The variation was the basis

Count the avoided crossings of an n = 3 alkali shell in a tilted electric field by pairwise two-state estimates, and the answer is three or six depending on the tilt. Diagonalise the five coupled levels exactly and the answer is one, at every tilt. The two-state estimates answer a real question — the one a two-level reduction is built for — but not the one being asked, which is where the levels of the whole problem actually come closest.

That leaves an obvious question. Five functions are not the shell: does the single shallow crossing stay single when the four remaining functions are added back? Nine levels give eight adjacent gaps to follow.

It stays single, and it does so for a reason that makes the whole tilt sweep unnecessary. It also stops moving, and the second half is the finding: the whole shell’s spectrum does not depend on which way the field points at all, and the thirty-four per cent variation the five-function spectrum shows, with a plausible explanation attached, is not a property of the problem.

Why three functions were left out, and what it cost #

A field in the xz plane leaves reflection in that plane a symmetry, so, by the theorem every selection rule in this subject is, the three functions odd under y → −y — $3p_y$, $3d_{yz}$ and $3d_{xy}$ — have no matrix element to any of the others. They do not couple. Excluding them is not an approximation in the usual sense: the five-by-five block is an exact block of the nine-by-nine.

What it is, is a subspace that is not closed under rotation. The five functions retained are picked out by a plane, and rotating the field out of that plane is a rotation of the problem — so the retained set stops being the right set, and the answer inherits a dependence on the tilt that the whole problem does not have.

The block is exact at one tilt and one only. With the field along z, the m = ±1 pair is decoupled by a stronger argument than the plane’s — a field along z conserves m — and the five-function answer is the nine-function answer.

Everywhere else it is not, and how far is measurable.

The five-function crossing field is 2.0876 × 10⁻⁵ at zero tilt, 2.0253 at ten degrees, 1.8612 at twenty, 1.6769 at thirty, falls to a minimum of 1.6435 × 10⁻⁵ near forty degrees, and rises again through 1.6808 at forty-five, 1.8098 at fifty-five and 1.8903 at sixty to 2.2014 × 10⁻⁵ at eighty — the factor of 1.34 the five-function spectrum gives, with a shape that reads as a smooth monotone rise over a narrower range of tilts and is in fact a shallow valley.

The nine-function answer is 2.0876 × 10⁻⁵ at every tilt. The two agree to every digit at zero and the truncation’s error is the whole of the variation.

One field, thirteen directions #

Sampling the sphere rather than a line — five tilts in the plane and eight directions out of it, up to a hundred and twenty degrees from z and two hundred and ten in azimuth — gives the same field at every one.

That is not a numerical coincidence and it is not a limit. The zero-field Hamiltonian is the screened energies on the diagonal, and the screening depends on l alone, so the matrix is a multiple of the identity within each l and is therefore invariant under every rotation. The field’s coupling is F(n̂ · r̂), and rotating n̂ is the same as rotating the whole problem. So the spectrum is a function of the field’s magnitude and of nothing else.

Checked against a reference direction, the nine levels agree to 1.8 × 10⁻¹¹ against a level scale of 5.6 × 10⁻², which is a relative agreement of 3 × 10⁻¹⁰ — the Jacobi eigensolver’s own tolerance rather than anything physical. The statement is exact and the arithmetic reproduces it as far as the arithmetic goes.

The second crossing was the truncation too #

In five functions there is a second avoided crossing with the field exactly along x, and it makes a natural control: a search that returns one crossing everywhere is worth being suspicious of, and finding two somewhere shows it can. That second crossing’s gap closes to 2.07 × 10⁻⁷ against 5.0 × 10⁻⁴ — three and a half orders deeper than the shallow dip present at every tilt, and on its face unmistakably a real avoided crossing.

It is not in the whole shell. At ninety degrees the nine-level count is one, as it is everywhere else.

The reason is the same truncation, seen from the other side. The five-function basis retains two of the five d functions — $3d_{z^2}$ and $3d_{xz}$ — and with the field along x those two are left degenerate, because the coupling that would separate them runs through functions the basis has thrown away. A gap that starts at zero and stays there is not an avoided crossing; it is a degeneracy the calculation cannot lift.

The whole shell’s field lifts the entire d quintet, at every direction, by the same amounts. There is nothing to cross.

