Two orbitals, 4 electrons, S = 0 and S = 0.25

One of the figures on bonding models: Valence bond, molecular orbital and hybrids, drawn as descriptions of one thing rather than as competing pictures.

Nine 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.

The antibonding level goes up more

Two orbitals with four electrons in them, drawn twice. On the left the overlap is dropped and the shifts are exactly equal, which is the diagram usually taught. On the right the overlap is kept at 0.25: the lower level falls by 0.800 and the upper rises by 1.333, so four electrons end up 1.067 above where they started. That is not a failure to bond; it is a repulsion.

The same four electrons at an overlap of 0.5, where the asymmetry is unmistakable: the bonding level falls by 0.667 and the antibonding rises by 2.000, three times as far. Large overlap makes the antibonding orbital very expensive, which is why two atoms with filled shells resist being pushed together long before their nuclei come near each other.

Two electrons in the same pair of levels. Both go into the bonding orbital, the net is 1.600 below the starting point, and the molecule is bound. The difference between this figure and the one at the top of the page is two electrons and nothing else.

Overlap is not interaction

Two levels four units apart with a coupling of one, drawn with the overlap kept at 0.2 rather than dropped. The lower combination falls well short of the whole unit of coupling; the algebra in the text, which sets the overlap to zero, puts the shortfall at 0.236 against 1.000. The mismatch is doing most of the work either way.

The badly matched pair: a coupling half again as large, across a gap four times worse. The bonding combination is barely below the lower original, and almost all of it is still the lower atom’s own orbital — which is what an interaction that has not achieved much looks like. The overlap is kept here too, which is why the antibonding partner rises further than the bonding one falls.

The same pair at half the energy mismatch. Nothing about the overlap has changed and the stabilisation has more than doubled, because what a pair gains depends on the gap between the two levels as well as on the integral between them — and the gap is the variable a rule of thumb about overlap has no place for.

The same overlap, a different bond

What three pairs are stabilised by at an imposed overlap. Two orbitals at the same energy are bonded nearly in proportion to their overlap; a 1s with a 2s is bonded by an amount that barely notices it, because the gap between their energies rather than the coupling is setting the size.

The other input, held at its worst, as in the comparison that separated the two: a pair eight units apart in energy, coupled at −1.5, which is a coupling half again as large as the matched case and buys a small fraction of what it buys there. Everything in this essay is one pair moving along the overlap axis at a fixed gap, and it is the gap that decides how much the movement is worth.

A regime that belongs to the neighbours

And the regime itself, measured as a ratio: how much the pair’s bonding responds to its own overlap, taken between an overlap of 0.4 and one of 0.1. Alone the ratio is 11.83, which is the regime in which bonding tracks overlap. With a third orbital coupled to both it falls to 1.46, 0.92 and 0.74 — and a ratio below one means a fourfold increase in overlap leaves the pair less bonded than it was.

The sharpest form of the claim: a pair bonded by a negative amount. What the A–B interaction is worth is computed as the energy of the trio less the energy of the same trio with that one interaction removed and nothing else changed. With the third orbital decoupled the pair gains 0.539 eV. With it coupled to both, the same interaction costs between 1.142 and 3.096 eV at every placement tried.

The two-electron and four-electron cases of the same pair, which is where the sign comes from. With both levels filled the interaction costs rather than pays, and the third centre is what makes the second level filled.

A bond order between atoms that do not interact

The three levels of a trio in which the two outer orbitals have exactly no overlap with each other, as the third orbital’s energy is swept. One of them does not move.

The three levels at three third-orbital energies, with the free-atom energy marked. One line lands on it every time.

The A–B bond order along the same sweep, with the A–C one beside it. Nothing connects A and B.

A filled shell is not an empty statement

The bond order between the two orbitals that do not interact, as the third orbital’s energy is swept, at every electron count the trio can hold.

The two open-shell counts as the third orbital is taken far below the pair: to zero at two electrons, to minus one at four.

The filled-shell bond orders, computed by diagonalising the trio and by inverting its overlap matrix, for the pair that does not overlap and the pair that does.

A symmetry holds or it does not

The three levels as one of the two outer orbitals is raised. The middle one is exact at zero detuning and leaves linearly.

The level’s displacement divided by the detuning that caused it, against the detuning.

The level’s distance from the mean of the two site energies, against the detuning, on logarithms.

A level no symmetry was protecting

The three levels as the A–C overlap is raised by up to a fifth. The outer two move by electronvolts; the middle one does not move.

How far the level moves against the size of three different changes. Two of them move it linearly; the third leaves it at rounding.

The share of the exact level on A, on B and on C as the A–C overlap is raised, solved (points) against $S_{BC}A - S_{AC}B$ (lines). It turns towards B and never touches C.

The spin the count does not hold

The trio’s three levels, and the unpaired spin on A and on B, as A’s overlap with C is raised from 0.25 to 0.45.

Three changes of a chemically ordinary size, each read for how far it moves the singly occupied level and how far it moves the spin on A.

With overlaps of 0.45 and 0.25, the spin on A and on C as C’s energy runs from −24 to +2 eV, at three coupling constants.