Anisotropic displacement parameters

One of the most popular refinement programs is the state-of-the-art package Refmac (Murshudov et ah, 1997). Refmac uses atomic parameters (xyz, B, occ) but also offers optimization of TLS and anisotropic displacement parameters. The objective function is a maximum likelihood derived residual that is available for structure factor amplitudes but can also include experimental phase information. Refmac boasts a sparse-matrix approximation to the normal matrix and also full matrix calculation. The program is extremely fast, very robust, and is capable of delivering excellent results over a wide range of resolutions. 

Capelli SC, Fortsch M, Biirgi HB (2000) Dynamics of molecules in crystals from multitemperature anisotropic displacement parameters. II. Application to benzene (C Dg) and urea [0C(NH)2]. Acta Crystallogr A 56 413 24... 

Munshi P, Madsen A0, Spackman MA, Larsen S, Destro R (2008) Estimated H-atom anisotropic displacement parameters a comparison between different methods and with neutron diffraction results. Acta Crystallogr A 64 465 75... 

Madsen A0 (2006) SHADE web server for estimation of hydrogen anisotropic displacement parameters. J Appl Crystallogr 39 757-758... 

A special kind of dynamic information can be supplied by careful analysis of anisotropic displacement parameters. Dunitz and others have used the thermal parameters from X-ray to provide valuable information on intra-and intermolecular mobility [36]. 

Table 2. Positional and anisotropic displacement parameters for PrMnO,. ...

Atoms in crystals seldom have isotropic environments, and a better approximation (but still an 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, the anisotropic vibration or displacement parameters, are introduced for each atom. Three of these parameters per atom give the orientations of the principal axes of the ellipsoid with respect to the unit cell axes. One of these principal axes is the direction of maximum displacement and the other two are perpendicular to this and also to each other. The other three parameters per atom represent the amounts of displacement along these three ellipsoidal axes. Some equations used to express anisotropic displacement parameters, which may be reported as 71, Uij, or jdjj, axe listed in Table 13.1. Most crystal structure determinations of all but the largest molecules include anisotropic temperature parameters for all atoms, except hydrogen, in the least-squares refinement. Usually, for brevity, the equivalent isotropic displacement factor Ueq, is published. This is expressed as ... 

Disorder in a crystal structure is frequently revealed by the shapes of the thermal ellipsoids obtained from the least-squares refinement of the anisotropic displacement parameters. An example is provided by the crystal structure determination of potassium dihydrogen isocitrate. One carboxyl oxygen atom is very anisotropic as a result of two possible hydrogen bonding schemes in which it can take part (Figure 13.10). 

The crystal structure found represents an average of these two possibilities. If the anisotropic displacement parameters do not correspond to ellipsoids but to other quadratic surfaces that are not everywhere positive, the atomic displacement parameters may lose their physical significance (they become nonpositive definite). 

The set of anisotropic displacement parameters, obtained from the least-squares refinement of the crystal structure (as described by Chapter 10) can be analyzed to obtain T, L and S. It has been assumed that there is no correlation between the motion of different atoms. Values of Uij are analyzed (again by an additional least-squares analysis) in such a way that good agreement is obtained between the refined values and those predicted when constants have been obtained for the T, L, and S tensors. The total number of anisotropic displacement parameters (6 per atom) is the input, and a total of 12 parameters for a centrosymmetric structure, or 20 parameters for a noncentrosymmetric structure, is the output of this least-squares analysis. The results consist of the molecular translational (T), librational (L), and screw (S) tensors. This treatment leads to estimates of corrections that should be made to bond distances. On the other hand, this type of analysis cannot be used for intermolec-ular distances because the correlation between the motion of different molecules is not known. 

Analyses of anisotropic displacement parameters give several pieces of information of interest to the chemist. 

If the mean-square amplitude of libration has been determined from a TLS analysis of the anisotropic displacement parameters, the force constant (for a harmonic oscillator) is... 

Emily Maverick and Jack Dunitz used the anisotropic displacement parameters from crystal structure analyses of metallocenes at various temperatures to estimate the barriers for the rotation of an individual C5H5 ring in the crystal about its five fold axis. The average mean-square amplitude of libration ( ) is is found to be 28° at 101 K, corresponding to an energy barrier of approximately 2 kcal moE (9.3 kJ... 

Highly anisotropic displacement parameters can also indicate incipient reactions. This has been illustrated by a reaction in which a boron-carbon bond is reversibly made and broken. The anisotropic displacement parameters and conformation of an intermediate give a good indication of the stereochemistry of the reaction. This will be discussed in Chapter 18. 

In order to determine whether a decrease in scattering at high angles is due to vibration effects or to disorder, the data should be measured at a series of temperatures. Only the vibration effects should show a strong temperature dependence. Displacements of atoms from their equilibrium positions can be anisotropic and are represented by anisotropic displacement parameters which, are refined by least-squares techniques together with the atomic coordinates (see Chapter 10). A further analysis of these anisotropic displacement parameters in terms of translation T, libration L, and screw S motions can give information on the nature of the molecular motion. 

Detailed analyses of anisotropic displacement parameters can give information on barriers to the rotation of groups, on the characteristic temperature, and on entropy. 

As follows from Eqs. 2.94 and 2.95, the relationships between By and Uij are identical to that given in Eq. 2.92 and both are measured in A. The P,y parameters in Eq. 2.93 are dimensionless but may be easily converted into By or Uy. Very high quality powder diffraction data are needed to obtain dependable anisotropic displacement parameters and even then, they may be reliable only for those atoms that have large scattering factors (see next section). On the other hand, the refinement of anisotropic displacement parameters is essential for those crystal structures, where strongly scattering atoms are distinctly anisotropic. 

The anisotropic displacement parameters can also be represented in a format of a tensor, Ty, i.e. a square matrix symmetrical with respect to its principal diagonal. For By it is given as... 

All nine elements of the tensor Ty establish the orientation of the ellipsoid in the coordinate basis of the crystal lattice. Hence, any or all nondiagonal elements may be positive or negative but certain relationships between them and the diagonal parameters should be observed as shown in Eq. 2.97 for By. If any of these three relationships between the anisotropic displacement parameters is violated, then the set of parameters has no physical meaning. 

The anisotropic displacement parameters can be visualized as ellipsoids Figure 2.54) that delineate the volume where atoms are located most of the time, typically at the 50 % probability level. The magnitude of the anisotropy and the orientations of the ellipsoids may be used to validate the model of the crystal structure and the quality of refinement by comparing thermal motions of atoms with their bonding states. Because of this, when new structural data are published, the ellipsoid plot is usually required when the results are based on single crystal diffraction data. 

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