Fourier Transformation of the Coulomb Potential

The Coulomb potential, or more precisely, the Coulomb potential energy V between two charged particles with charges and 2 h three dimensions in 

For the discussion of the Douglas-Kroll-Hess transformation in chapter 12 it has been advantageous to consider the momentum-space representation of the Coulomb potential, which may be obtained via a Fourier transformation of V (r). It is given by 

However, Eq. (E.8) cannot be evaluated directly in a straightforward manner. We thus introduce a suitable cutc which damps the Coulomb potential sufficiently and define the family of cutoff potentials 

For ji 0 we finally arrive at the momentum-space representation of the Coulomb potential. 

The Coulomb potential thus features a l/fc -dependence in momentum space. 

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The proof is a straightforward application of the fundamental properties of the Fourier transform, namely, its linearity, and how it intertwines differentiation, multiplication and convolution. This material is available in any introduction to Fourier transforms for example, see [DyM, Chapter 2]. The only tricky part is the calculation of the Fourier transform of the Coulomb potential. See Exercise 9.3. 

Following the procedure used in electrostatics where it is found that the Fourier transform of the Coulomb potential yields a k dependence, we can invert the expression found above rather easily as... 

In the following sections, particularly for the calculation of phonon spectra, the correct small-wave-vector limit of the pseudopotential is required for the internal consistency of the calculations. This limit is easily calculated here. The ionic contribution for —> 0, is the Fourier transform of the Coulomb potential of an ion with charge —Ze acting on an electron with charge e. Therefore ... 

The simple treatment of the nearly free ion was based on the use of a fast Fourier transform of the effective potential measured along the helical axis. It is possible, however, to evaluate matrix elements of both the Coulomb interaction and the Morse potential in terms of basis functions that are bound in the x,y-plane and a plane wave along the helical axis. The purpose of this appendix is to outline these evaluations. 

The site-site direct correlation function c y(r) often contains nonanalytic terms which come directly from the particular site-site interaction potential and the assumed closure relation. The most common such term comes from the site-site Coulomb interaction, in combination with any closure relation whose long-range form is — )3u(r). In general, the Fourier transform of c, r) is of the form... 

Bohm and Pines transformed the electron-repulsion terms in the classical many-electron Hamiltonian into its Fourier components. This Fourier transform has the effect of interpreting the familiar Coulomb repulsion potential term as a series of momentum-transfers between the states of the electrons. They showed how an important series of results could be obtained by the assumption that the terms in this Fourier transform which depended on a non-zero phase difference in the k-vector could be neglected. The idea behind this approximation is that these terms, having random phases, have a zero mean value and contribute only to random fluctuations in the electron plasma which are negligible under the circumstances of their study, or, what amounts to the same thing, the momentum transfers could be replaced by their ensemble average. 

Much as for the ID solvent-solvent correlations, the renormalization of the 3D-RISM/HNC equations is not necessary in respect to convergence. Nor is it required for the 3D fast Fourier transform (3D-FFT) employed to evaluate the convolution in Eq. (4.A.41). For a periodic solute neutralized by a compensating background charge, the Coulomb potential of the solute charge is screened at a supercell length. Therefore the 3D site direct, and hence total correlation functions, are free from the Coulomb singularity at fc = 0 and can be transformed directly by the 3D-FFT. 

The solutions to this approximation are obtained numerically. Fast Fourier transform methods and a reformulation of the HNC (and other integral equation approximations) in terms of the screened Coulomb potential by Allnatt [64] are especially useful in the numerical solution. Figure A2.3.12 compares the osmotic coefficient of a 1-1 RPM electrolyte at 25°C with each of the available Monte Carlo calculations of Card and Valleau [M]. 

W i) is an integral operator with kernel W p, p ) and V p, p ) is the Fourier-transformed external potential. Hess s code has been implemented in several program codes like TURBOMOLE or MOLCAS3. Most of the applications are carried out scalar (one-component) without spin-orbit coupling and usually the two-electron operator is chosen as the simple Coulomb operator. This scheme (extended to spin-orbit coupling it necessary) leads to very accurate molecular properties for even the heaviest elements. A large number of applications in the chemistry of heavy elements are carried out by using either the scalar relativistic pseudopotential or density functional approximations. The pseudopotentials most widely used are linear... 

Lastly, an inverse FT leads back to the functions yij r). In the presence of Coulombic, 1/r or 1/q, diverging potentials and correlation functions, reference functions of same divergence at large r or small q are subtracted from Cij r) or yij q) before Fourier transform, the numerical FT is performed for the non-diverging difference and the analytical FT of the reference is added afterwards. 

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