In general, Patterson methods favour a small number of "outstanding" scatterers which dominate the vector map. Typically this is one or more heavy atoms in the presence of light atoms. In fact, the heavy-atom technique for structure solution is rather robust with incomplete and resolution-limited diffraction data such as from macromolecular single-crystals or polycrystalline samples, where limits in attainable resolution (min d-spacing) or the number of intensities suffering from reflection overlap can seriously degrade the quality of the diffraction data.

In fact, the position of one or more heavy atoms can often be solved by hand from the Patterson map. But the identification and analysis of the interatomic vectors by hand becomes increasingly difficult with Patterson maps of lesser quality and structures of higher complexity. Interpreting the cross-word puzzle of interatomic vectors can become cumbersome and may not discriminate sufficiently between atomic positions. In contrast, SHAPE utilises all the possible atomic positions for a given map, and not just the identifiable Patterson peaks. The program SHAPE offers an automatic and objective way to quantify the likelihood of a single atom sitting on a particular position by means of the symmetry minimum function.

Limits of attainable resolution (min d-spacing) will affect both the resolution of atomic peaks in the crystal unit cell as well as peaks in the Patterson cell. The width of Patterson peaks will also increase with increasing thermal motion. Consequently, peak position are less reliable or may be lost altogether. The interpretability of the Patterson map can be improved by either sharpening (Patterson, 1935), origin removal (Karle and Hauptmann, 1964) or maximum entropy image reconstruction techniques (David, 1990).

In the XTAL system, it is strongly recommended to use the "epat" option when calculating the Patterson map with the program FOURR. The "epat" option offers a good compromise between (a) sharpening the Patterson and (b) keeping the Fourier series truncation effect under control.

In the early 1950's, systematic Patterson vector-search techniques for structure solution were developed which were based on the superposition of shifted Patterson maps. The symmetry minimum function (SMF) is a natural extension of these methods (Kraut, 1961; Simpson, 1965). The SMF includes the space group symmetry and uses the entire Patterson map rather than just the identifiable peaks. The SMF can actually be justified on rigid Bayesian and statistical grounds (Bricogne, 1992). The program SHAPE is a computer implementation of the ideas of Kraut (1961) and Simpson (1965) in direct space.

For the applications of SHAPE to structure solution from powder diffraction data, see, for instance, Estermann (1995) and Hofman et al. (1995).

The symmetry minimum function is defined as

          p
SMF(r) = min  (1/m(i)) P(r - C(i)*r)
         i=1

where a trial atomic position r in the crystal cell is ranked by comparing the heights of all unique Harker vectors H(r) = r - C(i)*r, (i=1, ...,p) in the Patterson cell (the origin vector is not included). The symmetry operator of the space group is given by C(i) where C(i)*r = R(i)*r + T(i) is the operator applied to a position r, R(i) is the rotational and T(i) the translational part of the symmetry operator. The multiplicity of the Harker vectors is given by m(i). For every grid point in the input map, the co-ordinate r is calculated and the value SMF(r) derived. The minimum function in the equation for the SMF ensures a continuously low value for any position r unless all Harker vectors H(r) are above background.

The enantiomorph structure and the structure shifted by a permissible origin shift have exactly the same vector set (homometric structures). Consequently, all the vector sets of these strictures are correct solutions of the Patterson map and are present in the SMF. Therefore the maxima in the SMF are single-site solutions, since the different maxima may not relate to the same origin. Therefore the maxima cannot be used directly as input for a structure refinement.

In the next step, a single image will be selected from the Patterson map with the help of the image seeking minimum function.

To retrieve a single image of the structure from the Patterson map, it is necessary to search the Patterson map for vectors between atoms which are not related by symmetry, the so-called cross-vectors. A maxima from the SMF map at position r' is selected as an origin fixing atom. This fixed pivot position r', and its symmetry related copies C(i)*r', are then tested against all other possible atomic positions r in the crystal cell by analysing the height of all cross vectors r - C(i)*r' (i=1, ..., number of symops) with the minimum function:

          n
IMF(r) = min  P(r - C(i)*r')
         i=1

It is named the image seeking minimum function (IMF) because it locates a single image out of all the ambiguous origin-shifted images. One of the two enantiomorphic structures may be eliminated by using a second pivotal position.

The size of the input Patterson map defines the size of the output map. Even, if only a partial map is used, e.g. an asymmetric unit (ASU), the backmapping of vectors outside the ASU is correctly done. However, the ASU of the Patterson map is not necessarily the same as the one of the image seeking minimum function. It is therefore strongly recommended to use the "full" option in the program FOURR in order to work with a complete unit cell.

If the symmetry of the output map does not agree with the symmetry of the space group then it is probably due to an unsuitable grid division. Preferably use multiples of 2, 3, 4, 6 as grid divisions in the program FOURR.

Deconvolution with the symmetry minimum function, Harker vectors

: calculate an |E*F| Patterson map, full cell, add I(000) term
FOURR epat full
grid 48 48 152
layout layer down across
fzero 5332.
end
: deconvolute voxel-wise with the symmetry minimum function
: inspecting Harker sections only
: single-sites
SHAPE
smf
end
: copy the deconvoluted map "smf" onto standard input "map" for
PEKPIK
COPYBDF smf map
: search for maxima in the SMF map
: for a large number of maxima, adjust the parameters in PEKPIK
PEKPIK punch
plimit *2 0.1  20  0.4
end

Deconvolution with the image seeking minimum function, cross vectors

: calculate an |E*F| Patterson map, full cell, add I(000) term
FOURR epat full
grid 48 48 152
layout layer down across
fzero 5332.
end
: select a strong peak from the previous SHAPE run (smf option)
: inspecting all possible cross-vectors for a given atom position
: selects a single image from the Patterson map
SHAPE
imf .7500   .0104   .7434 : pivot selected from PEKPIK output
end
: copy the deconvoluted map "imf" onto standard input "map" for
PEKPIK
COPYBDF imf map
: search for maxima in the IMF map
: PEKPIK maxima can be used as input for
PEKPIK punch
plimit *2 0.1  20  0.4
end