The indexing of the lattice in an integration program (such as Mosflm, or the indexing and integration stages of XDS and DIALS) is based on the lattice geometry, with no regard for the symmetry of the diffraction pattern, which can only be determined after integration. The default option here scores potential symmetry operators in the diffraction pattern, and ranks the possible rotational groups (Laue group, Patterson group or point group, see Footnote), and also inspects axial reflections for possible systematic absences which may indicate a likely space group. Be careful, the presence of pseudo-symmetry may suggest a higher symmetry than the truth. POINTLESS tries to allow for this possibility, but inspection of the scores for individual symmetry elements may help to indicate the correct space group in difficult cases (fig 3). Conversely, severe radiation damage may obscure the true symmetry. Weak high resolution data does not contain reliable information for the determination of rotational symmetry (the Laue or point group), so POINTLESS may cut the resolution just for this purpose. If possible it cuts the data at the point where CC(½) falls below 0.6 (calculated using only the Friedel symmetry (ie in P1)), or I/σ(I)>6 if there are insufficent data to calculate CC(½).
This first stage of the task has 4 options, with different result reports:
Scoring individual symmetry elements. Each possible symmetry operator in the lattice is scored separately, by a pairwise correlation coefficient (CC) between E 2 for observations related by that operator, and also by an R-factor, Rmeas. The CC is used to estimate a probability, allowing for possible small samples by comparing it with the distribution CCs of equal sized samples of unrelated observation pairs. This separate scoring is useful in cases of pseudo-symmetry to indicate the true symmetry, in cases where the program gets it wrong. In the example in figure 3, the true symmetry is C222, but because of the accidental combination of cell lengths, b ≈ √3 a, the lattice can be indexed as hexagonal. The scores show that three orthogonal dyads are present, but the other potential operators of the hexagonal lattice are absent.
Scoring Laue groups. The scores from the individual elements are then combined into a joint score for all possible Laue groups which are subgroups of the lattice group. A high score for a symmetry element which is present in the lattice group but not in the Laue group will count against that group (the CC-, Zc- and R- columns in the table). Note that in Figure 4 the pseudo-hexagonal lattice can accommodate three possible Cmmm settings, 60° apart: the one chosen randomly by the original indexing (Reindex operator [h,k,l]) was wrong here.
Systematic absence scores. Within a chosen Laue group, space groups (all chiral groups apart from the pairs I222 and I212121, and I23 and I213) may be distinguished by the presence or absence of screw axes along the cell edges. A screw axis leads to systematic absences along an axis of the reciprocal lattice eg a 21 screw along b in space group P21 makes axial reflections 0k0 present only when k is even. Detecting systematic absences may be unreliable because the axial reflections may be few in number (as along the a axis in figure 5), or missing from the dataset if they lie along the spindle rotation axis in the data collection (in the blind region), or may be misleading if there are approximate non-crystallographic screw axes (tNCS), but in many cases they can suggest the space group, to be confirmed later: the space group remains a hypothesis until the structure is satisfactorily solved. In POINTLESS, a Fourier analysis is used to detect periodicity, on I’/σ: the intensity I’ used here is adjusted by subtraction of a small fraction (default 0.02) of the intensity of the neighbouring reflection along the axis, to allow for possible overlap of a nearby strong reflection (figure 5).
Choice of space group or point group. Possible space groups are ranked according to their total probability = Laue group probability × Systematic absence probability. If there is a unique solution with the highest total probability, this will be chosen as a “space group” solution. If some of the potential systematic absence data is missing, then more than one space group has the same score, and the space group without translations will be chosen, eg P 2, P 2 2 2, P 4 2 2, P 6 2 2 etc. Enantiomorphic space groups will have the same score, so the first one is chosen as a space group solution (see figure 6), unless the other one matches that in the input file. As well as the scores, a “confidence” score is calculated for both the Laue group and the space group, defined as √[Score × (Score - NextBestScore)]: these values are printed in a summary table as the Best Solution. If you prefer a different solution to that chosen by the program, you can rerun the program with the Choose a previous solution option.
In some point-groups there are more than one (typically two or four) valid but non-equivalent indexing possibilities. For your first crystal, you may choose any of these, but subsequent crystals must match the first. This problem generally arises in cases where the point group symmetry is less than the lattice symmetry: the alternative indexing schemes are related by the symmetry operators present in the lattice but not in the point group. These are the same point groups which may lead to merohedral twinning, eg point group P3 has four possible indexing schemes in the lattice point-group P622 (further explanation). Within multiple datasets from the same crystal (eg MAD), you can avoid this problem by only autoindexing one set, & using the same indexing matrix for the others, but different crystals must be explicitly checked for consistent indexing. Reindexing ambiguities may also arise in lower-symmetry point-groups in case of accidental coincidences or relationships between cell dimensions (eg a ≈ b in orthorhombic or the pseudo-hexagonal C222 example in figure 3).
A ranked table of scores for each possible indexing scheme is printed. The example in figure 7 is unusually complicated: the true space group is rhombohedral, R3, but the lattice is pseudo-cubic; in the rhombohedral axis system (with a = b = c, α = β = γ) the angles are close to 90°. This means that in addition to the usual ambiguity in R3 with alternative indexing [k,h,-l] (in the hexagonal setting aka H3 or R3:H) , there are four possible directions for true 3-fold axis, along the body-diagonals of the cubic lattice, leading to a total of eight possible indexing schemes.
This option allows an explicit choice of indexing or symmetry. The Choose solution from search options perform the usual search, but selects the given solution even if it does not have the highest score. The Specify Laue group name or Specify space group name use the given groups without doing the searches, ie the program just changes the group, with an option to reindex.
This option just sorts one or more input files, usually because there are more than one, and you know the space group.
Note that strictly a Laue group is the point group symmetry of the diffraction pattern, ie the crystal point group plus a centre of inversion at the origin of the reciprocal lattice: here we include the lattice centring type (P, I, F, C, R) in the “Laue group” definition, equivalent to the Patterson space group.
