Effective Hamiltonian¶

In conventional perturbation theory, each electronic state is treated independently. This approach works well when the states are well separated in energy.

Many photochemical systems, however, contain several electronic states that lie close together and interact strongly. In such cases, applying perturbation theory to each state separately can lead to inaccurate energies and unstable results.

XMCQDPT overcomes this limitation by constructing an effective Hamiltonian that describes all reference states simultaneously.

Why Is an Effective Hamiltonian Needed?¶

Consider three CASSCF reference states,

\[ \Psi_1,\; \Psi_2,\; \Psi_3, \]

with similar energies.

Because these states are close in energy, electron correlation can mix them together.

Instead of correcting each state independently,

State 1  → Perturbation

State 2  → Perturbation

State 3  → Perturbation

XMCQDPT allows all states to interact simultaneously,

            ┌──────────────┐
State 1 ───►│              │
            │              │
State 2 ───►│ H_eff        │──► Corrected States
            │              │
State 3 ───►│              │
            └──────────────┘

The perturbation therefore acts on the entire state manifold, rather than on individual states.

Effective Hamiltonian¶

The multistate interaction is represented by the effective Hamiltonian

\[ \boxed{ \mathbf H_{\mathrm{eff}} = \mathbf H^{(0)} + \Delta\mathbf H } \]

where

\(\mathbf H^{(0)}\) is the zeroth-order Hamiltonian obtained from CASSCF,
\(\Delta\mathbf H\) contains the perturbative corrections.

Unlike the ordinary Hamiltonian, the effective Hamiltonian is a matrix whose elements describe the interactions between different electronic states.

Matrix Representation¶

For three electronic states,

\[ \mathbf H_{\mathrm{eff}} = \begin{pmatrix} H_{11} & H_{12} & H_{13} \\ H_{21} & H_{22} & H_{23} \\ H_{31} & H_{32} & H_{33} \end{pmatrix}. \]

Each element has a physical meaning.

Diagonal Elements¶

The diagonal elements

\[ H_{11},\; H_{22},\; H_{33} \]

represent the energies of the individual electronic states after perturbative corrections.

These values become the corrected state energies.

Off-Diagonal Elements¶

The off-diagonal elements

\[ H_{12},\; H_{13},\; H_{23}, \]

describe the interaction between different electronic states.

Large off-diagonal elements indicate strong mixing between states.

If the states are nearly independent, these elements are very small.

Zeroth-Order Effective Hamiltonian¶

Before second-order corrections are added, XMCQDPT first constructs the zeroth-order effective Hamiltonian.

In the GAMESS output this appears as

*** HEFF(0): THE 0TH ORDER XMCQDPT HAMILTONIAN ***

For the example calculation,

               1            2            3
1   -4.780009E+02
2   -2.450246E-02 -4.778120E+02
3    4.265494E-03 -7.737128E-02 -4.776616E+02

Several observations can be made.

The diagonal elements correspond to the zeroth-order energies.
The off-diagonal elements are small but non-zero.
The three electronic states are therefore weakly coupled even before perturbation corrections are applied.

Why Diagonalization Is Necessary¶

Because the electronic states interact,

the effective Hamiltonian is not diagonal.

To obtain physically meaningful electronic states, XMCQDPT diagonalizes the effective Hamiltonian,

\[ \boxed{ \mathbf H_{\mathrm{eff}} \mathbf C = \mathbf C \mathbf E } \]

where

\(\mathbf C\) contains the mixing coefficients,
\(\mathbf E\) contains the corrected energies.

The resulting eigenvectors define new electronic states that already include state interaction.

These states are referred to as the intermediate states.

Connection with the GAMESS Output¶

Immediately after printing the zeroth-order Hamiltonian,

GAMESS prints

*** ROTATION FROM CAS-CI TO INTERMEDIATE BASIS ***

This matrix contains the eigenvectors obtained by diagonalizing the effective Hamiltonian.

For the example,

            1          2          3

1     0.9908850  0.1297189  0.0363299
2     0.1335261 -0.9101029 -0.3922800
3     0.0178222 -0.3935554  0.9191281

Each column represents one intermediate state as a linear combination of the original CASSCF states.

For example,

Intermediate State 1 is composed primarily of CAS-CI State 1.
Intermediate State 2 contains significant contributions from States 2 and 3.
Intermediate State 3 also mixes States 2 and 3.

Thus, before any second-order energy correction is applied, XMCQDPT has already transformed the CASSCF states into a basis that better represents the interacting electronic manifold.

Key Points¶

Ordinary perturbation theory treats each state independently.
XMCQDPT treats all selected states simultaneously.
The effective Hamiltonian contains both state energies and state interactions.
Diagonal elements represent corrected state energies.
Off-diagonal elements describe interstate coupling.
Diagonalizing the effective Hamiltonian produces the intermediate-state basis used throughout the remainder of the XMCQDPT calculation.