Molecular Geometry & Bonding
what the shape decides
Concept

Quantum numbers — where it appears

The labels an eigenstate carries, one for each conserved quantity. Where a symmetry is broken the corresponding label stops being good, which is why an asymmetric top's states carry two K labels and no valid one.

Named by 11 essays across 4 fields — each of them below, with the objects they name alongside it.

The 2pz orbital. The 2pz orbital at the contour enclosing 90 per cent of its density — a level solved for by integration rather than chosen. The two colours are the two signs of the wavefunction, which is what distinguishes a bonding interaction from an antibonding one. Contours drawn: 2pz at 90% of its density, |ψ| = 9.48e-3.

What an orbital is

Not a region the electron occupies, not a path it follows, and for any atom but hydrogen not an exact anything. An orbital is a one-electron wavefunction, and almost every difficulty in this subject comes from forgetting that.

orbitals · Orbital The radial function of 3s. The radial part of the wavefunction, which changes sign at each node, and the radial distribution, which is the probability of finding the electron in a shell at that radius. The second vanishes at the nucleus and the first does not.

Nodes

An orbital with quantum numbers n and l has exactly n−l−1 radial nodes and l angular ones. That is a count, it is exact, and it is the fastest way to catch a drawing that is wrong.

orbitals · Orbital Orbitals at the 90 per cent contour. Several orbitals drawn at the same enclosed fraction and, unless stated otherwise, at the same scale — so the sizes on the page are the sizes. Each contour was solved for separately by integrating that orbital's own density. Contours drawn: 2pz at 90% of its density, |ψ| = 9.48e-3; 2px at 90% of its density, |ψ| = 9.49e-3; 2py at 90% of its density, |ψ| = 9.49e-3.

Complex harmonics against real ones

The p orbitals every chemist draws are not eigenfunctions of anything. They are real combinations of the complex solutions, chosen because they point along axes — and the choice is invisible until a magnetic field makes it matter.

orbitals · Orbital 4s and 3d from K to Zn. The mean radius of the 4s and 3d orbitals across the elements K to Zn, at the nuclear charge each shell actually feels. The screening is Slater's, which is fitted; the radius that follows from it is the closed form for a hydrogen-like orbital, which is not.

The aufbau order is not a property of the atom

Iron's 3d orbital is more than four times smaller than its 4s and, by every one-electron estimate available, far lower in energy. The 4s fills first anyway, and it empties first too — which is not a paradox but a sign that the filling order was never a list of orbital energies.

wrong · Approximation 4s and 3d from Sc to Zn. The mean radius of the 4s and 3d orbitals across the elements Sc to Zn, at the nuclear charge each shell actually feels. The screening is Slater's, which is fitted; the radius that follows from it is the closed form for a hydrogen-like orbital, which is not.

What an electron actually feels

A 3d orbital is less than half the size of the 4s beside it and fills second anyway. The charge an electron feels is not the nuclear charge, the correction is a fit rather than a derivation, and the two facts together explain the shape of the periodic table.

orbitals · Orbital 2s and 2p from B to Ne. The mean radius of the 2s and 2p orbitals across the elements B to Ne, at the nuclear charge each shell actually feels. The screening is Slater's, which is fitted; the radius that follows from it is the closed form for a hydrogen-like orbital, which is not.

What the screening model cannot see

Slater's rules put the 2s and the 2p in one group, so they give both orbitals exactly the same effective nuclear charge at every element from boron to neon. The two are separated by several electronvolts in all six, and the model has no term that could produce it.

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 water: every level up to J = 4, from a matrix. The rotational levels of water at κ = -0.4322, each J diagonalised in the symmetric-top basis. A symmetric top would show one level per K with everything above K = 0 doubly degenerate; here every degeneracy is split, and the size of each splitting is what the third constant is measured from.

The top that reports all three

A symmetric top hides one of its two rotational constants in every line of its spectrum. Break the symmetry and the hiding stops: for water, twenty-three of the twenty-five levels up to J = 4 move when A is changed, and the two that do not are the ground state and the one at B + C. There is no formula for any of them.

spectra · Rotation How nearly a broken symmetry survives. A screened potential splits the n = 2 shell and destroys the degeneracy the linear Stark effect depends on. The field needed to overcome the splitting and restore the linear behaviour runs from 4.3e+4 volts a centimetre at a quantum defect of 0.00040 to 2.9e+7 at a defect of 0.208. The dipole between the states is 3.000 throughout, so the field is exactly the splitting divided by twice it.

How nearly a broken symmetry survives

The hydrogen shell's extra symmetry is what makes its Stark effect linear, and a real atom does not have it. Screening splits the shell, and the field needed to overcome the splitting and restore the linear behaviour is a curve — from forty thousand volts a centimetre at a quantum defect of 0.0004 to thirty million at a defect of 0.21.

symmetry · Representation A fixed offset against a rising threshold. The offset at which a shell's coupled minimum disappears, against the principal quantum number, with the offset every shell's own d level actually has. The threshold is 2(n² − 4)/(5(n² − 1)) — zero at the second shell, a quarter at the third, and rising to two fifths. The shell's own offset is one fifth whatever the shell, because it comes from the reciprocal of l plus a half and carries no n at all. So the comparison is a constant against a curve, and it changes answer exactly once.

The quarter, generalised

A shell's one coupled minimum exists because its d level sits within a quarter of the s–p gap, and the quarter was found by bisecting a numerical search on one shell. It is exactly 2(n² − 4)/(5(n² − 1)) for every shell — zero at the second, a quarter at the third, two fifths in the limit — while the offset it is compared against is one fifth whatever the shell.

symmetry · Representation The same count in both pictures, and no rule between them. Every hydrogenic orbital with a radial node, drawn twice: its position nodes on the left axis and its momentum nodes on the right, at the same nuclear charge. The counts are identical and exact — n − l − 1 in each — because the two radial functions are polynomials of the same degree. The positions are unrelated: a node three quarters of the way out in one picture is not three quarters of the way out, or anywhere in particular, in the other.

The nodes in the other variable

An orbital has n − l − 1 radial nodes, and it has exactly that many in momentum too — the two radial functions are polynomials of the same degree. Nothing pairs one node with another: they are zeros of two different classical families. What is exact is the product over all of them, which is a ratio of factorials and does not depend on the nuclear charge at all.

orbitals · Orbital

Named alongside it

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

One-electron modelsClosed formDegeneracyEffective nuclear chargeModel limitthe aufbau principleContour levelExpectation valueNodePenetrationRadial nodeShielding

All concepts