Efficient spatial transformations

The density here refers to the spatial coordinate, i.e. the concentration of the reaction product, and is not to be confused with the D(vx,vy,vz) in previous sections which refers to the center-of-mass velocity space. Laser spectroscopic detection methods in general measure the number of product particles within the detection volume rather than a flux, which is proportional to the reaction rate, emerging from it. Thus, products recoiling at low laboratory velocities will be detected more efficiently than those with higher velocities. The correction for this laboratory velocity-dependent detection efficiency is called a density-to-flux transformation.40 It is a 3D space- and time-resolved problem and is usually treated by a Monte Carlo simulation.41,42... 

A special case is the application domain of discrete functions (e.g., measurements on some spatial grid). The Fourier transform of a discrete function can be computed quite efficiently by a special algorithm (Fast Fourier Transform) at discrete points in Fourier space [132]. 

We discovered in Chapter 9 that the spatial function as given by the discrete Fourier transform (DFT) is a discrete Fourier series. Letting u(k) denote the (known) series consisting of only low-frequency terms and v(k) the series consisting of only high-frequency terms, we want to determine the unknown coefficients in v(k) that best satisfy the constraints. Expressing deviations of the total function forbidden by the constraints as some function of u(k) + v(k), we shall try to determine the coefficients of v(k) that minimize these deviations. Sum-of-squares expressions for these measures of the error have been found to result in the most efficient computational schemes. 

Heather, R.W. and Metiu, H. (1987). An efficient procedure for calculating the evolution of the wave function by fast Fourier transform methods for systems with spatially extended wave function and localized potential, J. Chem. Phys. 86, 5009-5017. 

One of the leading tendencies in chemistry of nanomaterials is their modification for obtaining new properties. Halogenation as one of efficient methods of modification is of great interest because both chlorine- and fluoro- derivatives should serve molecular intermediates for further modification. Chemical transformations of the attached functional groups give the base for the creation of principally new spatial structures based on carbonic nanomaterials. 

Besides the scientific questions related to the coupling of models, the interaction of the numerical models is a big technical challenge. The transformation of data at different temporal and spatial resolutions as well as computational efficiency, memory consumption, data storage capacity, meta-data communication and code management are issues which have to be addressed. 

Development of powerful spectroscopy and microscopy techniques, which allow us to study underlying chemical transformations that govern the performance of catalysts, including reaction mechanisms and the evolution of catalyst structure, with high spatial and temporal resolutions and at relevant conditions [2-6]. Development of density functional theory (DFT) methodology, which is utilized to study chemical transformations at the elementary step level with reasonable accuracy and efficiency [7]. DFT is particularly well suited for the treatment of extended metallic structures, which are often ideal model systems for heterogeneous catalytic processes [8-11]. 

Instead of polarized noble gases, thermally polarized NMR microimaging was used to study of liquid and gas flow in monolithic catalysts. Two-dimensional spatial maps of flow velocity distributions for acetylene, propane, and butane flowing along the transport channels of shaped monolithic alumina catalysts were obtained at 7 T by NMR, with true in-plane resolution of 400 xm and reasonable detection times. The flow maps reveal the highly nonuniform spatial distribution of shear rates within the monolith channels of square cross-section, the kind of information essential for evaluation and improvement of the efficiency of mass transfer in shaped catalysts. The water flow imaging, for comparison, demonstrates the transformation of a transient flow pattern observed closer to the inflow edge of a monolith into a fully developed one further downstream. 

---