Perturbation theory connection with operators

The solution of the Schrddinger equation by means of the partitioning technique and the concept of reduced resolvents is then treated. It is shown that the expressions obtained are most conveniently interpreted in terms of inhomogeneous differential equations. A study of the connection with the first approach reveals that the two methods are essentially equivalent, but also that the use of reduced resolvents and inverse operators may give an altemative insight in the mathematical structure of perturbation theory, particularly with respect to the bracketing theorem and the use of power series expansions with a remainder. In conclusion, it is emphasized that the combined use of the two methods provides a simpler and more powerful tool than any one of them taken separately. 

Coupled cluster is closely connected with Mpller-Plesset perturbation theory, as mentioned at the start of this section. The infinite Taylor expansion of the exponential operator (eq. (4.46)) ensures that the contributions from a given excitation level are included to infinite order. Perturbation theory indicates that doubles are the most important, they are the only contributors to MP2 and MP3. At fourth order, there are contributions from singles, doubles, triples and quadruples. The MP4 quadruples... 

All of the terms in eqs. (8.29-8.34) may be used as perturbation operators in connection with non-relativistic theory, as discussed in more detail in Chapter 10. It should be noted, however, that some of the operators are inherently divergent, and should not be used beyond a first-order perturbation correction. 

The coupled cluster (CC) method is actually related to both the perturbation (Section 5.4.2) and the Cl approaches (Section 5.4.3). Like perturbation theory, CC theory is connected to the linked cluster theorem (linked diagram theorem) [101], which proves that MP calculations are size-consistent (see below). Like standard Cl it expresses the correlated wavefunction as a sum of the HF ground state determinant and determinants representing the promotion of electrons from this into virtual MOs. As with the Mpller-Plesset equations, the derivation of the CC equations is complicated. The basic idea is to express the correlated wave-function Tasa sum of determinants by allowing a series of operators 7), 73,... to act on the HF wavefunction ... 

Connection with perturbation theory the wave and reaction operators in the generd case. 

Connection with perturbation theory the wave and reaction operator in the general case.- Let us now consider the case when the Hamiltonian H consists of two terms H = + V, where the perturbation V does not... 

The generalized Bloch equation (12) is the basis of the RS perturbation theory. This equation determines the wave operator and, together with Eq. (11), the energy corrections for all states of interest especially, it leads to perturbation expansions which are independent of the energy of the individual states, just referring the unperturbed basis states. Another form, better suitable for computations, is to cast this equation into a recursive form which connects the wave operators of two consecutive orders in the perturbation V. To obtain this form, let us start from the standard representation of the Bloch equation (16) in intermediate normalization and define... 

In obtaining (14.2.23), we have used the fact that the cluster operator (14.2.22) gives zero when applied to the bra state (HF. The connection with coupled-cluster theory is a close one, which we shall explore in Section 14.3. At this stage, the reader may wish to compare the expression for the Mpller-Plesset anq)litudes (14.2.23) with the expression (13.4.10) used in the perturbation-based optimization of coupled-cluster wave functions. 

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