Linear-Scaling Evaluation of the Local Energy in Quantum Monte Carlo

Abstract

For atomic and molecular quantum Monte Carlo calculations, most of the computational effort is spent in the evaluation of the local energy. We describe a scheme for reducing the computational cost of the evaluation of the Slater determinants and correlation function for the correlated molecular orbital (CMO) ansatz. A sparse representation of the Slater determinants makes possible efficient evaluation of molecular orbitals. A modification to the scaled distance function facilitates a linear scaling implementation of the Schmidt-Moskowitz-Boys-Handy (SMBH) correlation function that preserves the efficient matrix multiplication structure of the SMBH function. For the evaluation of the local energy, these two methods lead to asymptotic linear scaling with respect to the size of the molecule.In the quantum Monte Carlo (QMC) method [1][2][3][4][5][6][7], the expectation value of the Hamiltonian H is computed as a statistical average of the local energy of a trial wave-function, Ψ T (R), where R denotes the 3N coordinates of the N-particle system.

Quantum Monte Carlo and Related Approaches

“…A linear scaling algorithm for evaluation of three-body terms in the BH expansion has been described by Austin et al Rewriting each term as a trace over a matrix product, U mno A = ∑ i ≠ j r̅ Ai m r̅ ij o r̅ jA n , permitted the use of fast matrix multiplication libraries. Linear scaling was then achieved by taking advantage of the sparsity in the r̅ matrices.…”