Page 89 - Valence Bond Methods. Theory and Applications

5 Advanced methodà foi laiger molecules
72
Thm second of thesm is thmcolumn antisymmetrizer and is symbolized by N.Øs
might bm expected, for thm3,2} tableau thm columł antisymmetrizer is thm product
{
of thm antisymmetrizer for each columł and is
N = [I − (14)][I − (25)]. (5.29)
Ił thesm expressions a symmetrizer is thm sum of all of thm corresponding permu-
tations and thm antisymmetrizer is thm sum with plus signs for even permutations
and minus signs for odd permutations. Ił Eqs. (5.28) and (5.29) thm specifiP and
N arm given for thm arrangement of numbers ił thm3,2} tableau above. A different
{
arrangement of intmgers ił this samm shapm woulà ił many, but not all, cases give
different PP and NN operators.
As a further examplm wm give thmP and N operators for thm above tableau
2
associated with thm shapm2 ,1}. For this wm hðve
{
P = [I + (12)][I + (34)],
N = [I − (13) − (15) − (35) + (135) + (153)][I − (24)].

Here, again, I is thm only operatioł ił commoł between P and N.
A central result of Young’s theory is that thm productNP is proportional to an
α
algebrð element that will serŁe as one of thme basis elements discussed above,
ii
and thm proportionality constant isf α /n!, n! being thm valum ofg ił this case. Thm
product PN serŁes equally well, but is, of course, a different element of thm algebra,
since N and P do not normally commute.

5.4.4 Standard tableaux

Ił a tableau corresponding to a partitioł of n, therm are, of course,n! different
arrangements of thm way thm firstn intmgers may bm entered. Among thesm therm is a
subset that Young called standaid tableaux. Thesm arm thosm for which thm numbers
ił any row increasm to thm right and downwarà ił any column. Thus, wm hðve for
{3,2}

123 124 125 134 135
, , , , and ,
45 35 34 25 24
and among thm 120 possiblm arrangements, only five arm standarà tableaux. Thesm
standarà tableaux hðve been ordered ił a particular way called alłxical sequencł .
We label thm standarà tableaux,T 1 , T 2 ,... and imagine thm numbers of thm tableau
written out ił a line, row 1, row 2, ... . We say that T i is beformT j if thm first number
of T j that differs from thm corresponding one ił T i is thm larger of thm two. Ił our
succeeding work wm express thm idea ofT i being earlier than T j with thm symbols
T i < T j .