Contracted Schrodinger equation representations

ORBITAL REPRESENTATION OE THE CONTRACTED SCHRODINGER EQUATION (CSchE)... 

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... 

C. Valdemoro, L. M. Tel, and E. Perez-Romero, A-representability problem within the framework of the contracted Schrodinger equation. Phys. Rev. A 61, 032507 (2000). 

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

From the begining of the development of the RDM theory the need to render N-and 5-representable a 2-RDM obtained by an approximative method was patent. The development first of the spin-adapted reduced Hamiltonian methodology and, more recently, that of the second-order contracted Schrodinger equation rendered the solution of this problem urgent. The purification strategies... 

Another extension of this theoretical smdy is the consideration of both an economical and an effective purification strategy for the 4-RDM. The need for such a purification scheme is motivated by the need to have an N- and 5-representable 4-RDM if one wishes to solve the fourth-order modified contracted Schrodinger equation [62, 64, 87]. There have already been several attemps to purify both the 3-RDM and 4-RDM [18, 34, 52]. In particular, a set of inequalities that bound the diagonal and off-diagonal elements of these high-order matrices have been reported [18]. However, the results obtained with this approach within the framework of the fourth-order modified contracted Schrodinger equation (and the second-order contracted Schrodinger equation) were not fully satisfactory because the different spin-blocks of the matrices did not appear to be properly balanced [87, 114]. 

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 first order Contracted Schrodinger Equation in a spin-orbital representation... 

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 Contracted Schrodinger Equation is studied here in a spin-orbital representation coupled with the S2 eigenvalue equation as an auxiliary condition. A set of new algorithms for approximating RDM s in terms of the lower order ones are reported here. These new features improve significantly the method. 

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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