Final XMCQDPT Energies¶

After evaluating all second-order perturbative corrections, XMCQDPT constructs the second-order effective Hamiltonian. The final step of the calculation is to diagonalize this Hamiltonian to obtain the corrected electronic energies and wavefunctions.

Unlike CASSCF, where the reported energies are purely variational, the final XMCQDPT energies include both static and dynamic electron correlation.

The Second-Order Effective Hamiltonian¶

Following the perturbative treatment, 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.

The diagonal elements represent the corrected energies of each electronic state, while the off-diagonal elements describe the residual coupling between different states.

Final Diagonalization¶

The corrected electronic states are obtained by solving

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

where

\(\mathbf{H}^{(2)}\) is the second-order effective Hamiltonian,
\(\mathbf{C}\) contains the eigenvectors,
\(\mathbf{E}\) contains the final XMCQDPT energies.

The eigenvectors define the final electronic wavefunctions, while the eigenvalues are the final correlated energies reported by GAMESS.

The Final Wavefunction¶

The XMCQDPT wavefunction can be expressed as

\[ \boxed{ \Psi_i^{\mathrm{XMCQDPT}} = \sum_j C_{ji}\Phi_j } \]

where

\(\Phi_j\) are the intermediate states,
\(C_{ji}\) are the coefficients obtained from diagonalizing the second-order effective Hamiltonian.

Thus, the final wavefunction is not simply the original CASSCF wavefunction but a correlated multiconfigurational state that incorporates dynamic electron correlation.

Energy Evolution During the Calculation¶

The complete energy refinement proceeds in three stages.

Hartree–Fock
      │
      ▼
Single-determinant wavefunction

      │
      ▼
CASSCF
      │
      ▼
Static correlation included

      │
      ▼
XMCQDPT
      │
      ▼
Dynamic correlation added

Each step improves the description of the electronic structure.

Connection with the GAMESS Output¶

Search for

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

Within this section, the most important results appear under

*** MCQDPT2 ENERGIES ***

For the present calculation,

STATE                       1ST ORDER                       2ND ORDER

1   E(MCSCF)= -2043.6513774204
    E(MP2)=   -2047.3009641749

2   E(MCSCF)= -2043.4818059159
    E(MP2)=   -2047.1996167857

3   E(MCSCF)= -2043.4624067890
    E(MP2)=   -2047.1534558976

These values summarize the entire XMCQDPT calculation.

The 1st ORDER column corresponds to the converged CASSCF energies, while the 2nd ORDER column contains the final XMCQDPT energies after dynamic correlation has been included.

Correlation Energy¶

The dynamic correlation recovered by XMCQDPT can be estimated as

\[ \Delta E_{\mathrm{corr}} = E_{\mathrm{XMCQDPT}} - E_{\mathrm{CASSCF}}. \]

For the ground state,

\[ \Delta E_{\mathrm{corr}} = -2047.300964175 - (-2043.651377420) = -3.649586754\ \mathrm{Hartree}. \]

Similarly,

State	CASSCF Energy	XMCQDPT Energy	Correlation Energy
S₀	-2043.651377420	-2047.300964175	-3.649587
S₁	-2043.481805916	-2047.199616786	-3.717811
S₂	-2043.462406789	-2047.153455898	-3.691049

These energy differences represent the stabilization arising from dynamic electron correlation.

Perturbed Electronic States¶

Immediately after the energy table, GAMESS prints

*** PERTURBED MCQDPT STATES ***

followed by the final Configuration State Function (CSF) expansions.

These wavefunctions describe the electronic states after

state interaction,
intermediate-state transformation,
second-order perturbation correction,

have all been incorporated.

Although the dominant electronic configurations usually remain similar to those obtained from CASSCF, the coefficients change slightly because of dynamic correlation.

Interpretation¶

The final XMCQDPT energies should be regarded as the best electronic energies obtained in this workflow.

For most applications, these energies are used to compute

excitation energies,
potential energy surfaces,
energy gaps,
conical intersections,
reaction profiles,
diabatic Hamiltonians.

The XMCQDPT wavefunctions also serve as the starting point for subsequent calculations such as diabatization, spin–orbit coupling, and spectroscopic property calculations.

Key Points¶

The second-order effective Hamiltonian is diagonalized to obtain the final electronic states.
The resulting eigenvalues are the XMCQDPT energies reported by GAMESS.
These energies include both static and dynamic electron correlation.
The difference between the CASSCF and XMCQDPT energies represents the dynamic correlation recovered through perturbation theory.
The final XMCQDPT wavefunctions provide the reference for higher-level analyses and post-processing calculations.