development and applications of advanced electronic structure methods · 2018. 10. 10. · with...

Development and Applications of Advanced Electronic

Structure Methods

By

Franziska Bell

A dissertation submitted in partial satisfaction of the

requirements for the degree of

Doctor in Philosophy

in

Chemistry

in the

Graduate Division

of the

University of California, Berkeley

Committee in charge:

Professor Martin Head-Gordon, ChairProfessor William H. Miller

Professor Alexis T. Bell

Spring 2012

Development and Applications of Advanced Electronic Structure Methods

Copyright 2012

by

Franziska Bell

Abstract

Development and Applications of Advanced Electronic

Structure Methods

by

Franziska Bell

Doctor of Philosophy in Chemistry

University of California, Berkeley

Professor Martin Head-Gordon, Chair

This dissertation contributes to three different areas in electronic struc-ture theory. The first part of this thesis advances the fundamentals of orbitalactive spaces. Orbital active spaces are not only essential in multi-referenceapproaches, but have also become of interest in single-reference methods asthey allow otherwise intractably large systems to be studied. However, de-spite their great importance, the optimal choice and, more importantly, theirphysical significance are still not fully understood. In order to address thisproblem, we studied the higher-order singular value decomposition (HOSVD)in the context of electronic structure methods. We were able to gain a phys-ical understanding of the resulting orbitals and proved a connection to un-relaxed natural orbitals in the case of Møller-Plesset perturbation theory tosecond order (MP2). In the quest to find the optimal choice of the activespace, we proposed a HOSVD for energy-weighted integrals, which yieldedthe fastest convergence in MP2 correlation energy for small- to medium-sizedactive spaces to date, and is also potentially transferable to coupled-clustertheory.

In the second part, we studied monomeric and dimeric glycerol radi-cal cations and their photo-induced dissociation in collaboration with Prof.Leone and his group. Understanding the mechanistic details involved in theseprocesses are essential for further studies on the combustion of glycerol and

1

carbohydrates. To our surprise, we found that in most cases, the experi-mentally observed appearance energies arise from the separation of productfragments from one another rather than rearrangement to products.

The final chapters of this work focus on the development, assessment, andapplication of the spin-flip method, which is a single-reference approach, butcapable of describing multi-reference problems. Systems exhibiting multi-reference character, which arises from the (near-) degeneracy of orbital en-ergies, are amongst the most interesting in chemistry, biology and materialsscience, yet amongst the most challenging to study with electronic structuremethods. In particular, we explored a substituted dimeric BPBP moleculewith potential tetraradical character, which gained attention as one of themost promising candidates for an organic conductor. Furthermore, we ex-tended the spin-flip approach to include variable orbital active spaces andmultiple spin-flips. This allowed us to perform wave-function-based studiesof ground- and excited-states of polynuclear metal complexes, polyradicals,and bond-dissociation processes involving three or more bonds.

2

.

To allwho walked this path with me,

but most of all,my beloved mother.

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Contents

1 Introduction 11.1 Elementary Electronic Structure Theory . . . . . . . . . . . . 11.2 Wavefunction based Methods . . . . . . . . . . . . . . . . . . 3

1.2.1 Hartree-Fock Theory . . . . . . . . . . . . . . . . . . . 31.2.2 Weak Correlation . . . . . . . . . . . . . . . . . . . . . 5

1.2.2.1 Møller Plesset Perturbation Theory . . . . . . 51.2.2.2 MP2 Theory . . . . . . . . . . . . . . . . . . 61.2.2.3 Configuration Interaction Theory . . . . . . . 71.2.2.4 Coupled-Cluster Theory . . . . . . . . . . . . 8

1.2.3 Strong Correlation Methods . . . . . . . . . . . . . . . 91.2.3.1 MCSCF, CASSCF and RASSCF . . . . . . . 9

1.2.4 Single Reference-based Approaches for Strong Corre-lation Problems . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Reduction of the Parameter Space . . . . . . . . . . . . . . . . 131.4 Density Functional Theory . . . . . . . . . . . . . . . . . . . . 15

2 HOSVD in Quantum Chemistry 172.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 182.1.2 Notation and Basic Definitions . . . . . . . . . . . . . 182.1.3 Overview of Tensor Decompositions and Their Rela-

tion to Quantum Chemistry . . . . . . . . . . . . . . . 192.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.1 Review of HOSVD without any numerical approxima-tions such as rank truncation (untruncated HOSVD) . 22

2.2.2 rank-(R1, R2, ..., Rd) truncated HOSVD . . . . . . . . . 232.2.3 Computational cost of HOSVD . . . . . . . . . . . . . 24

2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

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2.3.1 Numerical Results . . . . . . . . . . . . . . . . . . . . 242.3.2 Connection between HOSVD T2 and MP2 Natural Or-

bitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.3.3 Higher Order Orthogonal Iterations (HOOI) vs. HOSVD 40

2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 HOSVD of Energy-Weighted Integrals 453.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.2 Notation and Basic Definitions . . . . . . . . . . . . . . . . . . 473.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3.1 Overview of the HOSVD . . . . . . . . . . . . . . . . . 473.3.2 HOOI vs. Orbital Optimization for MP2 . . . . . . . . 493.3.3 Proposed Coordinate Transformation for MP2 . . . . . 52

3.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 573.5 Transferability to Coupled-Cluster Methods . . . . . . . . . . 603.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4 Glycerol Photodissociation 654.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2.1 Experimental . . . . . . . . . . . . . . . . . . . . . . . 674.2.2 Computational . . . . . . . . . . . . . . . . . . . . . . 68

4.2.2.1 Neutral and Radical Conformers . . . . . . . 684.2.2.2 Ionization Potentials . . . . . . . . . . . . . . 694.2.2.3 Transition States . . . . . . . . . . . . . . . . 694.2.2.4 Choice of Functional . . . . . . . . . . . . . . 704.2.2.5 Dimeric Glycerol . . . . . . . . . . . . . . . . 70

4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 704.3.1 Experimental Measurements . . . . . . . . . . . . . . . 704.3.2 Monomeric Glycerol . . . . . . . . . . . . . . . . . . . 75

4.3.2.1 Neutral Conformers . . . . . . . . . . . . . . 754.3.2.2 Radical Cation Conformers . . . . . . . . . . 764.3.2.3 Water Loss (74 m/z) . . . . . . . . . . . . . . 834.3.2.4 62 m/z . . . . . . . . . . . . . . . . . . . . . 874.3.2.5 61 m/z . . . . . . . . . . . . . . . . . . . . . 894.3.2.6 60 m/z . . . . . . . . . . . . . . . . . . . . . 904.3.2.7 45 m/z . . . . . . . . . . . . . . . . . . . . . 914.3.2.8 44 m/z . . . . . . . . . . . . . . . . . . . . . 92

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4.3.2.9 43 m/z . . . . . . . . . . . . . . . . . . . . . 934.3.2.10 Formation of CHxO

+ and smaller fragment ions 944.3.2.11 Comparison to Neutral/Protonated Glycerol . 94

4.3.3 Dimeric Glycerol . . . . . . . . . . . . . . . . . . . . . 954.3.3.1 153 m/z . . . . . . . . . . . . . . . . . . . . . 974.3.3.2 136 m/z . . . . . . . . . . . . . . . . . . . . . 974.3.3.3 93 m/z and 185 m/z . . . . . . . . . . . . . . 98

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5 Substituted PBPB Dimers 1015.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.2 Computational Details . . . . . . . . . . . . . . . . . . . . . . 1045.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 105

5.3.1 Molecular Geometries . . . . . . . . . . . . . . . . . . . 1055.3.2 Relative Stability . . . . . . . . . . . . . . . . . . . . . 1105.3.3 Communication between Diradical Sites . . . . . . . . 1115.3.4 Radical Character . . . . . . . . . . . . . . . . . . . . . 1135.3.5 Low-lying Excited States and Magnetic Couplings . . . 1175.3.6 Magnetic Couplings . . . . . . . . . . . . . . . . . . . . 118

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6 Restricted Active Space Spin-Flip (RAS-SF) with ArbitraryNumber of Spin-Flips 1276.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.2.1 FCI in the active space, A (RAS-II) . . . . . . . . . . . 1316.2.2 Properties of RAS-SF . . . . . . . . . . . . . . . . . . . 132

6.3 Computational Details . . . . . . . . . . . . . . . . . . . . . . 1326.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 133

6.4.1 Bond Dissociation . . . . . . . . . . . . . . . . . . . . . 1336.4.2 Binuclear Metal Complexes . . . . . . . . . . . . . . . 134

6.4.2.1 µ-hydroxo-bis[pentaaminechromium(III)] Cation1356.4.2.2 µ-oxo-bis[pentaamminechromium-(III)] . . . . 1356.4.2.3 trans-[HO-Cr(cyclam)-NC-Cr(CN)5]

− . . . . . 1376.4.2.4 Co2O4 . . . . . . . . . . . . . . . . . . . . . . 1386.4.2.5 [(TPA*)Co(II)(DHBQ2−)Co(II)(TPA*)]2+ . . 139

6.4.3 Organic Polyradicals . . . . . . . . . . . . . . . . . . . 1406.4.3.1 Linear Carbenes . . . . . . . . . . . . . . . . 141

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6.4.3.2 Branched Carbenes . . . . . . . . . . . . . . . 1436.4.3.2.1 Polycarbene m = 3 . . . . . . . . . . . . . . . . . . . . 1436.4.3.2.2 Catenated Closs Radicals . . . . . . . . . . . . . . . . 144

6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7 Appendix: Glycerol Photodissociation 179

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List of Figures

2.1 HOSVD procedure. Adapted from [1]. . . . . . . . . . . . . . 232.2 Comparison of the original T2 amplitudes and those decom-

posed using the HOSVD (H2/6-31G(3df,3pd)). . . . . . . . . 252.3 Percent MP2 correlation energy vs. number of amplitudes

included (H2/6-31G**) . . . . . . . . . . . . . . . . . . . . . . 262.4 Percent MP2 correlation energy vs. number of amplitudes

included (H2/6-31++G**) . . . . . . . . . . . . . . . . . . . . 272.5 recovery of % MP2 correlation energy vs. number of virtual

orbitals included for H2 at equilibrium bond distance in variousbasis sets. Box: the four most correlating virtual orbitals. . . . 32

2.6 recovery of % MP2 correlation energy vs. number of virtualorbitals included for linear H4 (R1 = 0.76Å, R2 = 1.2 Å) invarious basis sets. Box: the eight most correlating virtualorbitals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.7 recovery of % MP2 correlation energy vs. active space (oc-cupied, virtual) included for linear N2 at equilibrium bonddistance in various basis sets. Boxes: the five most correlat-ing virtual orbitals for HOSVD and HF. (HOSVD order inbrackets) (cutoff: 0.3) . . . . . . . . . . . . . . . . . . . . . . 36

