Solving from the vector set

The map that needs no phases establishes what an experiment can always compute: the transform of the intensities with every phase set to zero, whose peaks are not atoms but the vectors between them. Where symmetry stacks the vectors establishes that symmetry piles those vectors onto sections, which is what makes a large map searchable.

Neither says how anybody gets a structure back out. The answer is older and considerably simpler than the direct methods that replaced it, and it is a piece of arithmetic that can be watched happening.

The map is the structure, convolved with its own inverse #

The property that makes the method work is one line and it is worth stating in the form that makes the method obvious.

The Patterson function is the autocorrelation of the electron density: P(u) = ∫ ρ® ρ(r + u) dr. Written as a convolution, it is ρ convolved with ρ̅, where ρ̅ is the density inverted through the origin. And a convolution with a set of point atoms is a sum of copies: the Patterson map is a copy of the structure laid over every atom of the inverted structure, all superposed.

For n atoms that is n copies of an n-atom structure, so n² peaks, of which n sit on top of one another at the origin — which is why the origin peak weighs n and every other peak weighs one, a count the vector set makes exactly.

The consequence is the method. If one interatomic vector u is known, then the map contains a copy of the structure with its origin at 0 and another with its origin at u. Shift the whole map by −u and the second copy moves to the origin, where the first already is. Both maps now contain a copy of the structure at the same place — and elsewhere they contain copies at different places, because the other copies moved too.

Reading a small map by hand #

Before the machinery, it is worth seeing that a small map can be read directly, because that is what makes the superposition step believable rather than magical.

A structure of four atoms has sixteen interatomic vectors, of which four are zero and land on the origin, and the remaining twelve come in six ± pairs. So the map has an origin peak of weight four and six pairs of peaks of weight one — and the arrangement of those six pairs is exactly the shape of the four-atom structure seen from each of its atoms in turn, superposed.

The reader’s task, doing this by hand, is to find four positions whose mutual differences reproduce the twelve. That is a jigsaw with a small number of pieces and it was routinely done with a pencil. The superposition method is that jigsaw mechanised: instead of trying arrangements, it uses one correct piece — one vector — and lets the map cancel the rest.

Take the minimum, and the mismatches vanish #

The minimum function is M® = min(P®, P(r − u)), and it is the crudest possible way of intersecting two maps: keep at each point whichever of the two is smaller. Where both maps have a peak, the minimum has a peak. Where only one does, the minimum has whatever the other map’s background is, which is nearly nothing.

So the minimum function contains the copies the two maps share and not the rest. If u is a genuine interatomic vector between atoms A and B, the shared copies are the one at the origin and — because the map contains the inverted structure convolved in — one more, the inverse through the midpoint of A and B.

That is the whole method, published by Martin Buerger in the 1950s and used for two decades before direct methods made it unnecessary for small structures. Its only input is one vector.

Which vector, and how the choice can go wrong #

The vector is found rather than supplied: take the strongest peak of the map that is not the origin. That is exactly what a crystallographer does with a heavy-atom structure, where the strongest non-origin peak is the vector between the two heaviest atoms, because a Patterson peak’s weight is the product of the two atoms’ scattering powers.

There is a trap in the “not the origin” part, and it is quiet. The origin peak is the tallest thing in any Patterson map and it is wide: its skirt covers several grid steps, so a candidate taken one or two steps away from the origin is not an interatomic vector at all but the same peak seen again. Superposing by such a vector shifts the map by almost nothing, the minimum is nearly the map itself, and what comes back is a beautifully self-consistent picture of the Patterson function rather than of the structure.

The exclusion radius is therefore a fraction of the cell rather than a few samples — and this is the sort of parameter that has to be named where it is used, since a figure drawn with a smaller one produces something that looks like a result and is not.

What comes back #

Four atoms in, four atoms out, from a map computed with no phases whatever. The recovered arrangement sits at an arbitrary place in the cell, which is not a defect of the method but a statement about what the data contain: the Patterson function of a structure and of the same structure translated are identical, so nothing computed from it can say where the structure is.

What also comes back, unavoidably, is the inverse. The vector set of a structure and of its mirror image are identical — every vector r_i − r_j becomes r_j − r_i, and the set contains both already — so no computation on the vector set alone can prefer one hand over the other. That is Friedel’s law in real space, and it is the same ambiguity the law that hides handedness breaks with a different measurement entirely.

