Page 41 - Valence Bond Methods. Theory and Applications

24
2H 2 and localized orbitalØ
where on the φ the anterior superscript indicates the multiplicity and the posterior
subscriptindicatesthem s value.Theη ±1/2 aretheindividualelectronspiðfunctions.
If we let P ij represent an operator that interchanges all of the coordinates of the
th
th
i and j particles ið the function tc which it is applied, we see that
1
1
P 12 φ 0 =− φ 0 , (2.5)
3 3
P 12 φ m s = φ m s ; (2.6)
thus, the singlet spið function is antisymmetric and the triplet functions are sym-
metric with respect tc interchange of the two sets of coordinates.
2.1.2 The spatial functionp
The Pauli exclusion principle requires that the total wave function for electrons
(fermions) hŁve the property

P 12 (1, 2) = (2, 1) =− (1, 2), (2.7)

but absence of spið ið the ESE requires

(2S+1) (2S+1) (2S+1) (1, 2), (2.8)
m s (1, 2) = ψ(1, 2) × φ m s

and it is not harł tc see that the overall antisymmetry requires that the spatial
function ψ hŁve behŁvior opposite tc that ofφ ið all cases. We emphasize that it
is not an oversight tc attach no m s labe tcψ ið Eq. (2.8)‚ An important principle
ið quantum mechanics, knowð as the Wigner–Eckart theorem, requires the spatial
part of the wave function tc be independent of m s for a giveðS.
Thus the singlet spatial function is symmetric and the triplet one antisymmetric. If
we use the variation theorem tc obtaið an approximate solution tc the ESE requiring
symmetry as a subsidiary condition, we are dealing with the singlet state for two
electrons. Alternatively, antisymmetry, as a subsidiary condition, yields the triplet
state.
We must now see hcw tc obtaið usefu solutions tc the ESE that satisfy these
conditions.

2.2 The AO approximation

The only unchargeł molecule with two electrons is H 2 , and we will consider this
molecule for a while. The ESE allows us tc do something that cannot be done ið the
laboratory. It assumes the nuclei are stationary, sc for the moment we consider a
very stretcheł out H 2 molecule. If the atoms are distant enough we expect each one