论文标题
解决任意周期性电荷密度的修改后的Helmholtz方程的解决方案
Solution to the modified Helmholtz equation for arbitrary periodic charge densities
论文作者
论文摘要
我们提出了一种解决修饰的Helmholtz方程的通用方法，没有形状近似，用于任意周期性电荷分布，该分布被称为Yukawa电位或筛选的库仑电位。该方法是Weinert的伪电荷方法的扩展[M. Weinert，J。Math。物理。 22，2433（1981）]用于解决相同类别的电荷密度分布的泊松方程。泊松和修饰的Helmholtz方程之间的固有差异在其各自的径向溶液中。对于修改的Helmholtz方程，这些是用于泊松方程的多项式函数，而修饰的球形贝塞尔函数。这导致了修改后的伪电荷密度和修改后的多极力矩的定义。我们已经表明，Weinert对伪电荷密度的绝对和均匀收敛的傅立叶序列的收敛分析被转移到修改的伪电荷密度上。最后，我们通过说明将Poisson求解器的可用实现变成修改后的Helmholtz方程所需的算法更改所需的算法更改。
We present a general method for solving the modified Helmholtz equation without shape approximation for an arbitrary periodic charge distribution, whose solution is known as the Yukawa potential or the screened Coulomb potential. The method is an extension of Weinert's pseudo-charge method [M. Weinert, J. Math. Phys. 22, 2433 (1981)] for solving the Poisson equation for the same class of charge density distributions. The inherent differences between the Poisson and the modified Helmholtz equation are in their respective radial solutions. These are polynomial functions, for the Poisson equation, and modified spherical Bessel functions, for the modified Helmholtz equation. This leads to a definition of a modified pseudo-charge density and modified multipole moments. We have shown that Weinert's convergence analysis of an absolutely and uniformly convergent Fourier series of the pseudo-charge density is transferred to the modified pseudo-charge density. We conclude by illustrating the algorithmic changes necessary to turn an available implementation of the Poisson solver into a solver for the modified Helmholtz equation.
