Perturbational Decomposition of the Cluster Operators

Two essential concepts underlie the construction of MBPT from basic Rayleigh-Schrodinger perturbation theory  

The zeroth-order component of the electronic Hamiltonian is taken to be the Fock operator such that the perturbation operator (sometimes called the fluctuation potential) is then the remaining two-electron operator, 

This partitioning, when applied in conjunction with the set of canonical Hartree-Fock orbitals (in which is diagonal), corresponds to the Moller-Plesset variant of many-body perturbation theory. A Hartree-Fock determinant, which is an eigenfunction of Pjq, is therefore the natural choice for the zeroth-order wavefunctiond 

Each perturbed wavefunction, , is expanded in a Cl-like fashion as a linear combination of excited determinants. 

As discussed in detail in Refs. 77 and 82, for example, this expansion is not N-fold (where N is the number of electrons in the system) for the lower perturbational orders, but truncates to include only modest excitation levels. For example, the first-order wavefunction, which may be used to compute both the second- and third-order energies, contains contributions from doubly excited determinants only, whereas the second-order wavefunction, which contributes to the fourth- and fifth-order perturbed energies, contains contributions from singly, doubly, triply, and quadruply excited determinants. Furthermore, the sum of the zeroth- and first order energies is equal to the SCF energy. This determinantal expansion of the perturbed wavefunctions suggests that we may also decompose the cluster operators, T , by orders of perturbation theory  

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