Ground state densities from electron propagators: Optimized Thomas–Fermi approximation for short wavelength modes

Abstract

Electron densities obtained from a ground state path integral approach to density functional theory using a primitive Monte Carlo method display large statistical uncertainties when short wavelength fluctuations of the paths are considered directly. An optimized Thomas–Fermi approximation is developed to eliminate these degrees of freedom in a systematic and physically motivated fashion. Beyond improving the precision of the numerical results, this theoretical development permits a simple qualitative discussion of how the calculation pushes electron density into tunneling regions by iteratively renormalizing the Fermi wavelength appropriate to the remaining integrations which involve only long wavelength coordinates.

Statistical theory of electron densities at nonzero temperatures

“…These developments build from recent work on statistical theories of electron densities in extended systems (12)(13)(14)(15)(16)(17). The principal objective here is to work out the finite temperature version of the optimized Thomas-Fermi theory (14)(15)(16)(17).…”

“…The principal objective here is to work out the finite temperature version of the optimized Thomas-Fermi theory (14)(15)(16)(17). The new result is indeed presented below.…”

“…Therefore, [6] shows that the finite /3 result corresponds to an averaging of ground state densities according to a distribution of Fermi energies. Because of this, the Monte-Carlo methods devised elsewhere (13,15,16) to implement the ground state theory are directly applicable to the finite temperature theory as well. In particular, the thermal result can be implemented by considering cyclic paths {r,r,, .…”

“…For personal use only. rable with the Fermi wavelength ( 8~r / 3~) "~ then the ThomasFermi approximation for the ground state density, would be appropriate (14)(15)(16)(17)26). When P is finite the familiar thermal Thomas-Fermi approximation obtains…”

“…More physically, the density depends on the potential in the neighborhood of the point in question. In the interest of simplicity, we can agree to retain the simple form, [9], but we can make it more accurate by replacing z~(r;p) with a parameter u, chosen so as to make the first-order corrections to [9] vanish (14)(15)(16)(17). Using [6] …”