2.8 recovery of % MP2 correlation energy vs. number of virtualorbitals included for N2 at equilibrium bond distance in variousbasis sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.9 HF and HOSVD orbitals for H2/6-31G**, as well as MO dif-ferences (cutoff: 0.15) . . . . . . . . . . . . . . . . . . . . . . . 38

2.10 HF and HOSVD orbitals for H4/6-31G** (R1 = 0.76 Å, R2 =1.2Å), as well as MO differences (cutoff: 0.15) . . . . . . . . . 39

2.11 HF and HOSVD occupied orbitals for N2/6-31G* (at equilib-rium), as well as MO differences (cutoff: 0.15) . . . . . . . . . 42

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2.12 HF and HOSVD virtual orbitals for N2/6-31G* (at equilib-rium), as well as differences in their coefficients (cutoff: 0.15) . 43

2.13 HOSVD occupied-virtual correlating orbital pairs for CH4 (cut-off: 0.2) (6-31G**; MP2 optimized geometry) . . . . . . . . . 44

2.14 Adapted from [1]. Alternating least squares algorithm to com-pute a rank-(R1, R2, ..., Rd) Tucker decomposition for a tensor,T ∈ RI1×I2×...×Id . Also known as the higher-order orthogonaliteration (HOOI). . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.1 Comparison of HOOI of MP2 amplitudes and norm-preservingorbital-optimized MP2 (H2O, cc-pVDZ) . . . . . . . . . . . . . 50

3.2 Extension to Figure 3.1: Energetic differences (a.u.) in theHOOI of MP2 amplitudes and norm-preserving orbital-optimizedMP2 (H2O, cc-pVDZ) . . . . . . . . . . . . . . . . . . . . . . 51

3.3 HOOI procedure. Adapted from [2]. . . . . . . . . . . . . . . 53

3.4 Recovery of percent MP2 correlation energy vs. number ofvirtual orbitals included for CH4 at its equilibrium geometry(cc-pVDZ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.5 Recovery of percent MP2 correlation energy vs. number ofvirtual orbitals included for H2O at its equilibrium geometry(cc-pVDZ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.6 Recovery of percent MP2 correlation energy vs. number ofvirtual orbitals included for H2O at its equilibrium geometry(cc-pVTZ). For clarity only the first 20 of 53 virtual orbitalsare shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.7 Recovery of percent MP2 correlation energy vs. number ofvirtual orbitals included for CO at its equilibrium geometry(cc-pVDZ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.1 Photoionization TOF mass spectrum of glycerol in (a) super-sonic expansion, (b) effusive source and (c) D5-glycerol in aeffusive source at 10.5 eV . . . . . . . . . . . . . . . . . . . . . 72

4.2 (a) Normalized PIE curves for the parent ion C3H8O+3 . The

adiabatic IE is found to be 9.4±0.1 eV (supersonic expan-sion). (b)-(d) Normalized PIE curves for C3H6O

+2 , C2H6O

+2

and C2H4O+, respectively. Red circles: supersonic expansion,

black triangles: effusive conditions. AEs shown are from thesupersonic expansion. . . . . . . . . . . . . . . . . . . . . . . 73

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4.3 The energetically two lowest neutral gas-phase conformers.Left: conformer 95, right: conformer 100 . . . . . . . . . . . . 76

4.4 Representative conformers from each of the main sub-classesfor gas-phase monomeric radical glycerol. . . . . . . . . . . . . 80

4.5 Proposed mechanisms for water loss (74 m/z), concerted reac-tion involving a 6-membered proton-transfer . . . . . . . . . . 85

4.6 Other mechanisms considered for water loss (74 m/z), 5-memberedtransition state . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.7 Other mechanisms considered for water loss (74 m/z), directepoxide formation, ether intermediate, and C-H abstraction . . 86

4.8 Other mechanisms considered for water loss (74 m/z), via C-2bound H2O intermediate . . . . . . . . . . . . . . . . . . . . . 87

4.9 Energetic overview of the excluded water loss channels (ωB97X/6-31+G∗∗) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.10 intermediate 1, INT1 . . . . . . . . . . . . . . . . . . . . . . . 89

4.11 Proposed mechanisms for formation of fragment 62 m/z. . . . 89

4.12 IRC following reactant (radical conformer 100) to the productcomplex. Full separation to products also does not feature anexit barrier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.13 Proposed mechanism for formation of fragment 61 m/z. . . . . 91

4.14 Proposed mechanisms for formation of fragment 60 m/z. . . . 91


4.16 Proposed mechanisms for water loss (74 m/z), formaldehydeloss (62 m/z) and formation of 44 m/z (vinyl alcohol) . . . . . 93


4.18 Cartoon of the lowest energy dimer structure (conformer 1),highlighting the hydrogen bonding network . . . . . . . . . . . 96

4.19 Spin density in dimeric glycerol radical cation (cutoff: 0.02) . 97

5.1 Overview of the most important structures. Para and metaconformations are specified by p- and m- prefixes. A and B la-belling indicates planar (1,3-diborata-2,4-diphosphoniocyclobutane-1,3-diyl) and bicycle (1,3-diborata-2,4-diphosphoniobicyclo[1.1.0]butane)rearrangements of the PBPB moieties, respectively. Num-bers 1, 2 and 3 correspond to PBPB monomer, phenyl-PBPBmolecule and PBPB dimer, respectively. . . . . . . . . . . . . 103

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5.2 Atomic labeling of PBPB used in the description of geomet-rical parameters (Tables 7.1-5.3). Although the B-B bond isnot represented, labels are applicable for both, planar (A) andbicyclic (B) molecules. . . . . . . . . . . . . . . . . . . . . . . 106

5.3 Delocalization energy in para isomers. Delocalization energyis defined as the difference between the energy cost of twistingthe phenyl ring (a) and two times the twisting energy withoutone of the PBPB units (b). The same scheme is applied forfully (p-1A, R = iPr, R = tBu) and H-substituted (p-1AH,R, R = H) molecules. . . . . . . . . . . . . . . . . . . . . . . . 112

5.4 B3LYP/6-31G(d) optimized geometry of m-3A with some shortH· · ·H distances indicated (in Å). . . . . . . . . . . . . . . . . 113

5.5 Diagrammatic representation of the quadruple cluster excita-tion from the HOMO-1 and HOMO to the LUMO and LUMO+1orbitals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.6 State energy diagram (in kcal/mol) computed at RAS-2SF/6-31G(d) computational level for p-3A, p-3AH, m-3A and m-3AH. All energies are given with the respective ground stateas reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.7 RAS-2SF natural orbitals of p-3AH and m-3AH. RAS-2SFand CASSCF(4,4) (in italics) orbital occupations are also in-dicated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.8 Four paramagnetic center model for the para and meta sys-tems. First neighbors exchange constants (σ and λ) are indi-cated by arrows between centers. . . . . . . . . . . . . . . . . 124

5.9 Short (σ) and long (λ) range energy splitting in the four para-magnetic centers models of Figure 5.3.6. A and B indicatesymmetrical and asymmetrical rotation around the principalaxis, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.1 Orbital subspaces in the RAS-SF approach. Adapted from [3]. 1316.2 Dissociation curve for the N2 molecule (cc-pVTZ) . . . . . . . 1336.3 Binuclear Cr(III)-Cr(III) complex, [Cr2(NH3)10(OH)]

5+. Hy-drogen atoms omitted for clarity. . . . . . . . . . . . . . . . . 136

6.4 Binuclear Cr(III)-Cr(III) complex, [Cr2(NH3)10(O)]4+ . . . . . 137

6.5 trans-[HO-Cr(cyclam)-NC-Cr(CN)5]− . . . . . . . . . . . . . . 138

6.6 Co2O4. Structure from [4]. . . . . . . . . . . . . . . . . . . . . 1396.7 Binuclear Co2(II) complex, [(TPA*)Co(II)(DHBQ

2−)CO(II)(TPA*)]2+140

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6.8 One- and two-dimensional carbenes. (n = 1− 5, m = 3, 6, 9) . 1436.9 Catenated Closs Radicals: Structure 1 . . . . . . . . . . . . . 1496.10 Model systems used in [5] to study relative stabilities of cate-

nated Closs radicals. . . . . . . . . . . . . . . . . . . . . . . . 149

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List of Tables

2.1 Computational cost of HOSVD . . . . . . . . . . . . . . . . . 24

2.2 HOSVD vs. Original compression factors γOrig/HOSVD for agiven % correlation energy - Pople style bases; Hn: R1 = 0.76,R2 = 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3 HOSVD vs. Original compression factors γOrig/HOSVD for agiven % correlation energy - Dunning style bases; Hn: R1 =0.76, R2 = 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4 HOSVD compression factors, γHOSVD/HOSVDfull for a given per-cent correlation energy - Dunning bases; Hn: R1 = 0.76, R2 = 1.2 31

2.5 The 14 largest amplitudes for N2 (6-31G*; MP2 optimizedgeometry) in HOSVD and Original T2. Orbital labellings ac-cording to HOSVD. For open shell singlets only one of the twopossible configurations is listed. A blank field indicates thatthis amplitude is not amongst the 14 largest ones. Amplitudesare ordered according to HOSVD results. Values in bracketsindicate the ordering for original amplitudes. . . . . . . . . . . 33

2.6 The 9 largest amplitudes for H2 (6-31G** ; MP2 optimizedgeometry) in HOSVD and Original T2. Orbital labellings ac-cording to HOSVD. For open shell singlets only one of the twopossible configurations is listed. A blank field indicates thatthis amplitude is not amongst the 9 largest ones. Amplitudesare ordered according to HOSVD results. Values in bracketsindicate the ordering for original amplitudes . . . . . . . . . . 33

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2.7 The 16 largest amplitudes for CH4 (6-31G**; MP2 optimizedgeometry) in HOSVD and Original T2. Orbital labellings ac-cording to HOSVD. For open shell singlets only one of the twopossible configurations is listed. A blank field indicates thatthis amplitude is not amongst the 16 largest ones. Amplitudesare ordered according to HOSVD results. Values in bracketsindicate the ordering for original amplitudes. . . . . . . . . . . 34

2.8 %MP2 energy for H2 [(1, 1) active space] and two H2 moleculesat large distance (R = 5.0 Å), [(2, 2) active space] . . . . . . . 37

