Molecular Geometry & Bonding
what the shape decides
Concept

Quantum defect — where it appears

How far an orbital's effective principal quantum number falls short of its real one, because the electron penetrates the core and feels more of the nuclear charge. It is largest for an s orbital and smallest for a d, and the differences between the defects of one shell are what screening leaves of a Coulomb degeneracy.

Named by 9 essays across one field — each of them below, with the objects they name alongside it.

Two events, not one — and both below the n = 2 shell's. The field at which each coupled pair's shift stops being quadratic and starts being linear, for the n = 3 shell and for the n = 2 shell. The n = 3 shell has two, a factor of 3.66 apart, and both are far below n = 2's single one — so the linear effect returns is two events in a shell with a d, and it happens at a field thirty times weaker than in a shell without one.

Two events where there was one

A broken symmetry returns at a field where the coupling matches the gap it has to overcome, and in a shell with two levels that field is a single number. The next shell up has three, two gaps and two dipoles — so there are two crossovers, a factor of 3.66 apart, both of them below the single one of the shell below, and the exponent takes more than a decade of field to travel between them.

symmetry · Representation Every crossover a shell has. The field at which each coupled pair's linear behaviour returns — the gap between the two levels divided by twice the dipole joining them — on a logarithmic axis. The n = 3 shell has 3 distinct fields rather than four, because its m = ±1 half is a two-level ladder with one coupled pair. All three are below the n = 2 shell's single one.

Three events, and a ratio of two dipoles

The m = 0 half of a shell has two crossovers because it has three levels and two coupled pairs. The other half has two levels and one, so the whole shell has three distinct fields rather than four — and two of the three share a gap exactly, which makes the ratio between them a ratio of two dipoles, 2/√3.

symmetry · Representation One and eight, over two decades of defect. The number of avoided crossings the whole shell has, against the number of two-state crossover fields its coupled pairs supply, as the quantum defect is swept towards zero. The question is whether the estimated count falls to meet the exact one as the l degeneracy closes. It does not move: one against eight at every defect tried, from 0.02 down to 0.0002, with the estimates spanning a factor of 7.08 throughout.

Consistently wrong is not a limit

Does the two-state picture of a tilted Stark shell become right as the quantum defect closes the l degeneracy? Swept over two decades it does not move: one avoided crossing against eight estimates at every defect. The reason is that every dimensionless quantity settles — the crossing sits at 0.04000 of the zero-field gap and the nearest estimate at 0.9067 of the crossing, and neither is heading anywhere.

symmetry · Representation Two levels of different symmetry, closest at 0.0399865. The lowest m = 0 level and the lowest |m| = 1 level of the zero-defect shell, in units of the s–p gap, against the field in the same units. Both fall. Their separation is smallest at 0.039986525, where it is 0.97648 of the zero-field gap, and they never meet. The field along z commutes with the angular momentum about z, the two levels belong to different values of it, and no element of the field connects them.

The crossing nothing couples

A tilted Stark shell's one avoided crossing settled at 0.04000 of the zero-field gap, and the natural guess was a ratio of angular integrals. With the quantum defect taken out the limit is 0.0399865, not four hundredths — and the two levels at that minimum belong to different symmetries about the field, which no element of the field connects. It was never an avoided crossing. The one minimum between levels that do interact sits at half the field, behind a level of the other kind.

symmetry · Representation One spectrum along three directions, and three different sets of estimates. The two-state estimate gap ÷ 2d for every pair of the shell's functions the field couples, with the field along z, along x and at the tilt the defect sweep used. Along z there are three distinct estimates, along x three different ones and at the tilt eight, none equal to any of the axial three. The exact spectrum is identical along all three directions, and its two minima are drawn as vertical lines: the one between coupled levels at 0.0196 and the tangency of uncoupled levels at 0.0400. An estimate is a property of the axes the functions were written along, and a feature is not.

