Applications of the Ewald method. I. Calculation of multipole lattice sums

General principles of the Ewald method for evaluating multipole lattice sums are reviewed. The method is used to derive an expression for the Lorentz-factor dipole tensor sum in a form convenient for computation, and comparisons are made with the direct and plane-wise summation methods. Expressions are also given for computing quadrupole and octopole sums by the Ewald method. The effect of crystal symmetry on lattice sums is outlined; the number of independent sums relating different pairs of equivalent sublattices does not exceed the total number of such sublattices. Numerical results are given for the dipole lattice sums of hydrogen cyanide, benzene, durene, anthracene and pyrene. Quadrupole sums are given for cuprous chloride and pyrene, and octupole sums are given for hydrogen cyanide, benzene and anthracene. For dipole lattice sums, the Ewald method converges much faster than direct summation; for higher multipole sums, the Ewald method has no special advantage in speed, but may prove convenient, especially when sums are required for strained lattices.