Diagonalization and Excited States

In the previous chapter, we derived the CIS secular equation,

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

where

• \(\mathbf H\) is the Configuration Interaction Hamiltonian,
• \(\mathbf c\) is the vector of CI coefficients,
• \(E\) is the energy of an excited state.

The final step of a CIS calculation is to diagonalize the Hamiltonian matrix. This single operation determines both the excitation energies and the electronic character of each excited state.

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Why Diagonalize the Hamiltonian?

The Hamiltonian matrix contains all the interactions between the singly excited electronic configurations.

Each basis function corresponds to one singly excited Slater determinant,

\[ \Phi_1,\; \Phi_2,\; \Phi_3,\; \dots \]

Before diagonalization, these determinants are only possible electronic configurations.

Diagonalization determines how these configurations combine to form the actual excited states of the molecule.

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The Eigenvalue Problem

Mathematically, the CIS calculation solves

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

This is the standard matrix eigenvalue problem encountered throughout quantum mechanics.

Solving this equation produces

• a set of eigenvalues, and
• a corresponding set of eigenvectors.

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The Eigenvalues

The eigenvalues are

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

Each eigenvalue corresponds to the energy of one electronic excited state.

The ordering is

\[ E_1 < E_2 < E_3 < \cdots \]

where

• \(E_1\) is the first excited state,
• \(E_2\) is the second excited state,
• \(E_3\) is the third excited state,
• and so on.

The excitation energy reported by GAMESS is simply the difference between an excited-state energy and the Hartree–Fock ground-state energy,

\[ \boxed{ \Delta E = E_{\mathrm{excited}} - E_{\mathrm{ground}}. } \]

These excitation energies are commonly reported in

• Hartree,
• electron volts (eV),
• or wavenumbers (cm\(^{-1}\)).

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The Eigenvectors

Each eigenvalue has an associated eigenvector,

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

The components of each eigenvector are the CI coefficients.

For example,

\[ \mathbf c^{(1)} = \begin{pmatrix} 0.91\\ 0.27\\ 0.05 \end{pmatrix} \]

means that the first excited state is

\[ \Psi_1 = 0.91\Phi_1 + 0.27\Phi_2 + 0.05\Phi_3. \]

Thus, the eigenvector describes the composition of the excited-state wavefunction.

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Configuration Mixing

One of the most important consequences of diagonalization is configuration mixing.

Suppose two singly excited determinants have similar energies.

Instead of remaining independent,

Φ₁

Φ₂

they interact through the Hamiltonian and form new excited states,

Ψ₁ = c₁Φ₁ + c₂Φ₂

Ψ₂ = c₃Φ₁ + c₄Φ₂

Each excited state therefore becomes a mixture of several electronic configurations.

This mixing is responsible for the CI coefficients reported in the output.

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Dominant Electronic Transitions

Although many configurations contribute to an excited state, only a few usually have significant coefficients.

For example,

Excitation	Coefficient
HOMO → LUMO	0.89
HOMO−1 → LUMO	0.24
HOMO → LUMO+1	0.06

This tells us that the excited state is primarily a

HOMO → LUMO

transition.

The smaller coefficients represent secondary contributions.

This is why most quantum chemistry programs report only the dominant configurations.

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Connection with GAMESS Output

During a CIS calculation, GAMESS diagonalizes the CI Hamiltonian and prints the resulting eigenvectors in a compact form.

A typical output might contain

Excited State 1

5 -> 6     0.87
4 -> 6     0.23
5 -> 7     0.08

This corresponds to the wavefunction

\[ \Psi_1 = 0.87\Phi_5^6 + 0.23\Phi_4^6 + 0.08\Phi_5^7. \]

The reported coefficients are simply the components of the first eigenvector.

Similarly,

Excited State 2

5 -> 7     0.82
4 -> 6     0.31

represents the second eigenvector obtained during diagonalization.

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Interpreting the Results

After diagonalization, every excited state can be characterized by

• its excitation energy,
• its dominant orbital transitions,
• its CI coefficients,
• and its electronic wavefunction.

These quantities allow us to

• identify the nature of the excited state,
• assign UV–Visible absorption bands,
• compare different excited states,
• and select orbitals for multiconfigurational calculations such as CASSCF.

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Summary of the CIS Procedure

The complete mathematical workflow of a CIS calculation is

Hartree–Fock Calculation
            │
            ▼
Generate Singly Excited Determinants
            │
            ▼
Construct the CI Wavefunction
            │
            ▼
Build the Hamiltonian Matrix
            │
            ▼
Solve

      Hc = Ec

            │
            ▼
Diagonalize the Hamiltonian
            │
            ▼
Obtain

• Excitation Energies
• CI Coefficients
• Excited-State Wavefunctions

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Key Takeaways

• Diagonalizing the CIS Hamiltonian produces the excited-state energies.
• Each eigenvalue corresponds to one electronic excited state.
• Each eigenvector contains the CI coefficients describing that state.
• Large CI coefficients indicate the dominant electronic transitions.
• The excitation energies and coefficients printed by GAMESS are direct results of this matrix diagonalization.
• These results provide the foundation for interpreting excited-state properties and constructing active spaces for multiconfigurational methods.

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Looking Ahead

CIS provides a computationally efficient description of electronically excited states by considering only single excitations.

While this approximation is useful for many systems, it also introduces important limitations.

The next chapter discusses the strengths and weaknesses of the CIS method and explains why more advanced approaches such as TDDFT, CASSCF, CASPT2, and XMCQDPT are often required for quantitative excited-state studies.

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