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Manuscript Title: Vacuum-polarization potentials of extended nuclear charges.
Authors: V. Hnizdo
Program title: VACPOL
Catalogue identifier: ACVM_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 83(1994)95
Programming language: Fortran.
Computer: IBM 3081.
Operating system: VM/XA SP 2.1.
RAM: 34K words
Word size: 32
Keywords: Atomic physics, Structure, Vacuum polarization, Extended charge Distribution, Fusion muon-catalyzed, Fourier-bessel expansion.
Classification: 2.1.

Nature of problem:
The first-order vacuum-polarization correction of quantum electrodynamics to the Coulomb potential of a point charge is the so-called Uehling potential. The first-order vacuum-polarization potential due to two extended charges is obtained by folding the Uehling potential with the density distributions of the two charges. The program calculates such a potential as a function of the distance between the centres of the two density distributions, which can have the functional forms that are most often used to model the distribution of charge in atomic nuclei.

Solution method:
The vacuum-polarization potential is calculated using the Fourier-Bessel expansion method for the evaluation of folding integrals as applied to the present problem in [1,2]. Analytical methods are employed throughout, thus enabling a high accuracy and efficiency of the calculations.

Spherically symmetric charge distributions are assumed, which can have the modified harmonic oscillator, Fermi density, uniform, or point-like (i.e., the Dirac delta function) functional forms. The vacuum- polarization potential can be calculated easily to a more than 10-digit accuracy for distances where the effects of the finite size of the charges are appreciable. At greater distances, i.e., distances where there is no significant overlap of the two charge distributions, the vacuum-polarization potential goes over into the Uehling potential of point charges, which can be calculated using an exact closed-form expression.

Running time:
The running time is approximately proportional to the number of radial steps at which the vacuum-polarization potential is calculated and the number of terms employed in the Fourier-Bessel expansion (a calculation using closed-form expressions only is much faster). On an IBM 3081 computer, the test calculation of section 4, which comprises a closed- form expression case and two cases of 50 radial steps with Fourier- Bessel expansions of 400 and 300 terms, takes about 1.4 s.

[1] V. Hnizdo, J. Phys. A: Math. Gen. 21 (1988) 1629.
[2] V. Hnizdo, J. Phys. A: Math. Gen. submitted for publication.