Anisotropic displacement

Definitive proof of the structure of porphine in the solid state awaits a variable-temperature crystallographic (X-ray or neutron diffraction) study the analysis of the anisotropic displacement factors (ADP) should disclose any rotational motion or its absence as well as determine the positions of the inner hydrogens. A search in the September 1998 version of the Cambridge Structural Database [CSD (91MI187)] showed that the only structures of porphine (codename PORPIN) were obtained in 1965 and 1972. 

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]. 

Estimated standard deviation given in brackets to same significance level as parameter. x,y,z given as decimal fraction of cell edge. Anisotropic displacement factor given as 103 exp - 2tt. a a h.h. Fixed to define origin. J J J... 

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

Supplementary Material Available Tables SI and SlI, listing crystal data and details ofdaia collection and anisotropic displacement coefficients for RuFj (3 pages). Ordering information is given on any current masthead page. 

As the resolution of the Bragg reflection data is improved, it becomes possible to obtain information on the more minute details of electron density in a molecule. At high enough resolution information can be obtained on the redistribution of electron density (deformation density) around atoms when they combine to form a molecule. Electrons in molecules ma -form bonds or exist as lone pairs, thereby distorting the electron density around each atom and requiring a more complicated function to describe this overall electron density than normally used, in which it is treated as if it were spherically symmetrical (deformed to an ellipsoid in order to account for anisotropic displacements). This assumption is inherent in the use of spherically-symmetrical scattering factors although the elec-... 

Atomic parameters A set of numbers that specifies the position of an atom in the unit cell (atomic coordinates), the extent of its displacement about an equilibrium position (vibration), and an occupancy factor (generally 1.0). Three parameters define position, one parameter can be used to define isotropic displacements, or six to define anisotropic displacements and one parameter defines the site occupancy. 

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... 

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