ Dgj(r|al|z;ac) ^ ^ " (C.5) = Y1 ^sj{r\ws) = X , ^ s jw s if) . s s

Or, if we consider the collection of orbitals in determinant \(f)m) as a vector, v{(f>m) =

ZÜ • D, with D = D(<^m), where zn is the collection of ail orbitals in the basis set — occupied and virtual. In particular, for the SCF determinant written in terms of the SCF orbitals one has D(<^o);j = 5,-j.

193 The orbitals of a given determinant are orthonormal in our formulation, which says something more about the structure of the D s: ^i,j = (“i(r)|«j(r)) =ydru*{r)uj{r)

= ^ {wp{r)Dp^i)* Wq{r)Dqj = ^ I dr Wp{r)wq{r) P,9 P,9 = = D*D = I P,9 P The overlap of the orbitals in two determinants is written as a matrix, and by taking the determinant of this matrix the overlap between the two determinants is found to be {n\Tn) = |A|, with elements

= ^[D(«>„)i;,<(D(ÿ™)l<,i, (C.7) i To show this, take u, and uj to be the occupied orbitals of the determinant D(<^n) and D(^rn) with field operators q (c|) and bj (&j), respectively. We then have

(uiluj) = {{v\u))ij = ^(uac|c6^|(;oc)^_

s,t = T,m9n)]lsmm)]tj8s,t (C.S) s,t

=> A = [D(ÿ.)r|D(ÿm)l The calculation of matrix elements requires the inverse of the overlap matrix, how­ ever, for cases where the overlap is close to zero, the inverse of A is poorly defined.

Here we use a singular value decomposition, A = UdV*, where U and V are unitary and d is diagonal. With this we can write

A-^ |A| =VpU*, (C.9) where p is diagonal with elements pj = did2 ....dj^idj^i....dji^. 194 To calculate the matrix elements for the energy®® we first consider the one-

particle matrix elements between two determinants n and m, with

N l

km ) = n b\\vac) , 4 = 1 ] 4(D(^m))s,,- 1=1 Since the integrals are written in terms of the SCF field operators, their effect on the determinants must be established. The effect of a (SCF) destruction operator

Or on a determinant is, e.g., on

1 ^ ûrkn) = (Irÿ= 4|uac)

I ^ = -7^1](-l)''^(«r4)c|c^...ct_^ct_^i...c)^|uoc) Viv. (C .ll) 1 = ((D(^n))r,:) 4+1-"4

= £(-»'■ '^n4j I:"":). where the fact that the ct must obey the commutation relations is used and results in the factor ( — 1)*” ^. From the equation (C.ll) follows

W™|4 = -ÿL (D(ÿm))l.( m c | ( n ) • (C .12) t= i \ e ^ k J

With these results the one-particle matrix elements are,

Çrs — {n)

= (D(?>n)).,/ {vac\ ( n I ( n 4 I I""":). •fc=l«=l \ l : ^ k j ) (C.13) 195 The expression

{vac\ looks very much like the overlap matrix of equation (C.8), the difference being that one of each determinants’ orbitals is missing. This is reminiscent of the adjugate of a determinant, where each element of the determinant is replaced by that element’s cofactor. Using this and several steps of linear algebra, one finds

and hence N N

1=1 = |A| ( d (,^„)A-‘ (D(^„))')

Using the singular value decomposition (see equation (C.9)) this can be written as

5r, = (D(^„)VpU*(D(«>™))*)„.

Since we restrict ourselves to singlets and force all spatial orbitals to be doubly occupied, the overlap matrix is defined to be the spatial overlap matrix. Similarly the Çrs is defined between orbitals of one spin only, i.e., is calculated from the spatial parts of the orbitals. With this the matrix elements for the Hamiltonian become

{m\H\n) = 2 \ A \ " ^ h r s 9 s r + ^ (2(rs|rY) - (rs'|/s)) gsr^g'r', (C.14) r,s,r',s* where hrs are one-electron integrals, (rs|r's^) are two-electron integrals, the factors of 2 come about from the spatial orbitals being doubly occupied, and the minus sign results from exchange. 196 With the restriction of doubly occupied spatial orbitals and A the overlap ma­ trix between spatial orbitals, the overlap between the determinants becomes |A|^.

Finally, we can make a better connection between the as they appear in equation (C.3) and the structure of the new orbitals. In general, a m x n real matrix

A can be decomposed into singular values:

^k,p ^^k,j ^j,p = O'* A^ j = Uk p 0p Vjp (C.15)

We have > ^2 ^ .... > > 0, where p = min{m,n} = n (for our purposes, m = M — n is the number of unoccupied orbitals and n is the number of occupied orbitals). The dimensions of U and V are m and n, respectively. The matrices U and V are orthogonal, and A takes their vectors into each other, i.e..

