How is the Van der Waals interaction potential calculated?

[chikou24i]

Hello! In Van der Waals interaction, how to prove that : H= - (2*e^2*x1*x2) / R^3 ?

 

[Quantum Defect]

Hello! In Van der Waals interaction, how to prove that : H= - (2*e^2*x1*x2) / R^3 ?

This looks strange. The attractive part of the Van der Waals (what it looks like you are talking about) goes like 1/R^6, with the polarizabilities in the numerator).
A very complete discussion of the various kinds of forces between atoms/molecules can be found in Hirschfelder, Curtiss and Bird "Molecular Theory of Gases and Liquids"

 

[chikou24i]

I'm talking about the Coulomb interaction energy between two harmonic oscillator ( two atoms modelised by two harmonic oscillator)

 

[Quantum Defect]

I'm talking about the Coulomb interaction energy between two harmonic oscillator ( two atoms modelised by two harmonic oscillator)

This makes more sense. The 1/R^3 potential is one you get for two dipoles. To find this interaction you can sum the pair wise interactions for point charges on the different dipoles (the attraction between electron 1,2 and proton 1,2 can be ignored, since they are constant - just worry about the electron/proton on atom 1 interacting with the charges on atom 2.)

What you will do is approximate the 1/(R_1,2) terms in a Taylor approximation when R_1,2 >> r, where r is the length of the dipole. You will find that the terms that survive are the ones that look like 1/R^3. The sign (attractive, repulsive) and leading coefficient depend upon the orientation of the two dipoles.

I am nearly positive that Hirschfelder,Curtiss, and Bird show this. Probably a good e&m book will show this, too.

 

[chikou24i]

What you will do is approximate the 1/(R_1,2) terms in a Taylor approximation when R_1,2 >> r, where r is the length of the dipole. You will find that the terms that survive are the ones that look like 1/R^3. The sign (attractive, repulsive) and leading coefficient depend upon the orientation of the two dipoles.

Now you understand me, and this is what I'm looking for if you can help me.

 

[Quantum Defect]

The potential is easiest to see if you set up the two atoms, with the following orientations:

+ -.................... + -

The proton-electron separation in each atom is r, and the proton-proton separation is R (as is the electron-electron separation).

The potential is:

V = -e^2/r - e^2/r + e^2/R + e^2/R - e^2 /(R-r) - e^2/(R+r)

The first two terms are constants (assuming that r is fixed), so let's forget about those.

V = 2e^2/R - e^2/(R-r) - e^2/(R+r)

You are going to rearrange the 1/(R+/-r) into something that you can expand:

1/(R+/-r) = 1/R*(1/[1+/-x]) where x = r/R, a small number.

Use the binomial expansion for 1/(1+x) and 1/(1-x), and plug and chug...

You should find that the largest term looks like:

V = -2*mu^2/R^3, where mu = e*r