Reciprocal space methods

Cisneros GA, Piquemal J-P, Darden TA (2006) Generalization of the Gaussian electrostatic model extension to arbitrary angular momentum, distributed multipoles and speedup with reciprocal space methods. J Chem Phys 125 184101... 

Structure solution in powder diffraction is approached by two different methodologies. One is using the conventional reciprocal space methods. The second is by real space methods where all the known details about the sample (say, molecular details such as bond distances, angles, etc., for an organic molecule, and coordination spheres such as octahedral, tetrahedral etc., in case of inorganic compounds) in question are exploited to solve the structure. 

In theory single crystal methods can be effectively used for structure solution from powder diffraction X-ray data. However, in powder diffraction the number of peaks involved are limited, and hence the data-to-parameter ratio is very small, due to the transfer of the three-dimensional data to one dimension, namely, 29. In spite of the inherent shortcomings, conventional crystallographic methods such as Direct, Patterson and maximum entropy methods have been successfully applied to powder diffraction data. The most popular program, which uses reciprocal space methods for stmcture solution is EXPO. ... 

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. 

In any of the reciprocal space methods, which are based exclusively on the use of the observed structure factors, the powder diffraction pattern must be deconvoluted and the integrated intensities of all, or as many as possible, individual Bragg reflections determined with a maximum precision. Only then, Patterson or direct phase angle determination techniques may be employed to create a partial or compete structural model. Theoretical background supporting these two methods was reviewed in section 2.14. 

XRD (x-ray diffiaction)—a reciprocal-space method of structural analysis that measures the structure of a lattice based on the diffraction pattern generated when x-ray radiation passes through a regular crystal lattice. 

In this contribution, we present the theory behind the GEM method and recent advances and results on the application of two hybrid GEM potentials. In Section 8.2, we provide a brief review of the analytical and numerical density fitting methods and its implementation, including the methods employed to control numerical instabilities. This is followed by a review of the procedure to obtain distributed site multipoles from the fitted Hermite coefficients in Section 8.3. Section 8.4 describes the extension of reciprocal space methods for continuous densities. Section 8.5 describes the complete form for GEM and a novel hybrid force field, GEM, which combines term from GEM and AMOEBA for MD simulations. Finally, Section 8.6 describes the implementation and initial applications of a multi-scale program that combines GEM and SIBFA. 

The use of molecular densities results in the need to compute a large number of two center integrals for the intermolecular interaction. A significant computational speedup can be achieved by using reciprocal space methods based on Ewald sums. In this way, the integrals are calculated in direct or reciprocal space depending on the exponent of the Gaussian Hermites. 

For understanding and predicting material properties such as density, elasticity, magnetization or hardness from first principles quantum mechanical calculations, a reliable and efficient tool for electronic structure calculations is necessary. The reciprocal space methods, to which most attention has been dedicated so far, are very powerful and sophisticated but by their nature are suitable mostly for crystals. For systems without translational symmetry such as metallic clusters, defects, quantum dots, adsorbates and nanocrystals, use of real-space methods is more promising. 

Matsen and Schick [14] solved a set of self-consistent mean-fleld equations using the reciprocal space method. They started with the following form of the partition function Z for an AB-type diblock copolymer ... 

G. A. Cisneros, J. P. Piquemal, and T. A. Darden,/. Chem. Phys., 125(18), 184101 (2006). Generalization of the Gaussian Electrostatic Model Extension to Arbitrary Angular Momentum, Distributed Multipoles, and Speedup with Reciprocal Space Methods. 

Since the formulation of the SCFT for block copolymers by Helfand in 1975 [5], great effort has been devoted to the solution of the SCFT equations. To date, the most efficient and accurate method to solve the self-consistent mean-field equations is the reciprocal-space method developed by Matsen and Schick [14], which is based on the expansion in terms of plane wave-like basis functions. Recently, with the availability of increasing computing power and new numerical techniques, real-space methods have been developed to the level that they can be used to explore the possible phases for a given block copolymer architecture [17-19]. 

The reciprocal-space method described above is numerically efficient for a precise computation of free energies and phase diagrams. However, this method requires the space group of the ordered phase as an input. It is therefore desirable to develop methods that do not require prior knowledge of the symmetry of the phase. One possibihty of achieving this is to solve the SCMFT equations in real space. 

Within the mean-field approximation, the reciprocal-space method of Matsen and Schick [14] provides an efficient and accurate numerical technique to solve the SCMFT equations for given ordered structures. This has led to a comprehensive understanding of the equilibrium phase behavior of simple block copolymer systems. Valuable insights into the physics of the self-assembly in block copolymer systems have been obtained from the numerical solutions. In particular, the formation of different structures can be explained using the concepts of spontaneous interfadal curvature and packing frustration [3]. 

The reeiproeal method of SCMFT requires a priori knowledge of the structure of the phases. The recently introduced real-space methods have the potential to predict the phases without prior assumptions about the structures [17-19]. It is feasible that a combination of the real-space and reciprocal-space methods will lead to a numerical platform that is capable of predicting the phases and phase diagrams for complex block copolymers. In this scheme the real-space method can be used to earry out a combinatorial search for the possible candidate structures, and the reeiprocal-space method can then be used to obtain accurate free energies of these candidate structures. 

TRLS (i.e., a reciprocal-space method) has been used to measure the mean radius of curvature of the phase-separated structure [78, 79]. Since we evaluated (R) from the PSM (i.e., real-space method), it is worthwhile to compare the radius of curvature from both methods. The reciprocal-space method relies on the following Tomita s theory [80]. 

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