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

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 non-orthogonal. 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 image, picolinic acid N-oxide, trans-stilbene, a generalized proton transfer system, and selected model systems.