Thomas Fermi Dirac Equation Research Articles

Numerical Calculation of the Electron Density at the Wigner–Seitz Radius Based on the Thomas–Fermi–Dirac Equation

The fourth order Runge–Kutta method was used to solve the Thomas–Fermi–Dirac (TFD) equation. This method simplifies solving the TFD equation and improves the solution accuracy. The electron density of Cu at the Wigner–Seitz atomic radius was calculated as an example, using the TFD equation. The same method was used to calculate electron densities of other 24 elements at the Wigner–Seitz radius. These results demonstrate a successful application of the Thomas–Fermi– Dirac model in materials research.

Existence of boundary value problems for TFD equation in quantum mechanics

TFD (Thomas-Fermi-Dirac) equation in quantum mechanics is established. The existence theorems of the solutions are obtained by singular boundary value problem theory of ordinary differential equation and upper and lower solution method.

Analytic form of Thomas–Fermi–Dirac dielectric function for III–V compound semiconductors

AbstractThe nonlinear Thomas–Fermi–Dirac equation is solved variationally for positive and negative point charges in various III–V compound semiconductors. The parameters are presented, and the results are compared with those obtained numerically.

Cold equation of state from Thomas-Fermi-Dirac-Weizsacker theory.

The Thomas-Fermi-Dirac (TFD) electronic structure model with the Weizsacker gradient corrections (TFD-{lambda}W) is employed to calculate the cold equation of state in the Wigner-Seitz spherical-cell approximation. We demonstrate how inclusion of the Weizsacker term removes many of the unphysical features of the TFD lattice model. Results are summarized for seven elements: {sub 6}{sup 1}{sup 2}C, {sub 12}{sup 24}Mg, {sub 26}{sup 56}Fe, {sub 47}{sup 1}{sup 08}Ag, {sub 79}{sup 1}{sup 97}Au, {sub 82}{sup 2}{sup 07}Pb, and {sub 92}{sup 2}{sup 36}U. Our equation of state (computed using several values of the Weizsacker coupling coefficient) is compared with previous computations and with experimental data. The Weizsacker correction substantially improves the theoretical TFD equation of state at low densities. We also calculate low-mass, equilibrium stellar models constructed from the TFD-{lambda}W equation of state for carbon. We find that for {lambda}=1/9 the maximum radius of a carbon white dwarf star is {ital R}/{ital R}{sub {circle dot}}=3.9{times}10{sup {minus}2} at a mass {ital M}/{ital M}{sub {circle dot}}=2.3{times}10{sup {minus}3}.

Thomas–Fermi–Dirac-jellium model of the metal surface: Change of surface potential with charge

The Thomas–Fermi–Dirac equation is formulated for the electrons of a charged metal surface, the ionic background being represented by jellium. After transformation into dimensionless form, the equation is integrated to give the electron density and the surface potential as a function of surface charge. Calculated surface potentials for many (neutral) metals agree with the results of experiment and more sophisticated calculations. Coupling the model for the metal to a parametrized model for adsorbed water, we show that the metal can make a significant contribution to the capacitance of the interface, especially for negative surface charges.

Effect of exchange on shallow-impurity calculations

The Thomas-Fermi-Dirac (TFD) screening equation is constructed for the exchange-correlation potential expressed in the $X\ensuremath{\alpha}$ approximation. The TFD equation is then solved in closed form in the linearized TFD approximation. The shallow-donor impurity ionization energy is then calculated using the linearized TFD screening function and including the exchange potential in the $X\ensuremath{\alpha}$ approximation.

Thomas-Fermi-Dirac statistical theory of dispersive dielectric screening in undoped semiconductors at zero temperature

