XMCQDPT Algorithm¶

This chapter summarizes the complete XMCQDPT workflow discussed throughout the previous mathematical sections.

Unlike CASSCF, which determines the multiconfigurational reference wavefunction, XMCQDPT starts from a converged CASSCF solution and adds dynamic electron correlation using multistate second-order perturbation theory.

Complete XMCQDPT Workflow¶

The overall procedure can be represented as

Converged CASSCF Wavefunctions
            │
            ▼
 Canonicalize Active Orbitals
            │
            ▼
 Construct CAS-CI Reference States
            │
            ▼
 Build Zeroth-Order Hamiltonian H(0)
            │
            ▼
 Diagonalize H(0)
            │
            ▼
 Generate Intermediate States
            │
            ▼
 Evaluate Second-Order Perturbation Corrections
            │
            ▼
 Construct Effective Hamiltonian H(2)
            │
            ▼
 Diagonalize H(2)
            │
            ▼
 Final XMCQDPT Energies & Wavefunctions

Each step contributes to improving the description of the electronic states while maintaining computational efficiency.

Step 1, Converged CASSCF Reference¶

XMCQDPT begins with a fully converged CASSCF calculation.

The CASSCF wavefunctions provide

optimized molecular orbitals,
multiconfigurational reference states,
state-averaged orbitals (if multiple states are included).

These reference states contain the essential static electron correlation, which is a prerequisite for perturbation theory.

Step 2, Canonicalization of Active Orbitals¶

The optimized active-space orbitals are transformed into canonical orbitals.

Canonical orbitals diagonalize the generalized Fock matrix within each orbital subspace and provide a consistent reference for perturbation theory.

This transformation does not change the physical wavefunction but simplifies the subsequent calculations.

Step 3, CAS-CI Reference States¶

Using the canonical orbitals, GAMESS constructs the complete CAS-CI wavefunctions.

Each electronic state is expressed as

\[ \Psi_i=\sum_I c_I\Phi_I, \]

where

\(\Phi_I\) are Configuration State Functions (CSFs),
\(c_I\) are the CI coefficients.

These states form the reference space for XMCQDPT.

Step 4, Zeroth-Order Hamiltonian¶

The reference states are assembled into the zeroth-order Hamiltonian

\[ \mathbf H^{(0)}. \]

This Hamiltonian contains

diagonal elements corresponding to the CAS-CI energies,
off-diagonal elements describing interactions between electronic states.

Step 5, Intermediate-State Transformation¶

The zeroth-order Hamiltonian is diagonalized,

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

to obtain the intermediate states.

These states provide a stable orthogonal basis for evaluating perturbation theory and eliminate ambiguities associated with interacting CASSCF states.

Step 6, Second-Order Perturbation Theory¶

Dynamic electron correlation is incorporated through second-order perturbation theory.

For each intermediate state,

\[ E_i^{(2)} = \sum_r \frac {\left|\langle r|\hat V|i\rangle\right|^2} {E_i^{(0)}-E_r^{(0)}}. \]

External configurations are never included explicitly in the wavefunction.

Instead, their influence is added through perturbative energy corrections.

Step 7, Effective Hamiltonian¶

The perturbative corrections are assembled into the effective Hamiltonian

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

which contains both

static correlation from CASSCF,
dynamic correlation from perturbation theory.

Step 8, Final Diagonalization¶

The corrected Hamiltonian is diagonalized

\[ \mathbf H^{(2)} \mathbf C = \mathbf C \mathbf E, \]

yielding

final XMCQDPT energies,
final multiconfigurational wavefunctions.

These are the energies reported in the

MCQDPT2 RESULTS

section of the GAMESS output.

Overall Energy Evolution¶

The entire electronic structure calculation proceeds as

RHF
 │
 ▼
Single Slater Determinant
 │
 ▼
CASSCF
 │
 ▼
Static Correlation
 │
 ▼
CAS-CI
 │
 ▼
Reference States
 │
 ▼
Intermediate States
 │
 ▼
Second-Order Perturbation
 │
 ▼
Effective Hamiltonian
 │
 ▼
Final XMCQDPT Energies

Each stage refines the description of the electronic structure.

Computational Flow Inside GAMESS¶

Internally, GAMESS performs the calculation in approximately the following order:

Read CASSCF Orbitals
        │
        ▼
Generate CAS-CI States
        │
        ▼
Compute Density Matrices
        │
        ▼
Canonicalize Active Orbitals
        │
        ▼
Build Zeroth-Order Hamiltonian
        │
        ▼
Construct Intermediate States
        │
        ▼
Evaluate Perturbative Matrix Elements
        │
        ▼
Assemble Effective Hamiltonian
        │
        ▼
Diagonalize Effective Hamiltonian
        │
        ▼
Print Final MCQDPT2 Energies

This sequence closely matches the order of sections printed in the GAMESS output file, making it easier to follow the progress of a calculation.

Summary¶

XMCQDPT extends the CASSCF method by recovering dynamic electron correlation while preserving the multiconfigurational character of the reference wavefunction.

The method proceeds by

reading converged CASSCF orbitals,
constructing CAS-CI reference states,
building the zeroth-order Hamiltonian,
generating intermediate states,
evaluating second-order perturbation corrections,
constructing the effective Hamiltonian,
diagonalizing the corrected Hamiltonian,
obtaining the final correlated electronic energies and wavefunctions.

Because it combines the strengths of multireference wavefunctions with perturbation theory, XMCQDPT provides highly accurate energies for excited states, near-degenerate systems, conical intersections, and photochemical processes, making it one of the most powerful post-CASSCF methods available in GAMESS.

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