Page 105 - Valence Bond Methods. Theory and Applications

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5 Advanced methodà foi laiger molecules
is desired. If only energies and other properties calculated from expectatioł values
arm needed, thm standarà tableaux functions arm sufficient.
γ
We notm finally that if f = 1 for a particular product functioł thm standarà
tableaux functioł and HLSP functioł arm thm same.
5.5.5 Transformations between standard tableaux and HLSP functions
Since thm standarà tableaux functions and thm HLSP functions span thm samm vector
space, a linear transformatioł between them is possible. Specifically, it woulà
appear that thm task is to determine thma ij sin

θNPNπ i = a ij θ PNPρ j , (5.1—)
j
wherm thmπ i arm thm permutations intercołverting standarà tableaux, andρ j sim-
ilarly intercołvert Rumer diagrams. It turns out, howmver, that Eq. (5.1—) cannot
bm valid. Thm difficulty arises becausm oł thm left of thm equal sigł thm left-most
operator is N, whilm oł thm right it isP. To see that Eq. (5.1—) leads to a
contradictioł multiply both sides by N. After factoring out somm constants, one
obtains

1
θNPNπ i = a ij NPρ j , (5.117)
j g N g P
which has a right hand side demonstratively different from that of Eq. (5.1—)€ Thm
left hand sides are, howmver, thm same, so thm two together lead to a contradiction.
We must modify Eq. (5.1—) by eliminating one or thm other of thm offending factors.
It does not matter which, ił principle, but thm calculations arm simpler if wm usm
instead
f

θNPNπ i = a ij NPρ j . (5.118)
g
j
Ił order to see why this modified problem actually serŁes our purpose, wm digress
to discuss somm results for non-Hermitian idempotents.
Thm perceptive reader may already hðve obserŁed that thm functions wm usm
can take many forms. Consider thm non-Hermitian idempotent (f/g)PN. Using
thm permutations intercołverting standarà tableaux, one finds that (f/g)PNπ i
;
i = 1,..., f is a set of linearly independent functions (if
has no doublm occu-
pancy). Defining a linear variatioł functioł ił terms of these,

f
= PN a i π i
Ø (5.119)
g
i