Linear Vibronic Coupling¶

The electronic Hamiltonian described in the previous section corresponds to the reference molecular geometry, where the nuclei are located at their equilibrium positions. In reality, however, the electronic energies change continuously as the nuclei vibrate. This dependence of the electronic Hamiltonian on the nuclear coordinates is known as vibronic coupling.

The simplest approximation assumes that the electronic Hamiltonian varies linearly with the normal coordinates. This is referred to as the Linear Vibronic Coupling (LVC) model and forms the basis of many MCTDH calculations involving coupled electronic states.

Expansion About the Reference Geometry¶

The electronic Hamiltonian can be expanded as a Taylor series around the equilibrium geometry,

\[ \hat{H}_{el}(Q) = \hat{H}_{el}(0) + \sum_{\alpha} \left( \frac{\partial \hat{H}_{el}} {\partial Q_\alpha} \right)_0 Q_\alpha +\cdots \]

where

\(Q_\alpha\) is the normal coordinate of vibrational mode \(\alpha\),
\(\hat{H}_{el}(0)\) is the electronic Hamiltonian at the reference geometry.

Neglecting higher-order terms gives the linear approximation,

\[ \boxed{ \hat{H}_{LVC} = \hat{H}_{el}(0) + \sum_\alpha \mathbf{K}^{(\alpha)}Q_\alpha } \]

where \(\mathbf{K}^{(\alpha)}\) is the matrix of linear vibronic coupling constants.

Intrastate Linear Coupling¶

The diagonal elements describe how the energy of an electronic state changes with displacement along a normal coordinate.

For electronic state \(i\),

\[ \boxed{ V_{ii}^{(1)} = k_{ii}^{(\alpha)}Q_\alpha } \]

where

\(k_{ii}^{(\alpha)}\) is the linear intrastate coupling constant,
\(Q_\alpha\) is the normal coordinate.

These terms shift the minimum of the harmonic potential and are responsible for the displacement between potential energy surfaces.

Interstate Linear Coupling¶

The off-diagonal elements describe the interaction between different electronic states induced by nuclear motion,

\[ \boxed{ V_{ij}^{(1)} = k_{ij}^{(\alpha)}Q_\alpha, \qquad i\neq j. } \]

These coupling terms allow population transfer between electronic states and are responsible for nonadiabatic transitions during wavepacket propagation.

Linear Vibronic Hamiltonian¶

Combining both diagonal and off-diagonal contributions gives

\[ \boxed{ \hat{H}_{LVC} = \sum_{\alpha} \sum_{i,j} k_{ij}^{(\alpha)} Q_\alpha |i\rangle\langle j|. } \]

Each vibrational mode contributes an independent coupling matrix between the electronic states.

Matrix Representation¶

For three electronic states, the linear coupling matrix associated with vibrational mode \(Q_\alpha\) is

\[ \mathbf{K}^{(\alpha)} = \begin{pmatrix} k_{11}^{(\alpha)} & k_{12}^{(\alpha)} & k_{13}^{(\alpha)} \\ k_{12}^{(\alpha)} & k_{22}^{(\alpha)} & k_{23}^{(\alpha)} \\ k_{13}^{(\alpha)} & k_{23}^{(\alpha)} & k_{33}^{(\alpha)} \end{pmatrix}. \]

The complete linear vibronic contribution is simply this matrix multiplied by the normal coordinate \(Q_\alpha\).

Representation in the Operator File¶

The linear vibronic coupling constants appear in the Parameter Section as

h11_k1
h12_k1
h13_k1
h22_k1
h23_k1
h33_k1

where

the first two indices identify the interacting electronic states,
k denotes a linear coupling,
the final number indicates the vibrational mode.

For example,

h12_k5

represents the linear coupling between electronic states 1 and 2 arising from vibrational mode 5.

Hamiltonian Representation¶

Within the Hamiltonian section, these parameters appear as

h11_k1      |1 q  |el S1&1
h12_k1      |1 q  |el S1&2
h22_k1      |1 q  |el S2&2

The notation indicates that the coupling parameter multiplies the normal coordinate operator \(q\) for the corresponding vibrational mode while acting on the specified electronic states.

Each selected vibrational mode contributes a similar set of coupling terms.

Physical Interpretation¶

Linear vibronic coupling describes the first-order response of the electronic Hamiltonian to nuclear displacements.

If the coupling constant is zero,

\[ k_{ij}^{(\alpha)}=0, \]

the corresponding vibrational mode does not influence the interaction between those electronic states.

Larger coupling constants indicate stronger interaction between electronic and vibrational motion, leading to more efficient nonadiabatic population transfer.

The magnitude of these coupling constants is determined from the nonlinear fitting of the distorted-geometry calculations discussed in the prerequisite section.

Summary¶

Linear vibronic coupling introduces the dependence of the electronic Hamiltonian on the nuclear coordinates through first-order terms in the normal coordinates. These coupling constants connect the vibrational and electronic degrees of freedom and are represented in the operator file by parameters such as h11_k1, h12_k1, and h23_k5. They constitute the first-order vibronic interaction responsible for much of the nonadiabatic dynamics described by MCTDH.

The next section extends this model by introducing quadratic vibronic coupling, which accounts for second-order changes of the electronic Hamiltonian with respect to the nuclear coordinates.