4.1 Appearance energies (in eV ±0.1) measured in the dissocia-tive photoionization of glycerol and the corresponding frag-ment ions of its isotopologues. Because of background waterin the chamber, the other m/z = 18 isomer CD+3 could not bemeasured from the PIE curve. . . . . . . . . . . . . . . . . . . 74

4.2 Relative energies of the lowest neutral gas-phase conform-ers. Conformer labelling adopted from [6]. The MP2-TQ-extrapolation was carried out on B3LYP/6-311++G(p,d) ge-ometries. (freq) indicates frequency corrected MP2-TQ-extrapolatedvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.3 Ratio of the two lowest lying conformers 100 : 95 at the ex-perimental temperatures . . . . . . . . . . . . . . . . . . . . . 75

4.4 Vertical ionization energies (VIEs) for gas phase monomericglycerol conformers 100 and 95 (in eV). Geometry for the neu-tral species was optimized at B3LYP/6-311++G(p,d) . . . . . 76

4.5 Adiabatic ionization energies (eV) for monomeric glycerol con-formers 100 and 95 (Difference between vertical and adiabaticionization energies (eV)). Geometries optimized at B3LYP/6-311++G(p,d). . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.6 Structural features of representative gas-phase monomeric rad-ical glycerol conformers for each of the sub-classes, as well asthe lowest 10 conformers. Basis set: 6-311++G(p,d). Rel. Edenotes the energies relative to radical conformer 100 (kcal/mol).O...H lists the shortest hydrogen bond. CT/TT indicateswhether the shortest hydrogen bond occurs between a Cen-tral and a Terminal (CT) or two Terminal (TT) OH-groups. . 79

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4.7 Relative energies (kcal/mol) for representative gas-phase radi-cal glycerol conformers for each of the sub-classes as well as thelowest 10 conformers. Geometries optimized at the B3LYP/6-311++G(p,d) level of theory, except for ωB97X(MP2), whichwere optimized with MP2/6-311++G(p,d). Unless stated, thebasis set for the single point calculations is 6-311++G(2df,2pd). 81

4.8 Summary of activation barriers (in eV) for the photodissoci-ation of monomeric gas-phase glycerol. Bold: predominantpathway proposed. . . . . . . . . . . . . . . . . . . . . . . . . 82

4.9 Relative energies (in kcal/mol) and key structural parame-ters for dimeric gas-phase glycerol. Structures were optimizedat the B3LYP/6-311++G(p,d) level of theory. The type ofhydrogen bonds is either TT (between the two terminal OHgroups or CT (between a terminal and a central OH group) . . 99

4.10 Key structural parameters for gas-phase radical glycerol dimer,conformer 1. Radical indicates the glycerol molecule that dis-plays most of the spin density in the dimer. TT O...H is thehydrogen bond distance between the two terminal OH groups. 99

4.11 Appearance energies (AEs) (in eV ±0.1) measured in the dis-sociative photoionization of glycerol between 93 m/z and 185m/z (supersonic beam) . . . . . . . . . . . . . . . . . . . . . . 100

5.1 Experimental and B3LYP/6-31G(d) optimized most relevantatomic distances (in Å) and the BPPB dihedral angle (δ indegrees) of planar singlet structures (A). Values in parenthesiscorrespond to the H-substituted models. Atomic labeling isindicated in Figure 5.2. . . . . . . . . . . . . . . . . . . . . . . 107

5.2 Experimental and B3LYP/6-31G(d) optimized most relevantatomic distances (in Å), and the BPPB dihedral angle (δ indegrees) of bicycle singlet structures (B). Values in parenthesiscorrespond to the H-substituted models. Atomic labeling isindicated in Figure 5.2. . . . . . . . . . . . . . . . . . . . . . . 108

5.3 B3LYP/6-31G(d) optimized most relevant atomic distances(in Å) and the BPPB dihedral angle (δ in degrees) of H-substituted triplet and quintet structures. Values in paren-thesis for the p-3H and and m-3H triplet structures corre-spond to the non-planar PBPB unit (B). Atomic labeling isindicated in Figure 5.2. . . . . . . . . . . . . . . . . . . . . . . 109

xv

5.4 B3LYP and PP single point energies (in kcal/mol) of full andH-substituted 3 isomers relative to p-3A and p-3AH, respec-tively. All energies have been computed with the 6-31G(d)basis set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.5 Planar to bicyclic SCF energy and thermodynamic potentialof 1 and 3 compounds and their H-substituted analogues. Allvalues were computed at the B3LYP/6-31G* level. Energiesare given in kcal mol−1 and entropy values in cal mol−1K−1.∆G has been computed at 298K. . . . . . . . . . . . . . . . . 111

5.6 Computed effective unpaired electrons NU (Eq. 5.1) of planardimer molecules. . . . . . . . . . . . . . . . . . . . . . . . . . 114

5.7 Cluster quadruple amplitude excitations (t4) from PQ, CASSCF(4,4)and RAS-2SF computations of p-3AH and m-3AH. Interme-diate normalization of the wave function has been considered. 117

5.8 Vertical excitation energies (in kcal/mol) of p-3AH/p-3A andm-3AH/m-3A low-lying singlets, triplets and quintet statescomputed by RAS-2SF/6-31G(d). . . . . . . . . . . . . . . . 117

5.9 Magnetic coupling constants (in kcal/mol) computed by RAS-2SF/6-31G(d) of the p-3A (p-3AH) and m-3A (m-3AH)molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.1 J12 coupling constants for [Cr2(NH3)10(OH)]5+ in cm−1. . . . . 136

6.2 J12 coupling constants for [Cr2(NH3)10(O)]4+ in cm−1. . . . . . 137

6.3 J12 coupling constants for trans-[HO-Cr(cyclam)-NC-Cr(CN)5]−

in cm−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386.4 Ground and excited states for structure X of Co2O4 (in in

kcal/mol) (Figure 6.6). BS-DFT results from [4]. . . . . . . . . 1396.5 J12 coupling constants for [(TPA*)Co(II)(DHBQ

2−)Co(II)(TPA*)]2+

in cm−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.6 Relative energies (kcal/mol) between S=n and S=0 states of

linear n-carbenes n = 1−4 (Figure 6.8). Geometries restrictedto the C2v point group. . . . . . . . . . . . . . . . . . . . . . 143

6.7 Relative energies (kcal/mol) of linear carbene n = 1 (Figure6.8) RAS-SF(2,2)/6-31G∗. Geometries optimized at the spinstates indicated in the first row using UB3LYP/6-31G∗. . . . . 144


xvi


6.10 Relative energies (kcal/mol) of linear carbene n = 4 (Figure6.8) RAS-SF(8,8)/6-31G∗. Geometries optimized at 〈S2〉 = 20using UB3LYP/6-31G∗. 〈S2〉 given in parentheses for unre-stricted calculations. . . . . . . . . . . . . . . . . . . . . . . . 146

6.11 Relative energies (kcal/mol) of linear carbene n = 3 (Figure6.8) UB3LYP/6-31G∗. 〈S2〉 given in parentheses. Geometriesoptimized at the spin states indicated in the first row usingUB3LYP/6-31G∗. . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.12 Relative energies (kcal/mol) of linear carbene n = 3 (Figure6.8) UBP86/6-31G∗. 〈S2〉 given in parentheses. Geometriesoptimized at the spin states indicated in the first row usingUB3LYP/6-31G∗. . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.13 Relative energies (kcal/mol) of two-dimensional carbene m =3 (Figure 6.8). Geometries optimized at the spin states indi-cated in the first row using UB3LYP/6-31G∗. Single point cal-culations carried out using the 6-31G∗ basis. 〈S2〉 for UB3LYPenergies given in parentheses. . . . . . . . . . . . . . . . . . . 148

6.14 Relative energies (kcal/mol) of various spin states for the hexarad-ical shown in Figure 6.9. Geometry optimized for the high-spinseptet with UB3LYP/6-31G∗. Single point calculations arecarried out with the 6-31G∗ basis. 〈S2〉 for unrestricted cal-culations given in parentheses. 200 of the 612 virtual orbitalswere frozen in the RAS-SF calculation. . . . . . . . . . . . . . 150

7.1 Structural features of gas-phase monomeric radical glycerolconformers. Basis set: 6-311++G(p,d). Rel. E denotes theenergies relative to radical conformer 100 (kcal/mol). O...Hlists the shortest hydrogen bond. CT/TT indicates whetherthe shortest hydrogen bond occurs between a Central and aTerminal (CT) or two Terminal (TT) OH-groups. . . . . . . . 179

7.2 Relative energies (kcal/mol) for gas-phase radical glycerol con-formers. Geometries optimized at the B3LYP/6-311++G(p,d)level of theory, except for ωB97X(MP2), which were optimizedwith MP2/6-311++G(p,d). Unless stated, the basis set for thesingle point calculations is 6-311++G(2df,2pd) . . . . . . . . . 182

xvii

Acknowledgments

I would like to thank everyone who supported me during this PhD. Firstand foremost, Professor Martin Head-Gordon, a wonderful advisor, who gaveme the opportunity to work in his group and shared many invaluable dis-cussions and insights with me. I am very much indebted to the entire Head-Gordon group, not only for their academic support, but also for many unfor-gettable fun moments. In particular, I would like to acknowledge Dr. DanielLambrecht, Dr. John Parkhill, Dr. David Casanova, Dr. Alex Thom andDr. Keith Lawler. Many thanks also to Leslie Silvers for all her help andconversations. I also greatly appreciate the inspiring graduate classes givenby Professor William H. Miller, Professor Martin Head-Gordon, ProfessorPhillip Geissler, Professor Stephen R. Leone and Professor Daniel M. Neu-mark. I would also like to express my gratitude to my family, especially mymother, Sabine, and my friends, in particular Dr. Dmitry Zubarev, TammyVinson, Chamila Sumanasekera, and Otto and Inge Klopka.

xviii

Chapter 1

Introduction

1.1 Elementary Electronic Structure Theory

The main application of electronic structure theory is to compute the prop-erties of atoms and molecules, such as ground- and excited-state energies,forces, force constants, dipole moments, magnetic susceptibilities, and manyother quantities. For molecules, this requires finding approximations to themany-body, non-relativistic, time-independent Schrödinger equation [7]

H Ψ(r,R) = E Ψ(r,R), (1.1)

where {r} and {R} are the collective electronic and nuclear coordinates,respectively, Ψ(r,R) is the many-body wavefunction, E is the correspondingenergy, and H is the molecular Hamiltonian for N electrons and M nuclei.In atomic units, it takes the form

H =− 12

N∑i

∇2i −M∑A

1

2MA∇2A −

N∑i

M∑A

ZA|ri −RA|

+N∑i

N∑j>i

1

|ri − rj|+

M∑A

M∑B>A

ZAZB|RA −RB|

.

