Hooke's atom

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.^[1]^[2] This system is of significance as it is, for certain values of the force constant defining the harmonic containment, an exactly solvable^[3] 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.^[4]^[5] 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

{\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:

-{\frac {2}{r}}\rightarrow {\frac {1}{2}}kr^{2}.

Solution

The equation to be solved is the two electron Schrödinger equation:

{\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.^[5] 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, $(r 1, r 2)$ , to the center of mass coordinates, $(R, u)$ , defined as

\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 $| r 1 - r 2 |$ 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 $\Psi (\mathbf {r} _{1},\mathbf {r} _{2})=\chi (\mathbf {R} )\Phi (\mathbf {u} )$ . The original Schrödinger equation is then replaced by:

\left(-{\frac {1}{4}}\nabla _{\mathbf {R} }^{2}+kR^{2}\right)\chi (\mathbf {R} )=E_{\mathbf {R} }\chi (\mathbf {R} ),

\left(-\nabla _{\mathbf {u} }^{2}+{\frac {1}{4}}ku^{2}+{\frac {1}{u}}\right)\Phi (\mathbf {u} )=E_{\mathbf {u} }\Phi (\mathbf {u} ).

The first equation for $\chi (\mathbf {R} )$ is the Schrödinger equation for an isotropic quantum harmonic oscillator with ground-state energy $E_{\mathbf {R} }=(3/2){\sqrt {k}}E_{\mathrm {h} }$ and (unnormalized) wave function

\chi (\mathbf {R} )=e^{-{\sqrt {k}}R^{2}}.

Asymptotically, the second equation again behaves as a harmonic oscillator of the form $\exp(-({\sqrt {k}}/4)u^{2})\,$ and the rotationally invariant ground state can be expressed, in general, as $\Phi (\mathbf {u} )=f(u)\exp(-({\sqrt {k}}/4)u^{2})\,$ for some function $f(u)\,$ . It was long noted that $f (u)$ is very well approximated by a linear function in $u$ .^[2] Thirty years after the proposal of the model an exact solution was discovered for $k =¼$ ,^[3] and it was seen that $f (u)=1+ u /2$ . It was later shown that there are many values of $k$ which lead to an exact solution for the ground state,^[5] as will be shown in the following.

Decomposing $\Phi (\mathbf {u} )=R_{l}(u)Y_{lm}$ and expressing the Laplacian in spherical coordinates,

\left(-{\frac {1}{u^{2}}}{\frac {\partial }{\partial u}}\left(u^{2}{\frac {\partial }{\partial u}}\right)+{\frac {{\hat {L}}^{2}}{u^{2}}}+{\frac {1}{4}}ku^{2}+{\frac {1}{u}}\right)R_{l}(u)Y_{lm}({\hat {\mathbf {u} }})=E_{l}R_{l}(u)Y_{lm}({\hat {\mathbf {u} }}),

one further decomposes the radial wave function as $R_{l}(u)=S_{l}(u)/u\,$ which removes the first derivative to yield

-{\frac {\partial ^{2}S_{l}(u)}{\partial u^{2}}}+\left({\frac {l(l+1)}{u^{2}}}+{\frac {1}{4}}ku^{2}+{\frac {1}{u}}\right)S_{l}(u)=E_{l}S_{l}(u).

The asymptotic behavior $S_{l}(u)\sim e^{-{\frac {\sqrt {k}}{4}}u^{2}}\,$ encourages a solution of the form

S_{l}(u)=e^{-{\frac {\sqrt {k}}{4}}u^{2}}T_{l}(u).

The differential equation satisfied by $T_{l}(u)\,$ is

-{\frac {\partial ^{2}T_{l}(u)}{\partial u^{2}}}+{\sqrt {k}}u{\frac {\partial T_{l}(u)}{\partial u}}+\left({\frac {l(l+1)}{u^{2}}}+{\frac {1}{u}}+\left({\frac {\sqrt {k}}{2}}-E_{l}\right)\right)T_{l}(u)=0.

This equation lends itself to a solution by way of the Frobenius method. That is, $T_{l}(u)\,$ is expressed as

T_{l}(u)=u^{m}\sum _{k=0}^{\infty }\ a_{k}u^{k}.

for some $m\,$ and $\{a_{k}\}_{k=0}^{k=\infty }\,$ which satisfy:

m(m-1)=l(l+1)\,,

a_{0}\neq 0\,

a_{1}={\frac {a_{0}}{2(l+1)}},

a_{2}={\frac {a_{1}+\left({\sqrt {k}}(l+{\frac {3}{2}})-E_{l}\right)a_{0}}{2(2l+3)}}={\frac {a_{0}}{2(2l+3)}}\left({\frac {1}{2(l+1)}}+{\sqrt {k}}\left(l+{\frac {3}{2}}\right)-E_{l}\right),

a_{3}={\frac {a_{2}+\left({\sqrt {k}}(l+{\frac {5}{2}})-E_{l}\right)a_{1}}{6(l+2)}},

a_{n+1}={\frac {a_{n}+\left({\sqrt {k}}(l+{\frac {1}{2}}+n)-E_{l}\right)a_{n-1}}{(n+1)(2l+2+n)}}.