Using symmetry instead of a heavy atom #

The superposition vector does not have to come from a heavy atom. If the space group has symmetry, the vectors between symmetry-related atoms are constrained to lie in particular places — the Harker sections where symmetry stacks the vectors computes — and a peak on a Harker section is a vector whose form is known in advance.

For a twofold screw along b, the vector between an atom at (x, y, z) and its screw image is (2x, ½, 2z): every such vector lies on the section v = ½, and reading a peak’s position there gives x and z immediately. The atom’s own position is then known up to the origin choice, and a superposition on it does the rest.

That variant is the symmetry minimum function, and it is what made the method practical for structures with no heavy atom at all. In a group with several symmetry operations there are several Harker sections, each giving a partial constraint, and taking the minimum over all of them at once concentrates the answer very sharply.

What a second superposition does #

One superposition leaves two arrangements standing, and a second one removes one of them. Which one it removes is the interesting part, and it is not what the textbook sentence says.

After the first superposition the map holds the structure S at the origin and its inverse S̅ through the midpoint of the vector used, sharing some of their peaks. Laying a third copy of the map down at a second interatomic vector v and taking the minimum of all three keeps what every copy has. S survives if v is a vector of S; S̅ survives if v is a vector of S̅; and every vector of one is a vector of the other, reversed, so the choice of v decides which of the two comes through and nothing decides the choice of v.

So the usual phrasing — that a second superposition removes the inverse — asserts more than the arithmetic allows, and it contradicts the paragraph above it. A vector set is identical for a structure and for its mirror image; nothing computed from it can prefer one; so a sieve that leaves exactly one arrangement standing has no way of reporting which arrangement that is. Running the second superposition over every candidate vector the map offers, and keeping whichever leaves the most peaks standing, gives four peaks and one complete arrangement — and two different candidates of equal standing keep different arrangements. The hand is still open at the end, exactly as it was at the beginning.

What the second step really buys is a smaller answer. Six peaks holding two overlapping arrangements is a puzzle; four peaks holding one is a structure, up to a translation and up to the hand. The classical practice was to keep going until the map stopped changing, each step one shift and one pointwise minimum, which is why the method was practical on the computers of the 1950s and on none of the arithmetic that preceded them.

The limit of that process has a name — the map converges to the structure convolved with the intersection of all the shifted copies — and the useful way to think of it is as a sieve: every superposition throws away the peaks that cannot be atoms, and what is left when everything that can be thrown away has been is an arrangement, with the choice between it and its mirror image left to a measurement of a different kind.

Why it stopped being used, and where it still is #

Direct methods — which recover phases from statistical relations among the intensities themselves — solved small structures faster and without a heavy atom, and by the 1980s a small-molecule structure was solved by pressing a button. The superposition method went from being the technique to being a historical note.

Two places kept it. Protein crystallography never got direct methods to work at ordinary resolution, so heavy-atom methods, and with them the Patterson map, remained central for decades — the modern route via anomalous scattering is a descendant with the same shape: find a few marker atoms first, then everything else. And structures from powder data, where the phase problem is compounded by what a powder pattern loses, still use Patterson methods because the alternatives need better data than a powder gives.

There is also a conceptual reason to keep it in view. The method is the only route from measurement to structure that can be drawn — three pictures, and the argument is complete. Direct methods are statistics about phase relations, and no figure shows them happening.

The heavy-atom method, which is the same idea #

There is a second classical route from a Patterson map to a structure, and it is worth putting beside the first because the two are usually taught as alternatives and are the same observation used differently.

If a structure contains one atom much heavier than the rest, its contribution dominates the structure factors, so the phases of the reflections are approximately those the heavy atom alone would give. Find the heavy atom — from the Patterson map, whose strongest non-origin peak is the heavy–heavy vector — compute phases from it, and use those phases with the measured amplitudes to make a Fourier map. The heavy atom appears, and so do the light ones, weakly, because the phases are approximately right.

That is the heavy-atom method, and its dependence on the Patterson map is exactly the superposition method’s: both start by reading one vector off the map. Where they differ is what they do next — one works in real space with minima, the other in reciprocal space with phases — and the second scales better, which is why it is the ancestor of what protein crystallography does now.

