Approximate Variational Solution of the Thomas-Fermi Equation for Atoms

Csavinszky, P.

doi:10.1103/physrev.166.53

Cited by 73 publications

(14 citation statements)

References 7 publications

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“…Csa\·inszky (1968Csa\·inszky ( , 1973 has chosen the following four-parameter trial function Thus, the lower-bound variational principle is reduced to the simple minimization procedure given by expression (5.3-70) relative to two independent parameters. Instead of trying to obtain the exact solution to the Thomas-Fermi equation which, anyway, yields a density whose behavior is not correct, one can use the set of trial functions {ifJ( x)} chosen in some physically plausible way and search for the extremum of G[ifJ] or 1 [difJ/ dx 1 by a suitable choice of parameters.…”

Section: 3d Formulation Of Equivalent Variational Principles: In mentioning

confidence: 99%

Energy Density Functional Theory of Many-Electron Systems

1990

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Section: 3d Formulation Of Equivalent Variational Principles: In mentioning

confidence: 99%

Energy Density Functional Theory of Many-Electron Systems

1990

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“…This has the consequence that the TFA electron density of a singly charged negative ion runs out to infinity, thereby exhibiting the same behavior as that of the TF electron density of a neutral atom. Equation (13), for neutral atoms, has been solved approximately by an equivalent variational principle [3,4].…”

Section: (4)mentioning

confidence: 99%

Approximate variational solution of the thomas-fermi-amaldi equation for singly charged negative ions

Csavinszky

2009

Int. J. Quantum Chem.

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In the present work, approximate analytical solutions of the Thomas-Fermi-Amaldi (TFA) equation are obtained for the singly charged negative ions F-, CI-, Br-, and I-by making use of an equivalent variational principle. This approach is made possible by the fact that the TFA model for these ions can be expressed by a differential equation of the same form as the one which describes the Thomas-Fermi model for neutral atoms. (Formally, the two differential equations differ only in the definition of the independent variable.) This situation permits one to use the same variational principle in connection with the TFA equation for singly charged negative ions as that used in obtaining an approximate analytical solution of the TF equation for neutral atoms. The approximate solutions obtained for F-, CI-, Br-, and I-have been tested by calculating the diamagnetic susceptibilities of these ions and comparing them (1) with experimental values, (2) with values obtained by Jensen using a variational principle based on the density functional formalism, and (3) with values obtained by the author using a perturbation technique for relating the electron densities in the negative ions to those in neutral atoms.

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“…To determine the coefficient a, we use a variational principle which has been described in some detail by Csavinszky [19]. Choosing For the particular energy expression (31) the value of coefficient a could be found by means of a known variational principle.…”

Section: Expressions Of the Tf-energymentioning

confidence: 99%

On optimization procedures based upon the atomic thomas‐fermi energy

Hoor

1986

Int J of Quantum Chemistry

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For the total atomic Thomas-Fed (TF) energy many expressions in terms of the kinetic and potential energy contributions can be given. Thirty of these expressions exhibit either a maximum or a minimum if some variational approximation to the TF function is used. Three of these expressions, to note E, G, and J (see text) have been used in an optimization procedure, in which four two-parameter and two threeparameter approximate TP functions have been considered. One-parameter functions cannot be optimized, as the one parameter must be fixed to ensure proper normalization. It is found that optimization of E and G give reasonable and similar results, whereas the results of optimization of J are generally not very impressive. Where possible, expectation values of the type (r") (with n = -1, 1, 2, and 3) have been calculated from ten approximate TF functions. A new estimate of the exact atomic TF energy, as well as of the derivative of the TF function at the origin, has been obtained.

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Approximate Variational Solution of the Thomas-Fermi Equation for Atoms

Cited by 73 publications

References 7 publications

Energy Density Functional Theory of Many-Electron Systems

Energy Density Functional Theory of Many-Electron Systems

Approximate variational solution of the thomas-fermi-amaldi equation for singly charged negative ions

On optimization procedures based upon the atomic thomas‐fermi energy

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