The "spherium" model consists of two electrons trapped on the surface of a sphere of radius . It has been used by Berry and collaborators [1] 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.[2]

Definition and solution

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The electronic Hamiltonian in atomic units is

 

where   is the interelectronic distance. For the singlet S states, it can be then shown[3] that the wave function   satisfies the Schrödinger equation

 

By introducing the dimensionless variable  , this becomes a Heun equation with singular points at  . Based on the known solutions of the Heun equation, we seek wave functions of the form

 

and substitution into the previous equation yields the recurrence relation

 

with the starting values  . Thus, the Kato cusp condition is

 .

The wave function reduces to the polynomial

 

(where   the number of roots between   and  ) if, and only if,  . Thus, the energy   is a root of the polynomial equation   (where  ) and the corresponding radius   is found from the previous equation which yields

 

  is the exact wave function of the  -th excited state of singlet S symmetry for the radius  .

We know from the work of Loos and Gill [3] that the HF energy of the lowest singlet S state is  . It follows that the exact correlation energy for   is   which is much larger than the limiting correlation energies of the helium-like ions ( ) or Hooke's atoms ( ). 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

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Loos and Gill[4] considered the case of two electrons confined to a 3-sphere repelling Coulombically. They report a ground state energy of ( ).

See also

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References

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  1. ^ Ezra, G. S.; Berry, R. S. (1982), "Correlation of two particles on a sphere", Physical Review A, 25 (3): 1513–1527, Bibcode:1982PhRvA..25.1513E, doi:10.1103/PhysRevA.25.1513
  2. ^ Seidl, M. (2007), "Adiabatic connection in density-functional theory: Two electrons on the surface of a sphere", Physical Review A, 75 (6): 062506, Bibcode:2007PhRvA..75a2506P, doi:10.1103/PhysRevA.75.062506
  3. ^ a b Loos, P.-F.; Gill, P. M. W. (2009), "Ground state of two electrons on a sphere", Physical Review A, 79 (6): 062517, arXiv:1002.3398, Bibcode:2009PhRvA..79f2517L, doi:10.1103/PhysRevA.79.062517, S2CID 59364477
  4. ^ Loos, P.-F.; Gill, P. M. W. (2010), "Excited states of spherium", Molecular Physics, 108 (19–20): 2527–2532, arXiv:1004.3641, Bibcode:2010MolPh.108.2527L, doi:10.1080/00268976.2010.508472, S2CID 43949268

Further reading

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