Calculating vibrational spectra of molecules using tensor train decomposition

We propose a new algorithm for calculation of vibrational spectra of molecules using tensor train decomposition. Under the assumption that eigenfunctions lie on a low-parametric manifold of low-rank tensors we suggest using well-known iterative methods that utilize matrix inversion (LOBPCG, inverse iteration) and solve corresponding linear systems inexactly along this manifold. As an application, we accurately compute vibrational spectra (84 states) of acetonitrile molecule CH$_3$CN on a laptop in one hour using only $100$ MB of memory to represent all computed eigenfunctions.