Molecular Geometry & Bonding
what the shape decides
Concept

Quadrature — where it appears

Evaluating an integral by summing weighted values at chosen points. Its accuracy depends on the points, the weights and the domain, and an error in the domain is invisible to a check that varies only the resolution.

Named by 11 essays across 5 fields — each of them below, with the objects they name alongside it.

The same d shell in four fields. The five d energies in octahedral, tetrahedral, cubic, square planar coordinations, computed and drawn in units of the octahedral splitting. The dashed line is the barycentre, which the five levels sum to whatever the arrangement.

Two models, one ratio

A tetrahedron splits a d shell by four ninths of what an octahedron does. Two models that share nothing but the ligand directions — an integral over a point-charge potential and a rotated diagonal matrix — both produce that number to eight decimal places, and neither was told it.

applied · Ligand field The exponent the molecule chooses, and what it buys. The 1s exponent that minimises the energy of a one-electron diatomic, against the separation of the nuclei, with the binding curves at that exponent and at the free atom's. Held at ζ = 1 the bond comes out at 2.49 bohr and binds 0.0648 hartree; with the exponent free it comes out at 2.00 bohr at ζ = 1.238 and binds 0.0865. The exact answer for this molecule is 2.00 bohr and 0.1026.

The atom does not bring its own orbital

Build a one-electron diatomic from two hydrogen 1s functions and it comes out 25 per cent too long and 37 per cent too weakly bound. Let the molecule choose how large those functions are and the bond length is right to three figures, at an exponent of 1.238 — the orbital contracts by a quarter when the bond forms.

orbitals · Orbital One isovalue, many fractions — and one fraction, many isovalues. What each of 4 conventional isovalues encloses, for 7 orbitals of hydrogen, and in the last column the level each one needs to enclose 90 per cent. At 0.02 atomic units the fractions run from 0.4 to 96.2 per cent, and the levels in the last column differ by a factor of 32. A plate of orbitals drawn at one value is not a comparison of sizes. Contours drawn: 1s at 90% of its density, |ψ| = 3.94e-2; 2s at 90% of its density, |ψ| = 7.40e-3; 2pz at 90% of its density, |ψ| = 9.48e-3; 3s at 90% of its density, |ψ| = 2.64e-3; 3pz at 90% of its density, |ψ| = 3.03e-3; 3dz2 at 90% of its density, |ψ| = 3.60e-3; 4s at 90% of its density, |ψ| = 1.24e-3.

The isovalue nobody chose

Every program that draws an orbital asks for a number, and almost every user accepts the default. At 0.02 atomic units that default encloses 96 per cent of hydrogen's 1s, 52 per cent of its 2s and four tenths of one per cent of its 4s — three pictures drawn to one rule, meaning three completely different things.

orbitals · Contour The wrong shape, fitted as well as it can be. The exact hydrogen 1s orbital and the best sums of one, two, three and six Gaussians, each with its exponents optimised for the energy. Three of them already reproduce the exact function to 99.94 per cent by overlap, which is why the method works at all — and the two places it goes wrong, at the nucleus and far out, are exactly where the other faces of this figure look.

A Gaussian is the wrong shape

Sixty years of molecular calculation are built on functions that get the two ends of an orbital wrong. A Gaussian has no cusp at the nucleus and dies too fast far away, and no number of them fixes either — while three of them already reproduce hydrogen's 1s to better than 99.9 per cent by overlap, and that is why the method works.

orbitals · Basis The surface-honest curve and the page-honest curve. The 2pz orbital in its xz plane, drawn twice. The outer curve is the contour whose SURFACE encloses 90.0 per cent of the density in space — the claim every orbital figure here makes — and in this plane it encloses 96.7 per cent. The inner curve is the one that encloses 90 per cent in the plane, which is what a reader looking at a flat picture would take the caption to mean. The two levels differ by a factor of 1.87 and the curves differ visibly; the caption does not. Contours drawn: 2pz at 90% of its density, |ψ| = 9.48e-3.

