Dirac -function charge distribution

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

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. 

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]... 

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. 

The effect of the Breit interaction on the wave function can be conveniently studied by comparing radial moments (r) of shells calculated with Dirac-Coulomb and Dirac-Coulomb-Breit Hamiltonians. The effect is not very large as can be seen for Li- and Be-like ions in Figure 9.4. From the plot we note that the effect of the Breit interaction on the radial functions is small, but increases linearly with the nuclear charge number Z. Moreover, the four different models to describe the positive nuclear charge distribution (point-like, exponential, Gaussian shaped and Fermi) can hardly be distinguished. 

A key distinguishing characteristic of atomic charge definitions is their intended usage in describing some aspect of the electronic charge distribution, i.e., the spatial variation of electron density p(r). In the most superficial usages, this distribution is replaced by supposed point charges (Dirac delta functions) at each nucleus, namely. 

Visscher L and Dyall K 1997 Dirac-Fock atomic electronic structure calculations using different nuclear charge distributions. At. Data Nucl. Data Tables 67(2), 207-224. Autschbach J 2009 Magnitude of finite nucleus size effects in relativistic density functional computations of indirect nmr nuclear spin-spin coupling tensors. ChemPhysChem 10, 2274-2283. 

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