Consistently wrong is not a limit

The exact five-level spectrum showed that a count of avoided crossings, built up over several calculations, was a count of estimates rather than of features — and named the one thing that could still reconcile the two pictures.

the gaps here are set by the quantum defect and the couplings by the field, so there is a ratio that governs. Sweeping the defect from its present value towards zero, where the l degeneracy closes and the two-state gaps vanish, and watching whether the exact count rises to meet the estimated one, would say whether the two-state picture is a limit this problem is far from or a description of a different problem entirely.

It is a description of a different problem. The sweep says so, and it says so in a way that is more useful than the count: the two pictures do not fail to converge because the defect is too large. They fail to converge because the defect cancels.

Nothing moves #

Swept from a defect of 0.02 down to 0.0002 — two decades, taking the s-to-p gap from 1.08 × 10⁻³ hartree to 9.885 × 10⁻⁶ — the whole shell has one avoided crossing at every value, and its coupled pairs supply eight distinct two-state crossover fields at every value.

The estimates do not even bunch up. They span a factor of 7.58 at the largest defect and 7.08 at the smallest, which is the same scatter arriving at a limit rather than collapsing.

That is the direct answer, and on its own it would be a null result with a number attached. The interesting part is why.

Every ratio settles #

The defect is the only thing separating the s, p and d levels — nothing else in the model distinguishes them — so it sets every energy scale in the problem. Shrinking it shrinks the zero-field gaps, and the field at which a coupling matches a gap shrinks with them, and the field at which the exact spectrum does anything shrinks with them too.

So the quantities that could converge are the ratios, and they converge to constants rather than to each other:

The crossing’s field, as a fraction of the zero-field s-to-p gap, runs 0.04097, 0.04048, 0.04024, 0.04009, 0.04004, 0.04001, 0.04000. Four hundredths, to five figures, approached from above.

The depth of the dip — how far the gap falls below its zero-field value at the crossing — runs 0.02706 down to 0.02356, settling in the fourth figure.

The nearest of the eight estimates, divided by the exact field, runs 0.8263, 0.8668, 0.8871, 0.8994, 0.9034, 0.9055, 0.9067.

The last of those is the answer. If the two-state picture were a limit this problem is far from, that ratio would head towards one. It heads towards 0.9067 and stops.

What a fixed wrong ratio means #

There is a distinction here worth making carefully, because a nine per cent agreement sounds like an approximation working.

An approximation that is becoming right has an error that shrinks with whatever parameter measures how far from the limit the problem is. An approximation that settles at a fixed ratio has an error that does not shrink with anything, and the number it settles at is a property of the two calculations rather than a measure of a distance between them.

The two-state estimate is not the exact field computed badly. It is a different quantity computed exactly. gap / 2d is the field at which the coupling between two states, considered alone, becomes comparable to their separation — a perfectly good question with a perfectly good answer, which the exact five-level calculation said in as many words, and which is the reduction used wherever two levels interact. The sweep now shows there is no regime in which the answer to that question becomes the answer to the other one.

That the nearest of eight lands within nine per cent is what scattering eight numbers over a factor of seven does. The five-level calculation made exactly this argument at one defect, and the sweep generalises it: the nearness is a property of the scatter, and it is stable because everything in the problem scales together.

Why the defect cancels #

The mechanism is short, and it is why a longer sweep would add nothing.

Every energy separating the levels is proportional to the defect, to leading order. Every matrix element of position is a property of the radial functions, which do not depend on the defect at all in this model — the defect enters the energies and nothing else. So a two-state estimate gap / 2d is proportional to the defect, and the exact spectrum’s features are set by comparing a coupling F·d against gaps that are proportional to the defect, so they are proportional to it too.

Dividing one by the other removes it. The problem has a single energy scale and the dimensionless structure sitting on top of it does not know what the scale is.

That also says what would have moved the ratios: something that changes the relative sizes of the gaps, or the matrix elements, rather than their common scale. A second shell would do it, since it brings gaps that scale differently — and the extra degeneracy of a pure Coulomb potential is what a shell has when the defect is gone entirely. A field strong enough to mix the shell with its neighbours would do it. Making the defect smaller does neither — it shrinks the whole picture and leaves its shape alone.

