The Physics Department, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom.
J Chem Phys. 2013 Jan 21;138(3):034504. doi: 10.1063/1.4775367.
This work is concerned with the lattice energy of periodic assemblies of mass and charge distributions of the form, exp (-αp(2)), where α is an adjustable positive variable and p(̱) is the vector from the lattice site or average position. The energy of interaction between two distributions is the density-weighted integral of the interactions between the volume elements of each distribution. Reciprocal space lattice summation formulas derived for particles represented by gaussian smeared-out density distributions are applied to the gaussian potential and a bounded version of the soft-sphere potential for a range of exponents. Two types of spatial broadening are considered, continuous or physical broadening (PB) and broadening resulting from the time average of point particle positions, so-called "time" broadening (TB). For neutral mass distributions a reciprocal space lattice summation formula is derived which is applied to the bounded soft-sphere potential. For the charged systems, the methodology described in Heyes [J. Chem. Phys. 74, 1924 (1981)] is used, which for the PB case gives the Ewald-like formulas derived by Gingrich and Wilson [Chem. Phys. Lett. 500, 178 (2010)] using a different method. Another expression for the lattice energy of the spread out charge distributions is derived which is cast entirely in terms of a summation over the reciprocal lattice vectors, without the arbitrary charge spreading function used in the Ewald method. The effects of charge spreading on a generalized definition of the Madelung constant (M) for a selection of crystal lattices are shown to be insignificant for route mean square displacements up to values typical of melting of an ionic crystal. When the length scale of the charge distribution becomes comparable to or greater than the mean inter particle spacing, however, the effects of charge broadening on the lattice energy are shown to be significant. In the PB case, M → 0 for the uniform charge density or α → 0 limit, and M ultimately becomes negative in the TB case for a large enough root mean square displacement (or small enough α).
这项工作关注的是质量和电荷分布的周期性组装的晶格能，其形式为 exp(-αp(2))，其中 α 是一个可调的正变量，p(̱)是从晶格点或平均位置到的向量。两个分布之间的相互作用能是每个分布的体积元之间相互作用的密度加权积分。对于用高斯模糊密度分布表示的粒子，推导出了倒易空间晶格求和公式，并将其应用于高斯势和软球势的有界版本，涵盖了一系列指数。考虑了两种类型的空间展宽，连续或物理展宽 (PB) 和由于点粒子位置的时间平均而导致的展宽，即所谓的“时间”展宽 (TB)。对于中性质量分布，推导出了一个倒易空间晶格求和公式，并将其应用于有界软球势。对于带电系统，使用了 Heyes [J. Chem. Phys. 74, 1924 (1981)] 中描述的方法，对于 PB 情况，该方法给出了 Gingrich 和 Wilson [Chem. Phys. Lett. 500, 178 (2010)] 使用不同方法推导的类似 Ewald 的公式。还推导出了扩展电荷分布的晶格能的另一个表达式，该表达式完全是在倒易晶格矢量的求和中表示的，而没有在 Ewald 方法中使用任意的电荷扩展函数。对于选择的晶体晶格，电荷扩展对广义 Madelung 常数 (M) 定义的影响在直至离子晶体熔化的典型均方根位移值范围内可以忽略不计。然而，当电荷分布的长度尺度变得与平均粒子间距离相当或更大时，电荷展宽对晶格能的影响变得显著。在 PB 情况下，对于均匀电荷密度或 α→0 极限，M→0，并且对于足够大的均方根位移 (或足够小的 α)，在 TB 情况下 M 最终变为负值。
