Page 84 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

REDUCED DENSITY MATRIX VERSUS WAVE FUNCTION 69
• The second order condition
From the commutation of two annihilator with two creator operators follows the
also well known Q-condition [9] for the 2-RDM. In our notation, this condition
takes the form:

and replacing the Krönecker deltas by their value according to relation (41) one
finds [46]:

Note, that in this last relation, the part involving HRDM’s and that involving
RDM’s have the same structure.

• General N-order condition
The aim of the following discussion under this heading is not to describs the
formalism but merely to outline the ideas on which the method for approximat-
ing a p-RDM from the q-RDM’s with q < p is based. Nevertheless, in order to
avoid using vague or imprecise arguments the essential theoretical background
supporting the leading ideas must also be included here. The reader interested
in going beyond this sketchy discussion is referred to a recent paper [47] where
all the details are reported.
The result of commuting/anticommuting (for N even/odd) N annihilator op-
erators with N creator operators is:

where the symbols are N-electron configurations. This relation is very
elegant and compact but the following, in the orbital representation (obtained
by inference [47]), is more practical for our purpose:

where: