Second-order contracted Schrodinger equation

The outlook given in this chapter on the theory of the second-order contracted Schrodinger equation and on its methodology has been aimed mostly at convincing the reader that this theory is not difficult to understand and that its methodology is now ready to be applied. That is, in the author s opinion, this methodology can be considered as accurate and probably more economical than the best standard quantum chemical computational methods for the study of states where the occupation number of spin orbitals is close to one or zero. 

F. Colmenero and C. Valdemoro, Self-consistent approximate solution of the second-order contracted Schrodinger equation. Int. J. Quantum Chem. 51, 369 (1994). 

V. Purification Within the Framework of the Second-Order Contracted Schrodinger Equation... 

A. The Second-Order Contracted Schrodinger Equation and Its Iterative Solution... 

V. PURIFICATION WITHIN THE FRAMEWORK OF THE SECOND-ORDER CONTRACTED SCHRODINGER EQUATION... 

Among the several 2-RDM-oriented methods that have been developed for the study of chemical systems, one of the most recent and promising techniques is based on the iterative solution of the second-order contracted Schrodinger equation (2-CSE) [1, 6, 15, 18, 36, 45-60, 62-65, 68, 70, 79-85, 103-111]. The 2-CSE was initially derived in 1976 in first quantization in the works of Cho [103], Cohen and Erishberg [104, 105], and Nakatsuji [106] and later on deduced in second quantization by Valdemoro [45] through the contraction of... 

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]. 

My second aim here is directly related with the research project which is being developed in our laboratory at present the iterative solution of the first and second order contracted Schrodinger equations, (1- and 2-CSE respectively). 

Importantly, the anti-Hermitian CSE may be evaluated through second order of a renormalized perturbation theory even when the cumulant 3-RDM is neglected in the reconstruction. The anti-Hermitian part of the CSE [27, 31, 63] is the stationary condition for two-body unitary transformations of the A-particle wave-function [31, 32], and hence the two-body unitary transformations may easily be evaluated with the anti-Hermitian CSE and RDM reconstruction without the many-electron Schrodinger equation. The contracted Schrodinger equation in conjunction with the concepts of reconstruction and purification provides a new, important approach to computing the 2-RDM directly without the many-electron wavefunction. 

C. Valdemoro, D. R. Alcoba, and L. M. Tel, Recent developments in the contracted Schrodinger equation method controlling the Ai-representabihty of the second-order reduced density matrix. hit. J. Quantum Chem. 93, 212 (2003). 

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