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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 ${\hat S}^2$ (i.e., a CSF). This is easy to verify by direct application of the ${\hat S}^2$ operator, which is
$\displaystyle {\hat S}^2$ = $\displaystyle {\hat S} \cdot {\hat S}
= \sum_i^N \sum_j^N {\hat s}(i) \cdot {\hat s}(j)$  
  = $\displaystyle {\hat S}_{-} {\hat S}_{+} + {\hat S}_{z} + {\hat S}_z^2,$ (46)

$\displaystyle {\hat S}_z$ = $\displaystyle \sum_i^N {\hat s}_z(i)$ (47)
$\displaystyle {\hat S}_z^2$ = $\displaystyle \sum_i^N {\hat s}_z^2(i)$ (48)
$\displaystyle {\hat S}_{\pm}$ = $\displaystyle \sum_i^N {\hat s}_{\pm}(i)$ (49)

$\displaystyle \begin{array}{ccc}
{\hat s}_z(i) a_{i}^{\dagger} \vert \rangle= \...
...verline{i}}^{\dagger} \vert \rangle= a_{i}^{\dagger} \vert \rangle.
\end{array}$     (50)

The high-spin ROHF wavefunction can be written as

\begin{displaymath}\vert \Phi_{ROHF} \rangle =
\prod_{i}^{\rm docc} a_{i}^{\da...
\prod_{t}^{\rm socc} a_{t}^{\dagger} \vert \rangle,
\end{displaymath} (51)

where t and u will denote open-shells. Applying ${\hat S}^2$, this becomes

\begin{displaymath}{\hat S}^2 \vert \Phi_{ROHF} \rangle =
{\hat S}_{-}...
\prod_{t}^{\rm socc} a_{t}^{\dagger} \vert \rangle.
\end{displaymath} (52)

This is easy to evaluate:
$\displaystyle {\hat S}_z \prod_{i}^{\rm docc} a_{i}^{\dagger} a_{\overline{i}}^{\dagger}
\prod_{t}^{\rm socc} a_{t}^{\dagger} \vert \rangle$ = $\displaystyle \left[ \frac{1}{2}(N_{\alpha} - N_{\beta}) \right]
...} a_{\overline{i}}^{\dagger}
\prod_{t}^{\rm socc} a_{t}^{\dagger} \vert \rangle$ (53)
$\displaystyle {\hat S}_z^2 \prod_{i}^{\rm docc} a_{i}^{\dagger} a_{\overline{i}}^{\dagger}
\prod_{t}^{\rm socc} a_{t}^{\dagger} \vert \rangle$ = $\displaystyle \left[ \frac{1}{2}(N_{\alpha} - N_{\beta}) \right]^2
... a_{\overline{i}}^{\dagger}
\prod_{t}^{\rm socc} a_{t}^{\dagger} \vert \rangle.$ (54)

The raising operator ${\hat S}_{+}$ yields zero when acting on $\vert \Phi_{ROHF} \rangle$: raising operators applied to $\alpha$ electrons always yield zero, and raising operators applied to the $\beta$electrons yield $\alpha$ spin orbitals which are already occupied (so the determinant vanishes by the Pauli principle). Hence the final result is
$\displaystyle {\hat S}^2 \vert \Phi_{ROHF} \rangle$ = $\displaystyle \left[ \frac{1}{4} (N_{\alpha} - N_{\beta})^2
+ \frac{1}{2} (N_{\alpha} - N_{\beta})
\right] \vert \Phi_{ROHF} \rangle$  
  = $\displaystyle \left[ \frac{1}{2} N_s \left( \frac{1}{2} N_s + 1 \right) \right]
\vert \Phi_{ROHF} \rangle,$ (55)

where $N_s = N_{\alpha} - N_{\beta}$.

