# Modal Backflow Neural Quantum States for Anharmonic Vibrational Calculations

## TL;DR

This paper introduces a new neural network design for solving quantum problems involving vibrations with high accuracy.

## Contribution

The novel contribution is the development of modal backflow neural quantum states for efficient and accurate vibrational calculations.

## Key findings

- The MBF network achieves spectroscopically accurate results for zero-point energies and vibrational transitions.
- A selected-configuration scheme improves accuracy in spectroscopic calculations over traditional Monte Carlo methods.
- The MBF design enables efficient pretraining via vibrational self-consistent field calculations.

## Abstract

Neural quantum states (NQS) are a promising ansatz for
solving
many-body quantum problems due to their inherent expressiveness. Yet
this expressiveness can only be harnessed efficiently for treating
identical particles if the suitable physical knowledge is hardwired
into the neural network itself. For electronic structure, NQS based
on backflow determinants have been shown to be a powerful ansatz for
capturing strong correlation. By contrast, the analogue for bosons,
backflow permanents, is unpractical due to the steep cost of computing
the matrix permanent and due to the lack of particle conservation
in common bosonic problems. To circumvent these obstacles, we introduce
a modal backflow (MBF) NQS design and demonstrate its efficacy by
solving the anharmonic vibrational problem. To accommodate the demand
of high accuracy in spectroscopic calculations, we implement a selected-configuration
scheme for evaluating physical observables and gradients, replacing
the standard stochastic approach based on Monte Carlo sampling. A
vibrational self-consistent field calculation is conveniently carried
out within the MBF network, which serves as a pretraining step to
accelerate and stabilize the optimization. In applications to both
artificial and ab initio Hamiltonians, we find that the MBF network
is capable of delivering spectroscopically accurate zero-point energies
and vibrational transitions in all anharmonic regimes.