So the control was a good instrument pointed at an artefact. Its logic holds — a search that finds two crossings where there are two is not blind — but the case that demonstrated it turns out not to be a crossing at all.

What survives, and it is most of it #

It would be easy to read this as overturning the five-function result, and it does not.

Its central finding stands. The two-state count is a count of estimates rather than of features; the exact spectrum has one avoided crossing where the estimates give three to six; the four coincidence angles are properties of the estimates and mark nothing. All of that is about the relationship between a pairwise reduction and a full diagonalisation, and none of it depends on how many levels the full diagonalisation has, which is why the ratio of two dipoles it found is untouched as an exact property of the estimates. The nine-level problem has one crossing and the two-state estimates would give more of them, so the finding is if anything cleaner.

Its reasoning about degeneracy stands, and explains the nine-level result too. Two eigenvalues of a real symmetric matrix with no symmetry left meet only on a set of codimension two, so a one-parameter sweep does not find one. That is exactly why the nine-level spectrum has no crossings either, and it is why the second crossing at ninety degrees was suspicious: a gap closing to a part in two thousand of its neighbours is a symmetry that has not been broken, not a near-miss.

What does not survive is one number and one example. The 1.34-fold variation, which it reported as “a real feature of the tilt”, and the second crossing, which it reported as a control. Both are the truncation.

The five-function treatment is explicit that five functions is not the shell and that the nine-dimensional problem is still there. That was right about which quantity was at risk; what it could not have predicted is that the quantity does not exist.

It is worth saying what that does to the earlier results about the tilt, since two of them are about angles rather than fields. The four coincidence angles — 45°, 49.107°, 60° and 63.435° — are arctangents of ratios of angular integrals over the five retained functions, and they are exactly that whatever else is true. The exact five-level spectrum already showed that nothing happens at them; the whole shell shows that nothing in the spectrum happens as a function of the tilt at all, so the question of whether a feature coincides with an angle no longer has a subject. The angles remain correct arithmetic about a truncated basis, and quoting them to five figures was always a statement about the integrals rather than about a molecule.

The degeneracy the field lifts, and the one it does not #

There is a structure in the nine-level problem worth reading off, because it is what makes the direction-independence visible rather than merely provable.

At zero field the shell has three distinct energies with degeneracies one, three and five, set by the quantum defect and by nothing else. A field lifts them completely — the same breaking that can be measured against the quantum defect: at any non-zero field the nine levels are generically distinct, so six of the eight adjacent gaps open from zero and two — the s-to-p and the p-to-d — start open.

Those two are the only gaps a crossing can live in, and it is worth being clear that this is a statement about what is being counted rather than an approximation. A gap that opens from zero has no interior minimum to find; it rises monotonically from the moment the field is turned on. The two gaps that begin open can narrow and reopen, and exactly one of them does.

The five-function truncation has the same structure with the wrong multiplicities: one s, two p and two d, so two of its four gaps start at zero and two start open. The arithmetic of what can be counted is unchanged and the arithmetic of what is degenerate is not, and that difference is where the ninety-degree artefact lives — the two retained d functions have nothing to separate them when the field lies along x, while five of them always have each other.

That is the kind of degeneracy no group predicts, seen from an unusual angle: not one a group forces, but one a truncation fails to lift.

What was computed, and how #

The radial functions are hydrogenic, with a quantum defect on each l standing in for the screening, and the position matrix elements between the nine functions come from those radial functions and the real spherical harmonics. All five d functions are needed, including $3d_{yz}$, which the in-plane problem never touches: a field out of the xz plane means nothing until the shell is complete, because the plane is what excluded three of the nine functions in the first place.

The Hamiltonian in a field of arbitrary direction is the same screened diagonal plus $F$ times $\cos\theta\,Z + \sin\theta\cos\phi\,X + \sin\theta\sin\phi\,Y$, diagonalised at each field strength.

Crossings are interior minima of adjacent gaps — the definition used for the five-function spectrum, chosen because it is stable under refinement — with one restriction that is not a convenience. Only gaps that are open at zero field are followed. Six of the eight adjacent gaps here are between levels the quantum defect leaves degenerate, so they start at exactly zero and are opened by the field; a minimum of such a gap is the eigensolver’s ordering of two numbers that agree to their last bits. Following all eight gives between three hundred and six hundred spurious crossings per direction, and the relative depth they would be measured against is a division by zero.

