Developments of the projector configuration interaction method and the multireference driven similarity renormalization group translation missing: zh.hyrax.visibility.files_restricted.text

Zhang, Tianyuan (Spring 2019)

Permanent URL: https://etd.library.emory.edu/concern/etds/2z10wr28t?locale=zh
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Abstract

In this dissertation, I have developed efficient methods for both static and dynamical correlation. For static correlation, we propose the projector configuration interaction (PCI) approach to solving the Schrödinger equation with a determinant coupling filtering scheme. In contrast to selected configuration interaction (SCI) methods, in which an important subset of determinants are selected and the Hamiltonian is diagonalized exactly, in PCI we filter the most important determinant couplings, and truncate the determinant space accordingly. The PCI approach realizes a deterministic version of the full configuration interaction quantum Monte Carlo (FCIQMC) method [Booth, G. H.; Thom, A. J. W.; Alavi, A. J. Chem. Phys. 2009, 131, 054106], where a trial vector is projected onto the ground state wave function with an optimal polynomial projection scheme. Important determinant couplings are selected by the path filtering algorithm integrated into the wave function projection. We show that PCI is able to compute more accurate wave function than general SCI methods with less theoretical computational cost, and chemical accuracy are usually achieved using only a small fraction of the determinants in full CI space. The other topic of this dissertation is the development of a low cost linear multireference driven similarity renormalization group with singles and doubles [MR-LDSRG(2)] method for dynamical correlation. The goal is achieved without compromising the accuracy of the original MR-LDSRG(2) method by using a combination of 1) a sequential unitary transformation, 2) density fitting (DF) of the two-electron integrals, and 3) the non-interacting virtual orbital (NIVO) operator approximation. The scaling of memory requirement is reduced, and the computation is also accelerated. We report improved MR-LDSRG(2) study on the cyclobutadiene automerization reaction using a quintuple-ζ basis set.

Table of Contents

1 Introduction and literature review................ 1

1.1 Introduction ................................ 1

1.2 Sparsity in the full configuration interaction Hamiltonian . . . . . . . 5

1.3 Selected configuration interaction methods . . . . . . . . . . . . . . . 7

1.4 Cost in exact diagonalization of Hamiltonian. . . . . . . . . . . . . . 8

1.5 Full configuration interaction quantum Monte Carlo . . . . . . . . . . 10

1.6 Problems in multireference coupled cluster methods . . . . . . . . . . 12

1.7 Driven similarity renormalization group methods. . . . . . . . . . . . 16

1.8 Prospectus................................. 20

2 A deterministic projector configuration interaction approach for the ground state of quantum many-body systems ...................................... 30

2.1 Introduction ................................ 31

2.2 Theory................................... 34

2.2.1 General formalism of ground state projection . . . . . . . . . . 34

2.2.2 Rate of convergence of generators................. 36

2.2.3 Taylor and Chebyshev expansions of the imaginary-time propagator................................. 37

2.2.4 An improved generator: the wall generator and its Chebyshev expansion............................... 40

2.2.5 Determinant selection via path filtering.. . . . . . . . . . . . . 42

2.2.6 Sources of errors in the PCI method. . . . . . . . . . . . . . . 42

2.3 Implementation............................... 45

2.3.1 The PCI algorithm......................... 45

2.4 Results................................... 48

2.4.1 N2.................................. 48

2.4.2 C2.................................. 52

2.4.3 Size consistency and molecular orbital comparison . . . . . . . 54

2.5 Summary and conclusions......................... 56

Appendices.................................... 59

2.A Path filtering for polynomial generators. . . . . . . . . . . . . . . . . 59

Acknowledgments................................ 60

3 Hermitian projected configuration interaction method: Filtering the most important determinant couplings . . 67

3.1 Introduction ................................ 67

3.2 Theory................................... 70

3.2.1 Introducing determinant coupling space . . . . . . . . . . . . . 70

3.2.2 Review of PCI ........................... 74

3.2.3 Improving the Approximate Hamiltonian . . . . . . . . . . . . 76

3.3 Implementation............................... 79

3.3.1 Davidson–Liu diagonalization schemes . . . . . . . . . . . . . . 79

3.4 Results and Discussion .......................... 82

3.4.1 Accuracy improvements in Hermitian PCI. . . . . . . . . . . . 82

3.4.2 Comparison of the convergence rate . . . . . . . . . . . . . . . 83

3.4.3 Connection to the Heat-bath CI method. . . . . . . . . . . . . 85

3.4.4 Efficiency in computing the wave function. . . . . . . . . . . . 86

3.4.5 Ground state energy of Cr2 .................... 87

3.5 Conclusion and Future work ....................... 89

Acknowledgments................................ 90

4 Improving the efficiency of the multireference driven similarity renormalization group via sequential transformation, density fitting, and the non-interacting virtual orbital approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.1 Introduction ................................ 95

4.2 Theory ................................... 98

4.2.1 Review of the MR-DSRG method ................ 99

4.2.2 Simplifying the MR-DSRG equations: Sequential transformation 102

4.2.3 Alleviating the memory bottleneck: The non-interacting virtual orbital (NIVO) approximation .................. 103

4.3 Implementation............................... 105

4.3.1 Sequential transformation..................... 105

4.3.2 Batched tensor contraction for the DF algorithm . . . . . . . . 107

4.3.3 Computational cost reduction................... 108

4.4 Results and Discussion .......................... 111

4.4.1 First row diatomic molecules................... 111

4.4.2 Cyclobutadiene........................... 115

4.5 Conclusion................................. 119

Acknowledgments................................ 121

5 Concluding remarks and outlook................. 132

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