How many symmetry operations are needed to generate a space group?

1 Introduction

Some subset of elements U of group G is referred to as generating subset if the least subgroup of G containing U, denoted as 〈U〉, is G itself, i.e. 〈U〉 = G. A generating subset U is referred to as a minimal generating subset if it does not contain any generating subset of G except for itself. ¹ As all crystallographic groups are finitely-generated, each of them has a minimal generating subset with a finite cardinality |U|. The minimal generating subset of space group has its own applications to studying space-group properties, ^{2 , 3} however, the above definition of a minimal generating subset does not impose restrictions on the value of |U|, this is why it may vary as was shown specifically for crystallographic groups. ^{4 , 5 , 6} This work is focused on those U with minimal cardinality usually referred to as the rank d of a group: ⁷

The rank of the space group is interrelated with algorithmic complexity ⁸ and implicit hierarchical depth, ⁹ a variety of ordinary hierarchical depth, ¹⁰ of a crystal structure. More than 2/3 of homomolecular organic crystals consist of molecules whose centers of mass occupy a sole general position ^{11 , 12} ; in such structures d(G) accounts for the least required number or structure-forming intermolecular contacts. ¹³ Recently, d(G) was utilized to analyze the complexity of synthetic analogues of minerals and other compounds. ^{14 , 15 , 16 , 17} The lower limit of d(G) cannot be less than 2.

For the 17 plane-group types, Coxeter and Moser ¹⁸ has listed one of the minimal generating subsets for which cardinality equals the rank of a group, except for the plane-group type p6mm. ¹⁹ Finding the ranks of the 230 space-group types is even more a non-trivial problem. The powerful tool IDENTIFY GROUP implemented in Bilbao Crystallographic Server, ²⁰ is, unfortunately, of limited application to this problem, since it forcibly adds translations into any selected pool of generators. International Tables for Crystallography, Volume A (ITA) and contemporary services based on them ²¹ also lack subsets of generators with a minimal cardinality. Furthermore, the set of generators used by ITA is not minimal on purpose, since its aim is to accentuate important subgroups of space groups as much as possible. The classical work of Zassehaus ²² employed the modern matrix method of deriving all the 230 space-group types, but despite handling generators and generating relations with a vast toolset of algorithms ²³ arisen within more than half a century since that time, this method was never focused on finding a minimal generating subset. Lord and Banaru ¹³ manually obtained the ranks of biaxial (triclinic, monoclinic and orthorhombic) groups, and these results were further approved by the computer-aided minimization of a conventional (with regard to ITA ²⁴ ) generating subset of the group (generators selected), except for the space-group type No. 70 (Fddd), which had an overestimated value of d(G) (5 instead of 3) according to a computer-aided minimization. Meanwhile, for uniaxial (tetragonal, trigonal, hexagonal) ⁴ and isotropic (cubic) ⁵ groups the results of the computer-aided minimization were more often contradictory to those obtained manually. ²⁵ One of the reasons of contradictions were certain ITA conventions preventing the generating subset from being minimized into that with a minimal cardinality, e.g. including 3-fold and 2-fold rotations along the same axis instead of a 6-fold one. In this work, the initial pool of group generators was modified and put into the updated algorithm ⁶ performed in GAP ²⁶ to obtain true ranks of all the 230 space-group types. Aside from ITA, basic terminology regarding space groups in this work was briefly outlined by Nespolo et al. ²⁷

2 Methods

The minimization was performed in multi-flow PC calculations (up to 8 flows each time) in GAP standard package extended by additional libraries Cryst and CrystCat. The calculation included the following steps: (i) The input for each space-group type the initial pool of 4 × 4 matrices corresponding to symmetry operations. (ii) Next, a trial generation of a group by a subset of any pair of operations was made. (iii) Once the generated group coincided with the initial one, i.e. the group generated by the initial pool of operations, the subset of matrices thus found was considered a minimal generating subset, and the calculation passed to the next space-group type. If no generating subset was found among all the pairs of the initial operations, the step (ii) was repeated with triplets of operations instead of pairs. In case of unsuccessful trials to generate the group by all the possible triplets of operations, then the quadruplets were searched for, etc., until the first successful trial to generate the group. (iv) The number of operations in the minimal generating subset first found was considered the supremum of d(G). Obviously, if sup{d(G)} does not change upon extensions of the initial pool operations, it is a true value of d(G).

