Point nucleus

To give a simple classical model for frequency-dependent polarizabilities, let me return to Figure 17.1 and now consider the positive charge as a point nucleus and the negative sphere as an electron cloud. In the static case, the restoring force on the displaced nucleus is d)/ AtteQO ) which corresponds to a simple harmonic oscillator with force constant... 

The Coulomb interaction of the (point) nucleus with the potential Vo, which is also part of the monopole interaction, was neglected in (4.5) because it yields only an offset of the total energy. The subscript u in is introduced to distinguish the radius of the uniformly charged sphere from the usual mean square radius which can be obtained from scattering experiments. 

The unperturbed system is, of course, the hydrogen atom with a point nucleus. (Inside the nuclear sphere, the exponential... 

For spherically symmetric nuclear charge distribution (Gaussian, Fermi, or point nucleus), the electric field at a point r outside the nucleus can be evaluated from Gauss law as... 

The non-relativistic Hamiltonian for an idealized system made up of a point nucleus of infinite mass and charge Ze, surrounded by N electrons of mass m and charge - e is ... 

Calculation of the values and for the states nlj of //-like ions with a point nucleus (in accordance with the Zommerfeld formula) ... 

We start from the Dirac one-particle Hamiltonian for an electron bound to a point nucleus... 

The present calculation takes into account all these effects for Coulomb-Coulomb and Coulomb-Breit interaction and omits effects l)-3) for Breit-Breit interaction. On the other hand we, in contrast to [2], carry out all the calculations in the point nucleus approximation. The corrections to the finite size of nucleus for the bound energy as well as the corrections to the finite size of nucleus for electron interaction are considered separately (see Table 1). 

One of the main goals of the calculations of Refs. [25,26,33] was to evaluate the nuclear recoil correction for highly charged ions. In the case of the ground state of hydrogenlike uranium these calculations yield -0.51 eV for the point nucleus case [25] and -0.46 eV for the extended nucleus case [33], This correction... 

The original discussion of this problem was based on the solution of the ground state solution of the Dirac equation for the hydrogenic atom consisting of a point nucleus of charge Z and a single electron ... 

The total energy of a system consisting of a point nucleus with an infinite mass, surrounded by N electrons can be represented by the Hamiltonian (19),... 

Isomer Shift (IS). The shift observed in the Mossbauer lines with respect to zero velocity is produced by the electrostatic interaction of the nuclear and electron charge distributions inside the nuclear region. One assumes the nucleus is a uniformly charged sphere of radius R, and the electronic charge density is taken to be uniformly distributed over the nucleus. Then the difference between the electrostatic interaction of a point nucleus and a nucleus with radius R is given by... 

In the relativistic treatment, however, the situation is much different. A relativistic four-component Dirac wave function in a point-nucleus approximation diverges at the nuclei. We should recall that the two-component ZORA wave function is proportional to the large components of the respective four-component wave function, therefore showing similar divergences for -type orbitals in a point-nucleus approximation. However, the factor K becomes... 

Gaussian functions are appropriate functions for electronic structure calculations not only because of the widely recognized fact that they lead to molecular integrals which can be evaluated efficiently and accurately but also because such functions do not introduce a cusp into the approximation for the wave function at a physically inappropriate point when off-nuclei functions are employed. Furthermore, Gaussian functions are suitable for the description of wave functions in the vicinity of nuclei once the point nucleus model is abandoned in favour of a more realistic finite nucleus model. 

The most accurate calculations of the SE correction were carried out in Mohr (1974a, 1992) and in Indelicato and Mohr (1998) for the point nucleus, and in Mohr and Soff (1993) for the extended nucleus. For heavy systems (Z > 50) the dependence of the self-energy correction FSE on the nuclear radius R also Ahas to be taken into account (Soff 1993). 

To obtain the total binding energies, the (point-nucleus) Dirac eigenvalues of — 132279.92(1) eV for uranium and —101 581.37(1) eV for lead should be added to the Lamb-shift contribution quoted in Table 1.3. We may note, for example, that in... 

For the nucleus-electron interaction the Coulomb potential, either for a point nucleus or a finite nucleus, is used directly. The relativistic contributions to the interaction operator are obtained approximately, using the Pauli approximation. They are... 

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]... 

Changing the rms-radius yields the size effects. The experimental values for each nucleus can be taken from a number of tables [46 - 50]. For uranium, the difference to the point nucleus binding energy value of the lsi/2-state amounts to 198.82 eV. This number is obtained by assuming a Fermi distribution with — 5.860 fm... 

Fig. 22. The magnetic loop splitting energy Eml of the ground state, normalized to a(Za)AE ° , which is the leading term in a non-relativistic expansion. The dashed line refers to a relativistic point nucleus calculation whereas the solid line indicates the corresponding value for extended nuclei. The difference between both charge distribution models is notable. Note the logarithmic scale of the ordinate where 10° indicates 1.

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