Second-Order Energy Corrections¶

After constructing the intermediate-state basis, XMCQDPT evaluates the contribution of dynamic electron correlation using second-order perturbation theory.

Unlike CASSCF, which explicitly includes only the configurations within the active space, XMCQDPT estimates the influence of the vast number of external configurations without adding them directly to the wavefunction.

This approach provides most of the missing correlation energy while keeping the computational cost manageable.

The Second-Order Energy¶

For each intermediate state, the second-order energy correction is given by

\[ \boxed{ E_i^{(2)} = \sum_r \frac { \left| \left< r \left| \hat V \right| i \right> \right|^2 } { E_i^{(0)} - E_r^{(0)} } } \]

where

\(i\) denotes one of the intermediate reference states,
\(r\) represents an external electronic configuration,
\(E_i^{(0)}\) is the zeroth-order energy,
\(E_r^{(0)}\) is the energy of an external configuration,
\(\hat V\) is the perturbation operator.

The numerator measures the strength of interaction between the reference state and an external configuration, while the denominator depends on their energy separation.

Configurations that interact strongly with the reference state contribute the most to the correlation energy.

What Are External Configurations?¶

During the CASSCF calculation,

only configurations within the chosen active space are included explicitly.

For example,

Inactive      Active      Virtual

↓↓↓↓↓↓    ↑↓ ↑  ↑    □ □ □ □ □

The perturbation considers additional excitations such as

Inactive  → Virtual

Active    → Virtual

Double excitations

Higher-order excitations

These external configurations are never added to the CI expansion.

Instead, their effects are evaluated mathematically through the second-order correction.

Why Is the Energy Lowered?¶

Electron correlation allows electrons to avoid one another more effectively.

As a result,

the correlated wavefunction is always more stable than the reference wavefunction.

Therefore,

\[ E^{(2)}<0 \]

for most electronic states.

The second-order correction lowers the total electronic energy and brings it closer to the exact solution.

Perturbation Evaluation in GAMESS¶

After constructing the intermediate states,

GAMESS begins evaluating the perturbation corrections.

Search for

EVALUATING PERTURBATION THEORY CORRECTIONS TO HAMILTONIAN ELEMENTS

The calculation proceeds one electronic state at a time.

For example,

EVALUATING MATRIX ELEMENTS BETWEEN STATE 1

followed by

TIME FOR 0-BODY FORMULAE
TIME FOR 1-BODY FORMULAE
TIME FOR 2-BODY FORMULAE

These messages indicate that GAMESS is computing all perturbative contributions required for the second-order effective Hamiltonian.

The same procedure is then repeated for

STATE 2

and

STATE 3.

Computational Flow¶

The perturbative calculation follows the sequence

Intermediate State

        │
        ▼

Generate External Configurations

        │
        ▼

Evaluate Matrix Elements

        │
        ▼

Compute Second-Order Corrections

        │
        ▼

Construct Effective Hamiltonian

Each intermediate state interacts with a very large number of external configurations.

The resulting corrections are accumulated to produce the second-order effective Hamiltonian.

Building the Second-Order Hamiltonian¶

After all perturbative contributions have been evaluated,

the effective Hamiltonian becomes

\[ \boxed{ \mathbf H^{(2)} = \mathbf H^{(0)} + \Delta\mathbf H^{(2)} } \]

where

\(\mathbf H^{(0)}\) is the zeroth-order Hamiltonian,
\(\Delta\mathbf H^{(2)}\) contains the second-order perturbative corrections.

This corrected Hamiltonian is subsequently diagonalized to obtain the final XMCQDPT energies.

Connection with the GAMESS Output¶

Once all perturbation corrections have been evaluated,

GAMESS prints

###########################
###   MCQDPT2 RESULTS   ###
###########################

Immediately below this section appears

*** EFFECTIVE HAMILTONIAN (AT SECOND ORDER) ***

For the present calculation,

                   1            2            3
1     -2.047301E+03
2      1.980548E-04 -2.047194E+03
3     -6.166512E-04 -1.468216E-02 -2.047159E+03

Compared with the zeroth-order Hamiltonian,

the diagonal elements become significantly lower because dynamic correlation has been added,
the off-diagonal elements describe the remaining interaction between electronic states after the perturbative correction.

This matrix is the central quantity of the XMCQDPT calculation.

Key Points¶

XMCQDPT estimates dynamic electron correlation using second-order perturbation theory.
External configurations are treated implicitly rather than being included in the CI expansion.
The second-order correction generally lowers the electronic energy.
Perturbative contributions are evaluated separately for each intermediate state.
The resulting corrections are assembled into the second-order effective Hamiltonian, which is finally diagonalized to obtain the corrected electronic states.