Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA.
J Chem Phys. 2011 Feb 14;134(6):064111. doi: 10.1063/1.3553716.
We propose an efficient, accurate method to integrate the basins of attraction of a smooth function defined on a general discrete grid and apply it to the Bader charge partitioning for the electron charge density. Starting with the evolution of trajectories in space following the gradient of charge density, we derive an expression for the fraction of space neighboring each grid point that flows to its neighbors. This serves as the basis to compute the fraction of each grid volume that belongs to a basin (Bader volume) and as a weight for the discrete integration of functions over the Bader volume. Compared with other grid-based algorithms, our approach is robust, more computationally efficient with linear computational effort, accurate, and has quadratic convergence. Moreover, it is straightforward to extend to nonuniform grids, such as from a mesh-refinement approach, and can be used to both identify basins of attraction of fixed points and integrate functions over the basins.
我们提出了一种高效、准确的方法，用于整合定义在一般离散网格上的光滑函数的吸引域，并将其应用于电子电荷密度的 Bader 电荷划分。从沿着电荷密度梯度的轨迹在空间中的演化开始，我们推导出了一个表达式，用于表示流向相邻网格点的每个网格点相邻空间的分数。这是计算属于一个盆地（Bader 体积）的每个网格体积的分数的基础，也是在 Bader 体积上对函数进行离散积分的权重。与其他基于网格的算法相比，我们的方法具有鲁棒性、更高的计算效率（具有线性计算复杂度）、准确性和二次收敛性。此外，它可以很容易地扩展到非均匀网格，例如从网格细化方法，并且可以用于识别固定点的吸引域和在吸引域上积分函数。
