Modal Backflow Neural Quantum States for Anharmonic Vibrational Calculations

TL;DR

This work introduces modal backflow (MBF) neural quantum states to tackle anharmonic vibrational problems by embedding occupation-number dependent modals into a bosonic wavefunction, thereby avoiding the computational burden of bosonic permanents. A selected-configuration scheme for observables and gradients, together with vibrational self-consistent field (VSCF) pretraining, enables accurate ZPEs and vibrational transitions across varying anharmonicity. Tested on randomly generated Watson Hamiltonians and ab initio molecular systems (ClO$_2$, H$_2$CO, CH$_3$CN), MBF achieves spectroscopic accuracy (approximately 1 cm$^{-1}$) for ZPE and low-lying transitions and demonstrates faster convergence than standard feedforward networks, though it is not yet superior to tensor-network approaches in all cases. The MBF framework lays groundwork for integrating physically informed structures into NQS for vibrational spectroscopy and invites future optimization and basis-expansion enhancements, as well as extensions toward pre-Born-Oppenheimer treatments.

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 has 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.

Figures (6)

• Figure 1:
• Figure 2: Distribution of the anharmonic correction of sampled anharmonic 4-mode Hamiltonians with $N_{\rm max}=9$. In each distribution, $10^3$ Hamiltonians were sampled, and both $\kappa^{(3)}_{ijk}$ and $\kappa^{(4)}_{ijkl}$ were set to the weak, moderate, and strong regime of anharmonicity.
• Figure 3: (Top) Comparison of the optimization between FNN and MBF for targeting the ground state of a randomly generated 4-mode Watson Hamiltonian with the moderate anharmonicity setting. For both networks we set $\alpha=1$. (Bottom) Comparison of the final error between FNN and MBF after 2000 iterations, for $\alpha = 1,2,4,$ and 8. $N_s=128$ and $N_{\rm max}=6$ for all calculations.
• Figure 4: Error (cm$^{-1}$) of the ground state optimization with MBF wavefunctions. Each 3$\times$3 block correspond to a combination of $N_{\rm max}$ and selected space dimension $N_s$. Within each block, different combinations of anharmonic strength (w: weak, m: moderate, s: strong) of third- ($\bm{\kappa}^{(3)}$) and fourth-order partial derivatives ($\bm{\kappa}^{(4)}$) of the PES are used to generate 100 4-mode Watson Hamiltonians, over which the final errors of the MBF energy are averaged and color-coded on a log-scale, with red, white, and blue to represent above, at, and below spectroscopic accuracy of 1 cm${-1}$, respectively. All calculations used the same hyperparameters for the optimization and ran for 1000 iterations. A VSCF pretraining was used before each optimization.
• Figure 5: (Top) Optimization of the lowest eight vibrational levels (shifted by the ZPE) of $\rm ClO_2$ using an MBF with $\alpha = 2$ and $N_s = 64$. Energy levels obtained by vDMRG are plotted as reference. $N_{\rm max}=9$. (Bottom) Comparison of the average error of the three lowest eigenstates between optimizations with and without VSCF pretraining. The shaded area marks the range of the standard deviation of the three states.