Extensions of CIS to Include Certain Double Substitutions (XCIS)

The ROCIS method appears to be superior to UCIS for open-shell molecules [2]. Nevertheless, ROCIS is not as reliable for open-shell cases as RCIS is for closed-shell cases. As explained by Maurice and Head-Gordon [7], a careful analysis of the failures of ROCIS indicated that certain double substitutions which are neglected in UCIS and ROCIS can of crucial importance in open-shell systems. The spin-adapted configurations are of the form:

$$\vert {\tilde \Phi}_i^a(1) \rangle = \frac{1}{\sqrt{6}} \l... ...sqrt{6}} \vert \Phi_{t \overline{i}}^{a \overline{t}} \rangle.$$ (86)

Although the third determinant is a double substitution as far as spin-orbitals are concerned, it is only a single substitution when spatial orbital occupations are considered. Hence, it is very reasonable to assume that this CSF may be of comparable importance to the singles included in ROCIS. The extended CIS method (XCIS) is a spin-adapted CI method including these ``extended single'' substitutions.

It is helpful to first verify that eq. (86) is indeed an eigenfunction of ${\hat S}^2$. Using previous results for ROCIS, it is trivial to see that

$${\hat S}^2 \left( \vert \Phi_{\overline{i}}^{\overline{a}} ... ...t^{socc} \vert \Phi_{t \overline{i}}^{a \overline{t}} \rangle.$$ (87)

It remains to be seen what is the effect of ${\hat S}^2$ acting on $\vert \Phi_{t \overline{i}}^{a \overline{t}} \rangle$.

$${\hat S}^2 \vert \Phi_{t \overline{i}}^{a \overline{t}} \rangle {\hat S}^2 a_{a}^{\dagger} a_{\overline{t}}^{\dagger} a_{\overlin... ...dagger} a_{\overline{i}}^{\dagger} \prod_u^{socc} a_{u}^{\dagger} \vert \rangle$$ =

$${\hat S}_{-} {\hat S}_{+} \vert \Phi_{t \overline{i}}^{a \overlin... ...s}{2} + 1 \right) \right] \vert \Phi_{t \overline{i}}^{a \overline{t}} \rangle.$$ = (88)

The factor ${\hat S}_{+}$ acts on unpaired $\beta$ spins; thus,

$${\hat S}_{+} a_{a}^{\dagger} a_{\overline{t}}^{\dagger} a_{\overline{i}} a_{t} \vert \Phi_0 \rangle {\hat s}_{+}(\overline{t}) a_{a}^{\dagger} a_{\overline{t}}^{\dagger} a_{\overline{i}} a_{t} \vert \Phi_0 \rangle$$ =

$$- a_{t}^{\dagger} a_{a}^{\dagger} a_{\overline{i}} a_{t} \vert \Phi_0 \rangle$$ =

$$- a_{a}^{\dagger} a_{\overline{i}} \vert \Phi_0 \rangle.$$ = (89)

The result of $- {\hat S}_{-} a_{a}^{\dagger} a_{\overline{i}} \vert \Phi_0 \rangle$ has already been worked out in eq. (60). Thus overall,

$${\hat S}^2 \vert \Phi_{t \overline{i}}^{a \overline{t}} \rang... ...+ \sum_u \vert \Phi_{u \overline{i}}^{a \overline{u}} \rangle,$$ (90)

and it is easy to see that

$${\hat S}^2 \left( \vert \Phi_{\overline{i}}^{\overline{a}... ...\vert \Phi_{t \overline{i}}^{a \overline{t}} \rangle \right).$$ (91)

Hence $\vert {\tilde \Phi}_i^a(1) \rangle$ (eq. 86) is a CSF.