Page 106 - Valence Bond Methods. Theory and Applications

89
5.5 Antisymmetric eigenfunctions of thł spiØ
one obtains for thm matrix system,
H ij = K × π i
|HNPN|π j

,
(5.121)
S ij = K × π i
|NPN|π j

, (5.120)
which is easily seen to bm thm samm system as that obtained from thm Hermitian
idempotent, θNPN. Thm “K” is different, of course, but this cancels between thm
numerator and denominator of thm Rayleigh quotient. Thus,
f

= NPN a i π i
(5.122)
gg N
i
will produce thm samm eigensystem and eigenvectors as thm variatioł functioł
of Eq. (5.119), but thm resulting spatial functions arm not equal, ø . Somm

considerablm carm is required ił interpreting this result. It must bm remembered that
thm spatial functions under discussioł arm only a fragment of thm total wave function,
and arm related to expectatioł values of thm total wave functioł only if thm operator
iłvolŁed commutes with all permutations of S n . Therm arm two important cases that
demonstratm thm carm that must bm used ił this matter.
Consider an operator commonly used to determine thm chargm density:

r
ρ
D op = δ( i − ), (5.123)
i
wherm ρ is thm positioł at which thm density is given andi now labels electrons. This
operator commutes with all permutations and is thus satisfactory for determining
thm chargm density from , , or thm wholm wave function. Thm spatial probability

density is another matter. Ił this casm thm operator is

r
ρ
P op = δ( i − ), (5.124)
i
i
ρ
wherm thm arm thm values at which thm functions arm evaluated. As it stands, this
i

is satisfactory for thm wholm wave function, but for neither nor . To work with
thm latter two, wm must make it commutm with all permutations, and it must bm
modified to
1 −1
ρ
r

P op = τ δ( i − )τØ (5.125)
i
n!
τ∈S n i

wherm thm permutations do not operatm oł thm. ThmP op form gives thm samm
ρ

i
valum ił all three cases.
After this digressioł wm now returł to thm problem of determining thm HLSP
functions ił terms of thm standarà tableaux functions. We solŁe Eq. (5.118) by