(1.2)

In the above equation MA denotes the mass ratio of nucleus A and anelectron, ZA is the atomic number of nucleus A, and ri and RA are thecoordinates of the ith electron and Ath nucleus, respectively. Equation (1.2)

1

can be more compactly written as

H = T̂elec(r) + T̂nuc(R) + V̂elec,nuc(r,R) + V̂nuc,nuc(R) + V̂elec,elec(r). (1.3)

The first term in equations (1.2) and (1.3) is the kinetic energy opera-tor affiliated with the electrons, the second term refers to the kinetic energyoperator of the nuclei, and the third term describes the Coulomb attractionbetween electrons and nuclei. The remaining two terms represent the repul-sion between electrons and between nuclei, respectively.

The term V̂elec,nuc(r,R) prohibits the Hamiltonian, H , from being sepa-rated into electronic and nuclear parts which would permit the overall wave-function to be written as a product of electronic and nuclear terms

Ψ(r,R) = ψ(r; R)Θ(R). (1.4)

The commonly used Born-Oppenheimer approximation [8] allows such aseparation by assuming that due to the large mass difference between protonsand electrons (1 : 1836), electron motion will be very rapid compared tonuclear motion. Thus, as a first approximation the nuclei can be assumedto be fixed. The Born-Oppenheimer approximation is therefore also oftenreferred to as the clamped nuclei approximation. Under this assumption, theconditions for the electronic and nuclear wavefunction become

Helecψ(r; R) = Eelecψ(r; R) (1.5)

where

Helec = −N∑i

1

2∇2i −

N∑i

M∑A

ZA|RA − ri|

+N∑i,j

1

|ri − rj|, (1.6)

and

H Ψ(r,R) = (T̂nuc(R) + Helec)ψ(r; R)Θ(R). (1.7)

Equation (1.6) shows that the electronic wavefunction, ψ(r; R) ∈ C3N ,making the exact solution to the Schrödinger equation intractable, except forthe simplest cases.

2

So far, we have omitted the requirement that, for a full description ofan electron, both its spatial, r and spin coordinates, ω must be specified,which are collectively termed x = {r, ω}. In the non-relativistic Schrödingerequation, however, the electronic Hamiltonian is independent of spin. In or-der to account for the spin dependence, two spin functions α(ω) and β(ω)are introduced. Furthermore, the constraint that identical particles are in-distinguishable has to be imposed. From this it follows, that many-electronwavefunctions must be antisymmetric with respect to the interchange of anytwo coordinates, xi and xj, often referred to as the antisymmetry or Pauliexclusion principle. In a mathematical context, this requirement is achievedby re-writing the wavefunction as (linear combinations of) Slater determi-nants [9, 10].

1.2 Wavefunction based Methods

1.2.1 Hartree-Fock Theory

The starting point for most wavefunction based approaches is the mean-field or Hartree-Fock approximation. This method employs a single Slaterdeterminant composed of one-electron spin orbitals, φi(xi), and defines thesimplest antisymmetric wavefunction for describing an N -electron system

ψHF(x1,x2, ...,xN) = |φ1(x1)φ2(x2) · · ·φN(xN)|. (1.8)

In order to derive the Hartree-Fock equations, the expectation value

EHF =〈ψHF|Helec|ψHF〉〈ψHF|ψHF〉

(1.9)

has to be minimized with respect to the spin orbitals φi(xi), subject to theconstraint that the orbitals remain orthonormal. In equation (1.9), Helecdenotes the electronic Hamiltonian (equation (1.6)).

It has been shown [11], that the following eigenvalue equation providesthe solution to this constrained optimization problem:

f(x1)φi(x1) = �iφi(x1) (1.10)

with the one-electron Fock operator, f , and the energy of the ith orbital, �i.

3

Next, the set of orthonormal one-electron orbitals in equation (1.8) areexpanded in a known basis,

φi(x) =K∑µ

Cωµiχµ i = 1, 2, ..., K (1.11)

This expansion would be exact, if {χµ(r)} formed a complete set. However,in practice, only a small subset of Hilbert space is computationally feasibleand for molecular systems an atom-centered Gaussian basis is typically used.

Expansion in a basis allows the integro-differential equation shown in(1.10) to be written in matrix form as follows

F(C)C = SC�. (1.12)

where

Fµν = −1

2〈µ|∇2|ν〉+

nuc∑A

〈µ| 1|RA − r|

|ν〉+∑i

〈φiµ||φiν〉 (1.13)

Sµν = 〈µ|ν〉 (1.14)

〈pq||rs〉 =∫ ∫

φp(x1)φq(x2)1

|x1 − x2|[φr(x1)φs(x2)− φs(x1)φr(x2)] (1.15)

and the diagonal matrix, �, contains the orbital energies as entries. As indi-cated in equation (1.12), the so-called Roothaan-Hall equations [12, 13] arenonlinear and must be solved iteratively.

Although the Hartree-Fock approximation typically recovers more than99% of the total energy, the remaining 1% is crucial for quantitatively deter-mining relative energies and thus correctly predicting chemical events. Thedifference between the mean-field Hartree-Fock energy, EHF, and the exactnonrelativistic energy, Eexact, in a given (finite) basis defines the (basis set)correlation energy, Ecorr [14]:

Ecorr = Eexact − EHF (1.16)

4

In the following sections various different methods that attempt to re-cover the correlation energy will be discussed. In particular, the distinctionbetween weak and strong correlation problems will be made. In the for-mer case, the wavefunction can be described to a first approximation by theHartree-Fock wavefunction (cHF ≈ 0.9 or greater, equation (1.17))

|ψ〉 = cHF|ψHF〉+ ccorr|ψcorr〉 (1.17)

whereas in the strong correlation case this assumption will not hold andmultiple determinants will contribute considerably, or even equally.

1.2.2 Weak Correlation

1.2.2.1 Møller Plesset Perturbation Theory

Møller-Plesset perturbation theory derives from the Rayleigh-Schrödingermany-body perturbation theory (MBPT) [15, 16], in which the unperturbedzero-order wavefunction is the Hartree-Fock wavefunction [17]. If a singleconfiguration adequately approximates the full solution and the perturbationis small, the exact Hamiltonian H can be treated as a small perturbation, Vfrom the mean-field Hamiltonian, which is the sum of the one-electron Fockoperators (equation 1.19).

H |Ψi〉 = (H0 + λV )|Ψi〉 = Ei|Ψi〉 (1.18)

H0 =N∑i

f(i) (1.19)

Performing a Taylor series expansion of the exact eigenfunctions andeigenvalues in λ, yields

Ei =∞∑k=0

λkE(k)i (1.20)

and

|Ψi〉 =∞∑k=0

λk|Φ(k)i 〉 (1.21)

5

Substituting equations (1.20) and (1.21) into the Schrödinger equation(equation (1.18)) and then collecting powers of λ, yields a recursive equation,which generates progressively higher corrections to the wavefunction andenergy.

(H0 − E(0)i )Φ(λ)i = −V |Φ

(λ−1)i 〉+

λ∑k=1

E(k)i |Φ

(λ−k)i 〉 (1.22)

Corrections to the zero-order energy are obtained from equation (1.22)by projecting from the left with the zero-order (Hartree-Fock) wavefunction

E(λ) = 〈Φ(0)|V |Φ(λ−1)〉 λ > 0 (1.23)

Wavefunction corrections can be obtained from equation (1.22) on left-

multiplying by the inverse of (H0−E(0)i ), which is well-defined if intermediatenormalization is invoked.

|Φ(λ)i 〉 = −(H0 − E(0)i )−1

(V |Φ(λ−1)i 〉 −

λ∑k=1

E(k)i |Φ

(λ−k)i 〉

)(1.24)

1.2.2.2 MP2 Theory

Møller Plesset Theory to second order (MP2) is the most widely used form ofMøller Plesset Perturbation Theory in electronic structure theory. It requiresthe computation of the second-order correction to the energy, E(2). Dueto the two-particle nature of the Hamiltonian, only double excitations willcontribute to the MP1 wavefunction and its final form for canonical orbitalscan be derived as

|Φ(1)i 〉 = −1

4

∑ijab

tijaba†aaia

†baj|ΦHF〉 (1.25)

tijab corresponds to an order-4 tensor comprising amplitudes for all doubleexcitations from occupied orbitals i and j to virtual orbitals a and b and canbe expressed in terms of the antisymmetrized two-electron integrals 〈ij||ab〉defined in equation (1.15) and orbital energies �

6

tijab = −〈ab||ij〉

�a + �b − �i − �j(1.26)

Using the above expression for the MP1 wavefunction and the energycorrection (equation (1.23)), it can be shown that the second order correctionto the energy is given by

E(2) = −14

∑ijab

tijab〈ij||ab〉 = −1

4

∑ijab

|〈ij||ab〉|2

�a + �b − �i − �j. (1.27)

1.2.2.3 Configuration Interaction Theory

The configuration interaction (CI) wavefunction [18], is written as a linearcombination of Slater determinants, involving, to nth order n-tuple excita-tions with respect to a reference wavefunction, typically the Hartree-Fockwavefunction (section (1.2.1)). The expansion coefficients in equation (1.28),but not the molecular orbital coefficients, are variationally determined.

|ψCI〉 =c0|Φ0〉+∑ia

cai |Φai 〉+∑i

Typically, one approximates the FCI expansion by including a limitedsubset of the excitation space, which gives rise to a hierarchy of methods, suchas configuration interaction singles (CIS, Ĉ1), configuration interaction sin-gles and doubles (CISD, Ĉ1+Ĉ2), etc. However, as for other weak-correlationmethods, the configuration interaction method when restricted to only thefirst few terms, requires that the reference wavefunction, |Φ0〉, is an adequateapproximation to |ψCI〉. Another major problem occurs in the configurationinteraction method, with the exception of FCI and CIS, is that it is not size-extensive [11].

In practice, two major forms of CI implementations exist. Spin- andspatially adapted configuration state functions (CSFs) [19] form the basis ofone of the methods, whereas the other involves Slater determinants [20–22].Although the former requires fewer terms, the latter is often used since con-siderable computational speed-ups are possible as the problem can be formu-lated in terms of vector-vector and matrix-vector algebra.

