The map that needs no phases

The phase problem is the central obstruction of structure determination: a diffraction experiment measures |F| and destroys the phase of F, and the electron density needs both. Without phases there is no map, and without a map there are no atoms.

There is one map that survives. Take the intensities — which are measured — and transform them with every phase set to zero. That is a perfectly well-defined function, it needs nothing that was not measured, and in 1934 Arthur Lindo Patterson worked out what it is: the autocorrelation of the electron density.

Its peaks are not atoms. They are the vectors between atoms, every ordered pair, all brought to a common origin. A structure of n atoms gives n² peaks, of which n pile up at the origin. That is a much less convenient object than a structure — and it is computable from a measurement, which the structure is not.

Two definitions, one object #

The Patterson function can be defined two ways, and their agreement is the theorem.

As a transform. P(u) is the Fourier series with coefficients |F(h)|² and all phases zero. Every term is measured; nothing is assumed.

As an autocorrelation. P(u) is the integral of ρ(x)·ρ(x + u) over the cell — the overlap of the density with a copy of itself displaced by u. It is large where the displacement carries atoms onto atoms.

That the two are the same function is the convolution theorem: the transform of a product of a function with its own conjugate is the correlation. And the second form says immediately what the peaks are, since ρ is concentrated at atoms: the overlap is large exactly when u is a vector from one atom to another.

Why n² is a problem #

The count is the whole difficulty. Four atoms give thirteen distinct peaks; ten atoms give ninety-one; a hundred atoms give nine thousand nine hundred and one, in a cell that has room for a hundred.

The peaks also become weaker as they multiply, since the total scattering is fixed: the same intensity spread over more peaks makes each one smaller relative to the origin peak, which keeps growing. So a Patterson map of anything but a very small structure is a continuous blur. The peaks overlap, individual vectors cannot be picked out, and the information is present but not accessible. Interpreting one is not reading it — it is solving a puzzle: propose a structure, compute its vector set, compare.

Two escapes exist and both are exploited.

Make some atoms much heavier than the rest. The height of a Patterson peak goes as the product of the two atomic numbers, so a mercury–mercury vector in a protein stands twenty times above a carbon–carbon one. The map is then dominated by a handful of peaks, which can be read. That is the heavy-atom method, and it was how nearly every structure before the 1970s was solved.

Use the symmetry. A space group forces the vectors between symmetry-related atoms onto planes, so the map’s information concentrates where it can be searched. That is the Harker sections, and it is the next rung.

It is always centrosymmetric, and that is a loss #

Every Patterson map has a centre of symmetry, whatever the crystal has.

The reason is one line: for every pair of atoms A and B, the vector set contains u = B − A and also u = A − B, which is its negative. So the set is closed under u → −u, exactly, with matching weights. There is no way to build a structure whose vector set lacks that symmetry.

That is the real-space form of a fact this site has already made in reciprocal space. Friedel’s law says |F(h)| = |F(−h)|, so a diffraction pattern is centrosymmetric whether or not the crystal is; the Patterson map is the transform of those intensities and inherits the property. Two statements of one loss, in the two spaces, and the loss is exactly what makes the eleven Laue classes rather than thirty-two the thing a measurement sees.

The consequence is not only philosophical. A structure and its mirror image give identical Patterson maps, so a Patterson-based solution comes with a handedness that has to be decided by other means — and getting it wrong produces a structure that refines perfectly and is the wrong enantiomer, which has happened often enough in the pharmaceutical literature to be a standing caution.

The origin peak, and what to do about it #

The peak at the origin is the sum of every atom paired with itself. Its height is Σ Z², which for anything real is many times the tallest genuine peak, and it carries no information: it is there for any structure whatever.

Removing it is standard and slightly delicate. Subtracting the mean of |F|² takes out the origin peak’s contribution, at the cost of introducing a negative ring around it — the truncation ripple that every Fourier series has, which this site has met in the phase problem’s figures and in what a powder pattern loses. Every real Patterson map is computed with the origin peak removed and a sharpening function applied, and every one of those operations is a trade of one artefact for another.

