PHYSICAL RE VIEW B VOLVME 11, NUMBER 10 15 MA Y 1975 Cohesive energy and the P V relatipn for fcc neon in two variants of the Xa model~ J. R. Sabin, Joseph P. Worth, and S. B. Trickey Quantum Theory Project, Department of Physics df: Astronomy, University of Florida, Gainesville, Florida 32611 (Received 18 November 1974) The cohesive energy and associated PV relation for fcc neon have been calculated in the muNn-tin approximation via the augmented-plane-wave statistical-exchange (APW-Xa) method. Exchange parameters of 0. 72997 (Schwarz's avT), 0.66667 (the Kohn-Sham-Gaspar value) and 0.500 were used. In all cases, the static lattice is significantly overbound, the amount of overbinding lessening with decreasing a. The introduction of a simple radial independence (a = 0. 7772 for r g r„a = 2/3 for r p r, with r, some suitable change point inside the APW sphere) into the model does not cure the overbinding. I. BACKGROUND AND INTRODUCTORY REMARKS Over the last several years, the Xn statistical exchange model' has enjoyed considerable suc- cess as a procedure for the calculation of me- tallic and ionic crystalline cohesive energies and zero-temperature equations of state in the static-lattice limit. Recently, the model has been applied to the fcc rare-gas crystals' Ar, Kr, and Xe. In this latter work (hereafter re- ferred to as I), it was found that a popular crite- ~ rion for selecting the value of n [the "virial-theo- rem" o. , see Sec. II of I] enabled the calculation of static-lattice constants which were in surpris- ingly good agreement with experiments. The calculated PV relations were in fairly good agree- ment with experiment. A trend noted in I was that the best system so far as agreement of calcu- lated cohesive energy and lattice constant with experiment was argon, while the worst was xenon (about 30% underbound). However, the best calcu- lated PV relation was that for xenon. Extrapolation of the trends found in I suggests that fcc neon should be overbound when. the same criterion for the choice of n is used. In this paper we report confirmation of the prediction and, fur- ther, the inability of either a simple adjustment in the value of n or a simple radial dependence of a (Ref. 11) to improve matters. II. SUMMARY OF THE AI.&uMENTED-PLANE-WAVE STATISTICAL-EXCHANGE (APW-Xn ) METHOD The Xe scheme takes the total energy of the electrons in a solid (in the static-lattice approxi- mation) to be the same as the exact expression except for the replacement of the exchange-correla- tion contribution by a local exchange-correlation operator which, in the non-spin-polarized case, is assumed to be U„(1) =-9n(3/8m)' ~ p' ~(1) (1) Here p is the charge density (see I for details). It is this assumption which determines all the prop- r& r. + &c (2) Here r is the distance from the nuclear site and r, is the solitary parameter. This alternative was intended to provide a more appropriate Z dependence of a than that provided by Xn/1. We found that direct calculations of z, (by matching the Xo/2 atomic total energy to the Xo/I result) yields r, =0. 37 bohr, the value we used. [Wood'~ has reported z, = 3. 30/(Z — l. 65) which gives 0. 395 bohr for neon. ] Whether we employ the Xn/I or Xo/2 choice of local- exchange- correlation potential, the method of solution of the ensuing one-electron equations for the orbitals u; which we have used is the APW method in its traditional muffin-tin form (see Ap- pendix), with the computational details virtually the same as in I. Note that since the publication of I, Janak' has completed a self-consistent Kor- ringa-Kohn-Rostoker (KKR} calculation on fcc argon at a=9. 5 a. u. and @=0. 72100. The total energy he calculates is only slightly different erties of the method, details of which are eluci- dated elsewhere 'a'a In the cases of homogeneous crystals, homo- nuclear molecules, or i.solated atoms, the stan- dard practice to date has been to set the parameter o. 'in (1) to some fixed "optimum" value which is constant throughout space. There are a variety of suggestions for the choice of the optimum value; earlier we employed the so-called virial-theorem value o. vT (see I for details). The result of this procedure is a wel1. -defi. ned value of n for each value of the atomi. c number. For brevity, we shall refer to this variant of the Xn method (o. constant spatially, va, rying with Z) as Xo/l. In accordance with a suggestion by Slater' we have tried, for the first time, a variant of the method for which n has the radial dependence of a step function (which we will call Xo/2} 3658