Exchange energy density of an atom as a functional of the electron density

An electron-density functional for the conventionally defined exchange ener gy density of an atom is constructed using Becke's inhomogeneity parameter Q(B) based on the density matrix expansion of the exchange hole. The propos ed functional (the energy density meta-generalized gradient approximation o r EDMGGA) has the following properties: (i) The exchange energy density eps ilon (EDMGGA)(x)(r) has correct nuclear cusp and density-tail behaviors. (i i) The corresponding exchange potential deltaE(x)[n]/deltan(r) is finite ne ar the nucleus and decays asymptotically as -k/r in the tail. Numerical res ults show that our functional yields total exchange energies for atoms with about the same accuracy as Becke's widely used functional B88, but signifi cantly improves the local description of the exchange energy density. In on e Appendix, by introducing a general coordinate transformation, we show tha t the asymptotic behavior of the conventionally defined exchange energy den sity depends upon the choice of the coordinate transformation and the estab lished tail behavior, -1/2r, for a finite system is only a special case in the general coordinate transformation. In another Appendix, we discuss alte rnative definitions of the exchange energy density. (C) 2001 American Insti tute of Physics.