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Quadratic Vibronic Coupling (QVC)

The Linear Vibronic Coupling (LVC) model assumes that the electronic potential energy surfaces vary linearly with the nuclear displacements. While this approximation accurately describes dynamics close to the Franck–Condon geometry, it becomes less reliable when the wavepacket explores regions further from equilibrium.

The Quadratic Vibronic Coupling (QVC) model improves upon LVC by including second-order terms in the normal coordinates. These additional terms allow changes in vibrational frequencies, mode mixing, and anharmonic effects to be incorporated into the vibronic Hamiltonian.

Why are Quadratic Terms Needed?

After photoexcitation, the nuclear wavepacket moves away from the equilibrium geometry. As the displacement increases,

  • the potential energy surfaces become less harmonic,
  • vibrational frequencies may change,
  • different normal modes can interact,
  • the electronic coupling may become coordinate dependent.

The LVC approximation cannot describe these effects because it retains only first-order terms.

Including quadratic terms extends the validity of the Hamiltonian over a much larger region of the potential energy surface.

Taylor Expansion of the Potential

Expanding the diabatic potential around the reference geometry gives

\[ V(Q) = V_0 + \sum_k \left( \frac{\partial V}{\partial Q_k} \right)_0 Q_k + \frac12 \sum_{kl} \left( \frac{\partial^2V} {\partial Q_k\partial Q_l} \right)_0 Q_kQ_l +\cdots \]

The three terms represent

  • constant electronic energy,
  • linear vibronic coupling,
  • quadratic vibronic coupling.

The QVC model simply retains the second-order contribution of this expansion.

Diagonal Quadratic Terms

For an electronic state \(i\), the diabatic potential becomes

\[ V_i(Q) = E_i + \sum_k \kappa_k^{(i)}Q_k + \frac12 \sum_k \omega_k^2Q_k^2 + \frac12 \sum_{kl} \gamma_{kl}^{(i)} Q_kQ_l \]

where

  • \(E_i\) is the electronic energy,
  • \(\kappa_k^{(i)}\) are the linear intrastate coupling constants,
  • \(\gamma_{kl}^{(i)}\) are the quadratic coupling constants.

The parameters \(\gamma_{kl}\) describe how the curvature of the potential changes with nuclear displacement.

Off-Diagonal Quadratic Coupling

The interstate coupling may also depend quadratically on the nuclear coordinates,

\[ V_{ij}(Q) = \sum_k \lambda_k^{(ij)}Q_k + \frac12 \sum_{kl} \eta_{kl}^{(ij)} Q_kQ_l \]

where

  • \(\lambda_k^{(ij)}\) are the linear interstate couplings,
  • \(\eta_{kl}^{(ij)}\) are quadratic interstate couplings.

These terms become important when the electronic interaction changes significantly along the vibrational coordinates.

Complete QVC Hamiltonian

Including both diagonal and off-diagonal quadratic contributions gives

\[ \boxed{ \hat H = \hat T_{nuc} + \sum_i |i\rangle \left( E_i + \sum_k\kappa_kQ_k + \frac12\sum_k\omega_k^2Q_k^2 + \frac12\sum_{kl}\gamma_{kl}Q_kQ_l \right) \langle i| + \sum_{i\neq j} |i\rangle \left( \sum_k\lambda_kQ_k + \frac12\sum_{kl}\eta_{kl}Q_kQ_l \right) \langle j| } \]

The LVC Hamiltonian is obtained simply by setting

\[ \gamma_{kl}=0, \qquad \eta_{kl}=0. \]

Thus, QVC is a direct generalization of the Linear Vibronic Coupling model.

Physical Interpretation

Each quadratic term represents a correction to the harmonic approximation.

Different types of quadratic terms have different physical meanings.

Term Physical interpretation
\(Q_k^2\) Change in potential curvature
\(Q_kQ_l\) Coupling between two vibrational modes
\(\gamma_{kl}\) Intrastate quadratic coupling
\(\eta_{kl}\) Interstate quadratic coupling

These terms become increasingly important for large-amplitude nuclear motion.

Connection with the Operator File

When quadratic couplings are included, additional terms appear in the MCTDH operator file.

Typical examples are

q1^2
q3*q7
q12*q18

or, depending on the operator generator,

v3 | q^2
v5 | q*q

Each of these represents one quadratic contribution to the Hamiltonian.

Compared with the LVC Hamiltonian, the QVC operator file contains additional products of vibrational coordinates.

When is QVC Necessary?

The QVC model is generally preferred when

  • large nuclear displacements occur,
  • conical intersections are sampled far from the Franck–Condon point,
  • vibrational frequencies change significantly,
  • strong mode mixing is present,
  • highly accurate excited-state dynamics are required.

For systems whose dynamics remain close to the equilibrium geometry, the LVC approximation is often sufficient.

Relation to This Tutorial

The operator-generation workflow presented in this documentation is based primarily on the Linear Vibronic Coupling (LVC) model. The fitting procedure extracts linear coupling constants from quantum chemistry calculations and constructs an operator file suitable for MCTDH propagation.

Nevertheless, the mathematical framework naturally extends to the Quadratic Vibronic Coupling (QVC) model by including second-order coupling parameters obtained from higher-order fitting of the potential energy surfaces.

Understanding QVC is therefore important, even if only the LVC approximation is employed in the present calculations.

Next Step

The Hamiltonian has now been completely defined. The next question is how this Hamiltonian acts on the molecular wavefunction during propagation.

The next section introduces the MCTDH Wavefunction Expansion, which explains how the multidimensional wavefunction is represented efficiently using time-dependent basis functions.