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Vibronic Hamiltonian

The molecular Hamiltonian introduced in the previous chapter provides the general framework for quantum dynamics. However, it is still too complicated to use directly in an MCTDH calculation because the exact potential energy surfaces are not known analytically.

Instead, MCTDH represents the potential energy surfaces using a Taylor expansion around a reference geometry, usually the Franck–Condon equilibrium structure. The resulting Hamiltonian is known as the vibronic Hamiltonian, since it simultaneously describes

  • electronic states,
  • nuclear vibrations,
  • and their mutual coupling.

This is the Hamiltonian that is ultimately written into the operator file and used during wavepacket propagation.

Expanding the Potential Energy Surface

For each electronic state, the potential energy is expanded around the equilibrium geometry using normal-mode coordinates,

\[ Q=(Q_1,Q_2,\ldots,Q_f). \]

The expansion has the form

\[ V(Q) = V_0 + \sum_i \left( \frac{\partial V}{\partial Q_i} \right)_0 Q_i + \frac12 \sum_i \left( \frac{\partial^2V}{\partial Q_i^2} \right)_0 Q_i^2 +\cdots \]

where

  • \(V_0\) is the electronic energy at equilibrium,
  • the first derivatives produce linear coupling terms,
  • the second derivatives describe harmonic restoring forces,
  • higher-order derivatives describe anharmonicity.

For most vibronic dynamics calculations only the first few terms are retained.

Harmonic Approximation

Near the equilibrium geometry, every normal mode behaves approximately as a harmonic oscillator.

The potential becomes

\[ V(Q) = V_0 + \frac12 \sum_i \omega_i^2Q_i^2 \]

where

  • \(\omega_i\) is the vibrational frequency of mode \(i\).

These frequencies are obtained directly from the frequency calculation performed in GAMESS.

Consequently, every quadratic term appearing in the operator file originates from the normal-mode analysis.

Multiple Electronic States

Unlike ordinary vibrational calculations, MCTDH simultaneously treats several electronic states.

The Hamiltonian therefore becomes a matrix,

\[ \hat H = \begin{pmatrix} \hat H_{11} & \hat H_{12} & \cdots \\ \hat H_{21} & \hat H_{22} & \cdots \\ \vdots & \vdots & \ddots \end{pmatrix}. \]

Each diagonal element describes motion on one electronic potential-energy surface,

while the off-diagonal elements couple different electronic states and permit population transfer during propagation.

For example,

\[ \hat H_{12} \]

couples electronic states 1 and 2,

while

\[ \hat H_{23} \]

couples states 2 and 3.

These coupling operators appear in the operator file as terms such as

S1&2
S2&3
S3&4

which multiply the corresponding vibronic coupling constants.

Electronic-State Hamiltonian

The diagonal Hamiltonian for electronic state \(n\) is

\[ \boxed{ \hat H_n = E_n + \hat T + \frac12 \sum_i \omega_i^2Q_i^2 } \]

where

  • \(E_n\) is the electronic energy,
  • \(\hat T\) is the nuclear kinetic-energy operator,
  • the final term is the harmonic vibrational potential.

This represents nuclear motion confined to a single electronic state.

Coupled-State Hamiltonian

When several electronic states interact, the Hamiltonian becomes

\[ \boxed{ \hat H = \sum_n |n\rangle \hat H_n \langle n| + \sum_{m\neq n} |m\rangle V_{mn}(Q) \langle n| } \]

where

  • the first term describes motion on each electronic surface,
  • the second term allows transitions between different electronic states.

These interstate coupling operators are responsible for the population transfer observed during MCTDH propagation.

Connection with the Operator File

The operator file is simply a numerical implementation of this Hamiltonian.

For example,

modes
v1
v2
...
end-modes

defines the coordinates \(Q_i\),

while

1.2350 |1 S1&1

represents the electronic energy

\[ E_1|1\rangle\langle1|. \]

Similarly,

0.0214 |5 q S1&2

represents a coordinate-dependent interstate coupling,

\[ \lambda_5Q_5 |1\rangle \langle2|. \]

Thus every numerical coefficient appearing in the operator file corresponds directly to one term of the vibronic Hamiltonian.

Connection with the Previous Tutorial

During operator generation, the following quantities were obtained from quantum chemistry calculations:

  • equilibrium electronic energies,
  • normal-mode frequencies,
  • fitted linear coupling constants,
  • fitted quadratic coupling constants.

The bash script and Fortran program assembled these quantities into the Hamiltonian described in this chapter, producing the final operator file required by MCTDH.

What Comes Next?

The vibronic Hamiltonian still contains unknown coupling functions,

\[ V_{mn}(Q). \]

The next chapter introduces the Linear Vibronic Coupling (LVC) model, where these coupling functions are approximated by terms that are linear in the normal coordinates. This approximation forms the basis of most practical MCTDH simulations and explains the origin of the linear coefficients appearing throughout the operator file.