Translational and rotational invariance requisites for density functional derivatives

Translational and rotational invariance of functionals lead to hierarchies of equations between successive derivatives. These hierarchies allow alternating series expansions of some density functionals in terms of functional derivatives and charge density. Translational and rotational invariance also give rise to integrodifferential equations that link derivatives of all orders. From the minimal properties of the kinetic energy functional T_sρ] and the functional Fρ] = min_Ψ→ρ <Ψ|T + V_ee|Ψ>, it follows that $int d?3 r d?3 r?primef({bold r}) {del?2 T―srho]}over{delrho({bold r})delrho({bold r?prime})}f({bold r?prime}) geq 0hspace{1cm}hbox{and}hspace{1cm}int d?3 r d?3 r?primef({bold r}) {del?2 Frho]}over{delrho({bold r})delrho({bold r?prime})}f({bold r?prime}) geq 0$ for all ∫ d³ r d³ f′ f(r) = 0. This property combined with constraints on functionals due to translational invariance lead to inequalities satisfied by first derivatives of selected density functionals. © 1997 John Wiley & Sons, Inc.