Why Molecules Have Shapes

Chapter 1 Electron domains and VSEPR

Molecular geometry

Count the electron domains around a central atom, let them push apart, and the shape follows. Lone pairs take part in the pushing but stay invisible in the shape.

13 min read · 5 interactive figures · 2 questions

The builder starts as methane: four bonded atoms and no lone pairs around carbon, every H–C–H angle 109.5°. Use − and + to change the number of bonded atoms and lone pairs; the domains push apart and settle, and the text under the model names the electron geometry and the shape. With Measured angles on, the atoms then move to the measured geometry of a real molecule with the same counts. Drag to turn the model; pull an atom away and let go.

Electron geometry is the arrangement of every electron domain around a central atom: its bonds and its lone pairs. Molecular geometry, the shape, is the arrangement of the atoms alone. In water the oxygen atom has four domains, two bonds and two lone pairs, so its electron geometry is tetrahedral; the lone pairs carry no atoms, so the molecule you would see is bent, at a measured 104.5°[1]. Both geometries come from one rule: the domains spread out as far from one another as they can.

The model built on that rule is called valence shell electron pair repulsion, or VSEPR. It needs nothing but a count. Tell it how many atoms are bonded to the central atom and how many lone pairs it carries, and it returns a shape and a set of bond angles, usually within a few degrees of experiment. This chapter builds the model from the repulsion up, then measures where it succeeds, where it bends, and where it breaks.

What counts as an electron domain

An electron domain (also called a region of electron density) is any place around the central atom where electrons are concentrated. There are three kinds[2]:

  • a bond to another atom. A single, double or triple bond counts as one domain, because all its electrons sit between the same two atoms;
  • a lone pair, two electrons that belong to the central atom and are not shared;
  • rarely, a single unpaired electron, as in nitrogen dioxide.

Carbon dioxide therefore has two domains on carbon (two double bonds), ammonia has four on nitrogen (three bonds, one lone pair), and sulfur hexafluoride six. The number of domains is the steric number. The next chapter shows how to read it off a Lewis structure; here we start from the count and ask what it does.

Domains push apart

VSEPR's one physical assumption is that electron domains repel each other, so the lowest-energy arrangement is the one that keeps them as far apart as possible[2][3]. Everything else is geometry. The question becomes: how do you place two, three, four, five or six points on a sphere so that they are as far apart as possible?

The figure below answers it by doing it. Each domain is a point on a sphere around the central atom, and every pair of points pushes on each other with a strength that falls off with distance. Start them anywhere and let them move.

Figure 1 Five electron domains start at random places on a sphere and push each other apart until they stop. The curve under the sphere is the total repulsion energy at each step; the dashed line is the lowest value five points can reach. Press Scatter again for a different random start, change the number of domains, or pull one lobe away and let go. Drag elsewhere on the sphere to turn it.#

Whatever the start, the points finish in the same arrangement, only turned a different way. Two domains end on opposite sides (180°). Three make an equilateral triangle in a plane (120°). Four make a tetrahedron, with every angle 109.47°, the angle whose cosine is −1/3. Five make a trigonal bipyramid: two "axial" points at the poles and three "equatorial" points around the middle, with angles of 90°, 120° and 180°. Six make an octahedron, every neighbour at 90°. These are the five electron geometries. established

DomainsElectron geometryAngles between domains
2linear180°
3trigonal planar120°
4tetrahedral109.47°
5trigonal bipyramidal90°, 120°, 180°
6octahedral90°, 180°

The tetrahedron deserves a second look, because a flat drawing of methane suggests 90°. Four bonds drawn on paper as a cross are 90° apart, but nothing confines a molecule to paper. Lifting two of the bonds out of the plane lets all four open to 109.47°, which is as far apart as four directions in three dimensions can get. The energy curve in the figure makes the same point numerically: no arrangement of four points scores lower than the tetrahedron's 3.674.

Electron geometry and molecular geometry

Now give the domains identities. Some hold a bonded atom; some are lone pairs. Lone pairs take part in the pushing, so they help set the electron geometry, but they carry no atom, so they are invisible in the molecular geometry. The shape is named after the atoms only.

The builder at the top of the page does exactly this. It starts as methane, four bonded atoms and no lone pairs: tetrahedral electron geometry and tetrahedral shape. Turn one bonded atom into a lone pair (one fewer bonded atom, one more lone pair) and the electron geometry stays tetrahedral, but the shape you would see in the atoms is a trigonal pyramid, the shape of ammonia. Turn a second one into a lone pair and the atoms make a bent molecule, the shape of water. Uncheck Show lone pairs to see only what the atoms trace out.

