Molecular Geometry & Bonding
what the shape decides
Concept

Long-range interaction — where it appears

An interaction falling off slowly enough that distant partners still matter. Sums over such interactions converge conditionally, so the answer depends on how the terms are grouped.

Named by 11 essays across 6 fields — each of them below, with the objects they name alongside it.

What holds matter together, per pair. Four kinds of interaction between two units of matter, each computed from the model named beside it, on a logarithmic energy scale. The range from top to bottom is a factor of several hundred, which is the number behind why a molecular solid melts hundreds of degrees below a covalent one.

The lattice sum that depends on the order of adding

An ionic solid's binding is the sum of every pair of charges in it, and the series does not converge absolutely — rearranged, it gives a different answer. That is a genuine mathematical difficulty rather than a technicality, and it is the clearest example of something a real-space, neighbour-by-neighbour method cannot compute at all.

solids · Cohesion 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.

shape · Dipole The same angles, the same torsions, and not the same molecule. twelve closed conformers of a ring of 10 at a bond angle of 111.5 degrees. Every one has exactly the same bond angles, so an account built from angles and torsions places them all on the horizontal axis alone. The vertical axis is the closest approach of two atoms four or more bonds apart, which no term in that account mentions: two of these differ by 0.08 kilojoules in torsional energy and by 0.75 ångström in how close they come.

The atoms that meet across a ring

Twelve closed conformers of a ten-membered ring at one bond angle, so every one has identical angle strain by construction. Two of them differ by 0.026 kilojoules a mole in torsional energy and by 0.80 ångström in how close two atoms on opposite sides of the ring come — 2.331 against 3.131, where two carbons are in contact at about 3.4. An account built from angles and torsions calls those two structures the same.

shape · Strain The part of the bond that is outside the picture. The share of the overlap integral between two 1s orbitals that lies outside both of their 90 per cent contours, against how far apart the atoms are. At a bond length it is 8.3 per cent; by 7 bohr it is 53.4. The two drawn surfaces stop touching at 5.32 bohr, where the overlap is still 0.08 — so the picture separates well before the interaction does.

The tenth that is not drawn

A ninety per cent contour of a hydrogen 1s orbital is a sphere of radius 2.661 bohr, and two of them stop touching at 5.322 bohr — where the overlap between the two orbitals is still 0.0768 and rising in importance. At a three-ångström contact, 36.9 per cent of the overlap integral lies outside both drawn surfaces, and holding nine tenths of it inside the picture would take a contour enclosing 97.28 per cent.

orbitals · Contour Where two He atoms stop, with no contact distance put in. The repulsion between two He atoms, computed from the overlap of their filled valence orbitals — four electrons in a bonding and an antibonding pair, of which the antibonding one rises further — against London's dispersion attraction from the measured polarisability and ionisation energy. The minimum is at 3.14 ångström where the tabulated van der Waals contact is 2.8, and nothing anywhere in the calculation is a length.

The radius that was tabulated

A van der Waals radius is a fitted number that every structural argument in chemistry uses. Computing it instead — a repulsion taken from computed overlap integrals, an attraction taken from two measured scalars, and no length anywhere — puts helium's contact at 3.140 ångström against a tabulated 2.80, neon's at 3.226 against 3.08 and argon's at 3.996 against 3.76. And the surface two atoms actually stop at encloses 99.4 per cent of the density, not ninety.

orbitals · Contour The bonding follows the overlap over, and turns where it turns. The stabilisation of three pairs of orbitals against how far apart they are, each scaled to its own largest, with the coupling taken from the computed overlap. Two orbitals with no radial node are stabilised less at every separation further out; the two with one turn over — and the maximum of the bonding is at exactly the separation of maximum overlap, to five decimal places, because a coupling proportional to the overlap makes the stabilisation a strictly increasing function of it.

The same overlap, a different bond

A 1s and a 2s change their overlap by two thirds between two and seven bohr, and change what they are bonded by by 0.154 per cent. The overlap turns over and the bonding turns over with it, at exactly the same separation to five decimal places — the arithmetic refused the expectation that the secular denominators would move it — and what separates one pair from another is not where the maximum is but how little of it there is.

wrong · Overlap The 6-ring at 111.5°: the alternating form and what a search finds. The alternating ring — every atom displaced above or below the plane in turn — with its dihedral angles and its two strain terms, beside the lowest-torsion member a search constrained only by bond angles and closure returns. Both satisfy every geometric constraint; only one of them is staggered.

