Restricted Active Space CI

The Restricted Active Space (RAS) CI method was introduced by Olsen,  Roos,  Jø rgensen,  and Aa. Jensen  [16] in 1988. The RAS method calls for the partitioning of the one-electron basis into four subsets. The first subset consists of the core orbitals, which are constrained to remain doubly-occupied. The remaining three subsets are labelled I, II, and III, and the CI space is limited by requiring a minimum of p electrons in RAS I and a maximum of q electrons in RAS III. There are no restrictions on the number of electrons in RAS II, and thus it is analogous to the complete active space (CAS).  The full CI can be obtained as the maximum limit of the RAS space. Interestingly, although the main focus of the paper is on the utility of the RAS method in limiting the size of CI calculations, the maximum impact of this paper has been on the development of determinant-based full CI   algorithms [19,20].

The RAS CI method relies on Handy's separation of determinants into alpha and beta strings  (see section 6.2). As in other determinant-based CI methods, the basis determinants are restricted to those having a given value of M_s. Since the number of electrons N is also fixed, this means that the alpha and beta strings always have constant lengths of $n_{\alpha}$ and $n_{\beta}$, respectively. For a full CI,  one forms all possible alpha and beta strings for a given $n_{\alpha}$ and $n_{\beta}$, and the basis determinants are all possible combinations of these alpha and beta strings. In a RAS CI, the CI space is restricted in two ways: first, not all alpha and beta strings are allowed, and secondly, not all combinations of alpha and beta strings to form determinants are accepted. This is best seen from an example: consider the case of 6 orbitals, with $n_{\alpha} = n_{\beta} = 3$. If orbitals 4, 5, and 6 constitute RAS III, with a maximum of 2 electrons allowed, then clearly alpha strings such as $a_{4\alpha}^{\dagger} a_{5\alpha}^{\dagger} a_{6\alpha}^{\dagger}$ are not allowed. Similarly, even though $a_{1\alpha}^{\dagger} a_{4\alpha}^{\dagger} a_{5\alpha}^{\dagger}$ and $a_{1\beta}^{\dagger} a_{4\beta}^{\dagger} a_{5\beta}^{\dagger}$ are allowed alpha and beta strings, these strings cannot be combined with each other because the resulting determinant would represent a quadruple excition into RAS III.

If we employ a graphical description of the alpha and beta strings as described in section 6.2.1, in general we require one graph for alpha strings and one graph for beta strings. However, in the case that M_s = 0, only one graph is needed because the alpha string and beta string graphs are identical. As previously mentioned, for a RAS space not all alpha and beta strings can be freely combined. While it would be possible to create a table listing all allowed combinations of alpha and beta strings, there is a more efficient way around this difficulty. Instead of using a single graph to represent all alpha (or beta) strings, instead we use several graphs. For example, we might use one graph for all strings with no electrons in RAS III, one graph for all strings with one electron in RAS III, and one graph for all strings with two electrons in RAS III. In this case, the restrictions on combinations of strings become restrictions on combinations of graphs--a more efficient treatment computationally. Figure 4 displays string graphs for $n_{\alpha} = n_{\beta} = 3$ and n = 6 for at most 2 electrons in RAS III, orbitals 4-6. Graph (a) represents all walks with two electrons in RAS III; graph (b) gives all walks with one electron in RAS III; and graph (c) gives the one walk with no electrons in RAS III. If only two electrons are allowed in RAS III, it is clear that alpha strings of graph a can be combined only with the beta string from graph (c), and alpha strings of graph (b) can be combined with beta strings of graphs (b) and (c). An alpha string from graph (c) can be combined with beta strings from graphs (a), (b), and (c).

  

Figure: String graphs for $n_{\alpha / \beta } = 3, n = 6$ with at most two electrons in RAS III of orbitals 4-6. Vertex weights and arc weights are given for lexical ordering.