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

Abstract

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(n 4) and O­(n 5) of molecular size (n is the number of electrons) with and without the diagonal approximation, respectively, as opposed to O­(n 5) and O­(n 6) 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. 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.

Monte Carlo MP2-F12 for Noncovalent Interactions: The C60 Dimer

“…This provision not only compresses the cost of propagating electron pairs (walkers) but also dramatically reduces the variance, as proven below. On the other hand, the distributions (weight functions) for the imaginary-time coordinates for dimer AB, monomer A, and monomer B are kept unchanged because they depend on the highest-occupied and lowest-unoccupied molecular orbital energies, 43 which differ from one subsystem to another. In the direct-sampling algorithm, 37 random imaginary times are generated directly from the distribution using a random number seed, p τ [n] .…”

“…All good practices of the state-of-the-art MC-MP2­(-F12) implementation − were adopted in the supermolecular and concurrent MC-MP2­(-F12) methods for noncovalent interaction energies, including the redundant-walker, control-variate, and direct-sampling algorithms, as well as the stochastic imaginary-time integration . The reader is referred to these articles for the algorithm details.…”

“…Unlike the deterministic counterparts, they do away with the evaluation, transformation, or storage of two-electron and geminal integrals, reducing the size-dependence of cost by one to two ranks. In contrast to QMC, these methods have no bias, ECP, or Fermion sign problems (at second order at least), and can directly compute energy differences (such as correlation correction, ,, F12 correction, ,, electron binding energies, , or quasiparticle energy bands). Therefore, this method is an example of the burgeoning class of stochastic ab initio electronic structure methods. − …”

The sign problem in density matrix quantum Monte Carlo

“…DMQMC joins a growing set of methods including other quantum Monte Carlo methods, [6][7][8][9][10][11] many body theories [12][13][14] and others in attempting to solve the finite temperature problem that has attracted recent attention amongst quantum chemists. [15][16][17][18][19] Many of these methods, like DMQMC, continue to undergo development. [20][21][22][23][24][25][26][27][28] Widespread adoption of all methods in the FCIQMC family, including DMQMC, is hindered, in part, due to the sign problem.…”

Finite-temperature many-body perturbation theory for electrons: Algebraic recursive definitions, second-quantized derivation, linked-diagram theorem, general-order algorithms, grand canonical and canonical ensembles

“…This study along with the one reported in Ref. 59 have been conducted to lay a firm mathematical foundation of perturbation theories, 87,88 which continue to be a workhorse for efficient, converging ab initio electronic structure calculations [89][90][91][92][93][94][95][96][97][98][99][100][101] for large complex molecules and solids.…”