Optimized Orbitals and Natural Orbitals¶
After the Configuration Interaction (CI) coefficients have converged, GAMESS prints the final optimized molecular orbitals and several useful quantities derived from the multiconfigurational wavefunction.
Unlike RHF, these orbitals are not the orbitals obtained directly from the Hartree–Fock calculation. Instead, they have been optimized together with the CI coefficients to produce a balanced description of all electronic states included in the state-averaged CASSCF calculation.
This final section explains how to interpret these orbitals and evaluate the quality of the chosen active space.
1. Optimized Molecular Orbitals¶
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What are these orbitals?¶
At the beginning of the calculation, the molecular orbitals were imported from the previous RHF calculation using
During the CASSCF optimization, these orbitals were repeatedly rotated and optimized until they became self-consistent with the multiconfigurational wavefunction.
The orbitals printed here are therefore the final optimized molecular orbitals.
Why are they different from RHF orbitals?¶
In RHF,
The orbitals are optimized assuming that only one electronic configuration exists.
In CASSCF,
Initial RHF Orbitals
↓
CI Calculation
↓
Orbital Rotation
↓
New CI Calculation
↓
Repeat
↓
Optimized CASSCF Orbitals
The orbitals are optimized for many electronic configurations simultaneously, making them much more suitable for excited-state calculations.
2. Reading the Orbital Coefficients¶
Each molecular orbital is written as a linear combination of atomic basis functions.
A simplified example is
These coefficients describe
- where the orbital is located,
- which atoms contribute,
- whether the orbital is bonding or antibonding.
For large molecules, these coefficients are usually visualized using molecular orbital plotting software such as Avogadro, IQmol, Molden, or VMD.
3. Natural Orbitals¶
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This is one of the most useful sections of the CASSCF output.
Natural orbitals are obtained by diagonalizing the one-particle density matrix.
Unlike canonical molecular orbitals, they possess occupation numbers that provide direct information about electron correlation.
4. Natural Orbital Occupation Numbers¶
A typical section appears as
The occupation number indicates the average number of electrons occupying each orbital.
Interpretation¶
Occupation ≈ 2¶
These orbitals are essentially doubly occupied.
They belong to the inactive core.
Occupation ≈ 0¶
These orbitals remain essentially empty.
They correspond to virtual orbitals.
Fractional Occupations¶
These orbitals are only partially occupied.
Fractional occupations are the hallmark of a multiconfigurational wavefunction.
They indicate that electrons are distributed over several electronic configurations.
5. Why Are Fractional Occupations Important?¶
In RHF,
every orbital has either
There is no possibility of partial occupation.
In CASSCF,
electrons are allowed to redistribute among the active orbitals.
Consequently,
the average occupation becomes fractional.
This behaviour reflects static electron correlation, which RHF cannot describe.
6. Evaluating the Active Space¶
One of the main purposes of inspecting the natural occupations is to determine whether the chosen active space is appropriate.
A useful guideline is
| Occupation Number | Interpretation |
|---|---|
| ≈ 2.00 | Inactive orbital |
| 1.98 – 1.80 | Mostly occupied |
| 1.80 – 0.20 | Strongly active |
| 0.20 – 0.02 | Mostly virtual |
| ≈ 0.00 | External virtual orbital |
The active orbitals should generally possess fractional occupations, indicating that they are actively participating in the electronic structure.
7. Should the Active Space Be Modified?¶
Natural occupations are often used to refine the active space.
Case 1 — Good Active Space¶
All active orbitals have significant fractional occupation.
This usually indicates a well-balanced active space.
Case 2 — Active Space Too Large¶
The active orbitals behave like ordinary core or virtual orbitals.
These orbitals contribute very little to the multiconfigurational wavefunction and could potentially be removed.
Case 3 — Active Space Too Small¶
Suppose neighbouring orbitals outside the active space also show fractional occupation.
This suggests that important electron correlation has been omitted.
Additional orbitals should be added to the active space.
8. Final Checklist¶
Before accepting a CASSCF calculation, always verify
✅ The calculation converged successfully.
✅ The correct number of electronic states was optimized.
✅ The active space contains chemically important orbitals.
✅ The dominant CI coefficients are chemically reasonable.
✅ The natural occupation numbers indicate genuine multiconfigurational character.
Only after these checks should the wavefunction be used for subsequent calculations such as
- XMCQDPT,
- CASPT2,
- spin–orbit coupling,
- diabatization,
- or nonadiabatic dynamics.
Summary of the Complete CASSCF Workflow¶
The complete workflow presented throughout this tutorial is
Geometry Optimization
│
▼
RHF Calculation
│
▼
CIS Calculation
│
▼
Choose Active Space
│
▼
State-Averaged CASSCF
│
▼
Analyse CI Expansion
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Inspect Natural Orbitals
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Validate Active Space
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Proceed to XMCQDPT
The CASSCF wavefunction now serves as the reference for higher-level multireference methods.
Key Takeaways¶
- CASSCF optimizes both the molecular orbitals and the CI coefficients.
- The optimized molecular orbitals differ from the initial RHF orbitals.
- Natural orbitals provide a compact representation of the multiconfigurational wavefunction.
- Fractional occupation numbers indicate static electron correlation.
- Natural occupations are the primary tool for evaluating the quality of the active space.
- A well-chosen active space is essential for obtaining reliable excited-state energies and wavefunctions.