Static dielectric screening in undoped semiconductors at zero temperature is formulated within the framework of the Thomas-Fermi-Dirac (TFD) model of a homogeneous and isotropic solid. At each point in the solid the valence electrons are treated as a degenerate gas in statistical equilibrium in the space-varying self-consistent potential of a point-charge impurity. The theory involves the electrostatic, kinetic, and exchange energies of the electrons in the development of a nonlinear TFD equation for the screened potential. The Thomas-Fermi (TF) theory of dielectric screening is recovered when exchange effects are neglected. Closed analytical expressions for the wave-vector-dependent dielectric function and the spatial dielectric function are obtained by linearization of the TFD equation and the range of validity of approximation investigated. Numerical solutions of the nonlinear TFD equation for point-charge screening show an increasing departure from linear behavior with impurity charge. These properties of the nonlinear TFD theory are already manifest in the TF scheme. A comparison between TFD- and TF-model dielectric functions shows important differences due to exchange. In the linear screening regime, it is found that impurity potentials are more effectively reduced when exchange effects are included. As a result, the TF theory compares more favorably with accurate band-structure calculations of the dielectric functions for silicon and germanium. It is expected that improvement in the TFD dielectric functions depends on extending the treatment to include correlation and/or the quantum correction. In the nonlinear regime, attractive potentials are more effectively screened in the TFD theory, while the opposite is not generally true for repulsive potentials. Finally, it is seen that donor-acceptor asymmetry is stronger in the presence of exchange effects.

Thomas–Fermi–Dirac interaction potentials for rare gas systems

The energy components and the Thomas–Fermi–Dirac interatomic interaction potentials of the Ar–Ar, Kr–Kr, and Xe–Xe systems are calculated for several internuclear distances, solving exactly the Thomas–Fermi–Dirac equation for the diatomic systems. Comparisons are made to results from other density functional approaches. The dependence of the energy components on the nuclear charge is discussed, and it is suggested that (EαTFD −EαTF)/EαTF, the relative deviation of a TFD energy component from that calculated from the Thomas–Fermi density, is proportional to Z−2/3. A possible analytic form of the interatomic potential is proposed.

Note on the Thomas–Fermi and Thomas–Fermi–Dirac model calculation of atomic polarizabilities

This paper discusses a recent calculation by Bruch and Lehnen of the static electric dipole polarizabilities of neutral inert gas atoms. These authors obtained the polarizabilities within the framework of statistical models by making use of an energy functional method. The method, as applied by Bruch and Lehnen, makes use of the exact solutions of either the Thomas–Fermi (TF) or of the Thomas–Fermi–Dirac (TFD) equations. This paper points out that a better agreement than the one obtained by Bruch and Lehnen between calculated and experimental values of the atomic polarizabilities can be obtained upon resorting to approximate (analytical) variational solutions of the TF and TFD equations. The improved agreement, in the case of the TF model, is attributed to the fact that the variational solution of the TF equation permits one to construct a radial electron density for a neutral atom that decreases with the distance from the nucleus exponentially. That this is an improvement is seen from the fact that in the exact TF model the radial electron density drops off with the distance from the nucleus as the inverse fourth power of this quantity. Essentially the same situation prevails in the case of the approximate (analytical) solution of the TFD equation, which makes use of the variational solution for a TF atom. In this case, the improved radial electron density manifests itself in a contraction of the atomic radius, relative to that which is obtained within the exact TFD model.

Accurate Thomas–Fermi–Dirac calculations for diatomic systems

AbstractThe Thomas–Fermi–Dirac equation for the electron density in a homonuclear diatomic system is solved numerically. The problems arising and methods of dealing with them are discussed and attention is given to the accuracy of the calculated energies. Comparisons are made to previously published work employing approximations, permitting a discussion of their utility.

Effect of Quantum Inhomogeneity Corrections in Semi-Statistical Thomas–Fermi Calculations for Atoms

Approximate solutions to the Thomas–Fermi equation with so-called 'quantum correction terms' have been obtained by the use of a variational method. The results for krypton support the conclusions of Tomishima and Yonei that the coefficient of the gradient term should be 9/5 of the value derived by Kirzhnits. On the other hand, when the use of these equations is restricted to a region of the atom where the gradient expansion of Kirzhnits might be expected to be valid, the Kirzhnits value of the constant gives the better results, but the best results are obtained with no correction term at all (i.e. with the Thomas–Fermi–Dirac equation).

An Extended Thomas-Fermi-Dirac Theory for Diatomic Molecule

In order to remedy the well-known defect of the Thomas-Fermi-Dirac (TFD) theory which in its usual treatment fails to give a stable binding state of a molecule, the Weizsäcker correction multiplied by a constant weighting factor λ (=1/5) is introduced as a quantum correction. Numerical calculations to solve this extended TFD equation are carried out for the nitrogen molecule, and it is shown that the TFD model with the modified Weizsäcker correction surely yields an energetically stable molecule. The calculated values of the dissociation energy D e and of the equilibrium internuclear distance R e are 0.342 a.u. and 2.425 a.u., respectively. These values are in good agreement with experiment.

