Coordinate Scaling Requirements for Approximating Exchange and Correlation

Abstract

In order to continue to better approx. the exact exchange-correlation d. functional, Exc[n], for electronic structure calcns., sets of exact coordinate scaling and convexity requirements are reviewed with 53 refs. for the purpose of testing approxns. The scaling requirements dictate how Exc[n] must transform when n(x,y,z) is replaced by .alpha..beta..gamma.n(.alpha.X,.beta.Y,.gamma.Z). High-d. and low-d. limits are explored. Simple formulas are shown which isolate the kinetic component of Exc[n] and which generate Exc[n] from just its electron-electron repulsion component, or exchange-correlation hole, at a coupling const. of unity. The scaling and convexity conditions are derived with help from the constrained-search formulation and through the development of a coupling-const. perturbation expansion. The local d. approxn. and various gradient approxns. are compared against the exact conditions, with emphasis upon where the conditions are respected and where the conditions are violated. Numerical results are presented.