Vibrational Hamiltonian¶

The foundation of every MCTDH calculation is the description of nuclear motion. Within the harmonic approximation, the vibrations of a molecule are represented as a collection of independent normal modes, each behaving as a one-dimensional quantum harmonic oscillator.

Instead of treating the motion of individual atoms, the molecular vibrations are transformed into mass-weighted normal coordinates \(Q_i\), where each normal mode oscillates independently with its own harmonic frequency \(\omega_i\). This transformation greatly simplifies the molecular Hamiltonian and forms the basis for the vibronic model used in MCTDH.

Harmonic Approximation¶

Near the equilibrium geometry, the molecular potential energy surface can be approximated by a quadratic expansion,

\[ V(Q) \approx \frac{1}{2}\sum_i \omega_i^2 Q_i^2, \]

where

\(Q_i\) is the normal coordinate of the \(i^{\text{th}}\) vibrational mode,
\(\omega_i\) is its harmonic frequency.

Under this approximation, each vibrational mode behaves as an independent harmonic oscillator.

Kinetic Energy¶

The kinetic energy operator associated with the \(i^{\text{th}}\) normal mode is

\[ \hat{T}_i = -\frac{1}{2} \frac{\partial^2}{\partial Q_i^2}. \]

This operator describes the quantum mechanical motion of the nuclei along the corresponding normal coordinate.

For a molecule containing \(N\) active vibrational modes,

\[ \boxed{ \hat{T} = \sum_{i=1}^{N} -\frac{1}{2} \frac{\partial^2}{\partial Q_i^2} } \]

is the total vibrational kinetic energy.

Harmonic Potential Energy¶

Each normal mode experiences a restoring force that is represented by the harmonic potential

\[ \hat{V}_i = \frac{1}{2}\omega_i^2Q_i^2. \]

Summing over all active modes gives

\[ \boxed{ \hat{V} = \sum_{i=1}^{N} \frac{1}{2}\omega_i^2Q_i^2. } \]

The harmonic approximation assumes that the vibrational modes are uncoupled and that the potential energy surface is quadratic around the equilibrium geometry.

Vibrational Hamiltonian¶

Combining the kinetic and potential energy operators gives the vibrational Hamiltonian,

\[ \boxed{ \hat{H}_{\mathrm{vib}} = \sum_{i=1}^{N} \left( -\frac12 \frac{\partial^2}{\partial Q_i^2} + \frac12 \omega_i^2Q_i^2 \right). } \]

This Hamiltonian describes the vibrational motion of the molecule in the absence of electronic coupling.

Representation in the Operator File¶

The vibrational Hamiltonian appears directly in the generated MCTDH operator file.

The kinetic energy of each mode is written as

w1            |1 KE
w2            |2 KE
...

while the harmonic potential is represented as

0.5*w1        |1 q^2
0.5*w2        |2 q^2
...

Here,

KE denotes the kinetic energy operator,
q represents the normal coordinate,
q² represents the harmonic potential,
w1, w2, ... are the vibrational frequencies defined earlier in the parameter section.

The frequencies themselves are obtained from the harmonic vibrational analysis and are converted into electron volts before being written to the operator file.

Physical Interpretation¶

Each active vibrational mode behaves as an independent quantum harmonic oscillator. The kinetic energy term accounts for the motion of the nuclei, while the harmonic potential confines the nuclei around the equilibrium geometry.

These terms form the vibrational backbone of the vibronic Hamiltonian. In the following sections, electronic energies and vibronic coupling terms are added to this vibrational Hamiltonian, ultimately producing the complete Hamiltonian used for MCTDH wavepacket propagation.

Summary¶

The vibrational Hamiltonian represents the uncoupled nuclear motion of the molecule within the harmonic approximation. It consists of the kinetic energy and harmonic potential energy of every selected normal mode and forms the first contribution to the complete vibronic Hamiltonian. In the MCTDH operator file, these terms are implemented through the KE and q² operators together with the vibrational frequencies defined in the parameter section.