Convergence of the Series Expansion Solution to the Thomas–Fermi–Dirac Equation

Convergence of the Series Expansion Solution to the Thomas–Fermi–Dirac Equation

Asymptotic Relations between the Thomas–Fermi–Dirac and Thomas–Fermi Atom Models. II. Extension of the Case of Low Atomic Number

The asymptotic relation between the pressure with exchange in the Thomas–Fermi–Dirac atom model and the corresponding pressure without exchange in the Thomas–Fermi model for the case of low atomic number considered in the first paper of this series is generalized by including a correction corresponding to the first-order Coulomb contribution to the energy of the atom, in addition to the kinetic energy of the electrons alone. Comparison is made with numerical results obtained by direct solution of the Thomas–Fermi–Dirac equation.

Asymptotic Relations between the Thomas—Fermi—Dirac and Thomas—Fermi Atom Models. I. Case of Low Atomic Number

An asymptotic relation between the pressure with exchange on the Thomas—Fermi—Dirac atom model and the corresponding pressure without exchange on the Thomas—Fermi model, obtained by Gilvarry for the limit of vanishing atomic number Z, is derived by a direct physical argument. As compared to the original derivation, this procedure obviates the need for solutions of the differential equations of the models and for discussion of termination of the series so obtained. The virial theorem is used to delimit the domain of validity of the result, and thus to indicate the circumstances under which correction terms are necessary from the solution of March and the general asymptotic solution of Rijnierse for the Thomas—Fermi—Dirac equation at small atom radius. The limiting form for the radius of an isolated Thomas—Fermi—Dirac atom as a function of the atomic number in the case where Z approaches zero is determined on the basis of the Jensen boundary condition. This expression yields an immediate interpretation of the parameter entering the connection between the pressures with and without exchange. The results are discussed in terms of homology transformations of the Thomas—Fermi—Dirac and Thomas—Fermi equations. Corresponding asymptotic forms for other thermodynamic functions (parameters associated with the energy and the equation of state, as well as the compressibility) are considered also for the case of vanishing atomic number.

Solutions of the Thomas—Fermi—Dirac Statistical Model of Atoms

Numerical integration of the zero-temperature Thomas—Fermi—Dirac equation has been carried out on the IBM 701 for atomic numbers Z = 1 to 100. Compressed, expanded, and isolated atoms were treated, the latter using the boundary condition suggested by Jensen. Physical properties of interest for compressed and expanded atoms are presented graphically and, in the case of isolated atoms, tabularly as well.

Statistical Electron Density Distributions and Thomas-Fermi-Dirac Screening Functions for Neutral Atoms

Statistical electron density distributions and Thomas-Fermi-Dirac (TFD) screening functions have been obtained for the 104 elements corresponding to integral atomic numbers $Z=2$ to $Z=105$, for 117 values of radial distance from the atomic center in each case. These results were calculated, in part, by using Thomas' solutions of the TFD equation in terms of the statistical electron distributions for nonintegral values of $Z$ and Jensen's boundary conditions. Values for argon and copper are given here. The complete set of tables for all 104 elements has been deposited with the American Documentation Institute Auxiliary Publications Project.

Solution of the Thomas-Fermi-Dirac Equation

THE fundamental equation of the Thomas–Fermi–Dirac statistical atom model1 is the so-called Thomas–Fermi–Dirac equation where ψ″ denotes the second derivative of ψ by x, β represents the constant β = 31/3/(321/3π2/3Z2/3) and Z the atomic number. This equation was solved by Umeda2 for all neutral atoms with the boundary conditions derived by Brillouin3: where x0 corresponds to the border of the atom. The second and the third boundary conditions are equivalent to the minimization of the energy of any single electron in the statistical atom. This requirement and the boundary conditions resulting from it are, however—as has been already mentioned by Jensen4—not in agreement with the statistical point of view, according to which the total energy of the statistical atom has to be a minimum, which is not equivalent to the minimization of the energy of the single electrons.