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 $\beta$ 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 ${\hat S}^2$ on the determinant $\vert \Phi_i^a \rangle$:

\begin{displaymath}\left[ \hat{S}_{-} \hat{S}_{+} + {\hat S}_z + {\hat S}_z^2 \r...
\prod_t^{\rm socc} a_{t}^{\dagger} \vert \rangle.
\end{displaymath} (56)

The result of ${\hat S}_z + {\hat S}_z^2$ is easily determined to be

\begin{displaymath}\left[ {\hat S}_z + {\hat S}_z^2 \right] \vert \Phi_i^a \rang...
...ft( \frac{N_s}{2} + 1 \right) \right]
\vert \Phi_i^a \rangle.
\end{displaymath} (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.

\begin{displaymath}{\hat S}_{+} \vert \Phi_i^a \rangle = {\hat S}_{+} a_{a}^{\da...
\prod_{t}^{\rm socc} a_{t}^{\dagger} \vert \rangle.
\end{displaymath} (58)

By arguments similar to those presented above, all raising operators yield zero except for ${\hat s}_{+}(\overline{i})$. Using the anticommutation relations for creation and annihilation operators,
$\displaystyle {\hat s}_{+}(\overline{i}) a_{a}^{\dagger} a_{i} \vert \Phi_0 \rangle$ = $\displaystyle {\hat s}_{+}(\overline{i})
a_{\overline{i}}^{\dagger} a_{\...
...} a_{\overline{i}}^{\dagger}
a_{a}^{\dagger} a_{i} \vert \Phi_0 \rangle$  
  = $\displaystyle {\hat s}_{+}(\overline{i}) a_{\overline{i}}^{\dagger} a_{\overline{i}}
a_{a}^{\dagger} a_{i} \vert \Phi_0 \rangle$  
  = $\displaystyle a_{i}^{\dagger} a_{\overline{i}} a_{a}^{\dagger} a_{i} \vert \Phi_0 \rangle$  
  = $\displaystyle - a_{a}^{\dagger} a_{\overline{i}} \vert \Phi_0 \rangle.$ (59)

The ${\hat S}_{-}$ operator can now affect electrons in any of the following orbitals: a, i, and any of the open-shell orbitals.
$\displaystyle {\hat S}_{-} \left[ - a_{a}^{\dagger} a_{\overline{i}} \vert \Phi_0 \rangle
\right]$ = $\displaystyle - a_{\overline{a}}^{\dagger} a_{\overline{i}} \vert \Phi_0 \rangl...
...\dagger} a_{t} a_{\overline{t}}^{\dagger}
a_{\overline{i}} \vert \Phi_0 \rangle$  
  = $\displaystyle \vert \Phi_i^a \rangle - \vert \Phi_{\overline{i}}^{\overline{a}} \rangle
+ \sum_t^{\rm socc} \vert \Phi_{t \overline{i}}^{a
\overline{t}} \rangle.$ (60)

Thus overall,

\begin{displaymath}{\hat S}^2 \vert \Phi_i^a \rangle =
\left[ \frac{N_s}{2} \l...
...rm socc} \vert \Phi_{t \overline{i}}^{a \overline{t}} \rangle.
\end{displaymath} (61)

The analogous equation for $\vert \Phi_{\overline{i}}^{\overline{a}} \rangle$is

\begin{displaymath}{\hat S}^2 \vert \Phi_{\overline{i}}^{\overline{a}} \rangle =...
...rm socc} \vert \Phi_{t \overline{i}}^{a \overline{t}} \rangle.
\end{displaymath} (62)

Clearly a spin eigenfunction can be constructed as

\begin{displaymath}\vert ^{(N_s+1)}\Phi_i^a \rangle =
\frac{1}{\sqrt{2}} \left...
...e + \vert \Phi_{\overline{i}}^{\overline{a}} \rangle
\end{displaymath} (63)

Then the operation of ${\hat S}^2$ is

\begin{displaymath}{\hat S}^2 \vert ^{(N_s+1)}\Phi_i^a \rangle
= \left[ \frac{N...
...N_s}{2} + 1 \right) \right]
\vert ^{(N_s+1)}\Phi_i^a \rangle.
\end{displaymath} (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

$\displaystyle \begin{array}{cc}
\langle \Phi_{ROHF} \vert {\hat H} \vert \Phi_t...
...N_s+1)}\Phi_i^a \rangle = 0 &
\mbox{docc $\rightarrow$\space virt},
\end{array}$     (65)

which implies the following conditions on the Fock matrix elements
Fta = 0 (66)
$\displaystyle F_{\overline{i} \overline{t}}$ = 0 (67)
Fia = $\displaystyle -F_{\overline{i} \overline{a}}.$ (68)

Using these results, we can write down expressions for the $\sigma$vectors. Since determinants $\vert \Phi_i^a \rangle$ must enter with the same coefficients as $\vert \Phi_{\overline{i}}^{\overline{a}} \rangle$, $\sigma_0=0$ once again. Furthermore, since the ROHF reference cannot mix with any other singly excited configurations, the c0 contribution to $\sigma_{ia}$ and $\sigma_{\overline{i}\overline{a}}$ may be safely ignored. We will therefore consider four cases: $\sigma_i^a$, $\sigma_{\overline{i}}^{\overline{a}}$, $\sigma_t^a$, and $\sigma_{\overline{i}}^{\overline{t}}$, where once again t,u represent singly occupied orbitals.

