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.
Definition Employing atomic units, the Hamiltonian defining the Hooke's atom is
H ^ = − 1 2 ∇ 1 2 − 1 2 ∇ 2 2 + 1 2 k ( r 1 2 + r 2 2 ) + 1 | r 1 − r 2 | . {\displaystyle {\hat {H}}=-{\frac {1}{2}}\nabla _{1}^{2}-{\frac {1}{2}}\nabla _{2}^{2}+{\frac {1}{2}}k(r_{1}^{2}+r_{2}^{2})+{\frac {1}{|\mathbf {r} _{1}-\mathbf {r} _{2}|}}.}
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:
− 2 r → 1 2 k r 2 . {\displaystyle -{\frac {2}{r}}\rightarrow {\frac {1}{2}}kr^{2}.}
Solution The equation to be solved is the two electron Schrödinger equation:
H ^ Ψ ( r 1 , r 2 ) = E Ψ ( r 1 , r 2 ) . {\displaystyle {\hat {H}}\Psi (\mathbf {r} _{1},\mathbf {r} _{2})=E\Psi (\mathbf {r} _{1},\mathbf {r} _{2}).}
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 from the Cartesian electronic coordinates, (r1,r2), to the center of mass coordinates, (R,u), defined as
R = 1 2 ( r 1 + r 2 ) , u = r 2 − r 1 . {\displaystyle \mathbf {R} ={\frac {1}{2}}(\mathbf {r} _{1}+\mathbf {r} _{2}),\mathbf {u} =\mathbf {r} _{2}-\mathbf {r} _{1}.}
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 Ψ ( r 1 , r 2 ) = χ ( R ) Φ ( u ) {\displaystyle \Psi (\mathbf {r} _{1},\mathbf {r} _{2})=\chi (\mathbf {R} )\Phi (\mathbf {u} )} . The original Schrödinger equation is then replaced by:
… excerpt ends here. Continue reading the full article.
