A Theoretical Framework for an Efficient Normalizing Flow-Based Solution to the Electronic Schrodinger Equation

TL;DR

This work develops a rigorous normalizing-flow framework to solve the Electronic Schrödinger Equation with high sampling efficiency. The core idea is to represent the ground-state wavefunction as $\\psi(x)=\\kappa(x)\\sqrt{\\rho(x)}$, where the base density $\\rho$ is constructed to be symmetric under a spin-related permutation group and study its equivariant flow transformations. The authors prove key theorems ensuring (R1)-(R2) invariances, implement both continuous and discrete $ obreak\mathbb{G}$-equivariant flows, and address practicalities such as phase selection, cusps, and induction across multiple molecules. The resulting framework enables constant-time sampling and SGD-based optimization for Variational Monte Carlo in electronic structure calculations, offering a theoretically grounded and scalable route to accurate ground-state energies in molecules and materials.

Abstract

A central problem in quantum mechanics involves solving the Electronic Schrodinger Equation for a molecule or material. The Variational Monte Carlo approach to this problem approximates a particular variational objective via sampling, and then optimizes this approximated objective over a chosen parameterized family of wavefunctions, known as the ansatz. Recently neural networks have been used as the ansatz, with accompanying success. However, sampling from such wavefunctions has required the use of a Markov Chain Monte Carlo approach, which is inherently inefficient. In this work, we propose a solution to this problem via an ansatz which is cheap to sample from, yet satisfies the requisite quantum mechanical properties. We prove that a normalizing flow using the following two essential ingredients satisfies our requirements: (a) a base distribution which is constructed from Determinantal Point Processes; (b) flow layers which are equivariant to a particular subgroup of the permutation group. We then show how to construct both continuous and discrete normalizing flows which satisfy the requisite equivariance. We further demonstrate the manner in which the non-smooth nature ("cusps") of the wavefunction may be captured, and how the framework may be generalized to provide induction across multiple molecules. The resulting theoretical framework entails an efficient approach to solving the Electronic Schrodinger Equation.

Theorems & Definitions (35)

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