N-derivative of Shannon entropy of shape function for atoms

The Shannon entropy of the ratio of electron density and the number of electrons, shape function entropy, is reported for the atoms He-Ac within the non-relativistic exchange-only optimized effective potential model. The derivative of the shape function entropy with electron number at constant external potential is related to an integral containing the difference between the average Fukui function and the shape function weighted by the logarithm of electron density. The trends in the shape function entropy, its spin analogue and the corresponding derivatives with electron number reveal interesting periodic behaviour.