A multimode classical hierarchical Fokker–Planck equations approach to molecular vibrations: Simulating two-dimensional spectra

Abstract

The multimode Brownian model with nonlinear system–bath coupling offers a flexible framework for studying both intra- and intermolecular vibrational modes in condensed-phase molecular systems. This approach allows us to calculate linear and nonlinear spectra of molecular vibrations and to examine thermal effects—such as anharmonicity, energy relaxation, and dephasing—as reflected in the spectral peak profiles. In this study, we present computer software based on classical hierarchical Fokker–Planck equations applied to three vibrational modes of a molecular liquid. The primary objective of developing this code was to simulate the two-dimensional correlation spectrum of the intramolecular modes of liquid water [Hoshino and Tanimura, J. Chem. Phys. 162, 044105 (2025)]. The code has been further refined to optimize grid selection and numerical integration routines for graphics processing units. As a demonstration, we apply this setup to simulate three interacting modes representing intermolecular vibrations in water and calculate the resulting two-dimensional terahertz–Raman signals. The code and example routines are available in the supplementary material.