S^{2} = S_{x}^{2} + S_{y}^{2}
+ S_{z}^{2} 
I^{+} = (S_{x}
+ iS_{y}) 
I^{+}β =
α 
I^{+}α =
0 
I^{} = (S_{x}
 iS_{y}) 
I^{}α = β 
I^{}β =
0 
S_{x}^{2}+S_{y}^{2} 
= 
(I^{+}I^{})+i(S_{x}S_{y}S_{y}S_{x}) 

= 
(I^{}I^{+})+i(S_{y}S_{x}S_{x}S_{y}) 

= 
½(I^{+}I^{
}+ I^{}I^{+}) 
and finally
i(S_{y}S_{x}  S_{x} S_{y} ) = S_{z}.
For any microstate Ψ,
the expectation value of the S^{2} operator is
given by
<S^{2}> = <ΨS_{z}^{2}
+ S_{y}^{2} + S_{x}^{2}Ψ>
The first part of this expression is obvious, vis:
<ΨS_{z}^{2}Ψ>
= ¼(N^{α}
+ N^{β})
However, the effect of
S_{y}^{2}+
S_{x}^{2} is not so simple. By making use of the
fact that the operators involve two electrons, a large number of integrals
resulting from the expansion of the Slater determinants can be readily
eliminated. The only integrals which are not zero due to the orthogonality of
the eigenvectors, i.e., those which may be finite due to the spin operators,
are
Using the relationships already defined, this expression
simplifies [
65] as follows:
or,
Recall that p is the number of α
electrons, and q, the number of β
electrons. This expression simplifies to yield
For the general case, in which the state function Φ,
is a
linear combination of microstates, the expectation value of
S is more complicated:
As with the construction of the C.I. matrix, the elements of this expression
can be divided into a small number of different types:
 1. Ψ_{a}=Ψ_{b}:
Since the two wavefunctions are the same, this
corresponds to the expectation value of a microstate, and has already been
derived.
 2. Except for ψ_{i}
in Ψ_{a}
and ψ_{j}
in Ψ_{b};
Ψ_{a}=Ψ_{b}:
Assuming ψ_{i}
and ψ_{j}
to have alphaspin the expectation
value is
The effect of the spin operator is to change the spin of the electrons but
leave the space part unchanged. All integrals vanish identically due to one or
more of the following identities:
<ψ_{i}ψ_{j}> 
= 
<m_{i}m_{i}> 
= 
Δ(i,j) 
<ψ_{i}ψ_{k}> 
= 
Δ(i,k) 


<ψ_{i}ψ_{k}> 
= 
Δ(j,k) 


Therefore, <Ψ_{a}S^{2}Ψ_{b}>
= 0.  3. Except for ψ_{i}
and ψ_{j}
in Ψ_{a}
and ψ_{k}
and ψ_{l }in
Ψ_{b};
Ψ_{a=}Ψ_{b}.
Two situations exist: (a) when all four M.O.s
are of the same spin; and (b) when two are of each spin.
When all four M.O.s have the same spin, the effect of the spin operator is to
reverse the spin of two M.O.s in the ket half of the integral. By spin
orthogonality this results in an integral value of zero.
In the case where two M.O.s are of α
spin and two are of β
spin,
the matrix elements, after elimination of those terms which are zero due to
space orthogonality, are
The effect of S^{2} on ψ_{k}
and ψ_{l}
is to reverse the spin of these
functions; this gives
where ψ'
has the opposite spin to that of ψ.
Thus, only if ψ_{i}
and ψ_{j}
are spatially identical with ψ_{k}
and ψ_{l }will
<Ψ_{a}S^{2}Ψ_{b}>
be nonzero. The phasefactor W is such
that if i=k and j=l then W=1, and if i=l and j=k then W=1; for all
other cases the matrix element is zero, so the phase of W is irrelevant. For
these two cases, the matrix element is
<Ψ_{a}S^{2}Ψ_{b}>
= 1
if
,
otherwise
<Ψ_{a}S^{2}Ψ_{b}>
= 0.
 4.
If more than two differences exist,
<Ψ_{a}S^{2}Ψ_{b}>
= 0.