Why molecules have shapes

Water is bent at 104.5 degrees and methane is a tetrahedron. Both angles were measured before either was explained — and the first evidence that water is bent came from the dielectric constant of a gas, with no picture of a molecule involved anywhere.

1874 – 195716 min

By the end you should be able to

Explain how a dipole moment forces a shape, without any picture of a molecule
Say what a microwave spectrum measures, and how a bond length comes out of it
State the VSEPR rule and place it correctly in time — after the measurements, not before
Name two molecules VSEPR gets badly wrong, and say by how much

Get the order right before anything else, because the usual telling reverses it. By the mid-1930s a chemist could tell you that water is bent at about 105°, that methane is a tetrahedron, that the O–H bond is 0.958 Å long and the C–H bond 1.087 Å. Those were numbers in tables, obtained from instruments, with error bars. What the same chemist could not tell you was why. The explanations — hybrid orbitals in 1931, electron-pair repulsion in 1940, the molecular-orbital treatment properly in the 1930s and after — all arrived to account for angles that were already measured. This is not a criticism of them. It is the normal shape of the subject, and it is worth noticing because a lesson that opens with "electron pairs repel, therefore water is bent" has quietly told you that the theory came first. It did not. The measurement came first, and one of the measurements is so indirect it is worth the whole lesson on its own.

Start with the oldest of the shapes, which was deduced by counting bottles. Suppose carbon puts its four bonds at the corners of a square, with the carbon in the middle. Take CH₂Cl₂ — two hydrogens and two chlorines on one carbon. On a square you can put the two chlorines on adjacent corners, or on opposite corners, and those are genuinely different arrangements. They would be two different substances, with different boiling points, separable by distillation. Nobody had ever found two. One substance, one boiling point, however hard anyone looked. The same held for every CH₂X₂ anyone made. Now put the four bonds at the corners of a tetrahedron. Every way of choosing two corners out of four is equivalent to every other way — a tetrahedron has no notion of "adjacent" versus "opposite". So there is exactly one CH₂Cl₂. There is exactly one. Jacobus Henricus van ’t Hoff, in Utrecht, and Joseph Le Bel, in Paris, published this independently within two months of each other in 1874. Neither had seen a molecule and neither had an instrument that could. They had a count of substances, and the count only came out right for one arrangement in space.

A black and white photographic portrait of Peter Debye as a young man in a high collar and tie, facing slightly left. — Debye worked out, in 1912, how the dielectric constant of a gas depends on the permanent dipole moments of its molecules — and therefore how to measure one. The unit is named after him. Nothing in the procedure involves seeing, imaging or otherwise inspecting a molecule: it is a capacitance, measured at several temperatures.

A dipole moment is a vector sum, and that is the whole argument

Bonds between unlike atoms are polar: one end carries a little more electron density than the other. That gives each bond a small electric dipole, with a size and — this is the part that matters — a direction. The molecule’s dipole moment is the vector sum of them. So two identical polar bonds can add to zero, but only in one circumstance: when they point in exactly opposite directions. Any other angle leaves something over. That turns a shape question into an arithmetic question, and the arithmetic runs on a number you can measure in a capacitor: CO₂ has two very polar bonds and a dipole moment of zero. The only arrangement of two equal opposed vectors that sums to nothing is a straight line. Carbon dioxide is linear. H₂O has two very polar bonds and a dipole moment of 1.85 D, which is one of the largest of any small molecule. The vectors do not cancel. Water is not linear. No microscope, no diffraction, no spectrum. A gas, a capacitor, a thermometer, and a conclusion about geometry.

Start with carbon dioxide and drag the picture around. Watch the bottom line: the resultant is zero from every direction you can look at it from. Then pick H₂O from the row of formulae and watch it appear.

Loading the molecule…

The resultant of the bond moments, drawn as one arrow. For CO₂, BF₃, CH₄, XeF₂ and XeF₄ it is exactly zero — symmetry cancels it, however polar the individual bonds are, so the reading survives any argument about electronegativity. For water and ammonia it is not zero, and it is not small. That single readout is the difference between a molecule that is straight and one that is not.

The experiment

Molecular dipole moments from the dielectric constant of a gas

Peter Debye, and the many people who used his method through the 1920s · Theory 1912; the measurements that mattered here, 1923 – 1930 · Zurich, Leipzig, and eventually everywhere with a capacitance bridge

The question: A molecule’s shape is not observable. Is there any bulk property of a gas that depends on it?
The apparatus: A gas between the plates of a capacitor, and a bridge to measure the capacitance. Repeat at a series of temperatures. That is all the apparatus there is.

Theory predicted

A gas increases the capacitance because its molecules polarise in the field. If that were the whole story, the effect would not depend on temperature at all.

They measured

For some gases it does not depend on temperature. For others it falls steadily as the gas is warmed, and the molar polarisation plotted against 1/T is a straight line. The slope of that line gives μ², the square of a permanent dipole moment the molecule carries whether or not a field is applied. Warmth randomises the alignment, which is why the effect dies away with temperature.

