Universal mathematical identities in density functional theory: Results from three different spin-resolved representations

This paper supersedes previous theoretical approaches to conceptual DFT because it provides a unified and systematic approach to all of the commonly considered formulations of conceptual DFT, and even provides the essential mathematical framework for new formulations. Global, local, and nonlocal chemical reactivity indicators associated with the "closed-system representation" (N(alpha),N(beta),nu(alpha)(r),nu(beta)(r)) of spin-polarized density functional theory (SP-DFT) are derived. The links between these indicators and the ones associated with the "open-system representation" (mu(alpha),mu(beta),nu(alpha)(r),nu(beta)(r)) are derived, including the spin-resolved Berkowitz-Parr identity. The Legendre transform to the "density representation" (rho(alpha)(r),rho(beta)(r)) is performed, and the spin-resolved Harbola-Chattaraj-Cedillo-Parr identities linking the density representation to the closed-system and open-system representations are derived. Taken together, these results provide the framework for understanding chemical reactions from both the electron-following perspective (using either the closed-system or the open-system representation) and electron-preceding perspective (density representation). A powerful matrix-vector notation is developed; with this notation, identities in conceptual DFT become universal. Specifically, this notation allows the fundamental identities in conventional (spin-free) conceptual DFT, the N(alpha),N(beta) representation, and the N=N(alpha)+N(beta),N(S)=N(alpha)-N(beta) representation to be written in exactly the same forms. In cases where spin transfer and electron transfer are coupled (e.g., radical+molecule reactions), we believe that the N(alpha),N(beta) representation may be more useful than the more common N,N(S) representation.