Essentials of Computational Chemistry

Appendix C
Spin Algebra
C.1 Spin Operators
Electrons (and many other particles) have associated with them an intrinsic angular momentum
that has come to be called ‘spin’. One of the greatest successes of relativistic quantum
mechanics is that spin is seen to arise naturally within the relativistic formalism, and does
not need to be added post facto as it is in non-relativistic treatments. As with orbital angular
momentum, spin angular momentum has x, y, andz components, and the operators Sx, Sy,
and Sz, together with orthonormal eigenfunctions α and β of electron spin, are defined from
Sxα = 1
2¯hβ (C.1)
Sxβ = 1
2¯hα (C.2)
Syα = 1i¯hβ
2 (C.3)
Syβ =−1i¯hα 2 (C.4)
Szα = 1
2¯hα (C.5)
Szβ =−1 2¯hβ (C.6)
where i = √ −1.
Thus, α and β are eigenfunctions of the operator Sz, with eigenvalues of 1/2 and −1/2,
respectively, in atomic units (recall that the value of ¯h is 1 in atomic units, see Table 1.1).
The spin operator S is defined by
S = Sx + Sy + Sz
and repeated application of Eqs. (C.1) through (C.6) reveals that
Essentials of Computational Chemistry, 2nd Edition Christopher J. Cramer
© 2004 John Wiley & Sons, Ltd ISBNs: 0-470-09181-9 (cased); 0-470-09182-7 (pbk)
(C.7)
S 2 α = 1 1
( 2 2 + 1)¯h2 α (C.8)