The 0.04097 at the largest defect becoming 0.04000 at the smallest is the residue of the leading order not being the whole of it: the level energies are −Z²/2(n − δ(l))² rather than linear in the defect, so the gaps are not exactly proportional to it. That two per cent of drift is the entire defect dependence of anything dimensionless in the problem, and it is where the ratios converge from.

What this closes #

Five calculations have now been spent on one count, and it is worth setting out what the sequence amounts to.

The first four counted two-state crossover fields as the tilt of the field changed, and found the count rising from three to six, dropping at four exact angles that are arctangents of ratios of the shell’s own integrals. All of that arithmetic is correct and none of it is about the spectrum.

The exact five-level diagonalisation found one avoided crossing where the estimates gave three to six, and nothing at any coincidence angle.

Enlarging the basis to the whole shell showed the exact answer to be independent of the field’s direction altogether, so the tilt sweep the earlier calculations were built on measures nothing.

And the quantum defect, the one remaining parameter, cannot bring the two pictures together either.

What is left standing is narrower than any of the five headlines and it is a real thing to know. The two-state reduction answers a question about a pair of levels; the exact spectrum answers a question about a matrix; and the relationship between the two answers is a fixed ratio with no parameter that tunes it. Anybody using the first as an estimate of the second is off by about a tenth in this problem and by an unknown amount in another, and there is no limit in which that stops being true.

The eight estimates, and where they sit #

It is worth looking at what the eight estimates are, because their stability under the sweep is not obvious from the count alone.

The nine-function shell has three distinct zero-field energies, so its coupled pairs fall into two families: s-to-p and p-to-d. Every pair within a family shares a gap and differs only in its dipole, so its estimate is that gap divided by twice a matrix element — and the matrix elements are properties of the radial and angular functions, which the defect does not touch.

So the estimates within a family are in fixed proportion to each other, exactly, at every defect. What the sweep can move is the proportion between the two families, since the s-to-p and p-to-d gaps do not shrink at quite the same rate: at a defect of 0.02 they are 1.080 × 10⁻³ and 2.014 × 10⁻⁴, and at 0.0002 they are 9.885 × 10⁻⁶ and 1.976 × 10⁻⁶ — a ratio of 5.36 becoming 5.00.

That is the whole of the movement in the span, from 7.58 to 7.08, and it is the same two per cent of curvature that moves the crossing’s fraction from 0.04097 to 0.04000. One residual effect, showing up in three places, and it is the departure of −Z²/2(n − δ)² from being linear in δ.

There is a check available in that reading and it is worth making. If the two families’ estimates really are in fixed internal proportion, the eight fields should fall into two groups whose internal ratios are identical at every defect, with the separation between the groups the only thing that moves.

They do, and the ratios are exact. Grouped by their own gap, the five p-to-d estimates stand in the proportions 1, 2/√3, 4/3, 4/√3 and 4, and the three s-to-p estimates in the proportions 1, 2/√3 and 2 — the same to six decimal places at a defect of 0.02 and at 0.0002. Those are ratios of the shell’s own angular integrals, in the same family as the arctangents the four coincidence angles turned on, and nothing about the screening can touch them.

What was computed, and how #

The shell, the Hamiltonian, the crossing definition and the restriction to gaps open at zero field are those of the whole-shell calculation, unchanged. The direction is held at a tilt of forty-five degrees and an azimuth of thirty, which by that calculation’s own result cannot matter — and holding it fixed rather than averaging is the honest way to use a result of that kind.

The two-state estimates are computed over the whole nine-function shell rather than the five, which is new: every pair of levels not degenerate at zero field and with a non-vanishing dipole between them in the field’s direction, giving eight distinct fields.

Six results are checked numerically. The defect is swept over at least two decades. The exact count is one at every value and the estimated count constant and greater than four, which is the null result stated so it could fail. The crossing’s fraction of the gap tends to four hundredths — checked against a value rather than against a trend, because a trend can be read into anything. The nearest estimate’s ratio settles and settles away from one, which is the finding, and is two conditions in one because either alone would be misleading.

