Many-body inner shell

The electrostatic energy is calculated using the distributed multipolar expansion introduced by Stone [39,40], with the expansion carried out through octopoles. The expansion centers are taken to be the atom centers and the bond midpoints. So, for water, there are five expansion points (three at the atom centers and two at the O-H bond midpoints), while in benzene there are 24 expansion points. The induction or polarization term is represented by the interaction of the induced dipole on one fragment with the static multipolar field on another fragment, expressed in terms of the distributed localized molecular orbital (LMO) dipole polarizabilities. That is, the number of polarizability points is equal to the number of bonds and lone pairs in the molecule. One can opt to include inner shells as well, but this is usually not useful. The induced dipoles are iterated to self-consistency, so some many body effects are included. 

One of the necessary conditions for a many-body description is the validity of the decomposition of the system under consideration on separate subsystems. In the case of very large collective effects we cannot separate the individual parts of the system and only the total energy of the system can be defined. However, in atomic systems the inner-shell electrons are to a great extent localized. Therefore, even in metals with strong collective valence-electron interactions, atoms (or ions) can be identified as individuals and we can define many-body interactions. The important role in this separation plays the validity for atom- molecular systems the adiabatic or the Born-Oppenheimer approximations which allow to describe the potential energy of an N-atom systeni as a functional of the positions of atomic nuclei. 

When homoconjugation leads to electron or bond delocalization, and thereby to a change in the properties of a molecule, homoconjugation becomes chemically relevant. In fact, electrons are always delocalized over the space of a molecule. However, it has turned out that it is extremely useful to consider bonding, lone-pair and inner-shell electrons to be essentially localized in the bond, lone-pair or core region, respectively. This assumption is the basis of the concept of electron or bond localization and reflects the fact that many properties of a molecule can be reproduced in terms of bond or atom contributions. Of course, neither bond localization nor electron localization refers to any observable molecular property. They simply suggest that most molecules behave as if their bonds were localized and that their properties can be reproduced with the help of bond increments. With the concept of bond localization a large body of experimental data on molecular properties can be rationalized, i.e. bond or electron localization is a heuristic concept ... 

It is quite common in correlated methods (including many-body perturbation theory, coupled-cluster, etc., as well as configuration interaction) to invoke the frozen core approximation, whereby the lowest-lying molecular orbitals, occupied by the inner-shell electrons, are constrained to remain doubly-occupied in all configurations. The frozen core for atoms lithium to neon typically consists of the Is atomic orbital, while that for atoms sodium to argon consists of the atomic orbitals Is, 2s, 2px, 2py and 2pz. The frozen molecular orbitals are those made primarily from these inner-shell atomic orbitals. 

Fig. 7.1. Typical photoelectron spectra as computed from many-body theory (a) for valence excitation (optical spectra) (b) for subvalence shell excitation (vacuum ultraviolet spectra) and (c) for deep inner-shell excitation (X-ray spectra) (after J.-P. Connerade [355]).

Similarly, quantum defect theory plays a very important role in modern descriptions of atomic physics, and should be included at a less rudimentary level than is found in most texts. Again, its modern developments provide an excellent illustration of many fundamental principles of scattering theory. The principles underlying the Lu-Fano graph are easily grasped, and provide excellent insight into an important aspect of the many-body problem, namely interchannel coupling. Likewise double- and inner-shell excitation are hardly discussed in textbooks, structure in the continuum receives little attention, etc, etc. 

The interpretation of experimental results based on the one electron picture also raises fundamental questions. It has been shown that the low energy elementary excitations in metals can be described as quasi-particles. By making suitable many-body corrections one can convert the one electron states into quasi-particle states. For excitations from inner shells, which become possible when the excitation energy is high, the change of state of one electron is accompanied by a rearrangement of the states of many other electrons in the same core. This is a complicated many-body problem that can not be handled by the simple methods of band calculation. To what extent should one include many electron effects when an electron is excited from a deep band state remains an open question. 

In the paragraphs below we review some of the recent progress on relativi tlc many-body calculations which provide partial answers to the first of these questions and we also describe work on the Brelt Interaction and QED corrections which addresses the second question. We begin in Section IT with a review of applications of the DF approximation to treat inner-shell problems, where correlation corrections are insignificant, but where the Breit Interaction and QED corrections are important. Next, we discuss, in Section III, the multiconfiguration Dirac-Fock (MCDF) approximation which is a many-body technique appropriate for treating correlation effects in outer shells. Finally, in Section IV, we turn to applications of the relativistic random-phase approximation (RRPA) to treat correlation effects, especially in systems involving continuum states. 

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