Volume 16, number 1 CHEMICAL PHYSICS LETTERS 15 September 1972 FEW-ELECTRON CORRECTIONS OF STATISTICAL EXCHANGE POTENTIAL IN LOW-ENERGY ELkCTRON SCATTERING W.H. Eugen SCHWARZ Lehrstuhl fir 7’heoretische Chemie der Lbiversitiit, 53 Born,, Gerrna~~y Received 28 April 1972 Revised manuscript received 2G June 1972 In order to allow for the relative increase of self-interaction in few-electron systems, a reduction factor is intro- duced into the statistical freeelectron exchange potential. This netexchange potential has been used in the calcula- tion of elastic electron scattering by rare-gas atoms in the lo~-energy region from 0 to 10 eV. U’hereas former st- tempts to treat thew phenomena as simple potential scattering have failed, in this work at least a crude qualitative ageement with experimental data has been achieved in most uses. 1. Introduction Quantum-chemical ab-initio many-electron calcula- tions in the numerical Hartree-Fock regime may be considerably simplified. if the exchange operator is ap- proximated by a local density-dependent potential, the so-called exchange potential. Slates [I] and G&&p& [2] replaced the “gross- exchange” operator ciicP which is invariant under orthogorlal transformations of the one-electron orbi- tals, by corresponding free-electron gas formulae. This Hartree-Fock-Hater (I-IFS) method has since then been applied with increasing success to atoms and solids [3] and recently to molecules (41, too. Contrary to this situation, the simple statistical free-electron formula for the Coulomb-correlation en- ergy leads to values too large by a factor of about two, if applied to the inhomogeneous electron systems in atoms [5]. Schwarz [6] has pointed out that nearly the same factor is obtained, if one compares the free- electron exchange ener,v with the Fermi-correlation or “net-exchange” energy Ci>jKij (in the basis of canonical orbit&). In other. words, the self-interaction amounts to about 50% of the gross-exchange energy in heavy atoms and molecules, and to even more in a Riidenberg-localized basis or in systems of very few electrons. It has ttzrefore been suggested by Comb& [7], Cowan [8], Schwarz [9] and Lindgren and Rose’n [IO] only to approximate the net-exchange operator r,j+;Kj by a free-electron approximation, leading to an exchange corrected I&tree method (“enveitcrte Hartree-Methode”[9]) instead of the usual HFS method. Very satisfactory results have recently been obtained by Lindgren [ 1 11, 2. Effective one-electron problems Whereas both methods are competitive in many- electron calculations, this is not the case with effec- tive one-electron calculations, of which two examples are to be mentioned: (i) pseudo-potential calculations on a single valence or Rydberg electron, (ii) elastic electron scattering treated as potential scat- tering. In these probiems, an electron moves in the electro- static plus polarization potential of an atomic or mo- lecular core or target, which should be augmented by a ,tet-exchange potential in order to allow for elec- tron-core exchange. As is known, the self-interaction energy density of the (infinitely-extended) homogeneous electron gas 89