Direct-space approach

In 2000 Tremayne and Glidewell claimed the first instance of a structure determination of a cocrystal material by the direct-space approach [62]. Their work involved a 1 1 cocrystal of 1,2,3-trihydroxybenzene 37 and hexamethyl-... 

Techniques for structure solution from powder XRD data can be subdivided into two categories the traditional and direct-space approaches. 

We emphasize that an important feature contributing to the success of the direct-space approach is that it makes maximal use of information on molecular geometry that is already known reliably, independently of the powder XRD data, prior to commencing the stmcture solution calculation. The traditional approach for strac-ture solution, on the other hand, does not (in general) utilize prior knowledge of features of molecular geometry. 

For stmcture solution by the direct-space approach, the complexity of the stmcture solution problem is dictated largely by the dimensionality of the hypersurface to be explored (i.e. the total number of stmctural variables in the set F) rather than the number of atoms in the asymmetric unit. Thus, the greatest challenges in the application of direct-space techniques arise when the number of stmctural variables is large this situation occurs when there is considerable molecular flexibility (i.e. when the molecular geometry is defined by a large number of variable torsion angles) and/or when there are several independent molecules in the asymmetric unit. 

Determined by X-ray crystallographic analysis. b Solved by the direct-space approach using the Monte Carlo algorithm with the subsequent Rietveld refinement. 

The metastable y-form crystal structure prior to the slow phase transitions was determined by the X-ray crystallographic analysis, but the crystal structures of three other new polymorphs ( -, r)- and e-forms) were undetermined by the same X-ray analysis, because these polymorphs were obtained as powder samples. Since the nearly racemic e-form crystals could be obtained as the monophasic powder sample by just leaving the crystallization mixture untouched, its crystal structure was solved from its X-ray diffraction (XRD) data by the direct-space approach using the Monte Carlo algorithm with the subsequent Rietveld refinement.26,27 Consequently, the Rwp value has been satisfactorily converged to 9.1%. 

In the direct-space approach [26-28] for solving crystal structures from powder diffraction data, trial crystal structures are generated in direct space, independently of the experimental powder diffraction data. The powder diffraction pattern for the trial structure is calculated automatically using Eq. (1) in Sect. 2.2 [the structure factor amplitudes F(h) obtained using this equation are used to determine the relative intensities 1(h) of the diffraction maxima in the powder diffraction pattern]. The suitability of each trial structure is then assessed by direct comparison between the experimental powder diffraction pattern and the powder diffraction pattern calculated for the trial structure. The comparison between the experimental and calculated powder diffraction patterns is quanti-... 

Once the content of the unit cell has been established, a model of the crystal structure should be created using either direct or reciprocal space techniques, or a combination of both. Direct space approaches do not mandate immediate use of the observed integrated intensities, while reciprocal space methods are based on them. 

There are many ways to build a model of the crystal structure of a polycrystalline material without first using the intensities of individual Bragg reflections, which are hidden in powder diffraction due to partial or complete overlapping. Most of the direct space approaches are, in effect, trial-and-error methods and they include some or all of the following components ... 

These findings have an immediate return on the computational performances of the direct space approach. As we have previously shown, the decay of Fpi is not only affected by the localized nature of the involved oasis set but also, through its exchange part, by the convergence of i p ... 

N.2 Computational speedup for the direct and reciprocal sums Computational speedups can be obtained for both the direct and reciprocal contributions. In the direct space sum, the issue is the efficient evaluation of the erfc function. One method proposed by Sagui et al. [64] relies on the McMurchie-Davidson [57] recursion to calculate the required erfc and higher derivatives for the multipoles. This same approach has been used by the authors for GEM [15]. This approach has been shown to be applicable not only for the Coulomb operator but to other types of operators such as overlap [15, 62],... 

To discuss the influence of aberrations in the HRTEM image formation process in more detail, it is convenient to work in Fourier space, where the real-space quantities I r), T (r), V r), and T r) are related to their counterparts 7(g), T (g), Vp(g) and T(g) by a Fourier Transformation. Distances, d, in direct space correspond to spatial frequencies, g, in Fourier space. With this approach, the electron wave can be expressed as... 

Chowdhury K., Bhattacharya, S. and Mukherjee, M. (2005). Ab initio structure solution of nucleic acids and proteins by direct methods reciprocal-space and real-space approach. /. Appl. Cryst. 38,217-222. 

It is at once obvious that Fourier transformation of equation (2.2) should yield information about all the j shells in direct space that contribute to the EXAFS. The Rjs so obtained are, however, shortened by the k-dependent part of /k). Since the intensity of the outgoing spherical wave decreases very rapidly with increasing R, distant atoms contribute very little to the fine structure. Multiple scattering effects are also relatively unimportant and these have indeed been ignored in the derivation of equation (2.2). EXAFS should contain no information about shadowed or eclipsed atoms, but there are exceptions to this. Other theoretical approaches also use similar effects to explain the EXAFS. 

Electronic structure methods for studies of nanostructures can be divided broadly into supercell methods and real-space methods. Supercell methods use standard k-space electronic structure techniques separating periodically repeated nanostructures by distances large enough to neglect their interactions. Direct space methods do not need to use periodic boundary conditions. Various electronic structure methods are developed and applied using both approaches. In this section we will shortly discuss few popular but powerful electronic structure methods the pseudopotential method, linear muffin-tin orbital and related methods, and tight-binding methods. 

This PWE was used in [18] to obtain the numerical results. For the numerical implementation the B-spline approximation [21] was chosen that represents actually the refined version of the space discretization approach. In Table 1 the convergence of the PWE approach with the multicommutator expansion is presented for the lowest-order SE correction for the ground state of hydrogenlike ions with Z = 10. The minimal set of parameters for the numerical spline calcuations was chosen to be the number of grid points N = 20, the number of splines k = 9. This minimal set allowed to keep a controlled inaccuracy below 10%. What is most important for the further generalization of the PWE approach to the second-order SESE calculation is that with Zmax = 3 the inaccuracy is already below 10% (see Table 1). The same picture holds with even higher accuracy for larger Z values. The direct renormalization approach is not necessarily connected with the PWE. In [19] this approach in the form of the multicommutator expansion (Eq. (16)) was employed in combination with the Taylor expansion in powers of (Ea — En>)r 12 The numerical procedure with the use of B-splines and 3 terms of Taylor series yielded an accuracy comparable with the PWE-expansion with Zmax = 3. 

As already described, the GA approach has proven very successful as a tool for direct-space crystal structure solution from powder diffraction data. However, there remain many opportunities for optimizing GA methodologies. In this section, we discuss one approach for increasing the power and scope of GA methodologies which takes advantage of modern computational techniques. 

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