How Configuration Interaction Singles (CIS) Works¶

The Hartree–Fock (RHF) method provides a good description of the electronic ground state of a molecule, but it cannot directly describe excited electronic states. The CIS method extends the Hartree–Fock wavefunction by considering configurations in which one electron is promoted from an occupied molecular orbital to a virtual (unoccupied) molecular orbital.

Rather than optimizing a completely new set of molecular orbitals, CIS uses the converged Hartree–Fock orbitals as a reference and constructs excited-state wavefunctions by combining many possible single-electron excitations.

The overall CIS procedure can be summarized in six steps.

Step 1 – Perform an RHF Calculation¶

A CIS calculation always begins with a converged Restricted Hartree–Fock calculation.

The RHF calculation provides

the optimized molecular orbitals,
orbital energies,
and the ground-state electronic configuration.

These molecular orbitals serve as the reference for all subsequent excited-state calculations.

Step 2 – Read the Hartree–Fock Orbitals¶

Instead of generating new molecular orbitals, CIS reads the converged RHF orbitals from the $VEC section of the input file.

These orbitals define the reference Slater determinant representing the ground state.

Every excited configuration generated during the CIS calculation is built from this reference.

Step 3 – Generate Single Excitations¶

CIS systematically constructs all possible single excitations by promoting one electron from an occupied orbital to a virtual orbital.

For example,

Ground State

Occupied        Virtual

1   ↑↓
2   ↑↓
3   ↑↓
-------------
4   (empty)
5   (empty)

Possible single excitations include

3 → 4
3 → 5
2 → 4
2 → 5
1 → 4
...

Each of these excited electronic configurations is represented by a new Slater determinant.

Step 4 – Build the CIS Hamiltonian¶

The excited determinants generated in the previous step are used as a basis for constructing the Configuration Interaction (CI) Hamiltonian.

Each matrix element describes the interaction between two excited configurations.

Unlike RHF, which solves for molecular orbitals, CIS solves for the optimal combination of these excited configurations.

Step 5 – Diagonalize the CI Matrix¶

The CIS Hamiltonian is diagonalized to obtain

excitation energies,
excited-state wavefunctions,
and the contribution of each electronic configuration.

Each excited state is expressed as a linear combination of singly excited determinants.

The resulting coefficients indicate how important each electronic transition is in describing the excited state.

Step 6 – Analyze the Excited States¶

Finally, GAMESS prints the properties of each excited state, including

excitation energies,
dominant orbital transitions,
CI coefficients,
transition dipole moments (when requested),
oscillator strengths (when requested),
and state symmetry.

These results help identify the electronic transitions responsible for UV–Visible absorption and provide valuable information for selecting active spaces in multiconfigurational calculations.

CIS Workflow¶

The complete CIS procedure can be summarized as

Perform RHF Calculation
          │
          ▼
Read RHF Molecular Orbitals
          │
          ▼
Generate All Single Excitations
          │
          ▼
Construct the CIS Hamiltonian
          │
          ▼
Diagonalize the CI Matrix
          │
          ▼
Obtain Excited States
          │
          ▼
Analyze Excitation Energies
and Dominant Transitions

Each excited state obtained from this procedure represents a different way in which an electron can be promoted from an occupied molecular orbital to a virtual molecular orbital.

Key Takeaways¶

CIS starts from a converged Hartree–Fock wavefunction.
The Hartree–Fock molecular orbitals remain fixed during the calculation.
All possible single-electron excitations are generated from the RHF reference.
These excited configurations are combined to form the CIS Hamiltonian.
Diagonalizing the CIS Hamiltonian yields excitation energies and excited-state wavefunctions.
The dominant CI coefficients reveal the most important electronic transitions contributing to each excited state.

Looking Ahead¶

The workflow described above explains how a CIS calculation proceeds computationally.

In the next section, we will examine the mathematical foundations of CIS, including the construction of singly excited Slater determinants, the Configuration Interaction wavefunction, and the CIS secular equations used to obtain the excited-state energies.