MCSCF Optimization and Convergence¶

Once the active space has been constructed and the Configuration State Functions (CSFs) have been generated, GAMESS begins the Multiconfigurational Self-Consistent Field (MCSCF) optimization.

Unlike an RHF calculation, where only the molecular orbitals are optimized, CASSCF optimizes two quantities simultaneously:

the molecular orbitals,
the CI coefficients describing the electronic wavefunction.

Both must converge before the calculation is complete.

1. Beginning the Optimization¶

Search for

ITER     TOTAL ENERGY

or simply search for

ITER

You will find a table similar to

ITER     TOTAL ENERGY            DELTA E        GRADIENT
--------------------------------------------------------
1
2
3
...
...
13

This table records every optimization cycle performed during the calculation.

What Happens During Each Iteration?¶

Every iteration consists of four major steps.

Current Molecular Orbitals
            │
            ▼
Solve CI Problem
            │
            ▼
Update CI Coefficients
            │
            ▼
Optimize Molecular Orbitals
            │
            ▼
Compute New Energy
            │
            ▼
Check Convergence

If the convergence criteria are not satisfied,

the entire process repeats.

2. Simultaneous Optimization¶

One of the defining characteristics of CASSCF is that both the orbitals and the electronic wavefunction are optimized together.

This is fundamentally different from RHF.

RHF	CASSCF
Optimize molecular orbitals only	Optimize molecular orbitals and CI coefficients
Single determinant	Multiple configurations
One SCF loop	Nested orbital + CI optimization

Because two different quantities must converge simultaneously, CASSCF calculations generally require more iterations than RHF calculations.

3. Energy Convergence¶

The second column of the iteration table lists the electronic energy.

During the optimization,

Iteration 1
↓

Iteration 2

↓

Iteration 3

↓

...

↓

Iteration 13

the total energy gradually decreases.

This behaviour is expected.

Every optimization step attempts to find a lower-energy wavefunction by improving both

the orbital shapes,
and the CI expansion.

As the calculation approaches convergence,

the energy changes become progressively smaller.

4. Gradient Convergence¶

Another important quantity printed in the iteration table is the orbital gradient.

The gradient measures how far the current molecular orbitals are from the optimal solution.

Large gradients indicate that significant orbital rotations are still required.

As the optimization proceeds,

Large Gradient

↓

Medium Gradient

↓

Small Gradient

↓

≈ 0

the orbitals become self-consistent with the CI wavefunction.

Only when the gradient falls below the convergence threshold specified by

ACURCY = 1.0 × 10⁻⁵

is the calculation considered converged.

5. Orbital Rotations¶

Unlike RHF,

CASSCF repeatedly rotates the molecular orbitals.

A simplified view is

Old Orbitals

↓

Small Orbital Rotation

↓

New Orbitals

↓

Solve CI Again

↓

Repeat

Each rotation changes the Hamiltonian,

which changes the CI coefficients,

which then changes the optimal orbitals once again.

This mutual dependence is the reason CASSCF is often described as a self-consistent multiconfigurational method.

6. Why Does CASSCF Require More Time?¶

Compared with RHF,

each CASSCF iteration is computationally much more expensive.

During every iteration, GAMESS must

construct the CI Hamiltonian,
diagonalize the CI matrix,
optimize the molecular orbitals,
compute new one- and two-particle density matrices,
evaluate orbital gradients,
update the wavefunction.

Consequently,

even a small active space requires substantially more computational effort than a Hartree–Fock calculation.

7. Convergence¶

Towards the end of the optimization,

GAMESS prints a message similar to

MCSCF HAS CONVERGED

or

LAGRANGIAN CONVERGED

(depending on the version of GAMESS).

This indicates that

the molecular orbitals,
the CI coefficients,
and the electronic energy

have all become self-consistent.

At this stage,

the multiconfigurational wavefunction is fully optimized.

8. Final State-Averaged Energy¶

Immediately after convergence,

search for

FINAL MCSCF ENERGY

Your calculation reports

FINAL MCSCF ENERGY IS

-2043.5318635286 Hartree

This is the final converged CASSCF energy.

Unlike RHF,

this energy corresponds to the optimized multiconfigurational wavefunction, rather than a single Slater determinant.

Why Is This Called a State-Averaged Energy?¶

Recall that the input contained

$GUGDIA
NSTATE=3
$END

together with

$GUGDM2
WSTATE(1)=1,1,1
$END

This means that the molecular orbitals were optimized simultaneously for three electronic states.

The reported energy therefore corresponds to the orbital optimization performed using the weighted average of these three states.

The individual energies of

S₀,
S₁,
S₂,

are printed later in the output.

9. Why State Averaging?¶

Suppose the orbitals were optimized only for the ground state.

Ground-State Orbitals

↓

Excellent for S₀

Poor for S₁

Poor for S₂

Instead,

state averaging produces

Common Orbitals

↓

Good for S₀

Good for S₁

Good for S₂

This balanced description is essential for

excited-state dynamics,
conical intersections,
photoisomerization,
nonadiabatic transitions.

For these applications,

state-averaged CASSCF is almost always preferred.

Key Takeaways¶

CASSCF optimizes both molecular orbitals and CI coefficients.
Each iteration alternates between solving the CI problem and updating the orbitals.
The total energy decreases as the wavefunction improves.
The orbital gradient approaches zero as convergence is reached.
Your calculation converged successfully after 13 iterations.
The final converged state-averaged energy is

−2043.5318635286 Hartree

The next section examines the optimized electronic states and the Configuration State Functions (CSFs) that make up the multiconfigurational wavefunction.