Where does this (hydrogen molecule energy) graph come from?

Question

I was thinking about the good old question of 'Why do molecules have lower energy than the atoms?'

And in a video (around 6:15), this good old energy graph is shown, which is stated as the 'answer' to the question.

But, I still dont undertand how this graph comes about.

The video showed this for two hydrogen atoms, and how quantum mechanics is also involved. So, please tell me answers corresponding to a simple hydrogen molecule.

My list of questions is this -

• Is this graph just a result of all the coulombic forces involved? Or there are other factors at play too?
• When we consider the coulombic forces between electron of one atom with the proton of other atom, are we considering electron as "particle"(with some specific location) or a "cloud of charge"(spread uniformly)?
• Most importantly, Can you provide me a link of a book or other study material which can help me(a B.Sc. Physics student) to understand the proper mathematics involved in a hydrogen molecule formation, pertaining to the original question(like how the equation of the graph comes out, what quantum mechanics is happening, what equations are involved etc.)

Answer 1

I think this is a great question, and I agree that this concept is often introduced improperly, and students are just expected to accept it without knowing what the graph even means. It looks like you haven't yet gotten to quantum mechanics in your courses, so as for resources, I'll just say wait for your quantum mechanics class or read the book early.

This plot is the potential energy of the repulsion of the two protons plus the ground state energy of the quantum mechanical system for the electrons if we assume the two protons are classical point particles with a given distance.

Assuming you haven't gotten to quantum mechanics yet I'll be really surface level. But basically, solving an equation called the time independent Schrodinger equation tells us the energy levels of a quantum mechanical system (and differences in energy between different levels are, for example, the spectral emission lines of hot gasses that I'm sure you've heard of). The state with the lowest energy is called the "groundstate" and many systems at reasonable tempeartures are just in the groundstate at all times.

So supposing you had two electrons and two point particles at a variable distance, the total energy of the system would be the electrostatic potential of the two point particles plus the groundstate energy you find once you solve the schrodinger equation for the two electrons in the presence of the potential produced by the point particles.

Of course, protons are also particles, and they are also quantum mechanical, so at one level it seems ridiculous to treat the proton as a point particle but the electrons as quantum objects. But actually this is justified by the Born-Oppenheimer approximation, a more advanced topic in quantum mechanics.

Answer 2

I don't know what those two graphs are purporting to show. The relative positions of two hydrogen atoms, either close together or far apart, are two points on a single curve. Far apart is as close as you can come to describing the potential energy of something by itself, since potential energy always describes the relationship between a system of things.

The characteristic shape of the curve comes from the sum of:

...A $(1/r)^{12}$ dependent potential from a repulsive force that emerges from the Pauli Exclusion Principle, which prohibits two electrons from sharing the same quantized state. See: Wiki: Exchange Interaction. This is not a Coulomb interaction. While this effect is unambiguously a force in the classical physics sense (an influence which causes a change in momentum), the electron exchange interaction doesn't have a mediating force carrier like electromagnetism or the weak and strong nuclear forces.

... And a $-(1/r)^6$ dependent potential from an attractive force that emerges from the Coulomb interaction.

The larger exponent dominates for small $r$, while the smaller exponent dominates for large $r$, hence the characteristic potential well.

See Wiki: Van der Waals force and links therein for a discussion of these and other intermolecular forces.