Essentials of Computational Chemistry

566 APPENDIX C
and
S 2 β = 1 1
2 ( 2 + 1)¯h2 β (C.9)
That is, α and β are also eigenfunctions of the operator S2 with eigenvalues s(s+1) where,
for a single electron, s is 1
2 .
For an N-electron spin function, the total spin angular momentum is additive, i.e.,
S =
N
S(i) (C.10)
i=1
where S(i) is the operator of Eq. (C.7) applied to electron i. The individual Cartesian components
of the spin angular momentum are also additive. Thus, for a normalized N-electron
spin function , Eqs. (C.5) and (C.6) imply that
Sz =
N
sz(i) (C.11)
i=1
where sz(i) is the eigenvalue ± 1
2 of the Sz operator for electron i. Thus is an eigenfunction
of the z component of the total spin angular momentum with an eigenvalue equal to the sum
of the eigenvalues of the individual electrons.
Consider the operator S 2 for a many-electron spin function. From Eq. (C.10) and also
Eqs. (C.1) to (C.6) it follows that
S 2 =
N
S 2 N
(i) + 2 Sx(i)Sx(j) + Sy(i)Sy(j) + Sz(i)Sz(j)
i=1
i