How sure could they be? CO₂, CS₂, CH₄ and CCl₄ gave horizontal lines — no permanent dipole, to the accuracy of the bridge. H₂O gave 1.85 D, NH₃ 1.47 D, SO₂ 1.63 D. The zeros are the load-bearing measurements, and a zero is exactly the kind of result an instrument can report honestly.

Why it mattered

Geometry, out of a capacitor. Two polar bonds summing to nothing forces a straight line; two polar bonds summing to 1.85 D forbids one. This is the earliest evidence that water is bent, and it predates any spectroscopic measurement of the angle by roughly a decade.

It is worth pausing on how strange that inference is. The experiment is a capacitance. The molecule never appears in it. There is no image, no scattering pattern, nothing spatial anywhere in the apparatus. What comes out is a single number in units of charge times distance. And yet, given only that the two bonds are equivalent — which follows from the substance having one kind of O–H bond, not two — that one number rules out a straight molecule absolutely. Not "makes it unlikely". Rules it out, because two equal vectors cannot sum to 1.85 D while pointing opposite ways. What the dipole moment cannot do on its own is tell you which angle. For that you need the bond moment as well, and the bond moment is calibrated on molecules like this one, so the circle does not quite close. What closes it is a different instrument entirely.

Work along the row of formulae. Watch the two things that change together: the shape name, and whether the bond moments cancel. Then use the sliders to dial an arrangement — four bonding pairs and two lone pairs is a square, and squares cancel.

Loading the molecule…

Every angle and length in this picture is a measured value, mostly from microwave spectroscopy. The idealised VSEPR angle sits beside it, and the two disagree by 17.4° for hydrogen sulphide and 16.2° for phosphine — a good deal more than the rule’s reputation suggests, and far more than the 5.0° squeeze that takes water down from the tetrahedral angle.

The experiment

Bond lengths and angles from rotational spectra

C. E. Cleeton and N. H. Williams first; then most of a generation, after 1946 · Ammonia’s inversion measured at 1.25 cm in 1934; microwave spectroscopy general from 1946 · Ann Arbor, then everywhere that inherited a radar set

The question: A dipole moment says a molecule is bent. What actually measures the angle, and the lengths of the bonds?
The apparatus: A gas at low pressure in a waveguide, a tunable source of centimetre-wave radiation, and a detector. Sweep the frequency and record where the gas absorbs. Rotational transitions in small molecules land between roughly 1 and 300 GHz — which is exactly the band radar was built for, and exactly why nobody could do this before the war.

Theory predicted

A molecule free to rotate should absorb at frequencies set by its moments of inertia, and those moments are nothing but the masses of the atoms multiplied by the squares of their distances from the rotation axes. Get the frequencies and you have the moments; get enough moments and you have the positions.

They measured

Lines so sharp that bond lengths could be quoted to five figures. Water: O–H = 0.9578 Å, H–O–H = 104.48°. Ammonia: N–H = 1.0124 Å, H–N–H = 106.67°. Ammonia also showed the molecule turning itself inside out 24 billion times a second, an inversion that Cleeton and Williams had caught in 1934 and that the first maser would run on in 1954.

How sure could they be? Substituting deuterium for hydrogen changes the masses without changing the geometry, which gives independent equations for the same positions. That is how the coordinates are pinned down rather than merely fitted, and it is why these numbers have stood for seventy years.

Why it mattered

The angles in every textbook come from here. They are measurements, with quoted uncertainties, taken before anybody had a rule that predicted them — and they agree with what the dipole moments had already implied about which molecules are straight.

The rule, and where it sits in the story

Count the electron pairs around the central atom — bonding pairs and lone pairs together — and put them as far apart on a sphere as they will go. Two pairs: opposite ends of a line, 180°. Three: a flat triangle, 120°. Four: a tetrahedron, 109.5°. Five: a trigonal bipyramid. Six: an octahedron. Then name the shape by where the atoms are, ignoring the lone pairs, because a lone pair does not show up in a diffraction pattern or a moment of inertia. Four pairs with one of them lone is a pyramid, not a tetrahedron. Four pairs with two lone is bent. And squeeze the angles a little, because a lone pair is held by one nucleus rather than two and spreads out more: 109.5° becomes 107° in ammonia and 104.5° in water. It is a good rule. It gets the shape of most main-group molecules right on the back of an envelope, which is why it is taught. But note the dates. Sidgwick and Powell, 1940. Gillespie and Nyholm, 1957. Every angle in the paragraph above had been measured before either paper appeared. VSEPR did not predict the geometry of water; it organised a table somebody else had filled in.

You might think

The orbitals hybridise into sp³, and that is what makes water bent.

Actually

Hybridisation is not a process. Nothing mixes, nothing happens, and there is no moment at which an atom "becomes" sp³. It is a change of mathematical basis — a different, equally valid set of combinations of the same atomic orbitals — chosen because it makes a known geometry easy to write down. Pauling introduced it in 1931, sixty years after the tetrahedron was inferred from isomer counting and roughly a decade after the dipole moments were measured. It explains nothing that was not already known, and it was never meant to; it is a bookkeeping device, and an extremely useful one. The place it visibly fails is water’s lone pairs. The sp³ picture gives two equivalent lone pairs, and the photoelectron spectrum of water — which measures the energy needed to remove an electron from each occupied orbital in turn — shows them at plainly different energies, around 12.6 and 14.7 eV. Two equivalent things do not give two different numbers. The geometry is right; the story about why is a convenient fiction.

