MB-RSPT

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... 

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. 

The first term in the curled brackets can be decomposed to quantities which have already been presented, i.e. ionization potentials, electron affinities, excitation energies, double ionization energies as well as certain remaining terms. This composite property of MB-RSPT might be of value for practical applications because the mentioned quantities can be calculated separately or even substituted for with experimental values. Therefore, to interpret such complicated spectra it is sufficient to calculate a particular class of diagrams. We believe that this is more economic than a Cl calculation of comparable accuracy. Moreover, it gives us a microscopic view of the problem in contrast to the global nature of the Cl calculation. 

In this chapter we shall present the fundamental ideas of the MB-RSPT in its application to the interaction of two closed-shell ground state molecular systems. Remarkable features of the approach are the following ... 

In order to apply the MB-RSPT we have to change our Hamiltonian (224) to take the form... 

The other remaining terms are taken as perturbation. Now we can apply the MB-RSPT expansion from Chapter IV. 

Thus, we obtain the energy of the supersystem, EAB, from which it is possible to extract exact energies EA and EB for systems A and B and to obtain the expression for the interaction energy up to a particular order of the perturbation expansion. It was not the aim of this chapter to present the details of the MB-RSPT treatment of intermolecular interactions but rather to point out the fundamental ideas involved. Detailed derivation and explicit formulas as well as the physical interpretation of individual terms are presented in Refs.138,139 ... 

Valence shell correlation energy E/E,) in HpO given by MB-RSPT... 

Equation (4.106) leads directly to the conclusion that the energy in the fourth order of MB-RSPT is given solely by the connected diagrams A-D, This finding is, of course, just what follows from the formula-... 

Separation to EPV and NEPV contributions is of great importance for finding approximate relations between different methods. Listed below are the quantities that permit us a deeper insight into the problem. All these quantities may be taken as contributions to the correlation energy given by MB-RSPT through fourth order. 

Within the frame of MB-RSPT through the fourth order, a rigorous expression for the contribution from doubly excited configurations... 

The subscript R is used here for pointing out that the second terra in eqn. (4,114) comes from the fourth-order renormalization terra. Equations (4,113) and (4,114) show clearly differing meanings of the concept "effect of double excitations" in CI-D and MB-RSPT,... 

EPV contributions also permit us to find relationships between MB-RSPT and CEPA, It may be shown that differences in several existing variants of CEPA may be assigned just to the differing extent of inclusion of EPV contributions. A detailed discussion on this problem... 

Correlation energy contributions given by MB-RSPT through fourth order for H O with the contracted Gaussian DZ basis set ... 

Cl approach, the above noted cancellation would mean numerical (a posteriori) justification of Davidson s correction. The last entry listed in the family of "various quantities" in Table 4,10, mimics the effect of augmenting the Cl wave function with the quadruply excited configurations. Agreement between the fourth-order MB-RSPT and Cl is remarkable on this point. It should be realized, however, that in MB-RSPT a part of disconnected clusters (those from higher orders) is mis-... 

From eqn. (4.138) a very important consequence follows viz. that a Cl treatment limited to singly and doubly excited configurations cannot be applied rigorously to comparisons of molecules of different size. The Cl approach becomes size consistent only if quadruple excitations are involved or if some correction such as Davidson s expression is applied. As regards the other methods noted in this chapter, lEPA, MET, CPMET, CEPA, and various MB-RSPT approaches, they are all size-consistent. 

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