Phase-space approach to the exchange-energy functional of density-functional theory

The phase-space distribution function corresponding to a ground-state density of a many-electron system proposed earlier [S. K. Ghosh, M. Berkowitz, and R. G. Parr, Proc. Natl. Acad. Sci. USA 81, 8028 (1984)] is employed to obtain a new approximate exchange-energy functional. This is K[{ρ}]=({π}/2) F ρ²(r){β}(r)dr, with {β}(r)=1/kT(r), where T(r) is the local temperature previously defined; (3/2)kT(r) is the kinetic energy per electron at r. In Thomas-Fermi theory, {β}=5(3π²{ρ})^-2/3, and this formula gives (10/9) times the classical Dirac formula. This shows why the {α} parameter in X{α} theory is normally close to ((10/9))((2/3))=0.74. Numerical calculations on atoms are performed, giving excellent results, and the exchange hole associated with the new formula is studied in detail.