The Restricted Active Space (RAS) CI method was introduced by Olsen, Roos, Jø rgensen, and Aa. Jensen [17] 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 
. Since the number of electrons N is
also fixed, this means that the alpha and beta strings always have constant
lengths of 
and 
, respectively. For a
full CI, one
forms all possible alpha and beta strings for a given and
, 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 
. If orbitals 4, 5, and 6
constitute
RAS III, with a maximum of 2 electrons allowed, then clearly alpha strings
such as 
are not allowed.
Similarly, even though
and 
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 
, 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 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 4: String graphs for 
with at most two
electrons in RAS III of orbitals 4-6. Vertex weights and arc weights
are given for lexical ordering.
Equations