Hellman-Feynman Theorem: Coulomb Repulsion & Attraction

[Septim]

I have came across an interesting question in the book Laser Physics by Milonni, the question is as follows:

The binding energy of the ion H₂⁺ ( the energy required to separate to infinity the two protons and the electron) is -16.3 eV at the equilibrium separation 0.106 nm.
(a) What is the contribution to the energy from the Coulomb repulsion of the nuclei?
(b) What is the contribution to the energy from the Coulomb attraction of the electron to the nuclei?
(c) The Hellman-Feynman theorem says, in effect, that the force between the nuclei in a molecule can be calculated from the electrostatic repulsion between the nuclei and the electrostatic attraction of the nuclei to the electron distribution. According to this theorem, where must the squared modulus of the electron wave function in H₂⁺ have its maximum value?
(d) Estimate the rotational constant B_e for H₂⁺, and compare your result with the value 29.8 cm^-1 tabulated in Herzberg's Spectra of Diatomic Molecules.

I have found in the part (a)to be +13.58 eV by the familiar formula for Coulomb potential and part (b) to be -29.88 eV. By subtracting the result of part (a) from the given binding energy. Firstly I would like to ask if these are correct and secondly I have no idea on how to proceed in part (c) as I have only taken an introductory course on quantum physics. Can you give me some hints? Any help is appreciated.

Thanks

 

[Cthugha]

You get the implied binding energy, so the values seem to be correct.

With respect to c), the idea is that you can calculate the force on some given nucleus (or nuclei) along some coordinate by checking the derivative of the total energy with respect to that coordinate. Then you can go on and assume a general electron distribution and check what the resulting force looks like. You will get some force depending on the electron distribution. You are looking for an equilibrium situation, so the resulting force has to vanish.

So you just solve that equation and find the electron distribution for which the resulting force vanishes and you will find the position where the squared modulus will have its maximum.

 

[Septim]

I did not comprehend this to full extend, can you explain it more precisely preferably with equations ?

 

[Cthugha]

Hmm, doing the whole math will get quite lengthy.

Did you by chance check the wikipedia entry on the Hellmann-Feynman theorem (which is not as bad as some articles in wiki are) and have a look at the molecular forces application example?

If there is something unclear with that, it might be easier to discuss that special point directly.

 

[Septim]

Thanks for the reply, I will have a look at the article on Wikipedia and state the points unclear to me.

 

[Septim]

Sorry to invoke this thread again, but should the electron cloud mostly reside halfway between the two nuclei. I inferred this reason from the reasoning that the force on the electron should be zero at equilibrium classically (treating it as a point particle), so switching to QM the wave function modulus square should be maximum at this particular point.