Spherium

The "spherium" model consists of two electrons trapped on the surface of a sphere of radius ${\displaystyle R}$. 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

The electronic Hamiltonian in atomic units is

${\displaystyle {\hat {H}}=-{\frac {\nabla _{1}^{2}}{2}}-{\frac {\nabla _{2}^{2}}{2}}+{\frac {1}{u}}}$

where ${\displaystyle u}$ is the interelectronic distance. For the singlet S states, it can be then shown[3] that the wave function ${\displaystyle S(u)}$ satisfies the Schrödinger equation

${\displaystyle \left({\frac {u^{2}}{4R^{2}}}-1\right){\frac {d^{2}S(u)}{du^{2}}}+\left({\frac {3u}{4R^{2}}}-{\frac {1}{u}}\right){\frac {dS(u)}{du}}+{\frac {1}{u}}S(u)=ES(u)}$

By introducing the dimensionless variable ${\displaystyle x=u/2R}$, this becomes a Heun equation with singular points at ${\displaystyle x=-1,0,+1}$. Based on the known solutions of the Heun equation, we seek wave functions of the form

${\displaystyle S(u)=\sum _{k=0}^{\infty }s_{k}\,u^{k}}$

and substitution into the previous equation yields the recurrence relation

${\displaystyle s_{k+2}={\frac {s_{k+1}+\left[k(k+2){\frac {1}{4R^{2}}}-E\right]s_{k}}{(k+2)^{2}}}}$

with the starting values ${\displaystyle s_{0}=s_{1}=1}$. Thus, the Kato cusp condition is

${\displaystyle {\frac {S'(0)}{S(0)}}=1}$.

The wave function reduces to the polynomial

${\displaystyle S_{n,m}(u)=\sum _{k=0}^{n}s_{k}\,u^{k}}$

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

${\displaystyle R_{n,m}^{2}E_{n,m}={\frac {n}{2}}\left({\frac {n}{2}}+1\right)}$

${\displaystyle S_{n,m}(u)}$ is the exact wave function of the ${\displaystyle m}$-th excited state of singlet S symmetry for the radius ${\displaystyle R_{n,m}}$.

We know from the work of Loos and Gill [3] that the HF energy of the lowest singlet S state is ${\displaystyle E_{\rm {HF}}=1/R}$. It follows that the exact correlation energy for ${\displaystyle R={\sqrt {3}}/2}$ is ${\displaystyle E_{\rm {corr}}=1-2/{\sqrt {3}}\approx -0.1547}$ which is much larger than the limiting correlation energies of the helium-like ions (${\displaystyle -0.0467}$) or Hooke's atoms (${\displaystyle -0.0497}$). 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

Loos and Gill[4] considered the case of two electrons confined to a 3-sphere repelling Coulombically. They report a ground state energy of (${\displaystyle -.0476}$).

See also

• List of quantum-mechanical systems with analytical solutions

References

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. 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

• Loos, P.-F.; Gill, P. M. W. (2009), "Two electrons on a hypersphere: a quasiexactly solvable model", Physical Review Letters, 103 (12): 123008, arXiv:1002.3400, Bibcode:2009PhRvL.103l3008L, doi:10.1103/physrevlett.103.123008, PMID 19792435, S2CID 11611242