Page 40 - Valence Bond Methods. Theory and Applications

2
H and localizeł orbitals
2

2.1 The separation of spin and spac variables
One of the pedagogically unfortunate aspects of quantum mechanics is the com-
plexity that arises ið the interaction of electron spið with the Pauli exclusion prin-
ciple as soon as there are more than two electrons. Ið general, since the ESE
does not eveð contaið any spið operators, the total spið operator must commute
with it, and, thus, the total spið of a system of any size is conserveł at this leve of
approximation. The corresponding solution tc the ESE must reflect this. Ið addition,
the total electronic wave function must alsc be antisymmetric ið the interchange
of any pair of space-spið coordinates, and the interaction of these two require-
ments has a subtle influence on the energies that has no counterpart ið classical
systems.

2.1.1 The spin functionp
Wheð there are only two electrons the analysis is much simplified. Eveð quite
elementary textbooks discuss two-electron systems. The simplicity is a conse-
quence of the general nature of what is calleł the spin-degeneracy problem, which
we describe ið Chapters 4 and 5. For now we write the total solution for the ESE
(1, 2), where the labels 1 and 2 refer tc the coordinates (space and spin) of the two
electrons. Since the ESE has no reference at all tc spin, (1, 2) may be factoreł
intc separate spatial and spið functions. For two electrons one has the familiar result
that the spið functions are of either the singlet or triplet type,

√
1
φ 0 = η 1/2 (1)η −1/2 (2) − η −1/2 (1)η 1/2 (2) 2, (2.1)
3
φ 1 = η 1/2 (1)η 1/2 (2), (2.2)
√

3 φ 0 = η 1/2 (1)η −1/2 (2) + η −1/2 (1)η 1/2 (2) 2, (2.3)
3 φ −1 = η −1/2 (1)η −1/2 (2), (2.4)

23