Adaptive sparse grid expansions of the vibrational Hamiltonian

The vibrational Hamiltonian involves two high dimensional operators, the kinetic energy operator (KEO), and the potential energy surface (PES). Both must be approximated for systems involving more than a few atoms. Adaptive approximation schemes are not only superior to truncated Taylor or many-body expansions (MBE), they also allow for error estimates, and thus operators of predefined precision. To this end, modified sparse grids (SG) are developed that can be combined with adaptive MBEs. This MBE/SG hybrid approach yields a unified, fully adaptive representation of the KEO and the PES. Refinement criteria, based on the vibrational self-consistent field (VSCF) and vibrational configuration interaction (VCI) methods, are presented. The combination of the adaptive MBE/SG approach and the VSCF plus VCI methods yields a black box like procedure to compute accurate vibrational spectra. This is demonstrated on a test set of molecules, comprising water, formaldehyde, methanimine, and ethylene. The test set is first employed to prove convergence for semi-empirical PM3-PESs and subsequently to compute accurate vibrational spectra from CCSD(T)-PESs that agree well with experimental values.