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REDUCED DENSITY MATRIX VERSUS WAVE FUNCTION 71
terms is a product of (N – i) 1-RDM elements with an i-RDM element, (i.e.
the 1-RDM plays a similar role to the Krönecker deltas). Thus, for instance:

is spin-forbidden and must be eliminated.
The inference process leading to equations (45) and (49) was carried out with the help
of a set of graphs specially suited for operating with RO’s. This graphical method has
been described in several recent publications [26,27,47] and would excessively lengthen
this paper; therefore, I would like to mention its usefulness without elaborating.
The interest of relation (49) lies in that the holes and the particle parts of the
equation, have the same structure.
Equations (45) and (49) stress the direct connexion existing between the elements
and classes of the Symmetric Group of Permutations and the terms derived by com-
muting/anticommuting groups of fermion operators after summing with respect to
the spin variables.
Two important facts concerning the set of relations given above are that all the
N-representability relations known to us, can be derived from (45) (or (44) in a spin-
space representation) by varying the value of N and relation (49) condenses them
all.
It is interesting to note that relation (45) guaranties that the N-electron state of refer-
ence (whose superindex has been omitted) is antisymmetric since the RO's involved
on the l.h.s of these equations operate on N-electron states. Now, by contracting
this equation to a p-electron space an N-representable equation is obtained (by con-
struction). In view of this, I hoped that a relation obtained in such a way would
be a sufficient N-representability condition or at least more stringent than the (45)
equation for N = p. Now the contraction of equation (45) gives exactly the same
equation where N has been replaced by p. On the other hand the contraction of
equation (49) gives a very complicated equation where partial traces of RDM’s of
orders (p + l),(p + 2),....(N – 1) appear. This equation although difficult to analyse
may prove to be useful and it is being studied at the moment.

5.4.2. Approximation proposed
The method for approximating an RDM in terms of the lower order ones is based
on equation (49). The working hypothesis which has been put forward [46] is:
”Let us assume that Holes and Particles are totally different objects. If this as-
sumption were true, equation (49) could be exactly decoupled into two equations,
one involving RDM’s of different orders and the other, of similar structure, linking
HRDM’s of different orders”.
This hypothesis, given that Holes and Particles are related through the N-repre-
sentability conditions, is not true. On the other hand, by taking into account several