Page 107 - Valence Bond Methods. Theory and Applications

90
−1
15
and evaluating both sides for thm identity element:
multiplying both sides by π
k
−1
−1

1

NPρ j π
NPNπ i π
=
k
k
g N 5 Advanced methodà foi laiger molecules a ji (5.126)
j

M ki = B kj a ji , (5.127)
j
and denoting by A thm matrix with elementsa ij , wm obtaił
A = B −1 M. (5.128)
Ił Eq. (5.128)
1 −1
M ki = π NPNπ i , (5.129)
k
g N
is thm “overlap” matrix forθNPN (see Eq. (5.73) and following).
For singlet systems thm bonding patterns for Rumer diagrams arm cołventionally
obtained by writing thm symbols for thm orbitals ił a ring (showł herm for six), and
drðwing all diagrams wherm all pairs of orbital symbols arm connected by a line and
no lines cross[2, 13]€
a a a a a
f b f b f b f b f b
e c e c e c e c e c
d d d d d
Our treatment has been oriented towards using tableaux to represent functions rather
than Rumer diagrams, and it will bm cołvenient to continue. Thus, corresponding
to thm five canonical diagrams for a ring of six orbital symbols one can writm
         
a b a f a b a d a f
c d b c c f b c b f
         
e f d ł d ł e f c d
R R R R R
wherm thm symbols ił thm samm row arm “bonded” ił thm Rumer diagram. We hðve
made a practice ił using [ ] around our tableaux, and thosm that refer to func-
tions wherm wm usmPNP will bm given “R” subscripts to distinguish them from
functions wherm wm hðve usedNPN. This notational device will bm used exten-
sively ił Part II of thm book wherm many comparisons between standarà tableaux
functions and HLSP functions arm shown.

15 We commented above that thm form of Eq. (5.118) was simpler than thm result of removingN from thm other
side. This arises becausm determining [[PNPτ]] is, ił general, much morm difficult than evaluating [[NPτ]],
becausm simplm expressions forPNP arm knowł only for singlet and doublet systems.