Monte Carlo explicitly correlated many-body Green’s function theory

Abstract

A highly scalable stochastic algorithm is proposed and implemented for computing the basis-set-incompleteness correction to the diagonal, frequency-independent self-energy of the second-order many-body Green’s function (GF2) theory within the explicitly correlated (F12) formalism. The 6-, 9-, 12-, and 15-dimensional integrals comprising the F12 correction are directly evaluated by the Monte Carlo method using appropriate weight functions for importance sampling. The method is naturally and easily parallelized, involves minimal memory space and no disk I/O, and can use virtually any mathematical form of a correlation factor. Its computational cost to correct all ionization energies (IEs) is observed to increase as the fourth power of system size, as opposed to the fifth power in the case of the deterministic counterparts. The GF2 calculations and their F12 corrections for the first IEs of C60 and C70 were executed on 128 graphical processing units (GF2) and 896 central processing units (F12), respectively, to reach the results with statistical errors of 0.04 eV or less. They showed that the basis-set-incompleteness (from aug-cc-pVDZ) accounts for only 50%–60% of the deviations from experiments, suggesting the significance of higher-order perturbation corrections.

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

“…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. − …”

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.…”

Density-Based Basis-Set Incompleteness Correction for GW Methods

“…56 As shown in recent studies on both groundand excited-state properties, 57,58 similar to F12 methods, it significantly speeds up the convergence of energetics towards the CBS limit while avoiding the usage of the large auxiliary basis sets that are used in F12 methods to avoid the numerous three-and four-electron integrals. [50][51][52][53][54][59][60][61] Explicitly correlated F12 correction schemes have been derived for second-order Green function methods (GF2) 35,[62][63][64][65][66][67][68][69][70][71][72] by Ten-no and coworkers 73,74 and Valeev and coworkers. 75,76 However, to the best of our knowledge, a F12-based correction for GW has not been designed yet.…”