Gaussian functions/distribution computer simulation

An additional issue in the development of the density functional theory is the parameterization of the trial function for the one-body density. Early applications followed the Kirkwood-Monroe [17,18] idea of using a Fourier expansion [115-117,133]. More recent work has used a Gaussian distribution centered about each lattice site [122]. It is believed that the latter approach removes questions about the influence of truncating the Fourier expansion upon the DFT results, although departures from Gaussian shape in the one-body density can also be important as has been demonstrated in computer simulations [134,135]. 

The intensity, I(Hr) is proportional to the number of radicals contributing to the spectrum at the field Hr. This number was given by a distribution function P(a), for instance assumed to be a Gaussian distribution, with the distribution function, exp(asin a). A computer simulation was carried out for the orientation function, fc and the angles Xi, X2, and X3. The ESR spectrum, 1(H) can be calculated from the convolution of P(a) and the line shape function G as follows. 

Figure 1.41. Distribution of the end-to-end distance. R W(K) is plotted as a function of R/Rp. Symbols were obtained in computer simulation the sohd line represents the distribution for the Gaussian chain. (From Ref. 6.)...

Finally, there is the possibility to simulate an experimental curve (spectrum) by a mathematical algorithm, e.g., by a polynomial, a Fourier transform expression, or the superposition of Gaussian or other suitable distribution curves (cf. Sec. 2.3.4, Eq. (2-41) (2-47)). In this case, one must keep in mind that for simulation of real spectra it is also necessary to add a noise function, produced by a random generator, to the PC-computed curve. Otherwise, it is not possible to transfer the results of the investigations to real signals produced by any apparatus. Of course, it is much easier to get useful derivatives from undisturbed curves than from real spectra containing noise. 

Figure 9.14 Simulated recombination rate R(f) of geminate e...h pairs. Computer printouts show (a) the coincidence of the data obtained at temperatures 77, 100, 120, 240, 375, and 600 K, (b) the virtual independence of the normalized rate on the wave function overlap factor 2ya, and (c) the influence of the initial pair separation in units of the lattice constant a. The computer system was a sample of cubic symmetry in which the energy of the hopping sites was distributed according to a Gaussian distribution of variance 0.1 eV. (From Ries, B. and Bassler, H., J. Mol. Electron., 3, 15, 1987. With permission.)...

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