Space lattices symmetry operations

The results of a crystal structure are usually expressed in the primary literature as a numerical table of positional coordinates and vibrational parameters for the atoms contained in an asymmetric unit of structure. The asymmetric unit is repeated by the appropriate combination of space-group symmetry operations and lattice translations to give the crystal structure. 

Bravais also showed that when the full range of lattice symmetry operations is taken into account, there are a maximum of 230 distinct varieties of crystal symmetry. The ensemble of these 230 configurations is the space groups, and the structure of any given crystal must be described by one of these. A detailed description of all 230 space groups is beyond the scope of this discussion, but such treatments are available [7]. 

To chemists, it makes perfect sense to think of electrons as being assigned to particular orbitals, and therefore the task of modeling the electronic ground state means building a wavefunction that describes these orbitals. The better the description of the orbitals (i.e. the better the basis set), the better the resulting calculation. This type of basis set is referred to as a localized basis set, and it can equally well be used to describe molecular crystals as well as isolated molecules, provided that the basis set is replicated by the space-group symmetry operations that describe how all molecules are located with respect to one another in the crystalline lattice. 

We have introduced an important concept here - the unit cell. In crystallography, the unit cell represents the budding block from which the infinite three-dimensional crystal lattice is built. If we are to model solid-state systems we must make use of a similar concept, from which we can build an infinite array of rephcas positioned in accordance with the crystallographic space-group symmetry operations. 

In Point Groups, one point of the lattice remains invarient under symmetry operations, i.e.- there is no translation involved. Space Groups are so-named because in each group all three- dimensional space remains invarient under operations of the group. That is, they contain translation components as well as the three symmetiy operations. We will not dwell upon the 231 Space Groups since these relate to determining the exact structure of the solid. However, we will show how the 32 Point Groups relate to crystal structure of solids. 

Commonly, only the atomic coordinates for the atoms in one asymmetric unit are listed. Atoms that can be generated from these by symmetry operations are not listed. Which symmetry operations are to be applied is revealed by stating the space group (cf Section 3.3). When the lattice parameters, the space group, and the atomic coordinates are known, all structural details can be deduced. In particular, all interatomic distances and angles can be calculated. 

The unit cell considered here is a primitive (P) unit cell that is, each unit cell has one lattice point. Nonprimitive cells contain two or more lattice points per unit cell. If the unit cell is centered in the (010) planes, this cell becomes a B unit cell for the (100) planes, an A cell for the (001) planes a C cell. Body-centered unit cells are designated I, and face-centered cells are called F. Regular packing of molecules into a crystal lattice often leads to symmetry relationships between the molecules. Common symmetry operations are two- or three-fold screw (rotation) axes, mirror planes, inversion centers (centers of symmetry), and rotation followed by inversion. There are 230 different ways to combine allowed symmetry operations in a crystal leading to 230 space groups.12 Not all of these are allowed for protein crystals because of amino acid asymmetry (only L-amino acids are found in proteins). Only those space groups without symmetry (triclinic) or with rotation or screw axes are allowed. However, mirror lines and inversion centers may occur in protein structures along an axis. 

Crystal lattices can be depicted not only by the lattice translation defined in Eq. (7.2), but also by the performance of various point symmetry operations. A symmetry operation is defined as an operation that moves the system into a new configuration that is equivalent to and indistinguishable from the original one. A symmetry element is a point, line, or plane with respect to which a symmetry operation is performed. The complete ensemble of symmetry operations that define the spatial properties of a molecule or its crystal are referred to as its group. In addition to the fundamental symmetry operations associated with molecular species that define the point group of the molecule, there are additional symmetry operations necessary to define the space group of its crystal. These will only be briefly outlined here, but additional information on molecular symmetry [10] and solid-state symmetry [11] is available. 

Notice that the symmetry operations of each point group by continued repetition always bring us back to the point from which we started. Considering, however, a space crystalline pattern, additional symmetry operations can be observed. These involve translation and therefore do not occur in point groups (or crystal classes). These additional operations are glide planes which correspond to a simultaneous reflection and translation and screw axis involving simultaneous rotation and translation. With subsequent application of these operations we do not obtain the point from which we started but another, equivalent, point of the lattice. The symbols used for such operations are exemplified as follows ... 

Taking into account these symmetry operations together with those corresponding to the translations characteristic of the different lattice types (see Fig. 3.4), it is possible to obtain 230 different combinations corresponding to the 230 space groups which describe the spatial symmetry of the structure on a microscopic... 

The collection of all symmetry operations that leave a crystalline lattice invariant forms a space group. Each type of crystal lattice has its specific space group. The problem of enumerating and describing all possible space groups, both two dimensional three dimensional, is a pure mathematical problem. It was completely resolved in the mid-nineteenth century. A contemporary tabulation of the properties of all space groups can be found in Hahn (1987). Bums and Glazer (1990) wrote an introductory book to that colossal table. 

The symmetry operations, G, of the space group acting on an atom placed at an arbitrary point in space will generate a set of mo equivalent atoms in the unit cell. Operation of the lattice translations, R, acting on this set generates an infinite array of such atoms, with the finite set of ma atoms being repeated at each point on the lattice. This is illustrated in Fig. 10.1 in which nia = 4 and each of the rectangles defined by the horizontal and vertical lines represents a unit cell that is identical with the one outlined with heavy lines. 

But this does nob end the tale of possible arrangements. Hitherto we have considered only those symmetry operations which carry us from one atom in the crystal to another associated with the same lattice point—the symmetry operations (rotation, reflection, or inversion through a point) which by continued repetition always bring us back to the atom from which we started. These are the point-group symmetries which were already familiar to us in crystal shapes. Now in "many space-patterns two additional types of symmetry operations can be discerned--types which involve translation and therefore do not occur in point-groups or crystal shapes. 

The symmetry of a unit cell is described by its space group, which is represented by a cryptic symbol (like P212121), in which a capital letter indicates the lattice type and the other symbols represent symmetry operations that can be carried out on the unit cell without changing its appearance. Mathematicians in the late 1800s showed that there are exactly 230 possible space groups. 

Matter is composed of spherical-like atoms. No two atomic cores—the nuclei plus inner shell electrons—can occupy the same volume of space, and it is impossible for spheres to fill all space completely. Consequently, spherical atoms coalesce into a solid with void spaces called interstices. A mathematical construct known as a space lattice may be envisioned, which is comprised of equidistant lattice points representing the geometric centers of structural motifs. The lattice points are equidistant since a lattice possesses translational invariance. A motif may be a single atom, a collection of atoms, an entire molecule, some fraction of a molecule, or an assembly of molecules. The motif is also referred to as the basis or, sometimes, the asymmetric unit, since it has no symmetry of its own. For example, in rock salt a sodium and chloride ion pair constitutes the asymmetric unit. This ion pair is repeated systematically, using point symmetry and translational symmetry operations, to form the space lattice of the crystal. 

We delay the presentation of the Bravais lattices and the space groups, and we first deal with the symmetry operators and the point groups. 

The 32 crystallographic point groups, first mentioned in Table 7.1, are now described in Table 7.8 (ordered by principal symmetry axes and also by the crystal system to which they belong). The 230 space groups of Schonflies and Fedorov were generated systematically by combining the 14 Bravais lattices with the intra-unit cell symmetry operations for the 32 crystallographic point... 

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