(2) |

where the Fock matrix element

(3) |

The other relevant matrix elements are of the form
.
The singly excited determinants may differ
from each other by two spin orbitals if
and .
If
so, the determinants are already in maximum coincidence and the matrix
element is of the form

(4) |

or

(5) |

The matrix elements are and , respectively, and these antisymmetrized integrals are equal to each other (and also to and ).

For the case *i*=*j*, ,
the matrix elements are

(6) |

Likewise, for the case ,

(7) |

Finally, when

= | (8) | ||

= |

where ; if is obtained by an SCF procedure, then this is the SCF energy.

Using the permutational symmetries of the antisymmetrized two-electron
integrals, the two-electron terms for the preceding four cases
can be combined (rearranging the integral
requires
the assumption that the orbitals are real). This yields the final,
compact result

(9) |

This is equation (11) of Maurice and Head-Gordon [2], who extended the CIS method to the case of restricted open-shell (ROHF) and unrestricted (UHF) reference determinants. Note that

Given the above matrix elements, it remains to write down the CIS energy
expression. Recall that the CIS wavefunction is expanded as

(10) |

Assuming real CI coefficients, the energy is given by

(11) |

For a closed-shell SCF reference , off-diagonal terms of the Fock matrix vanish, and the expression becomes

(12) |

which matches equation (2.15) of Foresman

(13) |

One vector must be computed for each

= | (14) | ||

= | (15) |

where the bar over

= | (16) | ||

= | (17) |

The above equations make it clear that can be computed directly from the one- and two-electron integrals without the need to explicitly compute or store the one- and two-electron coupling coefficients and as separate quantities; this makes the CIS method a

= | (18) | ||

= | (19) |

where are the coefficients defining the transformation from atomic orbitals (AOs) to molecular orbitals (MOs). Evaluation of can be carried out as a series of matrix multiplies. The pseudodensity matrix,

can be multiplied by the two-electron integrals as they are formed in the atomic orbital basis to yield the AO Fock-like matrix,

which is transformed back into the MO basis by

- Restricted Hartree-Fock References
- Unrestricted Hartree-Fock References
- Restricted Open-Shell Hartree-Fock References