Self-Consistent Fields in Atoms. Hartree and Thomas–Fermi Atoms


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Self-Consistent Field

The present analytical approximation can be of great value to improve the description of elastic scattering events in Monte Carlo simulation of electron transport. Finally, Sec. IV is dedicated to the analysis of the reliability of IPM results based on these screening functions, including comparisons with other analytical IPM potentials. Screening functions adopted in the literature are usually based on the TF model and its refinements.


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The most elemental TF model provides a universal screening function satisfying the following differential equation see, e. Function 8 differs from the exact solution of 7 by less than 0. A similar analytical approximation has been proposed by Csavinszki4 on the basis of a variational solution of Eq. This feature makes expression 8 more reliable for practical calculations than the exact TF screening function.

Simple analytical screening functions have also been given by Gross and Dreizler3 on the basis of the variational formulation of the Thomas-Fermi-Dirac-Weizsacker theory. Although the screening functions of Bonham and Strand and of Gross and Dreizler are undoubtedly more reliable than the TF one, the scaling properties of the simple TF theory are lost when exchange and gradient corrections are introduced.

Bestselling in Hartree–Fock Method

Once we renounce to these scaling properties, self-consistent methods that are more accurate than the statistical ones can be adopted to derive analytical screening functions. The parameters H and d have been obtained by Green, Sellin, and Zachor6 from a least-squares fitting of the po-. More recently, the parameters of 9 have also been determined from a variational procedure in which the total binding energy of the atom, computed from a Slater determinant composed of one-electron orbitals, is minimized. The DHFS self-consistent calculations provide us with reliable screening functions accounting for relativistic effects which are difficult to be included in statistical or nonrelativistic self-consistent models.

Central field orbitals are obtained by solving the Dirac radial equations for the HFS potential with Latter's correction [Eqs. The solid curves are the DHFS results. The dashed dotted, dotted, and dashed curves correspond to the analytical screening functions 8 , 10 , and 11 , respectively. The corresponding electrostatic potential 1 is given as the superposition of three Yukawa potentials and the atomic density takes the expression.

Potential energy shift for a screened Coulomb potential

In principle, the parameters in 11 could be obtained by numerical fitting to the numerical DHFS screening function. To do this, a conventional least-squares-fitting procedure can be used to select the optimum parameters. However, proceeding in this way and using a standard minimization method, different sets of parameters were obtained from different initial estimates reflecting the existence of local minima in the function being minimized. To avoid these uncertainties in the determination of the parameters, we have adopted the procedure described below which generalizes a more crude approach previously proposed to obtain analytical Born cross sections for elastic scattering of electrons by atoms.

Except for the factor n -hi! It can be easily seen that. This leads to the following relations:. This last feature makes Born cross sections derived from 11 to practically coincide with those computed from the DHFS screening function see Sec. For neutral atoms, Eqs. However, it should be noticed that, as the a, values have to be positive, DHFS radial ex-. Elements indicated with an asterisk have DHFS radial expected values inconsistent with conditions Naturally, the analytical density 12 can only partially reproduce the oscillations of the DHFS density associated with different shell contributions Fig.

One may expect that an even better approximation to the DHFS screening function could be obtained from a numerical least-squares fitting using a standard minimization procedure with the parameters in Table I as initial estimates. Such a procedure has not been pursued here because the resulting parameters do not appreciably improve the quality of the fit, i. Moreover, as mentioned above, conditions 15 , even in the. The solid curve is the DHFS density. The dash-dotted curve is the TF density derived from Moliere's screening function 8.

The dotted curve corresponds to the density obtained from the analytical screening function The dashed curve is the analytical density The Born cross section for scattering of a fast particle which, for the sake of simplification, is assumed to have unit charge and unit mass in the atomic field 1 can be written as see, e. For the density associated with our and Moliere's analytical screening functions, the form factor takes the simple expression.

The form factors 18 are compared with the ones derived from the DHFS density and from other analytical screening functions in Fig.

Our analytical results differ from the DHFS numerical calculations only for relatively large momentum transfers. Although these differences might seem important, they do not appreciably. The dash-dotted and dashed curves have been obtained from the analytical screening functions 8 and 10 , respectively. The solid curve is the DHFS cross section. The cross section derived from the analytical density 12 coincides with the DHFS one at the drawing level. The correspondence between the curves and the theoretical models from which they have been obtained is the same as in Fig.

The solid curve is the DHFS form factor. The dash-dotted, long-dashed, and short-dashed curves correspond to form factors derived from the analytical screening functions 8 , 10 , and 11 , respectively. The correspondence between the curves and the different screening functions from which they have been derived is the same as in Fig.

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DHFS: Relativistic self-consistent results. IPM: Relativistic eigenvalues from the analytical screening function.

1st Edition

GSZ: derived from the analytical potential 9. Wheeler, of the division of invertebrate zoology at the American Museum of Natural History for clarification. It was a amusing acronym. So, scientists do have sense of humor. As for people with vulgar surnames … they do exist. In Italy there are thousands.

I will tell their stories in the next post. I n Scopus, professor Stronzo Bestiale appears to be working at the Institute of Experimental Physics , at the University of Vienna , the institutional affiliation attributed to him in the paper published by the Journal of chemical physics. Some readers of this blog report a Prin Research project of relevant national interest , presented in by researchers from INFN and University of Torino. The acronym of the system was Vaffa an abbreviation of vaffanculo , fuck off : Virtual Analysis Facility For the Alice experiment.

