Modified Weizsäcker Corrections in Thomas-Fermi Theories Modified Weizsäcker Corrections in Thomas-Fermi Theories

in which s kinetic-energy correction (Vs /32m m) f(V'p) /p] (p = density) is added to the usual Thomas-Fermi term. A treatment based on the WKB method and expected to be valid for large ~ shows 1=1 here, as found experimentally. For small x, A. =1 is needed to give proper behavior of p, but other arguments suggest that the Thomas-Fermi term be dropped here.

R. H, Hunt, C. W. Robertson, and E. K. Plyler, Appl. Opt. 6, 1295 (1967). 5W. Boyd, P. J. Brannon, and N. M. Gailar, Appl. Phys. Letters 16, 135 (1970). P. A. Jansson, J. Opt. Soc. Am. 60, 184 (1970 It seems that simply by adding the Weizsacker term to the usual Thomas-Fermi kinetic-energy term,~~p gives too much kinetic energy. The original derivation of U has been questioned, and it was suggested by Berg and Wilets that a term XU be used with X&1. It was found that X-8 works well for the harmonic oscillator, and '=(::::::)"' && exp -' p"dx+ p, dy+ p, dz +8, (&) where v" is the classical frequency, given by the size of the energy quantum divided by Il, and v" =P"/m is the speed at the point in question. The density, making the correspondence Zz»k f B"vzvz/v"vzvzdPzdPzdPz is p=2Z"@"-2k Jd p=8mp'/8k where it is assumed that the distribution in momentum is spherically symmetric at each point, P (r) being the maximum value of momentum.
In the zeroth approximation, the speeds are taken as constant in (8) and the kinetic-energy density, using (5), is yansions in @, and some of these derive terms of the form XU . For instance, Golden'found A. =~4, and several Russian authors & = g It seems, 8 1 however, that the expansions a,re not valid ' for very large or very small x, just where the correction is important. In fact, poor results are obtained'"' for energies with these corrected theories. Taking an experimental point of view, Yonei and Timoshima considered noninteracting electrons in a Coulombic field using the correction X U with X varying from 0 to 1 in steps of 0. 2.
They found that X = 1 led to densities in good accord with quantum-mechanical ones for large~, while X = 0. 2 gave the best results with respect to the small-x density. Later work on the rare gases with X= 0. 2 by these authors' confirmed this: p was accurate near r =0 but not for x-~. In this payer we wish to make several observations which suggest a slightly different way of using the Weizsacker correction.
We first consider large r, using a "derivation" of the Weizsacker term given by F6nyes' in terms of the WKB method. We employ the forma. lism of Brillouin' here. At some point in space, let the potential be locally separable along axes x, y, z.
Then the WKB one-electron wave functions are written in the form where w=P"P, P, . We again obtain ('7) but also an additional term, which we may write Turning to small x, we note that the singularity of the Coulomb field places a restriction on the behavior of the correct density. It is reasonable to demand that the modified Thomas-Fermi theory give a density obeying this restriction. One can show' that the correct density obeys where ao is the Bohr radius. This follows from a theorem by Kato concerning the wave functions, and is valid for the n-electron system, whether or not the electrons are interacting. It is easily understood when one realizes that, when very near a nucleus, the electrons see essentially a pure Coulombic field, and the electron density is essentially that of one or two 18 electrons: gZ8/ )e-2zziao One might as well have used plane waves here. In the next approximation, we must consider that dv"/dx is not zero (inhomogeneity), although our choice of axes means that v"does not depend on y or z, and so on. Then To summarize, our arguments suggest that the full Weizsacker correction (X= 1) be used throughout along with the Thomas-Fermi kinetic-energy term, except that the Thomas-Fermi term be dropped near the nucleus. Tomishima and Yonei' have given the electron density for calculations on the rare gases where UTF+ U"was used. One can use their results to get a rough estimate of the validity of our method, by calculating Jkso I~d v over a sphere of radius Z ao around the nucleus. We find this quantity to be roughly half of the difference between the calculated and correct energies. This suggests that f (v) be essentially zero out to x larger than Z~ao. The correct cutoff should in fact be such that the 1s density no longer dominates the total density. Gombas has in fact suggested"2 that the Thomas-Fermi term U» be multiplied by a correction factor. For the electrons with principal quantum number n, the kinetic energy would be the Weizsacker contribution plus (2n -2)/(2n+1) times the Fermi contribution [Eq. (7)]. In the present interpretation, the factor of X =0. 2 in the Weizsacker term near the nucleus (while keeping the full Fermi term) found by Yonei and Tomishima has no deep significance. It is suggested that 0 2 Uw+ UTF is roughly equivalent to Uw +f(x)UTr over the important region.