On the relationship between densities of Shannon entropy and Fisher information for atoms and molecules

An analytical relationship between the densities of the Shannon entropy and Fisher information for atomic and molecular systems has been established in this work. Two equivalent forms of the Fisher information density are introduced as well. It is found that for electron densities of atoms and molecules the Shannon entropy density is intrinsically related to the electron density and the two forms of the Fisher information density. The formulas have been confirmed by the numerical results for the first two-row atoms.