Expansion and Size-Extensivity

There is an intimate connection between the cluster-expansion of a wave-function and the property of size-extensivity. To describe this aspect in the simplest manner,it is pertinent to recall first the closed-shell ground state. The ways to encompass the open-shell states can then be indicated as appropriate extensions and generalizations of the closed-shell cluster expansion strategy. 

For an N-electron closed-shell state with a dominant reference function, the exact wave-function 4 can be written as a superposition of various n-fold excited determinants on 4.4 may thought to be generated from by anwave operator 

In the many-body description, is usually taken as the vacuum, and holes and particles are appropriately defined as those occupied and vacant in 

T introduces true n-particle correlation, and products like T T etc., arising out of the expansion eq.(3.2 ), generate simultaneous presence of k-particle amd m-particle correlations in a (k+m)-fold excited determinants etc. The truncation of T == T then corresponds to the pair—correlation model of Sinanoglu, while incorporating higher excited states with several disjoint pair excitations induced through the powers T - The amplitudes for T may be called linked or connected clusters for n electrons. The difficulty of a linear variation method such as Cl lies in its inability to realize the cluster expansion structure eq.(3.2),in a simple and practicable manner. 

The operators with different labels commute in the limit of no interactions between the groups, and we have 

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