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

TL;DR

The paper tackles the challenge of incorporating anharmonic vibrational effects into electronic structure calculations by developing a Gaussian-based, non-orthogonal basis that yields analytic overlap and kinetic energy matrix elements, while computing the anharmonic potential with Gauss–Hermite quadrature. Framed as a VCI-like approach, it extends naturally to multiple coupled vibrational modes via a product basis and 2D Hermite grids, enabling accurate predictions of fundamental frequencies, anharmonic corrections to bond lengths, and coupling-induced phenomena. Demonstrations on diatomic molecules show rapid convergence with modest basis sizes, while coupled-oscillator cases such as NH3 and CO2 reveal essential physics like N tunneling in umbrella modes and Fermi resonances, respectively. The framework offers a practical path to integrate anharmonic vibrational effects into density functional theory workflows and other electronic structure methods, with clear guidance on convergence and opportunities for reduced-scaling strategies.

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$_2$).

Figures (7)

• Figure 1: Fundamental frequencies of selected diatomic molecules calculated using different basis set sizes for the anharmonic Hamiltonian.
• Figure 2: Anharmonic corrections to the bond lengths of selected diatomic molecules using different basis set sizes for the anharmonic Hamiltonian.
• Figure 3: Energy surface of NH$_3$ with respect to the displacements along the umbrella mode (x) and symmetric stretch mode(y) on the two-dimensional Hermite grid. Note that the square root of the energy ($E(x,y)-E_{min}$) is used for the plot to enhance color variation. Therefore, the actual unit for the color map is eV$^{1/2}$.
• Figure 4: The ground-state vibrational energy (in cm$^{-1}$) of bending and symmetric stretch modes in NH$_3$ with different number of basis functions.
• Figure 5: Ground-state (top) and first excited-state (bottom) coupled-anharmonic wavefunctions for the umbrella and symmetric stretch modes were computed using 14 basis functions ($l = 0$ to $13$) per mode.