Exact Exchange in Density Functional Calculations

8.3 Miscellaneous 69And the expression for the Coulomb integral becomes∫dk n ∗(n 1 |n 2 ) = 4π1(k)n 2 (k)(2π) 3 k 2 (1 − cos(kr c )) (8.12)The Tuckerman potential does not suffer from the divergence at k = 0 as:lim v T (k) = 2πrc 2 (8.13)k→0Thus both the problem with non charge-neutral densities, and with the artificial interactions withperiodic images are avoided. The cure however is rather expensive, as the unit-cell must be twiceas big as the extend of the density in all three dimensions.Gaussian neutralizationAnother way of coping with the non-zero total charge, is to just subtract a density of the sametotal charge as the original, but with a a well known potential, and then add this afterward. Takefor example the Gaussian density distribution (of unit total charge) described by( a) 3/2n g (r) = e−ar 2 ,πn g (k) = e −k2 /4awith the potentialv g (r) = erf(√ ar),rv g (k) = 4π /4ak 2 e−k2To determine the potential of a density n with total charge Z, we can then calculate(8.14)(8.15)v[n](r) = v[n − Zn g ](r) + Zv g (r) (8.16)where v[n](r) is the potential of the density n.To do a symmetric neutralization of both densities in a Coulomb integral, one could do(n 1 |n 2 ) = (n 1 − Z 1 n g |n 2 − Z 2 n g ) + (Z ∗ 2 n 1 + Z 1 n ∗ 2 − Z ∗ 1 Z 2 n g |n g ) (8.17)This procedure solves the problem with non-zero total charge, but there can still be artificialinteractions from higher order moments (than the monopole) with the periodic images. To takecare of these, one could simply proceed with subtracting generalized gaussians of higher order:v[n](r) = v [n − ∑ L Q Lg L (r)] (r) + ∑ LQ L v L (r) (8.18)where Q L are the multipole moments of n, g L are generalized gaussians, and v L are the potential ofthe gaussians. The potentials v L of the gaussians (7.18), can be determined analytically in termsof Whittaker functions. These proved difficult to implement,as they satisfied quite complicatedrecursion relations.This method is applicable to both real and reciprocal space solvers. Presently I only neutralizethe monopole.Real SpaceIn principle densities of non-zero total charge is not a problem when solving the Poisson equationby some real-space differential equation solver. The specific choice of solver used in GPAW howevercan not handle charged systems, so when using the real space solver, I neutralize the density usinggaussians, as explained in the previous paragraph.A different problem in real space is that if one applies Dirichlet BC’s, the potential must becompletely localized within the unit cell to be described properly. Alternatively, one could do amultipole expansion of the density, as described in appendix D.2, and then apply the correct BC’s.In this case only the density itself should be completely contained within the unit cell, for thepotential to be described correctly.