Stochastic sampling of quadrature grids for the evaluation of vibrational expectation values

TL;DR

This paper reformulates the thermal lines method as a stochastic quadrature approach for vibrational expectation values, analyzes its bias, and extends it to multiple point grids, reducing bias and computational cost.

Contribution

It introduces a stochastic quadrature grid formulation of thermal lines, analyzes bias, and extends the method to multiple point grids for improved accuracy and efficiency.

Findings

Finer quadrature grids reduce bias in expectation value estimates.

The extended method achieves ~30% lower computational cost than Monte Carlo.

Bias depends on the local form of the expectation value.

Abstract

The thermal lines method for the evaluation of vibrational expectation values of electronic observables [B. Monserrat, Phys. Rev. B 93, 014302 (2016)] was recently proposed as a physically motivated approximation offering balance between the accuracy of direct Monte Carlo integration and the low computational cost of using local quadratic approximations. In this paper we reformulate thermal lines as a stochastic implementation of quadrature grid integration, analyze the analytical form of its bias, and extend the method to multiple point quadrature grids applicable to any factorizable harmonic or anharmonic nuclear wave function. The bias incurred by thermal lines is found to depend on the local form of the expectation value, and we demonstrate that the use of finer quadrature grids along selected modes can eliminate this bias, while still offering a ~30% lower computational cost than direct Monte Carlo integration in our tests.