A cubic site is blind to a degree-two error

Better counting, and the wrong model set two disorder models against each other with a smooth error in the data, the kind an uncorrected absorption leaves: a scale that varies over the sphere of reflection directions like the second Legendre polynomial about a direction along no axis of the cell. With the random error counted away, perfect counting chose the wrong model for one pair in eighteen at sixty-four atoms, three at a hundred and twenty-eight and four at two hundred and fifty-six. That essay measured how many went wrong and not which, and it proposed a way to tell in advance. The pairs at risk should be those whose difference correlates with the error surface, and that correlation can be computed from the two models’ ideal structures before any data are refined.

The computation here makes that prediction for all ninety pairs and sizes, and it predicts none of the eight wrong verdicts. Every pair that goes wrong has a correlation that, on its own, would have pushed the verdict the right way or left it alone. What turns the verdict is how much of the error each model’s refinement can absorb, and in every wrong case it is the wrong model, with fewer parameters, that absorbs more.

The same computation does predict something exactly, and in advance, from the site’s symmetry alone. Of the eighteen pairs, six sit at a cubic site, and for those the error’s correlation with the models’ difference is zero to fifteen decimal places, and so is what the right model absorbs. A cubic site cannot see a smooth error of degree two at all, because a cubic group keeps no harmonic of degree two, and a degree-four error, of which it keeps one, reaches the cubic pairs at once and turns one of them.

The prediction that was proposed #

Write $F_A$ for the true structure’s amplitudes on each reflection, from model A, and $e$ for the smooth error the data carry, $e_r = \varepsilon g_r F_{A,r}$ with $g$ the error surface. Refined against error-free data, the wrong model B leaves a residual $D_r$ on each reflection, the part of the two models’ difference it cannot absorb. To first order in the error, with each model’s parameters held where the error-free refinement put them, the difference of the two sums of squares is

$$S_B - S_A \approx \sum_r D_r^2 + 2 \sum_r D_r e_r ,$$

and perfect counting decides for the wrong model when that is negative. So the prediction is that a pair goes wrong when the cross term $2\sum D e$ is negative and larger than the models’ difference $\sum D^2$. A naive version replaces $D$ by the unrefined difference $F_A - F_B$, which is what “the two models’ ideal structures” says if it is read literally.

The picture at the head of this essay is the result, and it has two halves. Under the degree-two error the eight wrong verdicts are two orthorhombic and six tetragonal, and none is predicted. Ninety cases is the eighteen pairs drawn from the orientations a disorder needs at each of five molecule sizes, the sizes the molecule size that hides a disorder showed are where the difference between models thins out. The refined version of the prediction calls all eight right; the naive version calls all eight right too. The cross terms of the wrong pairs are small and mostly positive. The tetragonal pair that goes wrong at sixty-four atoms has a models’ difference of 2.09 and a cross term of $+0.31$, in units where the error itself is 52.6, and the formula says its verdict is safe by 2.40. Perfect counting decides it the wrong way by 0.92.

What actually turns a verdict #

The first-order formula holds each model’s parameters fixed, and that is what it gets wrong. Each refinement moves its parameters to follow the error, and the full balance is

$$S_B - S_A = \sum_r D_r^2 + 2 \sum_r D_r e_r - \bigl(\text{absorbed by B} - \text{absorbed by A}\bigr),$$

where what a model absorbs is how much less its refined sum of squares is than the error it was handed. The identity is exact by construction; the question is which term decides.

The figure answers it. For each pair and size it plots how much more of the error the wrong model absorbs than the right one, divided by the difference plus the cross term. A verdict goes wrong exactly when the ratio passes one, and all eight wrong verdicts are there, at the orthorhombic and tetragonal sites, with ratios from just over one to over twenty. In the orthorhombic pair that goes wrong at a hundred and twenty-eight atoms, the right model absorbs 6.6 of an error of 378 and the wrong model 37.5, more than the models’ difference of 29.2 can survive.

