Understanding the XMCQDPT Output¶
Once the CASSCF reference wavefunction has converged, GAMESS proceeds with the Extended Multi-Configuration Quasi-Degenerate Perturbation Theory (XMCQDPT) calculation. The output is naturally divided into two major parts:
- CAS-CI Results – verification of the multiconfigurational reference wavefunctions.
- MCQDPT Results – inclusion of dynamic electron correlation through second-order perturbation theory.
Unlike RHF or CASSCF, XMCQDPT does not optimize molecular orbitals. Instead, it starts from the converged CASSCF wavefunction and computes perturbative corrections to obtain more accurate electronic state energies.
1. CAS-CI Reference States¶
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This section is identical to the final CAS-CI calculation performed in the CASSCF run. It is repeated because these wavefunctions serve as the reference states for the perturbation calculation.
Example:
STATE # 1 ENERGY = -2043.651377420
STATE # 2 ENERGY = -2043.481805916
STATE # 3 ENERGY = -2043.462406789
These are exactly the same state energies obtained at the end of the CASSCF calculation.
For this calculation,
| State | CASSCF Energy (Hartree) |
|---|---|
| S₀ | -2043.651377420 |
| S₁ | -2043.481805916 |
| S₂ | -2043.462406789 |
At this point, only static electron correlation has been included.
2. Configuration State Function (CSF) Analysis¶
Immediately below each state energy GAMESS prints the dominant CSFs.
Example
The coefficient indicates how much each electronic configuration contributes to the total wavefunction.
For the ground state,
- CSF 1 contributes 95.64%
- remaining configurations contribute only small corrections
This indicates that the ground state is largely single-reference with moderate multiconfigurational character.
The excited states contain larger contributions from multiple CSFs, confirming that they require a multireference description.
3. One-Particle Density Matrix¶
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Example
The one-particle density matrix contains the electronic occupation information for the optimized CASSCF wavefunction.
Diagonal elements represent orbital occupations, while off-diagonal elements measure orbital mixing.
This matrix is used internally during the perturbation calculation and is generally not interpreted directly unless performing advanced wavefunction analysis.
4. Canonical Orbital Energies¶
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The orbitals are divided into three groups:
For this calculation the active orbitals are
These correspond to the four orbitals chosen for the CAS(4,4) active space.
This section confirms that the orbitals used in the perturbation calculation are exactly those optimized during CASSCF.
5. Generator Reduction¶
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Example
GAMESS removes redundant excitation operators before starting the perturbation calculation.
Since the original and reduced spaces are identical, no generators were discarded.
6. Zeroth-Order Effective Hamiltonian¶
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This matrix defines the zeroth-order Hamiltonian used in perturbation theory.
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The off-diagonal elements represent coupling between different electronic states before perturbative corrections are included.
Unlike ordinary MP2, XMCQDPT treats multiple electronic states simultaneously.
7. Rotation to the Intermediate Basis¶
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Example
This transformation rotates the original CAS-CI states into an intermediate basis.
The purpose is to minimize interactions between nearly degenerate states and improve the numerical stability of perturbation theory.
This step is one of the defining features of XMCQDPT.
8. Intermediate CI State Basis¶
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The rotated states are printed once again with their dominant CSFs.
Although the numerical coefficients differ slightly from the original CAS-CI states, the physical character of each electronic state remains the same.
These intermediate states are used as the actual reference for the perturbation calculation.
9. First-Order Effective Hamiltonian¶
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This Hamiltonian contains the zeroth-order Hamiltonian plus first-order corrections.
It forms the starting point for evaluating the second-order perturbative contributions.
10. Perturbation Treatment¶
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For every electronic state GAMESS evaluates
Example
The same procedure is repeated for State 2 and State 3.
During this stage, dynamic electron correlation arising from excitations outside the active space is included through second-order perturbation theory.
11. MCQDPT2 Results¶
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This is the most important section of the entire output.
Effective Hamiltonian¶
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Example
This matrix contains the complete second-order perturbative corrections for all electronic states.
Diagonalizing this Hamiltonian produces the final XMCQDPT energies.
Zeroth-Order Energies¶
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These are the energies of the intermediate reference states before perturbation corrections are applied.
Diagonal Elements of the Effective Hamiltonian¶
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These values represent the state-specific second-order perturbation energies before diagonalization.
Final XMCQDPT Energies¶
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This is the section most users are interested in.
For this calculation,
| State | CASSCF (Hartree) | XMCQDPT (Hartree) |
|---|---|---|
| S₀ | -2043.6513774204 | -2047.3009641749 |
| S₁ | -2043.4818059159 | -2047.1996167857 |
| S₂ | -2043.4624067890 | -2047.1534558976 |
Notice that all three states become significantly lower in energy after dynamic correlation is included.
These are the energies that should be used for subsequent analysis such as excitation energies and potential energy surfaces.
12. Perturbed MCQDPT States¶
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The final corrected wavefunctions are printed with their updated CSF coefficients.
Example
The coefficients are very similar to the original CAS-CI wavefunctions, indicating that perturbation theory has mainly corrected the energies while preserving the electronic character of each state.
Large changes in these coefficients would indicate strong mixing between electronic states.
13. End of Calculation¶
The calculation finishes with
followed by
Always verify that this message is present before using any of the reported energies.
Summary¶
The XMCQDPT output can be interpreted in the following order:
- Verify the CAS-CI reference energies.
- Examine the dominant CSF coefficients.
- Confirm the active orbital energies.
- Locate the HEFF(0) matrix.
- Inspect the intermediate basis rotation.
- Verify that perturbation corrections completed successfully.
- Read the MCQDPT2 ENERGIES table.
- Analyze the final perturbed wavefunctions.
- Ensure the calculation ended with EXECUTION OF GAMESS TERMINATED NORMALLY.
For most practical applications, the MCQDPT2 ENERGIES section provides the final electronic energies that should be used in subsequent analyses, while the preceding sections explain how those corrected energies were obtained.