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Mathematics

The MCTDH operator file is a direct mathematical representation of the molecular vibronic Hamiltonian. Every parameter appearing in the operator file corresponds to a specific term in the Hamiltonian, including vibrational kinetic energy, harmonic potential energy, electronic state energies, and vibronic coupling terms.

While the previous sections focused on generating the operator file automatically, this section explains the mathematical meaning of the quantities that appear in it. Understanding these equations provides insight into how the Hamiltonian is constructed and how MCTDH interprets the operator file during wavepacket propagation and Hamiltonian diagonalization.

Rather than presenting the complete Hamiltonian immediately, we build it step by step, following the same order in which the terms appear in the generated operator file.

Learning Objectives

In this section, you will learn

  • the mathematical form of the vibronic Hamiltonian,
  • the origin of the vibrational kinetic and potential energy terms,
  • the representation of diabatic electronic state energies,
  • the meaning of interstate electronic couplings,
  • the mathematical form of linear and quadratic vibronic coupling terms,
  • and how each equation is translated into the syntax of the MCTDH operator file.

By the end of this section, every parameter appearing in the operator file will have a clear mathematical interpretation, providing a direct connection between the theoretical Hamiltonian and its implementation within MCTDH.

Mathematical Roadmap

The mathematical development follows the same structure as the generated operator file.

Vibrational Hamiltonian
          │
          ▼
Electronic Hamiltonian
          │
          ▼
Linear Vibronic Coupling
          │
          ▼
Quadratic Vibronic Coupling
          │
          ▼
Complete Vibronic Hamiltonian
          │
          ▼
Corresponding MCTDH Operator File

Each chapter introduces one component of the Hamiltonian before combining all contributions into the complete vibronic Hamiltonian used in MCTDH calculations.