The Complete CASSCF Algorithm

Throughout this tutorial, we have explored the individual components of the CASSCF method:

• Hartree–Fock molecular orbitals
• Active space selection
• Configuration State Functions (CSFs)
• Configuration Interaction (CI)
• Hamiltonian construction
• Orbital optimization
• State averaging
• Density matrices
• Natural orbitals

In this final chapter, we combine these ideas into a single workflow that illustrates how a CASSCF calculation is performed inside GAMESS.

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The Big Picture

A CASSCF calculation consists of two nested optimization procedures:

1. Configuration Interaction (CI) Optimization
2. Orbital Optimization

These two steps are repeated until both the molecular orbitals and the CI coefficients are self-consistent.

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Step 1 — Hartree–Fock Calculation

Every CASSCF calculation begins with a set of molecular orbitals.

These orbitals are usually obtained from a previous RHF calculation.

Geometry

↓

RHF Calculation

↓

Canonical Molecular Orbitals

These orbitals provide the starting point for the multiconfigurational calculation.

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Step 2 — Choose an Active Space

The user selects the chemically important orbitals that will participate in electron correlation.

Core Orbitals

↓

Always Occupied

──────────────

Active Orbitals

↓

Variable Occupation

──────────────

Virtual Orbitals

↓

Initially Empty

A carefully chosen active space is the foundation of a successful CASSCF calculation.

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Step 3 — Read Molecular Orbitals

Using

$GUESS
 GUESS=MOREAD
$END

GAMESS reads the molecular orbitals generated during the RHF calculation.

These orbitals become the initial guess for the CASSCF optimization.

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Step 4 — Generate Configuration State Functions

Using the selected active space,

GAMESS generates every allowed Configuration State Function (CSF).

Active Electrons

+

Active Orbitals

↓

Generate All Allowed CSFs

The GUGA algorithm ensures that every CSF has the correct spin and symmetry.

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Step 5 — Construct the Hamiltonian Matrix

For every pair of CSFs,

GAMESS evaluates

\[ H_{IJ} = \langle\Phi_I|\hat H|\Phi_J\rangle \]

producing the CI Hamiltonian matrix.

CSFs

↓

Hamiltonian Matrix

This matrix contains all interactions between the electronic configurations.

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Step 6 — Solve the CI Problem

The Hamiltonian matrix is diagonalized,

\[ \mathbf H\mathbf c = E\mathbf c \]

to obtain

• electronic energies,
• CI coefficients,
• multiconfigurational wavefunctions.

At this point,

the orbitals are still unchanged.

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Step 7 — Construct Density Matrices

Using the CI coefficients,

GAMESS builds

• the one-particle density matrix,
• the two-particle density matrix.

These matrices summarize the electronic distribution of the optimized wavefunction.

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Step 8 — Compute Orbital Gradients

The density matrices are used to evaluate how the total energy changes when the molecular orbitals are rotated.

If the gradient is non-zero,

the orbitals are not yet optimal.

Density Matrix

↓

Orbital Gradient

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Step 9 — Rotate Molecular Orbitals

The molecular orbitals are updated using a unitary rotation,

\[ \mathbf C' = \mathbf C e^{\kappa} \]

where

\(\kappa\)

contains the orbital rotation parameters.

These rotations lower the total electronic energy while preserving orbital orthogonality.

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Step 10 — Repeat the CI Calculation

Once the orbitals change,

the Hamiltonian also changes.

Therefore,

GAMESS returns to the CI calculation.

Rotate Orbitals

↓

New Hamiltonian

↓

New CI Solution

The CI coefficients are recalculated using the updated orbitals.

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Step 11 — Check Convergence

After every iteration,

GAMESS examines

• energy change,
• orbital gradient,
• wavefunction stability.

If the convergence criteria are satisfied,

Energy Change

↓

Very Small

AND

Orbital Gradient

↓

Very Small

the calculation terminates.

Otherwise,

the optimization cycle begins again.

