Ionization potential of large metallic clusters: explanation for the electrostatic paradox

The ionization potential of large metallic clusters is IP=WF+α(e2/4πε0R), where WF is the work function, R is the cluster radius, and α is a constant that is equal to 3/8 or 1/2 depending on the method the above formula was derived. This electrostatic paradox was resolved here by using proper values for the work function. In the image charge method WF refers to a plane (denoted as W∞), whereas in the spherical-capacitor approach WF refers to the surface of a grounded metallic ball, which is W∞−(1/8)(e2/4πε0R).