Electronic Hamiltonian¶

The vibrational Hamiltonian described in the previous section accounts only for the nuclear motion of the molecule. However, many photochemical and nonadiabatic processes involve several interacting electronic states whose energies change as the nuclei move.

To describe these processes, the electronic states are incorporated into the Hamiltonian using the diabatic representation. In this representation, each electronic state is treated as a basis function, while the interactions between different states are represented explicitly through electronic coupling terms.

Diabatic Electronic States¶

Consider a molecule described by \(M\) diabatic electronic states,

\[ |1\rangle,\; |2\rangle,\; |3\rangle,\; \ldots,\; |M\rangle. \]

Each state possesses an electronic energy evaluated at the reference (equilibrium) geometry.

These reference energies form the diagonal elements of the electronic Hamiltonian.

Diagonal Electronic Energies¶

If the energy of the \(i^{\text{th}}\) diabatic state is denoted by \(E_i\), the electronic Hamiltonian without interstate interactions is

\[ \boxed{ \hat{H}_{\mathrm{diag}} = \sum_{i=1}^{M} E_i |i\rangle\langle i| } \]

where

\(E_i\) is the energy of electronic state \(i\),
\(|i\rangle\langle i|\) projects onto that electronic state.

The lowest-energy state is usually taken as the reference, and its energy is shifted to zero. All remaining state energies are expressed relative to this reference.

Interstate Electronic Couplings¶

The diabatic electronic states are generally not independent. Interactions between different electronic states are represented by off-diagonal coupling elements,

\[ V_{ij}, \qquad i\neq j. \]

These couplings produce transitions between electronic states and are essential for describing nonadiabatic dynamics.

The corresponding Hamiltonian is

\[ \boxed{ \hat{H}_{\mathrm{coupling}} = \sum_{i<j} V_{ij} \left( |i\rangle\langle j| + |j\rangle\langle i| \right). } \]

The coupling matrix is symmetric,

\[ V_{ij}=V_{ji}, \]

ensuring that the total Hamiltonian remains Hermitian.

Complete Electronic Hamiltonian¶

Combining the diagonal energies and interstate couplings gives the complete electronic Hamiltonian,

\[ \boxed{ \hat{H}_{\mathrm{el}} = \sum_i E_i |i\rangle\langle i| + \sum_{i<j} V_{ij} \left( |i\rangle\langle j| + |j\rangle\langle i| \right). } \]

This Hamiltonian describes the electronic structure at the reference geometry before any vibrational coupling is introduced.

Matrix Representation¶

For three electronic states, the Hamiltonian can be written as

\[ \hat{H}_{\mathrm{el}} = \begin{pmatrix} E_1 & V_{12} & V_{13} \\ V_{12} & E_2 & V_{23} \\ V_{13} & V_{23} & E_3 \end{pmatrix}. \]

The diagonal elements correspond to the diabatic state energies, while the off-diagonal elements describe the electronic interactions between different states.

Representation in the Operator File¶

The electronic Hamiltonian appears in the Parameter Section of the operator file as

e11 = ...
e22 = ...
e33 = ...

e12 = ...
e13 = ...
e23 = ...

where

e11, e22, e33, ... are the diabatic electronic energies,
e12, e13, e23, ... are the interstate electronic couplings.

These parameters are extracted automatically from the reference diabatic GAMESS calculation by the operator-make.sh script.

Later, within the Hamiltonian section, these parameters are assigned to the corresponding electronic operators,

e11    | el S1&1
e22    | el S2&2
e33    | el S3&3

e12    | el S1&2
e13    | el S1&3
e23    | el S2&3

where S1&2, S2&3, etc., denote operators acting between the corresponding electronic states.

Physical Interpretation¶

The diagonal energies determine the relative positions of the diabatic potential energy surfaces, while the off-diagonal coupling elements govern the interaction between these surfaces.

If all interstate couplings are zero, each electronic state evolves independently. Non-zero coupling terms allow population transfer between electronic states and are therefore responsible for nonadiabatic transitions.

In the operator file, these electronic energies are independent of the nuclear coordinates. The dependence of the Hamiltonian on the vibrational motion is introduced through the linear and quadratic vibronic coupling terms discussed in the following sections.

Summary¶

The electronic Hamiltonian describes the energies of the diabatic electronic states together with the couplings between them. These quantities form the electronic component of the vibronic Hamiltonian and are represented in the operator file through the parameters e11, e22, e12, and related terms. In the next section, these electronic states are coupled to the vibrational motion through the linear vibronic coupling Hamiltonian.