Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

MO-HEOM: Extending Hierarchical Equations of Motion to Molecular Orbital Space

Yankai Zhang, Yoshitaka Tanimura, So Hirata

TL;DR

MO-HEOM extends hierarchical equations of motion to molecular orbitals by embedding a 3D rotationally invariant system-bath framework that explicitly includes intramolecular vibrational motion. The approach combines MS-LT-QFPE in Wigner space (for quantum transport), BO-HEOM (for harmonic vibrational baths), and AO-HEOM (for electromagnetic baths) into a unified MO-based open-quantum-dynamics method, preserving molecular symmetry. Demonstrated on H_2, the method yields absorption spectra that capture bath-induced broadening, vibronic sidebands, and temperature-dependent features in a nonperturbative, numerically exact manner. This framework provides a pathway for first-principles quantum thermodynamics of excited molecular states and can be extended to more complex systems, anisotropic environments, and nonlinear spectroscopies.

Abstract

Studies of quantum thermal effects on molecular excitation dynamics have often relied on oversimplified models, such as energy eigenstates or low-dimensional potentials, which fail to capture the complexity of real chemical systems. In reality, molecules are spatially extended and embedded in anisotropic environments, where molecular orbitals (MOs) play a central role in determining quantum behavior. To advance beyond these limitations, we propose a three-dimensional rotationally invariant system-bath (3D-RISB) model within the MO framework, with explicit inclusion of intramolecular vibrational motion. From this MO foundation, we derive numerically ``exact'' hierarchical equations of motion (MO-HEOM). As a demonstration, we analyze hydrogen molecules and hydrogen molecular ions with vibrational degrees of freedom, revealing their linear absorption spectra.

MO-HEOM: Extending Hierarchical Equations of Motion to Molecular Orbital Space

TL;DR

MO-HEOM extends hierarchical equations of motion to molecular orbitals by embedding a 3D rotationally invariant system-bath framework that explicitly includes intramolecular vibrational motion. The approach combines MS-LT-QFPE in Wigner space (for quantum transport), BO-HEOM (for harmonic vibrational baths), and AO-HEOM (for electromagnetic baths) into a unified MO-based open-quantum-dynamics method, preserving molecular symmetry. Demonstrated on H_2, the method yields absorption spectra that capture bath-induced broadening, vibronic sidebands, and temperature-dependent features in a nonperturbative, numerically exact manner. This framework provides a pathway for first-principles quantum thermodynamics of excited molecular states and can be extended to more complex systems, anisotropic environments, and nonlinear spectroscopies.

Abstract

Studies of quantum thermal effects on molecular excitation dynamics have often relied on oversimplified models, such as energy eigenstates or low-dimensional potentials, which fail to capture the complexity of real chemical systems. In reality, molecules are spatially extended and embedded in anisotropic environments, where molecular orbitals (MOs) play a central role in determining quantum behavior. To advance beyond these limitations, we propose a three-dimensional rotationally invariant system-bath (3D-RISB) model within the MO framework, with explicit inclusion of intramolecular vibrational motion. From this MO foundation, we derive numerically ``exact'' hierarchical equations of motion (MO-HEOM). As a demonstration, we analyze hydrogen molecules and hydrogen molecular ions with vibrational degrees of freedom, revealing their linear absorption spectra.
Paper Structure (22 sections, 52 equations, 2 figures, 2 tables)

This paper contains 22 sections, 52 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Molecular electronic systems in dissipative environments
  3. Electronic Hamiltonian for a Diatomic Molecule
  4. Linear absorption spectrum
  5. Effects of environments
  6. Molecular vibrational relaxation
  7. Electromagnetic relaxation
  8. MO-HEOM
  9. MS-LT-QFPE framework
  10. BO-HEOM framework
  11. Including AO-HEOM framework for electromagnetic bath
  12. Numerical demonstrations: absorption spectra of H$_2$ molecule
  13. Setting input parameters for MO-HEOM
  14. Absorption spectra
  15. Results
  16. ...and 7 more sections

Figures (2)

  • Figure 1: Linear absorption spectrum of H$_2$ for (a) $\beta = 40$ and (b) $\beta = 60$. For comparison with the MO-HEOM results, the spectral peaks calculated from the golden rule [Eq. \ref{['Goldenrule']}] are shown as green vertical lines. Each spectrum is normalized to its maximum value, whereas the results obtained from Fermi's golden rule are scaled to 0.5 so as not to obscure the MO-HEOM spectra.
  • Figure 2: The potential energy surfaces of the $\Sigma_{g}^{+}$ and $B^{1}\Sigma_{u}^{+}$ states, given in atomic units (a.u.).