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add a reference

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	~~The "'''spherium'''" model consists of two [[electron]]s trapped on the surface of a [[sphere]] of radius~~ <~~math~~>R</~~math>~~. It has been ~~used by Berry and collaborators <ref name="BerryPRA1982">{{citation \|author = G~~. ~~S. Ezra and R. S. Berry \|journal = Phys. Rev. A \|volume = 25 \|pages = 1513 \| year = 1982}}~~<~~/ref~~> 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 functional]]s within the [[adiabatic connection]].<ref name="SeidlPRA2007">{{citation \|author = M~~. ~~Seidl \|journal = Phys. Rev. A \|volume = 75 \|pages = 062506 \|year = 2007}}<~~/~~ref>~~

	~~The electronic [[Hamiltonian (quantum mechanics)\|Hamiltonian]~~] in ~~atomic units is~~

	~~:<math>\hat{H} = - \frac{\nabla_1^2}{2} - \frac{\nabla_2^2}{2} + \frac{1}{u}</math>~~

	~~where <math>u</math> is the interelectronic distance~~.
	~~For~~ the ~~singlet S states~~, ~~it can be then shown<ref name="LoosPRA2009">{{citation \|author = P~~.~~-F. Loos, P. M. W. Gill \|journal = Phys. Rev. A \|volume = 79 \|pages = 062517 \|year = 2009}}</ref> that~~ the ~~[[wave function]] <math>S(u)</math> satisfies~~ the ~~[[Schrödinger equation]]~~

	~~:<math>\left( \frac{u^2}{4R^2} - 1 \right) \frac{d^2S(u)}{du^2} + \left(\frac{3u}{4R^2} - \frac{1}{u} \right) \frac{dS(u)}{du} + \frac{1}{u} S(u)= E S(u)</math>~~

	~~By introducing the dimensionless variable <math>x = u/2R</math>, this becomes~~ a ~~[[Heun equation]] with singular points at <math>x =~~ -~~1, 0, +1</math>~~. ~~Based on the known solutions~~ of ~~the Heun equation~~, ~~we seek wave functions of~~ the ~~form~~

	~~:<math>S(u) = \sum_{k=0}^\infty s_k\,u^k</math>~~

	and ~~substitution into~~ the ~~previous equation yields~~ the ~~[[recurrence relation]]~~

	:<~~math~~>~~s_{k+2} = \frac{ s_{k+1} + \left[ k(k+2) \frac{1}{4R^2} - E \right] s_k }{(k+2)^2}~~<~~/math~~>

	~~with~~ the ~~starting values <math> s_0 = s_1 = 1 </math>~~. ~~Thus~~, the ~~[[Kato cusp condition]] is~~
	~~:<math> \frac{S'(0)}{S(0)} = 1 </math>~~.

	The ~~wave function reduces~~ to the ~~[[polynomial]]~~

	~~:<math>S_{n,m}(u) = \sum_{k=0}^n s_k\~~,~~u^k</math>~~

	~~(where <math>m</math> the number of roots between <math>0</math>~~ and <~~math~~>2R<~~/math~~>~~) if~~, ~~and only~~ if~~, <math>s_{n+1} = s_{n+2} = 0</math>~~. ~~Thus, the energy~~ <~~math>E_{n,m}</math~~> is a ~~root~~ of ~~the polynomial equation <math>s_{n+1} = 0<~~/~~math> (where <math>\deg s_{n+1} = \lfloor (n+1)~~/~~2 \rfloor<~~/~~math>) and the corresponding radius <math>R_{n,m}<~~/~~math>~~ is ~~found from the previous equation which yields~~

	:~~<math> R_{n,m}^2 E_{n,m} = \frac{n}{2}\left(\frac{n}{2}+1\right)<~~/~~math>~~

	~~<math>S_{n,m}(u)<~~/~~math>~~ is the ~~exact wave function of~~ the ~~<math>m</math>~~-~~th excited~~ state ~~of singlet S symmetry for the radius <math>R_{n~~,~~m}</math>.~~

	~~We know from~~ the ~~work of Loos and Gill <ref name="LoosPRA2009"/> that~~ the ~~HF energy of~~ the ~~lowest singlet S state is <math>E_{\rm HF} = 1~~/R</~~math>~~. ~~It follows that the exact correlation energy for <math>R = \sqrt{3}~~/2</~~math> is <math>E_{\rm corr} = 1-2~~/~~\sqrt{3} \approx -0~~.~~1547<~~/~~math> which~~ is ~~much larger than~~ the ~~limiting correlation energies~~ of ~~the helium-like ions (<math>-0~~.~~0467~~<~~/math~~>~~) or Hooke's atoms (~~<~~math~~>-0.~~0497</math>). 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~~.<~~ref name="LOOSArxiv2010"~~>~~{{citation \|author = P. Loos and P. M. W. Gill \|journal = Arxiv\| year = 2010\| url=~~http://~~arxiv~~.~~org~~/~~abs/1004~~.~~3641}}~~<~~/ref~~> ~~considered~~ the ~~case of two electrons confined to a [[3~~-~~sphere]] repelling Coulombically. They report a ground state energy~~ of ~~(<math>-.0476</math>).~~

	~~==See also==~~
	*[[List of ~~quantum-mechanical systems with analytical solutions]]~~

	~~==References==~~
	~~{{reflist}}~~

	~~==Further reading==~~

	* {{citation \| author = P.-F. ~~Loos~~ and P. ~~M. W. Gill \| title = Two electrons on a hypersphere:~~ a ~~quasiexactly solvable model \| journal = Physical Review Letters \| year = 2009 \| volume = 103 \| pages = 123008 \| url = http://scitation~~.~~aip~~.~~org/getabs/servlet/GetabsServlet?prog=normal&id=PRLTAO000103000012123008000001&idtype=cvips&gifs=Yes}}~~

	~~[[Category:Quantum chemistry]]~~
	~~[[Category:Quantum mechanics]]~~
	~~[[Category:Quantum models]]~~

en>True Pagan Warrior: Reverted good faith edits by 79.3.179.103 (talk): Obviously this link doesn't belong here, but it's unclear what it's referencing without an inline citation. (TW)

2013-12-11T19:35:09Z

Reverted good faith edits by 79.3.179.103 (talk): Obviously this link doesn't belong here, but it's unclear what it's referencing without an inline citation. (TW)

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en>Giftlite: mv .

2012-07-02T17:14:05Z

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en>Waldir: add a reference

en>True Pagan Warrior: Reverted good faith edits by 79.3.179.103 (talk): Obviously this link doesn't belong here, but it's unclear what it's referencing without an inline citation. (TW)

en>Giftlite: mv .