# SEARCH

** Fourier transform Jacquinot’s advantage **

** Fourier transforms Bragg’s law and **

** Green’s function Fourier transform **

In any case, as based on Fourier s transformations that connect the electronic density (as real measure) with the structure factor (as complex measure), the Parseval s theorem is formulated which stipulates that the mean square value on each side of the Fourier s transformations are proportional, i.e., it can be written like (Cantor Schimmel, 1980) [Pg.508]

Notice that Gq is the Fourier time transform of the one-electron energy-dependent Green s function Go(r,r ). Expansion of a particular wavefunction, in terms of site atomic orbitals, as [Pg.356]

Fourier s law of heat conduction, reservoirs, second entropy, 63-64 Fourier transform [Pg.280]

Equation (4-12) can be transformed into a more amenable form by several substitutions. The first of these is that (from Fourier s law), [Pg.159]

For apiecewise continuous function F(x) over a finite interval 0 = x S ji, finite Fourier cosine transform of F(x)is [Pg.2434]

Helmholtz theorem is a direct consequence of Fourier s formalism. In order to demonstrate this claim, we shall first consider the direct and inverse transforms of the B(r) function, respectively known as [Pg.561]

There is one added complication. The preceeding functionsf(x) and F(h)are one-dimensional. Fortunately, the Fourier transform applies to periodic functions in any number of dimensions. To restate Fourier s conclusion in three dimensions, for any function f(x,y,z) there exists the function F(h,k,l) such that [Pg.91]

The complex energies k (k = 1,2,..., 5) are the roots of the polynomial Q(z) and the/j s are generalized oscillator strengths. The inverse Fourier-Laplace transformation provides the survival probability P t) of the initial state Is [Pg.19]

V To examine photodissociation given this field requires, as shown below, the S t frequency-frequency correlation function (( (a X aq)) where T(m) is the Fourier V transform of eJt) for Jt) equal to a constant <5,.. Given Gaussian pulses [Eqs. (5.28) and (5.29)1 we have 11891 [Pg.107]

The reader will recall, from Appendix A, the Wiener-Khinchin theorem namely, the a.c.f. and the spectral density are each other s Fourier cosine transform. Thus [Pg.435]

I br the majority of physicaJ systems, the Lagrangian is invariant to any space translation. In this case, all the equations of the theory cam be simplified by means of Fourier s transformation, i.e. by considering the characteristic values in the momentum space. 1 he rules for drawing graphs of the characteristic values in the momentum space are given. [Pg.251]

For Eq. (11) S is the Bragg vector S = 2ttH, IT is the row vector (htk,l) and the scalar S - S = 4ir sin 0/A. The index / covers the N atoms in the unit cell. The atomic scattering factor f (S) is the Fourier-Bessel transform of the electronic, radial density function of the isolated atom. This density function is usually derived from a spin-restricted Hartree-Fock wave function for the atom in its ground state. The structure fac- [Pg.544]

The investigation of relaxation times and diffusion coefficients requires the determination of the eigenvalues of the matrix corresponding to the system of algebraic equations obtained from Eqs. (18) after Fourier-Laplace transformation (s, Laplace transform of time q, Fourier transform of the space coordinate). The roots of the secular equation are [Pg.105]

The superposition principle for heat flow as measured by power-compensated DSC should apply—just as it would be expected that the water flow into one tank from two pipes would be additive. Assuming Fourier s law holds (steady state heat flow proportional to temperature gradient), the temperature differences measured in DTA (and heat-flux DSC) are additive via contributions from multiple transformation sources within the sample material. [Pg.143]

For many-electron molecules, the Hartree-Fock wavefunction that is computed by conventional electronic structure packages, such as GAUSSIAN, can be expanded from singleparticle molecular orbitals, i /i(r), that are themselves constructed from atom-centered gaussians that are functions of coordinate-space variables. The phase information that is contained in the molecular orbitals is necessary to define the wavefunction in momentum-space. In other words, the density in coordinate-space cannot be Fourier transformed into the density in momentum-space. Rather, within the context of molecular orbital theory, the electron density in momentum space is obtained by a Fourier-Dirac transformation of all of the v /i (r) s, followed by reduction of the phase information, weighting by the orbital occupation numbers. [Pg.141]

** Fourier transform Jacquinot’s advantage **

** Fourier transforms Bragg’s law and **

** Green’s function Fourier transform **

© 2019 chempedia.info