1.2.2.4 Coupled-Cluster Theory

In (single-reference) coupled-cluster (CC) theory, first introduced to the the-oretical chemistry community by Cizek and Paldus [23–27], the wavefunctionis represented in the form of a cluster expansion

|ψCC〉 = eT̂ |ψ0〉 (1.30)

where |ψ0〉 typically refers to the HF wavefunction, and the cluster operatorT̂ up to rank κ can be written as

T̂ =N∑κ

T̂κ (1.31)

with

T̂κ =1

(n!)2

∑i,b

T b1,b2,...,bni1,i2,...,in a†bn...a†b1ai1 ...ain . (1.32)

Truncating operator in equation (1.31) yields methods such as coupled-cluster doubles, CCD, κ = 2 or coupled-cluster singles and doubles, CCSD,κ = 1, 2. In the case of the latter, often a perturbative triples correction,

8

indicated by (T ) is introduced. Today, CC defines one of the most suc-cessful and accurate correlated wavefunction approaches for single-referenceproblems [28]. In particular, the CCSD(T) [29, 30] approach has gained areputation as the ‘gold standard’ in quantum chemistry, because it produceshighly accurate energies. As a result of the exponential ansatz, CC theorypreserves size-extensivity, contrary to the CI method. Just as in the CI ap-proach, CC cannot describe strongly correlated systems without extendingthe excitation manifold to a computationally intractable length.

1.2.3 Strong Correlation Methods

Many problems encountered in chemistry feature electronic degeneracies ornear-degeneracies. These are referred to as multi-reference problems. Ex-amples include transition metals, bond dissociations, radicaloid systems andexcited states. (Near-)degeneracies lead to wavefunctions in which severaldeterminants contribute considerably, rather than just one. Since most elec-tronic structure methods are based on a single reference (c.f. sections 1.2.2.1,1.2.2.3, 1.2.2.4), systems featuring (near-)degeneracies continue to pose con-siderable challenges. Although some single-reference methods outlined in sec-tions 1.2.2.3 and 1.2.2.4 in principle could capture multi-reference character ifa sufficiently long expansion is used, this is typically not practical as the exci-tation level required would be intractably high. Therefore, many different al-ternative approaches have been developed to address the exponentially com-plex strong correlation problem. Noteworthy amongst these are the multi-configurational self-consistent field (MCSCF) approach and its variants, aswell as the density matrix renormalization group (DMRG) [31–38], valencebond methods, such as the spin-coupled (SC-VB) [39–41] and coupled-clustervalence bond (CC-VB) [42,43] theories. Also a broad class of methods existwhich attempt to describe strong correlation problems, but are based on asingle reference. The most pertinent methods to this thesis will be discussedin more detail in the following sections.

1.2.3.1 MCSCF, CASSCF and RASSCF

In multi-configurational self-consistent field (MCSCF) theory [44–49], boththe expansion coefficients, cI , as well as the corresponding orbital coefficients,C (section 1.2.1), are optimized self-consistently.

9

ψMCSCF =∑I

cIΦMCSCFI (1.33)

Here, the trial state function, ΦMCSCFI , may be a single Slater determi-nant or a configuration state function (CSF). CSFs are spin- and, possibly,symmetry-adapted linear combinations of Slater determinants. If the sum inequation (1.33) consists of only one single Slater determinant, the expressionreduces to Hartree-Fock or mean field theory (section 1.2.1).

Contrary to the MCSCF approach, in which individual configurations areselected, in a complete active space SCF (CASSCF) [50, 51] calculation, anactive space is determined by choosing a set of active orbitals and electronsand forming all possible configurations in this subspace. By convention, theactive space is labelled as (nelec, norb). This corresponds to performing a fullconfiguration interaction (FCI) in the active space and thus the CASSCFmethod scales exponentially with the size of the active space. Although theideal active space would be spanned by all valence electrons, in practice, thisis unfeasible for most systems under consideration, as the largest active spacecurrently possible is (16,16) [52]. This constraint requires the user to select anorbital active space, which means the results may strongly depend on the cor-rect chemical intuition for the system under consideration [53]. Even worse,sometimes the physically correct active space is not computationally feasible.Especially, for constructing continuous and smooth potential energy surfacesthe correct choice of active space can pose considerable challenges. In orderto verify the adequacy of the active space in terms of its size and choice, atrial and error approach often becomes necessary. However, for many cases,demonstrating energy convergence with active space size is found to be chal-lenging due to practical limitations [54,55]. State-averaging [56,57], which isused to minimize root flipping [58] also appears as an issue in MCSCF andCASSCF calculations, since the number of states to average over is oftensomewhat arbitrary.

By definition, the CASSCF wavefunction incorporates all static correla-tion (at least if the appropriate active space is possible) [59]. In order torecover the missing dynamical correlation, which comprises the energy dif-ference between FCI and CASSCF [59], various different approaches havebeen applied, such as multi-reference CI (MRCI) [60], CASPT2 [61] and

10

multireference coupled cluster (MRCC) methods [62–65]. All of these maysuffer from the so-called intruder state problem [66, 67], which can lead toconvergence issues or, much worse, unphysical behavior.

Since the computational cost rises so rapidly for the CASSCF method,the restricted acive space SCF (RASSCF) method has been introduced [20,68–70], in which the orbital space is divided into additional subspaces fromand to which truncated CI-type excitations are allowed. Although the di-mensionality compared to a conventional CASSCF calculation can be reducedsubstantially in this approach, its scaling is still considerable and does not re-solve the inherent difficulties of CASSCF. In particular, choosing the correctactive-space partitioning and maximum excitation level may pose additionalproblems. Furthermore the method is no longer size-consistent or orbitalinvariant.

1.2.4 Single Reference-based Approaches for StrongCorrelation Problems

Given the tremendous practical challenges when trying to perform multiref-erence calculations, several single-reference methods have been developedthat attempt to describe systems exhibiting multireference character withthe hope of retaining the simplicity and computational cost of the former.Among these are the active-space coupled-cluster approaches [71–75], orbital-optimized coupled-cluster schemes, such as valence orbital optimized coupled-cluster (VOO-CC) [76–78], and their local variants, including perfect pair-ing (PP) [79, 80], perfect quadruples (PQ) [80, 81], and perfect hextuples(PH) [80,82].

Another group of approaches involves a non-iterative energy correctionto single-reference coupled-cluster methods. These include higher-order, per-turbative corrections based on a similarity-transformed Hamiltonian [83–86],and the method-of-moments (MM) approach, which defines the basis forrenormalized and completely renormalized coupled-cluster (RenCC) meth-ods [87–94].

A very promising method that falls into this category is the spin-flip (SF)approach [3, 95–100], a size-extensive, variational, multistate method, which

11

can be extended to yield spin-eigenfunctions. In the spin-flip approach, ahigh-spin restricted open-shell Hartree-Fock (ROHF) reference is used as astarting point. The desired manifold is then accessed by spin-flipping. Bothsingle and double spin-flips have been explored within coupled cluster the-ory and in a RASSCF framework. The motivation behind the choice ofreference is that the appropriate high-spin wavefunction will be essentiallysingle-reference at all nuclear separations. This stands in contrast to theclosed-shell restricted wavefunction (RHF), which, as discussed, cannot ade-quately describe strong correlation problems. Furthermore, due to the Pauliexclusion principle, the dynamical correlation energy for same-spin electronswill be about an order of magnitude smaller than for opposite-spin elec-trons [101]. The improved description of the ground state reference alsostrongly influences the ability to correctly describe excitation energies [101].Besides this, the high-spin reference allows for an equal treatment of excitedand ground states. Also, contrary to e.g. CASSCF, the spin-flip method doesnot involve orbital optimizations and therefore avoids any attendant issues.Instead, orbital optimization is approximated through single excitations fromthe inactive spaces to the active space and vice versa. This allows the userto detect any inappropriate active space choices, which creates difficulty inCASSCF.

On the border between single- and multi-reference techniques lies a classof methods which makes use of a hybrid approach. In these approachesinformation from, e.g., valence-bond (VB) [102, 103], CASSCF [104] or CIcalculations, such as the important high-order cluster amplitudes, is used ina single-reference framework. Examples of such appraoches include the re-duced multireference coupled-cluster (RMRCC) method [105–110], and tai-lored CC [111,112]. Such methods, however, still depend on the success andcost of the initial multi-reference or VB calculations.

The unrestricted Hartree-Fock (UHF) method appears as the simplestamong all single-reference methods used for strong correlation. Although theenergetics often vastly improve by lifting the constraints of having the samespatial orbitals for α and β electrons, this approach often results in consid-erable spin contamination, which in turn can cause problems with propertycalculations [96]. Spin-projected methods have been introduced to removespin-contamination [113]. The so-called symmetry-projected Hartree-Fock-Bogoliubov (HFB) method has recently been implemented [114], which is

12

based on symmetry-breaking in an active space, followed by spin projection.The success of these methods, however, strongly depends on the amount ofspin contamination introduced.

1.3 Reduction of the Parameter Space

Including higher-order terms in the wavefunction expansions of MBPT, CI orCC, or increasing the number of electrons and orbitals in a CASSCF calcula-tion means that tensors of increasing order will enter the equations, causinga steep rise in computational cost and restricting such high-level calculationsto very small molecules. Even truncating the expansion at low order, theparameter space grows rapidly with the size of the molecule if no furtherconstraints are imposed. Thus, on the quest to push the limits of electronicstructure theory towards larger systems or more accurate treatments, low-parametric representations of the methods discussed above become essential.

The main basis for most of these approaches is to perform unitary orbitaltransformations so as to obtain a more compact description of the wavefunc-tion, which, upon truncation according to a certain parameter aims to max-imize the recovery in correlation energy. Often, such orbitals are determinedfrom computationally lower-scaling methods, such as MP2, and are subse-quently used to direct the choice of active space of higher-level methods, suchas the valence orbital optimized coupled-cluster (VOO-CC) method [76–78].

One research direction based on this concept focuses on exploiting wave-function locality [115]. The pioneering work by Pulay and Saebø, which com-bines the Boys [116] or Pipek-Mezey [117] localization schemes with numer-ical cutoffs determined by spatial distances, indicated substantial computa-tional savings with only small losses in numerical accuracy [115,118–121] andhas inspired local variants of MP2 [120, 122–124] and CC theory [125–128].Possible discontinuities in potential energy surfaces provide the main disad-vantage of this method [129–131]. Many researchers, including those in thegroups of Head-Gordon [130–135] and Scuseria [136,137] have formulated al-ternative theories to address this issue.

