Contracted Schrodinger equation cumulants

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. 

W. Kutzelnigg and D. Mukherjee, Irreducible Brillouin conditions and contracted Schrodinger equations for n-electron systems. I. The equations satisfied by the density cumulants. J. Chem. Phys. 114, 2047 (2001). 

CUMULANTS, EXTENSIVITY, AND THE CONNECTED FORMULATION OF THE CONTRACTED SCHRODINGER EQUATION... 

D. A. Mazziotti, Cumulants and the contracted Schrodinger equation, in Many-Electron Densities and Reduced Density Matrices, (J. Cioslowski, ed.), Plenum, New York, 2000, p. 139. 

Formulating conditions for the energy to be stationary with respect to variations of the wavefunction P in this generalized normal ordering, one is led to the irreducible Brillouin conditions and irreducible contracted Schrodinger equations, which are conditions on the one-particle density matrix and the fe-particle cumulants k, and which differ from their traditional counterparts (even after reconstruction [4]) in being strictly separable (size consistent) and describable in terms of connected diagrams only. 

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