Molecular Geometry & Bonding
what the shape decides
Where the atoms go

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.
8 min read 6 figures Taught confidently, and wrongCounted, not quoted

Water’s bond angle is 104.5 degrees. The tetrahedral angle is 109.47. The gap is about five degrees, the standard explanation accounts for its sign, and it does not account for its size.

water — C2vThe molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.HHOC2vprincipal axis C22 mirror planesno inversion centremay be polarcannot be chiralgroup recovered from the coordinates3 atoms
Fig. 1 Water at its measured geometry, in the point group its coordinates give. The angle is 104.5 degrees, and the question is what fixes it there rather than five degrees higher.

The standard account

Oxygen has four regions of electron density round it: two O–H bonds and two lone pairs. Four regions arrange tetrahedrally, as the minimisation shows, so the arrangement is roughly tetrahedral and the H–O–H angle should be roughly 109.5.

It is smaller because lone pairs are held closer to the nucleus than bonding pairs — a bonding pair is shared between two nuclei and pulled away — so a lone pair occupies more angular space and squeezes the bonding pairs together.

That is a good argument. It predicts the right sign, it predicts correctly that ammonia’s angle (107) is between water’s and the tetrahedral value, and it generalises to a large family of molecules.

4 sites, minimisedThe arrangement of 4 points on a sphere that minimises their mutual repulsion. The angles printed were measured off the result rather than quoted, and the shape was not assumed.tetrahedral109.47° × 6repulsion minimised, angles measured off the result4 sites
Fig. 2 The four-region arrangement the argument starts from: 109.47 degrees, from minimising repulsion rather than from a table. Water’s two bonds occupy two of these directions and its two lone pairs the other two.

Where it stops being quantitative

The account contains the phrase “more angular space”, and nothing in it says how much more.

That is the whole difficulty. The rule predicts smaller than tetrahedral, and every observed angle in the family is smaller than tetrahedral, so the rule is never wrong and never sharp. Ammonia at 107 and water at 104.5 are both consistent with it; so would be 108 and 100.

A rule that cannot be wrong is not making a prediction. It is organising a set of facts, which is useful and is a different thing.

What a sharper rule looks like

Henry Bent proposed one in 1961, and it does considerably better.

Bent’s rule: atomic s character concentrates in orbitals directed toward electropositive substituents, and p character concentrates toward electronegative ones.

The reasoning is that an s orbital has density at the nucleus and is lower in energy than a p; an atom will therefore keep s character for itself where it can. A bond to an electronegative partner has its electrons drawn away, so the central atom invests less s character there — and a lone pair, which is not shared at all, gets the most s character of anything.

More s character means a larger angle, because pure s is spherical and pure p orbitals are at 90 degrees to one another. So lone pairs, being s-rich, spread apart and push the s-poor bonding orbitals together.

Why it is sharper

Because electronegativity is a number, Bent’s rule is quantitative in the right direction and it predicts a trend rather than a single case.

The oxygen family bears it out. Water is 104.5. Replace the hydrogens with more electronegative fluorines and the angle should fall: OF₂ is 103.3. Replace them with less electronegative substituents and it should rise: in dimethyl ether the C–O–C angle is about 111, above tetrahedral.

That is a prediction with a direction and a rough magnitude, tested across a series, which the lone-pair argument on its own does not supply.

The heavier hydrides, where both accounts struggle

Now the case that shows the limits of everything above.

Water is 104.5. Hydrogen sulfide is 92.1. Hydrogen selenide is 91. Hydrogen telluride is about 90. The trend is large, systematic, and not a small correction to a tetrahedral starting point.

Neither VSEPR nor Bent’s rule predicts that on its own, and the usual explanation is different in kind: the heavier central atoms barely hybridise at all. Their valence s and p orbitals differ enough in energy and size that mixing them costs more than it gains, so the bonds use nearly pure p orbitals — and two p orbitals are at 90 degrees.

On that account water is the anomaly rather than H₂S. Oxygen is small enough for s–p mixing to be favourable, so its angle is pushed up from 90 toward tetrahedral; sulfur is not, so it stays near 90.

Which explanation to believe

An honest answer: all three are describing the same energy surface in different vocabularies, and none of them is the calculation.

The shape is whatever minimises the total electronic energy. That is a many-electron problem, computable to high accuracy by modern methods, and it does not decompose cleanly into “lone pair repulsion” or “s character” or anything else. The rules are heuristics for what the calculation will produce.

