Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 20

Page 20 of 42, continuing through the fields in the same order.

Orbitals Where the atoms go Bonding models What symmetry decides Beyond the octet What a spectrum settles When the molecule does not stop What the shape is for What is taught wrongly Series Named objects Orbitals Refutations Search

Where the atoms go

VSEPR as a repulsion minimisation rather than a table, the angles that fall out of it, and the case where five sites are not all alike.

The s character a bond angle requires. The fraction of the one s orbital each of two equivalent hybrids must carry in order to meet at a given angle, from pure p at 90 degrees to half s at 180. The marks are measured bond angles — H₂O, NH₃, H₂S, PH₃ — and the column beside them is the budget each implies: what one bond hybrid takes, and what is left for the lone pairs to share. The relation is orthogonality, not a model of bonding, and nothing here is fitted.

The angle does not fix the hybridisation

Two equivalent hybrids are orthogonal only at one relation between their s character and the angle between them. Run it backwards on measured bond angles and water's bonds come out sp³·⁹⁹, hydrogen sulfide's sp²⁷, and phosphine's lone pair takes eighty-four per cent of the one s orbital.

5 figures One dihedral, three point groups. Hydrogen peroxide built at 13 values of its dihedral angle, with the point group searched for from the coordinates at each. The group is C2v when the hydrogens eclipse, C2h when they are anti, and C2 at every angle strictly between. The rails below are what the group settles on its own: the molecule may be polar except at the anti arrangement, and it is chiral except at the two ends. No energy is computed anywhere, and the marked angle is the measured one rather than a minimum found here.

One coordinate, three point groups

Hydrogen peroxide has four atoms and one soft internal coordinate. Turning it from nought to a hundred and eighty degrees takes the molecule through C2v, C2 and C2h — so it is chiral at every angle but two, and forbidden a dipole at exactly one of them.

8 figures The moment that does not depend on where the origin is. For each molecule, the dipole and the largest quadrupole component about the centre, and the same two about an origin moved 1.6 bohr away. The lowest non-vanishing moment is unchanged in every row and the one above it moves in every row. The 4 non-polar molecules here have a quadrupole that is a property of the molecule; the polar ones have one that is a property of a choice.

What a dipole cannot tell apart

A dipole moment is three numbers extracted from a whole charge distribution, and enormously many distributions give the same three. The moment above it is not even a property of the molecule unless the one below it vanishes — which is why a quadrupole is quoted with an origin and a dipole is not.

4 figures How far two hybrids are from orthogonal. The overlap between two equivalent s–p hybrids of a stated label, against the angle between them. Each curve crosses zero at exactly one angle — sp3 at 109.47°, sp2 at 120.00°, sp at 180.00° — and a molecule whose measured angle is not that angle has hybrids that overlap. The largest here is cyclopropane at 0.63.

Hybrids that were never orthogonal

Two sp³ hybrids are orthogonal at 109.47° and nowhere else. Water's are drawn at 104.5° and overlap by 0.062; cyclopropane's are drawn at 60° and overlap by 0.625, which is not a small correction to a basis but a description that has stopped being one.

5 figures The largest angle each ring size can have. The ceiling on a bond angle in a closed ring of equal bonds, 180°(n−2)/n, which is the interior angle of the regular planar polygon and follows from a closed curve having to turn through a full circle. The line at 109.47° is the tetrahedral angle: rings of 3, 4, 5 atoms cannot reach it at any geometry whatever, and every larger ring can, by leaving the plane.

The angle a ring cannot have

A closed ring of equal bonds has to turn through a full circle, so its bond angles cannot average more than 180°(n−2)/n. Three, four and five atoms are below the tetrahedral angle at every geometry whatever; six is above it, and reaches it only by leaving the plane.

5 figures A 6-ring at 111°: the twist-boat. A closed ring of 6 equal bonds meeting at 111°, drawn from the coordinates the closure conditions produce. Its torsions are 16.9°, -63.6°, 44.2° and repeat; its puckering amplitude is 0.508 bond lengths at a phase of 344°, with q₃ exactly zero, so it lies on the equator. Turning the ring changes none of those numbers, which is what makes them a description of the shape rather than of the view.

