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