Chemists write these combinations in AXE notation: A is the central atom, Xn the n bonded atoms, Em the m lone pairs. Methane is AX4, ammonia AX3E, water AX2E2.

steric number = n bonded atoms + m lone pairs(1.1)

With two to six domains and any number of them lone pairs, there are thirteen common shapes. All of them, with their ideal angles and a real example of each, are in the VSEPR chart, which also works as a shape finder.

Ammonia, water and the tetrahedral angle

Ammonia and water both have four electron domains on their central atom, like methane. The difference is that some of those domains are lone pairs. What does that do to the angle between the bonds that remain?

Question 1 Prediction: ammonia’s angle#

The measured values[1]: methane 109.47°, ammonia 106.67°, water 104.48°. One lone pair takes about three degrees off the angle; two take about five. The chart shows those three measurements on the first line, and lets you add more families.

Figure 2 Measured bond angles of three molecules with four electron domains, plotted against the number of those domains that are lone pairs: CH₄, NH₃ and H₂O. The dashed line is the tetrahedral angle. Tick the other families to add them, and choose a molecule to read off its value. All values are experimental (NIST CCCBDB).#

VSEPR explains the downward slope with a size argument. A bonding pair is pulled out between two nuclei; a lone pair is held by one nucleus only, so it sits closer to the central atom and spreads over a wider angle. A larger domain crowds its neighbours. The usual statement is an order of repulsion: lone pair–lone pair > lone pair–bond > bond–bond[2]. Two lone pairs, as in water, squeeze harder than one. established

Now tick the other families, and the size argument starts to strain. The fluorides follow the pattern (CF4 109.5°, NF3 102.4°, OF2 103.1°), but they start lower than the hydrides, and NF3 sits below OF2 even though it has fewer lone pairs. The hydrides of the heavier elements fall off a cliff: phosphine is 93.3°, hydrogen sulfide 92.1°, and by hydrogen telluride the angle is 90.3°, almost a right angle[1]. Nothing in "lone pairs are bigger" predicts a 13-degree difference between NH3 and PH3.

Both effects point to something VSEPR does not model: how the central atom shares out its s and p orbital character between its bonds and its lone pairs. The usual explanation is that the central atom puts more p character into its bonds to electronegative neighbours such as fluorine, which narrows the angle between them, and that heavier central atoms keep their lone pair in a nearly pure s orbital, leaving nearly pure p orbitals, which lie 90° apart, for the bonds. The rule that ties hybridization to electronegativity is called Bent's rule, and it is where Clauss and co-authors propose that teaching should go next[5]. established VSEPR gets the direction of the squeeze right in every row of the chart; it does not get the size.

Where lone pairs go when the sites are not alike

In a tetrahedron all four corners are equivalent, so it does not matter which one holds the lone pair. A trigonal bipyramid is different. Its two axial sites each have three neighbours at 90°; its three equatorial sites each have only two. Put a lone pair in a bipyramid and you have to choose.

Question 2 Prediction: sulfur tetrafluoride#

The rule VSEPR uses is a count. At 90° two domains are close; at 120° they are much further apart, so the 90° contacts dominate the repulsion, and lone pairs, the largest domains, should have as few of them as possible[2]. The figure starts with the lone pair in an axial site. Tap an equatorial atom to move it there.

Figure 3 Five electron domains in a trigonal bipyramid, with one lone pair placed at an axial site. Short arcs between domains mark 90° contacts that involve a lone pair: orange for lone pair–bond, blue for lone pair–lone pair. The table counts them for this arrangement and for the best possible one. Tap a bonded atom to move a lone pair there; add lone pairs or switch to six domains with the controls. Drag to turn the model.#

Axial, the lone pair has three 90° contacts with bonds; equatorial, only two. So in SF4 the lone pair is equatorial, and the four fluorines form a seesaw. The measurements show the squeeze as well as the placement: the axial fluorines bend away from the lone pair, to 173.1° instead of 180°, and the two equatorial ones close up to 101.6° instead of 120°[1].

The same count settles every case with five or six domains[2]:

  • Two lone pairs in a bipyramid go equatorial too, leaving a T-shaped molecule. Chlorine trifluoride is the example: its axial fluorines lean toward the third fluorine, making the axial–equatorial angle 87.5°, not 90°[1].
  • Three lone pairs fill the equator, and the two bonded atoms sit at the poles: xenon difluoride is linear, with five domains on xenon.
  • In an octahedron every site is alike until the second lone pair arrives. Placed opposite the first, it makes no lone pair–lone pair contact at 90°; next to it, one. So xenon tetrafluoride is square planar, its lone pairs above and below the plane.
  • One lone pair in an octahedron leaves a square pyramid, as in bromine pentafluoride, where the four basal fluorines tilt away from the lone pair: 84.8° from the apex fluorine instead of 90°[1].

