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Understanding the CIS Output

After the RHF reference wavefunction has been read, GAMESS constructs all possible single excitations from the occupied molecular orbitals to the virtual molecular orbitals. It then diagonalizes the CIS Hamiltonian to obtain the excited-state energies and wavefunctions.

The most important part of the output is the list of excited states and the dominant orbital transitions contributing to each state.

A typical section of the output appears as

EXCITED STATE   1  ENERGY= ...
     5 -> 6     0.91
     4 -> 6     0.18
     3 -> 7     0.06

Each line has a specific meaning.

Excited State Number

EXCITED STATE 1

This simply labels the excited state being reported.

If the input contains

NSTATE=3

then GAMESS will print information for

  • Excited State 1
  • Excited State 2
  • Excited State 3

Each state has its own energy and electronic configuration.

Excitation Energy

Immediately following the state number, GAMESS reports the excitation energy.

For example,

EXCITED STATE 1

ENERGY = ...

This value represents the energy difference between the RHF ground state and the excited state.

Depending on the output options, the excitation energy may be printed in

  • Hartree
  • Electron volts (eV)
  • Wavenumbers (cm⁻¹)

The lowest excitation energy corresponds to the first electronically excited state.

Orbital Transitions

The most informative part of the CIS output is the list of orbital transitions.

For example,

5 -> 6      0.91

means

  • an electron is promoted from occupied orbital 5
  • into virtual orbital 6

The number following the transition is the CI coefficient.

CI Coefficients

The coefficient indicates how strongly a particular orbital excitation contributes to the excited-state wavefunction.

For example,

5 -> 6     0.91

means that the excited state is dominated by the

HOMO → LUMO

transition.

A second transition

4 -> 6     0.18

has a much smaller contribution.

Therefore,

the first excited state is primarily described by a single excitation from orbital 5 to orbital 6.

Physical Meaning of the Coefficients

The CIS wavefunction is written as

\[ \Psi_{CIS} = \sum_i c_i\Phi_i \]

where

  • \(\Phi_i\) is one singly excited determinant,
  • \(c_i\) is its CI coefficient.

Large coefficients correspond to important electronic configurations.

Small coefficients contribute very little to the excited state.

For example,

5 -> 6     0.92
4 -> 6     0.05
3 -> 7     0.03

clearly indicates that the excited state is almost entirely described by the

5 → 6

excitation.

Choosing an Active Space

One of the most common applications of a CIS calculation is to identify a suitable active space for subsequent multiconfigurational calculations such as

  • CASSCF
  • XMCQDPT
  • CASPT2
  • MRCI

Rather than selecting orbitals arbitrarily, the dominant CIS transitions provide a physically motivated choice.

Coefficient Threshold

A common practical approach is to consider only transitions whose absolute CI coefficients satisfy

\[ |c_i| > 0.2 \]

These transitions are regarded as making significant contributions to the excited-state wavefunction.

Transitions with coefficients smaller than this threshold usually contribute only weakly and are often neglected during active-space selection.

Note

The value 0.2 is not a strict theoretical limit, but it is widely used as a practical guideline for identifying important configurations.

Example

Suppose the CIS output contains

Excited State 1

5 -> 6    0.82
4 -> 6    0.28
5 -> 7    0.11
3 -> 7    0.05

Using a threshold of

\[ |c_i| > 0.2 \]

the important transitions are

  • 5 → 6
  • 4 → 6

The remaining transitions can generally be ignored because their contributions are comparatively small.

Constructing the Active Space

The active space should include every orbital participating in the significant transitions.

For the example above,

important orbitals are

  • 4
  • 5
  • 6

These orbitals should be included in the active space because they are directly involved in describing the excited state.

If another excited state contains

5 -> 7    0.54

then orbital 7 should also be included.

The resulting active space would contain

Orbitals:
4 5 6 7

rather than considering only the first excited state.

Using Multiple Excited States

When several excited states are of interest, the active space should be chosen by examining all computed excited states.

For example,

State Important Transition
S₁ 5 → 6
S₂ 4 → 6
S₃ 5 → 7

The combined active space should therefore include

4 5 6 7

This ensures that all important electronic configurations can be described during a subsequent CASSCF calculation.

Practical Guidelines

When selecting an active space from a CIS calculation:

  • Examine every excited state of interest.
  • Identify transitions with
\[ |c_i| > 0.2 \]
  • Include both occupied and virtual orbitals involved in those transitions.
  • Remove orbitals that never participate significantly.
  • Verify that the chosen active space can describe all important electronic excitations.

Key Takeaways

  • CIS expresses each excited state as a linear combination of singly excited configurations.
  • The CI coefficients measure the importance of each excitation.
  • Transitions with coefficients larger than 0.2 are commonly regarded as significant.
  • The occupied and virtual orbitals participating in these significant transitions form an excellent starting point for selecting the active space in multiconfigurational methods such as CASSCF.