Encyclopedia of Crystallographic Prototypes

Chiral Space Groups

Chiral Space GroupsI

Schematic of equipment needed to measure the optical rotation of a sample
Figure 1: Schematic of an experimental setup to measure optical rotation of sample.(Langdon)

If we place an amino acid in the sample chamber of the apparatus sketched by S. Langdon in Fig. 1, the polarized beam of light will rotate counter-clockwise. If, on the other hand, we use ordinary glucose the polarization will rotate in a clockwise direction, as shown in the figure.(Zhou, 2024) On the gripping hand, a dedicated researcher can find samples of amino-acids that cause clockwise rotation and glucose that rotates the light counter-clockwise, even though the samples are otherwise apparently identical to the original ones. The two different forms of the same material are called enantiomorphs.

Left Hand Screw Right Hand Screw
Figure 2: “Left” and “Right” handed screws. The screw on the left is driven into the wood by turning it counter-clockwise as seen from the top, while the more common screw on the right is turned clockwise, leading to the expression “lefty-loosey, righty-tighty.”

We can see enantiomorphs in a system as simple as a wood screw. At first glance there is nothing very different about the two screws in Fig. 2. Closer examination shows that they are in fact different. As with most screws, the one on the right is driven into the wood by turning it clockwise as seen from above, often referred to as “right-tighty”. The screw on the left requires a counter-clockwise motion to drive it into the wood. While my brother and my spouse would no doubt approve of such a screw, the majority of the human race is right-handed, and so nearly all screws are designed like the one on the right.II

gamma Se Space Group 152 gamma Se Space Group 154
Figure 3: γSe seen in space group $P3_{1}21$ #152 (left) and in space group $P3_{2}21$ #154 (right).

Since this site usually discusses atoms, we should ask if there are any examples of this left-/right-handedness in crystals. The answer is of course yes. Fig. 3 shows two different versions of γSe (Strukturbericht symbol $A8$). On the left we see the original experimentally-reported version in space group $P3_{1}21$ #152. The bonds wind counterclockwise going from the back of the picture to the front. The right-handed picture shows an alternative form of γSe in space group $P3_{2}21$ #154. Here the bond rotate clockwise.

Both versions of γSe have the same lattice constants, the same distances between atoms, and the same bond angles. Both structures can be seen experimentally. First-principles calculations would predict the same energy, elastic structure, elastic constants and phonon frequencies for each system. The only difference is how the bonds wind through the crystal. The structures are mirror images of each other.III

TlZn2Sb2 Reported Configuration TlZn2Sb2 Mirrored Configuration
Figure 4: Crystal structure of body-centered tetragonal TlZn$_{2}$Sb$_{2}$ seen looking down the $\hat{z}$-axis of the conventional cell. On the left is the structure reported in the literature. On the right is a structure obtained by mirroring the structure around the $z$ axis: $(x y z) \rightarrow (-x -y z)$. Both structures are in the same space group, $I4$ #79, and all interatomic distances and angles are the same, yet they are mirror images of one another, making the structure chiral.

Finally, consider body-centered tetragonal TlZn$_{2}$Sb$_{2}$, seen in Fig. 4. The structure on the left is the one reported in the literature and on our TlZn$_{2}$Sb$_{2}$ page. We created the structure on the right by mirroring the coordinates, $(x y z) \rightarrow (-x -y z)$. Just as in the previous examples the two crystals are mirror images of one another.IV All of the interatomic distances are the same, and the angle between any two bonds is the same in both cases. The energetics and mechanical properties are exactly the same, just as in the γSe example. The difference here is that the original structure and its mirror image are in the same space group, $I4$ #79.

Chirality

What we've shown above are examples of what are known as chiral systems. In 1894 Lord Kelvin defined chiral structures:

I call any geometrical figure, or group of points, chiral, and say that it has chirality, if its image in a plane mirror, ideally realized, cannot be brought to coincide with itself.

— Lord Kelvin, 1894

In other words, if you hold an object up to a mirror and it looks reversed, it's chiral. Otherwise we call it achiral. As an example, your shoes and gloves are chiral, but your tennis racket is (usually) achiral.

A periodic crystalline system must exist in one of the fourteen plane groups in two dimensions or two hundred thirty space groups in three dimensions. Here we want to find out which groups support chiral structures.

Obviously any group that contains a mirror operation cannot allow chiral structures, because all atoms will automatically be mirrored. In addition, any “rotation-reflection” operation, which we'll discuss below, eliminates chirality, as does an inversion in three dimensions.V.

