State Energy Functionals and Variational Equations in Density Functional Theory

In two recent papers [1,2] we have presented significant portions of an effort to provide a comprehensive mathematical basis for Density Functional Theory. The aim is to be both formally sound and to provide a structural framework for the development of improved approximations. The earlier paper addressed the relationship between symmetry breaking in Kohn-Sham solutions and the essential structure of the KS orbitals that emerges from general classification by spin-rotation and time-reversal symmetries. The later paper summarized a reformulation of the constrained-search construction of time-independent Hohenberg-Kohn functionals and their variational properties and, on that groundwork, presented a new formulation of time-dependent DFT based on the timedependent variational principle. New and/or improved results for several major aspects of the time-independent problem were stated there without proof. Here we address those issues. We pay particular attention to rigorous definition of the functionals (including careful distinctions among them that often seem to be ignored), rigorous conditions for the existence of well-defined functional derivatives, conditions under which a density is associated with a one body potential (V-representability), and formulation of various kinds of one-particle functional equations (including but not limited to Kohn-Sham equations).