The explanation with the wrong sign

Two methyl groups on one carbon make a ring easier to close, by up to eleven thousand-fold, and the textbook reason is that they compress the ring's internal angle. Computed from a standard bending term, that compression helps a three-ring and a four-ring and hinders every ring from five up — predicting a slowing of 0.754-fold for the five-ring measured to speed up 250-fold. The account with the right sign is about rotations rather than angles, and it has a ceiling of 36.5 that the measurement is already above.

shape · Strain A ring of 6 as the neighbour repulsion is turned up. At an on-site repulsion of 8, three quantities against the nearest-neighbour repulsion: the alternating structure factor, the double occupancy, and the nearest-neighbour opposite-spin pair distribution. The rise is steepest at V = 4.5, which is 0.563 times the on-site repulsion. Far past it the ring is charge ordered — nearly every electron paired on alternate sites, which is what a double occupancy approaching a half means.

The give-back that turned into a saving

Take a wavefunction optimised for an on-site repulsion, weight its correlation hole with an interaction that reaches one neighbour, and the enhancement at one bond gives back forty-four per cent of the on-site saving. Putting that neighbour term into the Hamiltonian and solving exactly does not shrink the give-back. It reverses its sign.

beyond · Correlation One sign change is a root and the other is not. The solved give-back against the neighbour repulsion on a ring of 6 at U = 8. It crosses zero at V = 3.895, where the price of everything beyond contact really is nothing, and changes sign again at V = 4.159, where the quantity it is a fraction of has vanished instead. The curve is broken at the second because it is an asymptote and not a crossing; four of the 12 points fall outside the band drawn here.

A sign change is not always a zero

The solved give-back changes sign somewhere between a neighbour repulsion of two and one of four, and a bisection looks like the way to find the value where the structure beyond contact contributes exactly nothing. It changes sign twice. One crossing is that value; at the other the quantity the fraction is a fraction of has vanished instead, and a bisection reports the two in identical words.

beyond · Correlation Two contributions of opposite sign, and where they cancel. The two terms of the give-back's numerator on a six-site ring, with the interaction reaching two bonds at half the strength of one. The first term is what the correlation hole returns at one bond and the second is what it costs at two — and they have opposite signs, because penalising pairs at two bonds depletes them where a one-bond interaction enhanced them. The numerator vanishes where the two are equal and opposite, which is not where either is zero.

A cancellation between two separations

The give-back's numerator has one term in it, so its root is the repulsion at which the pair count at one bond returns to its free value. Let the interaction reach two bonds and the numerator has two terms with opposite signs, and its root becomes a cancellation between an enhancement at one separation and a depletion at the next — at a repulsion where neither is at its free value. Which side of the old zero the new root falls on reverses between a reach of a third and a half, because whether two bonds are enhanced or depleted is the interaction's own doing.

bonding · Correlation Three terms in the numerator, and one place they cancel. The three contributions to the give-back's numerator on a six-site ring whose interaction falls as one over the distance round the ring — a half at two bonds, a third at the antipode — with their sum. The sum crosses zero once, at V = 5.9198, where the one-bond and antipodal terms are negative and the two-bond term positive: the two-bond enhancement is paying for depletions on either side of it.

A root that moves when nothing else does

Give the ring of six an interaction that falls off as one over the distance and the give-back's numerator has three terms. It still has one root, and at that root every separation's pair change has the opposite sign from the two-bond case. But the root has a weakness the earlier essays never tested: lower the interaction by the same amount at every separation and not one state changes, while the root moves — by as much as the reach of the interaction did, and out of existence altogether if the zero is put at one bond.

bonding · Correlation

Named alongside it

The objects these essays reach for when they reach for this one.

Model limitIntermolecular forceConventionCorrelation energyElectron correlationExact diagonalisationHubbard modelOn-site repulsionDouble occupancyMany-electron wavefunctionsOverlap integralReference state

All concepts