Complete Vibronic Hamiltonian¶

The previous sections introduced the individual components of the vibronic Hamiltonian:

the vibrational Hamiltonian,
the electronic Hamiltonian,
the linear vibronic coupling,
and the quadratic vibronic coupling.

These contributions are now combined to form the complete Hamiltonian used throughout the MCTDH calculations described in this documentation.

Total Hamiltonian¶

The complete vibronic Hamiltonian can be written as

\[ \boxed{ \hat H = \hat H_{\mathrm{vib}} + \hat H_{\mathrm{el}} + \hat H_{\mathrm{LVC}} + \hat H_{\mathrm{QVC}} } \]

Substituting the individual contributions gives

\[ \boxed{ \begin{aligned} \hat H = & \sum_{\alpha=1}^{N} \left( -\frac12 \frac{\partial^2}{\partial Q_\alpha^2} + \frac12 \omega_\alpha^2Q_\alpha^2 \right) \\[4mm] & + \sum_i E_i |i\rangle\langle i| + \sum_{i<j} V_{ij} \left( |i\rangle\langle j| + |j\rangle\langle i| \right) \\[4mm] & + \sum_{\alpha} \sum_{i,j} k_{ij}^{(\alpha)} Q_\alpha |i\rangle\langle j| \\[4mm] & + \frac12 \sum_{\alpha} \sum_{i,j} g_{ij}^{(\alpha)} Q_\alpha^2 |i\rangle\langle j|. \end{aligned} } \]

This Hamiltonian describes the coupled motion of the nuclei and electrons within the Linear Vibronic Coupling model including quadratic corrections.

Interpretation of Each Term¶

The complete Hamiltonian consists of four physically distinct contributions.

Vibrational Hamiltonian¶

\[ \sum_\alpha \left( -\frac12 \frac{\partial^2}{\partial Q_\alpha^2} + \frac12 \omega_\alpha^2Q_\alpha^2 \right) \]

describes the independent harmonic motion of every selected normal mode.

Electronic Hamiltonian¶

\[ \sum_i E_i|i\rangle\langle i| + \sum_{i<j} V_{ij} (|i\rangle\langle j| + |j\rangle\langle i|) \]

defines the diabatic electronic states together with the electronic couplings between them.

Linear Vibronic Coupling¶

\[ \sum_{\alpha} \sum_{i,j} k_{ij}^{(\alpha)} Q_\alpha |i\rangle\langle j| \]

describes the first-order dependence of the electronic Hamiltonian on the nuclear displacements.

These terms are responsible for most nonadiabatic population transfer between electronic states.

Quadratic Vibronic Coupling¶

\[ \frac12 \sum_{\alpha} \sum_{i,j} g_{ij}^{(\alpha)} Q_\alpha^2 |i\rangle\langle j| \]

introduces second-order corrections to the electronic Hamiltonian and improves the representation of the diabatic potential energy surfaces.

Relation to the Operator File¶

The MCTDH operator file is simply another representation of the Hamiltonian shown above.

Instead of writing the Hamiltonian as mathematical equations, each contribution is expressed using the syntax recognized by MCTDH.

For example,

Mathematical term	Operator file
\( -\frac12\dfrac{\partial^2}{\partial Q_1^2} \)	`w1 \|1 KE`
\( \frac12\omega_1Q_1^2 \)	`0.5*w1 \|1 q^2`
\(E_1\)	`e11`
\(V_{12}\)	`e12`
\(k_{12}^{(1)}Q_1\)	`h12_k1 \|1 q`
\(\frac12g_{12}^{(1)}Q_1^2\)	`0.5*h12_g1 \|1 q^2`

Thus, every line appearing in the generated operator file corresponds directly to one term in the mathematical Hamiltonian.

Hamiltonian Construction Workflow¶

The construction of the vibronic Hamiltonian follows the workflow below.

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

The operator generation scripts developed in this work automate the final step by converting the fitted vibronic coupling parameters into the syntax required by MCTDH.

Summary¶

The complete vibronic Hamiltonian combines the vibrational motion of the nuclei, the energies of the diabatic electronic states, and the interaction between them through linear and quadratic vibronic coupling terms. Rather than requiring the user to manually construct this Hamiltonian, the operator generation workflow automatically translates each mathematical term into the corresponding MCTDH operator syntax, producing a complete operator file ready for wavepacket propagation and Hamiltonian diagonalization.

The next section provides a detailed interpretation of the generated operator file, showing how every keyword and parameter corresponds to the mathematical quantities introduced throughout this chapter.