Ay,- = 0iU{ and A^u,- = 0,-y,-. (C.16)

In particular, a Hermitian matrix (operator) A can be diagonalized by a unitary matrix W :

A = W O W * (C.17)

We have M bt = exp(zA)4exp(-iA) = ^ ^ 4 [We*^W* sr 5 = 1 (C.IS) M M = ^ ai [Wcos 0^ * \s r [W sin 0 ^ \ r ■ 5 = 1 5 = 1 This suggests splitting the transformation into a part that describes the transforma­ tion of occupied and unoccupied orbitals separately with cosines and their mixing with sines. This is done by first defining two matrices A = W and B = —zW, i.e.,

B* = zW*, and then letting the appropriate block terms in A and B be zero. For 197 the (new) occupied orbitals (r = l..n) we then have,

7X M bl = Y,4\W coseW *l^ + i ^ oI[WsingW*]gr s = l s = n + l (C.19) n M = ^4[Bcos0B%,.+ ^ 4[Asinm*L; s = l s = n + l

For the (new) unoccupied orbitals (r = n + L.M ) we then have,

M n b l= Y , 4[Wcos^W%^ + t'^4[W sin0W *].r s = n + l s = l (C.20) M n = ^ [A cos ^ [B sin ; s = n + l s = l

Letting A and B act on the vacuum state leads to the vacuum state again,

hence we can write for the (new) occupied orbitais (j = L.n),

n M vj{f) = (r|uj) = ^ Ws{f)Bsj cos 9j + ^ Wg(r)Agj sin^y, (C.21) 3=1 s= n + I and for the (new) unoccupied orbitals {j = n + L.M),

M n uj{rj = {r\i/j) = Ws{f)Asj cos 9j -Y^Ws(f)Bsj sin9j. (C.22) 3=71+1 3 = 1

198 APPENDIX D THE RESTRICTED-EXCITATION ‘APPROXIMATION’

Beginning with the reference determinant, a determinant is written in terms of a unitary transformation,

|0m) = exp(iA(’^))|V>0>. (D.l)

The exponent is written in terms of the SCF creation and annihilation operators:

= - < ^]cr^kaY (D .2 ) k,i O'

The are taken to be independent of spin a.

In general, a unitary operator of the form U = exp(z Ylrs with Xrs =

A*g, will generate transformations of the field operators.

UCrU^ — aj- + 2 ^ ^ flfiAsr ~ ^ ^ ^ OsA^^ ^ ] o,\Ds,r- (D 3) st

Here we have Xrs = —zAr%^\ and so (D.3) becomes

U alu^ = aj^ + ^ a^Asr + 9 X / ^^Xst^tr + •••• = asF>s,r- (D.4) s “ si s 199 Consider the case where the mixing of only two spatial orbitals is permitted, /.e.,

\ _ K , if r = ro and s = so; “ 10, otherwise, then (D.4) becomes

= (D.6) y aj otherwise.

With that we have

\^rn) = exp{i^^^'>)\1po)

= exp(iA^”*^)a|a2....aji|uac)

= exp(zA^"*^)a|exp(—iA^’^^)exp(zA^’^^)a 2 exp(— (D.7)

....exp(zA^”*^)an exp(—iA^”’^)luac)

= a{4....(a!;^^ + 4oT^)(‘'roi +

Expanding this out, the result is four determinants:

I0m) = 1*0) + e ( l O + I* * )) + (D.8) where iV’rS) is the reference determinant |^ q) with the up-spin electron of orbital

TQ excited into orbital sg, is the same excitation for the down-spin electron, and |^ro%) is both excitations (up- and down-spin electron) at once.

With this result, the ‘traditional’ Cl expansion,

l^ci) = cqU'q) + Cl + 1^^)^ + C2lV'%%>, (D.9)

can be written in North C oin as

l«Nc) = 4 l* m ) + (D.IO)

200 with the coeflBcients obeying the equations,

CQ = Cl = (/q ■ ^ 1 aad C2 = do ' + <^2- (D .ll)

Table 18 shows the ten linearly independent terms that can be formed by taking double excitations out of two orbitals {i and (j)j) into two other orbitals [r and

term orbitals spin factors

1 ii(j>rr 2 (f>i(f>i(i>s(f>s 3 jjrr 4 4>j4>jss a^Q/5 5 (j>i4>i{(i>rs + (f>sr) a^o/3 6 7 8 4- 4>ji)(f>ss 9 a/3a^ 10 2(f>i(l)s4>j(i>r 4- 2(f>s(l)i4>rj 4- (f>isrj+ +s4>i4>jr 4- (f>i4>jr 4- j(f>i(i>rs aj3a0

