RSPT

An essential thing to stress concerning the above development of so-called Rayleigh-Schrodinger perturbation theory (RSPT) is that each of the energy corrections... 

Given this particular choice of H, it is possible to apply the general RSPT energy and wavefunction correction formulas developed above to generate explicit results in terms... 

The first- and second- order RSPT energy and first-order RSPT wavefunction correction expressions form not only a useful computational tool but are also of great use in understanding how strongly a perturbation will affect a particular state of the system. By... 

Rayleigh-Schrodinger many-body perturbation theory — RSPT). In this approach, the total Hamiltonian of the system is divided or partitioned into two parts a zeroth-order part, Hq (which has... 

Using the above partitioning into the Rayleigh-Schrddinger perturbation theory (RSPT) allows the perturbed reference function to be written as,... 

Then it follows from the inhomogeneous equations of RSPT that... 

The above development considers Ho to be an entity, and develops the nonsym-metric form of RSPT accordingly [18]. 

It should be apparent that the expressions for the wave functions after interaction [equations (3.38) and (3.39)] are equivalent to the Rayleigh-Schrodinger perturbation theory (RSPT) result for the perturbed wave function correct to first order [equation (A.109)]. Similarly, the parallel between the MO energies [equations (3.33) and (3.34)] and the RSPT energy correct to second order [equation (A. 110)] is obvious. The missing first-order correction emphasizes the correspondence of the first-order corrected wave function and the second-order corrected energy. Note that equations (3.33), (3.34), (3.38), and (3.39) are valid under the same conditions required for the application of perturbation theory, namely that the perturbation be weak compared to energy differences. 

IV. Many-Body Rayleigh-Schrodinger Perturbation Theory (MB-RSPT) A. Brief Description... 

Here we shall follow the derivation of time-independent MB-RSPT in its main features as is described in Ref.9 Let us assume that a perturbed Hamiltonian of an atomic or molecular system, K, may be split as... 

Furthermore we assume that the complete solution of Eq. (51) is known. Our goal is to find the solution of Eq. (50) under the assumption that I <> changes into the state I iff > if the perturbation W is switched on. We shall not go into details of the derivation. Instead, we state that for kt the following RSPT expansion holds ... 

In this section we shall discuss an approach which is neither variational nor perturba-tional. This approach also has its origin in nuclear physics and was introduced to quantum chemistry by Sinanoglu47, It is based on a cluster expansion of the wave function. A systematic method for the calculation of cluster expansion components of the exact wave function was developed by C ek48 The characteristic feature of this approach is the expansion of the wave function as a linear combination of Slater determinants. Formally, this expansion is similar to the ordinary Cl expansion. The cluster expansion, however, gives us not only the physical insight of the correlation energy but it also shows the connections between the variational approaches (Cl) and the perturbational approaches (e.g. MB-RSPT). 

This section focuses on two practical points the portion of the correlation energy covered by the perturbation expansion and the cost involved. We are not going to attempt to give a complete bibliography of papers dealing with MB-RSPT calculations of the correlation energy. Instead, we shall mention only those that are relevant to the problem noted, as well as, those that are closely related to the computational scheme outlined in Section IV. 

A systematic study devoted to the MB-RSPT was undertaken by Pople et al.11 s. The utility of the theory was demonstrated by calculations up to the second and third orders on the equilibrium geometries, dissociation energies and energy differences between the electronic states of different multiplicity. 

The comparison of the second and the third order of the MB-RSPT and the CI-SD depends on two errors ... 

Specifically, tetra-excited states are responsible for the cancellation of the incorrect N2 behaviour of the CI-SD, as well as, for the further reduction of the correlation energy. Due to truncation of the MB-RSPT at the third order the effect of doubly excited configurations are not included fully and, of course, the higher excited configurations are omitted completely. 

Table 5. Valence shell correlation energy in H2O1) given by MB-RSPT treatment116) and the INO-CI calculations96) including all singly and doubly excited configurations (Cl-SD)...

To complete the description of the utility of the MB-RSPT, we would like to note that for the basis used, and k calculations are relatively fast, e.g. for the largest basis set (H20) the integral transformation time is almost identical to that needed for the atomic integral calculation, while the time for the k and k calculation represents about 50% of what is needed for the SCF procedure. 

In this chapter we present the utility of the MB-RSPT for applications in different fields of spectroscopy. The theory will be demonstrated to be an excellent tool for interpreting various phenomena related to ionization, excitation and combinations of both. 

Here we shall demonstrate how to obtain the explicit expression129 13°1 for (188) by means of the MB-RSPT. Let us study the ionized state which we shall describe using I ki>. We shall limit ourselves to a state I for which the initial state li) is obtained from the neutral closed shell ground state in which we annihilate one particle. The state l< j) is therefore realized by... 

We have already shown that excitation energies can be diagrammatically decomposed to yield simpler quantities such as ionization potentials and electron affinities plus some remaining diagrams. MB-RSPT permits the use of this treatment for even more complex processes. In this section, we present the applicability of the theory to double ionizations observed in Auger spectra as well as excitations accompanying photoionization (shake-up processes) observed in ESCA and photoelectron spectroscopy. A detailed description of this approach is given in Refs.135,136. Here we shall present only the formal description. 

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