Hartree-Fock and density functional theory calculations are used to study the 1,3-intramolecular proton-transfer reactions in HXNY --> XNYH (X,Y = O,S). Energy and a number of reactivity descriptors such as chemical potential, hardness, and electrophilicity index have been studied along the proton-transfer reaction path. It has been found that the profiles of most properties of the 1,3-reordering HXNY --> XNYH can be described as a combination of the corresponding profiles of symmetric reactions in which X = Y. A good linear correlation between potential energy and the electrophilicity index in the HONO --> ONOH reaction has been found, indicating that any change of these properties during the reaction are well represented in terms of changes in both the electronic chemical potential and molecular hardness.