The other main school of thought restricts the excitation level withina certain orbital active space. Besides the aforementioned trial-and-error

13

approach guided by the user’s chemical intuition (section 1.2.3.1), many dif-ferent attempts have been made to determine how to most efficiently truncatethe orbital active space using quantitative descriptors. These mainly focuson the compression of the virtual space. The oldest and most commonly en-countered technique forms natural orbitals by diagonalizing the one-particledensity matrix of a particular method. The resulting eigenvalues, also knownas occupation numbers, lie between 0 and 1. These determine the importanceof the corresponding orbital. This often allows truncation of the virtualspace by up to 50%, yet only introducing errors of < 1 kcal/mol. In thefrozen natural orbital (FNO) approach, the virtual orbitals to be retainedin a CC [138–140], optimized doubles (OD) [76–78], or QCISD calculation,are obtained from the virtual-virtual block of the MP2 density, which scalesas O(N5). Recently, this method has been extended to ionized systems (IP-CC) [141]. After determining the FNOs, the final subset of virtual orbitalsare selected by specifying the percentage of the full space to be included(percentage of virtual orbitals, POVO) [139, 140] or by requiring the accu-mulative natural orbital occupation number to lie above a certain thresholdcompared the total number of electrons in the system under consideration(occupation threshold, OCCT) [141]. The resulting computational speed-upwill be 1/(1 − n)N , where n defines the fraction of the full virtual space re-tained in the calculation and the method scales as N .

Other commonly used forms of natural orbitals approximate the pair-natural orbitals, PNO [18, 142–145] (formerly known as pseudo-natural or-bitals [146]). These maximize the interaction of a pair of occupied orbitalswith a linear combination of virtual ones, which is equivalent to forming thenatural orbitals for this specific pair. This procedure introduces localizationwithout the need of selecting spatial domains. Another approach involvesoptimization of the virtual spaces (OVOS) [147–149] to select important vir-tual orbitals. In this approach, the virtual orbitals are divided into an activeand inactive space and the transformation of basis is achieved by performingself-consistent orbital rotations between those two subspaces through invok-ing the Hylleraas functional.

Besides these two main classes, there are several other approaches whichutilize mathematical rather than physical insight in order to reduce the com-putational cost. These include matrix or tensor decomposition methods,such as the Cholesky factorization [150–157], the resolution-of-the-identity

14

approach, also known as density fitting [158, 159], the Laplace transforma-tion [136,160–168], and methods based on matrix or tensor sparsity.

1.4 Density Functional Theory

Density functional theory (DFT) ranks as the most commonly used alterna-tive to wavefunction-based methods and today about 80-90% of all electronicstructure calculations use this approach. Although, unlike wavefunctionbased methods, density functional theory cannot be systematically improved,it often becomes the only available method for large systems due to its low-scaling computational cost (O(N3)). The Hohenberg-Kohn theorems [169]form the theoretical basis for density functional theory. Introduction of theKohn-Sham equations, which use the idea of expressing the problem as afictitious system of non-interacting electrons in an effective potential [170]then made this method viable.

Although the theory is formally exact, the form of the exact exchange-correlation functional is unknown and therefore numerous flavors of densityfunctionals exist to approximate it. Often, the different types of exchange-correlation functionals get grouped according to the so-called ‘Jacob’s lad-der’: local density approximation, LDA, generalized gradient approximation,GGA, that depend on the spin densities and their reduced gradient, the meta-GGAs described via the spin kinetic energy density, the hybrid functionalswhich contain a fraction of exact exchange, range-separated functionals anddouble-hybrids where the DFT orbitals are used in a non-self-consistent MP2-type correction.

Amongst these, the hybrid functional B3LYP [171] is still one of the mostwidely used functionals, despite its known drawbacks of inadequately describ-ing Rydberg [172–176] and charge-transfer excited states [177, 178], barrier-heights [179], and bond dissociation processes due to the self-interaction er-ror [180]. Different attempts have been made to partially resolve this is-sue [180–182], with one promising strategy being the development of range-separated functionals [183–191]. Their inability to describe dispersion inter-actions creates another limitation of most functionals. The simplest way tocorrect for this failure are the −D [192–194] and −XDM corrections [195].However, more elaborate schemes have also been proposed [196].

15

Furthermore, due to the linear response formalism in time-dependentDFT (TDDFT), these methods cannot capture excitations featuring pre-dominantly doubly excited character [176, 197], which become crucial in ex-plaining various phenomena, for example, the singlet fission process [198,199].

Although DFT describes dynamic correlation fairly well, strongly-correlatedsystems pose considerable challenges to DFT [200], as the reference is asingle Slater determinant. Recently, Grimme and Cremer proposed a com-bined DFT/multireference approach, in which DFT orbitals are used in anMRCI [201] or CASSCF [200] calculation to address this issue. Of course,such approaches will not reduce the inherent problems of multi-referencemethods (section 1.2.3.1).

16

Chapter 2

Higher Order Singular ValueDecomposition (HOSVD) inQuantum Chemistry

Higher Order Singular Value Decomposition (HOSVD) is studied in the con-text of quantum chemistry, with particular focus on the decomposition of theT2 amplitudes obtained from second order Møller Plesset Perturbation (MP2)theory calculations. Our test calculations reveal that HOSVD transformedamplitudes yield considerably faster convergence in MP2 correlation energy,both in terms of amplitude and orbital truncation. Also, HOSVD orbitalsdisplay increased bonding/antibonding character compared to Hartree-Fock(HF) orbitals. In contrast to canonical MP2 theory, the leading amplitudesare those between corresponding occupied-virtual orbital pairs. The HOSVDorbitals are paired up automatically around the Fermi level in decreasing im-portance, so that the strongest occupied virtual pair are the Highest Occu-pied Molecular Orbital and Lowest Unoccupied Molecular Orbital (HOMO-LUMO). We show that in the case of MP2 amplitudes, the HOSVD orbitalsare equivalent to the unrelaxed MP2 natural orbitals.

The least squares Higher Orthogonal Iteration (HOOI) algorithm yieldsonly minor improvements over the sub-optimal truncated orbital space ob-tained from the HOSVD.

17

2.1 Introduction

2.1.1 Motivation

Decomposition of multilinear arrays has important applications in numerousfields, such as signal processing [202–204], graph analysis [205–207], numeri-cal analysis [208–212], neuroscience [213–216] and computer vision [217–220].

In Quantum Chemistry, higher order tensors arise for example in thewavefunction-based description of electron correlation and are an essentialpart of any state-of-the-art wavefunction based calculation [221]. However,the total number of elements in an array grows exponentially with the num-ber of indices. This problem, known as the “curse of dimensionality” [222],restricts high-level calculations to molecules comprised of only a handful ofatoms. Low-parametric representations of multilinear arrays have the poten-tial of making such computations accessible to larger systems. In addition,such a compact representation may lend itself to natural truncations andthus novel active space methods [51, 77]. Despite great progress, for exam-ple [208], there is still room for improvement and it may be advantageous toconsider the problem from the perspective of numerical analysis.

Tensor decompositions may also be useful in data analysis. Multiple in-dex quantities are difficult to interpret and thus a one-particle description isdesirable to aid physical understanding [223–225].

In this chapter, we investigate the applicability of the Higher Order Sin-gular Value decomposition (HOSVD), a multilinear generalization of the sin-gular value decomposition (SVD) towards novel active space methods. In afirst attempt we apply this method to the MP2 T2 tensor.

The chapter is structured as follows: In the first part we give a briefoverview of the most important tensor decompositions before discussing theHOSVD in more detail in sections 2 (Theory) and 3 (Results).

2.1.2 Notation and Basic Definitions

The following notation will be used throughout the chapter: scalars are indi-cated by lower-case letters (a, b, ...;α, β, ...), vectors are denoted as capitals(A,B, ...) (italic shaped), matrices are written in bold-face capitals (A,B,

18

...), and tensors are indicated as calligraphic capitals (A,B,...). The only ex-ceptions are indices indicating upper bounds. For these cases, the followingletters are reserved: I, P,Q,R.

A tensor with d indices will be referred to as a d-mode tensor or an order-dtensor.

The scalar product 〈A,B〉 of two tensors A,B ∈ RI1×I2×...×Id is given as

〈A,B〉 def=∑i1

∑i2

...∑id

b∗i1i2...idai1i2...id (2.1)

in which * denotes the complex conjugate.

The Frobenious norm of a tensor A, ||A||F is defined as

||A||Fdef=√〈A,A〉 (2.2)

The n-mode product A×n U of a tensor A ∈ RI1×I2×...×Id with a matrixU ∈ RJn×In is an (I1×I2× ...×In−1×Jn×In+1× ...×Id) tensor with entries:

(A×n U)i1i2...in−1jnin+1...iddef=∑in

ai1i2...in−1inin+1...idujnin (2.3)

The outer product, denoted by ◦, defines the following operation:

(U (1) ◦ U (2) ◦ ... ◦ U (d))i1i2...iddef= U

(1)i1U

(2)i2...U

(d)id

∀ in ∈ {1, ..., In}(2.4)

2.1.3 Overview of Tensor Decompositions and TheirRelation to Quantum Chemistry

Tensor decompositions can be divided into two main classes. The first isreferred to as Tucker decomposition or principal component analysis [226–231], and takes the following form for a d-mode tensor:

T def= S ×1 U(1) ×2 U(2) ×3 ...×d U(d) =

=P∑p=1

Q∑q=1

...

R∑r=1

spq...rU(1)p ◦ U (2)q ◦ ... ◦ U (d)r

(2.5)

19

Here, the original tensor T is decomposed into the core-tensor S andmatrices U(n). The latter are known as Tucker factors or mode factors, whichact on the ith mode of S. If orthogonality constraints are introduced, thismethod is known as Higher-Order Singular Value Decomposition (HOSVD)or multilinear SVD [232].

In the context of quantum chemistry the Tucker decomposition may belinked to the complete active space self-consistent field (CASSCF) method[51]. Decomposition of amplitudes corresponding to a specific level of exci-tation yields optimized orbitals for each of these tensors. However, if theseTucker factors are optimized under the constraint that all of the mode factorshave to be the same, the CASSCF transformation matrices are obtained.

The second tensor decomposition is known as CANDECOMP/PARAFAC(CP) [233–239]. This approach approximates a tensor as a sum of Kroneckerproducts of rank-one tensors:

T def=R∑r=1

U (1)r ◦ U (2)r ◦ ... ◦ U (d)r (2.6)

Here, the number of summands R is called the canonical rank. It is oftenassumed that the columns of U(i) are normalized to length 1 and the vectorΛ absorbs the corresponding weights:

T =R∑r=1

ΛrU(1)r ◦ U (2)r ◦ ... ◦ U (n)r (2.7)

This form reveals that CP can be regarded as a special case of the Tuckerdecomposition, provided the core tensor is superdiagonal and P = Q = ... =R.

The relationship to quantum chemistry can be drawn by noting that thefull configuration interaction (FCI) method can be seen as a CP decompo-sition. In particular, the perfect pairing (PP) approach [240–242] can beviewed as a mode-1 approximation to the FCI tensor, as described in equa-tion (7), i.e.:

T PP =R∑r=1

ΛrU(1)r (2.8)

20

since the associated PP excitation operator takes the form:

T̂ PP2 |Φ0〉 =∑i

t̂i |Φi〉 (2.9)

Here |Φi〉 is the determinant |Φ0〉 with the occupied pair φiφ̄i replaced by theunoccupied (correlating) pair φ∗i φ̄

∗i .

