Solving the vibrational Schrödinger equation on an arbitrary multidimensional potential energy surface by the finite element method

A computational protocol has been developed to solve the bounded vibrational Schrödinger equation for up to three coupled coordinates on any given effective potential energy surface (PES). The dynamic Wilson G-matrix is evaluated from the discrete PES calculations, allowing the PES to be parametrized in terms of any complete, minimal set of coordinates, whether orthogonal or nonorthogonal. The partial differential equation is solved using the finite element method (FEM), to take advantage of its localized basis set structure and intrinsic scalability to multiple dimensions. A mixed programming paradigm takes advantage of existing libraries for constructing the FEM basis and carrying out the linear algebra. Results are presented from a series of calculations confirming the flexibility, accuracy, and efficiency of the protocol, including tests on FHF−, picolinic acid N -oxide, trans-stilbene, a generalized proton transfer system, and selected model systems.