Excited Pfaffians: Generalized Neural Wave Functions Across Structure and State

TL;DR

This paper introduces Excited Pfaffians and Multi-State Importance Sampling, enabling neural network wave functions to efficiently and accurately represent multiple quantum states with reduced computational cost.

Contribution

The authors develop Excited Pfaffians and MSIS, allowing neural network wave functions to model multiple states simultaneously with improved efficiency and scalability.

Findings

Achieved faster training and modeling of more states on the carbon dimer.

First neural network method to find all energy levels of the beryllium atom.

Demonstrated multi-state representation across various molecules.

Abstract

Neural-network wave functions in Variational Monte Carlo (VMC) have achieved great success in accurately representing both ground and excited states. However, achieving sufficient numerical accuracy in state overlaps requires increasing the number of Monte Carlo samples, and consequently the computational cost, with the number of states. We present a nearly constant sample-size approach, Multi-State Importance Sampling (MSIS), that leverages samples from all states to estimate pairwise overlap. To efficiently evaluate all states for all samples, we introduce Excited Pfaffians. Inspired by Hartree-Fock, this architecture represents many states within a single neural network. Excited Pfaffians also serve as generalized wave functions, allowing a single model to represent multi-state potential energy surfaces. On the carbon dimer, we match the $O(N_s^4)$-scaling natural excited states while training $>200\times$ faster and modeling 50\% more states. Our favorable scaling enables us to be the first to use neural networks to find all distinct energy levels of the beryllium atom. Finally, we demonstrate that a single wave function can represent excited states across various molecules.