The project, which you can read about here , was not approved. Oransky recalls a similar case : the immunologist Polly Matzinger , who in published a study co-authored by entering Galadriel Mirkwood : the name of her a fghan hound. It is named by its discoverers Douglas Hartree and Vladimir Fock. The latter is a Russian physicist with a surname that , in English, is likely to be crippled so vulgar fuck.

Difficult to determine whether it was a mistake or a joke.

Density based methods - Thomas-Fermi

Or a way to get more clicks on the Web even by those who make mistakes typing the name? Since this is a research written by Chinese authors , it was almost certainly an accident. I checked if one of the most prolific authors , a geologist from Oregon, Gerald G. Connard , really exists : in telephone directories he is present, and probably others. Moreover , people with a vulgar surname are many : among them there must be some scientist! Maybe one day prof. Do we candidate him? Both have no follower: but their existence testifies that the myth of prof. Bestiale remains alive … even in the professional field!

Among the authors there is a certain Robert Stadler, who qualifies as a member of the Stronzo Bestiale Institute of Technology , obviously non-existent. And Stadler is also likely to be a pseudonym, given that on the Web the only Stadler that is mentioned is an Austrian designer who has nothing to do with cryptocurrencies. In short, a fake name of a very false institution: among researchers dealing with digital currencies, anonymity is quite common.

And in this case there is a note of mocking. The third is a study by Samuel Moore et al. An explicit tribute to the story told by this blog. Thanks to Sara Scharf for the precious help in reviewing my translation. I would add to the story another strange case in scientific publishing. Stephen J. George A. Hagedorn , Crossing the interface between chemistry and mathematics , Notices Amer.

MR 82a Hunziker and I. Royal Soc. Edinburgh 96, pp. MR 85k S AAD and J. Tosio Kato , On the eigenfunctions of many-particle systems in quantum mechanics , Comm. Reprint of the edition. Acta 44, pp. Le Bris ed. Special volume: computational chemistry. Claude Le Bris , A general approach for multiconfiguration methods in quantum molecular chemistry , Ann.

Mathieu Lewin , The multiconfiguration methods in quantum chemistry: Palais-Smale condition and existence of minimizers , C. Mathieu Lewin , Solutions of the multiconfiguration equations in quantum chemistry , Arch.

Density_functional_theory

A 29, pp. Elliott H. Lieb , The stability of matter: from atoms to stars , Bull. Dreizler and J. MR 83g MR 83aa Lieb, B. Simon and A. Wightman, eds. Press, pp. A 34, pp. IX, Pitman Res. Notes Math. MR 90i CO;2-C MR 88e The limit case. I, II, Rev. MR 87c , MR 87j The locally compact case.

I and II, Ann. MR 87ea , MR 87eb Horstemeyer , Preface [Special issue on multiple scale methods for nanoscale mechanics and materials] , Comput. Methods Appl. Yvon Maday and Gabriel Turinici , Error bars and quadratically convergent methods for the numerical simulation of the Hartree-Fock equations , Numer. A, and W. MR 80i MR 83e B 51, pp. Pagano and R. Paroni , A simple model for phase transitions: from the discrete to the continuum problem , Quart. MR 83b Michael Reed and Barry Simon , Methods of modern mathematical physics. A general criterion, J. Applications to nonlinear partial differential equations and Hamiltonian systems.

MR 87i Herschel Rabitz , Gabriel Turinici , and Eric Brown , Control of quantum dynamics: concepts, procedures and future prospects , Handbook of numerical analysis, Vol. MR 95m 4. MR 94d 6. MR 92g 9. MR f MR j MR k A 23, pp. MR 91d Pisa 2, pp. Bennewitz, ed. MR 92m MR 93i Defranceschi, C. MR g MR h MR c MR i E, Vol. MR a MR m MR 95b MR 97d MR 97j MR 99d MR 99g Contemporary Math. Solids 7, no. MR 99b MR b MR 99j MR e

Self-Consistent Fields in Atoms. Hartree and Thomas–Fermi Atoms Self-Consistent Fields in Atoms. Hartree and Thomas–Fermi Atoms
Self-Consistent Fields in Atoms. Hartree and Thomas–Fermi Atoms Self-Consistent Fields in Atoms. Hartree and Thomas–Fermi Atoms
Self-Consistent Fields in Atoms. Hartree and Thomas–Fermi Atoms Self-Consistent Fields in Atoms. Hartree and Thomas–Fermi Atoms
Self-Consistent Fields in Atoms. Hartree and Thomas–Fermi Atoms Self-Consistent Fields in Atoms. Hartree and Thomas–Fermi Atoms
Self-Consistent Fields in Atoms. Hartree and Thomas–Fermi Atoms Self-Consistent Fields in Atoms. Hartree and Thomas–Fermi Atoms
Self-Consistent Fields in Atoms. Hartree and Thomas–Fermi Atoms Self-Consistent Fields in Atoms. Hartree and Thomas–Fermi Atoms
Self-Consistent Fields in Atoms. Hartree and Thomas–Fermi Atoms Self-Consistent Fields in Atoms. Hartree and Thomas–Fermi Atoms
Self-Consistent Fields in Atoms. Hartree and Thomas–Fermi Atoms Self-Consistent Fields in Atoms. Hartree and Thomas–Fermi Atoms

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