Ab initio computations of molecular systems by the auxiliary‐field quantum Monte Carlo method

Abstract

The auxiliary-field quantum Monte Carlo (AFQMC) method provides a computational framework for solving the time-independent Schrödinger equation in atoms, molecules, solids, and a variety of model systems. AFQMC has recently witnessed remarkable growth, especially as a tool for electronic structure computations in real materials. The method has demonstrated excellent accuracy across a variety of correlated electron systems. Taking the form of stochastic evolution in a manifold of nonorthogonal Slater determinants, the method resembles an ensemble of density-functional theory (DFT) calculations in the presence of fluctuating external potentials. Its computational cost scales as a low-power of system size, similar to the corresponding independentelectron calculations. Highly efficient and intrinsically parallel, AFQMC is able to take full advantage of contemporary high-performance computing platforms and numerical libraries. In this review, we provide a self-contained introduction to the exact and constrained variants of AFQMC, with emphasis on its applications to the electronic structure of molecular systems. Representative results are presented, and theoretical foundations and implementation details of the method are discussed. This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods Structure and Mechanism > Computational Materials Science Computer and Information Science > Computer Algorithms and Programming K E Y W O R D S ab initio methods, auxiliary-field quantum Monte Carlo, back-propagation, computational quantum chemistry, constrained path approximation, electronic structure, importance sampling, phase problem, phaseless approximation, quantum many-body computation, quantum Monte Carlo methods, sign problem 1 | INTRODUCTIONA central challenge in the fields of condensed matter physics, quantum chemistry (QC), and materials science is to determine the quantum mechanical behavior of many interacting electrons and nuclei. Often relativistic effects and the coupling between the dynamics of electrons and nuclei can be neglected, or treated separately. Within this approximation, the many-electron wave function can be found by solving the time-independent Schrödinger equation for the Born-Oppenheimer Hamiltonian

An auxiliary-Field quantum Monte Carlo perspective on the ground state of the dense uniform electron gas: An investigation with Hartree-Fock trial wavefunctions

“…AFQMC has been successfully applied in recent years to a number of challenging problems in both quantum chemistry [29][30][31][32][33] and solid state physics [34][35][36] . However, the broad applicability of the method is not as well under-stood as more traditional quantum chemistry approaches which have seen decades worth of sustained development and benchmarking.…”

Ab Initio Electronic Structure Calculations by Auxiliary-Field Quantum Monte Carlo

“…Each column of this matrix represents a single-particle orbital that is completely specified by its N-dimensional vector. For convenience, we will think of the different columns as all properly orthonormalized, which is straightforward to achieve by, for example, modified Gram-Schmidt (see e.g., Zhang (2003Zhang ( , 2013; Motta and Zhang (2018)). The mean-field Hartree-Fock (HF) solution is of course an example of a Slater determinant: |φ HF = ∏ σ |φ σ HF , where |φ σ HF is defined by a matrix Φ σ HF whose columns are the M σ lowest HF eigenstates.…”

Beyond Walkers in Stochastic Quantum Chemistry: Reducing Error Using Fast Randomized Iteration

“…This has motivated the development of discrete-space methods, e.g. full configuration interaction QMC (FCIQMC) and auxiliary-field QMC, [16][17][18][19] in which the antisymmetry is provided by a Slater determinant basis, thereby obviating the need to impose nodal constraints on the wave function. 8,16,20,21 A disadvantage of discrete-basis methods is that the basis is not complete, but this can be addressed using standard extrapolation techniques.…”