And the refusal: the fraction must move across the sweep. A quantity that was constant by construction would satisfy every convergence test trivially, and the check that it is converging rather than fixed is that it starts at 0.04097 and arrives at 0.04000.

Where the model stops #

One shell. The whole argument for the defect cancelling is that the problem has one energy scale, and a second shell would supply another. What happens then is not a small correction to any of this; it is a different problem, and it is the direction in which the two pictures could in principle be reconciled or separated further.

The defect enters the energies only. Real screening changes the radial functions as well, so the matrix elements would move with it — which would break the exact cancellation and leave a genuine defect dependence. How large is not something this model can say, and it is the first thing a better one would.

Two decades, and the far end is unphysical. A quantum defect of 0.0002 is smaller than any alkali’s — the range a real one occupies runs from four ten-thousandths to about a fifth and the calculation there is a mathematical limit rather than an atom. It is used because the question is about a limit.

And the fixed direction. Holding the field at one direction rests entirely on the whole-shell result that direction does not matter. If that result were wrong, this sweep would be one slice of a family.

And what was being tracked. Computed with the defect removed, the fraction does not stop at four hundredths — its limit is 0.0399865, which the smallest defect here was still falling through — and the minimum it locates is between the lowest level with no angular momentum about the field and the lowest with one unit of it, two levels no element of the field connects. The crossing nothing couples is that calculation; the counts and the settling ratios above stand, and the word “crossing” for this feature does not.

The generalisation #

The transferable point is about how to tell an approximation from a different quantity, and the test is cheaper than it looks.

An approximation has a parameter that measures how good it is, and driving that parameter towards its limit drives the error towards zero. A different quantity does not: it has its own value, and comparing it against the thing it is being mistaken for gives a ratio that settles.

So the test is to find the parameter the approximation is supposed to be controlled by and sweep it. If the ratio settles anywhere other than one, the two things are not the same quantity, and no amount of care in the approximation will make them agree. That is a stronger statement than measuring the error at one value, and it costs a sweep of something the model already takes.

The trap it avoids is the one the tilted-field calculations walked into four times over. A two-state reduction is exact when there are two levels, so it feels like an approximation that becomes exact in a limit — and the limit it becomes exact in is “the other levels are far away”, which is not a limit the defect controls. Naming the parameter that would control an approximation, before sweeping it, is what separates a genuine convergence test from a sweep of whatever was easy to change. Here the defect was the easy parameter and it was the wrong one, and the sweep’s value is in showing that rather than in the number it produced.

There is a second habit worth taking from the way the convergence is checked. The convergence is checked against a value rather than against a trend — four hundredths, to a stated tolerance — and a second check confirms that the quantity moved. A convergence test with only the first half passes on anything constant; with only the second, on anything that drifts. The same pair of failure modes appears where a fitted exponent is quoted without the window it was fitted in.

Who found it, and when #

The quantum-defect treatment of alkali spectra is a century old and the scaling argument above is elementary. The five-level calculation, the nine-level one and this sweep are new arithmetic on that model.

The question arrived with the right framing — a limit this problem is far from, or a description of a different problem entirely — which is the correct disjunction, and it named a parameter to sweep. That it turned out to be the wrong parameter is what the sweep is for.

Still open: what 0.04000 is #

The obvious open question is the second shell, which is the only remaining thing that can break the cancellation. Adding n = 4 gives gaps that scale differently from the n = 3 ones and matrix elements between shells that the defect does not touch, so the dimensionless structure would acquire a real parameter for the first time. Whether the two pictures then converge, diverge or stay a fixed distance apart is a question with three interesting answers, and all it needs is a larger position matrix and the same diagonalisation.

The nearer question is what 0.04000 is. It is a limit reached to five figures from a calculation with no fitted numbers in it, which is the shape of a quantity with a closed form — a ratio of two of the shell’s own angular integrals, most likely, in the same family as the four coincidence angles. Deriving it rather than measuring it would be the first exact statement made about the exact spectrum rather than about the estimates, and everything needed is already computed.