Next: Extensions of CIS to Up: CIS Energy Equations Previous: Unrestricted Hartree-Fock References

## Restricted Open-Shell Hartree-Fock References

A single-determinant restricted open-shell Hartree-Fock (ROHF) wavefunction describing a high-spin open-shell system will be an eigenfunction of (i.e., a CSF). This is easy to verify by direct application of the operator, which is
 = = (46)

where
 = (47) = (48) = (49)

and
 (50)

The high-spin ROHF wavefunction can be written as

 (51)

where t and u will denote open-shells. Applying , this becomes

 (52)

This is easy to evaluate:
 = (53) = (54)

The raising operator yields zero when acting on : raising operators applied to electrons always yield zero, and raising operators applied to the electrons yield spin orbitals which are already occupied (so the determinant vanishes by the Pauli principle). Hence the final result is
 = = (55)

where .

Now it is worthwhile to consider how to form CSFs for the single excitations. Determinants which promote electrons from the singly-occupied space to the virtual space, as well as determinants which promote electrons in the doubly-occupied orbitals to the singly-occupied orbitals, are already spin-adapted (the proof is completely analogous to that above for the ROHF reference determinant). The only other relevant single excitations are those which move an electron from a doubly-occupied orbital to a virtual orbital. These determinants are not spin-adapted, as we will proceed to demonstrate. Consider the action of on the determinant :

 (56)

The result of is easily determined to be

 (57)

Now all that remains is the raising and lowering operators. These are somewhat more involved and require that attention be paid to the sign.

 (58)

By arguments similar to those presented above, all raising operators yield zero except for . Using the anticommutation relations for creation and annihilation operators,
 = = = = (59)

The operator can now affect electrons in any of the following orbitals: a, i, and any of the open-shell orbitals.

 = = (60)

Thus overall,

 (61)

The analogous equation for is

 (62)

Clearly a spin eigenfunction can be constructed as

 (63)

Then the operation of is

 (64)

Now that the relevant CSFs have been obtained, they can be used to define the ROHF convergence criteria: the final ROHF wavefunction will not mix with any of the singly substituted CSFs. Thus

 (65)

which implies the following conditions on the Fock matrix elements
 Fta = 0 (66) = 0 (67) Fia = (68)

Using these results, we can write down expressions for the vectors. Since determinants must enter with the same coefficients as , once again. Furthermore, since the ROHF reference cannot mix with any other singly excited configurations, the c0 contribution to and may be safely ignored. We will therefore consider four cases: , , , and , where once again t,u represent singly occupied orbitals.

The equation for is readily seen to be

 = + (69)

Separating the Fock operator terms from the two-electron integrals, and integrating out spin, this yields
 = + (70)

where we have used the relation . The analogous equation for spins is
 = + (71)

If the equality is to be maintained, we must have . It is then computationally convenient to form these quantities as

 (72)

Thus
 = + + (73)

Now it is clear that the two-electron integrals can be treated all together. To begin condensing the notation once again, let us define the following quantities:
 = (74) = (75) = (76) = (77) = (78)

where the same as that defined in eq. (17) of Maurice and Head-Gordon [2]. Then can be evaluated as

 (79)

where
 = (80) = (81)

Next consider the term :

 = = + = (82)

It is computationally more efficient to evaluate at the same time as . The value of computed in this manner must of course be corrected for the difference in the formulas for and , but this correction scales as only (see Maurice and Head-Gordon [2]). The two-electron part is thus computed by

 (83)

Similar considerations apply to , which is

 = = + = (84)

where is actually evaluated according to

 (85)

Next: Extensions of CIS to Up: CIS Energy Equations Previous: Unrestricted Hartree-Fock References
C. David Sherrill
2000-04-18