Molecular geometry: why molecules have the shapes they do

Most molecules in this lesson have one central atom with other atoms round it. What decides the shape is the electrons in the central atom’s outer shell, its valence electrons, as on the Bohr atom page: carbon has 4, nitrogen 5, oxygen 6. Some of these electrons are shared with neighbouring atoms in bonds, and the rest stay with the central atom in pairs, called lone pairs.

Here is how to count them, with ammonia, NH₃, as the example. Nitrogen has 5 outer electrons. Each of its three bonds to hydrogen uses one of them; the hydrogen supplies the other electron of the shared pair. So 3 are used and $5 - 3 = 2$ are left, which is $(5 - 3) \div 2 = 1$ lone pair. Choose a molecule below to see the same count for it. A double bond uses two of the central atom’s electrons, a triple bond three.

Now the key definition. An electron group (also called an electron domain) is one region round the central atom where electrons are concentrated. Each lone pair is one group. Each neighbouring atom is also one group, whether it is joined by a single, double or triple bond: the two or three shared pairs of a multiple bond all lie between the same two atoms, so they all point the same way and count as one direction. Carbon dioxide therefore has 2 groups, even though it has 4 shared pairs.

Each electron group is a cloud of negative charge, and like charges repel, as the Coulomb’s law page shows: the closer two charges are, the harder they push. We use a simplified model: represent these regions by directions around the central atom and seek an arrangement with small overall repulsion. Moving one region farther from one neighbour may move it closer to another, so we must consider all pairs together. This is valence-shell electron-pair repulsion, VSEPR. The moving markers below illustrate that comparison; real electron clouds are not rigid beads travelling around a nucleus. Nuclear attractions and quantum effects also contribute to a real molecule’s structure.

Try it in the picture. Choose a number of groups and press Shake: the groups start in random places round the central atom, and each one is pushed away from every other one, harder when they are closer. Watch where they settle, and read the smallest angle between any two groups. Tied balloons do the same: switch to balloons and you will see the arrangement a bunch of balloons takes when its knots are tied together.

The ideal arrangements we compare are the following. The numerical relaxation is an illustration, so a finite run or an exactly symmetric starting state is not a proof of a unique minimum. Two groups end up on opposite sides, 180° apart: a straight line. Three end up in one flat plane, 120° apart, because three equal angles that go once round a full turn make $360^\circ \div 3 = 120^\circ$. Four groups do not stay flat. In a flat square they would be only 90° apart; by moving out of the plane they can all get further apart, and they settle at the corners of a tetrahedron, 109.5° apart. Try “start flat” to watch four groups leave the square. Five groups settle as a triangle with one group above and one below (90° and 120°), and six at the corners of an octahedron, 90° apart.

Why exactly 109.5°? We can work it out with nothing more than Pythagoras’ theorem and the sine of an angle. The trick is to put the tetrahedron inside a cube. Take every second corner of a cube, four corners in all, chosen so that no two of them are joined by an edge of the cube. Each pair of them is joined by a diagonal across one face, so all four are the same distance apart from one another, and the same distance from the centre. That is a tetrahedron. Put the carbon of methane at the centre and the four hydrogens at those corners, as in the picture.

Let the cube’s edge be $a$. Step 1: the diagonal across one face. It is the long side of a right-angled triangle whose two short sides are both $a$, so by Pythagoras it is $\sqrt{a^2 + a^2} = \sqrt{2} a$. Two hydrogens are exactly one face diagonal apart (the blue line), so their distance is $\sqrt{2} a$. Step 2: the diagonal through the cube, from a corner to the opposite corner. It is the long side of a right-angled triangle whose short sides are one edge, $a$, and one face diagonal, $\sqrt{2} a$, so it is $\sqrt{a^2 + 2a^2} = \sqrt{3} a$. The centre is halfway along it, so each bond (orange) has length $\sqrt{3} a \div 2$. Step 3: the two bonds and the blue line make a triangle with two equal sides. The green line from the centre to the middle of the blue line cuts it into two right-angled triangles, and cuts the angle $\theta$ between the bonds in half. In one of these triangles the long side is a bond, $\sqrt{3} a \div 2$, and the side opposite the half-angle is half the blue line, $\sqrt{2} a \div 2$. The sine of an angle is the opposite side divided by the long side, so

The $a$’s cancel, so the answer does not depend on the size of the cube. A calculator’s inverse sine $(\sin^{-1})$ gives the angle whose sine is 0.8165, which is 54.74°, so $\theta = 2 \times 54.74^\circ = 109.47^\circ$. This is the angle between any two bonds of methane, and it agrees with what is measured.

Ammonia and water also have 4 electron groups each, so their groups also point towards the corners of a tetrahedron. But the shape we name for a molecule is the shape made by its atoms, because lone pairs are not atoms and do not show up as atoms when the shape is measured. Ammonia has 3 atoms on the 4 corners, so it is a pyramid with a triangle of hydrogens as its base: trigonal pyramidal. Water has 2 atoms on 4 corners, so its three atoms make a bent shape. Switch the lone pairs on and off to see both pictures at once.

There is a second, smaller effect. A lone pair is held by only one nucleus, so it sits closer to the central atom and spreads out more widely than a bonding pair, which is stretched out between two nuclei. So a lone pair pushes harder on its neighbours than a bonding pair does, and squeezes the bonds a little closer together. Measurements agree with the order this predicts: 109.5° in methane (no lone pairs), 107° in ammonia (one), and 104.5° in water (two). VSEPR explains why the angles get smaller; the exact numbers come from measurement.

With five groups, the positions are not all alike. The two groups above and below (axial) each have three neighbours at 90°; the three round the middle (equatorial) each have only two neighbours at 90°, the others being further away at 120°. A lone pair spreads widest, so it takes an equatorial place, where it has fewest close neighbours. That gives sulfur tetrafluoride, SF₄ (one lone pair), a see-saw shape; chlorine trifluoride, ClF₃ (two lone pairs), a T shape; and xenon difluoride, XeF₂ (three lone pairs, all round the middle), a straight line.

With six groups every position is alike, so one lone pair can go anywhere. A second lone pair goes opposite the first, as far from it as possible, which leaves the four atoms in a flat square: xenon tetrafluoride, XeF₄, is square planar.

Now the opening question. In carbon dioxide the carbon has 2 groups, two double bonds and no lone pairs, so they point opposite ways: the molecule is straight. In water the oxygen has 4 groups, two bonds and two lone pairs, so its electron groups have an approximately tetrahedral arrangement. The lone pairs repel the bonding groups more strongly and compress the bond angle from the ideal tetrahedral value of about 109.5° to the observed value of about 104.5°: the molecule is bent.

Each O–H bond, and each C=O bond, has its shared electrons pulled towards the oxygen, which makes the oxygen end slightly negative. For the molecule as a whole, compare the centres of its positive and negative charge. Their separation gives a direction and a size called the dipole moment. In straight, symmetric carbon dioxide the two bond contributions cancel, so there is no permanent overall dipole. The oxygen atoms still carry local partial negative charge; their charges have not disappeared. In bent water the contributions do not cancel: the oxygen side is partly negative and the hydrogen side partly positive. This helps neighbouring water molecules attract, including through hydrogen bonding. These attractions help explain why water is liquid at ordinary room temperature and pressure; shape alone does not decide its state.