Bent's rule, against the substituent series

VSEPR predicts a direction and nothing more: bond angles fall below the ideal value when lone pairs are present, by an amount the rule does not name. Bent’s rule has a number in it, and the number is an electronegativity.

That makes it sharper and it makes it inherit a problem, which is the subject of the second half of this essay.

The rule #

Atomic s character concentrates in orbitals directed toward electropositive substituents; p character concentrates toward electronegative ones.

The reasoning is an energy argument about the central atom rather than about the bond. An s orbital has density at the nucleus and lies lower in energy than a p orbital on the same atom, so an atom would prefer to keep its s character where it does the atom most good.

A bond to an electronegative partner has its electron density drawn away from the central atom. Investing s character there is investing it in a region the central atom does not get to keep, so the atom invests p character instead — which it can spare, being higher in energy anyway. A lone pair is the extreme case: not shared at all, and correspondingly s-rich.

The geometric consequence follows from the hybrid composition alone.

Why composition fixes an angle #

Two hybrids built from one s and some p have an angle set entirely by their s fractions, and the relation is exact.

For two equivalent hybrids each with s fraction $a$, orthogonality requires

$$\cos\theta = \frac{-a}{1-a}.$$

Set $a = \tfrac14$ and the angle is $\arccos(-\tfrac13)$, which is 109.4712 degrees. Set $a = \tfrac13$ and it is 120. Set $a = \tfrac12$ and it is 180. Set $a = 0$ — pure p — and it is 90.

So the rule is quantitative in principle: get the s fractions and the angle follows. What it does not supply is how much s character a given substituent draws off, and that is where the electronegativity enters as a proxy.

The series it gets right #

The oxygen series is the standard test and the rule passes it.

Water’s angle is 104.5 degrees. Replace the hydrogens with fluorines, which are more electronegative, and the rule says the bonding hybrids should lose s character and the angle should fall: oxygen difluoride is 103.3. Replace them with less electronegative groups and the angle should rise: dimethyl ether’s C–O–C angle is about 111, above tetrahedral.

The carbon series does the same. Fluoromethane, difluoromethane, trifluoromethane: each substitution should pull s character out of the C–F bonds and concentrate it in the remaining C–H ones, so the H–C–H angle should rise and the F–C–F angle should fall. In difluoromethane the measured H–C–H angle is 113.7 and the F–C–F angle is 108.3, straddling the tetrahedral value in the directions predicted.

That is a genuine prediction with a direction and a rough magnitude, tested across a series, and neither VSEPR nor the lone-pair argument supplies it.

The rule also explains a fact usually presented as an oddity: in a cyclopropane ring the C–C bonds are p-rich, because the ring geometry forces the interorbital angle down toward 60 degrees, and the external C–H bonds are correspondingly s-rich. Their coupling constants and acidities both show it, and the rule anticipates both.

The quantity it depends on #

Now the problem, and it is not small.

Bent’s rule is quantitative because electronegativity is a number. There are four numbers.

The four scales are defined from four unrelated things — bond energies, the mean of ionisation energy and electron affinity, the field at a covalent radius, and the average valence-shell energy from spectra. Two are dimensionless numbers on a conventional footing and two are energies in electronvolts, so comparing them by value requires a linear fit whose coefficients are themselves a choice.

Comparing them by rank avoids that, and the ranks disagree. Pauling against Mulliken has twelve discordant pairs out of a hundred and fifty-three; Mulliken against Allen has fifteen. Hydrogen moves three positions in a list of eighteen.

And the disagreement is not confined to elements nobody bonds. C–H, S–H, C–S, C–I, N–Cl and N–Br are ordinary bonds whose polarity direction — which end is the more electronegative — depends on which of the four tables is consulted, and it is common for two of the four to appear on the same page of the same book. Bent’s rule says the s character goes to the less electronegative substituent; on those bonds it does not say which substituent that is.

What that does to the rule #

The carbon–hydrogen bond is the case that matters, and it matters directly for Bent’s rule.

Pauling, Allred–Rochow and Allen all place carbon above hydrogen. Mulliken places hydrogen above carbon, because the mean of hydrogen’s ionisation energy and electron affinity is 7.18 electronvolts against carbon’s 6.27.

Take the second and Bent’s rule reverses for every C–H bond in chemistry. The hydrogen would be the electronegative partner, the carbon should invest p character in its C–H bonds, and every angle prediction involving a hydrogen substituent flips sign.

Nobody does take the second, and that is the interesting part. In practice Bent’s rule is always applied with Pauling’s scale, or with whatever scale gives the familiar answer, and the choice is never stated. So the rule as used is not “s character follows electronegativity”; it is “s character follows Pauling’s electronegativity”, which is a narrower claim resting on a thermochemical fit from 1932.

