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

An electron-density functional for the conventionally defined exchange energy density of an atom is constructed using Becke’s inhomogeneity parameter QB based on the density matrix expansion of the exchange hole. The proposed functional (the energy density meta-generalized gradient approximation or EDMGGA) has the following properties: (i) The exchange energy density εxEDMGGA(r) has correct nuclear cusp and density-tail behaviors. (ii) The corresponding exchange potential δExn/δn(r) is finite near the nucleus and decays asymptotically as −k/r in the tail. Numerical results show that our functional yields total exchange energies for atoms with about the same accuracy as Becke’s widely used functional B88, but significantly improves the local description of the exchange energy density. In one Appendix, by introducing a general coordinate transformation, we show that the asymptotic behavior of the conventionally defined exchange energy density depends upon the choice of the coordinate transformation and the established tail behavior, −1/2r, for a finite system is only a special case in the general coordinate transformation. In another Appendix, we discuss alternative definitions of the exchange energy density.