Four kinds of pools of initial operations were dealt in this work with:

1. (A) A pool of all operations corresponding to a general position of a group in accordance with ITA (for all the sets of translationally inequivalent points), complemented by the basic translations t(1,0,0), t(0,1,0), t(0,0,1) and with no identity matrix. For instance, the space-group type F d 3 ‾ m has 4·48 = 192 matrices of a general position, consequently, 192 matrices + 3 translations – 1 identity matrix = 194 matrices in all were taken into consideration.

2. (B) A pool of all operations of a group corresponding to generators selected in accordance with ITA complemented by some representatives of TW _i and W _i T (the right and the left cosets of G with respect to the translation subgroup T). Each selected representative was a pairwise product of some initial operation W _i that is not a pure translation and some initial translation t _j . As T is a normal subgroup of G, the right coset TW _i and the left coset W _i T are the same, and there is no need to consider them separately. However, partial cosets taken into consideration may intersect partially rather than fully, hence considering representatives of both cosets is reasonable. Being represented as a matrix-column pair, W _i  = (W _i , w _i ), a right coset representative for some t _j will be (W _i , w _i  + t _j ), and the left coset representative (W _i , W _i t _j  + w _i ). For instance, space-group type C2 (No. 5) has 5 generators selected: t(1,0,0), t(0,1,0), t(0,0,1), t(0,1/2,1/2) and [ x ‾ , y ‾ , z ] (a 2-fold rotation along 0,0,z). Hence this set is complemented by operations: [ x ‾ + 1 , y ‾ , z ] , [ x ‾ , y ‾ + 1 , z ] , [ x ‾ , y ‾ , z + 1 ] , [ x ‾ , y ‾ + 1 / 2 , z + 1 / 2 ] (TW _i ) and [ x ‾ − 1 , y ‾ , z ] , [ x ‾ , y ‾ − 1 , z ] , [ x ‾ , y ‾ − 1 / 2 , z + 1 / 2 ] (W _i T, the operation [ x ‾ , y ‾ , z + 1 ] from TW _i is duplicated).

3. (C) A pool of all operations corresponding to a general position of a group (see (A)), among those W _i  = (W _i , 0) were complemented by the other representatives of solely TW _i (see (B)). All the groups, no matter symmorphic or not symmorphic they are, were treated in this way. A matrix-column pair W _i  = (W _i , 0) accounts for pure rotations or rotoinversions along some axis intersecting the origin O, which is conventionally chosen in a point with the highest site symmetry. Moving this axis to another place, i.e. choosing another representative of TW _i , will make a shift between the new axis and all the other rotation or rotoinversion axes intersecting O. This shift may potentially generate some basic translation and thus diminish the required number of group generators. Moving axes other than those mentioned above is unlikely to be necessary, as even without a moving they already contain a translational part thus potentially generating a basic translation parallel or perpendicular to the axis. However, an additional pool (D) was built up (see below) in order to examine, if this is the case or not. It is worth mentioning that, along with symmorphic groups, some W _i  = (W _i , 0) may be contained in the generating set of not symmorphic groups except the 13 Bieberbach group types with no special positions, e.g. P2₁2₁2₁.

4. (D) A pool of all operations corresponding to a general position of a group (see (A)), among those all W _i  = (W _i , w _i ) that are no pure translations (W _i ≠ I) were complemented by the other representatives of solely TW _i (see (B)). This pool was anticipated to give the same results as (C), despite a much more time-demanding calculation. In this calculation, minimal cardinalities n of a generating subset previously obtained for (C) were checked for redundancy in a downward fashion, i.e. cardinalities (n–1), (n–2), etc., were checked until the cardinality could not be minimized anymore.

Prior to each calculation start, matrix-column pairs of the initial pool were sorted in the following order: (I, w _i ), (W _i , 0), (W _i , w _i ) to promote simplicity of the first found generating subset of a minimal cardinality, as the search starts from the beginning of the list of operations. See the file of the Supplementary Material (readable in any Notepad) containing the GAP code for space-group type No. 5 to demonstrate the algorithm.

3 Discussion

General remarks. Even in a multi-flow regime the calculation on PC lasted from hours (pool (B)) to weeks (pool (D)). Calculations performed for pools of matrices (A) and (B) for some space-group types reduced the estimated value of d(G) with respect to that previously found for generators selected, ^{4 , 5} and/or to that listed by Lord. ²⁵ Unfortunately, for some other space-group types utilization of (A) and (B) resulted in an overestimated value of d(G) with respect to previous results. ^{4 , 5 , 25} However, the pool (C) did not lead to overestimating d(G) for any space-group type and either confirmed all previous minimization results, or further minimized them. The pool (D), in its turn, did not lead to a re-estimated d(G) for any group type, thus indicating (C) a reliable way of estimating d(G). The first found generating subsets of the minimal cardinality for each group type is listed in the Supplementary Material. The final estimated values of d(G) are listed in Table 1. The pool (C) looks excessive from a purely mathematical point of view; however, the work was aimed at finding d(G), not finding a reasonable fashion of selecting a generating subset. The known values of d(G) should facilitate developing conventions of listing a minimal generating subset for each of the space-group type in data tables like ITA.

