Nuclear charge distribution

A molecule contains a nuclear distribution and an electronic distribution there is nothing else in a molecule. The nuclear arrangement is fully reflected in the electronic density distribution, consequently, the electronic density and its changes are sufficient to derive all information on all molecular properties. Molecular bodies are the fuzzy bodies of electronic charge density distributions consequently, the shape and shape changes of these fuzzy bodies potentially describe all molecular properties. Modern computational methods of quantum chemistry provide practical means to describe molecular electron distributions, and sufficiently accurate quantum chemical representations of the fuzzy molecular bodies are of importance for many reasons. A detailed analysis and understanding of "static" molecular properties such as "equilibrium" structure, and the more important dynamic properties such as vibrations, conformational changes and chemical reactions are hardly possible without a description of the molecule itself that implies a description of molecular bodies. 

Hence, a simple, essentially classical model provides a useful approximation to the relations between the electronic and nuclear distributions one may think of the electron distribution as a formal charge cloud, and the nuclear distribution as an... 

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). 

The approximation of the nucleus as an infinitely heavy point charge makes possible analytical solution of the Dirac equation for the hydrogen-like problem. The resulting orbitals are, however, too tightly bound and clearly unphysical within the nucleus. A homogeneously charged nucleus is a significant improvement and is sufficient for many applications. For more detailed studies of nuclear properties, it is, however, desirable to use a more physical nuclear distribution, such as the Fermi and the Fourier-Bessel distributions described below. 

G-spinors are appropriate for distributed charge nuclear models, and are much the most convenient for relativistic molecular calculations. Whereas neither L-spinors nor S-spinors satisfy the matching criterion (140) for finite c (although they do in the nonrelativistic limit), G-spinors are matched according to (140) for all values of c. The radial functions can be written... 

Nuclear charge density distributions in quantum chemistry... 

The present chapter deals with the representation of extended nuclei, and the use of suitable models for extended nuclei in theoretical approaches to the problem of electronic structure, with emphasis on the nuclear charge density distribution. We begin with a brief general description of nuclear... 

The state-dependent nuclear charge density distribution, p r), can then be obtained from the particle density distributions through convolution with the charge density distributions of the single nucleons, Pp(r) and Pn(r) respectively ... 

The charge density distributions of a single neutron, p (r), and of all neutrons (the second integral in Elq. (3)) both integrate to zero. Usually, the contribution of the neutrons is omitted in Eq. (3). In any case, normalization of this nuclear charge density distribution correctly yields the total nuclear charge Q = J d r p r) = Ze. 

We can now summarize our discussion on nuclear structure as follows A stationary state of the atomic nucleus can be represented, in general, by a real-valued non-negative charge density distribution p r) (a scalar function of coordinates), and by a real-valued current density distribution j r) (a vector function of coordinates). The former can be expanded into a series with standard spherical harmonics i i(r) [ the unit vector r = r/r is equivalent to the angles Q = 9,(f)) ],... 

With given nuclear charge density distribution p(r) and nuclear current density distribution j r), we can now obtain all quantities required to describe the resulting electric and magnetic fields from basic relations of the theory of electromagnetism [11]. 

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