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The CASSCF Wavefunction

After selecting the active space and generating the Configuration State Functions (CSFs), the next objective is to construct the electronic wavefunction.

Unlike Hartree–Fock (RHF), which represents the entire electronic structure using a single Slater determinant, CASSCF expresses the wavefunction as a linear combination of many Configuration State Functions (CSFs).

This multiconfigurational description enables CASSCF to accurately describe systems where several electronic configurations contribute significantly to the electronic structure.

Review: The RHF Wavefunction

In RHF, the wavefunction is remarkably simple.

It consists of a single Slater determinant,

\[ \boxed{ \Psi_{\mathrm{RHF}}=\Phi_0 } \]

where

  • \(\Psi_{\mathrm{RHF}}\) is the Hartree–Fock wavefunction,
  • \(\Phi_0\) is the single Slater determinant constructed from the occupied molecular orbitals.

Conceptually,

Wavefunction

↓

One Determinant

↓

One Electronic Configuration

This approximation assumes that one electronic configuration dominates the molecule.

For many systems this works well.

However, when several configurations have comparable energies, this description becomes insufficient.

Why One Configuration Is Not Enough

Consider a molecule undergoing bond breaking.

Near equilibrium,

Configuration A

≈100%

Configuration B

≈0%

The RHF approximation is excellent.

As the bond stretches,

Configuration A

≈60%

Configuration B

≈40%

Now both configurations contribute significantly.

The exact wavefunction must therefore include both configurations.

This motivates the multiconfigurational expansion used in CASSCF.

The CASSCF Wavefunction

Instead of using a single determinant, CASSCF writes the wavefunction as

\[ \boxed{ \Psi= \sum_{I} c_I \Phi_I } \]

This is the fundamental equation of the CASSCF method.

Understanding the Equation

The equation

\[ \Psi= \sum_I c_I\Phi_I \]

contains three important quantities.

1. The Wavefunction

\[ \Psi \]

represents the complete electronic wavefunction of the molecule.

It contains all information about

  • electronic energy,
  • electron density,
  • molecular properties,
  • excited states,
  • transition moments.

Every observable quantity can ultimately be calculated from this wavefunction.

2. Configuration State Functions

\[ \Phi_I \]

Each \(\Phi_I\) is a Configuration State Function (CSF).

A CSF represents

  • one allowed electronic configuration,
  • with the correct spin symmetry,
  • constructed from one or more Slater determinants.

For example,

Φ₁

2200

↓

Two occupied orbitals

Another CSF might be

Φ₂

2110

↓

One electron promoted

Each CSF represents a different arrangement of electrons inside the active space.

3. CI Coefficients

The coefficients

\[ c_I \]

are called the Configuration Interaction (CI) coefficients.

These numbers determine the contribution of each CSF to the total wavefunction.

For example,

\[ \Psi = 0.96\Phi_1 + 0.21\Phi_2 - 0.12\Phi_3 +\cdots \]

Here,

  • \(\Phi_1\) dominates the wavefunction,
  • while the remaining configurations contribute smaller corrections.

During the CASSCF calculation, these coefficients are not known beforehand.

Determining them is one of the primary goals of the CI calculation.

Physical Meaning of the CI Coefficients

The magnitude of a coefficient indicates the importance of its corresponding configuration.

For example,

Coefficient Interpretation
0.98 Dominant configuration
0.80 Significant contribution
0.50 Strong multiconfigurational mixing
0.10 Minor contribution
≈0 Negligible contribution

The larger the coefficient,

the more important that electronic configuration becomes.

Normalization of the Wavefunction

Because the total probability of finding the electrons must equal one,

the wavefunction must satisfy the normalization condition

\[ \boxed{ \sum_I c_I^2 = 1 } \]

This means the squared CI coefficients behave like probabilities.

For example,

\[ \Psi = 0.95\Phi_1 + 0.22\Phi_2 + 0.12\Phi_3 \]

gives

\[ 0.95^2 + 0.22^2 + 0.12^2 \approx1 \]

Thus,

the larger the squared coefficient,

the greater the contribution of that configuration.

Why Do We Use Many Configurations?

Suppose the active space contains four orbitals.

Instead of

Configuration 1 only

CASSCF considers

Configuration 1

+

Configuration 2

+

Configuration 3

+

Configuration 4

+

...

Every allowed electronic arrangement contributes to the final wavefunction.

The exact contribution is determined by solving the CI problem.

Comparison with RHF

The difference between RHF and CASSCF can now be summarized mathematically.

Hartree–Fock

\[ \boxed{ \Psi=\Phi_0 } \]

One determinant.

One electronic configuration.

CASSCF

\[ \boxed{ \Psi= \sum_I c_I\Phi_I } \]

Many configurations.

One optimized wavefunction.

Why Is This More Accurate?

Suppose two configurations have nearly identical energies.

RHF chooses only one,

Configuration A

while CASSCF uses

0.71 × Configuration A

+

0.69 × Configuration B

Both configurations contribute naturally to the electronic structure.

This enables CASSCF to describe

  • excited states,
  • bond dissociation,
  • conical intersections,
  • transition-metal complexes,
  • photochromic reactions,
  • diradicals,

much more accurately than RHF.

How Are the CI Coefficients Obtained?

The coefficients

\[ c_I \]

are not guessed.

Instead,

GAMESS

  1. constructs the Hamiltonian matrix,
  2. evaluates the interaction between every pair of CSFs,
  3. diagonalizes the Hamiltonian,
  4. obtains the energies and the CI coefficients simultaneously.

This is the Configuration Interaction (CI) step of the CASSCF calculation.

The mathematical details of this procedure are presented in the next chapters.

Looking Ahead

Now that the wavefunction has been defined,

the next question is

How are the coefficients \(c_I\) determined?

To answer this,

we first need to construct the Hamiltonian matrix describing the interactions between all Configuration State Functions.

This leads naturally to the next chapter,

The CASSCF Hamiltonian.

Key Takeaways

  • RHF uses a single Slater determinant to represent the electronic wavefunction.
  • CASSCF expresses the wavefunction as a linear combination of Configuration State Functions.
  • Each CSF represents one spin-adapted electronic configuration.
  • The CI coefficients determine the contribution of each configuration.
  • The wavefunction is normalized so that the sum of the squared coefficients equals one.
  • Determining the CI coefficients is the central objective of the Configuration Interaction calculation.