Matrices representation, contracted Schrodinger equation

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

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... 

D. A. Mazziotti, Pursuit of A-representability for the contracted Schrodinger equation through density-matrix reconstruction. Phys. Rev. A 60, 3618 (1999). 

The A-representability constraints presented in this chapter can also be applied to computational methods based on the variational optimization of the reduced density matrix subject to necessary conditions for A-representability. Because of their hierarchical structure, the (g, R) conditions are also directly applicable to computational approaches based on the contracted Schrodinger equation. For example, consider the (2, 4) contracted Schrodinger equation. Requiring that the reconstmcted 4-matrix in the (2, 4) contracted Schrodinger equation satisfies the (4, 4) conditions is sufficient to ensure that the 2-matrix satisfies the rather stringent (2, 4) conditions. Conversely, if the 2-matrix does not satisfy the (2, 4) conditions, then it is impossible to construct a 4-matrix that is consistent with this 2-matrix and also satisfies the (4, 4) conditions. It seems that the (g, R) conditions provide important constraints for maintaining consistency at different levels of the contracted Schrodinger equation hierarchy. 

The question, what conditions are to be fulfilled by a density matrix to be the image of a wave function, that is, to describe a real physical system is opened till today. The contracted Schrodinger-equations derived for different order reduced density matrices by H. Nakatsui [1] give opportunity to determine density matrices by a non-variational way. The equations contain density matrices of different order, and the relationships needed for the exact solutions are not yet known in spite of the intensive research activity [2,3]. Recently perturbation theory corrections were published for correcting the error of the energy obtained by minimizing the density matrix directly applying the known conditions of N-representability [4], and... 

The interest of contracting the matrix form of the Schrodinger equation by employing the MCM, is that the resulting equation is easy to handle since only matrix operations are involved in it. Thus, when the MCM is employed up to the two electron space, the geminal representation of the CSchE has the form [35] ... 

This equation is the matrix representation of the Schrodinger equation in the N-electron space. In order to contract it into the two-electron space, we will apply the MCM to both sides of the equation and get... 

When contracting the matrix form of the Schrodinger equation into the two-electron space and transforming into normal form the resulting equation, one obtains the 2-CSE [45, 50]. In the spin-orbital representation, the 2-CSE splits... 

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