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We derive the next order correction to the Dirac exchange energy for the freeelectron gas in a box with zero boundary conditions in the thermodynamic limit.The correction is of the order of the surface area of the box, and comes fromthree different contributions: (i) a real-space boundary layer, (ii) aboundary-condition-induced small shift of Fermi momentum and bulk density, and(iii) a long-range electrostatic finite-size correction. Moreover we show thatthe LDA, in addition to capturing the bulk term exactly, also produces acorrection of the correct order but not the correct size. GGA corrections arefound to be capable of capturing the surface term exactly, provided thegradient enhancement factor satisfies a simple explicit integral constraint.For current GGAs such as B88 and PBE we find that the new constraint is notsatisfied and the size of the surface correction is overestimated by about tenpercent. The new constraint might thus be of interest for the design of futureexchange functionals.

Next-order correction to the Dirac exchange energy of the free electron gas in the thermodynamic limit and generalized gradient approximations