Both are defeated by the same thing. If no atom is heavy enough, the strongest peak of the map is not distinguishable from its neighbours, there is no vector to be confident about, and neither method has a place to start. The threshold is usually stated as the heavy atom contributing a comparable amount of scattering to all the light ones together, which for an organic structure means something around the atomic number of bromine.

What the method assumes about the structure #

Two assumptions are buried in “take the strongest non-origin peak”, and both fail in recognisable circumstances.

That the strongest peak is a single vector. In a structure with several equal atoms the map’s peaks overlap: a peak may be two or three coincident vectors, and superposing on it superposes on several shifts at once, which cuts away more than it should. The classical symptom is a minimum function with too few peaks, and the classical remedy is to try the second and third strongest peaks as well and compare.

That the peak is resolved from the origin. Discussed above as an exclusion radius, and it is worth restating as a property of the structure rather than of the code: a structure whose atoms are all close together has all its vectors close to the origin, sitting inside the origin peak’s skirt, and none of them can be read at all. Layer structures and stacked aromatic systems are the awkward cases.

Both failures are silent in the sense this site cares about: the method returns a map with peaks in it, and nothing about the result announces that the input vector was wrong. What catches it is the check that the recovered peaks are mutually consistent — the vectors between them must themselves be peaks of the original Patterson map, and if they are not, the answer is not a structure.

What is exact and what is not #

Four claims, in decreasing order of warrant.

The peak positions of the map are exactly the interatomic vectors. This is checked elsewhere on this site and holds here: the peaks of the computed map are matched against the vector set computed by subtracting positions, and every strong one is a vector.

The minimum function’s construction is exact. A pointwise minimum of two arrays is not an approximation to anything.

The recovery is a measurement. Four of four atoms is what this structure gives at this resolution with this superposition vector. A structure with atoms closer together than the map’s resolution would give fewer, and the figure reports what it recovered rather than asserting that the method always works.

The resolution is the map’s, not the structure’s. The map is computed from reflections out to a finite range and sampled on a finite grid, so its peaks have a width of order the reciprocal of that range. Two atoms closer than that width give one peak, and no amount of superposition separates them. Every number here is a property of the computation as much as of the structure.

Where the ladder goes next #

The Patterson anchor now has three rungs: the map an experiment can always compute, the sections symmetry stacks its vectors on, and the route from the map back to a structure. Two rungs are visible above.

The anomalous difference Patterson. Feed the map the differences between Friedel pairs rather than the intensities, and what comes out is the vector set of the anomalous scatterers alone — a handful of atoms rather than thousands. That is the point at which this anchor meets the law that hides handedness, and it is how a modern protein structure begins.

Direct methods, honestly. The statistical relations among intensities that replaced all of this are a different kind of argument — probabilistic rather than decidable — and putting them on a site whose whole proposition is exactness would take some care. The interesting question is what they are relations about, and the answer is the same vector set seen a third way.

Where the superposition survives #

The method is described here as classical, and it is — and it is also inside the programs that replaced it, which is worth knowing because a reader might otherwise take the account for history.

Modern structure solution for the difficult cases is dual-space recycling: alternate between a map and a set of amplitudes, imposing what is known in each — atomicity in real space, the measured amplitudes in reciprocal space — and iterate. The procedure needs a starting point, and a random one converges rarely.

A Patterson superposition is the standard way to make a good one. Compute the map, pick a strong non-origin peak, form the minimum function, take its highest peaks as candidate atoms, and hand those to the recycling. The starting set is then not random: it already satisfies the vector set, which is a real constraint from the data.

That is how the heavy-atom substructures of protein crystallography are found. The difference maps carry a few dozen sites among tens of thousands of atoms, direct methods alone converge poorly at that size, and the combination — superposition to start, dual-space recycling to refine — is what the standard programs run.

So the method’s status changed rather than ending. It stopped being a way of solving a structure and became a way of starting one, and the arithmetic in it is unchanged: a map, a shift, a pointwise minimum.

And the reason it works there is the reason it worked then. The Patterson map is the one thing an experiment always supplies, it needs no phases, and it is a hard constraint rather than a statistical one — so a starting set drawn from it is standing on the measurement rather than on a guess.