The ring that cannot hold still

Cyclohexane's chair is rigid and its boat is not, and that is a statement about the rank of a matrix rather than about strain. Hold every bond length and every bond angle fixed and count what is left: the chair has nothing, and the boat sits on a continuous loop of shapes with the same bonds and the same angles.

7 figures The bond sum, the measurement, and what is left over. For each pyramid: the vector sum of three bond moments estimated from the electronegativity difference and the measured geometry, the measured dipole, and the difference between them. Positive is towards the lone pair. Ammonia's bonds point that way and nitrogen trifluoride's point the other, which is why the trifluoride's far more polar bonds give it a dipole six times smaller. The leftover is 0.58 D for ammonia and at least 1.48 D for the trifluoride, so it is not one lone pair's property.

The lone pair is not the missing term

Ammonia's dipole is 1.47 debye and nitrogen trifluoride's is 0.235, although the N–F bonds are far more polar than the N–H ones. The bond sums explain the reversal exactly and point in opposite directions — and the lone pair that is supposed to make up the difference has to be worth 0.58 debye in one molecule and at least 1.48 in the other.

4 figures Which site a long bond takes, and it is not the one the rule says. The energy of the axial placement minus the energy of the equatorial one, against the odd bond's length. Above the line the equatorial placement wins and below it the axial one does. The textbook sentence is that a bulkier group goes equatorial where there is more room; given a longer bond and nothing else, the repulsion model puts it AXIAL, at every length, by a margin that grows. A site pushed further out interacts with everything less, so what is left to decide is the arrangement of the four sites left behind — and three equatorial with one axial is the better set of four.

The sites are not the same size

Every arrangement in this collection puts its sites on one sphere, which is an assumption about bond lengths made silently. Give the repulsion model a bond length and it predicts that the long bond goes axial — the opposite of the rule the model is always cited for.

7 figures The twist the ring forces on the bond. For each ring size, the largest torsion about the double bond that the ring will close on — 180° being a flat trans arrangement and 90° being a π bond broken outright. The six-ring will not close at any torsion tested. The eight-ring reaches 139.3°, against 136° measured in trans-cyclooctene by diffraction: a model with bond lengths and bond angles in it and no energy anywhere agrees with the crystal to a few degrees.

The double bond a ring cannot hold

A trans double bond needs a ring of nine carbons to sit flat, and the eight-ring will hold one twisted by 40.7 degrees — against 136 degrees measured in trans-cyclooctene. The same eight-ring threshold, applied by counting ring sizes, sorts nine bridgehead alkenes correctly with no bridgehead anywhere in the argument.

7 figures Face to face repels, edge to face attracts. Two benzene molecules 5 Å apart, one turned against the other, with only the quadrupole interaction between them. Stacked, the two negative faces meet and the energy is +8.8 kJ/mol; perpendicular, one molecule's positive rim meets the other's negative face and it is -4.6. The sign changes on the way, so the preference is not that one arrangement is weaker — it is that they are opposite. This is a molecule with no dipole moment at all.

Zero dipole is not no interaction

Benzene's dipole moment is exactly zero at every origin, and its quadrupole moment is large enough to decide a crystal structure. Two benzenes face to face repel by nine kilojoules a mole; edge to face they attract, and the sign changes on the way between.

7 figures water, drawn with its nuclei the size they are. The molecule with a disc round each nucleus whose radius is the computed root-mean-square displacement of that nucleus in the vibrational ground state. The hydrogens' discs are a substantial fraction of the bond length, and this is at no temperature at all.

The atoms are not at the points

Every structure in this collection is a set of points, and every bond angle it argues about is a property of that set. Computed from the fitted force fields it already has, water's hydrogens are 0.094 Å from where they are drawn and its bond angle has a spread of 8.9 degrees — larger than the difference between 104.5 and the tetrahedral value that half the essays here are about. This is at no temperature at all.

9 figures The dipole moment, and how many bands there are. For each of five molecules: the dipole moment of the point-charge model, the number of modes whose dipole derivative does not vanish, and the largest derivative. The molecules with no dipole at all have the most active bands, which is the whole of the argument.

A dipole is not what an infrared spectrum sees

Carbon dioxide has no dipole moment at all and three of its four modes are infrared active. Methane has none and six of nine; boron trifluoride none and five of six. Water, which has the largest dipole of the five, has three modes and three bands — and its dipole predicted neither number.

8 figures