Page 83 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

68 C. VALDEMORO
• From an initial 2-RDM the corresponding 3- and 4- order RDM’s are approx-
imated by using a method which will be described in the following section.

• Then, all the approximated RDM’s are replaced in the r.h.s. of equation (36) so
that its three terms are added into a matrix, say and relation (36) becomes:

• By taking the trace of both sides of equation (39) one obtains since

• The following step is to divide by which gives a new 2-RDM from which
the procedure can start again.

All these steps are built into an iterative procedure whose success pivots on the
approximation of the higher order RDM’s in terms of the 2-RDM. This important
part of the method will be addressed in the next section.

5.4. APPROXIMATING AN RDM IN TERMS OF THE LOWER ORDER ONES
As has been mentioned, the iterative procedure for solving the 2-CSchE will only
work if sufficiently precise approximations of the 3- and 4-order RDM’s in terms of
the 2-RDM can be obtained. Since the method is based on the N-representabili-
ty relations, the subsection 8.1 is dedicated to discuss these fundamental equations.
Then in 8.2 the method will be outlined and some examples will be given.

5.4.1. The N-representability conditions
The basic relations for studying the properties of the RDM’s are the anticommu-
tation/commutation relations of groups of fermion operators since their expectation
values give a set of N-representability conditions of the RDM’s. Thus,

• The first order condition
From the fundamental rule of anticommutation of an annihilator with a creator
operator it follows, in our orbital representation, that:

Since both the RDM’s and the HRDM’s are positive matrices, this relation
says that the eigen-value of the 1-RDM, must be which is the well
known ensemble N-representability condition for the 1-RDM [10] represented
in an orbital basis (in a spin-orbital representation the upper bound would be
1 instead of 2).