Linear Vibronic Coupling (LVC)¶

The vibronic Hamiltonian introduced in the previous section is completely general and can describe any interaction between electronic and vibrational motion. However, for most photochemical systems, only the lowest-order terms are required to accurately describe the coupling between electronic states.

The Linear Vibronic Coupling (LVC) model assumes that the potential energy surfaces can be expanded around a reference geometry (usually the Franck–Condon point), retaining only terms that are linear in the normal mode coordinates.

This approximation forms the foundation of most MCTDH calculations involving excited-state dynamics.

Potential Energy Expansion¶

Near the reference geometry, the diabatic potential of each electronic state can be expanded as

\[ V_i(\mathbf{Q}) = E_i + \sum_k \kappa_k^{(i)}Q_k + \frac12 \sum_k \omega_k^2Q_k^2 +\cdots \]

where

\(E_i\) is the electronic energy of state \(i\),
\(Q_k\) is the normal coordinate,
\(\omega_k\) is the vibrational frequency,
\(\kappa_k^{(i)}\) is the linear coupling constant.

The first term gives the vertical excitation energy.

The second term shifts the equilibrium position of the harmonic oscillator.

The third term represents the harmonic restoring force.

Off-Diagonal Coupling¶

Electronic transitions are produced through the off-diagonal elements of the Hamiltonian.

Within the LVC approximation,

\[ V_{ij}(Q) = \sum_k \lambda_k^{(ij)}Q_k \]

where

\(i\neq j\),
\(\lambda_k^{(ij)}\) is the interstate vibronic coupling constant.

Unlike the diagonal terms, these couplings mix different electronic states and allow population transfer during wavepacket propagation.

Complete LVC Hamiltonian¶

Combining the diagonal and off-diagonal contributions gives

\[ \boxed{ \hat H= \hat T_{nuc} + \sum_i |i\rangle \left( E_i + \sum_k\kappa_k^{(i)}Q_k + \frac12\omega_k^2Q_k^2 \right) \langle i| + \sum_{i\neq j} |i\rangle \left( \sum_k\lambda_k^{(ij)}Q_k \right) \langle j| } \]

This is the Hamiltonian implemented by the operator file generated earlier in this tutorial.

Physical Interpretation¶

Each vibrational mode contributes independently to the Hamiltonian.

For every mode,

the harmonic oscillator term keeps the nuclei bound,
the linear diagonal term shifts the equilibrium geometry,
the linear off-diagonal term transfers population between electronic states.

Modes with larger coupling constants have a greater influence on the excited-state dynamics.

Connection with the Operator File¶

The LVC model directly explains many of the terms appearing in the generated operator file.

Examples include

v1          KE
0.5*w1      q^2
k1_s1       q
l1_s1_s2    q

These correspond to

Operator file term	Mathematical meaning
`KE`	Nuclear kinetic energy
`0.5w²q²`	Harmonic potential
`κQ`	Linear intrastate coupling
`λQ`	Linear interstate coupling

Thus, every line of the operator file is simply one mathematical contribution to the Linear Vibronic Coupling Hamiltonian.

Why the LVC Model Works Well¶

The LVC approximation is widely used because

it accurately describes dynamics close to the Franck–Condon region,
only a limited number of fitting parameters are required,
the Hamiltonian can be generated directly from quantum chemistry calculations,
it is computationally efficient for MCTDH simulations,
it captures the dominant mechanism of ultrafast population transfer in many molecular systems.

For many photochromic molecules, the LVC Hamiltonian reproduces the essential features of excited-state dynamics while remaining computationally tractable.

Next Step¶

The Linear Vibronic Coupling model assumes that the diabatic potentials are linear in the normal coordinates.

For highly distorted geometries or strongly anharmonic systems, higher-order terms become important.

The next section introduces the Quadratic Vibronic Coupling (QVC) model, which extends the Hamiltonian by including second-order coupling terms.