Point-like charge density distribution

The term Lamb shift of a single atomic level usually refers to the difference between the Dirac energy for point-like nuclei and its observable value shifted by nuclear and QED effects. Nuclear effects include energy shifts due to static nuclear properties such as the size and shape of the nuclear charge density distribution and due to nuclear dynamics, i.e. recoil correction and nuclear polarization. To a zeroth approximation, the energy levels of a hydrogen-like atom are determined by the Dirac equation. For point-like nuclei the eigenvalues of the Dirac equation can be found analytically. In the case of extended nuclei, this equation can be solved either numerically or by means of successive analytical approximation (see Rose 1961 Shabaev 1993). 

This is the charge density distribution for the point-like nucleus case (PNC), which we include for completeness and because of the importance of this model as a reference for any work with an extended model of the atomic nucleus (finite nucleus case, FNC). The charge density distribution can be given in terms of the Dirac delta distribution as... 

The study of the electronic structure of diatomic species, which can nowadays be done most accurately with two-dimensional numerical finite difference techniques, both in the non-relativistic [90,91] and the relativistic framework [92-94], is still almost completely restricted to point-like representations of the atomic nuclei. An extension to allow the use of finite nucleus models (Gauss-type and Fermi-type model) in Hartree-Fock calculations has been made only very recently [95]. This extension faces the problem that different coordinate systems must be combined, the spherical one used to describe the charge density distribution p r) and the electrostatic potential V(r) of each of the two nuclei, and the prolate ellipsoidal one used to solve the electronic structure problem. 

In this last section we mention a few cases, where properties other than the energy of a system are considered, which are influenced in particular by the change from the point-like nucleus case (PNC) to the finite nucleus case (FNC) for the nuclear model. Firstly, we consider the electron-nuclear contact term (Darwin term), and turn then to higher quantum electrodynamic effects. In both cases the nuclear charge density distribution p r) is involved. The next item, parity non-conservation due to neutral weak interaction between electrons and nuclei, involves the nuclear proton and neutron density distributions, i.e., the particle density ditributions n r) and n (r). Finally, higher nuclear electric multipole moments, which involve the charge density distribution p r) again, are mentioned briefly. 

The use of extended nuclear charge density distributions, instead of the simple point-like Dirac delta distribution, is almost a standard in present-... 

Finally, in Sect. 6, we have briefly given some examples for physical properties or effects, which involve the nuclear charge density distribution or the nucleon distribution in a more direct way, such that the change from a point-like to an extended nucleus is not unimportant. These include the electron-nucleus Darwin term, QED effects like vacuum polarization, and parity non-conservation due to neutral weak interaction. Hyperfine interaction, i.e., the interaction between higher nuclear electric (and magnetic)... 

A modem description of a conventional hydrogen bond as well as its older, more accurate definition are based on Bader s theory of atoms in molecules (AIM theory) [4]. Bader considers matter a distribution of charge in real space of point-like nuclei embedded in the diffuse density of electron charge, p(r). All the properties of matter are manifested in the charge distribution and the topology... 

The Coulombic potential becomes infinitely negative when an electron and a nucleus coalesce and, because of this, the state function for an atom or molecule must exhibit a cusp at a nuclear position. That is, as shown by Kato (1957), the first derivative of the function is discontinuous at the position of a nucleus. Thus, while the charge density is a maximum at the position of a nucleus, this point is not a true critical point because Vp, like is discontinuous there. However, as discussed in Election E2.1, this is not a problem of practical import and the nuclear positions behave topologically as do (3, — 3) critical points in the charge distribution and hereafter they will be referred to as such. 

For our description of atoms and molecules, we rely on the orbital model, where atoms or molecules are described by one or more point-like positively charged nuclei surrounded by a cloud of negatively charged electrons, whose density is distributed in space in terms of atomic orbitals (one-centre, AOs) or molecular orbitals (multicentre, MOs) f/(r), one-electron wavefunctions, such that... 

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