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Title:Monte Carlo explicitly correlated methods
Author(s):Johnson, Cole M.
Director of Research:Hirata, So
Doctoral Committee Chair(s):Hirata, So
Doctoral Committee Member(s):Ceperley, David; Wagner, Lucas; Makri, Nancy
Department / Program:Chemistry
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):Electronic structure theory
Explicitly correlated methods
Monte Carlo
Abstract:Solving the non-relativistic time-independent electronic Schrödinger equation is in general difficult and requires approximation. For experimental accuracy, wave-function based methods require a large set of basis functions and inclusion of instantaneous correlation through expensive correlated methods. The methods that have been developed to account for the incompleteness of the basis set, the R12/F12 methods, create high dimensional integrals that need to be separated with the resolution of the identity, are limited in their form of the correlation factor due to analytical integration, and not highly parallel scalable. The solution to these drawbacks proposed in this work is Monte Carlo (MC). The stochastic second-order many-body perturbation theory, or the MC-MP2-F12 method, was developed for highly parallel evaluation of second-order many-body perturbation theory (MP2) energies near the complete basis set (CBS) limit. Single molecule energies were on average closer to the CBS limit than the corresponding method with a much larger basis set. Many different reaction energies for small molecules were computed showing a mean error from the CBS limit result within chemical accuracy. Two different methods were used the full variational MP2-F12 correction, MC-MP2-F12($VBX$), and a non-variational approximate form only satisfied at the minimum of the MC-MP2-F12($VBX$) formula, MC-MP2-F12($V$). Despite previous assumptions, the MC-MP2-F12($V$) formula is accurate not only for absolute energies but relative energies as well. Scaling for relative errors was shown to be $O(n^{4})$ where $n$ is the number of basis functions, one order lower than the corresponding deterministic method. Due to the MC-MP2-F12($V$) and more complete MC-MP2-F12($VBX$) having the same asymptotic scaling as $n$ increases, it is generally recommended that one use the $VBX$ method for larger molecules. Various correlation factors were tested but the Slater-type geminal (STG) developed by Ten-no was confirmed to be the best. A more extensive study of different functional forms of correlation factors was conducted using the MC-MP2-F12 method with a total of 17 correlation factors in order to elucidate qualities of the correlation hole and shape. Higher-order cusp conditions, or derivatives of the wavefunction, and their properties were also studied. It was found that every correlation factor that had the best convergence to the CBS limit had a very specific shape on the range of 0 to 1.5 Bohr. Despite having vastly differing long-range behavior, the best correlation factors gave very similar energies. This was found to be due to the decoupling of electrons at long distance, and the dominance of the orbital expansion at large inter-electron distance, $r_{12}$. While the importance of satisfying the cusp condition at $r_{12}=0$ could not be determined, the study confirmed that the intermediate region is of the most importance in general. Lastly, the MC-F12 algorithm developed for MC-MP2-F12 was extended to explicitly correlated second-order Green's function theory (GF2-F12) for basis-set corrected ionization potentials (IPs). The same set of benchmark organic molecules that were studied in the original GF2-F12 study were compared to verify the usefulness of the MC algorithm. Analogous to MC-MP2-F12, the two different methods MC-GF2-F12($V$) and MC-GF2-F12($VBX$) were tested. A mean average error of 0.049 eV and 0.018 eV was achieved for the MC-GF2-F12($V$) and MC-GF2-F12($VBX$) methods respectively. System size scaling was found to be $O(n^{4})$. As a demonstration of size scalability, the first IPs of fullerenes C$_{60}$ and C$_{70}$ were corrected from HF at the MC-GF2-F12($VBX$) level. Errors of 0.37 eV and 0.05 eV from experiment were achieved for C$_{60}$ and C$_{70}$ respectively. The sources of the large error in C$_{60}$ is unknown. Further accuracy can be expected from developing the full non-diagonal frequency-dependent formalism with MC, as well as combining MC-GF2-F12 with the MC-GF3 and MC-GF4 methods.
Issue Date:2018-04-17
Rights Information:Copyright 2018 Cole Johnson
Date Available in IDEALS:2018-09-04
Date Deposited:2018-05

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