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State-Averaged CASSCF

So far, we have assumed that the CASSCF calculation optimizes the wavefunction for a single electronic state.

For many applications, such as calculating the ground-state energy, this approach is entirely sufficient.

However, many photochemical and excited-state processes involve multiple electronic states simultaneously. In these situations, optimizing the molecular orbitals for only one state can lead to an unbalanced and sometimes incorrect description of the system.

To overcome this problem, CASSCF can optimize several electronic states at the same time. This approach is known as State-Averaged CASSCF (SA-CASSCF).

Why Isn't State-Specific CASSCF Enough?

Suppose we optimize only the ground state.

Optimize Orbitals

↓

Ground State

↓

Lowest Possible Energy

The molecular orbitals become ideal for describing the ground state.

Now consider the first excited state.

Those same orbitals may no longer provide the best description.

Ground-State Orbitals

↓

Excited State

↓

Poor Representation

As a result,

  • excitation energies may become inaccurate,
  • excited-state wavefunctions may be poorly described,
  • electronic states may become discontinuous along a reaction coordinate.

The Problem During Photochemistry

Imagine a molecule absorbing light.

Ground State (S₀)

↓

Photon

↓

Excited State (S₁)

↓

Relaxation

↓

Conical Intersection

↓

Ground State

Throughout this process, both the ground and excited states are equally important.

Optimizing only one of them introduces an imbalance.

Instead, we require one common set of orbitals capable of describing all important electronic states simultaneously.

The Idea Behind State Averaging

Instead of minimizing the energy of one state,

State-Averaged CASSCF minimizes the weighted average energy of several states.

Mathematically,

\[ \boxed{ E_{\text{SA}} = \sum_{I=1}^{N} w_I E_I } \]

where

  • \(E_I\) is the energy of state \(I\),
  • \(w_I\) is the weight assigned to that state,
  • \(N\) is the number of states included in the average.

The weights satisfy

\[ \boxed{ \sum_I w_I = 1 } \]

so that the average energy remains properly normalized.

Equal Weight Averaging

The most common choice is to assign equal importance to every state.

For three electronic states,

\[ w_1=w_2=w_3=\frac13 \]

giving

\[ E_{\text{SA}} = \frac13 (E_1+E_2+E_3) \]

The orbitals are therefore optimized for all three states simultaneously.

Unequal Weights

Sometimes one state is more important than the others.

For example,

Ground State

50%

First Excited State

30%

Second Excited State

20%

The weights become

\[ w=(0.5,\;0.3,\;0.2) \]

The optimized orbitals now favor the ground state while still providing a reasonable description of the excited states.

One Set of Orbitals, Many Wavefunctions

An important feature of SA-CASSCF is that all states share the same molecular orbitals.

Common Molecular Orbitals

│
├── State 1
├── State 2
├── State 3
└── State 4

Although the orbitals are identical,

each state has

  • different CI coefficients,
  • different electronic energy,
  • different multiconfigurational wavefunction.

Why Is This Important?

Using common orbitals provides several advantages.

  • The electronic states remain balanced.
  • Excitation energies become more reliable.
  • Potential energy surfaces become smoother.
  • State crossings are described correctly.
  • Conical intersections can be located accurately.

For this reason, SA-CASSCF has become the standard approach for excited-state calculations.

Connection with GAMESS

In GAMESS,

multiple electronic states are requested using

$GUGDIA
 NSTATE=3
$END

This tells GAMESS to compute three electronic states.

The averaging weights are specified using

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

Here,

1,1,1

means that all three states receive equal weight.

Internally,

GAMESS normalizes these weights,

so

1,1,1

↓

1/3, 1/3, 1/3

The orbitals are then optimized using the average energy of these three states.

Relation to Your Input File

In your CASSCF example,

$GUGDIA
 NSTATE=3
$END

requests

  • S₀
  • S₁
  • S₂

while

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

ensures that

  • S₀,
  • S₁,
  • and S₂

are treated equally during the orbital optimization.

This balanced description is particularly important before performing methods such as

  • XMCQDPT,
  • diabatization,
  • nonadiabatic dynamics,
  • excited-state geometry optimization.

Advantages of State Averaging

Compared with state-specific CASSCF,

State-Averaged CASSCF provides

State-Specific State-Averaged
Optimizes one state Optimizes multiple states
Best energy for one state Balanced energies for all states
Orbitals differ between states Common orbital set
Can produce discontinuities Smooth potential energy surfaces
Poor near crossings Stable near state crossings

Limitations

Although SA-CASSCF is extremely useful,

it also has some limitations.

Because the orbitals are shared,

no single state is described as accurately as in a dedicated state-specific optimization.

Instead,

the orbitals represent a compromise between all included states.

Choosing too many states can therefore reduce the accuracy of the individual states.

Choosing the Number of States

A practical guideline is

Application Recommended States
Ground-state chemistry 1
Lowest excitation 2–3
UV–Visible spectroscopy 3–6
Photochemistry 3–8
Conical intersections Include all interacting states

The chosen states should include every electronic state expected to interact strongly along the reaction pathway.

Looking Ahead

At this point, the CASSCF wavefunction has been fully optimized.

The next question becomes:

How can we determine whether our chosen active space was appropriate?

The answer lies in the density matrix.

By constructing the one-particle density matrix and diagonalizing it, we obtain natural orbitals and their occupation numbers, which provide valuable insight into electron correlation and active-space quality.

Key Takeaways

  • State-Specific CASSCF optimizes one electronic state.
  • State-Averaged CASSCF optimizes several states simultaneously.
  • The orbitals are optimized by minimizing the weighted average energy.
  • All electronic states share the same molecular orbitals but have different CI coefficients.
  • SA-CASSCF is essential for excited-state calculations, conical intersections, and photochemistry.
  • In GAMESS, NSTATE specifies the number of states, while WSTATE defines their relative weights.