This work is concerned with the accurate numerical simulation of the many-electron problem, which involves the modeling of the electron wavefunction, from which all of the properties of a chemical or condensed matter system can, in principle, be computed. This problem poses a number of challenges, including the effective parametrization of the wavefunction space. The combination of neural networks and quantum Monte Carlo methods has arisen as a promising path forward for highly accurate electronic structure calculations. Previous proposals have combined equivariant neural network layers with a final antisymmetric layer in order to satisfy the antisymmetry requirements of the electronic wavefunction. However, to date it is unclear if one can represent arbitrary antisymmetric functions of physical interest, and it is difficult to precisely measure the expressiveness of the antisymmetric layer.

In the first chapter of this dissertation, we begin by introducing the electronic structure problem and the variational nature of finding the lowest energy wavefunction, or ground state. We describe Metropolis Monte Carlo sampling techniques, as well as a simplifying reduction to the number of degrees of freedom. We conclude the first chapter with a brief discussion of some optimization techniques used to address the variational ground state problem once a trial parametrization has been established.

In the next chapter, we then introduce the form of some modern neural-network based trial wavefunctions. We attempt to investigate the expressiveness of the antisymmetric layers by proposing explicitly antisymmetrized universal neural network layers. This approach has a computational cost which increases factorially with respect to the system size, but we are nonetheless able to apply it to small systems to better understand how the structure of the antisymmetric layer affects its performance. We first introduce a generic antisymmetric (GA) neural network layer, which we use to replace the entire antisymmetric layer of the highly accurate ansatz known as the FermiNet. We also consider a factorized antisymmetric (FA) layer which more directly generalizes the FermiNet by replacing the products of determinants with products of antisymmetrized neural networks.

We next investigate the numerical performance of these explicitly antisymmetrized ansatzes. We demonstrate that the FermiNet-GA architecture can yield effectively the exact ground state energy for small atoms and molecules. We find, interestingly, that the resulting FermiNet-FA architecture does not outperform the FermiNet. This strongly suggests that the sum of products of antisymmetries is a key limiting aspect of the FermiNet architecture. To explore this further, we also investigate a slight modification of the FermiNet, called the full determinant mode, which replaces each product of determinants with a single combined determinant. We find that the full single-determinant FermiNet closes a large part of the gap between the standard single-determinant FermiNet and FermiNet-GA on small atomic and molecular problems. Surprisingly, on the nitrogen molecule at a dissociating bond length of 4.0 Bohr, the full single-determinant FermiNet can significantly outperform the largest standard FermiNet calculation with 64 determinants, yielding an energy within 0.4 kcal/mol of the best available computational benchmark.

In the final chapter, we introduce the VMCNet repository, which was used to implement the numerical experiments previously described. VMCNet is intended to be a flexible, general purpose VMC framework which interfaces natively with the JAX library for rapid prototyping, performance benefits due to just-in-time XLA compilation, and easy dispatch to multiple GPU systems. We describe both the Python API, intended for more complex use-cases that require customization of finer details of the VMC loop, and the command-line interface, which provides a more streamlined and encapsulated way to run variational Monte Carlo experiments, including those described in this dissertation.

We present an efficient parallel algorithm and its implementation for computing the diagonal of $H^-1$ where $H$ is a 2D Kohn-Sham Hamiltonian discretized on a rectangular domain using a standard second order finite difference scheme. This type of calculation can be used to obtain an accurate approximation to the diagonal of a Fermi-Dirac function of $H$ through a recently developed pole-expansion technique \cite{LinLuYingE2009}. The diagonal elements are needed in electronic structure calculations for quantum mechanical systems \citeHohenbergKohn1964, KohnSham 1965,DreizlerGross1990. We show how elimination tree is used to organize the parallel computation and how synchronization overhead is reduced by passing data level by level along this tree using the technique of local buffers and relative indices. We analyze the performance of our implementation by examining its load balance and communication overhead. We show that our implementation exhibits an excellent weak scaling on a large-scale high performance distributed parallel machine. When compared with standard approach for evaluating the diagonal a Fermi-Dirac function of a Kohn-Sham Hamiltonian associated a 2D electron quantum dot, the new pole-expansion technique that uses our algorithm to compute the diagonal of $(H-z_i I)^-1$ for a small number of poles $z_i$ is much faster, especially when the quantum dot contains many electrons.

We describe an efficient implementation of an algorithm for computing selected elements of a general sparse symmetric matrix A that can be decomposed as A = LDL^T, where L is lower triangular and D is diagonal. Our implementation, which is called SelInv, is built on top of an efficient supernodal left-looking LDL^T factorization of A. We discuss how computational efficiency can be gained by making use of a relative index array to handle indirect addressing. We report the performance of SelInv on a collection of sparse matrices of various sizes and nonzero structures. We also demonstrate how SelInv can be used in electronic structure calculations.