Crystallographic Structural Data P. Ugliengo

Summary

• Basis concepts and notations
• Evaluating the accuracy of experimental data
• Source of structural data
• Final remarks

The purposes of this short notes for non-crystallographers are:

• introducing the basic concepts and notations found in crystallographic literature
• evaluating the accuracy of the experimental data
• showing where and how crystal information is available

Basic concepts and notations

How crystal data are obtained?

When a crystal is exposed to a beam of monochromatic X rays, the electron density of any atom diffuses the X ray radiation in all directions. Because of the lattice repetition, the radiation diffused by one atom will interfere with that diffused by all other atoms belonging to the crystal. The result is a diffraction pattern:

Nuclei do not play any role in the diffraction process.
The lattice enhances the destructive interference between the diffused waves. There are, however, a number of directions along which constructive interference is possible, according to the well known Bragg formula.
 

Fractional coordinates

Fractional coordinates are dimensionless triplets (in 3D) of numbers which are used to identify atoms in the crystallographic unit cell. Let us consider a 2D unit cell with the following cell parameters;

The point P has fractional coordinates x=1/3 and y=1/2.
The cartesian coordinates (x' y') of P are related  to the fractional coordinates as:

Asymmetric unit

Let us consider the contents of a given unit cell with symmetry belonging to one of the 230 space groups.
Because of the presence of the symmetry elements not all atoms of the unit cell are independent. In other words, there is a subset of atoms from which we can generate all the remaining atoms of the unit cell by the action of the symmetry operators on their coordinates.

This subset is called asymmetric unit (see the objects within the grey area of the picture). Usually, only this set of atoms is given both in papers and crystallographic data bases. The space group and the cell parameters (a, b, c, α, β, γ) are then needed to build up the unit cell. Only non-redundant cell parameters are usually specified, i.e. for a cubic system only the cell size a is given whereas for a monoclinic system only a, b, c and β are specified. The number of asymmetric units which are generated by the action of all symmetry operators is indicated with the letter Z.
Because the CRYSTAL code is able to take advantage of both the space and point group symmetries to avoid redundant calculations of symmetry related integrals, it is the size of the asymmetric unit which should be considered carefully before starting any calculation. Indeed, some class of interesting materials may be simply out of reach with the actual technology because of a too large asymmetric unit.

Evaluating the accuracy of experimental data

Vibrations in crystals

Even at 0 K atoms are vibrating around their equilibrium positions with frequencies of the order of 10¹³ per second. The frequency of X rays are much higher, of the order of 2x10¹⁸per second. As a result, the atom may vibrate and be "viewed" by X rays as apparently stationary, but displaced in a random manner from its average location to some other location along one of its possible vibration pathways.

The graph shows an hypothetical potential energy surface (PES) of an atom vibrating around its equilibrium position. In this case the displacement is small because the energy increase for larger displacements is large. It is clear that larger displacements are possible at higher temperature.

The temperature factor

Due to the vibrational motion the ρ(r) of an atom is apparently spread over the region of the vibration and the scattered amplitude f_j being the Fourier Transform of the ρ(r) decreases much faster at high values of θ with respect to that resulting from a fixed atom:

As a result the atomic scattering factor at temperature T is written as:

where B = 8 π² < u² > = 8 π² U in which U is the mean square displacement of each atom from its average position. Experimental values of B are 2-6 Ų, i.e. atoms displacements are in the range 0.1-0.3 Å. The simplest assumption to make about atomic displacements is to consider them as isotropic. Only a single term is needed, named B_iso. However, atoms in crystals seldom have isotropic environments, and a better approximation is to describe the atomic motion in terms of an ellipsoid, with larger amplitudes of vibration in some directions than in others. Six parameters are then introduced for each atom.

ORTEP Diagrams and T dependence

In this picture are shown atomic displacement ellipsoids at different temperatures, drawn with ORTEP (Johnson, 1965) for naphthalene, studied by X-ray diffraction at 92 K and 239 K. Note the increase in the sizes of the ellipsoids at the higher temperature where the atomic motion is higher.

Librations in crystals

In some structures of molecular crystals, the motion of entire fragments within the same molecule become possible because of the small energy involved for changing torsion angles. A similar situation occurs when the molecule may rotate as a whole in the crystal due to very low rotation energy barriers. From the experimental point of view this means that the electron density associated with the librating atoms is spread over larger portion of the unit cell with respect to the case of quasi-harmonic vibrations. Consequently, larger and more anisotropic ellipsoids are seen.

The graph shows an hypothetical potential energy surface (PES) for an atomic group librating around two equivalent equilibrium positions. The energy required to surmount the barrier separating the two minima is very low. It is clear that larger displacements are possible at higher temperature. In such cases, the crystallographic data should be used with great caution in any q.m. calculations because of the larger standard deviations associated with the librating atoms.

Rotational motion

Atomic displacement ellipsoids of monoclinic ferrocene at 173 K and 293 K. From the anisotropic displacements parameters of metallocenes at various temperatures Maverick and Dunitz were able to estimate the barrier of the rotation of an individual C₅H₅ ring in the crystal about its five fold axis. From the estimated value of 9.3 kJ/mol it was possible to calculate that the ring rotates at a frequency of 3x10⁷ sec^-1 at 100 K.

