What repulsion keeps non-hydrogen atoms at a distance: between inner shells or between nuclei? 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?


The repulsion of the negatively-charged inner-shell electrons
The repulsion of the positively-charged nuclei (or the effective positive charge of nuclei plus inner-shell electrons)
Any classical explanation solely based on electrostatics falls short, and the increase in energy is related to the Pauli exclusion principle.
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.
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.

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.