Every essay — page 19
Orbitals Where the atoms go Bonding models What symmetry decides Beyond the octet What a spectrum settles When the molecule does not stop What the shape is for What is taught wrongly Series Named objects Orbitals Refutations Search
Orbitals
A one-electron wavefunction, drawn as a contour surface at a level somebody chose. Nodes, signs, and what the picture is a picture of.
The shortfall was a length, not a strength
With each third-row ion's orbital range set by its own contraction, an ionic model leaves all six rock-salt separations about fourteen per cent short, the same for every pair to 1.3 points. A shortfall common to every pair is what a strength constant is for, and one closes it at a factor of 6.15 — while spreading the six over 8.8 points, because a strength moves each equilibrium by a length, the repulsion's own decay length times the logarithm of the factor, and those lengths are not the same fraction of every separation. Scaling every ion's orbitals by one factor instead closes the mean with the six within a point of each other. It works because it is exactly a small strength change followed by a uniform stretch.
The third function fills the floor
A polar σ bond built from one s function per atom has a momentum profile whose minimum at π/R never reaches zero — for carbon monoxide, 1.35 per cent of the peak. The natural hope was that a p function mixed into the diffuse atom would reshape it towards the compact one and lower that floor. It cannot. At π/R an s function on one atom and a p function on the other are exactly out of phase, so a p function on one atom cancels nothing and only adds itself. With p on both atoms in the sense that makes the bond, the pair adds fifteen times as strongly as the s pair cancels: a quarter p takes the minimum to 12.5 per cent. That is eight times the whole range through which polarity moves the s-only bond.
The lone pair sits where the floor was
A polar σ bond's momentum profile has a minimum at π/R whose floor was proposed as a measure of polarity, and the essay that swept the p weight could not say whether carbon monoxide's own σ bond sat near the s-only floor of 1.35 per cent or the quarter-p value of 12.5. With the weights computed — carbon and oxygen 2s and 2pσ, extended Hückel — the answer is that carbon monoxide has no such bond. Its lowest σ orbital reads 2.5 per cent, its middle one 8.4 per cent of its own peak, and its highest, the orbital with the lone pair's sense, has no minimum at π/R at all: it peaks there. The six σ electrons together have no minimum anywhere, and the polar molecule reads lower at π/R than nitrogen does.
The minimum belongs to the heavy atom's node
Carbon monoxide had no single σ bond for the momentum-space polarity reading to read, so the next place to look was a hydride, where there is one. Lithium hydride's computed σ orbital is ninety-one per cent hydrogen and its profile falls straight through π/R without a minimum. Across all seven first-row hydrides one occupied orbital of thirteen has a minimum at all, and where a lithium-to-boron hydride could have one, it would not be the bond's fringe: it would be the heavy atom's 2s node in momentum, a flat point that a little hydrogen tips over.
The node a nodeless orbital has
Lithium hydride's would-be minimum in momentum was traced to the 2s function's node at 0.624 π/R, with a warning that the position belonged to the hydrogenic model and that a nodeless Slater function would have no node at all. Solving the lithium atom properly moves the node to 0.889 π/R; the nodeless function has one too, at 1.080, because a node in momentum is not a node in position but a disagreement between a function's sum and its slope at the nucleus. Lithium hydride's own bond has no minimum with any of the three. What the function decides is whether a minimum can be born at the node at all.
Where the atoms go
VSEPR as a repulsion minimisation rather than a table, the angles that fall out of it, and the case where five sites are not all alike.
VSEPR, computed
The tetrahedral angle is not 109.5 degrees because a textbook says so. It is arccos(−1/3), and it falls out of minimising the repulsion of four points on a sphere without ever being written down.
Five sites are not alike
Every other common arrangement has one or two distinct angles. Five has three, because two of its positions are on an axis and three are round an equator — and a molecule built that way does something about it.
Why water is bent
The standard answer is lone pair repulsion, it predicts the right direction, and it cannot predict the magnitude. A better rule can, and the heavier hydrides show where both accounts run out.
The shapes above six coordination
Eight points on a sphere do not arrange themselves in a cube. They twist one face by forty-five degrees, and above six the arrangements stop being the ones anybody would name and start being the ones a minimisation finds.
Bent's rule, against the substituent series
More electronegative substituents get more p character, so the angle between them shrinks. The rule predicts a trend, the trend is observed, and the quantity it depends on turns out not to be one quantity.
What a lone pair is worth
A lone pair repels more than a bonding pair, says VSEPR, without saying how much more. Put a number on it and fit that number to water, and the same number is wrong for hydrogen sulfide by a factor of two — which means it was never a property of a lone pair.
Which angles are symmetry and which are the model
VSEPR says electron pairs repel and never says by what law. For four, five and six domains it makes no difference whatever — change the exponent by a factor of twelve and not one angle moves. For seven it decides the answer.