Deriving equation of potential energy of H2

[zimo123]

Homework Statement

Derive the equation for the potential energy of a system of 2 hydrogen atoms as a function of internuclear distance.

Homework Equations

For electron/electron and nucleus/nucleus repulsion:

$V_E=\frac{e^2}{4\pi\epsilon_0 R}$

For electron/nucleus attraction:

$V_E=-\frac{e^2}{4\pi\epsilon_0 R}$

Schrodinger equation:

$-\frac{\hbar^2}{2m}\frac{\partial^2\psi (R)}{\partial R^2}+V(R)\psi (R)=E\psi (R)$

The Attempt at a Solution

I am really confused on what to take for V(R) ? There are 4 electric interactions (2 electrons and 2 positively charged nuclei). Also, I don't understand what I should graph, is it the wave function ? If someone could just give me a little hint, it would help a lot!

Thank you so much

 

[mark726]

for your question. The potential energy of a system of 2 hydrogen atoms can be derived using the Schrodinger equation. In this case, V(R) would represent the total potential energy, which is the sum of the electron/electron, nucleus/nucleus, and electron/nucleus interactions. Therefore, the equation for the potential energy of a system of 2 hydrogen atoms as a function of internuclear distance (R) would be:

V(R) = \frac{e^2}{4\pi\epsilon_0 R} + \frac{e^2}{4\pi\epsilon_0 R} - \frac{e^2}{4\pi\epsilon_0 R} = \frac{e^2}{2\pi\epsilon_0 R}

To graph this potential energy, you would plot V(R) on the y-axis and the internuclear distance (R) on the x-axis. The resulting graph would show the potential energy as a function of the internuclear distance, giving you an idea of the stability of the system at different distances.

I hope this helps clarify the concept for you. Good luck with your studies!