翻訳と辞書 Words near each other ・ Spherical space form conjecture ・ Spherical surface Acoustic Wave (SAW) sensor ・ Spherical tokamak ・ Spherical trigonometry ・ Spherical variety ・ Spherical wave transformation ・ Spherical wedge ・ Spherical Worlds ・ Spherically complete field ・ Spherically symmetric spacetime ・ Sphericity ・ Sphericity scale ・ Sphericon ・ Spherification ・ Spherinder ・ Spherium ・ Sphero ・ Spherocobaltite ・ Spherocytosis ・ Spheroid ・ Spheroid Hill ・ Spheroidal wave equation ・ Spheroidal wave function ・ Spheroidal weathering ・ Spheroidene monooxygenase ・ Spheromak ・ Spherometer ・ Spheroolithus ・ Spheroplast ・ Spherosome Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Spherium ： ウィキペディア英語版 Spherium The "spherium" model consists of two electrons trapped on the surface of a sphere of radius $R$. It has been used by Berry and collaborators 〔 〕 to understand both weakly and strongly correlated systems and to suggest an "alternating" version of Hund's rule. Seidl studies this system in the context of density functional theory (DFT) to develop new correlation functionals within the adiabatic connection.〔 〕 The electronic Hamiltonian in atomic units is :$\backslash hat\; =\; -\; \backslash frac\; -\; \backslash frac\; +\; \backslash frac$ where $u$ is the interelectronic distance. For the singlet S states, it can be then shown〔 〕 that the wave function $S(u)$ satisfies the Schrödinger equation :$\backslash left(\; \backslash frac\; -\; 1\; \backslash right)\; \backslash frac\; +\; \backslash left(\backslash frac\; -\; \backslash frac\; \backslash right)\; \backslash frac\; +\; \backslash frac\; S(u)=\; E\; S(u)$ By introducing the dimensionless variable $x\; =\; u/2R$, this becomes a Heun equation with singular points at $x\; =\; -1,\; 0,\; +1$. Based on the known solutions of the Heun equation, we seek wave functions of the form :$S(u)\; =\; \backslash sum\_^\backslash infty\; s\_k\backslash ,u^k$ and substitution into the previous equation yields the recurrence relation :$s\_\; =\; \backslash frac\; -\; E\; \backslash right\; )\; s\_k\; \}$ with the starting values $s\_0\; =\; s\_1\; =\; 1$. Thus, the Kato cusp condition is :$\backslash frac\; =\; 1$. The wave function reduces to the polynomial :$S\_(u)\; =\; \backslash sum\_^n\; s\_k\backslash ,u^k$ (where $m$ the number of roots between $0$ and $2R$) if, and only if, $s\_\; =\; s\_\; =\; 0$. Thus, the energy $E\_$ is a root of the polynomial equation $s\_\; =\; 0$ (where $\backslash deg\; s\_\; =\; \backslash lfloor\; (n+1)/2\; \backslash rfloor$) and the corresponding radius $R\_$ is found from the previous equation which yields :$R\_^2\; E\_\; =\; \backslash frac\backslash left(\backslash frac+1\backslash right)$ $S\_(u)$ is the exact wave function of the $m$-th excited state of singlet S symmetry for the radius $R\_$. We know from the work of Loos and Gill 〔 that the HF energy of the lowest singlet S state is $E\_\; =\; 1/R$. It follows that the exact correlation energy for $R\; =\; \backslash sqrt/2$ is $E\_\; =\; 1-2/\backslash sqrt\; \backslash approx\; -0.1547$ which is much larger than the limiting correlation energies of the helium-like ions ($-0.0467$) or Hooke's atoms ($-0.0497$). This confirms the view that electron correlation on the surface of a sphere is qualitatively different from that in three-dimensional physical space. ==Spherium on a 3-Sphere== Recent work by Loos et al.〔 〕 considered the case of two electrons confined to a 3-sphere repelling Coulombically. They report a ground state energy of ($-.0476$). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Spherium」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.