Hooke's atom, also known as harmonium or hookium, refers to an artificial helium-like atom where the Coulombic electron-nucleus interaction potential is replaced by a harmonic potential. This system is of significance as it is, for certain values of the force constant defining the harmonic containment, an exactly solvable ground-state many-electron problem that explicitly includes electron correlation. As such it can provide insight into quantum correlation (albeit in the presence of a non-physical nuclear potential) and can act as a test system for judging the accuracy of approximate quantum chemical methods for solving the Schrödinger equation. The name "Hooke's atom" arises because the harmonic potential used to describe the electron-nucleus interaction is a consequence of Hooke's law.
Contents
Definition
Employing atomic units, the Hamiltonian defining the Hooke's atom is
As written, the first two terms are the kinetic energy operators of the two electrons, the third term is the harmonic electron-nucleus potential, and the final term the electron-electron interaction potential. The non-relativistic Hamiltonian of the helium atom differs only in the replacement:
Solution
The equation to be solved is the two electron Schrödinger equation:
For arbitrary values of the force constant, k, the Schrödinger equation does not have an analytic solution. However, for a countably infinite number of values, such as k=¼, simple closed form solutions can be derived. Given the artificial nature of the system this restriction does not hinder the usefulness of the solution.
To solve, the system is first transformed form the Cartesian electronic coordinates, (r1,r2), to the center of mass coordinates, (R,u), defined as
Under this transformation, the Hamiltonian becomes separable – that is, the |r1 - r2| term coupling the two electrons is removed (and not replaced by some other form) allowing the general separation of variables technique to be applied to further a solution for the wave function in the form
The first equation for
Asymptotically, the second equation again behaves as a harmonic oscillator of the form
Decomposing
one further decomposes the radial wave function as
The asymptotic behavior
The differential equation satisfied by
This equation lends itself to a solution by way of the Frobenius method. That is,
for some
The two solutions to the indicial equation are
These directly force a2 = 0 and a3 = 0 respectively, and as a consequence of the three term recession, all higher coefficients also vanish. Solving for
and the radial wave function
Transforming back to
the ground-state (with
Combining, normalizing, and transforming back to the original coordinates yields the ground state wave function:
The corresponding ground-state total energy is then
Remarks
The exact ground state electronic density of the Hooke atom is
From this we see that the radial derivative of the density vanishes at the nucleus. This is in stark contrast to the real (non-relativistic) helium atom where the density displays a cusp at the nucleus as a result of the unbounded Coulomb potential.