A slice is not the surface

The page is flat, so every printed orbital is a slice through a contour rather than the contour itself — and a curve that leaves a tenth of the density outside it in space leaves about a thirtieth outside it on the page. Both claims are true of the same picture and only one of them is ever stated.

orbitals · Contour Overlap does not always fall as the atoms are pulled apart. The overlap integral of two pairs of orbitals against the separation between their centres, with the turning point and the sign change of each found by bisection on the integral itself. A pair with a radial node in it does not fall monotonically and does not keep one sign.

Closer is not more overlap

Two 1s orbitals overlap more the closer they are, and every curve drawn from that pair says the same thing. Put a radial node into one of them and the rule fails: a 1s with a 2s overlaps by 0.016 at contact, by 0.282 near four bohr, and less again beyond — and two 2p orbitals head-on change sign at 5.06 bohr and are more strongly coupled at eight bohr than at four.

bonding · Overlap Where the electron is, and how fast it is going. The radial distribution in position on the left and in momentum on the right, for the same orbitals. The two run opposite ways: the 1s is the most compact in space and the widest in momentum, and every excited orbital that spreads out in one narrows in the other. Both are normalised, both are the same function, and neither is more fundamental than the other — the transform loses nothing and adds nothing.

The orbital in momentum space

Every orbital has a second picture as complete as the first and almost never drawn. Nothing is added by taking it — it is the same function in the other variable — but the uncertainty product falls out of it, and the functions quantum chemistry is built from turn out to be the only ones that attain the bound.

orbitals · Orbital A field splits the n = 2 shell into whole numbers. The eigenvalues of z inside the shell, which are the shifts a uniform field produces to first order. There are three distinct ones and each is a whole number times (3/2)n, so the splitting is proportional to the field itself rather than to its square — which is what no other atom does.

The symmetry that is not a rotation

Hydrogen's n = 2 shell holds four states at one energy and its rotation group accounts for at most three. The operator that accounts for the fourth is built here out of computed integrals: three matrices whose commutators close into the rotations, whose product with the angular momentum vanishes, and whose Casimir comes out at exactly n² − 1.

symmetry · Representation A σ contour at 90 per cent, and the two atomic ones. The section through both nuclei of the surface enclosing 90 per cent of the bonding orbital's density at 2 bohr, with circles marking where two atomic contours of the same stated fraction would be. The two pictures are different shapes and enclose different amounts, and the atomic pair encloses 91.70 per cent of the molecular orbital's density. Contours drawn: 1s at 90% of its density, |ψ| = 3.94e-2.

A bond is not two atoms overlapping

The surface enclosing ninety per cent of a σ orbital's density is one closed surface with both nuclei inside it, at a level of 0.0359. The two atomic surfaces usually drawn instead sit at 0.0394, are a different shape, and enclose 91.70 per cent of the same orbital — and whether the molecular one is one object or two is decided by the fraction the caption claims, anywhere between 2.1 and 5.2 ångström.

orbitals · Contour 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 The alternation a spring buys, three ways. The alternation against the elastic constant, on a logarithmic axis. The middle line solves (2/π)(K − E)/(1 − δ²) = K for the complete elliptic integrals; the lower one is the exponential form every account of a Peierls distortion quotes, which is its own asymptote and is 6.5 per cent low at K = 1.2; the upper one is a ring of 40, which leaves the infinite chain as the spring stiffens because a smaller alternation is a longer coherence length.

The amplitude the collapse left behind

Five Peierls curves became one curve when each was divided by the alternation its ring settles at cold, so that amplitude is the whole of what distinguished them — and it was five golden-section searches over diagonalisations with no formula anywhere. It has one, exactly, as a sum of square roots; and writing it down says that one of the five rings was never measuring a long chain.

wrong · Metal

Named alongside it

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

One-electron modelsModel limitClosed formProbability densityContour levelConvergenceEnclosed probabilityIsosurfaceOverlap integralBasisConventionExpectation value

All concepts