Method of Increments Ground State

A method of increments [111, 159,160] is a wavefunction-based ab-initio correlation method for solids. This method is closely related to the ideas of the local ansatz (LA), [5] where local operators acting on the SCF wavefunction are used to admix suitable one- and two-particle excitations to the mean-field HF ground state. The many-electron Hamiltonian is split according to (5.44) and the ground-state Hamiltonian Hscf and the corresponding wavefunction scf = are assumed to be known. 

A product of two operators A and B in the LiouviUe space is defined as follows  

The superscript c indicates that the cumulant of the expectation value is taken, which 

By using (5.46) the exact ground-state energy E is written in the following way  

The following operators are dehned Ai, where i should be considered as a compact index that includes the bond i as weU as the one- and two-particle excitations of bond i, and Aij, which describes the two-particle excitations where one excitation is out of bond i while the other is out of bond j. Within the restricted operator subspace spanned by Ai and A j the operator Q can be written in the form 

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