That is not a refutation. Pauling’s scale may well be the right proxy for this purpose — it is derived from bond energies, which is closer to what a bond orbital cares about than a free-atom property is. What it is, is an unstated assumption doing load-bearing work, and stating it changes the rule’s status from a principle to a correlation with a specified scale.

What was computed, and how #

Two different things on this page are computed, and they are computed to different standards.

The hybrid angles are exact. Every angle printed on a hybrid figure is measured from the coefficient matrix rather than quoted, the matrix is checked orthogonal to $10^{-15}$ before the angle is read, and the directions are turned with the slider while the angle is measured again at every frame. The tetrahedral angle is never put in, and comes out as $\arccos(-\tfrac13)$.

The electronegativity comparison is a rank statistic on quoted data. The four scales are measurements or fits to measurements and are quoted, as they should be. What is computed is the comparison: ranks, Spearman’s coefficient for every pair, the count of discordant pairs, and the list of ordinary bonds whose direction is disputed.

The comparison carries two checks that can fail. A scale compared with itself must show no disagreement, which rules out the count fabricating inversions out of ties or arithmetic noise. And both halves of the claim are checked: every pair must correlate above 0.9, or these are not four measurements of anything like the same thing; and at least one pair must be discordant, or the second half of the claim is false and this section should be withdrawn.

Spearman’s coefficient is computed from the ranks rather than by the usual shortcut, because the shortcut assumes no ties and hydrogen ties with itself across Pauling and Allred–Rochow at exactly 2.20.

The surprise: the rule works better than its input deserves #

The uncomfortable observation is that Bent’s rule makes correct predictions while resting on a quantity that is not well defined.

The resolution is that the rule uses electronegativity only for ordering, and the orderings mostly agree. The scales disagree on twelve pairs out of a hundred and fifty-three, and the disagreements cluster in comparisons — carbon against sulfur, nitrogen against bromine — that rarely arise as substituents on the same centre.

Where a real Bent’s-rule argument is made, the substituents being compared are usually far apart on every scale: fluorine against hydrogen, oxygen against carbon. Any of the four scales gives the same ordering, so the ambiguity never surfaces.

That is a genuine explanation and it is also a warning about where the rule will fail. Applied to a case where the substituents are close in electronegativity — and therefore where the scales might disagree — the rule is predicting a small effect from an ambiguous input, and its direction is not reliable.

What it costs #

The hybrid computations cost nothing: a four-by-four matrix and an arc cosine. The electronegativity comparison costs a page of arithmetic over a table of seventy-two numbers.

What the rule costs to use is a decision nobody records. Applying it requires choosing a scale, the choice changes some predictions, and no source this essay is aware of states which scale was used. That is a small omission with a specific consequence: two chemists applying Bent’s rule to the same molecule can reach opposite conclusions and neither will be able to say why.

There is a second and larger cost, which is the same one hybridisation carries everywhere. The s character of a bond orbital is not observable. It is defined within a partitioning scheme, different schemes give different numbers for the same molecule, and “this carbon is sp²·³ hybridised” is a statement about a fitted description rather than about the carbon.

So Bent’s rule relates two quantities, neither of which is an observable, and predicts a third — the angle — which is. That it works is worth being surprised by rather than taking for granted.

Where the model stops #

Four limits.

It predicts a trend, never a value. Nothing in the rule says how many degrees. Water at 104.5 and oxygen difluoride at 103.3 differ by 1.2 degrees, and the rule predicts the sign of that difference and no part of its size.

The proxy is unspecified. Which electronegativity scale, and no answer is standard.

It says nothing about the heavy hydrides. The drop from water at 104.5 to hydrogen telluride at about 90 is nineteen degrees and is not a substituent effect at all — the hydrogens are the same in every case. That series needs the hybridisation-cost argument, which is a different mechanism.

Hybrid composition is not observable, so the intermediate quantity in the whole chain of reasoning cannot be checked against anything. Only the endpoint can.

The two ends of the chain, and the one in the middle #

It is worth laying the argument out as a chain, because that makes visible which link is weakest.

Substituent electronegativity → hybrid s character → bond angle.

The last link is exact. Given two s fractions, the angle follows from orthogonality with no approximation at all, and it is computed here rather than quoted.

The first link is the empirical one, and it is where the scale ambiguity bites. It is also where the rule’s rationalisation lives, which means the rationalisation and the ambiguity are attached to the same step.

The middle quantity is the one that cannot be measured. That is an awkward position for a chain to be in: both ends are firm and the joint between them is a quantity nobody can weigh. What makes the rule usable anyway is that the middle can be eliminated — the composition drops out and what is left is a direct correlation between substituent electronegativity and angle, which is testable and is tested above.

