Beiträge zur Quantenmechanik der Atome

Unsöld, Albrecht

doi:10.1002/andp.19273870304

Cited by 109 publications

(41 citation statements)

References 23 publications

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“…The Ψ lm obey Unsöld's theorem 64 , i.e., which mirrors the fact that spatial densities must be radially symmetric. In the same way, the Y lm 's obey the well known addition theorem i.e., We take the system to be filled with particles up to principal quantum number η = N shell , which means that it contains N = 2 N shell η=1 η 2 = (1/3)N shell (N shell + 1)(2N shell + 1) particles (including the spin degree of freedom).…”

Section: Positive Definite Ked Of Closed Radially Symmetric Systemsmentioning

confidence: 80%

Theoretical prediction of properties of atomistic systems : Density functional theory and machine learning

Lindmaa¹

2017

Linköping Studies in Science and Technology. Dissertations

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The prediction of ground state properties of atomistic systems is of vital importance in technological advances as well as in the physical sciences. Fundamentally, these predictions are based on a quantum-mechanical description of many-electron systems. One of the hitherto most prominent theories for the treatment of such systems is density functional theory (DFT). The main reason for its success is due to its balance of acceptable accuracy with computational efficiency. By now, DFT is applied routinely to compute the properties of atomic, molecular, and solid state systems.The general approach to solve the DFT equations is to use a densityfunctional approximation (DFA). In Kohn-Sham (KS) DFT, DFAs are applied to the unknown exchange-correlation (xc) energy. In orbitalfree DFT on the other hand, where the total energy is minimized directly with respect to the electron density, a DFA applied to the noninteracting kinetic energy is also required. Unfortunately, central DFAs in DFT fail to qualitatively capture many important aspects of electronic systems. Two prime examples are the description of localized electrons, and the description of systems where electronic edges are present.In this thesis, I use a model system approach to construct a DFA for the electron localization function (ELF). The very same approach is also taken to study the non-interacting kinetic energy density (KED) in the slowly varying limit of inhomogeneous electron densities, where the effect of electronic edges are effectively included. Apart from the work on model systems, extensions of an exchange energy functional with an improved KS orbital description are presented: a scheme for improving its description of energetics of solids, and a comparison of its description of an essential exact exchange feature known as the derivative discontinuity with numerical data for exact exchange.An emerging alternative route towards the prediction of the properties of atomistic systems is machine learning (ML). I present a number of ML methods for the prediction of solid formation energies, with an accuracy that is on par with KS DFT calculations, and with orders-ofmagnitude lower computational cost. V Svensk sammanfattningAtt kunna förutsäga egenskaper hos atomistiska system utgör en viktig del av vår teknologiska utveckling, samt spelar en betydande roll i de fysikaliska vetenskaperna. Sådana förutsägelser bygger på en kvantmekanisk beskrivning av mångelektronsystem. En av de mest framstående teorierna för att behandla den här typen av system är täthets-funktionalteorin (DFT). Den främsta orsaken till dess framgång är att den lyckas kombinera skaplig noggrannhet med en bra beräkningsef-fektivitet. DFT används numera rutinmässigt för att beräkna storheter hos atomer, molekyler, och fasta kroppar.Generellt sett löses ekvationerna inom DFT genom att man inför en täthetsfunktionalapproximation (DFA). I Kohn-Sham (KS) DFT, används DFAer för att approximera utbytes-korrelationsenergin. Inom orbitalfri DFT, där målet är att direkt minimera den totala ...

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Section: Positive Definite Ked Of Closed Radially Symmetric Systemsmentioning

confidence: 80%

Theoretical prediction of properties of atomistic systems : Density functional theory and machine learning

Lindmaa¹

2017

Linköping Studies in Science and Technology. Dissertations

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“…Physics: Pauling (17)]. This theorem, formulated 65 years ago, has to be accepted, and its acceptance requires that the moment of inertia of nuclei be attributed to the remaining nucleons.…”

Section: Discussionmentioning

confidence: 99%

Analysis of gamma-ray energies for 56 excited superdeformed rotational bands of nuclei of lanthanons La to Dy and of Hg, Tl, and Pb on the basis of the two-revolving-cluster model, with evaluation of moments of inertia and radii of revolution and assignment of nucleonic compositions to the clusters and the central sphere.

Pauling¹

1992

Proc. Natl. Acad. Sci. U.S.A.

