Density functional theory for open-shell systems using a local-scaling transformation scheme. II. Euler-Lagrange equation for f(r) versus that for ρ(r)

Following the previous article (Part I), we express the total nonrelativistic energy for spin manifolds of open-shell multielectronic systems, within an orbit θ^N induced by a model wave function (MWF) _Ψ using a single local-scaling transformation (LST) as an exact functional of the single-particle density ρ( r ) or, alternatively, of the LST scalar function f( r ). We derive the corresponding Euler–Lagrange variational equations: one implicit in ρ( r ), which can be solved iteratively through steps involving f( r ), and one explicit in f( r ), derived from the total energy as a functional of f( r ). Both equations fulfill the space and spin symmetries characterizing the system. The problems arising from the specificities of these two highly nonlinear integrodifferential equations are discussed. The optimal charge density ρ( r ) derived from these equations is N- and v-representable and determines the optimal spin density σ( r ) as well. Accurate optimal values of all observables can be derived from this scheme using standard procedures. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 257–268, 1997