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Complete CASSCF Flowchart

Geometry
    │
    ▼
Hartree–Fock Calculation
    │
    ▼
Canonical Molecular Orbitals
    │
    ▼
Choose Active Space
    │
    ▼
Generate Configuration State Functions
    │
    ▼
Construct Hamiltonian Matrix
    │
    ▼
Diagonalize Hamiltonian
    │
    ▼
Obtain CI Coefficients
    │
    ▼
Construct Density Matrices
    │
    ▼
Compute Orbital Gradient
    │
    ▼
Rotate Molecular Orbitals
    │
    ▼
Rebuild Hamiltonian
    │
    ▼
Repeat Until Converged
    │
    ▼
Natural Orbitals
    │
    ▼
Final CASSCF Wavefunction

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What Does GAMESS Print?

During a successful CASSCF calculation, the output typically contains the following major sections.

Output Section	Purpose
RHF Orbitals	Initial molecular orbitals
DRT Information	Active space definition
GUGA Information	Generated CSFs
CI Iterations	Optimization of CI coefficients
State Energies	Energies of all requested states
Orbital Optimization	SCF optimization of orbitals
Density Matrices	Electron distribution
Natural Orbitals	Orbital occupations
Final Energies	Converged electronic energies

Understanding these sections allows the user to quickly locate important information within the GAMESS output.

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Checklist Before Proceeding to XMCQDPT

Before using a converged CASSCF wavefunction as the reference for XMCQDPT, verify the following:

• ✅ The CASSCF calculation converged successfully.
• ✅ The active space includes all chemically important orbitals.
• ✅ State averaging includes all relevant electronic states.
• ✅ Natural occupation numbers are physically reasonable.
• ✅ No important orbitals have been omitted from the active space.
• ✅ The ordering of the electronic states is correct.
• ✅ The reference wavefunction is stable.

Only after these checks should the wavefunction be used for multireference perturbation theory.

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From CASSCF to XMCQDPT

Although CASSCF provides an excellent description of static electron correlation, it does not recover most of the dynamic electron correlation arising from the instantaneous motion of electrons.

To account for this missing correlation, the converged CASSCF wavefunction is used as the reference for Extended Multiconfiguration Quasi-Degenerate Perturbation Theory (XMCQDPT).

The workflow therefore becomes

RHF

↓

CIS (optional)

↓

Choose Active Space

↓

CASSCF

↓

Validate Active Space

↓

XMCQDPT

↓

Diabatization

↓

Spin–Orbit Coupling

↓

Spectroscopy / Dynamics

Each stage builds upon the previous one, making CASSCF a central component of the multireference workflow.

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Summary

The Complete Active Space Self-Consistent Field method combines two complementary ideas:

• Configuration Interaction, which determines the best mixture of electronic configurations for a fixed set of orbitals.
• Self-Consistent Orbital Optimization, which adjusts the molecular orbitals to better describe the correlated electronic wavefunction.

By iteratively optimizing both components, CASSCF provides a balanced and physically meaningful description of systems where a single electronic configuration is insufficient.

This makes it one of the most powerful and widely used methods for studying excited states, bond breaking, photochemical reactions, conical intersections, and strongly correlated molecules.

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Key Takeaways

• CASSCF begins with molecular orbitals obtained from an RHF calculation.
• An active space is selected to describe the important valence electrons.
• All Configuration State Functions are generated using the chosen active space.
• The CI Hamiltonian is constructed and diagonalized to obtain multiconfigurational wavefunctions.
• Density matrices are computed and used to optimize the molecular orbitals.
• CI optimization and orbital optimization are repeated until self-consistency is achieved.
• The final CASSCF wavefunction serves as the reference for XMCQDPT and other multireference methods.

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Where to go next

You now have a converged multiconfigurational wavefunction that accurately describes static electron correlation. The next stage is to recover the missing dynamic electron correlation using XMCQDPT, where the CASSCF wavefunction serves as the reference state for perturbation theory.