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{{expert-subject|Physics|date=October 2009}}
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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]]

Latest revision as of 20:11, 3 July 2014

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