In two dimensions there are only six groups which allow chiral structures. These are the groups that do not have an m (mirror operation) or g (glide) in their labels: $p1$ #1 , $p2$ #2 , $p3$ #13 , $p4$ #10 , and $p6$ #16 .

Sohncke (Chiral) Space Groups

Since three dimensions give us more ways to manipulate a crystal you might expect that the situation is more complicated. To begin with, we now have 230 groups. (Cockcroft, 1999) In 1879 Leonhard Sohncke found that only 65 of them support chiral structures.(Fecher, 2002) The remaining 165 space groups (Class I groups) all have an “improper” operation: inversion, a mirror plane, a glide plane, or what is known as an $S_{n}$ operation: a rotation followed by a reflection in a plane perpendicular to the axis of rotation. Any structure found in one of these space groups will be identical to its mirror image. The remaining 65 space groups only contain what are called operations of the first kind, or proper operations: pure rotations and screw axes, with no operation that does any kind of mirroring. As a result, these 65 groups are the only ones which can support chiral structures. We call them the Sohncke groups.

We can then put all 230 space groups in three classes: Class I contains the 165 achiral space groups. The 65 Sohncke groups are divided into two classes: Class II has 22 space groups, while Class III has the remaining 43.

Sohncke Class II Space Groups

Class II structures all have screw axes. (CCDC) Simply put, a screw axis is a rotation around an axis followed by a translation along the axis, just like when we drive one of the screws in Fig. 2. By convention we'll take the axis to be in the $\hat{z}$ direction when possible and put the third primitive vector of the conventional lattice along that direction, giving

$\large{\mathbf{a}_{3} = c \hat{z}}$   ,   (1)
although there are space groups which have more than one screw axis.

Screw Axes

A screw axis is labeled $n_{m}$, where $n$ and $m$ are integers, and

  • $n$ means we rotate a point through an angle of
    $\Large{\frac{360\degree}{n}}$       (2)
    counterclockwise as seen looking down the axis toward the original atomic position. This is clockwise along the direction the atom is traveling, so this is equivalent to the right-handed screw in Fig. 2.
  • We then translate the point by the distance
    $\large{d =} \Large{\frac{m}{n}} \large{\, c}$   ,   (3)
    along the vector describing the screw where $c$ is the length of the primitive vector in the direction of the screw (1). If this moves the atom out of the unit cell we use translational symmetry to bring it back into the cell.
  • We fill out the unit cell by repeating this operation $n$ times for every atom, bringing all the atoms back to their original positions.

    • An example is in order:

    • Start with an atom at, say,
      $\large{r \cos\theta \, \hat{x} + r \sin\theta \, \hat{y} + z \, \hat{z}}$   .   (4)
    • Now rotate this point by $\alpha$ and translate it by $d$:
      $\large{r \cos(\theta+\alpha) \, \hat{x} + r \sin(\theta+\alpha) \, \hat{y} + (z + \frac{m}{n} c)\, \hat{z}}$   .   (5)
      If the new $z$ coordinate of the atom is greater than $c$, we can use the periodicity of the lattice to translate it back into the original unit cell.
    • Place a copy of the original atom at this position.
    • Repeat for $n$ rotations, which will bring us back to the original position.
    • Repeat the entire process for every atom in the crystal.

    Fig. 4 shows a $4_{1}$ screw axis in the imaginary tetragonal compound AgAuCuZn. Starting with the square in the lower left we apply the above procedure three times to fill the cell. The numbers in the figure indicate how many screw operations have been applied to move the square to its new location. The fourth turn of the screw will bring the atoms back to square zero.

    4_1 rotation 4_1 rotation looking down z axis
    Figure 5: An imaginary AgAuCuZn crystal constructed with a $4_{1}$ screw axis. The view on the right is looking down the $z$ axis from the top. The 0, 1, 2, and 3 indicate the number of screw operations (2)-(5) that need to be applied to move each AgAuCuZn square to that point.

    Next look at a system with that starts with the same four atoms but with a $4_{3}$ screw axis. The result is in Fig. 6. This structure is nearly identical to the one in Fig. 5, but now the chain of atoms winds clockwise as we go up the cell, rather than counterclockwise.

    4_3 rotation 4_3 rotation looking down z axis
    Figure 6: An alternative, but still imaginary, AgAuCuZn crystal constructed with a $4_{3}$ screw axis. Again the view on the right is looking down the $z$ axis from the top. The numbers again indicate how many screw operations have been applied, with “0” and “2” leaving the atoms in the same positions as in Fig. 5. This structure can be viewed as a mirror image of that structure.

