Using an internal coordinate Gaussian basis and a space-fixed Cartesian coordinate kinetic energy operator to compute a vibrational spectrum with rectangular collocation

We demonstrate that it is possible to use basis functions that depend on curvilinear internal coordinates to compute vibrational energy levels without deriving a kinetic energy operator (KEO) and without numerically computing coefficients of a KEO. This is done by using a space-fixed KEO and computing KEO matrix elements numerically. Whenever one has an excellent basis, more accurate solutions to the Schrödinger equation can be obtained by computing the KEO, potential, and overlap matrix elements numerically. Using a Gaussian basis and bond coordinates, we compute vibrational energy levels of formaldehyde. We show, for the first time, that it is possible with a Gaussian basis to solve a six-dimensional vibrational Schrödinger equation. For the zero-point energy (ZPE) and the lowest 50 vibrational transitions of H₂CO, we obtain a mean absolute error of less than 1 cm⁻¹; with 200 000 collocation points and 40 000 basis functions, most errors are less than 0.4 cm⁻¹.