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Analysis of the vray energies of 28 excited superdeformed bands of lanthanon nuclei by application of the two-revolving-cluster model yields the result that the central sphere for all 28 has the semimagc-magic composition p4n-", with the range p8n'2 to p'4n'8 for the clusters and the radius of revolution increasing from 7.31 to 7.76 fin. Similar analysis of 28 excited bands of Hg, Tl, and Pb nuclei leads to p56n82 (semimagic-magic) for the central sphere of 24 bands, p64n82 (semimagic-magic) for 2, and p"n9' (doubly semimagic) for 2, with cluster range p8n'2 to p'4n"6 and values of the radius of revolution from 8.70 to 8.92 fm for 26 bands and 9.2 fm for 2.The first discussion of excited rotational bands of nuclei corresponding to superdeformed structures that I know about is that of Morinaga (1), who discussed bands in 160 and 24Mgwith unusually large values of the moment of inertia, suggested that they corresponded to the linear sequences a-aa-a and a-a-a-a-a-a, respectively, of a particles, and surmised that 12C and 20Ne should have similar excited bands. It is now known that his structures do not apply to the bands that he discussed: the 160 band corresponds to revolution of a helion about 12C, and the 24Mg band corresponds to revolution of two helions about 160. It is also now known that spheron-spheron distances are about 3.6 + 0.4 fm. An excited 0+ to 6+ band for 160 of 16.75 to 19.35 MeV corresponds to the linear structure (2, 3) with a-a = 4.0 fm, and a recently observed (4) excited band of 24Mg beginning at 32.5 MeV gives a moment of inertia of 950 Da fm2 and a-a = 3.7 fm for a linear arrangement of six helions. In 1965 I suggested that the fission of U, Pu, and heavier nuclei involves an intermediate superprolate structure (5), and a similar suggestion was made in 1966 by Strutinsky (6) on the basis of the double-humped potential function calculated by his microscopic-macroscopic method. An excited superdeformed band of 66Dy was reported in 1968 by Twin et al. (7), and during later years Twin et al. and many other investigators reported many more for lanthanon nuclei and isotopes of Hg, Tl, and Pb. A review article giving the -ray energies for 56 excited superdeformed bands has been compiled by Han and Wu (8). Ground-State-Band Superdeformed NucleiA few years ago, while I was calculating moments of inertia (I) for the levels J = 2+, 4+, 6+,. . . of the ground-state bands of the even-even lanthanon nuclei and assigning values ofthe nucleon number ofthe single revolving cluster to these levels, I noticed that at large values ofJ, usually beginning at 12+ to 20+, the values of the moment of inertia became too large to be explained by a single cluster revolving about a central sphere at a radius compatible with the known nuclear dimensions. The values could be accounted for, however, by the assumption that there has occurred a transition to a superdeformed structure consisting of a central sphere with two clusters, on opposite sides, revolving about it. The properties of 35 such ground-state two-...

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“…for each k = 2l contribution (19). However, the radial density ρ(r) will be more complicated than (33), involving mixed contributions of the type P n l (r)P nl (r) = r 2 R n l (r)R nl (r), as developed below.…”

Section: The Spherical Density Functionmentioning

confidence: 99%

Atomic Density Functions: Atomic Physics Calculations Analyzed with Methods from Quantum Chemistry

Borgoo

Godefroid

Geerlings

2011

Advances in the Theory of Quantum Systems in Chemistry and Physics

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This contribution reviews a selection of findings on atomic density functions and discusses ways for reading chemical information from them. First an expression for the density function for atoms in the multi-configuration Hartree-Fock scheme is established. The spherical harmonic content of the density function and ways to restore the spherical symmetry in a general open-shell case are treated. The evaluation of the density function is illustrated in a few examples. In the second part of the paper, atomic density functions are analyzed using quantum similarity measures. The comparison of atomic density functions is shown to be useful to obtain physical and chemical information. Finally, concepts from information theory are introduced and adopted for the comparison of density functions. In particular, based on the Kullback-Leibler form, a functional is constructed that reveals the periodicity in Mendeleev's table. Finally a quantum similarity measure is constructed, based on the integrand of the Kullback-Leibler expression and the periodicity is regained in a different way.

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Beiträge zur Quantenmechanik der Atome

Cited by 109 publications

References 23 publications

Theoretical prediction of properties of atomistic systems : Density functional theory and machine learning

Theoretical prediction of properties of atomistic systems : Density functional theory and machine learning

Atomic Density Functions: Atomic Physics Calculations Analyzed with Methods from Quantum Chemistry

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