Comparison of the CP and Tucker decompositions from a numericalstandpoint reveals that the canonical rank reduction is an unstable problemand none of the current algorithms can perform a reliable rank reductiongiven a prescribed accuracy [243], which is also known as recompression.There have been some advances in this field [208, 244, 245], however, thecanonical rank has to be known or guessed in advance. In addition, conver-gence may be slow [246].

On the other hand, the corresponding Tucker approach can be performedwith the reliability of the well-understood singular value decomposition (SVD).This makes the HOSVD a preferable candidate for active space techniques,especially since algorithms have been devised which determine the reducedtensor dimension automatically [247].

The Tucker decomposition, however, also has a disadvantage comparedto the CP approach; the cost scales exponentially with the number of modes,d. This is due to the core tensor, which contains Rd elements (if P = Q =... = R). At first glance the number of parameters in a CP decomposition(d · R · n + R) does not seem to suffer from this exponential dependence.However, R is found to increase rapidly and often depends on d exponen-tially [248]. In order to overcome memory issues, so-called cross approxima-tion techniques have been introduced [249–254], which make Tucker approx-imations feasible even for tensors with d ≥ 1000.

We are interested in studying this method in more detail, in light ofthe potential applications of HOSVD in quantum chemistry, in particularin Coupled-Cluster (CC) methods [255], many-body perturbation theory(MBPT) [256] or active-space approaches. Specifically, we analyzed decom-posed and rank truncated T2 tensors, which were obtained from MP2 calcu-lations. We were also able to show the connection between HOSVD T2 andMP2 natural orbitals.

21

2.2 Theory

2.2.1 Review of HOSVD without any numerical ap-proximations such as rank truncation (untrun-cated HOSVD)

The HOSVD is defined for complex tensors [257]. However, in the applica-tions under consideration, all tensors are real-valued and the correspondingnotations are adapted accordingly.

The main steps of the untruncated HOSVD are1.) Matricization (unfolding) of the mode-d tensor T ∈ RI1×I2×...×Id yields dmatrices, A(1), ...,A(d); A(n) ∈ RIn×(I1I2...In−1In+1...Id). Thereby, tensor element(T )i1,i2,...,id maps to matrix element (A(n))in,j(n) where

j(n) = in+1In+2In+3...IdI1I2...Id−1+

in+2In+3In+4...IdI1I2...Id−1 + ...

+ idI1I2...Id−1 + i1I2I3...In−1 + ...+ in−1

(2.10)

for in = 0, 1, ..., In − 1.2.) Determining the d left singular matrices, U(1), ...,U(d)

A(n) = U(n)Σ(n)V(n)T

for n = 1, ..., d (2.11)

where U(n) ∈ RIn×In . Σ(n) ∈ RIn×(I1I2...In−1In+1..Id) is a diagonal matrix,featuring the singular values in descending order and V(n) are the right sin-gular matrices. The singular matrices are orthonormal: U(n)

TU(n) = 1 and

V(n)TV(n) = 1.

3.) Determining the (compressed) core tensor, S ∈ RI1×I2×...×Id by contract-ing the original tensor T with the left singular matrices U(n) obtained in step2:

S = T ×1 U(1)T ×2 U(2)

T

...×d U(d)T

(2.12)

A pseudocode HOSVD algorithm is shown in Fig. 2.1.

One of the properties of the subtensors Sin=α of S ∈ RI1×I2×..×Id (whichare obtained by fixing the nth index to α) is ordering:

22

procedure HOSVD(T )for n = 1, ..., d do

U(n) ← In leading left singular vectors of A(n)end for

S ← T ×1 U(1)T ×2 U(2)

T × ...×d U(d)T

return S,U(1),U(2), ...,U(d)end procedure

Figure 2.1: HOSVD procedure. Adapted from [1].

||Sin=1||F = σ(n)1 ≥ ||Sin=2||F = σ

(n)2 ≥ ... (2.13)

≥ ||Sin=In||F = σ(n)In≥ 0

for all possible values of n, where the Frobenious norms σ(n)i = ||Sin=i||F ,

corresponds to the ith n-mode singular value of T .

2.2.2 rank-(R1, R2, ..., Rd) truncated HOSVD

For the above defined decomposition, the following holds [232]:

||T ||2F =R1∑i=1

(σ

(1)i

)2= ... =

Rd∑i=1

(σ

(d)i

)2= ||S||2F (2.14)

where Rn is the n−rank of S, i.e. the dimension of the vector space spannedby the n−mode vectors.

Let the n-mode rank of T be equal to Rn(1 ≤ n ≤ d). Define a tensorT̃ by keeping only the largest I ′n n-mode singular values and discarding theremaining values. Then the error due to rank truncation is bounded by [232]:

∣∣∣∣∣∣T − T̃ ∣∣∣∣∣∣ ≤ d∑n=1

R1∑i1=I′1+1

(σ

(n)in

)2. (2.15)

23

Table 2.1: Computational cost of HOSVDStep N − dim T2 (MP2)

Unfolding T O(I1I2...IN) O(o2v2)

Building A(n)A(n)T O(I2nI1I2...In−1In+1...IN) A(1), A(2): O(o3v2)A(3), A(4): O(o2v3)

Diag(A(n)A(n)T) to get U(n) O(I3n) A(1), A(2): O(o3)A(3), A(4) : O(v3)

Contract T with O(I2nI1I2...In−1In+1...IN) A(1), A(2): O(o3v2)U(n)’s to get S A(3), A(4): O(o2v3)

In practice, the rank-(R1, R2, ..., Rd) truncated core tensor, S̃, is found usingan analogous procedure to the one outlined in Fig. 2.1. Instead of keepingall left singular eigenvectors, only the Rn leading singular eigenvectors areused to build the transformation matrix Ũ(n). This requires the user toinput the ranks Ri (for i = 1, 2, ..., d) in addition to the original tensor T .Obviously, such an approach is only computationally feasible if the truncationis physically motivated, e.g. by introducing an orbital active space. Otherforms of truncation algorithms were also considered and implemented in thisstudy. These included truncations based on amplitude threshold, number ofretained amplitudes, as well as percent correlation energy.

2.2.3 Computational cost of HOSVD

Table 2.1 displays the computational cost involved in calculating the HOSVDorbitals and core tensor.

2.3 Results

2.3.1 Numerical Results

In the following section we present numerical results for atoms and vari-ous small molecules, including insulators and near degenerate systems. We

24

carried out computations with several Pople and Dunning style basis sets.

1,0E-16

1,0E-14

1,0E-12

1,0E-10

1,0E-08

1,0E-06

1,0E-04

1,0E-02

1,0E+00

1 10 100 1000

log(#Amplitudes)

log(

Am

plitu

de M

agni

tude

)

HOSVD

Original

Figure 2.2: Comparison of the original T2 amplitudes and those decomposedusing the HOSVD (H2/6-31G(3df,3pd)).

Fig. 2.2 shows a log-log plot of amplitude magnitude vs. number ofpermutationally unique amplitudes for both the original T2 amplitudes andthose decomposed using the HOSVD for H2/6-31G(3df,3pd). Although thenumber of non-zero amplitudes remains (nearly) equal in both cases, themost important amplitudes arising from the HOSVD are larger in magni-tude and fall off much faster than the original T2 tensor elements. On firstglance, this seems to indicate a faster recovery of MP2 correlation energy forHOSVD decomposed amplitudes. However, since the expression for the MP2correlation energy involves the transformed two-electron integrals,

EMP2 =1

4

∑ijab

tijab (Iijab − Iijba) (2.16)

25

we have also tested the performance of the HOSVD amplitudes when con-tracted to the MP2 energy.

The percent contribution to the MP2 correlation energy for the n largestamplitudes was computed for both the original T2 tensor and the HOSVDdecomposed analogue. Since the transformation matrices obtained fromHOSVD act on the occupied and virtual space separately (i.e. there areno orbital rotations that cause mixing of the occupied and virtual space),the overall density and therefore the energy is the same as for canonicalMP2. This allows direct comparison of the correlation energy values for thetwo methods. Figs. 2.3 and 2.4 show the percent correlation energy recoveryfor H2 in various basis sets.

H2,6-31G**

1

2

3

5

6

1

2

37

8 910

13 15 17

4 56 7

8 9

0

20

40

60

80

100

0 2 4 6 8 10 12 14 16 18

#Amplitudes included

%M

P2

Cor

rela

tion

Ene

rgy

OriginalHOSVD

Figure 2.3: Percent MP2 correlation energy vs. number of amplitudes in-cluded (H2/6-31G**)

26

H2,6-31++G**

1

2

3

5

6

1

2

37

28262321201917151413129

29

114 5

6 78 9 10 11

0

20

40

60

80

100

0 5 10 15 20 25 30 35

#Amplitudes included

%M

P2

Cor

rela

tion

Ene

rgy

OriginalHOSVD

Figure 2.4: Percent MP2 correlation energy vs. number of amplitudes in-cluded (H2/6-31++G**)

We also calculated compression factors γOrig/HOSVD for various systemsand basis sets (Tables 2.2 and 2.3). The compression factor γOrig/HOSVD isdefined as the ratio of original and HOSVD decomposed amplitudes neces-sary to reach a given percentage of correlation energy. This allows compactcomparison of the correlation energy convergence between the two represen-tations. Since the HOSVD is performed on amplitudes, this approach is notexpected to yield an optimal convergence of the correlation energy.

In almost all cases considerably fewer HOSVD transformed amplitudesare required to obtain a certain amount of correlation energy, even for 99 per-cent correlation energy recovery. The main exceptions are CH4 (cc-pVDZ)and Ne (6-311G).

