Page 78 - Strategies and Applications in Quantum Chemistry From Molecular Astrophysics to Molecular Engineer

REDUCED DENSITY MATRIX VERSUS WAVE FUNCTION 63

should be fulfilled.
Now, when a pair is described by a state I, its Density Matrix (DM) is

and by applying this working hypothesis, the 2 – RDM corresponding to the N-
electron eigen-state can be approximated by

It should be noted that this relation is formally identical to the spectral resolution
of the 2 – RDM. That is, in this model, all happens as if the eigen-vectors of the
2-SRH were natural geminals.
In order to determine the we proposed two main approximations:
• Let us start by assuming that in our N-electron state there is a
eigen-state, having a coefficient of a much higher absolute value than all the
rest i.e., is the dominant configuration in Then the are approximated
as follows:

This is called [35] Mixed Pair State approximation (MPS). The name, which
probably is not the best one, refers to the fact that the barring exceptions,
has a value smaller than one, which means that the electron pair is not in the
pure state I.
• A variant of relation (27) was initially proposed [34] where the were deter-
mined as follows. By definition:

where has the same meaning as before and is the orbital part of the eigen-
states of the operator. The two electron configurations, having a non zero
value are thus selected. Now, the eigen-vector whose highest coefficient (in
absolute value) is, is allocated the occupation number:

This approximation was denoted initially by the acronym IQG [34] and later on
by IP (Independent Pairs) [35]. It gave satisfactory results in the study of the
Beryllium atom and of its isoelectronic series as well as in the BeH system. The
drawback of this approximation is that when the eigen-vectors are diffuse, i.e.
there is more than one dominant two electron configuration per eigen-vector,
the determination of the corresponding is ambiguous. In order to avoid this
problem the MPS approximation, which does not have this drawback, was
proposed.