http://WWW.Q-CHEM.ORG FULL PAPER Some Formal Properties of Ensemble Density Functionals Daniel P.Joubert Formal properties of ensemble density functionals are examined. Expressions for the difference between energy functionals where the particle number differs by one are constructed in terms of their first functional derivatives for the universal energy functional, the electron–electron repulsion energy functional, and the interact- ing kinetic energy functional. Equations that must be satisfied by second and higher order functional derivatives are derived. It is also shown that the shape of δVee [ρ] δρ (r) and δK [ρ] δρ (r) , the functional derivatives of the mutual electron–electron repulsion, and kinetic energy, respectively, are separately particle number independent for particle numbers between successive integers. © 2012 Wiley Periodicals, Inc. DOI: 10.1002/qua.24150 Introduction Formal properties of density functionals play an essential role in the development of approximate density functionals. [1] The Kohn–Sham (KS) [2] implementation of density functional theory (DFT) [3] is a formidable tool in quantum chemistry and con- densed matter physics. The formulation is in principle exact and elegant in its simplicity. It allows the calculation of the ground state density and energy of an interacting system electrons by performing a self-consistent calculation for a fictitious nonin- teracting system of particles, a calculation that is vastly simpler than any attempt to calculate properties of an interacting sys- tem directly. In all practical applications of DFT, approximations to the exact functionals have to be made. [4–8] There is no con- sistent way to develop approximate functionals [1] and a large number of approximations have been proposed during the past two decades. [4–8] An appealing and successful approach to the design of improved approximate density functionals is by “constraint satisfaction,” [1] where the approximate functionals are required to satisfy properties of the exact functionals. The quality and accuracy of approximate functionals can be tested and benchmarked by comparing to experiment or accurate calculations. [4–9] Formal properties of density functionals can give insight into the internal structure of functionals and can guide the develop- ment of approximate functionals. In this article, formal properties of exact ensemble density functionals are examined. New results in this article include Eqs. (33), (53), and (54) for G[ρ], the uni- versal energy functional, V ee [ρ], the mutual electron–electron Coulomb interaction energy functional, and K [ρ], the interac- tion kinetic energy functional for which the difference between energy functionals where particle number differs by one are constructed in terms of their first functional derivatives. Equa- tion (50) was derived by the author in Ref. [10] for pure state DFT, but here it is generalized to functionals of non-integer particle numbers. Information on particle number invariance for the first functional derivative of V ee [ρ] is contained in Eq. (50). Particle number invariance of the functional derivatives of V ee [ρ], K [ρ] and G[ρ], is discussed in Discussion and Conclusion section. Equation (63) shows constraints on higher order func- tional derivatives. In Derivation of properties of Higher Order Functional Derivatives and the Shape Invariance of Function- als as a Function of Particle Numbers, it is shown that the shapes of δVee [ρ] δρ (r) and δK [ρ] δρ (r) are particle number independent for particle numbers between successive integers. The shape invari- ance of δG[ρ] δρ (r) , as a function of particle number, follows form the Euler-Lagrange equation (44). The results in this article are derived within the framework of a coupling constant ensemble formalism [11–15] where the electron–electron interaction strength is varied between zero and one through the coupling constant γ as defined in Eq. (3). When the coupling constant is zero, the electrons are treated as noninteracting Fermions and in the adiabatic coupling constant connection, where the density is constructed to be independent of the coupling constant, the zero coupling constant system corresponds to the KS independent particle system. When the coupling constant γ = 1, the fully interacting system is recov- ered. Adiabatic coupling constant studies have been successfully used in previous investigations to uncover formal properties of density functionals. For example, an exact perturbation expan- sion for the correlation energy was derived by Görling and Levy, [16, 17] while in Ref. [18], DFT ionization formulas were devel- oped. Other formal studies where use was made of the adiabatic coupling constant formalism include Refs. [19–24]. In this article, the adiabatic connection system for particle number N is used as an intermediate reference system to derive relationships between functionals. Results are derived for densi- ties of arbitrary particle numbers n with the external potential of the adiabatic connection system used for all values of γ . The final results are independent of the reference system and are valid for all ensemble v -representable densities [25] at all coupling constants. D. P. Joubert Centre for Theoretical Physics, School of Physics, University of Witmatersrand, PO Wits 2050, Johannesburg, South Africa E-mail: daniel.joubert2@wits.ac.za Contract grant sponsor: National Research Foundation; contract grant num- ber: 48543. © 2012 Wiley Periodicals, Inc. http://onlinelibrary.wiley.com International Journal of Quantum Chemistry 2012, DOI: 10.1002/qua.24150 1