    If we use AFLOW to determine the space groups we find that the structure in Fig. 5 is in space group $P4_{1}$ #76, and the one in Fig. 6 is in $P4_{3}$ #78. The origin of the group labels should be obvious.

    The two structures described above are identical in every way except for the rotation of the screw axis. The bond distance are identical, as are the angles between bonds. The structure are said to be enantiomorphic, and $P4_{1}$ and $P4_{3}$ are enantiomorphic space groups. The 22 Class II Sohncke space groups form eleven enantiomorphic pairs, which we've outlined in Table 1.

    Table 1 The eleven pairs of space groups forming Sohncke Class II. Each line has one set of enantiomorphically paired groups. The links lead to an index of the structures in the Encyclopedia which are in that space group. All Class II groups have a 3-, 4-, or 6-fold rotation axis, so there are no triclinic, monoclinic, or orthorhombic entries.
    Tetragonal
    $P4_{1}$   #76 $P4_{3}$   #78
    $P4_{1}22$   #91 $P4_{3}22$   #95
    $P4_{1}2_{1}2$   #92 $P4_{3}2_{1}2$   #96
    Trigonal
    $P3_{1}$   #144 $P3_{2}$   #145
    $P3_{1}12$   #151 $P3_{2}12$   #153
    $P3_{1}21$   #152 $P3_{2}21$   #154
    Hexagonal
    $P6_{1}$   #169 $P6_{5}$   #170
    $P6_{2}$   #171 $P6_{4}$   #172
    $P6_{1}22$   #178 $P6_{5}22$   #179
    $P6_{2}22$   #180 $P6_{4}22$   #181
    Cubic
    $P4_{3}32$   #212 $P4_{1}32$   #213

    Sohncke Class III Space Groups

    Table 1 has no entries for the $2_{1}$, $4_{2}$, or $6_{3}$ screw axes, which seems a bit surprising. We can see the reason for this in Fig. 7, which shows a another fictional AgAuCuZn structure, this time with a $4_{2}$ screw axis. Each turn of the screw will rotate the AgAuCuZn square by 90° and translate it by $c/2$ along the $z$-axis ($0 \rightarrow 1$ in the figure). The next operation ($1 \rightarrow 2$) rotates the square by 180° from its initial position and translates it back to the bottom of the unit cell. The next turn moves the square 270° from the original position and moves it to the top left of the figure ($2 \rightarrow 3$). A final turn restores the square to its original position.

    4_2 rotation 4_2 rotation, z-axis view
    Figure 7: Another invented AgAuCuZn crystal, here constructed with a $4_{2}$ screw axis. Again the view on the right is looking down the $z$ axis from the top. The square labeled “0” is in the same location as it is in Fig. 5 and Fig. 6. Unlike in those figures, rotating clockwise or counterclockwise does not change the crystal. If we mirror this structure the crystal will still be in the same space group, $P4_{2}$.

    We could repeat this operation using a 90° clockwise screw and we would get the same structure. A mirror image of the structure will look different than the original image, so the structure is still chiral, but belongs to Class III. We get similar results for $2_{1}$ and $6_{3}$, the other spin axes that translate the atoms by $c/2$ along the $z$-axis.

    The Class III groups are all the chiral groups that aren't in Class II. That means that they either have no screw axis, or, as we just showed, have screw axes that do not form enantiomorphic pairs: $2_{1}$, $4_{2}$, and $6_{3}$. The groups themselves are achiral: if we invert all the atomic positions the space group remains the same, even though the structures themselves are chiral, as seen in Fig. 4 and Fig. 7. The Class III space groups are listed in Table 2.

