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 is the position of the n charge in the simulation cell and 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 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 .
Figure: Schematic representation of the two components of the Ewald charge density. Blue indicates positive charge and red negative charge.
The potential due to is most conveniently calculated in reciprocal space. has non-zero Fourier components on supercell reciprocal lattice vectors, , given by
where is the volume of the supercell. Solving the corresponding reciprocal space version of Poisson's equation, , gives
To calculate , 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 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 is
where the last term, , is added so that the average potential in the supercell is zero. Combining the reciprocal space sum for and the real space sum for gives the final result for
The value of is independent of the half width, , of the Gaussian charges. However, the value of 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 , as defined in Eq.() 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. 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.