The equation for $\sigma_i^a$ is readily seen to be

$\displaystyle \sigma_i^a$ = $\displaystyle \sum_{jb} c_j^b
F_{ab} \delta_{ij} - F_{ij} \delta_{ab} + ...
\langle a\overline{j} \vert\vert i\overline{b} \rangle$  
  + $\displaystyle \sum_{tb} c_t^b
-F_{it} \delta_{ab} + \langle at \vert\ver...
\langle a\overline{j} \vert\vert i\overline{t} \rangle.$ (69)

Separating the Fock operator terms from the two-electron integrals, and integrating out spin, this yields
$\displaystyle \sigma_i^a$ = $\displaystyle \sum_{jb} c_j^b \left[ F_{ab} \delta_{ij} - F_{ij} \delta_{ab} \r...
...t} \delta_{ab}
+ \sum_{jb} c_j^b \left[ 2 (ai \vert jb) - (ab \vert ji) \right]$  
  + $\displaystyle \sum_{tb} c_t^b \left[ 2 (ai \vert tb) - (ab \vert ti) \right]
+ \sum_{\overline{j}\overline{t}} c_{\overline{j}}^{\overline{t}}
(ai \vert jt),$ (70)

where we have used the relation $c_j^b = c_{\overline{j}}^{\overline{b}}$. The analogous equation for $\beta$ spins is
$\displaystyle \sigma_{\overline{i}}^{\overline{a}}$ = $\displaystyle \sum_{\overline{j}\overline{b}} c_{\overline{j}}^{\overline{b}}
... c_{\overline{j}}^{\overline{b}}
\left[ 2 (ai \vert jb) - (ab \vert ji) \right]$  
  + $\displaystyle \sum_{tb} c_t^b (ai \vert tb)
+ \sum_{\overline{j}\overline{t}} c_{\overline{j}}^{\overline{t}}
\left[ 2 (ai \vert jt) - (at \vert ji) \right],$ (71)

If the equality $c_j^b = c_{\overline{j}}^{\overline{b}}$ is to be maintained, we must have $\sigma_i^a = \sigma_{\overline{i}}^{\overline{a}}$. It is then computationally convenient to form these quantities as

\begin{displaymath}\sigma_i^a = \sigma_{\overline{i}}^{\overline{a}} =
\frac{1}{2} ( \sigma_i^a + \sigma_{\overline{i}}^{\overline{a}}).
\end{displaymath} (72)

$\displaystyle 2 \sigma_i^a = 2 \sigma_{\overline{i}}^{\overline{a}}$ = $\displaystyle \sum_{b} c_i^b \left[ F_{ab} + F_{\overline{a}\overline{b}} \righ...
...sum_{\overline{t}} c_{\overline{i}}^{\overline{t}}
  + $\displaystyle \sum_{jb} c_j^b \left[ 2 (ai \vert jb) - (ab \vert ji) \right]
+ ...
... c_{\overline{j}}^{\overline{b}}
\left[ 2 (ai \vert jb) - (ab \vert ji) \right]$  
  + $\displaystyle \sum_{tb} c_t^b \left[ 2 (ai \vert tb) - (ab \vert ti) \right]
+ ...
\left[ 2 (ai \vert jt) - (at \vert ji) \right].$ (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:
$\displaystyle {\tilde P}_{\lambda \sigma}^{DV}$ = $\displaystyle \sum_{jb} C_{\lambda j}^* c_j^b C_{\sigma b}$ (74)
$\displaystyle {\tilde P}_{\lambda \sigma}^{\overline{DV}}$ = $\displaystyle \sum_{\overline{j}\overline{b}} C_{\lambda \overline{j}}^*
c_{\overline{j}}^{\overline{b}} C_{\sigma \overline{b}}$ (75)
$\displaystyle {\tilde P}_{\lambda \sigma}^{SV}$ = $\displaystyle \sum_{tb} C_{\lambda t}^* c_t^b C_{\sigma b}$ (76)
$\displaystyle {\tilde P}_{\lambda \sigma}^{\overline{DS}}$ = $\displaystyle \sum_{\overline{j}\overline{t}} C_{\lambda \overline{j}}^*
c_{\overline{j}}^{\overline{t}} C_{\sigma \overline{t}}$ (77)
$\displaystyle {\tilde P}_{\lambda \sigma}^{+}$ = $\displaystyle \frac{1}{2} \left[
{\tilde P}_{\lambda \sigma}^{DV}
+ {\tilde P}_...
... P}_{\lambda \sigma}^{SV}
+ {\tilde P}_{\lambda \sigma}^{\overline{DS}}
\right]$ (78)