TABLE D .l The ten linearly independent spin-adapted configurations con­ structed from double excitations from the orbitals i and

With the two spatial orbitals,

201 we can form a determinant, - 1(4^: + ^j4>jWi4>i + + ^s4>rWrr +

+ {}i4>jr(i>r) + \jirr)) + {\i(i>j(f>ss) + \4>j(f>i4>ss))

+ i\iir(l>s) + \(f>ii(t>s(l>r)) + {\j<}>jr(l>s) + \j(l>jsr))

+ iiCjir^s i\i((>jr4>s) + \jirs) + \ij(f>s4>r) + \j4>i(f>sr)) , (D.12) which has the first nine terms in it. Setting i{ = £r = 1 and the others to zero, e.g., gives the first term. The nine terms in (D.12) are not independent, since we have only two coefficients {ii,Cr with ij = (1 — & = (1 — to vary, however, such determinants with different coefficients can be combined to form the nine terms with independent coefficients.

The ninth and the tenth terms axe one representation of the two orthogonal singlet states with all four orbitals ^j, (^j, ÿr, and (f>s occupied. The ninth term consists of orbitals (j>i and j coupled as a singlet and the orbitals r and coupled as a singlet. The tenth term results from constructing a state with the orbitals and (f>s coupled as a singlet and the orbitals r coupled as a singlet, and orthogonalizing this state to the ninth term. In the same way that the ninth term is formed using spatial orbitals

(f>a = and b = {irr + ^ss)-, the tenth term is formally constructed using spatial orbitals

c = + f-s^s) and + ^p^r), and orthonolization to the ninth term. 202 APPENDIX E RECURSION RELATIONS FOR GAUSSIAN INTEGRALS

Obara and Saika have derived recursion relations for molecular integrals involv­

ing Cartesian Gaussian basis functions.®^ In accordance with their notation, we

write a general Cartesian Gaussian function as

(a,a, A) = (x — Ax)^^{y — Ay)^^{z — Az)°^ exp(—^a(r — A)^) . (E-1)

As examples of how the recursion relations are built up, considered the three

integrals J dr ya(r; Ca, A) exp(-/i(r - A)^) y&(r; (&, n&,B) , (E.2)

J dr ya(r; Ca, Ra, A) ex p (-/i(r - C)^) (&, n^, B) , and (E.3)

j dr v?a(r; Ca, n«. A) exp(-/i(r - C)^) y&(r; C&, n&, B) . (E.4)

The integral in (E.2) is a three-center overlap integral, i.e.,of the form

(a|c|b) = J dr

For s functions, the integral can be solved, resulting in

(O^IOclOs) = exp(-4(A - B)2) e x p ( - - ^ ( P - C)) , (E.6) C + SC c, C + sc 203 where

The recursion formula is

(a + l,-[c|b) = [Gi - A,-)(a|c|b) + ^ ^ ~ Ni{a.){a. - l,ic|b ) , + 1 (EJ)

+ 2(cTC c) - ^'1^) ’

where

Gi s . “ <1 Af,(a) = » i . Ca + C6 + Cc and a + 1,- symbolizes the Cartesian Gaussian function a with the i-th angular momentum index incremented by 1.

The integral in (E.3) is a nuclear attraction integral,

(a|.4(0)lb) = j dr(,oa(r;Ca,na,A) (,36(r;C6,nft,B) .

The Coulomb interaction can be rewritten as an integral over a gaussian, and so

9 /-oo (a|>t(0)|b) = (aiO clb ),

where

(ajOclb) = J dr v?a(r;Ca,na, A) exp(-u^(r - C)^) y&(r; C&, n^, B) .

The required recursion relation as given by Obara and Saika is

(a+ l,|>l{0)|b)<'") = (a|y((0)|b)M - (P; - C,)(aWO)|b)<’"+‘>

+ ^W ,.(a)((a-l,|>t(0)|b)(’")(a-li|.4(0)|b)('"+») (e .s)

+ ^jV,(b)((a|X(0)|b - lj)('")(a|vKO)|b - 1,)('"+»).

204 Here auxiliary nuclear attraction integrals are defined as

(=|^(0)|b)("') =

(0/i|^(0)|0s)<’"l = y exp(«A - B)2)f„(y) , (E.IO) where

Fm{U)= f\tê^ex^{-Ut^) , U = a ^ - C f. (E.ll) Jo We evaluated the integrals for the Fm{U) with a symbolic manipulator, which leads to expressions with error functions.

The integral in (E.4) can be handled with the recursion relation (E.8), the only difference being that the Fm{U) are replaced by

f 1 Gm{U)= / df (l-t2)”»exp(-17(l-t2)) . (E.I2) Jo

Evaluation of these integrals leads to similar expressions eis the Fm{U).

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