Electronic States and Configuration State Functions

After the MCSCF optimization has converged, GAMESS prints the properties of the optimized electronic states.

Unlike RHF, where the wavefunction consists of a single Slater determinant, a CASSCF wavefunction is expressed as a linear combination of many Configuration State Functions (CSFs).

This section explains how to interpret these electronic states and identify the dominant electronic configurations.

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1. State Energies

Search for

STATE # 1

You will find a section similar to

----------------------------------------
STATE # 1
----------------------------------------

Energy =  ...

----------------------------------------
STATE # 2
----------------------------------------

Energy = ...

----------------------------------------
STATE # 3
----------------------------------------

Energy = ...

These correspond to the three electronic states requested in the input file.

$GUGDIA
NSTATE=3
$END

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What do these states represent?

Because we requested

NSTATE = 3

GAMESS optimizes

State	Physical Meaning
State 1	Ground electronic state (S₀)
State 2	First excited singlet (S₁)
State 3	Second excited singlet (S₂)

Each state possesses

• its own electronic energy,
• its own CI expansion,
• and its own electronic structure,

while sharing the same optimized molecular orbitals obtained from the state-averaged calculation.

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2. Configuration State Functions (CSFs)

Immediately below each state, search for

CSF      COEF

You will find a table similar to

CSF      COEF

1      0.9648
2      0.0217
3     -0.1475
4      0.0318
...

This is one of the most important sections of the entire CASSCF output.

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What is a CSF?

A Configuration State Function (CSF) is a spin-adapted electronic configuration.

Instead of describing the wavefunction using individual Slater determinants,

GAMESS combines determinants with the correct spin symmetry to form CSFs.

The electronic wavefunction is written as

\[ \Psi = \sum_i c_i \Phi_i \]

where

• \( \Phi_i \) is a Configuration State Function,
• \( c_i \) is its CI coefficient.

Each coefficient indicates how much that configuration contributes to the electronic state.

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3. Interpreting the CI Coefficients

Consider the following example.

CSF      COEF

1      0.9648
2      0.0217
3     -0.1475
4      0.0318

The largest coefficient is

0.9648

This means

• CSF 1 dominates the wavefunction.
• The remaining configurations contribute only small corrections.

Such a wavefunction is still largely single-reference.

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Now consider another example.

CSF      COEF

1      0.63
2      0.51
3     -0.42
4      0.28

Now several configurations contribute significantly.

This indicates a strongly multiconfigurational wavefunction.

This is exactly the type of system where RHF is no longer adequate.

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4. When is a Molecule Multiconfigurational?

A useful guideline is

Largest CI Coefficient	Interpretation
> 0.95	Nearly single-reference
0.80–0.95	Moderate multiconfigurational character
< 0.80	Strong multiconfigurational character

The smaller the dominant coefficient,

the more important the remaining configurations become.

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5. Occupation Strings

Beside each CSF, GAMESS prints occupation strings such as

2200

or

2110

These numbers describe the occupation of the active orbitals.

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Example 1

2200

Suppose the active space contains four orbitals.

Orbital 1 → 2 electrons

Orbital 2 → 2 electrons

Orbital 3 → 0 electrons

Orbital 4 → 0 electrons

Graphically,

↑↓

↑↓

__

__

This is the dominant closed-shell configuration.

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Example 2

2110

Now the occupation becomes

Orbital 1 → 2 electrons

Orbital 2 → 1 electron

Orbital 3 → 1 electron

Orbital 4 → 0 electrons

Graphically,

↑↓

↑

↓

__

This represents an excited electronic configuration.

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Example 3

2020

This corresponds to

Orbital 1 → 2 electrons

Orbital 2 → 0 electrons

Orbital 3 → 2 electrons

Orbital 4 → 0 electrons

This configuration represents a double excitation inside the active space.

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6. Why Are There Many Configurations?

In RHF,

only one configuration exists.

Wavefunction

↓

One determinant

In CASSCF,

every allowed electron arrangement inside the active space is considered.

Configuration 1

+

Configuration 2

+

Configuration 3

+

...

↓

Final Wavefunction

The CI coefficients determine the importance of each configuration.

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7. Which Configurations Matter?

Usually,

only a few configurations possess large coefficients.

For practical interpretation,

it is common to examine only configurations satisfying

\[ |c| > 0.10 \]

or

\[ |c| > 0.20 \]

depending on the system.

Configurations with very small coefficients contribute little to the overall wavefunction.

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8. Why Is This Important?

The CI expansion tells us

• whether RHF would have been sufficient,
• whether strong static correlation exists,
• whether the chosen active space is reasonable,
• and which electronic configurations dominate each excited state.

This information is invaluable when analysing

• excited-state reactions,
• conical intersections,
• bond-breaking processes,
• photochemistry,
• and spin-state changes.

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9. Connection to the Active Space

Recall that the active space was selected from the CIS calculation.

Only those orbitals are allowed to change their occupation during the CASSCF calculation.

The occupation strings printed here demonstrate exactly how electrons move among these active orbitals.

Thus,

the CIS calculation determines which orbitals are important,

while the CASSCF calculation determines how electrons are distributed among them.

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Key Takeaways

• Every electronic state has its own CI expansion.
• A CASSCF wavefunction is a linear combination of Configuration State Functions.
• The CI coefficients measure the contribution of each configuration.
• Large coefficients correspond to dominant electronic configurations.
• Occupation strings describe how electrons populate the active orbitals.
• The CI expansion provides a direct measure of multiconfigurational character.

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