water — C2v

One of the figures on point groups from coordinates: A molecule turned in space with its group recovered from the turned coordinates, and what the group settles on its own.

27 essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

Hybridisation does not explain

Methane, point group Td, recovered from its coordinates. Four equivalent bonds by symmetry — and a spectrum with two bands, which four equivalent orbitals cannot produce.

Hypervalency without d orbitals

Sulfur hexafluoride, point group Oₕ. Six equivalent bonds, an octahedral arrangement, and twelve electrons in the valence region — none of which requires d orbitals to explain.

A second hypervalent molecule, four-coordinate rather than five or six. Xenon tetrafluoride needs the same accounting as sulfur hexafluoride and gets it from the same s and p orbitals — so whatever is happening is not a property of the coordination number, and the d-orbital account would have to be invoked at every one of them.

Phosphorus pentafluoride, D₃ₕ. The axial pair forms one three-centre system and the three equatorial bonds are ordinary two-centre ones — which predicts both the longer axial bonds and the preference of electronegative substituents for those positions.

Point groups from coordinates

Ammonia, with its point group recovered from its coordinates rather than looked up. Every candidate operation the structure suggests was applied and kept when it permuted the atoms among themselves.

Benzene at D₆ₕ. The sixfold axis, the six twofold axes in the plane, the horizontal mirror and the inversion centre were all found by search rather than assumed, and the symbol is what the search produced.

The easiest case to check by eye, and the one the two rules above can be read off. Water’s group is C₂ᵥ, recovered here from three sets of coordinates: the twofold axis is a direction every operation leaves alone, so water may be polar, and the two mirror planes are improper operations, so it cannot be chiral. Neither conclusion required knowing what the bonds are made of.

VSEPR, computed

Water, at its measured geometry. The angle is 104.5 degrees, not 109.5, and the difference is where the simple rule stops being quantitative.

Delocalisation

Benzene, point group D₆ₕ. Six equivalent carbons, six equivalent bonds, and a sixfold axis — recovered here from the coordinates. A structure with alternating single and double bonds would be D₃ₕ, and it is not.

Ethene for comparison: a genuine localised double bond, D₂ₕ, with a C–C length of 1.334 ångström. Benzene’s bonds at 1.397 sit between this and a single bond at 1.54, which is the measurement the whole discussion is about.

Symmetry forbids a dipole

Which molecules may be polar, from the group and nothing else. No bond dipoles were added, no electronegativities consulted, and the answers are exact.

Carbon dioxide, D∞ₕ. The inversion centre at the carbon reverses every vector, so the dipole must be its own negative — which only the zero vector can be. The C–O bonds are strongly polar and the molecule is not.

Methane, Td. Four threefold axes pointing in four different directions, so no direction is fixed by every operation — and the moment is exactly zero however polar the individual C–H bonds are.

Two models, one ratio

A tetrachloridonickelate ion, with Td recovered from its coordinates by the same search that recovers methane’s. The ligand-field calculation and the point-group reduction are done on the same structure, so a claim about the group and a claim about the splitting are claims about one object.

Chirality is a symmetry statement

A carbon with four different substituents: point group C₁, containing nothing but the identity. No mirror, no inversion centre, no improper rotation — so the molecule and its mirror image are different objects.

Chirality read off the symbol, alongside polarity. Two properties, both settled by the group, neither requiring any knowledge of the bonding.

Hydrogen peroxide in its gauche conformation: point group C₂, a rotation and nothing else. No stereocentre exists anywhere in the molecule and it is chiral, because chirality is a property of the whole object rather than of an atom.

The dipole is not a sum of bonds

The same question answered from the point group alone. Nothing here required an electronegativity, and where the answer is no, the moment is exactly zero rather than a sum that happens to cancel.

Ammonia, C₃ᵥ. The symmetry permits a moment along the threefold axis, which is all the group can say. What actually produces most of it is the lone pair, which the bond-vector method does not count.

The textbook case. Two identical polar bonds pointing opposite ways, vectors cancelling, moment zero. The method gets it right — and so does the symmetry argument, which additionally says the answer is exact rather than a cancellation that happens to work out.

Why water is bent

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.

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 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.

Site symmetry, and what it constrains

Phosphorus pentafluoride, D₃ₕ, with the group recovered from the coordinates and searched for again at every step of the slider. Five fluorines; two environments. The molecule turns and neither number changes.

The molecule whose fluorines are all one orbit, for contrast with the one whose are two. Forty-eight operations carry any fluorine of SF₆ onto any other, so every site in it has the same site group and there is nothing for an orbit split to distinguish — which is the case the essay’s arithmetic reduces to when the orbit count is one.

And a molecule where the orbit structure is decided by something other than the bonds. Xenon tetrafluoride’s four fluorines are one orbit and its two lone pairs occupy the axial positions, so the site group of a fluorine is what it is because of where the lone pairs are — which the orbit arithmetic sees and a bond count does not.

What a lone pair is worth

Water, with the point group recovered from its coordinates rather than assigned. The readout holds at C2v as the molecule turns, which is the check that the recovery is from the positions and not from the pose — and C2v is the whole of what symmetry can say here. Symmetry permits a bent geometry and says nothing at all about 104.5 against 92.1.