In general, addition of diffuse basis functions results in higher compression

27

Table 2.2: HOSVD vs. Original compression factors γOrig/HOSVD for a given% correlation energy - Pople style bases; Hn: R1 = 0.76, R2 = 1.2

Molecule Basis Compression Factor80% 90% 95% 99%

H2

6-31G 2.7 2.3 2.6 1.66-311G 5.5 4.9 4.4 3.66-31G** 1.9 1.8 1.7 1.7

6-31++G** 2.4 2.9 2.7 2.86-31G(3df,3pd) 5.8 4.5 4.1 3.8

6-31++G(3df,3pd) 7.9 5.8 5.1 4.4

linear H4

6-31G 2.2 2.0 1.8 1.66-311G 4.6 4.0 3.4 2.26-31G** 1.8 1.8 1.8 1.5

6-31++G** 2.6 3.0 3.0 2.3

linear H6

6-31G 2.2 1.9 1.7 1.46-311G 4.9 4.1 3.4 2.26-31G** 1.8 1.9 1.8 1.4

6-31++G** 3.2 3.4 3.0 2.1

N2

6-31G 2.7 2.1 1.8 1.26-311G 3.1 1.9 1.4 1.16-31G* 1.9 1.8 1.5 1.1

CH4

6-311G 2.5 2.1 1.7 1.36-31G* 2.1 1.8 1.5 1.26-31G** 2.0 1.6 1.5 1.2

6-31+G** 2.5 2.0 1.8 1.4

H2O

6-31G 3.4 2.5 2.0 1.26-311G 2.9 2.4 1.8 1.26-31G* 2.3 2.1 1.7 1.26-31G** 2.9 2.2 1.7 1.3

6-31+G** 4.9 3.4 2.4 1.66-31++G** 5.1 3.6 2.7 1.8

Ne

6-31G 1.0 1.1 1.2 1.56-311G 1.4 1.2 1.1 0.96-31G* 1.1 1.2 1.0 1.1

6-31+G* 1.6 2.1 1.8 1.4

28

Table 2.3: HOSVD vs. Original compression factors γOrig/HOSVD for a given% correlation energy - Dunning style bases; Hn: R1 = 0.76, R2 = 1.2


H2

cc-pVDZ 2.1 2.3 2.0 1.8aug-cc-pVDZ 5.7 6.6 5.7 5.1

cc-pVTZ 5.0 3.7 3.4 3.0

linear H4


cc-pVTZ 5.6 4.0 3.3 2.3

linear H6


cc-pVTZ 5.6 4.0 3.1 2.1

N2


cc-pVTZ 2.3 1.6 1.4 1.1

CH4cc-pVDZ 1.4 0.9 0.7 0.5

aug-cc-pVDZ 4.8 3.3 2.3 1.3

H2Occ-pVDZ 3.3 2.5 1.9 1.4

aug-cc-pVDZ 8.2 5.7 4.2 2.5cc-pVTZ 5.2 4.0 2.9 1.9

Ne


cc-pVTZ 1.6 1.4 1.3 1.1aug-cc-pVTZ 3.1 2.8 2.2 1.5

29

rates in these model systems. This is not surprising, as diffuse functions arenot essential to model these molecules. Increasing the size of the molecule ina linear fashion (H2 to H6) does not alter the compression factors significantly.

In order to identify the efficiency of using a subset of HOSVD ampli-tudes, compression factors for truncated vs. full HOSVD, γHOSVD/HOSVDfullwere computed (Table 2.4). Here the compression factor is defined as theratio of untruncated non-zero (T 2 > � = 2.2 × 10−16 ) HOSVD amplitudesto truncated HOSVD amplitudes to reach a given percentage of correlationenergy.

For large percent correlation energy recovery, most systems in mediumsized basis sets can be accurately described by incorporating only a tenthof the full set of non-zero amplitudes or even fewer. However, the questionremains whether these amplitudes form a systematic and/or physical subsetof the full HOSVD transformed amplitudes. To investigate the possibilityof utilizing HOSVD in the context of active space methods, the algorithmwas modified such as to involve truncation of the orbital space, rather thanamplitude space.

Figs. 2.5 to 2.7 indicate that orbital truncation yields a fast convergencein the MP2 correlation energy. In case of N2 in a 6-31G* basis, about 15percent more correlation energy is recovered by using HOSVD orbitals, if 7occupied and virtual orbitals, respectively are included ((7,7) space). Inter-estingly, the data suggests a natural division of the space into a (3, 3) anda (5, 5) subspace for N2 (Fig. 2.7). If the full occupied space is included, asimilar trend is observed (Fig. 2.8). The inclusion of the 3 most correlatingoccupied and virtual orbitals corresponds to the π / π* and lone pair orbitals,in correspondence with physical intuition. Similarly, in the case of H2 andH4 the largest contribution is recovered by including one and two pairs oforbitals, respectively.

Orbital pictures shown in Figs. 2.9 through 2.12 suggest that the bond-ing, as well as antibonding character is enhanced in the HOSVD orbitals,indicating that the bonding HOSVD orbitals are more localized than the HFones. In general, rotations in the virtual space are larger than in the occupiedspace. Most strikingly, however, is that HOSVD systematically pairs up oc-cupied and corresponding virtual orbitals around the Fermi level in order of

30

Table 2.4: HOSVD compression factors, γHOSVD/HOSVDfull for a given percentcorrelation energy - Dunning bases; Hn: R1 = 0.76, R2 = 1.2


H2



linear H4


cc-pVTZ 262.3 108.5 63.2 27.6

linear H6


cc-pVTZ 320.3 138.7 75.0 30.6

N2


cc-pVTZ 77.6 28.3 14.9 6.8

CH4cc-pVDZ 43.6 18.1 10.2 4.5

aug-cc-pVDZ 163.5 67.0 34.8 11.6

H2Occ-pVDZ 82.5 39.1 21.6 9.5

aug-cc-pVDZ 285.6 116.9 60.0 20.9cc-pVTZ 171.1 67.3 32.3 12.3

Ne



31

40

50

60

70

80

90

100%

MP

2 C

orr E

H2

6-31G 6-31G** 6-31++G** 6-31G(3df,3pd)

1 2

H

H

0

10

20

30

0 2 4 6 8 10 12 14 16number of virtuals included

3 4

cutoff: 0.1

Figure 2.5: recovery of % MP2 correlation energy vs. number of virtualorbitals included for H2 at equilibrium bond distance in various basis sets.Box: the four most correlating virtual orbitals.

decreasing importance in terms of amplitude magnitude (Figs. 2.12 to 2.13).To investigate the extent of this pairing, we studied the orbitals involved inthe largest HOSVD amplitudes.

As can be seen from Tables 2.5 to 2.7, the largest HOSVD and originalamplitude(s) arise between very similar orbitals. However, the HOSVD am-plitudes are generally larger in magnitude. Also, as mentioned previously, theHOSVD creates pairs of occupied and corresponding virtual orbitals, whichare arranged symmetrically around the Fermi level. The most important ex-citations occur only within those pairs. In this aspect, HOSVD resembles theImperfect Pairing Ansatz. However, as was mentioned above, HOSVD onlyincludes occupied-occupied and virtual-virtual orbital rotations, respectively,in contrast to Perfect Pairing and related methods. On the other hand, theoriginal excitations do not involve occupied-virtual pairs of orbitals and also

32

Table 2.5: The 14 largest amplitudes for N2 (6-31G*; MP2 optimized geom-etry) in HOSVD and Original T2. Orbital labellings according to HOSVD.For open shell singlets only one of the two possible configurations is listed. Ablank field indicates that this amplitude is not amongst the 14 largest ones.Amplitudes are ordered according to HOSVD results. Values in bracketsindicate the ordering for original amplitudes.

occ occ virt virt HOSVD (×10−2) Original (×10−2)7α 7β 8α 8β 9.94 8.75(1)6α 6β 9α 9β 9.94 8.75(2)6α 7β 8β 9α 8.01 7.01(3, 4)6β 7β 8β 9β 7.05 6.14(5, 6)5α 7β 8β 10α 4.355α 6β 9β 10α 4.355α 7α 8α 10α 4.19 3.11(9, 10)5α 6α 9α 10α 4.19 3.11(11, 12)4α 4β 9α 9β 3.78(7)4α 4β 8α 8β 3.78(8)5α 5β 8α 20β 3.05(13, 14)

Table 2.6: The 9 largest amplitudes for H2 (6-31G** ; MP2 optimized geom-etry) in HOSVD and Original T2. Orbital labellings according to HOSVD.For open shell singlets only one of the two possible configurations is listed.A blank field indicates that this amplitude is not amongst the 9 largest ones.Amplitudes are ordered according to HOSVD results. Values in bracketsindicate the ordering for original amplitudes

occ occ virt virt HOSVD (×10−2) Original (×10−2)1α 1β 2α 2β 6.93 4.75(1)1α 1β 3α 3β 4.39 4.19(2)1α 1β 4α 4β 2.42 2.42(6)1α 1β 5α 5β 2.42 2.42(7)1α 1β 6α 6β 1.24 3.24(3)1α 1β 7α 7β 1.14 1.33(8)1α 1β 8α 8β 0.91 0.91(9)1α 1β 9α 9β 0.91 0.91(10)1α 1β 10α 10β 0.421α 1β 2α 4β 2.80(4, 5)

33

Table 2.7: The 16 largest amplitudes for CH4 (6-31G**; MP2 optimized ge-ometry) in HOSVD and Original T2. Orbital labellings according to HOSVD.For open shell singlets only one of the two possible configurations is listed. Ablank field indicates that this amplitude is not amongst the 16 largest ones.Amplitudes are ordered according to HOSVD results. Values in bracketsindicate the ordering for original amplitudes.

occ occ virt virt HOSVD (×10−2) Original (×10−2)5α 5β 6α 6β 3.82 2.11(1)4α 4β 7α 7β 3.823α 3β 8α 8β 3.82 2.11(3)4α 5β 6β 7α 3.003α 5β 6β 8α 3.003α 4β 9β 8α 3.002α 5β 6β 9α 2.752α 4β 7β 9α 2.752α 3β 8β 9α 2.752α 2β 9α 9β 2.443α 3β 7α 7β 2.11(2)3α 3β 10α 10β 2.03(4)4α 4β 11α 11β 2.03(5)5α 5β 12α 12β 2.03(6)5α 5β 14α 14β 1.78(7)3α 3β 13α 13β 1.54(8)3α 4α 7α 12α 1.50(9, 10)3α 5α 6α 10α 1.50(11, 12)3α 4α 8α 11α 1.50(13, 14)

34

40

50

60

70

80

90

100

% M

P2

Co

rr E

H4, R2=1.2A

6-31G 6-31G** 6-31++G**

1 2

4

0

10

20

30

0 2 4 6 8 10 12 14 16 18 20 22

number of virtuals included

34

5 7

cutoff: 0.1

Figure 2.6: recovery of % MP2 correlation energy vs. number of virtualorbitals included for linear H4 (R1 = 0.76Å, R2 = 1.2 Å) in various basissets. Box: the eight most correlating virtual orbitals.

lack the symmetric arrangement observed for HOSVD (Tables 2.5 to 2.7).

Table 2.8 shows the percentage MP2 correlation energy that is recoveredfor H2 in a (1, 1) active space and two H2 molecules at large distance (R= 5.0 Å) in a (

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