Pauli exclusion is not a repulsive force, and yet…

“It’s not a force.” – Professor Steven Weinberg

I made a mistake.

In my book, Block by Block, I wrote about the attraction and repulsion forces between atoms. For the former, I stated that attraction results from the fact that atoms act like spinning magnets; they contain a positive charge (proton) that is separated from an orbiting negative charge (electron). The quickly varying dipole of one atom acts upon the same of another atom, thereby inducing dipoles that are in-phase with each other. The electrons of atoms attract the protons of other atoms, resulting in an attractive force between all atoms. This phenomenon is all very nicely laid out in a paper by F. W. London [1], who coined the term “dispersion effect.” But this wasn’t where I made a mistake.

The mistake was with my supposed understanding of the repulsion forces. On page 46 of my book I wrote that repulsion is caused by “the electromagnetic repulsion forces between electrons and between protons.” This might not look like a mistake to you, but it does to me, or at a minimum, as an incomplete answer, because I now know something I didn’t fully understand then. You see, I understood that Pauli exclusion was somehow involved in the repulsion effect but didn’t know why. I originally wondered whether or not it was a repulsive force itself. When I couldn’t find the answer I was seeking in the literature, I decided to go straight to the top for help. Professor Steven Weinberg at University of Texas at Austin. I was pleasantly shocked when he very kindly replied! In response to my question regarding whether or not Pauli exclusion is a force, Professor Weinberg replied, “It’s not a force.” To this day I remember that one line from his reply. He continued:

“There is no particle transmitted between the electrons in an atom other than the photons, which mediate the electromagnetic force. But the Pauli principle requires that the wave function of the atom be antisymmetric in the electron coordinates, and this has effects like a force – – – in particular, it prevents two electrons from occupying the same state.” [2]

As I recently began work on book #2, seeking to connect the micro-world of atoms to the macro-world of classical thermodynamics, I knew I had to resolve this issue as I consider it a very fundamental building block for the larger structure I want to create. Over the past two weeks I have done a deep dive into the literature, and here’s what I have found.

First things first. The Pauli exclusion principle states that no two particles can have the same four quantum numbers, which for orbital electrons translates into fact that no more than two electrons are permitted in any given orbit, and the two must have opposite spins.

Next. Electrostatic interactions, e.g., attraction and repulsion, are determined by the electron distribution. Again, the attraction force is caused by London dispersion, which is the induced shift in electron distributions that occurs when two atoms come near each other.

Regarding repulsion

Regarding repulsion. Let’s start with two simple hydrogen atoms approaching each other. Each atom consists of one proton and one electron. As they approach, their respective electron clouds overlap and the individual atomic orbitals of each merge into a bonding orbital. Both electrons populate this single orbital and then spend most of their time between the two nuclei, attracting both nuclei toward them and toward each other. A covalent bond results. No bond is stronger as emphasized by Richard Feynman: ‘‘It now becomes clear why the strongest and most important attractive forces arise when there is a concentration of charge between two nuclei. The nuclei on each side of the concentrated charge are each strongly attracted to it.’’ [3]

Now consider two helium atoms approaching each other. Each atom consists of two protons, two neutrons, and two electrons. As they approach, their respective electron clouds overlap, and their individual orbitals merge into a bonding orbital, which is then populated by two of the electrons, one from each atom.

Now comes the critical part of this post. What happens to the next two electrons? They can’t enter into this same bonding orbital with the others because of the Pauli exclusion principle. So they must enter into the next orbital available up on the energy ladder, which is an anti-bonding orbital. This orbital forms at the same time as the bonding orbital. The anti-bonding orbital is so named because the electrons in this orbital accumulate outside the region between the nuclei and so can’t contribute to bonding. Instead they contribute to anti-bonding since their location effectively decreases the pull of the two positively-charge nuclei toward the negative electrons between them, leaving the proton-proton repulsion force to dominate the situation.

The anti-bond negates the covalent-bond, leaving a very weak net bond. This is what happens when atoms “collide.” The bond between them is weak and they repel each other. Electron-electron interactions contribute to the net repulsion, but it’s the proton-proton interactions that dominates. Two helium atoms collide (and don’t react) because there is room in the bonding orbital for only one pair of electrons; the other pair must occupy the anti-bonding orbital.

The same general principles apply to all closed-shell molecules. Individual electron orbitals combine to form two orbitals, one bonding and one anti-bonding. Being closed-shell, two electrons are offered up by each atom. Two enter the bonding orbital, the other two the anti-bonding orbital, leaving a weak net bond that is easily broken by the strong proton-proton repulsion.

In the end, while Pauli exclusion doesn’t directly cause repulsion, and is not a repulsive force, it is because of Pauli exclusion that repulsion occurs.

I list three excellent references below that go into much greater detail on this matter: Henry Margenau [4], H.C. Longuet-Higgins [5], and Richard Bader [6]. I saw that Professor Bader’s 2007 paper was relatively recent and sought to contact him. I learned that he had passed in 2012 but noted that one of his students co-authored a relevant paper of his. It was through this path that I met this student, Chérif F. Matta. Chérif is Professor and Chair/Head of the Department of Chemistry and Physics at Mount Saint Vincent University in Halifax. He enthusiastically responded to my email inquiry and helped me with my understanding of the above topic; any mistakes above are all mine! I publicly thank him here for his contribution and look forward to further discussions with him.

(1) London, F. W., The General Theory of Molecular Forces, Trans. Faraday Soc., 1937,33, 8b-26. This paper, by the way, shows the origin of the r^6 attraction term in the Lennard-Jones potential equation.

(2) Weinberg, Steve, personal communication, 01 July 2009.

(3) Feynman, R. P., Forces in Molecules, Physical Review, 56, 15 August 1939, pp. 340-343.

(4) Margenau, Henry, The Nature of Physical Reality, McGraw Hill, 1950. Chapter 20, The Exclusion Principle, pp. 427-447.

(5) Longuet-Higgins, H. C., Intermolecular Forces, Spiers Memorial Lecture, The University, Cambridge, Received 23rd September, 1965, published in Discuss. Faraday Soc., 1965,40, 7-18

(6) Bader, R. F. W., J. Hernández-Trujillo, and F. Cortés-GuzmánBader, Chemical Bonding: From Lewis to Atoms in Molecules, J. Comput. Chem.. 2007 Jan 15;28(1):4-14

Thank you for reading my post. I go into much greater detail about the science and history of the atomic theory in Chapters 3 and 4, respectively, of my book Block by Block – The Historical and Theoretical Foundations of Thermodynamics.