Computing excited states of molecules using normalizing flows

Calculations of highly excited and delocalized molecular vibrational states is a computationally challenging task, which strongly depends on the choice of coordinates for describing vibrational motions. We introduce a new method that utilizes normalizing flows (parametrized invertible functions) to optimize vibrational coordinates to satisfy the variational principle. This approach produces coordinates specifically tailored to the vibrational problem at hand, significantly increasing the accuracy and enhancing basis set convergence of calculated energy spectrum. The efficiency of the method is demonstrated in calculations of the 100 lowest excited vibrational states of H$_2$S, H$_2$CO, and HCN/CNH. The method effectively captures the essential vibrational behavior of molecules by enhancing the separability of the Hamiltonian. We further 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.