The origin peak has one honest use: it is a check. Its height is a known function of the composition, so a map whose origin peak is the wrong height has a scaling error, and that is worth knowing before anything is interpreted.

What the weights say, and what they hide #

A peak’s height in a Patterson map is not one number but a sum, and knowing what goes into it is most of what interpreting one requires.

The height at u is Σ Z_i Z_j over every pair of atoms separated by u. Three consequences follow. Heavy atoms dominate, as noted, since the weight is a product. Vectors that coincide add, so a structure with several pairs at the same separation shows one tall peak rather than several ordinary ones — and reading that as a single heavy pair is a classic error. And a peak’s height says nothing about which atoms produced it, so a tall peak is ambiguous between one heavy pair and several light ones.

The figures on this page make the multiplicity visible by drawing a peak’s weight as its size, which a real map does by making it taller. In the four-atom example the only multiple peak is the origin, with weight four; in a real structure with any symmetry at all, whole classes of vectors coincide, and those coincidences are exactly what the next rung is about.

What can be read off one, in practice #

Three things, in increasing order of ambition.

Whether there is a heavy atom, and where. A single strong non-origin peak means two heavy atoms related by a symmetry operation, and its position gives their coordinates directly — which is the whole of the heavy-atom method and is still how most small structures with a metal in them are solved.

Whether there is local symmetry. Rotating the map and measuring the overlap finds rotations that map the vector set onto itself, which are the rotations relating copies of a molecule. That is the self-rotation function, and it works before anything about the structure is known.

Whether the crystal is twinned, and how. A twin superposes two orientations of one structure, so its vector set is the union of two vector sets plus the cross terms — and the extra peaks sit where the twin law puts them. A Patterson map is one of the standard places merohedral twinning announces itself, before any refinement has had a chance to absorb it into the wrong structure.

Whether a proposed structure is right. Compute the vector set of a model and compare it against the map. That comparison is the basis of molecular replacement, in which a known structure is rotated and translated until its Patterson matches the measured one — and it is now how the large majority of protein structures are solved, because a related structure is usually already known.

The symmetry of the map, which is not the symmetry of the crystal #

A Patterson map has a space group of its own, it is computable from the crystal’s, and it is never the same one.

Two things happen on the way from a structure to its vector set. The map acquires a centre of symmetry it may not have started with, as above. And every translation part is discarded: if an operation carries atom A to atom B, the vector between them depends on the operation’s rotation part and on where A is, but the set of all such vectors is unchanged by sliding the whole structure — so a screw axis and a plain rotation about the same direction produce vector sets with the same symmetry, and a glide and a mirror likewise.

The result is that the map’s symmetry is the crystal’s point group, together with an inversion, realised with no screws and no glides. P2₁ gives a map of symmetry P2; P2₁/c gives P2/m; Pna2₁ gives Pmmm. Those are the Laue classes written as space groups, and the Tables list one for every entry under the name Patterson symmetry for exactly this reason.

Two consequences follow and both are practical. The asymmetric unit of the map is larger than the crystal’s, sometimes by a factor of two, so a search over the map has more ground to cover than the structure’s own asymmetric unit suggests. And the map cannot distinguish groups that share a Patterson symmetry — which is the same statement as the one about absences, seen in real space: everything the vector set knows about the group is the point group and the lattice, and the translation parts are recoverable only from where the peaks sit rather than from the symmetry of the map itself.

A series with no phases is a series of cosines #

The definition sets every phase to zero, and that is a stronger statement about the arithmetic than it first appears.

A Fourier series with real, non-negative coefficients and zero phases is a sum of cosines: the sine terms of h and of −h cancel in pairs, because |F(h)|² = |F(−h)|² whatever the structure. So the Patterson is real and even by construction, before any property of any crystal is used, and computing it requires half the terms and none of the complex arithmetic that an electron-density synthesis needs.

