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Title:  Advancements in Monte Carlo many body methods 
Author(s):  Doran, Alexander 
Director of Research:  Hirata, So 
Doctoral Committee Chair(s):  Hirata, So 
Doctoral Committee Member(s):  Gruebele, Martin; Makri, Nancy; Schweizer, Kenneth S. 
Department / Program:  Chemistry 
Discipline:  Chemistry 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Manybody methods
Monte Carlo 
Abstract:  Manybody methods are one of the most powerful tools that may be brought to bear to solve the electronic structure problem. While both of the following theories provide a complete path to obtain any quantity, manybody perturbation theory is principally used to provide molecular energies, while manybody Green's function provides electron binding energies. Importantly, these methods are systematically improvable, which means there is a prescribed route to increase the accuracy of their results. They are also size consistent. Manybody methods are not without weakness, though, the chief of which is their considerable expense. For example, the operational cost to compute the korder energy correction with manybody perturbation theory (MBPT) is O (n k+3 ), with n being proportional to the molecular size. Additionally, manybody methods are typically cast into complex series of dense matrixmatrix multiplications, which is difficult to parallelize to millions of processors. Because of this, their enormous expense is difficult to mitigate with the highly parallel architecture of modern supercomputing resources. Recently, the Hirata lab has pioneered stochastic versions of MBPT and manybody Green's functions (MBGF), known as MCMP and MCGF, collectively MCMB, respectively, wherein the correlation energy or electron binding energies are obtained through Monte Carlo integration. The motivation to create stochastic implementations of MBPT and MBGF was to prioritize parallelizability, as Monte Carlo integration is trivially parallel, so that modern supercomputer may be used effectively in the challenge of applying manybody methods to chemical relevance. Here, an overview of the current state of the MCMP and MCGF methods, including the pioneering work, advancements that I have completed, and several new developments, is archived. Second and thirdorder Monte Carlo perturbation theory (MCMP2 and MCMP3) are two of the earliest MCMB methods to have been developed. The algorithmic details of these methods are considered due to their relevance within the MCMB family of methods. One significant methodological development for the MCMP methods, the implementation of fourthorder Monte Carlo perturbation theory (MCMP4), is presented. The development of MCMP4 represents a major milestone in the MCMB family of methods timeline, as MP4 is the lowest level of perturbation theory capable of producing benchmark quality energies. Two essential studies on the computational characterization of MCMP2 and MCMP3 are presented. In these studies, the cost of the methods is established by numerical experiments. The cost to obtain a result with a specified target relative statistical uncertainty is established to be O(n^3) and O(n 4) for MCMP2 and MCMP3, respectively. These cost functions are both two ranks lower than the cost functions of their conventional counterparts and establish the long term viability of MBMB methods. Secondorder Monte Carlo Green's function (MCGF2) was one of the methods pioneered during the early work on MCMB methods. However, the early implementation of MCGF2 was reserved to be only applicable within the frequencyindependent diagonal approximation. Recently, the MCGF family of methods was extended in several directions. The first was to increase the rank of the selfenergy to thirdorder producing the thirdorder Monte Carlo Green's function (MCGF3) method. The diagonal approximation for both the MCGF2 and MCGF3 methods have been lifted by implementing an efficient algorithm for the calculation of the selfenergy matrix. The frequencyindependent approximation has been lifted by expanding the selfenergy in a truncated Taylor series, which in turn is used to approximate the zeros of the inverse Dyson equation. The derivatives needed to construct the truncated Taylor series are efficiently computed by the MCGF algorithm. The computational characteristics of MCGF3 were extensively studied during this work. The most important among these properties was the establishment of the cost function to obtain binding energies within the diagonal frequencyindependent and frequencyindependent approximations. This experiment is extended here to establish the same for MCGF2. For the scaling studied, the MCGF methods are found to have a cost scaling one rank lower than their conventional counterparts. While MCMB methods have favorable cost scaling with respect to system size, the naive implementation of these methods is computationally demanding. As such, there has been a significant amount of work to accelerate the convergence of MCMB methods. Five algorithms that alter the original naive MCMB framework are discussed. (1) The redundantwalker algorithm accelerates the integration of the electronic coordinates, in which more than the minimally required number of electron pairs are propagated, and the relevant energy correction is sampled at all possible permutations of the electron pairs, for all MCMB methods. Because of disparities in cost between the various steps of MCMB methods, the redundantwalker algorithm is able to achieve a boost in the sampling rate. Recent analyses of the redundantwalker algorithm suggest that it is in part responsible for the low scaling of MCMB methods. (2) MCMB methods are particularly wellsuited to utilize many GPUs. This is because GPUs can push the redundantwalker algorithm harder, and thereby attain a higher than expected speedup. (3) Two strategies to integrate the imaginary timecoordinates are presented. In the first strategy, the imaginarytime coordinates are integrated by quadrature, while in the second strategy, they are integrated by direct Monte Carlo. The integration of these degrees of freedom by stochastic means proves to be significantly more efficient, with speedups of 8.4 and 221 for second and thirdorder methods being observed. (4) An algorithm to sample the electronic degrees of freedom directly is presented. This is in contrast to the use of the Metropolis algorithm which has thus far been used to sample these coordinates. The new algorithm eliminates autocorrelation in all integrated quantities, thereby reducing their uncertainties. Speedups of up to 3.21 and 1.38 are observed for MCMP2 and MCMP3. This algorithm also provides many qualitative improvements to the MCMB family of methods. (5) Control variates are applied to the MCMP family of methods. The application of these control variance produces speedups of 13.9, 17.11, and 58.29 for MCMP2, MCMP3, and MCMP4, respectively. Of these algorithms, the first is apart of the seminal work on MCMB methods. The last two are recently completed. 
Issue Date:  20191125 
Type:  Text 
URI:  http://hdl.handle.net/2142/106213 
Rights Information:  2019 by Alexander Doran. All rights reserved. 
Date Available in IDEALS:  20200302 
Date Deposited:  201912 
This item appears in the following Collection(s)

Dissertations and Theses  Chemistry

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois