An optimal iteration method with application to the Thomas-Fermi equation

The aim of this paper is to introduce a new approximate method, namely the Optimal Parametric Iteration Method (OPIM) to provide an analytical approximate solution to Thomas-Fermi equation. This new iteration approach provides us with a convenient way to optimally control the convergence of the approximate solution. A good agreement between the obtained solution and some well-known results has been demonstrated. The proposed technique can be easily applied to handle other strongly nonlinear problems.     

Rational Chebyshev pseudospectral approach for solving Thomas-Fermi equation

In this Letter we propose a pseudospectral method for solving Thomas-Fermi equation which is a nonlinear ordinary differential equation on semi-infinite interval. This approach is based on rational Chebyshev pseudospectral method. This method reduces the solution of this problem to the solution of a system of algebraic equations. Comparison with some numerical solutions shows that the present solution is highly accurate.     

Divergence-free WKB method

A new semiclassical approach to the linear and nonlinear one-dimensional Schr?dinger equation is presented. For both equations our zeroth-order solutions include nonperturbative quantum corrections to the WKB solution and the Thomas-Fermi solution, thereby allowing us to make uniformly converging perturbative expansions of the wave functions. Our method leads to a new quantization condition that yields exact eigenenergies for the harmonic-oscillator and Morse potentials.     

Time dependent Thomas Fermi approach to atomic collisions I.

A numerical solution of the time dependent Thomas-Fermi equations introduced by Bloch is presented for the case of the scattering of a proton from an argon atom. We discuss in detail the resulting time development of the many electron density distribution and velocity fields for proton bombarding energies of 0.4, 0.9 and 2.5 MeV. Possible foundations of the model, in particular the relation to the time dependent Hartree-Fock method are indicated.     

The imaginary time step method for Thomas-Fermi equations

The imaginary time step iteration procedure is extended to the solution of the Thomas-Fermi equations and their modifications. The convergence of the procedure is discussed and illustrated by a numerical example. The generalization to the finite temperature calculations is indicated.     

Semiclassical model of a one-dimensional quantum dot

A one-dimensional quantum dot at zero temperature is used as an example for developing a consistent semiclassical method. The method can also be applied to systems of higher dimension that admit separation of variables. For electrons confined by a quartic potential, the Thomas-Fermi approximation is used to calculate the self-consistent potential, the electron density distribution, and the total energy as a function of the electron number and the effective electron charge representing the strength of interaction between electrons. Use is made of scaling with respect to the electron number. An energy quantization condition is derived. The oscillating part of the electron density and both gradient and shell corrections to the total electron energy are calculated by using the results based on the Thomas-Fermi model and analytical expressions derived in this study. The dependence of the shell correction on the interaction strength is examined. Comparisons with results calculated by the density functional method are presented. The relationship between the results obtained and the Strutinsky correction is discussed.     

Solution of the Thomas-Fermi problem for continuous,spherical, positive charge distribution

A method to make anab initio calculation of the free energy of small drops of liquid metal is presented. The model chosen involves the replacing of the positive ion cores by an equivalent continuous spherical distribution of charge. The Thomas-Fermi potential is calculated as a starting point for a Hartree-Fock self-consistent field calculation. The results of the Thomas-Fermi calculation are reported as an example of the preliminary calculations.Research supported under NSF Grant #20884.     

An energy density functional-pseudopotential method for calculating the cohesive properties of metals

The great advantages of Thomas-Fermi approaches are their simplicity, flexibility and physical immediacy. However, it is generally accepted that such approaches are not capable of yielding quantitative results for the cohesive properties of metals. I demonstrate within the Thomas-Fermi approach that if one employs hard-core pseudopotentials derived from atomic charge densities, then accurate static properties for simple metals can be obtained.