Alpha and Beta Strings

Although Handy  was the first to use alpha and beta strings, we will use the notation of Olsen   et al. [16]. We define an alpha string as an ordered product of creation operators for spin orbitals with alpha spin. If $I_{\alpha}$ contains a list $\{i, j, \ldots n\}$ of the $n_{\alpha}$ occupied spin orbitals with alpha spin in determinant $\vert I \rangle$, then the alpha string $\alpha(I_{\alpha})$ is $a_{i \alpha}^{\dagger} a_{j \alpha}^{\dagger} \ldots a_{n \alpha}^{\dagger}$. A beta string is defined similarly. Thus we can rewrite Slater determinant $\vert I \rangle$ in terms of alpha and beta strings.

$$\vert I \rangle = \vert \alpha(I_{\alpha}) \beta(I_{\beta}) \rangle = \alpha(I_{\alpha}) \beta(I_{\beta}) \vert \rangle$$ (tex2html_deferredtex2html_deferred6.tex2html_deferred3 | I = | (I_) (I_) = (I_) (I_) | )

For instance, suppose we have the Slater determinant $\vert I \rangle = \vert \phi_{1\alpha} \phi_{2\alpha} \phi_{3\alpha} \phi_{1\beta} \phi_{2\beta} \phi_{4\beta} \rangle$. Then the alpha string $\alpha(I_{\alpha})$ is given by

$$\alpha(I_{\alpha}) = a_{1\alpha}^{\dagger} a_{2\alpha}^{\dagger} a_{3\alpha}^{\dagger}$$ (tex2html_deferredtex2html_deferred6.tex2html_deferred4(I_) = a_1^ a_2^ a_3^ )

and the beta string is given by

$$\beta(I_{\beta}) = a_{1\beta}^{\dagger} a_{2\beta}^{\dagger} a_{4\beta}^{\dagger}$$ (tex2html_deferredtex2html_deferred6.tex2html_deferred5(I_) = a_1^ a_2^ a_4^ )

Note that the order of the creation operators matters; if we swap the order of two creation operators within the alpha string (or within the beta string), then we introduce a sign change (see equation 5.2). Also, acting the alpha string on the vacuum first, rather than the beta string, may introduce a minus sign, depending on the number of alpha and beta electrons. Although the order of the orbitals and whether the alpha or beta string acts first is of no real consequence, we must be sure to keep our use of alpha and beta strings consistent, or sign problems will result. In most of the literature, and in these notes, the beta string will be placed to the right of the alpha string in equations like (6.3). Further, within each string, orbitals are listed in strictly increasing order.

Handy  realized the following advantages to alpha and beta strings:

1.

Direct CI  methods often require an index vector which points to a list of all allowed excitations from a given N-electron basis function. Using alpha and beta strings, the index vector need not be the length of the CI vector--its size is dictated by the number of alpha or beta strings, which is approximately the square root of the number of determinants. This results from the fact that (in determinant-based CI) electrons in alpha spin-orbitals can be excited only to other alpha spin-orbitals, and electrons in beta spin-orbitals can be excited only to other beta spin-orbitals.⁵

2.

To form $r(I_{\alpha}, I_{\beta})$ in equation (6.2), all functions $\vert \alpha(J_{\alpha}) \beta(J_{\beta}) \rangle$ which have non-zero matrix elements with $\vert \alpha(I_{\alpha}) \beta(I_{\beta}) \rangle$ are generated, one at a time, with the appropriate integral being looked up and multiplied by the appropriate CI coefficient. No time is wasted considering determinants which are noninteracting, and the coefficients of the integrals are simply $\pm 1$.

3.

Efficiency is increased by realizing that all integrals which enter the expression $\langle \alpha(I_{\alpha}) \beta(I_{\beta}) \vert \hat{H} \vert \alpha(J_{\alpha}) \beta(I_{\beta}) \rangle$ (equation 6.2), where $\alpha(J_{\alpha})$ differs from $\alpha(I_{\alpha})$ by two orbitals, are independent of $\beta(I_{\beta})$.

We will make these points more clear in our discussion of the RAS CI,  which is a direct extension of Handy's  observations concerning alpha and beta strings. However, at this point we will proceed to discuss the graphical representation of alpha and beta strings.