Two levels cannot make a minimum

The one minimum between coupled levels in a Stark shell sits at 0.0196 of the s–p gap, and the nearest two-state estimate at 0.0192 — two per cent away, which reads as the estimates having been aimed at the right feature all along. They were not. Two coupled levels only ever separate, so no estimate can be where its own pair is closest. The minimum belongs to a third level, exists only while the d level sits within a quarter of the s–p gap, and meets the estimate by crossing it.

symmetry · Representation Only lithium's d level sits close enough to p. Each alkali's offset — how far its d level sits above p, in units of its s–p gap — at every shell from its valence shell to the thirtieth, computed from measured quantum defects, against the threshold below which a shell's s and p levels have a coupled minimum in a field. The screening model's offset of one fifth is drawn for comparison. Lithium sits below the threshold at every shell. Sodium, potassium, rubidium and caesium sit above it everywhere, by factors of 2.6 to 7.

Four alkalis the model cannot hold

A shell's s and p levels have a coupled minimum in an electric field when its d level sits within a threshold fraction of the s–p gap above p, and a screening model put every shell's d level at one fifth — inside the threshold from the third shell on, with more room the larger the defect. Computed from measured quantum defects, lithium keeps the minimum at every shell. Sodium, potassium, rubidium and caesium lose it at every shell, and the screening model has no member that resembles any of them.

symmetry · Representation The threshold belongs to the atom, not to the shell. For each alkali's valence shell (lithium's third, the first with a d level): its offset — where d sits above p, in units of the s–p gap — against the threshold below which the s–p separation has a minimum in a field, computed first with hydrogen's dipole elements and then with the atom's own, on a logarithmic axis. Lithium's own threshold is half as large again as hydrogen's. Sodium's is three times hydrogen's, and its offset of 0.705 now sits below it. Potassium's, rubidium's and caesium's fall, by factors from one and a half to thirty, so they stay above for a new reason.

The threshold belongs to the atom

Whether an alkali shell's s and p levels have a coupled minimum in a field was decided by comparing where its d level sits with a threshold built from hydrogen's dipole elements. Computed from each atom's own quantum defects instead — with integrals that reproduce three measured resonance lines to one per cent — the threshold is not a property of the shell at all. Lithium's rises by half, sodium's third shell regains the minimum the hydrogenic comparison denied it, and the heavier alkalis keep their verdict for a reason nobody had given: their p–d integral cancels, and in potassium it changes sign between two shells.

symmetry · Representation Every level decides what three levels could not. For each alkali's valence shell, the polarisability of the s level and of the p level with no angular momentum about the field, each summed over every s, p and d level. Where the p bar is longer the s–p separation falls at first in a field. It is longer in sodium, potassium, rubidium and caesium — by 365, 506, 779, 1484 atomic units — and shorter only in lithium, whose valence shell has no d level of its own. The three-level threshold called potassium's, rubidium's and caesium's valence shells shells without a minimum.

The shell was the wrong unit

Whether an alkali's s and p levels approach each other in a weak field was decided from three levels of one shell. The criterion that holds every level is a difference of two polarisabilities, and polarisabilities are measured. Summed over every s, p and d level with integrals that reproduce five measured ground states to two per cent, the difference keeps sodium's verdict at thirty times the margin and reverses potassium's, rubidium's and caesium's at every shell — because the d level beside p belongs to the shell below.

symmetry · Representation With its measured Ritz series potassium's p level stops being the more polarisable at the thirty-fourth shell. Potassium's α(np₀)/α(ns), every s, p and d level summed, against the principal quantum number, with three readings of the defects. The four-decimal asymptotic defects put the turn below one at n = 36.93; the measured δ₀ alone at 36.99; the measured Ritz series, whose corrections at those shells are a ten-thousandth to a thousandth, at 33.72. The ratio falls by about 0.0074 a shell, which is why corrections that small move it three shells.

Potassium's turn was a fine-structure question

Summed over every level, potassium's p level stops being more polarisable than its s level somewhere past the thirtieth shell, and a thousandth in any quantum defect moved the turn by shells. With the Ritz series measured on ultracold ³⁹K in 2019 the inputs are fixed to a millionth, and the turn lands at the thirty-fourth shell, not the thirty-seventh. But the two fine-structure components of the p series differ by three thousandths: with either one in place of their average the turn is at the twenty-third or the forty-fifth. A sign decided by the spinless sum is decided by an interval a hundred and fiftieth of the s–p gap.

symmetry · Representation

Named alongside it

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

Model limitStark effectMatrix elementDegeneracyPerturbation theoryAvoided crossingClosed formScreeningSymmetry breakingConvergenceEigenvalueOne-electron models

All concepts