Natural variables for density functionals

Abstract

Various energy functionals in density-functional theory must possess the form of ^b f (z , h ,j1 ,j2 ,. . . ,jn)&, if one assumes that they can be expressed as ^g(l ,r,r(r),r ,r , . . . ,r )&. Here r(r) and r (m) are the electron density and its mth-order gradient; f and g are proper analytic functions of their variables; m is 2 when the functionals are Ts@r# , Tc @r# , and T@r# ~otherwise it equals 1!; l is the electron-electron interaction coupling constant in the adiabatic connection formulation; and $b ,z ,h ,j1 ,j2 ,. . . ,jn% is a set of independent variables ~called natural variables! in place of $l ,r,r ,r ,r , . . . ,r %. These variables are defined as $b(r)5r(r), z5b/l , h5rb , jm5r /bum51,2,.. . ,n%. Generalizations to more complex functional forms are also discussed. Some exact relationships are derived that should be useful for developing density functionals of the weighted-density approximation type. @S1050-2947~97!01806-4#