Potentials Uehling potential

This effect is usually taken into account to first order in PT by means of the Uehling potential. This potential is usually written as follows [1, 11] ... 

Here, for more preeise approximation of the Uehling potential, we make a complete numerical calculation of the function C(g) in Eq. (18). This procedure is described in detail in refs [11, 17], Using the precise values obtained for C(g)... 

The use of the new approximation for the Uehling potential permits to decrease the computation errors for this term down to 0.5- 1%. Besides, using such a simple analytical expression for approximating the Uehling potential allows its easy inclusion into the general system of differential equations. This system includes also the Dirac equation and the equations for the matrix elements. 

We have also performed the calculation of hyperfine coupling constants the electric quadrupole constant B and magnetic dipole constant A, with inclusion of nuclear finiteness and the Uehling potential for Li-like ions. Analogous calculations of the constant A for ns states of hydrogen-, lithium- and sodiumlike ions were made in refs [11, 22]. In those papers other bases were used for the relativistic orbitals, another model was adopted for the charge distribution in the nuclei, and another method of numerical calculation was used for the Uehling potential. 

It is not difficult to present an exact formula containing all corrections produced by the Uehling potential in Fig. 2.2 (compare with the respective expression for the self-energy operator above)... 

The logarithmic contribution is induced only by the Uehling potential in Fig. 3.10, and may easily be calculated exactly in the same way as the logarithmic contribution induced by the radiative photon in (3.97). The only difference is that now the role of the perturbation potential is played by the kernel which corresponds to the polarization contribution to the Lamb shift of order a(Za) m... 

It is not difficult to calculate anal3ftically nonlogarithmic corrections of order a(Zay generated by the Uehling potential. Using the formulae from [65[ one obtains for a few lower levels (see also [122[ for the case of 15-state)... 

There are no obstacles to exact numerical calculation of the Uehling potential contribution to the energy shift without expansion over Za and such calculations have been performed with high accuracy (see [65, 117] and references therein). The results of these calculations may be conveniently presented with the help of an auxiliary function Gu,7 Za) defined by the relationship... 

For the case of atoms with low Z (hydrogen and helium), values of the function Gu,7 Za) for the states with n = 1,2,4 are tabulated in [117] and respective contributions may easily be calculated for other states when needed. These numerical results may be used for comparison of the theory and experiment instead of the results of order a(Za) given above. We may also use the results of numerical calculations in order to make an estimate of uncalculated contributions of the Uehling potential of order a(Za) and higher. According to [117]... 

The other state-dependent corrections originate from accurate calculations of the Zemach term, which is proportional to the value of the wave function at the origin (i/ (0)). The wave function affected by the Uehling potential leads to a correction... 

The Uehling potential represents the dominant vacuum-polarization correction of order a(Za) to the one-photon exchange potential between the nuclear-chaige distribution and the bound electron. Finally, we derive the analytically known renormalized polarization function in terms of an integral representation. For spherically symmetric external charge distributions we obtain the renormalized Uehling potential (Klarsfeld 1977) ... 

The function H can be divided into a Uehling potential part and the higher-order remainder + -i called the Wichmann-Kroll part... 

The separation of the loop also implies a separation of the corresponding potential (24) into the Uehling potential and the Wichmann-Kroll potential, as the higher orders of the Za expansion were first considered by Wichmann and Kroll in 1956... 

The evaluation of the Uehling potential can be found in a number of articles [32 - 35]. For spherically symmetric extended charge densities p ( normalized to... 

The Uehling potential is shown in Fig. 7. The plot displays also the asymptotic behaviour for r —oo, which reads... 

In atoms, the dominating Uehling potential causes the vacuum polarization effect to be attractive instead of repulsive, as would be expected from classical polarization theory. The polarization of the vacuum can be imagined as in Fig. 8. This can be understood as a result of charge renormalization. The bare charge is unobservable... 

Fig. 7. Uehling potential of the uranium nu- Fig. 8. The vacuum polarization cleus. The specified radius corresponds to charge around a central nucleus.

The leading quantum electrodynamic effects to be accounted for in electronic structure calculations are the radiative corrections known as electron self-energy interaction and vacuum polarization. For the energy of electronic systems, the latter is usually small compared to the former, but only the latter can be expressed in terms of an effective additive potential to be included in the electronic structure calculations. The total vacuum polarization potential can be expanded into a double power series in the fine structure constant a and the external coupling constant Za. The lowest-order term, the Uehling potential, can be expressed as [110-112] ... 

D. J. Hylton, Finite-nuclear-size corrections to the Uehling potential, Phys. Rev. A 32 (1985) 1303-1309. 

The asymptotic behaviour of the Uehling potential is defined by the expressions ... 

Finally we present results of SE and VP calculations for ns valence electrons in heavy and superheavy atoms with n up to 8 and Z up to 119 [13], [14]. For the calculation of the SE contribution the PWR approach described in Sec.4.2 based on the multiple commutator expansion [71] was used. The corrections are given in Table 1. Since the B-spline approach requires the employment of the local potential, the local approximation to the DHF potential obtained by the direct parametrization [77] was used. The VP contribution was treated in the Uehling approximation. One can expect that the Uehling term will suffice not only for highly charged ions but in screened systems as well. The Uehling potential was corrected for the extended nucleus [78] - [80]. The Uehling potential for the point-like nucleus (233) was replaced by the expression ... 

The dominant contribution to Fyp can be obtained as the expectation value of the Uehling potential [17]... 

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