Complementary First and Second Derivative Methods for Ansatz Optimization in Variational Monte Carlo

We present a comparison between a number of recently introduced low-memory wave function optimization methods for variational Monte Carlo in which we find that first and second derivative methods possess strongly complementary relative advantages. While we find that low-memory variants of the linear method are vastly more efficient at bringing wave functions with disparate types of nonlinear parameters to the vicinity of the energy minimum, accelerated descent approaches are then able to locate the precise minimum with less bias and lower statistical uncertainty. By constructing a simple hybrid approach that combines these methodologies, we show that all of these advantages can be had at once when simultaneously optimizing large determinant expansions, molecular orbital shapes, traditional Jastrow correlation factors, and more nonlinear many-electron Jastrow factors

Density Functional Extension to Excited-State Mean-Field Theory.

We investigate an extension of excited-state mean-field theory in which the energy expression is augmented with density functional components in an effort to include the effects of weak electron correlations. The approach remains variational and entirely time independent, allowing it to avoid some of the difficulties associated with linear response and the adiabatic approximation. In particular, all of the electrons' orbitals are relaxed state specifically, and there is no reliance on Kohn-Sham orbital energy differences, both of which are important features in the context of charge transfer. Preliminary testing shows clear advantages for single-component charge transfer states, but the method, at least in its current form, is less reliable for states in which multiple particle-hole transitions contribute significantly

Reduced Scaling Hilbert Space Variational Monte Carlo

We show that for both single-Slater-Jastrow and Jastrow geminal power wave functions, the formal cost scaling of Hilbert space variational Monte Carlo can be reduced from fifth to fourth order in the system size, thus bringing it in line with the long-standing scaling of its real space counterpart. While traditional quantum chemistry methods can reduce costs related to the two-electron integral tensor through resolution of the identity and Cholesky decomposition approaches, we show that such approaches are ineffective in the presence of Hilbert space Jastrow factors. Instead, we develop a simple semi-stochastic approach that can take similar advantage of the near-sparsity of this four-index tensor. Through demonstrations on alkanes of increasing length, we show that accuracy and overall statistical uncertainty are not meaningfully affected and that a total cost crossover is reached as early as 50 electrons.Comment: 8 pages, 7 figure

Variational Excitations in Real Solids: Optical Gaps and Insights into Many-Body Perturbation Theory

We present an approach to studying optical band gaps in real solids in which quantum Monte Carlo methods allow for the application of a rigorous variational principle to both ground and excited state wave functions. In tests that include small, medium, and large band gap materials, optical gaps are predicted with a mean-absolute-deviation of 3.5% against experiment, less than half the equivalent errors for typical many-body perturbation theories. The approach is designed to be insensitive to the choice of density functional, a property we exploit in order to provide insight into how far different functionals are from satisfying the assumptions of many body perturbation theory. We explore this question most deeply in the challenging case of ZnO, where we show that although many commonly used functionals have shortcomings, there does exist a one particle basis in which perturbation theory's zeroth order picture is sound. Insights of this nature should be useful in guiding the future application and improvement of these widely used techniques.Comment: 8 pages, 5 figures, 2 table

An efficient variational principle for the direct optimization of excited states

We present a variational function that targets excited states directly based on their position in the energy spectrum, along with a Monte Carlo method for its evaluation and minimization whose cost scales polynomially for a wide class of approximate wave functions. Being compatible with both real and Fock space and open and periodic boundary conditions, the method has the potential to impact many areas of chemistry, physics, and materials science. Initial tests on doubly excited states show that using this method, the Hilbert space Jastrow antisymmetric geminal power ansatz can deliver order-of-magnitude improvements in accuracy relative to equation of motion coupled cluster theory, while a very modest real space multi-Slater Jastrow expansion can achieve accuracies within 0.1 eV of the best theoretical benchmarks for the carbon dimer.Comment: 6 pages, 4 figure

A Blocked Linear Method for Optimizing Large Parameter Sets in Variational Monte Carlo

We present a modification to variational Monte Carlo's linear method optimization scheme that addresses a critical memory bottleneck while maintaining compatibility with both the traditional ground state variational principle and our recently-introduced variational principle for excited states. For wave function ansatzes with tens of thousands of variables, our modification reduces the required memory per parallel process from tens of gigabytes to hundreds of megabytes, making the methodology a much better fit for modern supercomputer architectures in which data communication and per-process memory consumption are primary concerns. We verify the efficacy of the new optimization scheme in small molecule tests involving both the Hilbert space Jastrow antisymmetric geminal power ansatz and real space multi-Slater Jastrow expansions. Satisfied with its performance, we have added the optimizer to the QMCPACK software package, with which we demonstrate on a hydrogen ring a prototype approach for making systematically convergent, non-perturbative predictions of Mott-insulators' optical band gaps.Comment: 9 pages, 3 tables, 4 figure