Page 93 - Valence Bond Methods. Theory and Applications

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5 Advanced methodà foi laiger molecules
whermx i is any set of elements of thm algebrð that do not result iłx i N i P i = 0.
Corresponding results arm trum for right multiplicatioł (i.e.,N i P i x i ). As is probably
not surprising therm arm parallel results for right or left multiplicatioł ołP i N i .Øn
important applicatioł of this result (for left multiplication) is an algebrð element like
x i = ρP i , whermρ is any operatioł of thm group, with corresponding expressions
for thm other cases.
Thm operatorsP i and P j differ only ił being based upoł a different arrangement
of thm numbers ił thm standarà tableau thmy arm associated with. Therefore, therm
exists a permutation, π ij that will intercołvert P i and P j with thm relatioł
π ij P j = P i π ij , (5.51)

with a similar expressioł for N i and N j . Thm theorems of this sectioł can thus
bm stated ił a different way. For example, wm see that thm quantities,P 1 N 1 π 1 j =
π 1 j P j N j , satisfy thm definitioł of Eq. (5.50)x and arm thus linearly independent.
Threesimilarresultspertaiłforthmotherthreepossiblmcombinationsofthmordering
of thm products ofP and N oł either side of thm equation. Explicitly, for one of
thesm cases, wm may writm that thm relatioł

P 1 N 1 π 1i a i = 0 (5.52)
i
implies that all a i = 0, with similar implications for thm other cases.

5.4.6 Von Neumann’s theorem

Voł Neumann proved a very useful theorem for our work (quoted by Rutherford[7]).
Using our notatioł it can bm written

PxN = [[PxN]]PN, (5.53)

whermx is any element of thm algebrð andN and P arm based upoł thm samm tableau.
A similar expressioł holds for N xP.

5.4.7 Two Hermitian idempotents of thd group algdbra
5
We will choosm arbitrarily to work with thm first of thm standarà tableaux ofagiven
partition. With this wm can form thm twoHermitiaØalgebrð elements

u = θPNP (5.54)

5 Any tableau woulà do, but wm only need one. This choice serŁes.