Restricted Active Space CI

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
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 *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
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).