Department of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland.
J Chem Phys. 2013 Aug 7;139(5):054114. doi: 10.1063/1.4817002.
We analyze the accuracy of the Coulomb energy calculated using the Gaussian-and-finite-element-Coulomb (GFC) method. In this approach, the electrostatic potential associated with the molecular electronic density is obtained by solving the Poisson equation and then used to calculate matrix elements of the Coulomb operator. The molecular electrostatic potential is expanded in a mixed Gaussian-finite-element (GF) basis set consisting of Gaussian functions of s symmetry centered on the nuclei (with exponents obtained from a full optimization of the atomic potentials generated by the atomic densities from symmetry-averaged restricted open-shell Hartree-Fock theory) and shape functions defined on uniform finite elements. The quality of the GF basis is controlled by means of a small set of parameters; for a given width of the finite elements d, the highest accuracy is achieved at smallest computational cost when tricubic (n = 3) elements are used in combination with two (γ(H) = 2) and eight (γ(1st) = 8) Gaussians on hydrogen and first-row atoms, respectively, with exponents greater than a given threshold (αmin (G)=0.5). The error in the calculated Coulomb energy divided by the number of atoms in the system depends on the system type but is independent of the system size or the orbital basis set, vanishing approximately like d(4) with decreasing d. If the boundary conditions for the Poisson equation are calculated in an approximate way, the GFC method may lose its variational character when the finite elements are too small; with larger elements, it is less sensitive to inaccuracies in the boundary values. As it is possible to obtain accurate boundary conditions in linear time, the overall scaling of the GFC method for large systems is governed by another computational step-namely, the generation of the three-center overlap integrals with three Gaussian orbitals. The most unfavorable (nearly quadratic) scaling is observed for compact, truly three-dimensional systems; however, this scaling can be reduced to linear by introducing more effective techniques for recognizing significant three-center overlap distributions.
我们分析了使用高斯有限元库仑（GFC）方法计算库仑能的准确性。在这种方法中，通过求解泊松方程得到与分子电子密度相关的静电势，然后用它来计算库仑算符的矩阵元。分子静电势在一个混合的高斯有限元（GF）基组中展开，这个基组由位于原子核上的 s 对称高斯函数（指数是由全优化原子势得到的，原子势是由从对称平均限制开壳哈特里-福克理论产生的原子密度得到的）和定义在均匀有限元上的形状函数组成。GF 基的质量由一小部分参数控制；对于给定的有限元宽度 d，当使用三次（n = 3）元素与分别在氢原子和第一行原子上使用两个（γ（H）= 2）和八个（γ（1st）= 8）高斯函数（指数大于给定阈值αmin（G）= 0.5）时，在最小的计算成本下可以达到最高的精度。计算的库仑能除以系统中的原子数的误差取决于系统类型，但与系统大小或轨道基组无关，随着 d 的减小近似地像 d（4）那样消失。如果泊松方程的边界条件以近似的方式计算，则当有限元太小时，GFC 方法可能会失去变分特征；对于较大的元素，它对边界值的不准确性不太敏感。由于可以在线性时间内获得准确的边界条件，因此 GFC 方法对于大型系统的整体缩放由另一个计算步骤控制——即用三个高斯轨道生成三中心重叠积分。对于紧凑的、真正的三维系统，观察到最不利的（几乎二次的）缩放；然而，通过引入更有效的技术来识别显著的三中心重叠分布，可以将这种缩放降低到线性。
