A field splits the n = 2 shell into whole numbers

One of the figures on representations: Character tables generated from a molecule's own operations, bases reduced in them, and what a descent in symmetry does to a level.

Twelve essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

The symmetry that is not a rotation

The raw material: every element of the coordinate z between the four states of the n = 2 shell, each one an integral over the whole of space by the site’s mapped quadrature rule. Ten integrals, of which two are non-zero and eight vanish because the integrand is odd about a plane — at 10⁻¹⁸ rather than at something small.

Each identity computed from the integrals and multiplied out, with the largest entry left over. The first is exactly zero because those matrices are integers; the second is 4.6 × 10⁻¹⁴; the third is 1.2 × 10⁻⁸, which is the quadrature’s own noise carried through a product. A wrong construction leaves a residual of order one.

The same calculation for the m = 0 states of the n = 3 shell, where the matrix is three by three and the two off-diagonal elements are ⟨3s|z|3pz⟩ = −7.348 and ⟨3pz|z|3dz2⟩ = −5.196. The eigenvalues are −9, 0 and +9, so the whole numbers are −2, 0 and +2 and the shifts are (3/2)n times an integer again.

How nearly a broken symmetry survives

The field needed to restore the linear Stark effect, against how strongly the potential is screened. The defects marked beside each point are what a spectroscopist would quote; the open marks are where the first-order splitting has drifted more than a tenth from the exact one.

What the field does when it wins: the shell splits into a manifold of equally spaced levels with whole-number dipoles, which is the hydrogenic behaviour the degeneracy produces. Above the crossover field, a screened atom does this too.

The matrix the dipole comes from: the position operator inside one shell, whose off-diagonal elements are the coupling that a field uses. The screening does not touch this, which is why the crossover is proportional to the splitting alone.

Two events where there was one

Each coupled pair’s crossover, for both shells. The n = 3 shell has two and both are below the n = 2 shell’s one.

The three defects. An s penetrates the screening most and is shifted most; a d hardly penetrates at all.

The z matrix inside the shell. Two entries and a zero, and the zero is a selection rule rather than a small number.

Three events, and a ratio of two dipoles

Every field at which a coupled pair’s linear behaviour returns, in both halves of the n = 3 shell and in the n = 2 shell, on a logarithmic axis.

The matrix of ⟨i|z|j⟩ within each subshell, by quadrature. An entry symmetry forbids comes back below 10⁻⁹ and is written as an exact zero.

The zero-field levels of each subshell, with the pairs a field couples marked between them.

Four angles the shell chooses

The number of distinct fields with an event, against the angle between the field and the z axis. Hollow rings mark the angles where two events coincide.

Both dipole matrix elements for every pair the field can couple. Each pair has exactly one that is not zero.

Each pair’s crossover field against the tilt. The three the z component reaches rise; the three the x component reaches fall.

None of the six was a crossing

Every two-state crossover field at each tilt, with the field at which the exact spectrum’s one avoided crossing actually sits.

The five eigenvalues against the field at a tilt of forty-five degrees, fanning apart without crossing.

The number of distinct two-state events against the number of avoided crossings, at each tilt.

The variation was the basis

The field at the one avoided crossing against the tilt, in five functions and in nine. The two agree exactly at zero tilt and part company by twenty-one per cent by forty degrees.

The crossing field at thirteen directions spread over the sphere, including six with a component out of the xz plane. They are one number.

The count at each direction, in the whole shell and in the five functions. They agree at every tilt but ninety degrees.

Consistently wrong is not a limit

The exact count and the estimated count as the quantum defect falls by a factor of a hundred. Neither moves.

Three dimensionless quantities against the defect, each divided by its own value at the smallest one. All three settle, and the third settles away from one.

The same three ratios, drawn against the defect. Each settles; the third settles at 0.9067 rather than at one, and that is the answer.

The crossing nothing couples

The field at the counted minimum over the s–p gap, at seven defects and with the defect removed. The sequence crosses four hundredths and keeps going.

The lowest m = 0 level and the lowest |m| = 1 level of the zero-defect shell against the field. They are closest at 0.0399865 and never meet.

The size of each level’s induced dipole against the field. The two curves cross at 0.0399865, which is where the separation of the two levels is stationary.

Two levels cannot make a minimum

The two-state estimates with the field along z, along x and at the tilt the defect sweep used, against the exact spectrum’s coupled minimum and tangency. The spectrum is the same at all three directions; the estimates are not.

The separations of two coupled pairs on their own, each over its zero-field value: s with $p_0$, and $p_1$ with $d_1$. Neither falls anywhere. Each pair’s two-state estimate is marked on its curve.

The separation of the two lowest m = 0 levels with the d level in the block and with it removed. Only the three-level block has a minimum, and the p–d estimate is drawn beside it.

The quarter, generalised

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.

Every such element computable by quadrature, beside the closed form.

The m = 0 levels of the second shell and of the third, at a small quantum defect.

Four alkalis the model cannot hold

Each alkali’s offset from measured quantum defects at every shell from its valence shell to the thirtieth, against the threshold and against the model’s one fifth.

For each alkali, the ratio of its p–d defect difference to its s–p difference, which every shell’s offset approaches as the shell grows.

Each alkali’s measured s, p and d quantum defects, beside the defects the screening law’s proportions would give with the same s defect.