Intermediate-State Formalism¶

One of the defining features of XMCQDPT is that perturbation theory is not performed directly on the original CAS-CI states.

Instead, the reference states are first transformed into a new set of states called the intermediate states. These states provide a more stable reference for the perturbation calculation, especially when several electronic states are close in energy.

Why Are Intermediate States Needed?¶

Suppose the CASSCF calculation produces three electronic states,

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

If these states are well separated in energy, each can be treated independently.

However, in many molecules, especially those involved in photochemistry,

excited states lie very close together,
states often cross or become nearly degenerate,
electronic configurations mix strongly.

Applying perturbation theory directly to such states may lead to

unstable energy corrections,
incorrect state ordering,
excessive mixing between electronic states.

The intermediate-state formalism was developed to avoid these problems.

Constructing the Intermediate States¶

The first step is to diagonalize the zeroth-order effective Hamiltonian,

\[ \boxed{ \mathbf H^{(0)} \mathbf C = \mathbf C \mathbf E } \]

where

\(\mathbf H^{(0)}\) is the zeroth-order effective Hamiltonian,
\(\mathbf C\) is the rotation matrix,
\(\mathbf E\) contains the corresponding eigenvalues.

The eigenvectors define a new orthonormal basis,

\[ \boxed{ \Phi_i = \sum_j C_{ji}\Psi_j } \]

where

\(\Psi_j\) are the original CAS-CI states,
\(C_{ji}\) are the rotation coefficients,
\(\Phi_i\) are the intermediate states.

Thus, each intermediate state is a linear combination of the original CASSCF wavefunctions.

Physical Interpretation¶

The transformation can be viewed schematically as

CAS-CI States

Ψ₁
Ψ₂
Ψ₃

        │
        │  Diagonalize HEFF(0)
        ▼

Intermediate States

Φ₁
Φ₂
Φ₃

The intermediate states already include the most important interactions between nearby electronic states.

As a result,

perturbation corrections become more stable,
state crossings are treated more smoothly,
numerical instabilities are greatly reduced.

Rotation Matrix¶

The transformation between the CAS-CI and intermediate-state basis is given by

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

Each column represents one intermediate state expressed in terms of the original CAS-CI states.

For example,

\[ \Phi_1 = c_{11}\Psi_1 + c_{21}\Psi_2 + c_{31}\Psi_3. \]

If one coefficient dominates, the intermediate state closely resembles the corresponding CAS-CI state.

If several coefficients are significant, the state contains substantial multistate mixing.

Connection with the GAMESS Output¶

Search for

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

For the example calculation,

             1           2           3

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

This matrix defines the intermediate-state basis.

Intermediate State 1¶

The first column is

0.990885
0.133526
0.017822

or

\[ \Phi_1 = 0.990885\,\Psi_1 + 0.133526\,\Psi_2 + 0.017822\,\Psi_3. \]

Since the coefficient for \(\Psi_1\) is close to unity, the first intermediate state is dominated by the first CAS-CI state, with only minor contributions from the other two states.

Intermediate State 2¶

The second column is

0.129719
−0.910103
−0.393555

Here,

State 2 contributes most strongly,
State 3 also contributes significantly.

This indicates noticeable mixing between the second and third electronic states.

Intermediate State 3¶

The third column is

0.036330
−0.392280
0.919128

Again,

State 3 dominates,
State 2 contributes appreciably.

This behavior is expected for excited states that are energetically close.

Intermediate-State CI Expansion¶

Immediately after the rotation matrix, GAMESS prints

*** INTERMEDIATE CI STATE BASIS ***

followed by

STATE # 1 ENERGY = ...
STATE # 2 ENERGY = ...
STATE # 3 ENERGY = ...

These are not yet the final MCQDPT energies.

Instead, they represent the electronic states after

state interaction has been incorporated,
but before second-order perturbation corrections are evaluated.

The subsequent perturbation calculation uses these intermediate states as the reference wavefunctions.

Advantages of the Intermediate-State Formalism¶

Compared with performing perturbation theory directly on CAS-CI states, the intermediate-state approach

treats several electronic states simultaneously,
reduces numerical instabilities,
avoids discontinuities near avoided crossings,
provides smoother potential energy surfaces,
produces more reliable excited-state energies.

These properties make XMCQDPT particularly well suited for studies involving

excited states,
conical intersections,
photochemical reactions,
electronic state crossings.

Key Points¶

XMCQDPT first rotates the CAS-CI states into an intermediate-state basis.
The rotation is obtained by diagonalizing the zeroth-order effective Hamiltonian.
Each intermediate state is a linear combination of the original CAS-CI states.
Perturbation theory is performed in this new basis rather than on the original CASSCF states.
This transformation greatly improves the stability and accuracy of multistate perturbation theory.