A moment counts electrons, not orbitals

A hexafluoridocobaltate ion. Cobalt(III) is d⁶; fluoride is a weak-field ligand; the complex is high spin with four unpaired electrons and a moment near 5.3, where the hexaammine of the same metal is low spin and diamagnetic. Same metal, same count, same geometry, two magnetic states.

Counting electrons in an extended structure

The molecule the ring rule is usually stated about, with its group recovered from the coordinates. Six carbons, six π electrons, a closed shell — and the same count applied to a chain of six or a chain of six hundred gives a quite different answer, which is what makes the extended case a separate question rather than a limit of this one.

An octahedral complex, where the count that closes a shell is eighteen rather than eight. Three rules — the octet, 4n + 2 and eighteen — are three answers to one question about three kinds of structure, and each of them is a count of orbitals rather than a count of bonds.

The case that looks like a counting failure and is not. Sulfur has six neighbours and four valence orbitals, and the resolution is that not every line drawn to a neighbour is a shared pair — the electron count at the centre never exceeds what four orbitals can hold. The same distinction is what makes the extended count workable, since in a solid the number of neighbours routinely exceeds the number of orbitals.

One coordinate, three point groups

Hydrogen peroxide at its measured dihedral of 111.5°, with C2 recovered from the coordinates. Four atoms, no stereocentre, no substituents — and a handedness that belongs to the whole object.

A molecule with one shape and one group, for contrast with the one that has three. Water’s coordinates admit C₂ᵥ and nothing else at any tolerance worth using, and no internal coordinate of it can be turned to change that — which is the ordinary situation and the one the phrase the point group of a molecule is built for.

The planar case of the same skeleton. Ethene has the two ends locked together by a double bond, so the dihedral coordinate that hydrogen peroxide turns freely is frozen at zero — and with it frozen the group is D₂ₕ and stays D₂ₕ. What makes one molecule ambiguous and the other not is the barrier, not the geometry.

Six bonds and four orbitals

Sulfur hexafluoride with Oh recovered from its coordinates by the same search that recovers methane’s Td. Six equivalent bonds at 1.560 Å, forty-eight operations, and a valence shell that supplies four combinations to match six.

Xenon tetrafluoride, D4h, with four bonds in a plane and two lone pairs above and below. Its three bonding orbitals for four bonds is the same arithmetic as SF₆’s, and the two lone pairs occupy the xenon’s a₂u p orbital and its s — neither of which the ligand set can reach.

What a dipole cannot tell apart

The molecule the arithmetic above is about, with the operations that fix its axis. Everything the moment says is a statement about that axis; nothing it says distinguishes the charge distributions consistent with it.

The angle a ring cannot have

The other six-ring, which does not spend the margin. Benzene’s carbons want 120° and its ceiling is 120°, so it is planar exactly — the one ring size where the wanted angle and the ceiling coincide, and the reason a planar aromatic six-ring costs nothing in angle.

The moment that is the sum of the other two

Sulfur dioxide, the heavy bent triatomic. Same shape as water, same point group, defect zero — and κ = −0.941 against water’s −0.432, entirely because its atoms are heavier and its bond angle is a little wider.

A ranking is not a difference

Water, whose two O–H bonds are strongly polar on three scales and nearly non-polar on the fourth. Its measured dipole moment is not in dispute; what is in dispute is how much of it belongs to each bond, which is a question about a partition rather than about the molecule.

Mutual exclusion does not prove a centre

Eclipsed ferrocene, twenty-one atoms, with its group found from the positions. The search knows nothing about sandwich compounds; it finds a fivefold axis, five twofold axes perpendicular to it, a horizontal mirror and five vertical ones, and no inversion.

Zero dipole is not no interaction

Benzene with its symmetry elements marked and its point group recovered from the coordinates. Every element here is a reason the dipole vanishes, and none of them touches the quadrupole: the sixfold axis and the centre together force the vector to zero and leave the tensor with one independent component.

How much symmetry is left

The structure everything below is measured against: benzene, with its group found from its coordinates. Its twenty-four operations, closed under multiplication, are what the folding averages over — so the measure is a distance to this group in this orientation rather than to the nearest symmetric thing of any kind.

Four is all that s and p can match

The octahedron itself, with its symmetry elements recovered from the coordinates. Everything in this essay follows from these operations and from a count of four.

The table that could not have mattered

The molecule the correlation runs backwards on. Its charges are one number times the pattern (−3, +1, +1, +1) on every table there is, so nothing about the backwards correlation can be an artefact of which table was opened.

No length separates them

For each molecule, the longest pair that is a bond and the shortest pair that is not, on a logarithmic length axis.

The same longest bond and shortest non-bond for each molecule, with each length divided by the sum of its two atoms’ covalent radii.

Each of the twenty-three molecules at one tolerance at a time, from below the window to well above it. The shaded band is the window; drag the tolerance across it.

The census a bond rule was hiding

Every molecule’s internal coordinate set under the distance cutoff in use and under the radius rule.

For each molecule in the census under the radius rule: its totally symmetric count, the corrected formula and the usual one.

Eclipsed ferrocene’s coordinate census under the distance cutoff and under the radius rule.