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

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.