The same difference, at five repulsions

One of the figures on when repulsion is in the model: Hubbard systems small enough to diagonalise exactly: the singlet a one-electron model cannot find, the coupling between two spins, and a gap where band theory says there is none.

Two essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

A difference does not make a transfer

The check that licenses everything after it. With the repulsion switched off the computed transfer reproduces the two-level closed form at five differences, to nine decimal places. The calculation is therefore known to give the textbook answer in the case where the textbook answer is right.

The same difference at four repulsions, taken further than the figure above takes it. At no repulsion the charge that moves is the one-electron answer; by a repulsion of thirty-two it has fallen by most of itself, and the difference in orbital energy that produced it has not changed at all. One number in, four numbers out, and the extra variable is not on any electronegativity scale.

The charge transferred against the difference in orbital energy, at five strengths of the repulsion. The top curve is the two-level result. Every other one lies below it, and the lowest is nearly flat: at U = 16 a difference of eight moves 0.025 of an electron where the one-electron answer is 0.970.

The value that only exists in the bond

What resists the transfer: a difference in site energy moves a definite amount of charge, and how much depends on what opposes it. Hardness is the atomic version of that opposition, and the equalisation above is the same balance solved for a whole molecule at once.