Structure factor direct calculation

The absolute configuration of a-D-glucopyranose monohydrate has also been confirmed directly by the anomalous scattering of the oxygen atoms.70 After the atomic positions had been refined, the structure factors were calculated by using the oxygen anomalous scatter-... 

Figure 9.1 shows the simulated total neutron scattering functions obtained from both the compressed and quenched configurations and compared to those obtained from high pressure neutron scattering experiments [67]. The (Ashcroft-Langreth) structure factors are calculated directly from the atom positions via =... 

In the literature, several functional forms for S(qR ) can be found [51,52]. The difficulty lies in obtaining an expression for the structure factor that is valid for the whole range qR > 1. Lin et al. [53] calculated the aggregate structure factor directly from computer-generated clusters obtained under diffusion and reaction-limited conditions. They parameterized their result by fitting the polynomial expression ... 

There are four important techniques used in surmounting the phase problem trial-and-error, Patterson maps, direct methods, and isomorphous replacement. Most of the simple structures have been determined by trial and error. Structure factors are calculated from an assumed set of coordinates / exp[2jri(Iix -I- ky + IzJ]... 

The comparison with experiment can be made at several levels. The first, and most common, is in the comparison of derived quantities that are not directly measurable, for example, a set of average crystal coordinates or a diffusion constant. A comparison at this level is convenient in that the quantities involved describe directly the structure and dynamics of the system. However, the obtainment of these quantities, from experiment and/or simulation, may require approximation and model-dependent data analysis. For example, to obtain experimentally a set of average crystallographic coordinates, a physical model to interpret an electron density map must be imposed. To avoid these problems the comparison can be made at the level of the measured quantities themselves, such as diffraction intensities or dynamic structure factors. A comparison at this level still involves some approximation. For example, background corrections have to made in the experimental data reduction. However, fewer approximations are necessary for the structure and dynamics of the sample itself, and comparison with experiment is normally more direct. This approach requires a little more work on the part of the computer simulation team, because methods for calculating experimental intensities from simulation configurations must be developed. The comparisons made here are of experimentally measurable quantities. 

Much of what is knotm about the structure response of the ECD is based on empirical observations. Clearly, the ability to correlate the response of the detector to fundamental molecular parameters would be useful. Chen and Wentworth have shorn that the information required for this purpose is the electron affinity of the molecule, the rate constant for the electron attachment reaction and its activation energy, and the rate constant for the, ionic recombination reaction [117,141,142]. in general, the direct calculation of detector response factors have rarely Jseen carried j out, since the electron affinities and rate constants for most compounds of interest are unknown. 

In this work I choose a different constraint function. Instead of working with the charge density in real space, I prefer to work directly with the experimentally measured structure factors, Ft. These structure factors are directly related to the charge density by a Fourier transform, as will be shown in the next section. To constrain the calculated cell charge density to be the same as experiment, a Lagrange multiplier technique is used to minimise the x2 statistic,... 

The decay of the structural correlations measured by the static structure factor can be studied by dynamic scattering techniques. From the simulations, the decay of structural correlations is determined most directly by calculating the coherent intermediate scattering function, which differs from Eq. [1] by a time shift in one of the particle positions as defined in Eq. [2] ... 

Accurate measurements of low order structure factors are based on the refinement technique described in section 4. Using the small electron probe, a region of perfect crystal is selected for study. The measurements are made by comparing experimental intensity profiles across CBED disks (rocking curves) with calculations, as illustrated in fig. 5. The intensity was calculated using the Bloch wave method, with structure factors, absorption coefficients, the beam direction and thickness treated as refinement parameters. 

There are two approaches to map crystal charge density from the measured structure factors by inverse Fourier transform or by the multipole method [32]. Direct Fourier transform of experimental structure factors was not useful due to the missing reflections in the collected data set, so a multipole refinement is a better approach to map charge density from the measured structure factors. In the multipole method, the crystal charge density is expanded as a sum of non-spherical pseudo-atomic densities. These consist of a spherical-atom (or ion) charge density obtained from multi-configuration Dirac-Fock (MCDF) calculations [33] with variable orbital occupation factors to allow for charge transfer, and a small non-spherical part in which local symmetry-adapted spherical harmonic functions were used. 

In direct methods calculations we use normalised structure factors E(hkl), which are the structure factors compensated for the fall-off of the atomic scattering factors f hkl). In fact this procedure tries to simulate point-like scattering centres. 

When observed structure factors are used, the thermally averaged deformation density, often labeled the dynamic deformation density, is obtained. An attractive alternative is to replace the observed structure factors in Eq. (5.8) by those calculated with the multipole model. The resulting dynamic model deformation map is model dependent, but any noise not fitted by the muitipole functions will be eliminated. It is also possible to plot the model density directly using the model functions and the experimental charge density parameters. In that case, thermal motion can be eliminated (subject to the approximations of the thermal motion formalism ), and an image of the static model deformation density is obtained, as discussed further in section 5.2.4. 

A better alternative is to use the difference structure factor AF in the summations. The electrostatic properties of the procrystal are rapidly convergent and can therefore be easily evaluated in direct space. Stewart (1991) describes a series of model calculations on the diatomic molecules N2, CO, and SiO, placed in cubic crystal lattices and assigned realistic mean-square amplitudes of vibration. He reports that for an error tolerance level of 1%, (sin 0/2)max = 1-1.1 A-1 is adequate for the deformation electrostatic potential, 1.5 A-1 for the electric field, and 2.0 A 1 for the deformation density and the deformation electric field gradient (which both have Fourier coefficients proportional to H°). 

The electrostatic energy of a molecular crystal can be evaluated with summation over the structure factors in Eq. (9.15). But to obtain the cohesive energy of a molecular crystal with such a summation, we would have to subtract the molecular electrostatic energies, which are implicitly included in the result. An alternative is to perform the calculation in direct space. 

An X-ray atomic orbital (XAO) [77] method has also been adopted to refine electronic states directly. The method is applicable mainly to analyse the electron-density distribution in ionic solids of transition or rare earth metals, given that it is based on an atomic orbital assumption, neglecting molecular orbitals. The expansion coefficients of each atomic orbital are calculated with a perturbation theory and the coefficients of each orbital are refined to fit the observed structure factors keeping the orthonormal relationships among them. This model is somewhat similar to the valence orbital model (VOM), earlier introduced by Figgis et al. [78] to study transition metal complexes, within the Ligand field theory approach. The VOM could be applied in such complexes, within the assumption that the metal and the... 

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