Structure solution, direct method

Altomare A., Cascarano G., Giacovazzo C., Guagliardi A., Burta M. C., Polidori G. and Camalli M., SIRPOW-92 - a program for automatic solution of crystal structures by direct methods optimised for powder data, J. Appl. Cryst. (1994) 27 pp. 435-436. 

One of the specihc types of solutions for ab initio electronic structure was direct methods wherein intermediate quantities (two-electron integrals) normally stored on disk were recomputed when needed [7]. Binkley s report went on to say that the effort to adapt to the special features of vector and parallel architectures led to the production of better scalar algorithms. In other words, the basic ideas behind algorithms were influenced by the technology, in this case, computer architecture, and this is really a very significant and constant theme in the evolution of theoretical and computational chemistry. 

Figure 1, Computei-generaled molecular smiclure diagram after solution of a crystal structure by direct methods...

X-Ray diffraction from single crystals is the most direct and powerful experimental tool available to determine molecular structures and intermolecular interactions at atomic resolution. Monochromatic CuKa radiation of wavelength (X) 1.5418 A is commonly used to collect the X-ray intensities diffracted by the electrons in the crystal. The structure amplitudes, whose squares are the intensities of the reflections, coupled with their appropriate phases, are the basic ingredients to locate atomic positions. Because phases cannot be experimentally recorded, the phase problem has to be resolved by one of the well-known techniques the heavy-atom method, the direct method, anomalous dispersion, and isomorphous replacement.1 Once approximate phases of some strong reflections are obtained, the electron-density maps computed by Fourier summation, which requires both amplitudes and phases, lead to a partial solution of the crystal structure. Phases based on this initial structure can be used to include previously omitted reflections so that in a couple of trials, the entire structure is traced at a high resolution. Difference Fourier maps at this stage are helpful to locate ions and solvent molecules. Subsequent refinement of the crystal structure by well-known least-squares methods ensures reliable atomic coordinates and thermal parameters. 

X-ray structural analysis. Suitable crystals of compound 14 were obtained from toluene/ether solutions. X-ray data were collected on a STOE-IPDS diffractometer using graphite monochromated Mo-Ka radiation. The structure was solved by direct methods (SHELXS-86)16 and refined by full-matrix-least-squares techniques against F2 (SHELXL-93).17 Crystal dimensions 0.3 0.2 0.1 mm, yellow-orange prisms, 3612 reflections measured, 3612 were independent of symmetry and 1624 were observed (I > 2ct(7)), R1 = 0.048, wR2 (all data) = 0.151, 295 parameters. 

Sparse matrices are ones in which the majority of the elements are zero. If the structure of the matrix is exploited, the solution time on a computer is greatly reduced. See Duff, I. S., J. K. Reid, and A. M. Erisman (eds.), Direct Methods for Sparse Matrices, Clarendon Press, Oxford (1986) Saad, Y., Iterative Methods for Sparse Linear Systems, 2d ed., Society for Industrial and Applied Mathematics, Philadelphia (2003). The conjugate gradient method is one method for solving sparse matrix problems, since it only involves multiplication of a matrix times a vector. Thus the sparseness of the matrix is easy to exploit. The conjugate gradient method is an iterative method that converges for sure in n iterations where the matrix is an n x n matrix. 

The relatively recent development27 of the direct methods of crystal structure analysis has produced a great increase in the number of crystal structures reported in the literature, particularly with regard to the possible hydrogen bonds (also for biological molecules). Hence, the classical spectroscopic data on hydrogen bonding in solution are backed up by X-ray diffraction analysis data. 

Case study 2 Structural solution of zeolite from electron diffraction data, with a help of (a) Direct method, and (b) Patterson Map... 

Application of direct methods to macromolecular structure solution... 

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. 

The traditional approach for structure solution follows a close analogy to the analysis of single-crystal XRD data, in that the intensities 1(H) of individual reflections are extracted directly from the powder XRD pattern and are then used in the types of structure solution calculation (e.g. direct methods, Patterson methods or the recently developed charge-flipping methodology [32-34]) that are used for single-crystal XRD data. As discussed above, however, peak overlap in the powder XRD pattern can limit the reliability of the extracted intensities, and uncertainties in the intensities can lead to difficulties in subsequent attempts to solve the structure. As noted above, such problems may be particularly severe in cases of large unit cells and low symmetry, as encountered for most molecular solids. In spite of these intrinsic difficulties, however, there have been several reported successes in the application of traditional techniques for structure solution of molecular solids from powder XRD data. 

It is true that in some cases, the spectroscopic data on a reactive intermediate are so persuasive that the connection between structure and spectroscopic features is firm. However, in general this will not be the case, and additional spectroscopic or preparative criteria will have to be provided. So we are faced with the question How can we connect the information obtained, for example, from observations in matrices or in solution-phase fast kinetic studies, to molecular structure How do we know that the results of these experiments, using what we hopefully call direct methods, really pertain to the species we are trying to characterize I attempt to deal with this issue in what follows. Since the methods used vary from one class of non-Kekule species to another, specific classes are individually discussed, and special techniques are introduced as needed. Electron spin resonance spectroscopy has played such a pervasive role that it will be useful to give first a brief outline of that method. 

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