Choosing an Active Space¶

The active space is the most important user-defined component of a CASSCF calculation.

Unlike RHF or CIS, where the computational method automatically determines the wavefunction, CASSCF requires the user to decide which molecular orbitals should participate in the multiconfigurational treatment.

The quality of a CASSCF calculation depends much more on the choice of the active space than on any other input parameter.

What is an Active Space?¶

In CASSCF, the molecular orbitals are divided into three groups.

Higher Energy

Secondary (Virtual)

───────────────

Active Orbitals

───────────────

Inactive (Core)

Lower Energy

Inactive Orbitals¶

Always doubly occupied.
Do not participate in electronic excitations.
Remain unchanged throughout the calculation.

Active Orbitals¶

These orbitals are treated explicitly using Configuration Interaction.

Electrons are allowed to occupy these orbitals in every possible way consistent with the molecular symmetry and spin multiplicity.

This flexibility enables CASSCF to describe

excited states,
bond breaking,
near-degenerate electronic states,
conical intersections,
multiconfigurational wavefunctions.

Secondary Orbitals¶

These orbitals remain empty during the calculation.

They are optimized but are not included in the configuration interaction expansion.

Why is the Active Space Important?¶

Imagine a library.

Instead of searching every book in the building, you first choose a small shelf containing only the books relevant to your topic.

The active space works in exactly the same way.

Instead of allowing electrons to occupy every molecular orbital,

CASSCF considers only those orbitals that are chemically important.

This dramatically reduces the computational cost while retaining the essential physics of the electronic structure.

Where Do We Obtain the Active Space?¶

A common misconception is that the active space is chosen arbitrarily.

In practice, it is almost always selected using information obtained from earlier calculations.

The recommended workflow is

Geometry Optimization (RHF/DFT)
             │
             ▼
        RHF Calculation
             │
             ▼
       CIS Calculation
             │
             ▼
 Identify Important Transitions
             │
             ▼
 Choose Active Orbitals
             │
             ▼
      Perform CASSCF

The CIS calculation provides valuable information about which occupied and virtual orbitals contribute most strongly to the excited states.

Using the CIS Results¶

In the previous tutorial, we analyzed the CIS output for the water molecule.

The output listed the dominant electronic transitions together with their CI coefficients.

For example,

5 → 6      0.87

4 → 6      0.23

5 → 7      0.08

The CI coefficient measures the contribution of each electronic configuration to the excited-state wavefunction.

A larger coefficient indicates a more important excitation.

Review the CIS output here:
→ Water Molecule CIS Output

Choosing Important Orbitals¶

A practical strategy is to define a coefficient threshold.

Throughout this tutorial, we use

\[ |c| > 0.20 \]

as the criterion for selecting important transitions.

Applying this rule,

Transition	CI Coefficient	Include?
5 → 6	0.87	✅ Yes
4 → 6	0.23	✅ Yes
5 → 7	0.08	❌ No

The occupied and virtual orbitals involved in the important transitions should generally be included in the active space.

Constructing the Active Space¶

From the important transitions,

4 → 6

5 → 6

the orbitals involved are

Occupied

4
5

Virtual

6

Therefore,

the active orbitals should include

4
5
6

This is precisely the information required when defining the CASSCF active space.

Visual Representation¶

Higher Energy

7      Secondary

────────────────────

6      Active

5      Active

4      Active

────────────────────

3      Inactive
2      Inactive
1      Inactive

Only the active orbitals participate in the multiconfigurational wavefunction.

Chemical Considerations¶

Although CI coefficients provide an excellent starting point, chemical intuition should also be considered.

Active orbitals often include

π orbitals,
π* orbitals,
lone pairs,
antibonding orbitals,
partially occupied orbitals,
orbitals involved in bond breaking,
orbitals participating in photochemical reactions.

The goal is to include every orbital that may significantly change its occupation during the chemical process of interest.

How Large Should the Active Space Be?¶

Choosing too few orbitals may omit important electronic configurations.

Choosing too many orbitals greatly increases the computational cost.

A good active space should

include all chemically important orbitals,
exclude chemically inactive core orbitals,
remain computationally manageable.

Finding the smallest active space that captures the essential physics is often the best strategy.

Active Space in GAMESS¶

Once the orbitals have been selected, they are specified in the $DRT input group.

For example,

NMCC = 3
NDOC = 2
NVAL = 2

These keywords divide the molecular orbitals into

inactive orbitals,
active orbitals,
secondary orbitals.

The next section explains how these quantities are defined in the GAMESS input file.

Key Takeaways¶

The active space is the most important user-defined component of a CASSCF calculation.
Orbitals are divided into inactive, active, and secondary spaces.
The active space should contain orbitals involved in important electronic excitations.
CIS calculations provide an excellent guide for selecting active orbitals.
A practical rule is to include transitions with CI coefficients larger than approximately 0.20.
The selected active orbitals are defined in the $DRT section of the GAMESS input.