Solving the Configuration Interaction Problem

In the previous chapter, we showed that the CASSCF wavefunction is expressed as a linear combination of Configuration State Functions (CSFs),

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

and that the interactions between these CSFs are described by the Hamiltonian matrix,

\[ H_{IJ}=\langle\Phi_I|\hat H|\Phi_J\rangle. \]

The remaining question is:

How are the energies and the CI coefficients actually obtained?

The answer is by solving a matrix eigenvalue problem.

This chapter explains how the Hamiltonian matrix is diagonalized to obtain the electronic energies and the multiconfigurational wavefunctions.

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The Matrix Form of the Schrödinger Equation

After projecting the Schrödinger equation onto the CSF basis, we obtain

\[ \boxed{ \mathbf H\mathbf c=E\mathbf c } \]

This equation is known as the Configuration Interaction (CI) eigenvalue equation.

Although it appears compact, it contains the entire electronic structure problem.

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Understanding Each Quantity

The equation

\[ \mathbf H\mathbf c=E\mathbf c \]

contains three important quantities.

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1. Hamiltonian Matrix

\[ \mathbf H \]

contains all interactions between the Configuration State Functions.

For example,

\[ \mathbf H= \begin{pmatrix} H_{11} & H_{12} & H_{13}\\ H_{21} & H_{22} & H_{23}\\ H_{31} & H_{32} & H_{33} \end{pmatrix} \]

Each element represents

• the energy of a configuration (diagonal elements), or
• the coupling between two configurations (off-diagonal elements).

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2. CI Coefficient Vector

The unknown vector is

\[ \mathbf c= \begin{pmatrix} c_1\\ c_2\\ c_3\\ \vdots \end{pmatrix} \]

Each coefficient determines the contribution of one Configuration State Function to the final wavefunction.

For example,

\[ \Psi= 0.92\Phi_1 + 0.31\Phi_2 - 0.18\Phi_3 +\cdots \]

The coefficients are therefore one of the most important results of the CI calculation.

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3. Electronic Energy

The scalar

\[ E \]

is the electronic energy corresponding to one eigenstate.

Every solution of the CI equation produces

• one energy,
• one CI vector.

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Expanding the Matrix Equation

For three CSFs,

the eigenvalue equation becomes

\[ \begin{pmatrix} H_{11}&H_{12}&H_{13}\\ H_{21}&H_{22}&H_{23}\\ H_{31}&H_{32}&H_{33} \end{pmatrix} \begin{pmatrix} c_1\\ c_2\\ c_3 \end{pmatrix} = E \begin{pmatrix} c_1\\ c_2\\ c_3 \end{pmatrix} \]

This represents a set of coupled linear equations that must be solved simultaneously.

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What Does Diagonalization Mean?

The Hamiltonian matrix is generally not diagonal because different configurations interact with one another.

Configuration 1

↔

Configuration 2

↔

Configuration 3

Diagonalization finds a new basis in which these interactions disappear.

Instead of

H

↓

Coupled Configurations

we obtain

Diagonal H

↓

Independent Electronic States

Each diagonal element now corresponds to one electronic energy.

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Eigenvalues and Eigenvectors

Diagonalizing the Hamiltonian produces

Eigenvalues

\[ E_1,\; E_2,\; E_3,\; ... \]

These are the electronic energies.

For example,

Ground State

↓

E₁

First Excited State

↓

E₂

Second Excited State

↓

E₃

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Eigenvectors

Each eigenvalue has a corresponding eigenvector,

\[ \mathbf c^{(1)}, \mathbf c^{(2)}, \mathbf c^{(3)} \]

Each eigenvector contains the CI coefficients of one electronic state.

Thus,

Energy

↓

CI Coefficients

↓

Wavefunction

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Building the Wavefunctions

Once the coefficients have been obtained,

each electronic state becomes

Ground state

\[ \Psi_1 = \sum_I c_I^{(1)} \Phi_I \]

First excited state

\[ \Psi_2 = \sum_I c_I^{(2)} \Phi_I \]

Second excited state

\[ \Psi_3 = \sum_I c_I^{(3)} \Phi_I \]

Every electronic state therefore has its own multiconfigurational wavefunction.

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Physical Interpretation

The diagonalization process determines

• which configurations dominate each state,
• how strongly configurations mix,
• the ordering of electronic states,
• the energy separation between states.

For example,

Ground State

Configuration A

95%

Configuration B

5%

whereas

Excited State

Configuration A

20%

Configuration B

75%

Configuration C

5%

Different electronic states have different multiconfigurational character.

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Computational Considerations

The Hamiltonian matrix grows rapidly with the number of CSFs.

Active Space	Approximate CI Matrix Size
CAS(2,2)	Few × Few
CAS(4,4)	Tens × Tens
CAS(6,6)	Hundreds × Hundreds
CAS(8,8)	Thousands × Thousands
CAS(10,10)	Tens of Thousands × Tens of Thousands

Consequently,

• larger active spaces improve accuracy,
• but they also increase computational cost significantly.

Efficient diagonalization algorithms are therefore essential for practical CASSCF calculations.

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What Happens in GAMESS?

When the input contains

$MCSCF
 CISTEP=GUGA
$END

GAMESS performs the following operations:

1. Generates all valid Configuration State Functions (CSFs) using the GUGA algorithm.
2. Computes the Hamiltonian matrix elements between every pair of CSFs.
3. Constructs the CI Hamiltonian matrix.
4. Diagonalizes the matrix.
5. Obtains the electronic energies and CI coefficients.
6. Builds the multiconfigurational wavefunctions for all requested states.

At this stage, only the CI coefficients have been optimized. The molecular orbitals are still those from the previous iteration.

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Why Isn't This the Final Answer?

One might think that once the Hamiltonian has been diagonalized, the calculation is complete.

However, there is still a problem.

The CI calculation assumes that the molecular orbitals are fixed.

If the orbitals change,

• the Hamiltonian changes,
• the CI coefficients change,
• and therefore the electronic energy changes.

This means the molecular orbitals themselves must also be optimized.

This additional optimization distinguishes CASSCF from a simple Configuration Interaction calculation.

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Looking Ahead

So far, we have optimized only one part of the wavefunction:

Wavefunction

↓

Optimize CI Coefficients ✓

↓

Optimize Orbitals ✗

The next chapter explains how CASSCF rotates the molecular orbitals, reconstructs the Hamiltonian, and repeats the CI calculation until both the orbitals and the CI coefficients become self-consistent.

This iterative procedure is what gives the method its name:

Complete Active Space Self-Consistent Field (CASSCF).

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Key Takeaways

• The CI problem is expressed as the matrix equation \(\mathbf{Hc}=E\mathbf{c}\).
• Diagonalizing the Hamiltonian matrix yields the electronic energies (eigenvalues) and the CI coefficients (eigenvectors).
• Each eigenvector defines one multiconfigurational electronic state.
• The size of the CI matrix increases rapidly with the size of the active space.
• At this stage, only the CI coefficients are optimized.
• The molecular orbitals must still be optimized, leading to the self-consistent CASSCF procedure.

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