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Bonded or not, when two atoms or ions come closer than the bond distance or the van der Waals distance, the energy increases drastically. What is the underlying cause of that repulsion?

  1. The repulsion of the negatively-charged inner-shell electrons
  2. The repulsion of the positively-charged nuclei (or the effective positive charge of nuclei plus inner-shell electrons)
  3. Any classical explanation solely based on electrostatics falls short, and the increase in energy is related to the Pauli exclusion principle.
  4. There is no single underlying cause, it depends on the specific case. Sometimes repulsion of positive charges is the problem, sometimes repulsion of negative charges.

enter image description here

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    $\begingroup$ Option 3. $\endgroup$ – Mithoron Jan 11 at 21:15
  • $\begingroup$ ... enter at least 15 characters? @Mithoron $\endgroup$ – Karsten Theis Jan 11 at 22:15
  • $\begingroup$ Bypassing of this become like a way to show off quite some time ago :D $\endgroup$ – Mithoron Jan 11 at 22:36
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I found a nice figure and the relevant statement in a paper by Frenking and Krapp (Unicorns in the world of chemical bonding models, 2006, https://doi.org/10.1002/jcc.20543):

The crucial term which is responsible for repulsive interactions in chemical bonds except in two‐electron systems such as H2 is the Pauli repulsion. The three terms (a) quasiclassical electrostatic interaction, (b) resonance, and (c) Pauli repulsion give a complete picture of the interatomic interactions which yield a chemical bond.

The figure shows how the distance-dependent repulsion for one scenario including the Pauli repulsion and two scenarios ignoring the Pauli repulsion; the latter do not show the drastic increase in repulsion that is observed.

enter image description here

Source: https://onlinelibrary.wiley.com/cms/attachment/2b2dc015-ca3c-4fd3-b51c-f768a6a924b5/mfig003.jpg

The entire essay is great, connecting the picture of bonding that bench chemists have with the developments of theory and computation in the 90 years since quantum mechanics was first applied to chemistry.

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The Hamiltonian for the collection of charged particles (electrons and nuclei) comprising a pair of molecules or ions has terms describing the kinetic energy of individual particles and pair-potential terms describing Coulombic interactions, that is, all interactions between particles are Coulombic. Where a Pauli repulsion term can crop up is when evaluating the expectation value of the Hamiltonian and factoring the total energy into piece-wise contributions. A Pauli repulsion term appears because the wavefunction is antisymmetric, encoding how electrons avoid each other due to their fermionic behavior, but this does not invalidate the fundamentally Coulombic nature of the interaction, and all repulsion can be attributed to Coulombic repulsion. Nucleus-nucleus Coulombic interactions are always repulsive, as are electron-electron ones.

As emphasized in the OPs answer, presumably the question is not just about the origin of the repulsion, but also about the functional form of the potential. I would argue that this is a very broad question, since it depends on the specific species that are interacting, and on the value of r. One can agree however that at energies typically relevant to chemists the repulsion at short r is dominated by electron repulsion and specifically by a distribution determined by the fermionic nature of the electrons (Pauli exclusion) which forbids electrons from having the same wavefunction. This is sometimes roughly described by the orthogonality condition captured by the overlap term

$$S=\int\phi_a\phi_bd\tau$$

This term is often written as an exponentially decaying function.

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