Restricted Hartree-Fock References

Now consider the case of a closed-shell RHF reference determinant. In this case, $F_{pq} = \delta_{pq} \epsilon_{p}$, so that $\sigma_0$becomes zero. $\sigma_{ia}$ simplifies to

$$\sigma_{ia} = c_i^a (\epsilon_a - \epsilon_i) + \sum_{jb} c_j^b \langle aj \vert\vert ib \rangle.$$ (23)

Conserving M_s requires that the spins of j and b are equal. Therefore,

$$\sigma_{ia} = c_i^a (\epsilon_a - \epsilon_i) + \sum_{jb} c_... ...{b}} \langle a\overline{j} \vert\vert i\overline{b} \rangle.$$ (24)

After integrating over spin, this becomes in chemists' notation

$$\sigma_{ia} = c_i^a (\epsilon_a - \epsilon_i) + \sum_{jb} c_... ...}\overline{b}} c_{\overline{j}}^{\overline{b}} (ai \vert jb).$$ (25)

Time-reversal symmetry imposes certain conditions on the CI coefficients. In alpha and beta string notation [5], for M_s=0 cases,

$$c(I_{\alpha},I_{\beta}) = (-1)^S c(I_{\beta}, I_{\alpha}).$$ (26)

An analogous equation also holds for $\sigma$. This means that for a closed shell RHF reference, $c_i^a = c_{\overline{i}}^{\overline{a}}$( $\sigma_i^a = \sigma_{\overline{i}}^{\overline{a}}$) for singlets, and $c_i^a = -c_{\overline{i}}^{\overline{a}}$ ( $\sigma_i^a = -\sigma_{\overline{i}}^{\overline{a}}$) for triplets; thus only half of the CI coefficients must be computed explicitly. These sign rules are also evident from the observation that $\vert \Phi_i^a \rangle$ and $\vert \Phi_{\overline{i}}^{\overline{a}} \rangle$ are not spin eigenfunctions, but that the total CI wavefunction will be a spin eigenfunction (if the required determinants are present in the CI space). Using the determinant sign convention of Szabo and Ostlund [6], spin eigenfunctions (or CSFs) associated with the above determinants are

  

$$\vert ^1\Phi_i^a \rangle \frac{1}{\sqrt{2}} \left( \vert \Phi_i^a \rangle + \vert \Phi_{\overline{i}}^{\overline{a}} \rangle \right)$$ = (27)

$$\vert ^3\Phi_i^a \rangle \frac{1}{\sqrt{2}} \left( \vert \Phi_i^a \rangle - \vert \Phi_{\overline{i}}^{\overline{a}} \rangle \right).$$ = (28)

Using these relationships between CI coefficients, we obtain

$$^1\sigma_i^a({\rm RCIS}) c_i^a (\epsilon_a - \epsilon_i) + \sum_{jb} c_j^b \left[ 2 (ai \vert jb) - (ab \vert ji) \right]$$ = (29)

$$^3\sigma_i^a({\rm RCIS}) c_i^a (\epsilon_a - \epsilon_i) - \sum_{jb} c_j^b (ab \vert ji).$$ = (30)

Furthermore, $^1\sigma_{\overline{i}}^{\overline{a}}({\rm RCIS}) = ^1\sigma_i^a({\rm RCIS})$ and $^3\sigma_{\overline{i}}^{\overline{a}}({\rm RCIS}) = - ^3\sigma_i^a({\rm RCIS})$.

Consider how equations (20)-(22) change when spin is explicitly accounted for. There will be two pseudodensity matrices,

  

$${\tilde P}_{\lambda \sigma}^{\alpha} \sum_{jb} C_{\lambda j}^* c_j^b C_{\sigma b}$$ = (31)

$${\tilde P}_{\lambda \sigma}^{\beta} \sum_{\overline{j}\overline{b}} C_{\lambda j}^* c_{\overline{j}}^{\overline{b}} C_{\sigma b}.$$ = (32)

Due to equation (26), ${\tilde P}_{\lambda \sigma}^{\alpha} = {\tilde P}_{\lambda \sigma}^{\beta}$ for singlets, and ${\tilde P}_{\lambda \sigma}^{\alpha} = -{\tilde P}_{\lambda \sigma}^{\beta}$ for triplets. Thus it is necessary to form only one of the Fock-like matrices, ${\tilde F}_{\mu \nu}^{\alpha}$ or ${\tilde F}_{\mu \nu}^{\beta}$. The former is constructed as

$$^1{\tilde F}_{\mu \nu}^{\alpha} \sum_{\lambda \sigma} \left[ 2 (\mu \nu \vert \lambda \sigma) - (\mu \sigma \vert \lambda \nu) \right] {\tilde P}_{\lambda \sigma}^{\alpha}$$ = (33)

$$^3{\tilde F}_{\mu \nu}^{\alpha} - \sum_{\lambda \sigma} (\mu \sigma \vert \lambda \nu) {\tilde P}_{\lambda \sigma}^{\alpha}$$ = (34)

Finally ${\tilde F}_{ia}^{\alpha}$ is constructed according to eq (22). In terms of these quantities, the ${\mathbf \sigma}$ vector can be written

$$^1\sigma_i^a({\rm RCIS}) = ^1\sigma_{\overline{i}}^{\overline{a}}({\rm RCIS}) c_i^a (\epsilon_a - \epsilon_i) + {^1{\tilde F}_{ia}^{\alpha}}$$ = (35)

$$^3\sigma_i^a({\rm RCIS}) = - ^3\sigma_{\overline{i}}^{\overline{a}}({\rm RCIS}) c_i^a (\epsilon_a - \epsilon_i) + {^3{\tilde F}_{ia}^{\alpha}},$$ = (36)