Expansions of the inverse operator

Recall the effective eigenvalue equation (2.23) for the case of a single model function 

The key operator to be determined in order to construct the effective Hamiltonian //effective, the wavc Operator 12 or the reaction operator V, is the resolvent which can be loosely written as 

Different forms of perturbation theory can be obtained by expanding the inverse operator in eq. (2.79), i.e. ( - QSiQY. This operator is assumed to exist. It can be written as an infinite expansion using the operator identity 

Different forms of perturbation theory can be obtained according to the partition of - QS Q into the operator X and Y. 

We recognize that the more general recursion defined in eq. (2.85), facilitates the development of hybrid approaches. 

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In the development of the Pauli Hamiltonian in section 17.1, truncation of the power series expansion of the inverse operator after the first term yielded the nonrelativistic Hamiltonian. In (18.1), the zeroth-order term is the Hamiltonian first developed by Chang, Pelissier, and Durand (1986), often referred to as the CPD Hamiltonian. The name given by van Lenthe et al. is the zeroth-order regular approximation, ZORA, which we will adopt here. The zeroth-order Hamiltonian is... 

In the derivation of the ZORA equation, we made the assumption that < 2mc so that we could do the expansion of the inverse operator. However, the ZORA equation has a valid spectrum for all E. It is therefore not necessarily the case that the use of a truncated expansion is invalid outside the strict radius of convergence. We could... 

In the second section of this chapter, we shall employ the partitioning technique to develop various types of perturbation theory, including Rayleigh-Schrddinger perturbation theory and Brillouin-Wigner perturbation theory. This involves the expansion of the inverse operators which occur in the effective Hamiltonian operator and other operators obtained by the partitioning technique. Different expansions lead to different types of perturbation theory. 

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