Asymptotic Expansions of the Dirac Density Matrix

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Abstract
Asymptotic expansions are derived for the density operator ρN(x′, x)= ∑ n=1Nψn(x′)ψn(x)for small and large values of the relative distance x′ − x with a WKB expansion for bound-state wavefunctions in a plane, one-dimensional potential. In addition to some previously obtained corrections, the particle-density expansion includes a number of steady and oscillating correction terms to the zero-order Thomas-Fermi density which are related to the more recently derived corrections of Payne and of Kohn and Sham. A convenient and accurate approximation to the density operator for all arguments is also obtained from the first two terms of the expansion for large relative distances. The expansions become inadequate in the vicinity of the classical turning point of the highest energy state, as illustrated by numerical comparisons with the exact-density operator for the quadratic potential. A novel method is described for summing a finite series of oscillating terms such as occur in the density operator.