Switch the figure to six domains with two lone pairs and put them next to each other: the table shows one lone pair–lone pair contact that the opposite arrangement avoids.

Double bonds count once, and are fatter

A double bond holds four electrons, but they all sit between the same two atoms, so it is one domain, pointing one way. Carbon dioxide has two double bonds and nothing else on carbon: two domains, 180°, a straight line. Sulfur dioxide has two bonds and a lone pair on sulfur: three domains, so a trigonal-planar electron geometry and a bent molecule, 119.5°.

Figure 4 Carbon dioxide in its measured geometry: two double bonds, two domains, 180°. Choose other molecules to compare: formaldehyde (a double bond next to two single bonds), hydrogen cyanide (a triple bond), sulfur dioxide and ozone (a lone pair on the central atom), and nitrogen dioxide (a single unpaired electron). Angles are the measured values from NIST CCCBDB. Drag to turn.#

A double bond is still a bigger domain than a single bond: four electrons, not two. Formaldehyde shows it. Its carbon has three domains, ideal angle 120°, but the two H–C–H bonds are squeezed to 116.1° by the C=O double bond, and each H–C=O angle opens to 121.9°[1]. OpenStax ranks the space domains take as lone pair > triple bond > double bond > single bond[2].

Nitrogen dioxide is the reverse case. Its third domain on nitrogen is a single unpaired electron, half of a lone pair. A smaller domain should crowd its neighbours less, and it does: the O–N–O angle opens to 134.1°, wider than the ideal 120°, while sulfur dioxide, with a full lone pair in that place, stays just under 120°[1]. established

The method in five steps

  1. Draw the Lewis structure (chapter 2 shows how) and find the central atom.
  2. Count its domains: each bonded atom is one, whatever the bond order; each lone pair is one.
  3. The count gives the electron geometry: 2 linear, 3 trigonal planar, 4 tetrahedral, 5 trigonal bipyramidal, 6 octahedral.
  4. Place the lone pairs: anywhere for 2–4 domains; equatorial in a bipyramid; opposite each other in an octahedron.
  5. Name the shape from the atoms alone, and expect the angles next to lone pairs and double bonds to close up by a few degrees (by much more for central atoms below the second row).

For ammonia: three bonds and one lone pair, four domains, tetrahedral electron geometry, one corner empty of atoms, trigonal pyramidal shape, angles a little under 109.5° (106.7°). For xenon tetrafluoride: four bonds and two lone pairs, six domains, octahedral, lone pairs opposite, square planar, 90°.

Where the model came from

The idea that a chemical bond is a shared pair of electrons goes back to G. N. Lewis's 1916 paper "The atom and the molecule"[6]. documented In 1940 Nevil Sidgwick and Herbert Powell surveyed the known structures of inorganic molecules in a Bakerian Lecture to the Royal Society titled "Stereochemical types and valency groups", and connected the shape around an atom to the total number of electron pairs in its valence shell, shared or not[7]. In 1957 Ronald Gillespie and Ronald Nyholm turned that observation into a set of rules, including the extra size of lone pairs and multiple bonds, in a review titled "Inorganic stereochemistry"[3]. Half a century later Gillespie looked back on it in a review titled "Fifty years of the VSEPR model"[8].

What the model leaves out

VSEPR is a counting rule. It says nothing about bond lengths or energies, it is built for main-group central atoms, and, as the chart above showed, it gives the direction of the squeeze but not its size.

Its pictures can also mislead. Textbooks often draw water's two lone pairs as identical "rabbit ears" pointing above and below the molecular plane. In 2014 Clauss, Nelsen, Ayoub, Moore, Landis and Weinhold argued that this picture conflicts with experiment and with quantum-mechanical calculations, in which water's two lone pairs come out distinct and inequivalent, in shape and in energy, and that overemphasizing VSEPR leaves students with ideas they later have to unlearn[5]. Their article drew a published reply from Hiberty, Danovich and Shaik disputing it[9], so this is an argument among experts about how best to describe lone pairs, not about where the atoms are. contested The positions of the atoms, and the angles in this chapter, are measurements; the lobes drawn for lone pairs are a picture of the counting rule, not a picture of the electrons.

The counting itself survives that argument. Every shape in this chapter is a measured shape, and VSEPR predicts all of them from nothing but a Lewis structure. The next chapter is about getting that Lewis structure right.