Coulomb summation

VVe therefore return to the point-charge model for calculating electrostatic interactions. If sufficient point charges are used then all of the electric moments can be reproduced and the multipole interaction energy. Equation (4.30), is exactly equal to that calculated from the Coulomb summation. Equation (4.19). 

Most simulations of amorphous materials have described the structure in terms of a large simulation box to which periodic boundary conditions are applied. The periodicity is, of course, artificial, but if the simulation box is sufficiently large the effect of imposing periodicity should be unimportant. Given this approximation we may use the standard Ewald procedure (Chapter 4) for undertaking the Coulomb summation. 

Regardless of which algorithm is used for fast calculation of Ewald sums, the computational cost is now competitive with the cost of cutoff calculations, and there is no longer a need to employ cutoffs for purposes of efficiency. Since Ewald summation is the natural expression of Coulomb s law in periodic boundary conditions, it is the recommended approach if periodic boundary conditions are to be used in a simulation. 

Neglecting for simplicity the long-range character of the Coulomb force, the above summations yield (31) a bounded result (x) when extended to infinity. Bielectron integrals can thus be regarded as scaling like Nq", either in the thermodynamic limit (Nq °°), or (31) in the dissociation limit (aQ °°). 

Lekner J (1991) Summation of coulomb fields in computer-simulated disordered-systems. Physica A 176(3) 485-498... 

The terms on the right-hand side of eq. (11.41) denote the kinetic energy, the electron-nuclear potential energy, the Coulomb (J) and exchange (K) terms respectively. Together J and K describe an effective electron-electron interaction. The prime on the summation in the expression for K exchange term indicates summing only over pairs of electrons of the same spin. The Hartree-Fock equations (11.40) are solved iteratively since the Fock operator / itself depends on the orbitals iff,. 

The density functional theory (DFT) [32] represents the major alternative to methods based on the Hartree-Fock formalism. In DFT, the focus is not in the wavefunction, but in the electron density. The total energy of an n-electron system can in all generality be expressed as a summation of four terms (equation 4). The first three terms, making reference to the noninteracting kinetic energy, the electron-nucleus Coulomb attraction and the electron-electron Coulomb repulsion, can be computed in a straightforward way. The practical problem of this method is the calculation of the fourth term Exc, the exchange-correlation term, for which the exact expression is not known. 

As shown by Chang, Pelissier and Durand (CPD) [41] a regular expansion, however, can be deduced by isolating the Coulomb singularity by infinite summations. Let us rewrite the equation (38), when z — 0... 

The coulombic energy between interacting atoms A and B at distance R is the summation of four terms ... 

As a consequence, the summation over all occupied orbitals counts once again the 1-2 Coulomb interaction confined within the valence space, specifically, the dT2 dr term already found in (3.8), and adds the new integral... 

The first summation is over nuclei A. Z are atomic numbers and Rap are distances between the nuclei and the point charge. The second pair of summations is over basis functions, ( ). P is the density matrix (equation 16 in Chapter 2), and the integrals reflect Coulombic interactions between the electrons and the point charge, where rp is the distance separating the electron and the point charge. 

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