Static disorder

Usually a data collection can take from few hours to a few days to be completed. The diffraction pattern is due to the interference of all waves diffused by a very large number of unit cells (many billions) of the crystal. The observed intensities are then the result of an average process both in time and in space. If, for some reason, the crystal is made up of two different unit cells, the diffracted pattern will contain the scattered amplitudes from both configurations. An example is a crystal in which a group or a single atom occupies alternatively two different positions (1 and 2) in the unit cell. This clearly happens when the two positions can be occupied isoenergetically.
Configurational entropy of the whole crystal is then increased as a result of this form of static disorder.

In such cases the two positions are separated by high potential energy barriers so that dynamic disorder is not allowed. This also means that static disorder is independent from the temperature at which the data collection is carried out. If site occupancy p of the disordered atoms is fixed at 1 during the structure refinement, their B factors become abnormally large. A good example of fractional site occupancy is provided by the structures of many zeolites in which substitutional Al³⁺ and the corresponding charge balancing cation may occur in two or more equivalent tetrahedral sites.

Standard deviations of the atomic parameters

At the end of a structural determination one also get the standard deviations of all parameters.
For instance, the atomic coordinates are given with their standard deviations:

Their values are the best available parameters to evaluate the quality of a structure, but unfortunately they are not reported in most entries of the crystallographic databases. When the assessment of the quality of the data is important it is mandatory to check the original paper.

Form the σ's it is possible to compute the estimated standard deviations of all the derived geometrical parameters. In particular, for a good structure, we have:

In order to decrease the effect of the thermal motion and to increase the accuracy of structure determination, sometimes the diffraction measurements are carried out at low temperature (more often at 120 K with liquid nitrogen, and in few cases at around 10 K with liquid helium). Very accurate measurements at low temperatures are essential for meaningful charge density studies.

Source of structural data

Structural data papers

The usual way in which structural data are provided is by means of a paper. Shown below, is the case of a recent structural determination of Zeolite-N from synchrotron X-ray powder diffraction, published on Acta Chemica Scandinava. Data in Table 1 show the atom label, the site symmetry, the occupancy factor, the fractional atomic coordinates with the associated standard deviations as numbers enclosed in parentheses and the isotropic B factor. It is interesting to note the fractional occupancy of the water molecules OW1 and OW2 which is an indication of static disorder. If one use the coordinates as such non-chemical contacts between the disordered groups may arise as shown for the oxygen atoms of water in the enclosed picture.

Structural data from ICSD

The Inorganic Crystal Structure Database ICSD, is maintained through a collaboration of the Fachinformationszentrum Energie Physik Mathematik, Karlsruhe, Germany, and the Institute for Inorganic Chemistry of the University of Bonn, Germany, and is managed by Gunter Bergerhoff and I. David Brown. The ICSD contains information on all compounds containing at least one non-metallic element but non C-C or C-H bonds, which are covered by the Cambridge Structural Database. Each reported crystal structure has a separate entry. Information provided in the database includes the chemical name, phase designation, unit cell dimensions, density, space group and the oxidation state of the elements. Also listed are R value, temperature, pressure, method of measurement and the full journal reference Atomic information includes coordinates and displacement parameters as well as site occupancy.

Structural data from CSD

The Cambridge Structural DataBase (CSD) contains information on approximately 170.000 3D crystal structure determinations that have been studied by X-ray or neutron diffraction. All crystallographic structure determinations of carbon-containing compounds are included. The data contained in the CSD include: the atomic coordinates, information on the space group, chemical connectivity, and the literature reference to each structure determination The CSD was built up by 0. Kennard, F.H. Allen, D.G. Watson, W.D.S. Motherwell and R. Taylor and is maintained in Cambridge, England. Below is the result for the urea crystal and the picture shows the unit cell content with H-bonds highlighted.

Final remarks

When coordinates are needed to do some q.m. calculations try to follow these recipes:

• begin by searching the crystal coordinates in an electronic database (ICSD for inorganic and CSD for organometallic and organic structures)
• if possible, select those structures which have been solved after 1970. Before that date intensities were affected by larger errors with respect to the actual standards
• make a pre-screen of the retrieved structures upon the value of the agreement factor R. Structure with R > 0.1 are too inaccurate
• structures which have been solved at low temperature (liquid nitrogen) are to be preferred over those determined at room temperature. If no disorder is present they also give electron density which can be compared with that computed by q.m. methods
• check carefully in the original papers if disorder or other subtle effects are affecting the accuracy of the structural data
• to study hydrogen bonded crystals it should be kept in mind that coordinates of the hydrogen atoms determined by X-ray are too inaccurate. Try to find a redetermination of the structure by neutron diffraction or adjust the H positions following standard geometrical rules
• before doing any q.m. calculations run computer graphic programs to show the unit cell in order to check that symmetry and connectivity information are correct. Computer graphics is also useful to visualize crystal directions and atoms on exposed surfaces