Hooke atom — harmonium, two correlated electrons

Hooke atom (harmonium)

Two electrons in a harmonic trap with Coulomb repulsion (Hooke atom, harmonium). The relative motion is quasi-exactly solvable: closed forms only for discrete values of \(\omega\) (Taut 1993, the lowest being \(\omega=\tfrac12\)). It is the solvable model of electron correlation. Finite-difference computation in the Python backend (via gw2py).

The combination of harmonic confinement and Coulomb repulsion produces closed forms only when \(\omega\) takes discrete values (Taut): it is quasi-exact solvability in its most celebrated physical incarnation.

Relative coordinate \(r=|\mathbf r_1-\mathbf r_2|\); reduced units.

Centre-of-mass / relative separation

For two electrons in a common harmonic potential, the centre-of-mass/relative transformation decouples the problem. All the physics of correlation is in the relative motion:

\[ -u'' + \Big[\tfrac14\omega^2 r^2 + \tfrac{1}{r} + \tfrac{\ell(\ell+1)}{r^2}\Big]u = E_{\mathrm{rel}}\,u, \]

The solver maps the equation onto the standard form \(-\tfrac12 u''\) through the substitution \(x=r/\sqrt2\) (eigenvalues unchanged): at \(\omega=\tfrac12\) the ground level is exactly \(E_{\mathrm{rel}}=\tfrac54\) and the total energy, adding the centre-of-mass \(\tfrac32\omega\), is \(E_{\mathrm{tot}}=2\) hartree, as in the QMC suite.

with harmonic confinement \(+\) Coulomb repulsion \(1/r\).

Biconfluent Heun and quasi-exact solvability

This equation is of biconfluent-Heun type. Taut (1993) showed it admits closed-form solutions only for a discrete set of frequencies \(\omega\) (the lowest is \(\omega=\tfrac12\)): it is the prototypical quasi-exactly-solvable system.

For generic \(\omega\) the spectrum is only numerical; for Taut's special \(\omega\) some levels have an exact polynomial eigenfunction.

Why it matters

It is the solvable model of electron correlation: two interacting electrons treated exactly in the relative motion. A prototype of two-electron quantum dots.

State to represent

State type Level \(n_1\)

Level \(n_2\)

Eigenstates of the relative motion (for fixed \(\ell\)).

Physical parameters and grid

Trap frequency \(\omega\) = 0.5

\(\omega=0.5\) is the first exactly-solvable frequency (Taut 1993): relative ground level \(E_{\mathrm{rel}}=5/4\), total energy 2 Ha.

Relative angular momentum \(\ell\) \(r_{\max}\) = 18

Biconfluent Heun

The relative motion of harmonium maps onto the biconfluent-Heun equation (Karwowski–Witek): a regular singular point at \(0\) and a rank-2 irregular one at infinity — one step beyond the quartic.

Quasi-exact solvability

The combination of harmonic confinement and Coulomb repulsion produces closed forms only when \(\omega\) takes discrete values (Taut): there the Heun series terminates. It is the QES phenomenon in its most celebrated physical incarnation.

Computational evidence

The engine solves the relative motion for arbitrary \(\omega\); at \(\omega=\tfrac12\) (Taut's first case) the relative ground level isolates as expected. Orthonormality \(\sim10^{-15}\). The treatment is of the relative motion in reduced units.

References

M. Taut, «Two electrons in an external oscillator potential: particular analytic solutions», Phys. Rev. A 48, 3561 (1993). doi.
J. Karwowski, H. A. Witek, «Biconfluent Heun equation in quantum chemistry», Theor. Chem. Acc. 133, 1494 (2014).
A. V. Turbiner, Commun. Math. Phys. 118, 467 (1988). doi.
NIST DLMF, ch. 31 (Heun functions). dlmf.nist.gov/31.

WebNIR · CNR-IFAC | demo interface — numerical work is provided by the Python backend (gw2py).

Keywords: Hooke atom, harmonium, electron correlation, biconfluent Heun, Taut, quasi-exact solvability, quantum dot, quantum mechanics