That is the part no correlation computed before refinement can see, and it is not explained by size either. In every one of the eight, the wrong model has fewer free parameters than the right one: 16 against 19, 12 against 16, 19 against 23, 11 against 23. It absorbs more of the error with less freedom. What a model absorbs is the part of the error that lies along the directions its parameters can move the amplitudes, and a model with fewer parameters can still have them pointed better. What refinement takes back from a wrong model found the wrong model taking back most of the models’ difference by moving its atoms along their loci. Here the same freedom is spent on the error, and it is the direction of that freedom, not its amount, that decides.

The pairs that cannot go wrong #

The cubic pairs stand apart in every column of the census. Under the degree-two error their cross terms are below $10^{-13}$ of the error and the right model absorbs nothing to the same precision. None of them goes wrong at any size. That is not luck, and the reason can be stated before any refinement is run.

A disorder model’s averaged structure has the site’s symmetry, as the symmetry of an average showed: whatever orientations the molecule takes, the average over them is carried to itself by the site’s operations. So its amplitudes, on a reflection list that is itself symmetric, are unchanged when the reflections are moved by the site’s Laue group, and so are the derivatives of the amplitudes with respect to every refinable parameter, and so is the difference between two models. Everything a refinement at such a position can produce lives among the functions on the reflection list that the Laue group keeps.

The error does not have to. It is the true amplitudes multiplied by a smooth surface over the sphere of directions, and the surface is a sum of spherical harmonics of one degree. Its correlation with anything the site’s models produce is its correlation with the part of the surface the Laue group keeps, which is the surface averaged over the group. For a degree-two surface and a cubic group, that average is identically zero. There is no quadratic form in three variables that is unchanged by every cubic rotation except a multiple of $x^2 + y^2 + z^2$, and the degree-two harmonics are exactly the quadratic forms with no such part. So a cubic site’s models cannot correlate with a degree-two error, and the refinement cannot absorb any of it. Both terms that decide a verdict are zero, and the verdict is the one the models’ difference gives, whatever the error’s size and whatever its direction.

The count of kept harmonics is a character average, the same arithmetic as the shell that splits into kinds and what a group does to a function. The degree-$\ell$ harmonics have character $\sin((\ell + \tfrac12)\theta)/\sin(\theta/2)$ at a rotation by $\theta$, and averaging over the site’s rotations gives the number of harmonics its group leaves unchanged. For degree two the orthorhombic site keeps two, the tetragonal site one and the cubic site none. For degree four they keep three, two and one.

How much of the error each site can see #

The same argument says how much of a degree-two error the other sites see, and the answer depends on the error’s direction as well as the site. A degree-two surface about a direction $a$ is the traceless quadratic form $(3aa^T - I)/2$. Averaging it over a site’s Laue group keeps the part a model there can follow, and the share of the surface that survives is the squared size of that average over the squared size of the whole.

For the direction the earlier essay chose, $(1, 2, 3)$, along no symmetry element of the cell, the orthorhombic site keeps exactly a quarter of the surface: its averaging leaves the diagonal of the form and removes the off-diagonal part, and for this direction the diagonal is a quarter of the whole. The tetragonal site keeps $169/784$, about 0.216, the part that distinguishes the four-fold axis from the plane across it. The cubic site keeps nothing. Those are the fractions of the error that can correlate with a pair’s difference or be absorbed by a refinement, and the census’s cross terms are of the sizes they suggest. At sixty-four atoms the largest correlation between the error and a pair’s difference is 0.19 of the error’s size at the orthorhombic site, 0.08 at the tetragonal site and $6 \times 10^{-15}$ at the cubic one. The first two are of the order the visible shares allow, once the pair’s difference has taken its own share of the reflections; the third is rounding.

The direction matters as much as the site. An error surface about the four-fold axis is entirely visible to a tetragonal site, since the averaging then changes nothing. An error about a body diagonal of the cell, $(1, 1, 1)$, is entirely invisible to an orthorhombic site, because its diagonal part is a multiple of the identity, which the traceless surface does not contain. So a needle-shaped orthorhombic crystal elongated along a body diagonal of its cell would leave its disorder decisions untouched by the uncorrected absorption, and the same crystal elongated along an axis would not. The census here uses a single direction and does not test that second prediction, but it follows from the same two-line average as the first.

A degree-four error reaches the cubic site #

That count makes a prediction that can fail. An error of degree four should reach the cubic pairs, since the cubic group keeps one harmonic of degree four, the one whose sign follows $x^4 + y^4 + z^4$. So the census was run again with the error surface the fourth Legendre polynomial about the same general direction, at sixty-four, a hundred and twenty-eight and two hundred and fifty-six atoms.

