Coulomb Exchange Energy from Shell-Model Wave Functions

The Coulomb direct and exchange energies of several light nuclei have been calculated by using harmonic-oscillator wave functions and approximate wave functions for a realistic diffuse-boundary potential. The $Z$ dependence of the total exchange energy is expressed in terms of a convenient parameter $c$. Calculations show that for very low $Z$ this parameter is appreciably smaller than a prior estimate by Bethe and Bacher based upon the statistical model, but that it rapidly reaches their value $5{\left(\frac{3}{16\pi }\right)}^{\frac{2}{3}}$ for $Z\gtrsim 10$.