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Ewald Summation

The Ewald summation method[55, 56, 57] is a technique for evaluating the potential, subject to periodic boundary conditions, due to a lattice of point charges, plus a screening background,


where tex2html_wrap_inline6793 is the position of the n tex2html_wrap_inline6553 charge in the simulation cell and tex2html_wrap_inline6125 is the set of supercell translation vectors. To solve Poisson's equation for the Ewald potential,


due to the above charge density, the density is split up into two components, the background charge and the array of delta functions. An array of Gaussian functions, centred at tex2html_wrap_inline6799 is added to each component of the Ewald charge density. The Gaussians are normalised to ensure that both of the individual components of the Ewald charge density are neutral. The two charge density components can be written as




These charge densities are schematically represented in one dimension in figure gif.

Figure: Schematic representation of the two components of the Ewald charge density. Blue indicates positive charge and red negative charge.

The potential due to tex2html_wrap_inline6801 is most conveniently calculated in reciprocal space. tex2html_wrap_inline6801 has non-zero Fourier components on supercell reciprocal lattice vectors, tex2html_wrap_inline6677 , given by


where tex2html_wrap_inline6807 is the volume of the supercell. Solving the corresponding reciprocal space version of Poisson's equation, tex2html_wrap_inline6809 , gives


To calculate tex2html_wrap_inline6811 , the potential due to the array of point charges minus the screening Gaussians, evaluation in real space is more convenient since the coefficients in the Fourier expansion of a periodic array of delta functions do not decay for large tex2html_wrap_inline6813 vectors. The resultant potential is the summed potential of the delta function point charges minus the sum of potentials due to the Gaussian charge distributions. One can show that the potential of the Gaussian charge distribution is given by


where the error function is defined as


Consequently, the real space sum generating tex2html_wrap_inline6811 is


where the last term, tex2html_wrap_inline6817 , is added so that the average potential in the supercell is zero. Combining the reciprocal space sum for tex2html_wrap_inline6819 and the real space sum for tex2html_wrap_inline6811 gives the final result for tex2html_wrap_inline6823


The value of tex2html_wrap_inline6823 is independent of the half width, tex2html_wrap_inline5979 , of the Gaussian charges. However, the value of tex2html_wrap_inline5979 affects the speed of convergence of the above real and reciprocal space sums.

The full potential of a simulation cell containing N electrons and M ions is found by superposing all the potentials for each charge component, since the full charge distribution is the superposition of all the point charges and their cancelling backgrounds,


Therefore tex2html_wrap_inline6835 , as defined in Eq.(gif) is given by




is the self-image interaction, i.e. the potential at the unit point charge due to its own background and array of images. tex2html_wrap_inline6837 can be found in exactly the same way and the total electrostatic energy per simulation cell can then be written as


The charge neutrality of the simulation cell dictates that


Therefore the above expression for U is easily simplified to


In QMC solid calculations, the ionic coordinates are fixed throughout the calculation. Therefore the contribution to the total energy from the ion-ion Coulomb interaction need only be evaluated once at the beginning of the simulation.

As each of the electrons are moved in turn in a VMC calculation, the contribution to the total energy from the electron-electron and electron-ion Coulomb interactions needs to be recalculated for each electron after it is moved.

next up previous contents
Next: Electron-Ion Coulomb Interactions Up: Coulomb Interactions in Supercell Previous: Periodic Boundary Conditions

Andrew Williamson
Tue Nov 19 17:11:34 GMT 1996