Parameter displacement

The fitting parameters in the transfomi method are properties related to the two potential energy surfaces that define die electronic resonance. These curves are obtained when the two hypersurfaces are cut along theyth nomial mode coordinate. In order of increasing theoretical sophistication these properties are (i) the relative position of their minima (often called the displacement parameters), (ii) the force constant of the vibration (its frequency), (iii) nuclear coordinate dependence of the electronic transition moment and (iv) the issue of mode mixing upon excitation—known as the Duschinsky effect—requiring a multidimensional approach. 

Step 11. At this point a computer program refines the atomic parameters of the atoms that were assigned labels. The atomic parameters consist of the three position parameters x,j, and for each atom. Also one or six atomic displacement parameters that describe how the atom is "smeared" (due to thermal motion or disorder) are refined for each atom. The atomic parameters are varied so that the calculated reflection intensities are made to be as nearly equal as possible to the observed intensities. During this process, estimated phase angles are obtained for all of the reflections whose intensities were measured. A new three-dimensional electron density map is calculated using these calculated phase angles and the observed intensities. There is less false detail in this map than in the first map. 

Structure Determination from a Powder Pattern. In many cases it is possible to determine atomic positions and atomic displacement parameters from a powder pattern. The method is called the Rietveld method. Single-crystal stmcture deterrnination gives better results, but in many situations where it is impossible to obtain a suitable single crystal, the Rietveld method can produce adequate atomic and molecular stmctures from a powder pattern. 

Pt2,V and Pt y have been investigated at 1393 K and 1224 K respectively and we have explored the [100] and [110] planes of the reciprocal lattice. The measured Intensities have been Interpreted in a Sparks and Borie approach with first order displacements parameters and using a model Including 29 a(/ ) for PfsV and 21 for PtsV. In figure 1 is displayed the intensity distribution due to SRO a q) in the [100] plane. As for PdjV, the diffuse intensity of Pt V is spread along the (100) axes with maxima at the (100) positions, whereas the ground state is built on (1 j 0) concentration wave ( >022 phase). 

Roush, D. J., Gill, D. S., and Willson, R. C., Anion-exchange chromatographic behavior of recombinant rat cytochrome b5. Thermodynamic driving forces and temperature dependence of the stoichiometric displacement parameter Z, /. Chromatogr., 653, 207, 1993. 

In practice, the choice of parameters to be refined in the structural models requires a delicate balance between the risk of overfitting and the imposition of unnecessary bias from a rigidly constrained model. When the amount of experimental data is limited, and the model too flexible, high correlations between parameters arise during the least-squares fit, as is often the case with monopole populations and atomic displacement parameters [6], or with exponents for the various radial deformation functions [7]. 

Assuming spherical symmetric charge densities, the fourth group is forbidden. Hence, these reflections provide information about the anisotropic copper displacement parameters and chemical bonding, and a correct determination of these forbidden reflections gives evidence of the data quality. 

An isotropic extinction parameter, of type I and Lorentzian distribution (in the formalism of Becker and Coppens [16]), was also refined. The motions of the non-H atoms were described by anisotropic parameters, while those of the H atoms by isotropic B s. All these displacement parameters were included among the refinable quantities of the model, for a total of 1161 variables in a single least-squares matrix. 

With this structure model, the refinement readily converged to very good discrepancy factors RF2 = 5.4%, Rwp = 7.3%, Rp = 5.3%. More details concerning data collection and refinement are given in Table 1, whereas the atomic coordinate, occupancies and displacement parameters of ECS-2 are reported in Table 2. 

Table 2. Atomic coordinates, occupancies and displacement parameters for ECS-2.

Macromolecular crystallographic refinement is an example of a restrained optimization problem. Standard refinement programs adjust the atomic positions and, typically, also their atomic displacement parameters of a given model with the... 

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. 

High resolution (between 1.4 and 2.0 A) Automated model building with ARP/wARP should work with most phase sets. RESOLVE, which uses a template-based rather than atom-based approach, should also perform well but may be computationally more consuming. Refinement can best be carried out with REEMAC or PHENIX using isotropic ADPs since the amount of data is no longer sufficient for an anisotropic description of atomic displacement parameters. The use of TLS (Winn et ah, 2003) is highly recommended. A use of NCS restraints should be critically evaluated and in most cases the refinement can proceed without them. Double conformations of side chains should be visible and modelled. Ordered solvent can be modelled automatically. 

In practice, the probability distribution often includes static disorders in the crystal. The temperature parameter B in such cases is more properly described as a mean-square displacement parameter. 

S is the scattering vector, Mj is the atomic displacement parameter in this simplified notation assumed to be isotropic, 6 is the scattering angle, and 1 the wavelength of the incident radiation. The atomic displacement depends on the temperature, and hence so does the Debye-Waller factor. If an atom is modeled by a classical oscillator, then the atomic displacement would change linearly with temperature ... 

The typical behavior of an atomic displacement parameter is represented by the curve plotted in Fig. 2. This trend tells us that below the turn point (0e/2) atomic vibrations are not only smaller but also quite constant. 

As has become clear in previous sections, atomic thermal parameters refined from X-ray or neutron diffraction data contain information on the thermodynamics of a crystal, because they depend on the atom dynamics. However, as diffracted intensities (in kinematic approximation) provide magnitudes of structure factors, but not their phases, so atomic displacement parameters provide the mean amplitudes of atomic motion but not the phase of atomic displacement (i.e., the relative motion of atoms). This means that vibrational frequencies are not directly available from a model where Uij parameters are refined. However, Biirgi demonstrated [111] that such information is in fact available from sets of (7,yS refined on the same molecular crystals at different temperatures. 

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... 

Uncorrelated motion is most likely to be found when the bonds are weak, for example, when the cation is a large alkali metal. In this case the mean square amplitude, A, is given by the sum of the components of the atomic displacement parameters, U, of the two atoms along the bond direction, that is. 

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 2.2b Atomic Coordinates and Isotropic Displacement Parameters for si, si I, and sH Hydrates si Hydrate, CH4 5 75D2Oa ...

We calibrate the radial displacement parameter so that it coincides with the geodesic radius, and find the remarkable result that, on sufficiently large scales, the calibrated radius of a sphere centered on the chosen origin in the model universe then varies as the square root of the mass contained within the sphere. 

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