What separates them is how much they predict. VSEPR gives a family of shapes and a sign; Bent’s rule gives a trend with substituent electronegativity; the hybridisation-cost argument gives the heavy-hydride series. Each earns its place by covering cases the others do not, and treating any of them as the mechanism is a mistake the hybridisation essay takes up in general.

sp3 hybridsThe directions the hybrids point, with the angle between them computed from the coefficients rather than quoted. The set is an orthogonal transformation of the atomic orbitals, so it describes the same space in different coordinates.sp3109.471°between every pair4 hybridsworst off-diagonal 0e+0an orthogonal transformation of the atomic orbitalsone electron
Fig. 3 The sp³ hybrid directions, at 109.471 degrees — the geometry the simple account starts from. Water’s actual angle is five degrees below this, and every explanation above is an account of the five degrees rather than of the 109.

The thing water’s angle is often used to prove

One further caution, because water’s geometry is pressed into service for a claim it does not support.

It is common to see the 104.5 degrees offered as evidence that oxygen is “sp³ hybridised”. It is not evidence of that, and it could not be: hybridisation is a choice of basis, and a basis choice makes no prediction about a bond angle. The angle is an observable and the hybridisation is a description; one cannot confirm the other.

What the angle does support is that the arrangement is closer to tetrahedral than to 90 degrees, which is a statement about the molecule and a useful one.

What is measured, and how

The number itself deserves a word, since it is quoted to a tenth of a degree.

Water’s geometry comes from rotational spectroscopy — the microwave spectrum gives moments of inertia, and the moments give bond lengths and angles. The equilibrium angle is 104.48 degrees and the O–H length 0.9578 ångström.

Those are equilibrium values, meaning the bottom of the potential well. The molecule is never at rest there: even at absolute zero it has vibrational zero-point motion, and the average angle over that motion is slightly different. A quoted geometry always belongs to one of several definitions, and the differences are of the order of the effect this essay is about.

The comparison across the family

Setting the related molecules beside one another is what makes the trends legible.

ammonia — C3vThe molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.HHHNC3vprincipal axis C33 mirror planesno inversion centremay be polarcannot be chiralgroup recovered from the coordinates4 atoms
Fig. 4 Ammonia at 107 degrees: three bonds and one lone pair. One lone pair squeezes less than two, and the angle sits between water’s and the tetrahedral value — which is the lone-pair account’s best prediction.
methane — TdThe molecule with the point group found from its coordinates rather than looked up: every candidate operation was applied and kept when it permuted the atoms among themselves.HHCHHTdprincipal axis C36 mirror planesno inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates5 atoms
Fig. 5 Methane at exactly 109.47: four bonds and no lone pairs, so nothing to squeeze. The angle is the tetrahedral one to the precision of the measurement, and its point group Td requires it.
4 sites, minimisedThe arrangement of 4 points on a sphere that minimises their mutual repulsion. The angles printed were measured off the result rather than quoted, and the shape was not assumed.tetrahedral109.47° × 6repulsion minimised, angles measured off the result4 sites
Fig. 6 And the arrangement all three start from, produced by minimisation. Methane sits at this angle; ammonia and water sit below it, by amounts the simple rule orders correctly and cannot predict.

Methane, ammonia, water: 109.47, 107, 104.5, with zero, one and two lone pairs. The ordering is exactly what the rule says, the spacing is roughly even, and neither fact is derived from anything — the rule was constructed to fit them.

That is not a criticism so much as a description of what kind of thing the rule is. It organises a family correctly and predicts the next member’s ordering, which is genuinely useful, and it does not compute an angle. Bent’s rule does better because it has a number in it, and even that predicts a trend rather than a value.

Where the ladder goes next

The minimisation the argument starts from is VSEPR computed.

The status of hybridisation as a description rather than a cause is a basis is not a thing, with the sharper version in hybridisation does not explain.

And the case where an arrangement’s inequivalent sites do something visible is five sites.

What the pictures here cannot show. A drawn molecule is at one geometry, and a real one is vibrating through a range of them. Nothing on this page computes an energy, so none of these figures can adjudicate between the explanations — they show the geometries the explanations are about.

Where this goes next

tetrahedral109.47° × 6repulsion minimised, angles measured off the result4 sites 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. sp3109.471°between every pair4 hybridsworst off-diagonal 0e+0an orthogonal transformation of the atomic orbitalsone electron Hybrids are a basis An sp³ hybrid set is an orthogonal matrix applied to the atomic orbitals. Rotating a basis changes no observable, so asking whether the electrons are really in hybrids is asking which coordinate system nature prefers. axial — 2 sitesneighbours at 90°, 90°, 90°, 180°equatorial — 3 sitesneighbours at 90°, 90°, 120°, 120°the two are not equivalentneighbour angles measured, not assumed5 sites 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.