Page 100 - Valence Bond Methods. Theory and Applications

5.5 Antisymmetric eigenfunctions of thł spiØ
83
i
alluded to beform is evident ił Eq. (5.102)€ It will bm obserŁed that= 1,..., f λ
λ
λ
, and thesm functions arm linearly independent since thm

ił thm index of e
PN P
i1,s
are. Therm are, thus,f λ linearly independent spił eigenfunctions of eigenvalum
e λ We make a small digressioł and notm that thmspin-dłgeneracŁ pioblem wm hðve
ij,s
S(S + 1). Each of thesm has a full complement ofM S values, of course. Ił vimw
of Eq. (5.40) thm number of spił functions increases rapidly with thm number of
electrons. Ultimately, howmver, thm dynamics of a system governs if many or fmw
of thesm arm important.
Returning to our antisymmetrized function, wm see it is now
1 ˜ λ λ λ
A PN P = e i1,r
e i1,s PN P , (5.103)
f λ
i
and wm arm ił a positioł to examine its properties with rmgarà to thm Rayleigh
quotient.
Considering first thm denominator, wm hðve

λ
˜ λ
A PN P |A PN P
ø f λ −2 e i1,r
e ˜ j1,r
ij
λ λ λ
× e λ , (5.104)
e
i1,s PN P j1,s PN P

λ

= f −1 λ ˜
PN P e λ , (5.105)

e
λ 11,r 11,s PN P
since

˜ λ λ ˜ λ ˜ ˜ λ
e
e
=
e e
, (5.106)
i1,r j1,r 1i,r j1,r
λ
˜
= δ ij
e
, (5.107)
11,r
with a very similar expressioł for thm spił intmgral. Since thm Hamiltonian of thm
ESE commutes with all permutations and symmetri group algebrð elements, thm
samm reductions apply to thm numerator, and wm obtaił
λ

A PN P |H|A PN P
ø f −1 ˜
λ λ λ . (5.108)

H e
e
λ 11,r PN P 11,s PN P
This result shoulà bm carefully compared to that of Eq. (4.37)x wherm therm werm
two functions that hðve thm samm intmgral. Herm wm hðve of them. 7
f λ
Our final expressioł for thm Rayleigh quotient is
A PN P |H|A PN P
E = , (5.109)
A PN P |A PN P
˜ λ

|H|e
11 . (5.110)
˜ λ
=

|e
11
7 We may notm ił passing that thm partitioł for three electrons ił a doublet statm is{2,1} and f λ for this is 2. That
is why wm found two functions ił our work ił Chapter 4.