The CIS Secular Equation

In the previous chapter, we constructed the Configuration Interaction Hamiltonian matrix using singly excited Slater determinants as the basis functions.

The remaining task is to determine the unknown CI coefficients,

\[ c_i^a, \]

that define the excited-state wavefunction.

These coefficients are obtained by applying the variational principle, leading to the CIS secular equation, which is solved by diagonalizing the Hamiltonian matrix.

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The CIS Wavefunction

Recall that the CIS wavefunction is written as

\[ \Psi_{\mathrm{CIS}} = \sum_{i}^{occ} \sum_{a}^{virt} c_i^a \Phi_i^a, \]

where

• \(\Phi_i^a\) represents a singly excited Slater determinant,
• \(c_i^a\) is the corresponding CI coefficient.

The coefficients are initially unknown.

Our goal is to determine the values of these coefficients that best describe the excited state.

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Applying the Variational Principle

The Schrödinger equation is

\[ \hat H\Psi = E\Psi. \]

Substituting the CIS wavefunction gives

\[ \hat H \left( \sum_{i,a} c_i^a \Phi_i^a \right) = E \left( \sum_{i,a} c_i^a \Phi_i^a \right). \]

To isolate the unknown coefficients, we project this equation onto another singly excited determinant,

\[ \Phi_j^b. \]

Taking the inner product gives

\[ \left< \Phi_j^b \middle| \hat H \middle| \Psi \right> = E \left< \Phi_j^b \middle| \Psi \right>. \]

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Expanding the Equation

Substituting the wavefunction expansion,

\[ \sum_{i,a} c_i^a \left< \Phi_j^b \middle| \hat H \middle| \Phi_i^a \right> = E \sum_{i,a} c_i^a \left< \Phi_j^b \middle| \Phi_i^a \right>. \]

The singly excited determinants are orthonormal,

\[ \boxed{ \left< \Phi_j^b \middle| \Phi_i^a \right> = \delta_{ij} \delta_{ab} } \]

where

• \(\delta_{ij}\) is the Kronecker delta,
• \(\delta_{ab}\) is the Kronecker delta.

Therefore, the overlap matrix becomes the identity matrix.

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Defining the Hamiltonian Matrix

The Hamiltonian matrix elements are defined as

\[ \boxed{ H_{jb,ia} = \left< \Phi_j^b \middle| \hat H \middle| \Phi_i^a \right>. } \]

Substituting this definition simplifies the previous equation to

\[ \sum_{i,a} H_{jb,ia} c_i^a = E c_j^b. \]

This is the Configuration Interaction secular equation.

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Matrix Form

Collecting all coefficients into a vector,

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

the CIS equations become

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

This is an ordinary matrix eigenvalue equation.

The matrix

\[ \mathbf H \]

is the CIS Hamiltonian,

the vector

\[ \mathbf c \]

contains the CI coefficients,

and

\[ E \]

is the energy of the excited state.

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Physical Meaning

The eigenvalue equation tells us two things simultaneously.

The eigenvalues

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

are the electronic energies of the excited states.

The corresponding eigenvectors

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

contain the CI coefficients describing each excited state.

Thus,

one matrix diagonalization produces

• all excitation energies,
• all excited-state wavefunctions.

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Matrix Diagonalization

Suppose the Hamiltonian matrix is

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

Diagonalizing this matrix yields

\[ \mathbf H \longrightarrow \begin{pmatrix} E_1 & 0 & 0\\ 0 & E_2 & 0\\ 0 & 0 & E_3 \end{pmatrix}. \]

Each diagonal element corresponds to one excited-state energy.

The associated eigenvector contains the CI coefficients for that state.

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Connection to GAMESS Output

During a CIS calculation, GAMESS performs exactly this diagonalization.

For every excited state, it prints

• excitation energy,
• dominant orbital transitions,
• CI coefficients.

For example,

Excited State 1

5 -> 6    0.89
4 -> 6    0.27

These coefficients are simply components of the eigenvector

\[ \mathbf c^{(1)}. \]

Similarly,

the second excited state corresponds to another eigenvector,

the third excited state to a third eigenvector,

and so on.

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Comparison with Hartree–Fock

Readers who completed the RHF mathematics section will notice a close analogy.

Hartree–Fock	CIS
Roothaan–Hall equation	CI secular equation
Solve for molecular orbitals	Solve for excited states
Eigenvectors are MO coefficients	Eigenvectors are CI coefficients
Eigenvalues are orbital energies	Eigenvalues are excited-state energies

Both methods ultimately solve a matrix eigenvalue problem.

The difference lies in what the matrix represents.

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Key Takeaways

• The variational principle is used to determine the CI coefficients.
• Projecting the Schrödinger equation onto each singly excited determinant produces the CIS secular equation.
• The CIS secular equation is

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

• The eigenvalues are the excited-state energies.
• The eigenvectors contain the CI coefficients printed in GAMESS.
• CIS therefore reduces the excited-state problem to a matrix diagonalization.

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Looking Ahead

We now understand how the CIS eigenvalue problem is formulated mathematically.

The next chapter focuses on diagonalization and the interpretation of the resulting eigenvalues and eigenvectors, explaining how excitation energies, dominant electronic transitions, and excited-state wavefunctions are obtained from the solution of the CIS secular equation.

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