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Monte Carlo Second- and Third-Order Many-Body Green’s Function Methods with Frequency-Dependent, Nondiagonal Self-Energy

Journal Article · · Journal of Chemical Theory and Computation
 [1];  [1]
  1. University of Illinois at Urbana-Champaign, IL (United States)
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We fully develop the Monte Carlo many-body Green’s function (MC-GF) method with the following enhancements: (1) The truncation order of the perturbation expansion of the Dyson self-energy is raised from the second order (MC-GF2) to the third order (MC-GF3) with the aid of a computerized procedure to enumerate and transform all 84 third-order Goldstone diagrams into Monte Carlo integrable expressions and then into central processing unit (CPU)/graphical processing unit (GPU)-parallel computer codes. (2) An efficient algorithm is proposed that computes all off-diagonal and diagonal elements of the MC-GF2 and MC-GF3 self-energy matrices by common subexpression elimination. (3) The frequency-independent approximation is lifted by introducing a method that computes frequency derivatives of the MC-GF2 and MC-GF3 self-energies up to any arbitrarily high order at nearly no additional computational cost. (4) The imaginary-time integration in the Laplace-transformed expressions of the self-energy is carried out stochastically (instead of using a quadrature in the previous implementations), resulting in a 50- to 200-fold speedup. (5) The efficiency of the redundant-walker convergence acceleration scheme is analyzed numerically, and the guidelines are established to select an optimal number of walkers for maximal efficiency. When such an optimal number is used, the cost per sample is constant of molecular size on either many CPUs or many GPUs. (6) The computational cost to obtain a binding energy within a given statistical uncertainty is observed to increase as (tentatively) O(n4) and O(n5) of molecular size (n is the number of electrons) with and without the diagonal approximation, respectively, as opposed to O(n5) and O(n6) of the corresponding deterministic algorithms. With this method applied to the electron binding energies of C60, we show that the third-order corrections to the self-energies are much greater in electron binding energies than in ground-state energies. As a result, they display a sign of oscillatory convergence toward experimental results, not necessarily improving the agreement with increasing perturbation order, justifying MC-GF3 and motivating even higher-order methods.
Research Organization:
University of Illinois at Urbana-Champaign, IL (United States)
Sponsoring Organization:
National Science Foundation (NSF); USDOE Office of Science (SC), Basic Energy Sciences (BES). Chemical Sciences, Geosciences & Biosciences Division (CSGB)
Grant/Contract Number:
SC0006028
Other Award/Contract Number:
OCI-0725070
ACI-1238993
OSTI ID:
3021791
Alternate ID(s):
OSTI ID: 1801691
Journal Information:
Journal of Chemical Theory and Computation, Journal Name: Journal of Chemical Theory and Computation Journal Issue: 11 Vol. 15; ISSN 1549-9626; ISSN 1549-9618
Publisher:
American Chemical SocietyCopyright Statement
Country of Publication:
United States
Language:
English

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Related Subjects

Algorithms
Approximation
Binding energy
Chemical calculations
Hydrocarbons