Denote the MOs for a closed-shell system by 
. Then the total energy
is
where 
and 
is the kinetic
energy and nuclear attraction energy operator. 
and 
are
coulomb and exchange integrals:
where
Now the symmetry of the MOs is inserted. The MOs, 
,
are re-indexed by irreducible representation ( 
, 
, 
),
subspecies ( 
, 
, ) and a third index (i, j, )
to number MOs whose other two indices are the same. The indices
are analogous to the H atom 
indices except that i
always starts at one.
The expansion of the MOs in terms of basis functions is also put in at this
point. They are symmetry orbitals, 
, and are indexed by
irreducible representation, subspecies, and, again, a third index (p, q,
) to number functions whose other two indices are the same. The MO
coefficients are 
and, very importantly, don't depend on
.
The total energy is now
Since the i-type indices only appear in the form 
, the energy can be expressed in terms of the density matrix
where 
is the dimension of the irreducible representation
and, accordingly, 
is the occupation number of the 
shell.
The number of terms in the summations can be reduced by using the fact that the
density matrix 
is symmetric to interchange of p and q.
Thus, each of the pq-type sums in the energy expression may be subdivided
into p>q, p=q, and p<q terms. Then the p<q terms have their p and
q indices interchanged and are combined with the p>q terms. The p=q
terms are then incorporated into the same expression by the use of factors of
the form 
.
This is the energy expression in terms of the minimum number of independent integrals. Thus it is convenient to define notation for these (shell-) averaged one-electron, coulomb, exchange, and P integrals:
If these integrals are to be evaluated by actually performing the averaging
summations, some simplifications can be used. All the terms in the
sum are equal, and the coulomb and exchange integrals only
need to be averaged over either or , not both. The expressions
given above, however, have the useful property that, when the summations and
integrations are interchanged, the integrand contains only a totally symmetric
part. This is necessary in one method [13] of computing the integrals
when AO expansions are used.
For energy expressions it is simpler to write the summations as pq rather
than 
with factors. This will be done from this
point on with the understanding that when calculations are actually carried
out, elements will only be computed with , those with
p>q will be doubled, and the energy summations will be done for only.
