Ground State Frequency of Electron in H+

[joecoss]

I have seen the frequency of the electron in the ground state of the hydrogen atom as 6.58 x 10^15 Hz, however I have never been able to find the ground state frequency of the electron in the hydrogen molecular ion (H+). The equilibrium separation of the two protons is given as twice the Bohr radius. Does anyone have the value for this frequency?

 

[per.sundqvist]

You mean $$H_2^+=H+H^+$$ ? I don't think this molecule is stable, but you can evaluate the electronic energy when you keep the distance fixed -and the result will depend on that distance! There is no analytic answer as far as I know.

In semiconductor physics you could have thermally ionized impurities in nanostructures that could be localized close to each other and in that case this problem could be relevant (the ionized atoms are then approximate treated as H+).

Anyway its a linear PDF (eigen value problem) to solve in 2D ($$\Psi(z,\rho)$$) so you could find the value you want to any precision, by solving it numerically. Probably a lot of people have done that calculation and published it on the web. For example use Comsol/Femlab finite element software.

Note also that the total energy involves the proton-proton repulsion.

/Per

 

[joecoss]

Per
Thanks, I am interested in a diatomic Hydrogen Molecule that has lost one Electron (H2+). My three quantum texts list a stable equilibrium configuration that occurs when two Hydrogen Atom Wavefunctions overlap so that the distance between the two protons occurs at exactly twice the Bohr Radius. The reason that I am interested in the actual Electron Frequency is that I am trying to understand how Ionization Energies related to many-body geometry since for all single electron ions, the dissociation energy is the same as the value of the electron energy itself (-13.6 eV for the Hydrogen Atom, -54.4 eV for the Helium Ion, -122.4 eV for a single ELectron Lithium Ion, etc.). The Ionization Energy for the stable H2+ Ion is given as -16.3 eV including the repulsive Coulomb Potential of the two Protons, and I am curious how this linear rather than pointlike proton configuration behaves in terms of an Effective Atomic Number Z. Is there an Equation for the Electron Frequency as a function of Ionization Energy as an analytic solution of the combined hydrogenic wavefunctions? Thanks again.
JC