    Table 2 The forty-three Class III Sohncke space groups. The links lead to an index of the structures in the Encyclopedia which are in that space group. A screw axis is not required and $2_{1}$ screws are allowed in monoclinic and orthorhombic systems, so Class III structures can be all seven crystal systems.
    Triclinic
    $P1$   #1        
    Monoclinic
    $P2$   #3 $P2_{1}$   #4 $C2$   #5    
    Orthorhombic
    $P222$   #16 $P222_{1}$   #17 $P2_{1}2_{1}2$   #18 $P2_{1}2_{1}2_{1}$   #19 $C222_{1}$   #20
    $C222$   #21 $F222$   #22 $I222$   #23 $I2_{1}2_{1}2_{1}$   #24  
    Tetragonal
    $P4$   #75 $P4_{2}$   #77 $I4$   #79 $I4_{1}$   #80 $P422$   #89
    $P42_{1}2$   #90 $P4_{2}22$   #93 $P4_{2}2_{1}2$   #94 $I422$   #97 $I4_{1}22$   #98
    Trigonal
    $P3$   #143 $R3$   #146 $P312$   #149 $P321$   #150 $R32$   #155
    Hexagonal
    $P6$   #168 $P6_{3}$   #173 $P622$   #177 $P6_{3}22$   #182  
    Cubic
    $P23$   #195 $F23$   #196 $I23$   #197 $P2_{1}3$   #198 $I2_{1}3$   #199
    $P432$   #207 $P4_{2}32$   #208 $F432$   #209 $F4_{1}32$   #210 $I432$   #211
    $I4_{1}32$   #214        

    How Common Are Chiral Structures?

    Although the chiral compounds are particularly important for life, they are not as common as one might think. Table 3 shows the distribution of chiral structures (Sohncke Class II and III) for various databases, including one for only inorganic structures (ICSD, 2023), and another containing exclusively organic and metal-organic structures (CSD, 2024). For comparison we also include the data from this Encyclopedia in September, 2024. We see that the enantiomorphic space groups (Sohncke Class II) comprise only one percent of all structures, regardless of the source. The exception is this Encyclopedia, where we deliberately searched for structures in this class and often included both the reported structure and its enantiomorph if that space group had no other entry. Sohncke Class III structures, on the other hand, are quite common in organic systems, even more common than suggested by the number of Sohncke space groups. Inorganic materials are decidedly less chiral, as the ICSD has only 4% of its structures in Class III. This favoring of achirality can also be seen in the structures of elements, where there are only three known chiral structures: βMn ($A13$), in one of the enantiomorphic pairs $P4_{1}32$/$P4_{3}32$, γSe ($A8$), in $P3_{1}21$/$P3_{2}/21$ (Fig. 3), and a high pressure phase of Te, in the Class III group $P2_{1}$.

    Table 3 Distribution of achiral Class I and chiral Sohncke Class II and III structures found in several databases. “Encyclopedia” refers to the entries in this Encyclopedia as of September, 2004. The ICSD, 2023 is restricted to inorganic compounds appearing the the final 2023 distribution of the Inorganic Compounds Structural Database, while the CSD, 2024 has a list of organic and metal-organic crystals in identifiable space groups in the Cambridge Structural Database as of 1 January 2024. For comparison we also list the number of space groups in each class.
    Source Total Entries Class I Sohncke Class II Sohncke Class III
    Space Groups 230    165   (71.49%)    22   (9.53%)    43   (18.63%)   
    Encyclopedia 2,014     1,772   (87.98%)    66   (3.28%)    176     (8.74%)   
    ICSD 2023 216,428    205,199   (94.81%)    2,410   (1.11%)    8,819     (4.07%)   
    CSD 2024 1,294,724    1,083,566   (83.69%)    13,851   (1.07%)    197,307   (15.24%)   

    So if the question is “how many crystals have chiral structures?” the answer is “That depends on who you ask.” Table 3 shows this clearly: if you only look at inorganic solids you will find very few chiral structures, but if you look at organic systems you will find that nearly a eighth of them are chiral. There is certainly a bias in this Encyclopedia, as we wanted at least one example in every space group. The best we can say is that non-organic compounds appear to be mostly achiral, while a significant number of organic compounds are chiral, most of them in Class III.

    If there are any chiral (or achiral) compounds you would like us to add to the Encyclopedia, please contact us. In particular we would like an enantiomorph of α D-glucose, and possibly the solid crystal structure of some of the amino acids.

    Footnotes

    I This is an expansion of the discussion of enantiomorphic space groups found in D. Hicks et al., Comp. Mat. Sci. 161, S1-S1011 (2019). (doi=10.1016/j.commatsci.2018.10.043). A brief summary of that work can be found on the Enantiomorphic space groups page.

    II In fact, the screw on the left is just a mirror image of the screw on the right.

    III And, as you might have guessed, the image on the right is just a mirror image of the one on the left.

    IV Although this time we actually flipped the atoms in the structure to generate the figures, rather than simply creating a mirror image.

    V An inversion in two dimensions is just a 180° rotation about the origin, and so can allow a chiral structure.

    References