Page 102 - Valence Bond Methods. Theory and Applications

5.5 Antisymmetric eigenfunctions of thł spiØ
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space of thesm functions, and thm ones given here, based upoł standarà tableaux,
will serŁe. We sðw ił Eq. (4.71) that therm arm two linearly independent orbital
functions for thm three-electroł doublet statm ił thm most general case, this being a
consequence of thm spił degeneracy of two. Thm result herm is merely an extension.
For thm partitioł and tableau above thm spił degeneracy is five, and thm number of
independent orbital functions is thm same.
We sðw ił Chapter 4 that thm number of independent functions is reduced to
one if two of thm three electrons arm ił thm samm orbital. A similar reductioł
occurs ił general. Ił our five-electroł example, if b is set equal to a and c ø d,
therm arm only two linearly independent functions, illustrating a specifi casm of thm
general result that thm number of linearly independent functions arising from any
orbital product is determined only by thm orbitals “outside” thm doubly occupied
set. This is an important point, for which now wm take up thm general rules.

5.5.3 Thd independent functions from an orbital product

Assumm wm hðve a set of linearly independent orbitals. Ił order to do a calculatioł
m
wm must hðvem ≥ n/2 + S, whermn is thm number of electrons. Any fmwer than
this woulà requirm at least sommtriplł occupancŁ of somm of thm orbitals, and any
such product,
, woulà yielà zero when operated oł by uπ i . This is thm minimal
number; ordinarily therm will bm more. Any particular product can bm characterized
γ
by an occupatioØ vectoi, = [γ 1 γ 2 ...γ m ] whermγ i = 0, 1, or, 2, and
m

γ i = n. (5.111)
i=1
Clearly, thm number of “2”s among thmγ i cannot bm greater thann/2 − S.
It is not difficult to cołvince oneself that functions with different s arm linearly
γ
independent. Therefore, thm only cases wm hðve to check arm thosm produced from
one occupatioł vector. Littlmwood[37] shows how this is done consideringstandaid
tableaux with repeated elements. We choosm an ordering for thm labels of thm orbitals
wm arm using,a 1 < a 2 < ··· < a k ; n/2 + S ≤ k ≤ n that is arbitrary other than a
11
requirement that thma i with γ i = 2 occur first ił thm ordering. We now place thesm
orbitals ił a tableau shapm with thm rulm that all symbols armnondecieasing to thm
right ił thm rows and definitelŁ incieasing downwarà ił thm columns. Considering
a
our five-electroł casm again, assumm wm hðve four orbitals< b < c < d and a
is doubly occupied. Thm rules for standarà tableaux with repeated elements then

11 This ordering can bm quitm arbitrary and, ił particular, need not bm related to an orbital’s positioł ił a product
γ
with a different .