From Table 1 it follows that neither generators selected, nor even all operations of general position are, as a rule, fit for obtaining correct value of d(G) by just trying to generate G with a less number of generators. Namely, 18 of 73 group types with d(G) = 2, 11 of 99 those with d(G) = 3, and 1 of 48 those with d(G) = 4 would be overlooked, if it were not for a pool of generators extended by other coset representatives.

For some space-group types the found generating subset (see gVectors.xlsx) includes generators that one would hardly consider a generator “by default”. Let consider P31m (No. 157). The first found generating subset includes two generators: A = g (1/2,1/2,0) x+1/2,x,z; S = 3⁺ (0,0,1) 0,0,z. The first generator corresponds to an ordinary glide plane, but the second one corresponds to a screw axis with a period 3|c|. The translation c is not generated by this generator S, however, by a combination of S and A it is, indeed, generated. Denote A⁻¹ = g (–1/2,−1/2,0) x+1/2,x,z; S⁻¹ = 3^– (0,0,−1) 0,0,z. Then c = A⁻²SAS⁻¹AS (such compositions here and below should be read from the right to the left), a = SA⁻²S⁻¹, b = S⁻¹A⁻²S. Furthermore, the conventional generators Q = 3⁺ (0,0,0) 0,0,z (an ordinary rotation) and M = m x,x,z (an ordinary reflection) may be easily decomposed into A, A⁻¹, S, S⁻¹, as well: Q = A⁻¹SA⁻¹S⁻¹A², M = AS⁻¹A²S. Surprisingly, the plane-group type p31m is of exactly the same d(G) = 2 ¹⁹ despite a lower dimensionality. Analogously, P3m1 (No. 156) and p3m1 are of the same d(G) = 3.

For a group-subgroup pair, there is no simple regularity in d(G) values. Thus, even among minimal t-supergroups of index = 2 of P2₁ (d(G) = 3) there are those with d(G) = 2 (e.g. P2₁2₁2₁), those with d(G) = 3 (e.g. P2₁/c) and those with d(G) = 4 (e.g. P2₁/m).

The upper limit of d(G). For space-group types belonging to a certain arithmetic crystal class, d(G) cannot exceed the sum of the rank of the point group P of the space group and the rank of its translation subgroup d(T) = 3. Moreover, since some basic translations of the lattice may be equivalent with respect to P, the upper limit is further diminished:

Where d _P (T) is the number of non-equivalent basic translations of a primitive (non-conventional) cell of the lattice. The values of d _P (T) (Table 2) directly follow from Table 3.1.2.2 ²⁸ of ITA. For instance, in all cubic Bravais-lattice types cP, cI, cF, despite different centring types, there is only the equivalence class of basic translations building up a primitive cell. The rank of a crystallographic non-trivial point group ranges from 1 to 3 ²⁹ (Table 3), with all cubic geometric crystal classes having 2 generating elements. Consequently, for any space group d(G) ≤ 3 + 3 = 6. The only geometric crystal classes with 3 generating elements are mmm, 4/mmm and 6/mmm. Due to the value of d _P (T), the upper limit d(G) = 6 is realizable only for P = mmm. For groups of tetragonal and hexagonal crystal families d(G) ≤ 3 + 2 = 5, however, no hexagonal space group reaches this upper limit (Figure 1). For any cubic space group d(G) ≤ 2 + 1 = 3.

Figure 1:

The lowest, the highest, and the average value of d(G) for space groups of different crystal families.

The equality d(G) = d(P) + d _P (T) was presumed to more probably hold for symmorphic rather than non-symmorpic space groups, as the latter’s Hermann–Mauguin symbol explicitly displays screw axes and/or glide planes, which may generate a basic translation thus decreasing d(G). Indeed, this equality holds for 50 of 73 symmorphic space-group types and for just 17 of 157 non-symmorphic ones (see the Supplementary Material). Remark that all 23 symmorphic space-group types, for which the equality does not hold, fall in hexagonal and cubic crystal families. The value of (d(P) + d _P (T) – d(G)) shows how many symmetrically non-equivalent translations are redundant for U, while the value of (d(G) – d(P)), in fact, shows how many basic translations of a primitive cell may be contained in some U. Out of 230 space-group types, there were found 75 with d(G) – d(P) = 0, i.e. having no translations at all in any U. This fact once again illustrates a well-established concept ³⁰ that rather a local regularity of the structure results in a global one than vice versa.

