Many-body expansions, convergence

Hermann, A., Rrawczyk, R.P., Lein, M., Schwerdtfeger, P Hamilton, I.P. and Stewart, J.J.P. (2007) Convergence of the many-body expansion of interaction potentials From van der Waals to covalent and metallic systems. Physical Review A, 76, 013202-1-013202-10. 

As mentioned in Section 33.2, the many-body expansion cannot be expected to work for metals. One reason is that most atoms forming metals have open-shell ground states of symmetry other than S, therefore it is difficult to determine quantum states of the subsystems needed in the definition of the expansion, cf. Section 33.10. The second reason is that the complete delocalization of the conduction electrons results in the electronic structure of a metal that is very far from that of monomers. The first problem does not occur for alkaline-earth metals or for high-spin alkali-metal clusters, and the many-body expansion can be defined for such clusters. However, this expansion appears to be very slowly convergent [106-108]. For some specific information about the spin-polarized sodium trimer, see Section 33.10.2. 

Alreatty a vast experience has been accumulated and some generalizations are possible. The many-body expansion usually converges faster than in our fictitious example. For three argon atoms in an equilibrium configuration, the three-bo(ty... 

A direct transfer of this approach to metallic systems is not possible since localized orbitals become very long-range entities. Therefore, a many-body expansion in terms of such orbitals cannot be expected to have useful convergence characteristics. In this case we suggest to start from a system... 

The fourth order increments are of course much more numerous and we have not attempted to classify them in such a rigorous manner. Instead we have restricted our examination of four-body increments to a selected number of quite different geometries, with the aim of assuring ourselves that these are small enough to merit exclusion, and thereby confirming the convergence of the many-body expansion. The chosen geometries are shown in Fig. 23. For this purpose we use the cc-pVTZ basis set without the additional diffuse functions used for all smaller-order increments. 

The many-body expansion would not be of much use if we had to evaluate aU terms in O Eq. 6.2. But experience has shown that the expansion converges quickly and terms beyond those involving three bodies are not so important. This is fortunate as the two-body interactions are well understood and can be evaluated for moderate-sized molecules using a variety of methods, while good approximations are available for the terms involving three and more bodies, which usually arise from the effects of polarization in the cluster. 

In terms of basic physical effects included, the calculations of Drake and of Persson et al. are equivalent up to all terms of order a3 (assuming that the Many Body Perturbation Theory expansion has converged sufficiently well), and also terms of order a4Z6 and aAZb. Any difference between the two calculations should therefore scale as a4Z4, at least through the intermediate range of Z. 

In practice, one has to truncate the expansions (5). Recent studies of the convergence of the many-body perturbation expansions of the electrostatic (18), exchange (19, 20), induction (21), and dispersion (22) energies led to the development of approximation schemes which can be used to compute these components with controlled accuracy. See Refs. (1, 23) for a detailed discussion of this point. 

Recently, the convergence of the many-body sapt expansion for the interaction-induced dipole moment of He-H2 (24) and the polarizability of He2 (24, 32). has been checked by comparison with fci results in the same basis set. The numerical results for the helium dimer are summarized in Table 1 where we consider the anisotropy 7 and the trace a 7 = Aa — Aasx and a = (Act, + 2Aazz)/3, in which the z-axis is the molecular axis. An inspection of Table 1 shows that for all distances considered in Ref. (32) the many-body SAPT expansion reproduces the fci results to... 

The purpose of the preceding discussion of this particular transition in lithiumlike Bismuth was to show that use of the Furry representation firstly allows a consistent implementation of QED for the many-electron problem, with both correlation and radiative effects treated as Feynman diagrams, and secondly to show that when the extra expansion parameter 1/Z is present that extremely precise predictions result that agree well with experiment. There is no reason in principle, therefore, that QED cannot be applied to all atoms and molecules. In practice, however, without the rapid convergence provided by factors of 1/Z, the utility of this approach for neutral systems can be questioned. The best way to combine many-body methods and QED in this case is one of the forefront problems of the field. We now turn to a neutral system, the cesium atom, and describe the progress that has been made in the search for new physics in this much more challenging case. 

M-electron wavefunction can be expanded as a linear combination of an infinite set of Slater determinants that span the Hilbert space of electrons. These can be any complete set of M-electron antisymmetric functions. One such choice is obtained from the Hartree-Fock method by substituting all excited states for each MO in the determinant. This, of course, requires an infinite number of determinants, derived from an infinite AO basis set, possibly including continuum functions. As in Hartree-Fock, there are no many-body terms explicitly included in Cl expansions either. This failure results in an extremely slow convergence of Cl expansions [9]. Nevertheless, Cl is widely used, and has sparked numerous related schemes that may be used, in principle, to construct trial wavefunctions. 

The second aspect warranting special mention is the so-called intruder state problem. This problem has been found to plague practical many-body multireference formalisms. The presence of intruder states, which may be unphysical, can impair or even destroy the convergence of many-body, multireference expansions. HubaC and Wilson [95] explain ... 

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