We now turn our attention to the general formulation of the full
CI problem in terms of alpha and beta strings. Later on, we will
consider how to modify our results for RAS CI's.
We begin by describing Olsen's expressions for 
, which is the
action of the Hamiltonian 
on the CI vector (or matrix)
. First we express 
in
second-quantized
form (see section 5).
where 
is the shift operator
Again, 
is given by
As we will proceed to show, 
can be split up into three terms:
one involving only beta components of the linear group generators
(
),
one involving only alpha components of the generators (
), and one
involving mixtures of the two (
). Inserting equation
( 6.16) into equation ( 6.18) yields
Now expanding the shift operators according to equation ( 6.17),
we write 
as a sum of three terms
where
and
and
Obviously, these expressions can be simplified further.
First, we observe that the expression for 
contains no 
operators. Therefore, 
unless 
.
Using this fact, and integrating out the 
part, we obtain
Taking out the Kronecker delta term and rearranging, we have
Now the second term can be combined with the first to give equation (9a) of reference [16].
Similarly, 
can be simplified to
For efficient implementation, it is convenient to precompute the quantities
Finally, we simplify 
. It may be rewritten as
Since we sum over all ijkl, we can permute i and j with k and
l. We can also swap 
and 
as can easily be verified. This yields
equation (9c) from reference [16].
Thus we have written the action of the Hamiltonian on the current
CI vector in terms of alpha and beta strings and alpha and beta shift
operators. The product 
is written as a sum of three terms:
the first (
) involves only beta shift operators, the second
(
) involves only alpha shift operators, and the third
(
) involves both alpha and beta shift operators. Note that,
except for the factor 
, 
is
independent of 
so that the algorithm for computing
is vectorizable (see below). Analogous results hold for
. This situation does not obtain in the computation of
, however, which is the rate-limiting step.