Seen that way, Bent’s rule is a correlation with a mechanism attached, and the mechanism runs through an unobservable. That is a perfectly respectable thing for a chemical rule to be, provided nobody mistakes the middle term for an observation — which is the standing error this site’s wrong field is about.

It is worth adding that the rule’s most-cited successes and its unstated assumption sit in different places. Every case it is famous for — the oxygen series, the fluoromethanes, cyclopropane — compares substituents that all four scales order identically. So the successes do not test the assumption, and the assumption is never exercised where anyone is looking.

Who found it, and when #

Henry Bent published the rule in 1961, in a long review in Chemical Reviews that surveyed a great deal of structural data and extracted the correlation from it. It is an empirical generalisation given a plausible energetic rationalisation, in that order, and Bent presented it that way.

Its intellectual ancestry runs back to Walsh, who had noticed in the late 1940s that bond angles correlate with substituent electronegativity, and further back to Pauling’s hybridisation scheme itself — the scheme Bent’s rule adjusts had been in place for thirty years.

The rule’s staying power comes from its range. It accounts for the oxygen and carbon substituent series, for cyclopropane’s unusual bonds, for the coupling constants that measure s character in nuclear magnetic resonance, and for the anomeric effect in sugars. That is a wide span for a one-sentence rule with no arithmetic in it.

The unstated-scale problem seems never to have been raised in the original literature, which is unsurprising: in 1961 Pauling’s scale was overwhelmingly the one in use, Allen’s did not exist, and the question of which to choose had not yet acquired an answer to be ambiguous about.

The s fractions need not be fitted #

The rule is qualitative because it names no number, and the usual reason given for leaving it that way is that the s character of an inequivalent hybrid is a fitted quantity. For the commonest case it is not. It is determined by the measured angle and one conservation law.

Two orthogonal hybrids with s fractions $s_i$ and $s_j$ subtend an angle fixed by Coulson’s relation

$$\cos\theta_{ij} = -\sqrt{\frac{s_i\,s_j}{(1-s_i)(1-s_j)}}$$

which returns 109.47° at $s = 1/4$, 120° at $1/3$ and 180° at $1/2$ — the three familiar hybrids, with nothing put in but a fraction. The conservation law is that the four hybrids share one s orbital, so their s fractions sum to one.

For a molecule of type AX₂E₂ that is enough. The two bonds are equivalent, so the measured angle gives their common $s_b$ directly; the two lone pairs then take what is left, $s_\ell = (1 - 2s_b)/2$ each; and the angle between the lone pairs follows from the same relation with no further input.

Water’s 104.5° gives bonding hybrids that are 20.0% s and lone pairs that are 30.0% s, subtending 115.4°. Difluorine oxide’s 103.3° gives 18.7% and 31.3%, at 117.1°. Hydrogen sulfide’s 92.1° gives 3.5% — bonds that are very nearly pure p — with lone pairs at 46.5% s and 150.2° apart. Dimethyl ether’s 111° gives 26.4% and 23.6%, at 108.0°.

Three things fall out that no qualitative statement of the rule supplies.

The lone pairs really are fatter in s, and by a computable amount. Water’s are half again as rich in s as its bonds, which is the quantitative content of “a lone pair takes more s character” that the rule states and does not size.

The trend reverses. Dimethyl ether’s lone pairs are poorer in s than its bonds and closer together than they are — because its bond angle is above tetrahedral, and the sum rule then has no choice. The lone pair’s greed is not a property of lone pairs; it is a property of being on the low-angle side of 109.47°.

And hydrogen sulfide is the extreme rather than the exception. Bonds at 3.5% s is the arithmetic saying the sulfur is barely hybridising at all, which is the same conclusion the heavy-hydride series reaches by a different route, arrived at here from one measured angle.

Where to read on #

The case the rule was invented to sharpen is why water is bent.

The construction it adjusts is hybrids are a basis.

The measurement of how much the four scales disagree is electronegativity is not one quantity.

And the same ambiguity doing damage in a different argument is the dipole is not a sum of bonds.

Two habits follow from all this and both are cheap. State the scale whenever an electronegativity is used in an argument, since it costs three words and makes the argument reproducible. And treat a Bent’s-rule prediction between two similar substituents as no prediction at all, since that is precisely the regime where the input is ambiguous and the predicted effect is small.

The rule’s standing is best summarised as an empirical correlation with a mechanism attached and an unstated input. All three parts of that are worth keeping in view, and the last one is the only one that could be fixed by anybody writing it down.

What the pictures here cannot show. The hybrid figures on this page draw directions, not orbitals — a real hybrid has a large lobe, a small opposite one, and a shape depending on its composition, and none of that is drawn. More importantly, no figure here shows a molecule with inequivalent hybrids, which is what Bent’s rule is about — the s fractions for the two-bond, two-lone-pair case are computed in the section above from the measured angle and the sum rule, and nothing on this page draws them.