Optimizing the Energy with Quantum Monte Carlo: A Lower Numerical Scaling for Jastrow–Slater Expansions

Abstract

We present an improved formalism for quantum Monte Carlo calculations of energy derivatives and properties (e.g., the interatomic forces), with a multideterminant Jastrow-Slater function. As a function of the number N of Slater determinants, the numerical scaling of O(N) per derivative we have recently reported is here lowered to O(N) for the entire set of derivatives. As a function of the number of electrons N, the scaling to optimize the wave function and the geometry of a molecular system is lowered to O(N) + O(NN), the same as computing the energy alone in the sampling process. The scaling is demonstrated on linear polyenes up to CH and the efficiency of the method is illustrated with the structural optimization of butadiene and octatetraene with Jastrow-Slater wave functions comprising as many as 200 000 determinants and 60 000 parameters.

Variational and diffusion quantum Monte Carlo calculations with the CASINO code

“…There are a number of dedicated multideterminant wavefunction-evaluation algorithms. [90][91][92][93][94] CASINO implements the compression method of Weerasinghe et al 95 Although this method underperforms some of the other approaches, it can potentially be combined with them for additional efficiency since it does not require modification to the evaluation of the wave function. The compression algorithm is based on combining determinants differing by a single column, e.g.,…”

Excitation variance matching with limited configuration interaction expansions in variational Monte Carlo

“…These accuracies are achieved despite using ground state selective configuration interaction expansions that contain fewer than 40 configuration state functions and are thus very far from the exhaustive flexibility limit. As multi-Slater Jastrow optimizations can now handle hundreds of thousands of determinants and molecules containing over 100 atoms, [38,39] these preliminary results suggest that variance matching could play a major role in future high-accuracy work in molecular excited states.…”

“…Note that for the latter case, we have optimized the orbitals separately for the ground state and the first excited state (see Appendix B) using a recentlyimplemented combination of our direct targeting method and the efficient MSJ orbital optimization approach of Filippi and coworkers. [38,39] For each choice of basis, we first performed a ground state MSJ optimization with a short CSF expansion. Next, we performed excited state optimizations with different CSF expansion lengths so that we could estimate by interpolation the excited state expansion length (and its corresponding energy) at which the ground and excited state variances would match.…”

“…On the other hand, excitation energies based on CASSCF or VMC-optimized orbitals are within statistical uncertainty of the benchmark (cc-pVTZ, fullvalence-CAS) MRCI+Q results. Although one cannot expect to have access to CASSCF orbitals in large systems, it is possible [39] to produce VMC-optimized orbitals, making their excellent performance in conjunction with variance matching and short CSF expansions an extremely promising result. As a final note, and as we saw in other cases, CSF-number-matched results are …”

“…[35] Second, these multi-Slater Jastrow (MSJ) wave functions (whose computational efficiency has recently been greatly enhanced [36][37][38][39]) can be used to provide nodal surfaces for diffusion Monte Carlo (DMC), [40] which is highly effective at recovering the effects of weak correlation that can be missed by limited sCI expansions. Crucially, both MSJ wave functions [39] and DMC [41] have been demonstrated to scale to systems with over 100 atoms, implying that the advantages of variance matching should scale to a wide variety of chemical applications. In this paper, we will take a first step towards such applications by exploring the practical considerations that arise in working with this idea and by establishing its accuracy in a handful of small systems where high level benchmarks are available.…”