Irreducible Brillouin conditions

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). 

GENERALIZED NORMAL ORDERING, IRREDUCIBLE BRILLOUIN CONDITIONS, AND CONTRACTED SCHRODINGER EQUATIONS... 

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. 

Figure 2. Diagrammatic representation of the terms that contribute to the irreducible two-parti-cle Brillouin conditions, Eq. (169).

One can formulate a two-particle analogue of the Brillouin condition, the IBC2 ( I stands for irreducible, which essentially means connected. For details see Refs. [20, 29].)... 

If one formulates the conditions for stationarity of the energy expectation value in terms of generalized normal ordering, one is led to either the irreducible fc-particle Brillouin conditions IBCj or the irreducible A -particle contracted Schrodinger equations (IBC ), which are conditions to be satisfied by y = yj and the k. One gets a hierarchy of k-particle approximations that can be truncated at any desired order, without any need for a reconstruction, as is required for the reducible counterparts. 

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