In this article, we define a new mathematical object, called a pair
D=(A,C) of anti-commutation matrices (ACMP) based on the
anti-commutation relation aiβ βajβ+ajβaiβ β=Ξ΄ijβ
applied to the scalar product between the many-body wavefunctions. This ACMP
explicitly separates the different levels of correlation. The one-body
correlations are defined by a ACMP D0=(A0,C0) and the two-body
ones by a set of n ACMPs Di=(Ai,Ci) where n is the number
of states. We show that we can have a compact and exact parametrization with
n4 parameters of the two-body reduced density matrix (\TRDM) of any pure or
mixed N-body state to determine the ground state energy with a
O(n6) complexity