Five things are verified numerically rather than argued. The whole shell’s spectrum is direction-independent to under a part in a hundred million, which is the exact statement measured. The crossing therefore sits at one field, and there is exactly one of it at every direction. The truncated and the whole calculations agree about the count at every tilt but one, and the one disagreement is at ninety degrees and is the truncation’s extra crossing — a statement that agreement everywhere would contradict as surely as disagreement everywhere. And every gap minimum is strictly positive, so nothing actually crosses.

Where the model stops #

One electron. Hydrogenic radial functions with a quantum defect standing in for a many-electron screening, which is the caricature every Stark calculation here uses and which carries the same reservations in this one. What is exact is the direction-independence, and that is exact for any diagonal that depends only on l — including a real one.

The invariance is the shell’s. A real alkali atom has more than one shell, and a field mixes n as well as l. The nine functions here are one n, so the rotation argument holds inside them and says nothing about what a second shell would do. It would not restore a direction dependence, since the same argument applies to any collection of complete shells; it would move the crossing.

The quantum defect is one parameter, and the crossing’s field scales with the gaps it sets. Nothing here is a prediction about a spectrum.

And a nine-level problem is still small. The count is a topological statement about a one-parameter family of real symmetric matrices and does not depend on the size; the field it happens at does.

And the one crossing is a name for a shape. Resolved into the blocks the field cannot connect, the minimum counted here lies between the lowest level with no angular momentum about the field and the lowest with one unit of it, and nothing in the field couples those two — so it is a place where two independent levels are closest rather than a crossing any interaction avoided. The crossing nothing couples shows that, and finds the one minimum between levels that do interact hidden behind a level of the other kind. The direction-independence and the single field above are unaffected.

The generalisation #

The transferable point is that a basis chosen by a symmetry of the problem is not safe once the problem’s symmetry is the thing being varied.

The exclusion here was correct and was justified correctly: three functions have no matrix element to the rest, so dropping them changes no eigenvalue. That argument is airtight at a fixed field direction. What it does not survive is the field direction becoming the variable, because the set of functions the argument selects depends on the direction, and a calculation whose basis moves with its parameter is computing a different problem at each value of the parameter.

The tell is available before any arithmetic: is the retained subspace closed under the transformation being applied? Rotating the field is a rotation, the five functions are not closed under rotations, and that is the whole of it. A basis closed under the symmetry group of the sweep gives an answer that transforms correctly; one that is not gives an answer with a spurious dependence, and the spurious dependence looks exactly like physics — smooth, reproducible, with structure in it, and in this case with a minimum near forty degrees that had a plausible story attached.

There is a second half about how such an artefact gets caught, and it is not by being careful. The five-function treatment stated the limitation explicitly and in the right words — five functions is not the shell — without expecting it to matter. Stating a limitation is not the same as measuring it, and the distance between them here was a factor of 1.34 and a control. The same gap opens when a symmetry measure is quoted without the tolerance that produced it, and the tolerance turns out to admit amounts of asymmetry differing fifty-six-fold.

Who found it, and when #

The Stark effect, observed by Johannes Stark in 1913, the quantum-defect description of alkali spectra, and the rotational invariance of a central-field Hamiltonian are all long established; the last is elementary enough that stating it is a little awkward. The five-level calculation, the tilted sweep and the nine-level comparison are original to these essays rather than taken from the literature.

The question as first put — does the single crossing stay single? — has the answer yes. The answer to the question nobody put is better: the crossing field being tracked across the tilt is a constant, and a basis chosen by the plane of the field could never have suggested it.

Still open: the limit of zero quantum defect #

The obvious open question is the quantum defect, which is now the only parameter left that can move anything. The two-state estimates and the exact spectrum are set by the same gaps, and as the defect goes to zero those gaps close and the l degeneracy returns. Whether the estimated count falls to meet the exact one, or the exact spectrum acquires features it does not have at this defect, is one sweep of a single parameter — and it is the only remaining way the two pictures could be reconciled rather than merely compared.

The nearer question is what the direction-independence does to the earlier results about the tilt. Three calculations counted events as functions of the tilt, and two of them quoted angles to five figures. Those angles are exact properties of angular integrals and remain so; what they are properties of is a set of five functions rather than a shell. Re-deriving the four angles in the nine-function basis would say whether they survive as properties of anything at all.