Gaussian Fourier transform

If a gaussian function is chosen for the charge spread function, and the Poisson equation is solved by Fourier transformation (valid for periodic... 

It can be shown that the right-hand side of Eq. (3-208) is the -dimensional characteristic function of a -dimensional distribution function, and that the -dimensional distribution function of afn, , s n approaches this distribution function. Under suitable additional hypothesis, it can also be shown that the joint probability density function of s , , sjn approaches the joint probability density function whose characteristic function is given by the right-hand side of Eq. (3-208). To preserve the analogy with the one-dimensional case, this distribution (density) function is called the -dimensional, zero mean gaussian distribution (density) function. The explicit form of this density function can be obtained by taking the i-dimensional Fourier transform of e HsA, with the result.45... 

Keilson-Storer kernel 17-19 Fourier transform 18 Gaussian distribution 18 impact theory 102. /-diffusion model 199 non-adiabatic relaxation 19-23 parameter T 22, 48 Q-branch band shape 116-22 Keilson-Storer model definition of kernel 201 general kinetic equation 118 one-dimensional 15 weak collision limit 108 kinetic equations 128 appendix 273-4 Markovian simplification 96 Kubo, spectral narrowing 152... 

The time resolution of the instrument determines the wavenumber-dependent sensitivity of the Fourier-transformed, frequency-domain spectrum. A typical response of our spectrometer is 23 fs, and a Gaussian function having a half width... 

Fig. 3 The normalized Gaussian and its Fourier transform, a Gaussian in k space.

Show that the Fourier transform of a Gaussian is also a Gaussian. 

Fourier transforms boxcar function 274 Cauchy function 276 convolution 272-273 Dirac delta function 277-279 Gaussian function 275-276 Lorentzian function 276-277 shah function 277-279 triangle function 275 fraction, rational algebraic 47 foil width at half maximum (FWHM) 55, 303... 

Dice the Gaussian, discussed above, the function shah is its own Fourier transform. Thus,... 

Fusti-Molnar L, Pulay P (2002) The Fourier transform Coulomb method Efficient and accurate calculation of the Coulomb operator in a Gaussian basis. J Chem Phys 117 7827... 

It is also possible to play other mathematical tricks with the FID. For example, we may want to make our signals appear sharper so we can see small couplings. In this case, we want our FID to continue for longer (an infinite FID has infinitely thin lines when Fourier transformed). To do this we use Gaussian multiplication . This works exactly the same way as exponential multiplication but uses a different mathematical function (Figure 4.2). 

On the other hand, lattice distortions of the second kind are considered. Assuming [127] that ID paracrystalline lattice distortions are described by a Gaussian normal distribution go (standard deviation ay, its Fourier transform Gd (.S ) = exp (—2n2ols2) describes the line broadening in reciprocal space. Utilizing the analytical mathematical relation for the scattering intensity of a ID paracrys-tal (cf. Sect. 8.7.3 and [127,128]), a relation for the integral breadth as a function of the peak position s can be derived [127,129]... 

Fig. 12 C -detected C CSA patterns of the SHPrP109 i22 fibril sample. The upper and lower traces correspond to the experimental and simulated spectra, respectively. Simulations correspond to the evolution of a one-spin system under the ROCSA sequence. The only variables are the chemical shift anisotropy and the asymmetry parameter. A Gaussian window function of 400 Hz was applied to the simulated spectmm before the Fourier transformation. (Figure and caption adapted from [164], Copyright (2007), with permission from Elsevier)...

As a approaches zero, f(x) becomes oo at x = 0, when it is called a "unit impulse", or a delta function. Its transform g(k) = 1. Of equal importance is the Fourier transform of the Gaussian function... 

Figure 5.1 illustrates the Fourier transform (FT) of a simple function, viz., a Gaussian. The relatively sharp Gaussian function with the exponent a = 1 depicted in Figure 5.1a, yields a diffuse Gaussian (in dotted line) in momentum space. A flat Gaussian function in position space with a = 0.1, transforms to a sharp one (cf. Figure 5.1b). Connected by an FT, the wave functions in position and momentum...

The inverse Fourier transform of G again yields a Gaussian function g(t) = linewidth Tg of which can be numerically calculated from... 

The most obvious drawback of Fourier space approaches is the computational cost of the Fourier transformation itself. However, this can be circumvented in some virtual screening applications. Gaussian functions are frequently used to approximate electron densities. Interestingly, the Fourier transform of a Gaussian function is again a Gaussian function and hence amenable to analytic transformation. 

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