Quadratic Vibronic Coupling¶

The linear vibronic coupling model assumes that the electronic Hamiltonian varies linearly with the nuclear displacements. While this approximation is sufficient for many systems, it becomes inadequate when the potential energy surfaces exhibit significant curvature away from the equilibrium geometry.

To improve the description of the vibronic Hamiltonian, the Taylor expansion of the electronic Hamiltonian is extended to include second-order terms in the normal coordinates. These contributions are known as Quadratic Vibronic Coupling (QVC) terms.

Second-Order Expansion¶

Expanding the electronic Hamiltonian about the equilibrium geometry gives

\[ \hat{H}_{el}(Q) = \hat{H}_{el}(0) + \sum_\alpha \left( \frac{\partial \hat{H}_{el}} {\partial Q_\alpha} \right)_0 Q_\alpha + \frac12 \sum_\alpha \left( \frac{\partial^2 \hat{H}_{el}} {\partial Q_\alpha^2} \right)_0 Q_\alpha^2 +\cdots \]

The first-order derivatives produce the linear vibronic coupling discussed previously, while the second-order derivatives give rise to the quadratic coupling terms.

Intrastate Quadratic Coupling¶

The diagonal quadratic terms describe the curvature of an individual diabatic potential energy surface,

\[ \boxed{ V_{ii}^{(2)} = \frac12 g_{ii}^{(\alpha)} Q_\alpha^2 } \]

where

\(g_{ii}^{(\alpha)}\) is the quadratic coupling constant,
\(Q_\alpha\) is the normal coordinate.

These terms modify the shape of the harmonic potential associated with each electronic state.

Interstate Quadratic Coupling¶

Quadratic corrections may also occur between different electronic states,

\[ \boxed{ V_{ij}^{(2)} = \frac12 g_{ij}^{(\alpha)} Q_\alpha^2, \qquad i\neq j. } \]

These terms describe the second-order dependence of the interstate electronic coupling on the nuclear displacement.

Although they are generally smaller than the linear coupling terms, they can become important for strongly distorted geometries and highly anharmonic potential energy surfaces.

Quadratic Vibronic Hamiltonian¶

Including all quadratic contributions gives

\[ \boxed{ \hat{H}_{QVC} = \frac12 \sum_\alpha \sum_{i,j} g_{ij}^{(\alpha)} Q_\alpha^2 |i\rangle\langle j|. } \]

The factor of one-half arises naturally from the Taylor expansion.

Matrix Representation¶

For a system containing three electronic states, the quadratic coupling matrix for vibrational mode \(Q_\alpha\) is

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

The quadratic contribution to the Hamiltonian is obtained by multiplying this matrix by \(Q_\alpha^2\).

Representation in the Operator File¶

The quadratic coupling parameters are defined in the Parameter Section of the operator file as

h11_g1
h12_g1
h13_g1
h22_g1
h23_g1
h33_g1

where

the first two indices identify the interacting electronic states,
g denotes a quadratic vibronic coupling,
the final number specifies the vibrational mode.

For example,

h23_g7

represents the quadratic coupling between electronic states 2 and 3 associated with vibrational mode 7.

Hamiltonian Representation¶

Within the Hamiltonian section of the operator file, these parameters appear as

0.5*h11_g1    |1 q^2 |el S1&1
0.5*h12_g1    |1 q^2 |el S1&2
0.5*h22_g1    |1 q^2 |el S2&2

The factor of 0.5 corresponds directly to the coefficient appearing in the second-order Taylor expansion.

Each parameter multiplies the squared normal coordinate operator \(q^2\) and acts on the corresponding electronic states.

Physical Interpretation¶

Quadratic vibronic coupling accounts for changes in the curvature of the diabatic potential energy surfaces as the nuclei move away from the equilibrium geometry.

These terms become particularly important when

the harmonic approximation is insufficient,
the electronic energy varies nonlinearly with the nuclear displacement,
or accurate reproduction of the diabatic potential energy surfaces is required.

Including quadratic coupling generally leads to a more faithful representation of the vibronic Hamiltonian and improves the accuracy of wavepacket propagation.

Summary¶

Quadratic vibronic coupling extends the linear vibronic coupling model by incorporating second-order changes in the electronic Hamiltonian with respect to the normal coordinates. These contributions are represented in the operator file by parameters such as h11_g1, h12_g1, and h23_g5, which multiply the \(q^2\) operator in the Hamiltonian. Together with the linear coupling terms, they provide a more complete description of the interaction between electronic and vibrational motion.

The next section combines the vibrational Hamiltonian, electronic Hamiltonian, and both linear and quadratic vibronic couplings into the complete vibronic Hamiltonian implemented by the MCTDH operator file.