Where the rule of thumb stops working

VSEPR is usually presented with its successes. Its failures are more interesting, and they are not marginal. First, be fair about what the rule claims. It does not promise the idealised angle exactly — it says lone pairs squeeze it, so water coming in 5.0° below the tetrahedral 109.5° is a success, not an embarrassment. The question is how far a squeeze can be stretched. Hydrogen sulphide is 92.1°. It has exactly water’s arrangement — two bonding pairs, two lone pairs — so the rule predicts the same tetrahedral parent, squeezed. It is 17.4° below that parent, and 12.4° below water itself. Nothing in the rule licenses a squeeze three times the size of water’s, and the direction is awkward too: it sits nearer a right angle than a tetrahedron. Phosphine is 93.3°, which is 16.2° below the tetrahedral angle and 13.4° below ammonia. Same pattern, same size of error, one row down the table. Go further down and it gets worse, not better. The rule works best for the second row — carbon, nitrogen, oxygen, fluorine — and degrades below it, which is a suspicious way for a repulsion argument to behave, since nothing in the argument mentions which row you are on. And there is a whole family it gets simply backwards: the gas-phase alkaline-earth dihalides. CaF₂, SrF₂ and BaF₂ are AX₂ molecules with no lone pairs at all, so VSEPR says linear without hesitation — and they are measurably bent. None of this makes the rule useless. It makes it a rule of thumb, which is what its authors called it. The thing worth resisting is the slide from "this predicts most shapes" to "this is why molecules have shapes".

Problem

The bond moment that comes out impossible

Ammonia and nitrogen trifluoride are both trigonal pyramidal, with nearly the same bond angle. Their measured dipole moments are wildly different. Treat each molecule’s dipole as nothing but the vector sum of three equal bond moments. What N–F bond moment does NF₃’s measured dipole require? The identical calculation on ammonia gives 1.30 D for N–H, which is the accepted value — so the method itself is not the problem.

NF₃ dipole moment: 0.235 D
NF₃ bond angle, F–N–F: 102.3°
NH₃ dipole moment, for comparison: 1.472 D
NH₃ bond angle, H–N–H: 106.67°
Three bonds at β to the axis: cos θ = (3cos²β − 1)/2

A Dr. J. H. van ’t Hoff, of the Veterinary School at Utrecht, has no liking, apparently, for exact chemical investigation. He has considered it more convenient to mount Pegasus — evidently borrowed from the Veterinary School — and to proclaim how, on his bold flight to the top of the chemical Parnassus, the atoms appeared to him grouped in space.
Hermann KolbeJournal für Praktische Chemie, 1877. Kolbe was among the most distinguished chemists in Germany and had synthesised acetic acid from its elements; he was not a crank. Van ’t Hoff received the first Nobel Prize in Chemistry in 1901.

1874Van ’t Hoff and Le Bel, independently, put carbon’s four bonds at the corners of a tetrahedron — deduced from a count of isomers, not from any instrument.
1877Kolbe publishes his attack. The tetrahedron survives it.
1912Debye works out how a permanent dipole moment shows up in the dielectric constant of a gas, and how to extract it.
1920sCO₂, CS₂ and CCl₄ measure zero; H₂O measures 1.85 D. Some molecules are straight and symmetric, and some are not.
1931Pauling’s hybrid orbitals, offered as an account of shapes already in the tables.
1933The rotational structure of water’s infrared bands puts the angle near 105°.
1934Cleeton and Williams measure ammonia’s inversion at 1.25 cm — a microwave spectrum, twelve years early.
1940Sidgwick and Powell: count the electron pairs and the shape falls out.
1946Surplus radar hardware turns microwave spectroscopy into a routine method almost overnight.
1957Gillespie and Nyholm publish the version now taught as VSEPR.
1970Photoelectron spectra show water’s two lone pairs at different energies — not the two equivalent ears of the sp³ picture.

Where this leaves us

Shape is measured. A dipole moment tells you, from a capacitance, whether a molecule can possibly be straight. A rotational spectrum tells you the lengths and the angles to five figures. Both were in place before anyone had a rule that predicted them, and the rules that came afterwards — hybrid orbitals, electron-pair repulsion — organise the table rather than deriving it, and have real failures of their own. That is a comfortable position for a chemist in about 1940. You cannot say why water is 104.5° from first principles, but you know that it is, and you have a rule that gets most molecules close. Except for one, which the entire apparatus cannot touch. Six carbon atoms in a ring. Every rule of valence anyone had says the bonds must alternate — single, double, single, double, single, double — and a double bond is 0.20 Å shorter than a single one, which is twenty times any error bar in the business. When somebody finally measured all six, they were the same length.