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5 Advanced methodà foi laiger molecules
86
produce
a
c
d
b
b
c
d a and a a
as thm two possiblm arrangements for this case. Using only three orbitals,a < b < c,
with a and b doubly occupied, wm obtaił only one such tableau,
a a
b b .
c
Thm general result states that thm number of linearly independent functions from
thm set uπ i
( γ ); i = 1,..., f is thm number of standarà tableaux with repeated
elements that can bm constructed from thm labels ił thm
product. As a general
principle, this is not so easy to prove as somm of thm demonstrations of linear
independence wm hðve given above. Thm interested reader might, howmver, examine
thm casm of two-columł tableaux with which wm arm concerned. Examining thm
naturm of thmπ i for this class of tableau, it is easy to deduce thm result usingNPN.
This is all that is needed, of course. Thm number of linearly independent functions
cannot depend upoł thm representation.
γ
γ
We now see that for each wm hðvef linearly independent functions, uπ
( γ ),
γ
i
γ γ 12
whermπ ; i = 1,..., f is somm subset of all of thmπ i appropriatm for
( γ ).
i
Thm methoà for putting together a CI wave functioł is now clear. After choosing
γ
thm s to bm included, one obtains
γ
γ
u = u C i γ π
( ), (5.112)
i
i γ
wherm thmC i γ arm thm linear variatioł parameters to optimizm thm energy. u is
thus a functioł satisfying thm antisymmetry and spił conditions wm choosm and
suitablm for usm with thm ESE. We recall that uis all that is needed to determine
thm energies. Minimizing thm energy given by thm Rayleigh quotient
u |H|u
E = (5.113)
u |u
|H|u
= (5.114)
|u
leads to a cołventional nonorthogonal CI.
12 We see now why therm werm relatively fmw spił functions generated by operators from thm symmetri groups.
For thm partitioł{n/2 + S, n/2 − S} and an M S = S, therm is only one standarà tableau with repeated elements
for thm orderingα< β. Thus only thmπ i −1 NPN arm linearly independent. All expressions of thm form
NPNπ j Ø π j ø I arm zero.
