Unrestricted Hartree-Fock References

Now consider the case when $\vert \Phi_0 \rangle$ is obtained by the UHF procedure. Once again, the Fock matrix is diagonal: $F_{pq} = \delta_{pq} \epsilon_{p}$. However, it is customary to split the Fock matrix into two matrices, one for $\alpha$ and one for $\beta$ spin orbitals (mixed terms such as $F_{\overline{p}q}$ and $F_{p\overline{q}}$ are zero). As for the RHF case, $\sigma_0=0$ and $\sigma_i^a$ can be written

$$\sigma_i^a = c_i^a (\epsilon_a - \epsilon_i) + \sum_{jb} c_j... ...{b}} \langle a\overline{j} \vert\vert i\overline{b} \rangle.$$ (37)

Unfortunately, for a UHF reference equations (26)-(28) no longer hold, so that the above equation can be simplified only to

$$\sigma_i^a = c_i^a (\epsilon_a - \epsilon_i) + \sum_{jb} c_j... ...line{j}}^{\overline{b}} (ai \vert \overline{j}\overline{b}).$$ (38)

Even though spin has been integrated out, the overbars must be kept in the above equation because the spatial parts of spin orbitals i and $\overline{i}$ are not necessarily equal. An analogous equation holds for $\sigma_{\overline{i}}^{\overline{a}}$,

$$\sigma_{\overline{i}}^{\overline{a}} = c_{\overline{i}}^{\ove... ...{i}) ] + \sum_{jb} c_j^b (\overline{a}\overline{i} \vert jb),$$ (39)

and there is no general relationship between $\sigma_i^a$ and $\sigma_{\overline{i}}^{\overline{a}}$. The pseudodensity matrices ${\tilde P}_{\lambda \sigma}^{\alpha}$ and ${\tilde P}_{\lambda \sigma}^{\beta}$ are defined as in eq. (31) and (32), but they are no longer simply related. Thus it is necessary to compute two Fock-like matrices, according to

$${\tilde F}_{\mu \nu}^{\alpha} \sum_{\lambda \sigma} \left[ (\mu \nu \vert \lambda \sigma) - (\m... ...}^{\alpha} + (\mu \nu \vert \lambda \sigma) {\tilde P}_{\lambda \sigma}^{\beta}$$ = (40)

$${\tilde F}_{\mu \nu}^{\beta} \sum_{\lambda \sigma} \left[ (\mu \nu \vert \lambda \sigma) - (\m... ...^{\beta} + (\mu \nu \vert \lambda \sigma) {\tilde P}_{\lambda \sigma}^{\alpha}.$$ = (41)

In contrast to RCIS, there are no longer separate singlet and triplet cases, since the UCIS eigenfunctions are not CSFs. After transforming to the MO basis by

$${\tilde F}_{ia}^{\alpha} \sum_{\mu \nu} C_{\mu a}^* {\tilde F}_{\mu \nu}^{\alpha} C_{\nu i}$$ = (42)

$${\tilde F}_{\overline{i}\overline{a}}^{\beta} \sum_{\mu \nu} C_{\mu \overline{a}}^* {\tilde F}_{\mu \nu}^{\beta} C_{\nu \overline{i}},$$ = (43)

the expressions for ${\mathbf \sigma}$ become

$$\sigma_i^a({\rm UCIS}) c_i^a(\epsilon_a - \epsilon_i) + {\tilde F}_{ia}^{\alpha}$$ = (44)

$$\sigma_{\overline{i}}^{\overline{a}}({\rm UCIS}) c_{\overline{i}}^{\overline{a}} (\epsilon_{\overline{a}} - \epsilon_{\overline{i}}) + {\tilde F}_{\overline{i}\overline{a}}^{\beta}.$$ = (45)