Point charge lattice sums

High-temperature measurements of the paramagnetic quadrupole splittings have been made in Y, Sm, Gd, Dy, Yb, and Lu garnets, but comparison with estimates from point-charge lattice-sum calculations gives only poor agreement [136]. 

It is known that the y integrals tend to R as the distance of the centers involved increases, therefore the long-range contribution to G b can be evaluated from usual point charge lattice sums. It can be supposed that short-range corrections can be neglected for R > R (= 1.5 a.u.) and the effective interaction kernel can be calculated exactly in the CNDO/INDO approximation as... 

Abstract The history of computations at Namur and elsewhere on the electronic structures of stereoregular polymers is briefly reviewed to place the work reported here in the context of related efforts. Our earlier publications described methods for the formal inclusion of Ewald-type convergence acceleration in band-structure computations based on Gaussian-type orbitals, and that work is here extended to include a discussion of the calculation of total energies. It is noted that the continuous nature of the electronic density leads to different functional forms than are encountered for point-charge lattice sums. Examples are provided to document the correctness and convergence properties of the formulation. 

For each point charge qi involved in Eq. (6.4). the resulting electrastatic potential decays in proportion to the inverse distance [see Eq. (6.2)], such that the lattice sum buried in the expression for the total energy Uc [see Eq. (6.1)] converges rather slowly. In view of this dilemma, the central idea of the Ewald summation techniques is to rewrite the d -like charge density in Eq. (6.4) as a sum of three contributions, pf (r ), and p] (r ). Each... 

This section will be devoted to the calculation of the ground-state energy of an electron gas in a lattice formed by two types of point charges, and Vs, so arranged as to form a superlattice. The various sites of the A lattice will be denoted by a,. . ., while the various sites of the B lattice will be denoted by i,j,. The interaction between an electron at r and the whole lattice is given by the sum ... 

The basic ideas of the exchange-charge model of the crystal field as outlined in Ref. [40] are as follows The CFPs of any impurity ion, i.e., q(nT), are represented as a sum of two different contributions that arise from the point charges of aU crystal lattice ions and introduced fictitious exchange charges (placed in the space between an impurity ion and ligands) due to the spatial overlap of the impurity-ion and ligand wave functions ... 

The lattice sums in parameters Bpq absolutely converge and may be calculated by direct summation. Special care is needed only when calculating conditionally convergent sums in parameters of quadrupole components of the point charge field, which are proportional to the parameters Qap( ) of the Lorentz field ... 

---