Real‐space analysis of the exchange‐correlation energy

Abstract The exchange‐correlation energy of a many‐electron system may be written as the electrostatic interaction between the electron density at position r and the density of the exchange‐correlation hole at position r + u. If we average the hole over the entire system, we find that the energy is uniquely decomposed into contributions from various electronic separations u. We may also decompose the hole into contributions from parallel and antiparallel spins. We give several exact conditions which this system‐averaged, spindecomposed exchange‐correlation hole satisfies. Local spin density ( LSD ) and generalized gradient approximations ( GGAS ), are more appropriate for u → 0 than for large u and more trustworthy for antiparallel spins than for parallel spins. We illustrate how good LSD is as u = 0 with explicit examples, but also note that, contrary to expectation, LSD is not exact for u =0, except in certain limiting cases. We show that the dramatic failure of the second‐order gradient expansion for large u can be cured by a real‐space cutoff procedure which generates a nonempirical GGA, the Pw91 functional. We conclude with some thoughts about the search for greater accuracy in the next 30 years of density functional theory. © 1995 John Wiley & Sons, Inc.