<p>A comparative analysis between the procedure originally followed to establish the local hardness concept and a recent one based on a local chemical potential, defined within the grand canonical ensemble formalism, is done to get a better understanding of the main aspects involved in both procedures and to show that the local and non-local counterparts of global reactivity descriptors recently developed, constitute an excellent complement to analyze chemical reactivity within the framework of conceptual density functional theory.</p> Graphical abstract <p>A comparative analysis between the procedure originally followed to establish the local hardness and the hardness kernel concepts and a recent one based on a local chemical potential, defined within the grand canonical ensemble formalism, is done. The original local softness and softness kernel remain unchanged.</p> <p></p>