Page 85 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

70 C. VALDEMORO
–C N represents the classes of the Symmetric Group of Permutations S N
– are the permutations (of the indices of the annihilator operators) be-
longing to a particular class
– represents the parity of the permutations belonging to class C
– The symbol is given by:

– The symbol describes a sum of terms. Each of these terms is a
product of (N- i) Krönecker deltas with a i-RDM element. Now, the
terms whose addition is represented by a G symbol are those where the
indices are ordered according to the permutation of class
For instance, for N = 3 and the G symbols are:

Relations (44,45) describe the general form of the N-order condition; However,
some terms must be eliminated from relation (45) because they do not occur
when the anticommutation/commutation operations are carried out explicitly.
We call these terms spin – forbidden because in all of them the spin correspon-
dence which should exist between the creator and the annihilators forming the
p-RO (which generates the p-RDM) is not maintained. These spin-forbidden
terms are those having a transposition of at least two indices in their p-RDM.
For instance:

which is the third term of is spin-forbidden and must be eliminated.
An equivalent N-order equation having the same structure for the particle part
and for the holes part (in a similar way as in (43)), may also be inferred. This
equation has the form:

where all the symbols have the same meaning as in relation (45) except for
. This new symbol, like also describes a sum of terms. Each of these