Matrix contracting mapping

Properties of the 2-RDM and the V-Representability Ih-oblem The Matrix Contracting Mapping The Contracted Schrodinger Equation... 

Two different approaehes to this problem will be described in this work. They are based in quite different philosophies, but both are aimed at determining the RDM without a previous knowledge of the WF. Another common feature of these two approaches is that they both employ the discrete Matrix representation of the Contraction Mapping (MCM) [17,18]. Applying this MCM is the alternative, in discrete form, to integrating with respect to a set of electron variables and it is a much simpler tool to use. 

As mentioned in Section I, Cho [13], Cohen and Frishberg [14, 15], and Nakatsuji [16] integrated the Schrodinger equation and obtained an equation that they called the density equation. This equation was at the time also studied by Schlosser [44] for the 1-TRDM. In 1986 Valdemoro [17] applied a contracting mapping to the matrix representation of the Schrodinger equation and obtained the contracted Schrodinger equation (CSE). In 1986, at the Coleman Symposium where the CSE was first reported, Lowdin asked whether there was a connection between the CSE and the Nakatsuji s density equation. It came out that both... 

In many cases a simpler form of this mapping may be used. Thus, the RDM by itself, when it is not involved in matrix operations it can be contracted by using... 

Solution To decide whether an arbitrary two-dimensional map Zi+i = S(x ,y ) is area-contracting, we compute the determinant of its Jacobian matrix... 

Physically, the semidefinite conditions on the D, Q, and matrices restrict the probabilities of finding particle-particle, hole-hole, and particle-hole pairs to be nonnegative, respectively. Even though the nonnegativity constraints in Eqs. (6-8) are non-redundant, these matrices contain equivalent information as each matrix can be expressed in a one-to-one mapping of another by the fermionic anticommutation relations. The (2,2)-positivity conditions are often denoted as DQG. Contraction of the positive semidefinite D, Q, and matrices generates one-particle D and one-hole 2 matrices that are also positive semidefinite. 

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