Determining quantum molecular potentials from spectroscopic energy levels using parametric equations of motion

A system of differential equations is presented for evolving the quantum potential as a function of its energy levels. These inverse parametric equations of motion (i-PEM) offer a novel approach to determining quantum molecular potentials from spectroscopic energy levels. The technique uses singular-value decomposition to ensure that the chosen trajectory through energy space is representable by a smooth potential trajectory. The i-PEM are facilitated by discretizing the vibrational Schrödinger equation with a spectral element method which combines the features of Hamiltonian sparsity and exponential convergence of the wave function. Often, spectroscopic data significantly affect only a specific region of the potential, and the spectral elements offer a natural framework for identifying the appropriate portion of the potential. The i-PEM with spectral elements are applied in a simulation for determining the potential of hydrogen fluoride.