That mattered enormously in 1934 and it still explains why the map was tried first. A density map needs phases and cannot be computed; a Patterson map needs nothing but the measured numbers and a table of cosines, and in an era of Beevers–Lipson strips the difference between a real series and a complex one was the difference between a week and a fortnight.

It also removes a whole class of error. There is no phase to get wrong, no origin convention to violate, and no sign to choose: two people computing a Patterson from the same intensities get the same map to the precision of their arithmetic. Every disagreement about a Patterson map is a disagreement about how to read it, which is why the literature of the method is a literature of interpretation rather than of computation.

One caution goes with that, because the absence of phases is not the absence of assumptions. The coefficients are |F|² on whatever scale the measurement produced, and a map computed from unscaled intensities is the right map multiplied by a constant — harmless. What is not harmless is the set of reflections included: leaving out the strong low-angle terms changes the map’s shape rather than its scale, and a map computed to a different resolution limit is a different map with different peak widths. Two Pattersons of one crystal agree only when they were computed over the same reflections, which is a condition on the data rather than on the arithmetic.

Who Patterson was, and what he was trying to do #

Arthur Lindo Patterson published the function in 1934, in a paper about determining structures from powder data, and the mathematics was not new to him — the convolution theorem was standard, and he had learned Fourier methods working with Bragg.

What was new was the recognition that the unphased series is worth computing at all. The prevailing attitude was that intensities without phases are useless, and the correct response was that they are not useless, they are the transform of something else — a something else that is a poor description of a structure and a perfectly good measurement of it.

Two details of the reception are worth keeping. The first is that Patterson’s own paper is careful to say what the function is not — it is not the structure, and he says so in the abstract — which is a discipline the subject did not always maintain afterwards. The second is that the interpretation problem was recognised immediately: within two years there were papers on how to read a vector set back into a structure, and the general problem is still not solved.

The method’s high period was the two decades after 1945, when the heavy-atom method and Patterson interpretation between them solved essentially every structure that was solved. Direct methods displaced it for small molecules in the 1970s, and molecular replacement — which is Patterson matching under another name — displaced it again for large ones. So the function has been continuously in use for ninety years while being repeatedly described as superseded, which is a good sign about a piece of mathematics.

Where the exactness stops #

Three limits, and the first is about what these figures are.

The map here is exact and small. Four atoms, an 841-term series, and a comparison against a vector list — which is a demonstration of the theorem rather than a structure determination. A real map has thousands of peaks and the interpretation is a search, not a reading. The claim being made is the identity between the two calculations, and it holds at any size; the claim that a map can be read holds only at small ones.

A map is computed in a cell and read modulo it. Every vector is reduced into the cell, so a vector longer than the cell reappears at the other side and can land on top of a short one. In a large cell that is rare; in a small one it is routine, and it is one more way a peak can be a sum of things that have nothing to do with one another.

The series is truncated and the peaks have shape. A finite set of reflections gives peaks of finite width with negative rings around them, so a weak peak sitting in a strong one’s ripple can be missed or invented. Every choice made to combat that — sharpening, origin removal, resolution cutoff — trades one distortion for another, and none of them is neutral.

A vector set is not a structure and sometimes does not determine one. It contains the vector from atom A to atom B and the vector from C to D, with no label saying which is which. Reconstructing the structure from the vector set is a genuinely hard combinatorial problem, and in general it does not have a unique solution — there exist distinct structures with identical vector sets, called homometric, and no amount of care with the map distinguishes them.

Where the ladder goes next #

The way out of the n² problem is symmetry. If a crystal has a screw axis, the vector between an atom and its own symmetry image is confined to a plane — and searching a plane is a different order of task from searching a volume. Those planes are the Harker sections, they are computed from the operation’s matrix in integers, and they are the next rung.