Question on Hellman-Feynman Theorem

[Septim]

Greetings,

I have asked this question before, but since I tried to work it out once more and think that I will be able to get different insights; hence I am asking it again. The problem is in the end of chapter problems section for chapter 2 from the second edition of the book "Laser Physics" by Milonni and Eberly. The problem statement is as follows:

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 Be for H2+, and compare your result with the value 29.8 cm-1 tabulated in Herzberg's Spectra of Diatomic Molecules.

My findings:
(a) +13.58 eV by the familiar formula for Coulomb potential (positive since it is repulsive) (b) -29.88 eV. By subtracting the result of part (a) from the given binding energy.

At this point I would like to ask if my reasoning and results are correct or not. In addition to that I practically failed to understand part(c) but thinking that the force on the electron -treating the electron as a classical particle- should vanish at the equilibrium separatation it should be halfway between the two nuclei. However when I calculate the attractive potential energy this way I found a value much more negative than part(b) which makes me question my reasoning and result. The question asks where the electron is most likely to be found in my opinion and we will get this result by treating electron as a classical particle. Can you give me some hints? I really do not know where to start in part (c) and (d) and just need a little push. I think rereading the chapter for part (d) but have no clue on part (c). Please do not hesitate to post your suggestions and ideas; every suggestion is appreciated.

Thanks in advance

 

[DrClaude]

(a) +13.58 eV by the familiar formula for Coulomb potential (positive since it is repulsive) (b) -29.88 eV. By subtracting the result of part (a) from the given binding energy.

Correct.

However when I calculate the attractive potential energy this way I found a value much more negative than part(b) which makes me question my reasoning and result.

This is where QM comes into play. The electron is not localized half-way between the two nuclei, but its wave function is delocalized over the entire molecule, such that the average Coulomb energy is smaller (in absolute value).

The question asks where the electron is most likely to be found in my opinion and we will get this result by treating electron as a classical particle. Can you give me some hints?

In what direction are the repulsive forces on the nuclei acting? Where must the electron be to mitigate these forces?

For part (d), find the equation for the rotational constant (and think "moment of inertia").

 

[Septim]

Hmm I think the question does not ask for a quantum mechanical treatment, the expectation value of the Coulomb Hamiltonian will surely be less in magnitude since the wave function is delocalized over the entire molecule. My question is that the forces will mitigate just at the center and its a stable equilibrium point. So is my reasoning sort of correct that the wave function modulus squared has its maximum value halfway between the two nuclei(The expectation value is different of course)?

 

[DrClaude]

Hmm I think the question does not ask for a quantum mechanical treatment, the expectation value of the Coulomb Hamiltonian will surely be less in magnitude since the wave function is delocalized over the entire molecule.

I just wanted to point out why the value you calculated for a classical electron is much lower than the actual binding energy.

My question is that the forces will mitigate just at the center and its a stable equilibrium point. So is my reasoning sort of correct that the wave function modulus squared has its maximum value halfway between the two nuclei(The expectation value is different of course)?

Sounds correct.

 

[Septim]

Well thanks for your answer, I am now aware of the limitations of my answer and the reason behind it. For part (d) I think I will be able to tackle it on my own, I would consult here if I need further help.