Stochastic evaluation of four-component relativistic second-order many-body perturbation energies: A potentially quadratic-scaling correlation method

TL;DR

This paper introduces a stochastic method for evaluating relativistic second-order many-body perturbation energies that significantly reduces computational scaling from quintic to potentially quadratic, enabling efficient parallel computation for large molecules.

Contribution

The paper presents a novel stochastic approach using Monte Carlo integration for relativistic perturbation energies, achieving lower scaling and high parallel efficiency.

Findings

Computational cost scales no worse than cubic, possibly quadratic, with system size.

Achieved 92% strong scalability on 4096 processors.

Method significantly outperforms traditional deterministic approaches in scaling.

Abstract

A second-order many-body perturbation correction to the relativistic Dirac-Hartree-Fock energy is evaluated stochastically by integrating 13-dimensional products of four-component spinors and Coulomb potentials. The integration in the real space of electron coordinates is carried out by the Monte Carlo (MC) method with the Metropolis sampling, whereas the MC integration in the imaginary-time domain is performed by the inverse-CDF (cumulative distribution function) method. The computational cost to reach a given relative statistical error for spatially compact but heavy molecules is observed to be no worse than cubic and possibly quadratic with the number of electrons or basis functions. This is a vast improvement over the quintic scaling of the conventional, deterministic second-order many-body perturbation method. The algorithm is also easily and efficiently parallelized with demonstrated 92% strong scalability going from 64 to 4096 processors for a fixed job size.