Let us consider space-group types No. 195–199 of geometric crystal class 23. As it follows from Table 1, d(G) – d(P) = 3–2 = 1 for group types No. 195 and 196 (P23 and F23), but d(G) – d(P) = 0 for group types No. 197–199 (I23, P2₁3, I2₁3). For P2₁3 this result is not surprising, because this group type may be considered as the addition of a body-diagonal 3-fold rotation to P2₁2₁2₁, which is a typical group type with d(G) – d(P) = 2–2 = 0. Indeed, if S₁ = 2 (0,0,1/2) 1/4,0,z; S₂ = 2 (0,1/2,0) 0,y,1/4, then (S₂S₁)⁻¹ = 2 (1/2,0,0) x,1/4,0. Meanwhile, S₁² = t(0,0,1), S₂² = t(0,1,0), (S₂S₁)⁻² = t(1,0,0). However, a reduction of d(G) for I23 and I2₁3 is not so trivial and means that t(1/2,1/2,1/2) cannot be included in U. For group type I23 it can be shown that, if Q = 3⁺ x,x,x; S = 2 (0,0,1/2) 1/4,1/4,z, then t(1/2,1/2,1/2) = QSQS⁻¹QS. Analogously, for group type I2₁3 it can be shown that, if Q = 3⁺ x,x,x; R = 2 1/4,0,z, then t(1/2,1/2,1/2) = RQRQ⁻¹RQ⁻¹RQR. After t(1/2,1/2,1/2) is generated, t(1,0,0), t(0,1,0) and t(0,0,1) are easily generated, too.

Relation to reticular chemistry. Let a vertex occupy a general position of the space group. In this case, a simply connected graph linking vertices together must have at least as many classes of equivalent edges as the Cayley graph of the space group. For an actual crystal structure of this kind, the underlying net ³¹ of its building units may be reduced to the skeletal net ³² isomorphic to the Cayley graph of the space group (with multiple edges being neglected). For instance, for space group No. 19 (P2₁2₁2₁) such skeletal net can be of diamondoid (dia ³³ ) topological type (Figure 2). Crystallography is focused on studying Cayley graphs as such, ³⁴ as well as on some applications of Cayley graphs to the analysis of periodic nets, ³⁵ but no attention is usually paid to the minimal coloring of a Cayley graph. Table 1 manages to eliminate this drawback for all the space-group types.

Figure 2:

Space-group diagram for no. 19 (P2₁2₁2₁, d(G) = 2) (left) and a fragment of its Cayley graph (right).

Consider another example, space-group type No. 221 ( P m 3 ‾ m ) with d(G) = 3. Let the artificial structure be crystallized in such space-group with a = 10.0 Å and have some “atom” with coordinates (0.1,0.2,0.3) (Figure 3a). The next calculations are best performed in ToposPro program package. ³⁶ If one then links together those “atoms” that have adjacent Voronoi-Dirichlet polyhedra (such a graph is referred to as a Delone graph ³⁷ ), the 4-coordinated net of the zeolite type rho with four classes of equivalent edges is build up (Figure 3b). However, upon deleting any set of equivalent edges this net does not retain simple connectivity. Does that mean that the value d(G) = 3 for P m 3 ‾ m is underestimated? The answer is no. If one initially links together those “atoms” with a distance no more than 6 Å to obtain the complicated 43-coordinated net (Figure 3c), the obtained net is indeed able to be reduced to several simply connected subnets with exactly three classes of equivalent edges, e.g. to the subnet displayed in Figure 3d. The edges of this subnet correspond to the following symmetry operations generating the space group: m x,y,y (this edge was also present in the above rho net); 3 ‾ ^– x–1, x ‾ , x ‾ ; and 4 ‾ ⁺ −1/2,–1/2,z with the inversion point (−1/2,–1/2,0). The subnet has intersecting edges and thus cannot be realized in an actual crystal structure, but it confirms the value d(G) = 3 for P m 3 ‾ m .

Figure 3:

A hypothetic structure with the space group P m 3 ‾ m , a = 10.0 Å: (a) And “atom” in (0.1,0.2,0.3) and equivalent points; (b) a fragment of Delone graph; (c) a fragment of the latter’s supergraph containing all possible edges no longer than 6 Å; (d) a fragment of one of the latter’s subgraph with exactly three classes of equivalent edges highlighted by different colors.