Dirac charge distribution

ANGULAR CHARGE DISTRIBUTION FUNCTIONS ( FOR DIRAC ORBITALS 1 1... 

The aim of this section is to extract from the measurements the values of the Rydberg constant and Lamb shifts. This analysis is detailed in the references [50,61], More details on the theory of atomic hydrogen can be found in several review articles [62,63,34], It is convenient to express the energy levels in hydrogen as the sum of three terms the first is the well known hyperfine interaction. The second, given by the Dirac equation for a particle with the reduced mass and by the first relativistic correction due to the recoil of the proton, is known exactly, apart from the uncertainties in the physical constants involved (mainly the Rydberg constant R0c). The third term is the Lamb shift, which contains all the other corrections, i.e. the QED corrections, the other relativistic corrections due to the proton recoil and the effect of the proton charge distribution. Consequently, to extract i oo from the accurate measurements one needs to know the Lamb shifts. For this analysis, the theoretical values of the Lamb shifts are sufficiently precise, except for those of the 15 and 2S levels. 

The term containing Dirac s delta ZaS(r — Ra) represents the contribution from the positive point charge Za at the position Ra of the nucleus a, and -pA(r) is the electronic charge distribution, given by the diagonal element of the first-order density matrix normalized to the number of electrons in the monomer A. 

The finite difference HF scheme can also be used to solve the Schrodinger equation of a one-electron diatomic system with an arbitrary potential. Thus the approach can be applied, for example, to the construction of exchange-correlation potentials employed by the density functional methods. The eigenvalues of several GaF39+ states have been reported and the Th 79+ system has been used to search for the influence of the finite charge distribution on the potential energy curve. It has been also indicated that the machinery of the finite difference HF method could be used to find exact solutions of the Dirac-Hartree-Fock equations based on a second-order Dirac equation. 

A Dirac s va ue of the lsi/2 hyperfine-structure splitting including the nuclear-charge distribution,... 

Contrary to the mass of the nucleus, its size influences the binding energy considerably in heavy ions (Fig. 10). In studying nuclear size effects nowadays always a spherically symmetric charge distribution of the nucleus is assumed which allows a separation of the Dirac equation and corresponding wave function into an angular part and a radial part similar to the point nucleus case. The radial Dirac equation then reads [45]... 

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... 

A uniform distribution of charge over the surface of a sphere of radius R can be represented as charge density distribution in terms of the Dirac delta distribution as follows ... 

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

L. Visscher, K. G. Dyall, Dirac-Fock atomic electronic structure calculations using different nuclear charge distributions, At. Data Nucl. Data Tables 67 (1997) 207-224. 

When may such a concept as the Dirac delta function he useful Here is an example. Let us imagine that we have (in the 3-D space) two molecular charge distributions PA(r) and Pb(t). Each of the distributions consists of the electronic part and the nuclear part. 

How can such charge distributions be represented mathematically There is no problem with mathematical representation of the electronic parts-they are simply some functions of the position in space — pel, A(r) and —pei,B(r) for each molecule, respectively. The integrals of the corresponding electronic distributions yield, of course, —Na and —Nb (in a.u.), or the negative number of the electrons (because the electrons carry a negative charge). How, then, do you write the nuclear charge distribution as a function of r There is no way to do this without the Dirac... 

Thus, the Dirac delta function enables us to write the total charge distributions and their interactions in an elegant way ... 

To demonstrate the difference, let us write the electrostatic interaction of the two charge distributions both without the Dirac delta functions ... 

Of course, the two notations are equivalent because inserting the total charge distributions into the last integral, as well as using the properties of the Dirac delta function, gives the first expression for Einter-... 

The charge distribution in energy follows a quasi-equilibrium distribution and can be described as the DOS multiplied by a Fermi-Dirac function. 

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