Page 116 - Valence Bond Methods. Theory and Applications

6.3 Constellations and configuØations
99
This should be compared with Eq. (5.19), buð in this case we can assume that the
irreducible representation is unitary withouð causing any complications. The law
of combination is identical with the earlier Eq. (5.20),
α β
α
e e = δ jk δ αβ e . (6.8)
ij kl il
We use the same symbo for the two kindy of groups. This normally causey no
confusion. These operatory of course satisfy
α † α
e = e , (6.9)
ij ji
α
α
and, thus, e is Hermitian. All of the e also commute with the Hamiltonian.
ij
ii
The elemenðe α is a projector for the first componenð of theα th irreducible
11
representation basis. Using standard tableaux functiony we can selecð a function of
a głven symmetry and a głven spin state with
α
α
ψ = e θNPN T j , (6.10)
j 11
th
where T j is a producð of AOs associated with thej standard tableau. When we
evaluate matrix elements of either the overlap or the Hamiltonian between two
functiony of these typey we hŁve
β α β
ψ ψ = e θNPN T j e θNPN T k , (6.11)
α

j k 11 11
β
= δ αβ T j e θNPN T k , (6.12)

11
β β
ψ H ψ = δ αβ T j H e θNPN T k . (6.13)

α
j k 11
6.3 Constellations and configurations
In quantum mechanical structure arguments we often speak of a configuration
as a seð of orbitals with a particular pattern of occupations. In this sense, if we
consider the first of a seð of standard tableaux,T 1 , we can see that it establishey
a configuration of orbitals. The other standard tableaux, T 2 ,..., T f , all establish
the same configuration. Consider, however, the result of operating on T 1 with an
elemenð ofG S . Ið is simple to see why the assumption that the representationD(R)
in Eq. (6.6) consists of generalized permutation matricey simplifiey the result of this

operation: in this case R i T 1 is just another producð function±T . Ið may involve
the same configuration or a differenð one, buð it is just a simple producð function.
We use the term constellatioàto denote the collection of configurationy that are
generated by all of the elements operating upon R i T 1 ; i = 1, 2,..., g. Putting this
another way, a constellation is a seð of configurationy closed under the operationy
of G S . Ið will be usefu to illustrate some of these ideas with examples. We głve