Excited Pfaffians: Generalized Neural Wave Functions Across Structure and State

Abstract

Neural-network wave functions in Variational Monte Carlo (VMC) have achieved great success in accurately representing both ground and ex- cited states. However, achieving sufficient numer- ical accuracy in state overlaps requires increasing the number of Monte Carlo samples, and conse- quently 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 evalu- ate all states for all samples, we introduce Ex- cited Pfaffians. Inspired by Hartree-Fock, this architecture represents many states within a sin- gle 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(Ns4)-scaling natural excited states while train- ing > 200× 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 demon- strate that a single wave function can represent excited states across various molecules.