It does. Under the degree-four error the cubic pairs’ correlations with the error rise from the level of rounding to between two thousandths and six hundredths of the error’s size, the same order as at the other sites, and the right model absorbs a measurable share. One cubic pair whose models differ goes wrong at sixty-four atoms. Two other cubic verdicts come out wrong at the larger sizes, but they belong to the pair whose two models fit every structure the other fits, the pair no data can decide, and for it a wrong verdict only means the more flexible model followed the error further.

So the site’s symmetry gives a sharp, testable rule for when a smooth error can matter. A smooth error of degree $\ell$ can mislead a decision between two disorder models at a site only if the site’s Laue group keeps a harmonic of degree $\ell$. A cubic crystal is immune to the elongated shape that gives a degree-two absorption error and exposed to the faceted shape that gives a degree-four one. A tetragonal or orthorhombic crystal is exposed to both.

The rule says which pairs can go wrong, and not which will. Among the pairs at sites that keep the error’s harmonics, which ones turn depends on how the models’ parameters move the amplitudes along that harmonic, which is the absorption the first part of this essay found decisive.

What the account rests on #

A symmetric reflection list. The cancellation at the cubic site is exact because the reflection list is a full sphere of reflections to a resolution limit, which the cubic group maps to itself. A real data set with missing wedges is not symmetric, and the correlation would then be small rather than zero. How small depends on what is missing, and a crystal measured on one axis with a large blind region could lose much of the protection.

The averaged structure carries the site’s symmetry. That is true of every model here, because each is an orientational average over the site’s group, which is the setting the occupancy does not name the disorder built. A model refined with its symmetry relaxed, a lower-symmetry approximation of a disordered site, would not be protected.

The wrong model’s refit wanders. At the cubic site the wrong model’s refinement against error-bearing data can still end a little above or below where its error-free refinement ended, by up to 0.7 in units where the error is twenty. That is not absorption of the error, which first order forbids. It is the refinement settling into a slightly different minimum from its several starts. The test of immunity is made on the cross term and on the right model’s absorption, where first order is exact, and both are zero to rounding.

Perfect counting. Every verdict here is the sign of the difference in fit with no random error, the limit the earlier essay identified as what better counting approaches, and the limit that measuring every reflection several times over moves an experiment towards without ever reaching. With random error added, an immune site keeps its immunity, since the random error adds nothing that correlates with the models’ difference, and the other sites keep their exposure.

Who worked out the pieces #

The correction of absorption and other smooth errors by a sum of spherical harmonics in the reflection direction is Blessing’s empirical method of 1995, and it is the correction the earlier essay refined. The count of harmonics a point group keeps is the character theory of the rotation group restricted to a finite subgroup, which Bethe used in 1929 for the splitting of atomic levels in a crystal field. A cubic crystal field has no term of degree two and begins at degree four, for exactly the reason a cubic site cannot see a degree-two error. Applying it to the systematic errors of a disorder refinement is what this essay adds.

What the account must refuse #

The two refusals are the two natural predictions that fail. The correlation before refinement was the earlier essay’s proposal, and at sixty-four atoms the one pair that goes wrong is called right by both forms of it. The count of parameters is the obvious explanation of why a wrong model fits an error better, and it points the wrong way in every case. The tests the account passes are the ones the symmetry argument makes: a zero where the harmonic count is zero, and a non-zero, with a wrong verdict, where it is one.

Still open: the direction of a model’s freedom #

The decisive quantity at the sites that keep the error’s harmonics is how much of the error each model’s refinement can follow, and that is the projection of the error onto the directions in which the model’s parameters move its amplitudes. Those directions are the columns of the refinement’s Jacobian at the error-free solution, and they can be computed before any data exist, from the two models and the reflection list alone. A prediction built from them, the error’s projection onto the wrong model’s directions minus its projection onto the right model’s, would be a first-order account that holds the parameters free rather than fixed. Whether it catches the eight wrong verdicts, and whether it can be stated as a rule about which atoms of a model sit where, is the computation the symmetry rule leaves.