Computing Excited States of Molecules Using Normalizing Flows

Calculations of highly excited and delocalized molecular vibrational states are computationally challenging tasks, which strongly depend on the choice of coordinates for describing vibrational motions. We introduce a new method that leverages normalizing flows, i.e, parametrized invertible functions, to learn optimal vibrational coordinates that satisfy the variational principle. This approach produces coordinates tailored to the vibrational problem at hand, significantly increasing the accuracy and enhancing the basis set convergence of the calculated energy spectrum. The efficiency of the method is demonstrated in calculations of the 100 lowest excited vibrational states of H₂S, H₂CO, and HCN/HNC. The method effectively captures the essential vibrational behavior of molecules by enhancing the separability of the Hamiltonian and hence allows for an effective assignment of approximate quantum numbers. We demonstrate that the optimized coordinates are transferable across different levels of basis set truncation, enabling a cost-efficient protocol for computing vibrational spectra of high-dimensional systems.