Universal Approach for Determining Multi-Dimensional Anharmonic Vibrations from Electronic Quantum Methods

Abstract

We present a simple and efficient method to incorporate anharmonic effects in the vibrational \textcolor{black}{analyses} of molecules within density functional theory (DFT) calculations. This approach is closely related to the traditional vibrational \textcolor{black}{configuration} interaction (VCI) technique, which uses the harmonic oscillator wavefunctions as the basis. In our implementation, we employ Gaussian-type orbitals (GTOs), with polynomial prefactors, as the basis set to evaluate the anharmonic Hamiltonian. Although these basis functions are non-orthogonal, the matrix elements such as overlap, kinetic energy terms, and position moments can be evaluated analytically. The terms in the Hamiltonian due to the anharmonic potentials are numerically calculated on a Hermite-Quadrature grid. The potentials can be evaluated using any electronic structure method. This framework enables us to accurately calculate the anharmonicity-corrected vibrational frequencies, the fundamental frequencies, and the corrections to bond lengths in diatomic molecules. This method is also generalized to handle coupled anharmonic oscillators, which is essential to model more complex phenomena such as nitrogen tunneling in the umbrella mode of ammonia (NH$_3$) and Fermi resonances in carbon dioxide (CO$_2$).