The two solutions to the indicial equation are $m=l+1$ and $m=-l$ of which the former is taken as it yields the regular (bounded, normalizable) wave function. For a simple solution to exist, the infinite series is sought to terminate and it is here where particular values of $k$ are exploited for an exact closed-form solution. Terminating the polynomial at any particular order can be accomplished with different values of $k$ defining the Hamiltonian. As such there exists an infinite number of systems, differing only in the strength of the harmonic containment, with exact ground-state solutions. Most simply, to impose $a k = 0$ for $k \geq 2$ , two conditions must be satisfied:

{\frac {1}{2(l+1)}}+{\sqrt {k}}\left(l+{\frac {3}{2}}\right)-E_{l}=0,

{\sqrt {k}}(l+{\frac {5}{2}})=E_{l}.

These directly force $a 2 = 0$ and $a 3 = 0$ respectively, and as a consequence of the three term recession, all higher coefficients also vanish. Solving for ${\sqrt {k}}\,$ and $E_{l}\,$ yields

{\sqrt {k}}={\frac {1}{2(l+1)}},

E_{l}={\frac {2l+5}{4(l+1)}},

and the radial wave function

T_{l}=u^{l+1}\left(a_{0}+{\frac {a_{0}}{2(l+1)}}u\right).

Transforming back to $R_{l}(u)\,$

R_{l}(u)={\frac {T_{l}(u)e^{-{\frac {\sqrt {k}}{4}}u^{2}}}{u}}=u^{l}\left(1+{\frac {1}{2(l+1)}}u\right)e^{-{\frac {\sqrt {k}}{4}}u^{2}},

the ground-state (with $l=0\,$ and energy $5/4E_{\mathrm {h} }\,$ ) is finally

\Phi (\mathbf {u} )=\left(1+{\frac {u}{2}}\right)e^{-u^{2}/8}.

Combining, normalizing, and transforming back to the original coordinates yields the ground state wave function:

\Psi (\mathbf {r} _{1},\mathbf {r} _{2})={\frac {1}{2{\sqrt {8\pi ^{5/2}+5\pi ^{3}}}}}\left(1+{\frac {1}{2}}|\mathbf {r} _{1}-\mathbf {r} _{2}|\right)\exp \left(-{\frac {1}{4}}{\big (}r_{1}^{2}+r_{2}^{2}{\big )}\right).

The corresponding ground-state total energy is then $E=E_{R}+E_{u}={\frac {3}{4}}+{\frac {5}{4}}=2E_{\mathrm {h} }$ .

Remarks

The exact ground state electronic density of the Hooke atom for the special case $k=1/4$ is^[4]

\rho (\mathbf {r} )={\frac {2}{\pi ^{3/2}(8+5{\sqrt {\pi }})}}e^{-(1/2)r^{2}}\left(\left({\frac {\pi }{2}}\right)^{1/2}\left({\frac {7}{4}}+{\frac {1}{4}}r^{2}+\left(r+{\frac {1}{r}}\right)\mathrm {erf} \left({\frac {r}{\sqrt {2}}}\right)\right)+e^{-(1/2)r^{2}}\right).

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.

References

^ Lucjan, Piela (2007). Ideas of Quantum Chemistry. Amsterdam: Elsevier. pp. 185–188. ISBN 978-0-444-52227-6.
^ ^a ^b N. R. Kestner; O. Sinanoglu (1962). "Study of Electron Correlation in Helium-Like Systems Using an Exactly Soluble Model". Phys. Rev. 128 (6): 2687–2692. Bibcode:1962PhRv..128.2687K. doi:10.1103/PhysRev.128.2687.
^ ^a ^b S. Kais; D. R. Herschbach; R. D. Levine (1989). "Dimensional scaling as a symmetry operation". J. Chem. Phys. 91 (12): 7791. Bibcode:1989JChPh..91.7791K. doi:10.1063/1.457247.
^ ^a ^b S. Kais; D. R. Herschbach; N. C. Handy; C. W. Murray; G. J. Laming (1993). "Density functionals and dimensional renormalization for an exactly solvable model". J. Chem. Phys. 99 (1): 417–425. Bibcode:1993JChPh..99..417K. doi:10.1063/1.465765.
^ ^a ^b ^c M. Taut (1993). "Two electrons in an external oscillator potential: Particular analytic solutions of a Coulomb correlation problem". Phys. Rev. A. 48 (5): 3561–3566. Bibcode:1993PhRvA..48.3561T. doi:10.1103/PhysRevA.48.3561. PMID 9910020.

Fahrer / Team für die Suche eingeben:

Fahrer / Team für die Suche eingeben:

Langbahn Team – Weltmeisterschaft

Hooke's atom

Definition

Solution

Remarks

See also

References

Further reading