where ${\tilde P}_{\lambda \sigma}^{+}$ the same as that defined in eq. (17) of Maurice and Head-Gordon [2]. Then $\sigma_i^a$ can be evaluated as

\begin{displaymath}\sigma_i^a = \sigma_{\overline{i}}^{\overline{a}} =
+ {\tilde F}_{ia}^{+},
\end{displaymath} (79)

$\displaystyle {\tilde F}_{ia}^{+}$ = $\displaystyle \sum_{\mu \nu} C_{\mu i}^{*} {\tilde F}_{\mu \nu} C_{\nu a}$ (80)
$\displaystyle {\tilde F}_{\mu \nu}^{+}$ = $\displaystyle \sum_{\lambda \sigma}
2 (\mu \nu \vert \lambda \sigma)
- (\mu \sigma \vert \lambda \nu)
{\tilde P}_{\lambda \sigma}^{+}$ (81)

Next consider the term $\sigma_t^a$:

$\displaystyle \sigma_t^a$ = $\displaystyle \sum_{jb} \left[ \langle aj \vert\vert tb \rangle - F_{tj} \delta...
... a \overline{j} \vert\vert t \overline{u} \rangle
c_{\overline j}^{\overline u}$  
  = $\displaystyle \sum_{jb} c_j^b \left[ -F_{tj} \delta_{ab} + (at \vert jb)
- (ab ...
... \sum_{\overline{j} \overline{b}} c_{\overline{j}}^{\overline{b}}
(at \vert jb)$  
  + $\displaystyle \sum_{ub} c_u^b \left[ (at \vert ub) - (ab \vert ut)
+ F_{ab} \de...
... \sum_{\overline{j} \overline{u}} c_{\overline{j}}^{\overline{u}}
(at \vert ju)$  
  = $\displaystyle {\tilde F}_{ta} - \sum_{j} c_j^a F_{tj} + \sum_b c_t^b F_{ab}
- \sum_u c_u^a F_{tu}$ (82)

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

\begin{displaymath}{\tilde F}_{ta} = {\tilde F}_{ta}^{+}
- \frac{1}{2} \sum_{u...
...} \overline{u}}
c_{\overline{j}}^{\overline{u}} (at \vert ju)
\end{displaymath} (83)

Similar considerations apply to $\sigma_{\overline{i}}^{\overline{t}}$, which is

$\displaystyle \sigma_{\overline{i}}^{\overline{t}}$ = $\displaystyle \sum_{jb} c_j^b \langle \overline{t} j \vert\vert \overline{i} b ...
... \overline{t} \overline{j} \vert\vert \overline{i} \overline{u} \rangle
  = $\displaystyle \sum_{\overline{b}} c_{\overline{i}}^{\overline{b}}
...}} c_{\overline{j}}^{\overline{b}}
\left[ (ti \vert jb) - (tb \vert ji) \right]$  
  + $\displaystyle \sum_{ub} c_u^b (ti \vert ub)
+ \sum_{\overline{j} \overline{u}} c_{\overline{j}}^{\overline{u}}
\left[ (ti \vert ju) - (tu \vert ji) \right]$  
  = $\displaystyle \sum_{\overline{b}} c_{\overline{i}}^{\overline{b}}
F_{\overline{i} \overline{j}}
+ {\tilde F}_{\overline{i} \overline{t}}$ (84)

where ${\tilde F}_{\overline{i} \overline{t}}$ is actually evaluated according to

\begin{displaymath}{\tilde F}_{\overline{i} \overline{t}} =
{\tilde F}_{\overl...
... \overline{u}}
c_{\overline{j}}^{\overline{u}} (tu \vert ji)
\end{displaymath} (85)

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