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A Multimode Classical Hierarchical Fokker-Planck Equations Approach to Molecular Vibrations: Simulating Two-Dimensional Spectra

Ryotaro Hoshino, Yoshitaka Tanimura

TL;DR

This work develops a GPU-accelerated, classical CHFPE framework for three vibrational modes in a bath, enabling efficient computation of linear and nonlinear 1D/2D spectra, including 2D THz-Raman signals. By formulating a multimode Brownian oscillator model with cubic anharmonicity and LL/SL system–bath couplings, and representing the Wigner function through a Hermite-polynomial basis, the authors achieve a flexible, numerically stable route to simulate environmental effects on vibrational spectra. The code computes R^(1), R^(2), and R^(3) responses and corresponding spectra, demonstrating qualitative agreement with MD results for water by concentrating on three intermolecular modes (HR, HT, LT) and their couplings; the approach remains extensible to quantum treatments via QHFPE. The presented tool offers a modular, reusable platform for classical and future quantum 2D spectroscopy across condensed-phase environments, with potential applications to other solvents and systems beyond water.

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 a computer program 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. [R. Hoshino and Y. 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 (GPUs). 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.

A Multimode Classical Hierarchical Fokker-Planck Equations Approach to Molecular Vibrations: Simulating Two-Dimensional Spectra

TL;DR

This work develops a GPU-accelerated, classical CHFPE framework for three vibrational modes in a bath, enabling efficient computation of linear and nonlinear 1D/2D spectra, including 2D THz-Raman signals. By formulating a multimode Brownian oscillator model with cubic anharmonicity and LL/SL system–bath couplings, and representing the Wigner function through a Hermite-polynomial basis, the authors achieve a flexible, numerically stable route to simulate environmental effects on vibrational spectra. The code computes R^(1), R^(2), and R^(3) responses and corresponding spectra, demonstrating qualitative agreement with MD results for water by concentrating on three intermolecular modes (HR, HT, LT) and their couplings; the approach remains extensible to quantum treatments via QHFPE. The presented tool offers a modular, reusable platform for classical and future quantum 2D spectroscopy across condensed-phase environments, with potential applications to other solvents and systems beyond water.

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 a computer program 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. [R. Hoshino and Y. 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 (GPUs). 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.

Paper Structure

This paper contains 20 sections, 34 equations, 3 figures, 2 tables.

Table of Contents

  1. INTRODUCTION
  2. Formulation
  3. Multimode anharmonic Brownian model
  4. Multimode CHFPE
  5. Linear and non-Linear spectra
  6. NUMERICAL IMPLEMENTATION OF CHFPE
  7. Momentum space representation using Hermite polynomial
  8. Compact Finite Difference Scheme (CFDS)
  9. Parallel computing method for tridiagonal simultaneous differential equations
  10. Non-Uniform Mesh in coordinate space
  11. CHFPE code for simulating multi-dimensional spectra
  12. Initialization
  13. Equilibration
  14. Calculations of Response Functions and Spectra
  15. 1D IR and Raman spectra
  16. ...and 5 more sections

Figures (3)

  • Figure 1: Flowchart of the CHFPE code for computing 1D and 2D spectra
  • Figure 2: The computational results of (a) THz and (b) parallelly polarized (VV) Raman. In each figure, the red, blue, and green curves represent the contributions from hindered rotation (HR), translational (HT), and translational (LT) motion, respectively. The spectral intensities are normalized with respect to the absolute value of the peak intensities of the total contribution.
  • Figure 3: The 2D THz-Raman signals of water were calculated using the three-mode model in two cases: (A) when all mode-mode coupling was set to zero, and (B) in the case listed in Table \ref{['tab:FitAll2']}. For each case, (i) the 2D Raman-THz-THz (RTT), (ii) 2D THz-Raman-THz (TRT), and (iii) 2D THz-THz-Raman (TTR) signals were depicted. Red and blue shadings represent positive and negative signals, respectively. Signal intensities are normalized to the absolute value of the peak amplitude. The relative intensities are color-coded, with positive values shown in red and negative values in blue, as indicated by the color bar.