标题： 扩散基组的困境：准确性的祝福，但对稀疏性而言却是一场灾难 显示英文标题

标题： The Conundrum of Diffuse Basis Sets: A Blessing for Accuracy yet a Curse for Sparsity

摘要： 扩散的原子轨道基组已被证明对于获得准确的相互作用能至关重要，尤其是在非共价相互作用方面。 然而，它们也对单粒子密度矩阵（1-PDM）的稀疏性产生了有害的影响，这种影响程度比仅由基函数的空间范围所能解释的要强。 尽管如此，绝缘体（具有显著HOMO-LUMO间隙的系统）的1-PDM矩阵元预计会随着远离对角线的实空间距离呈指数衰减，且渐近衰减率预计有一个明确的基组极限。 观察到的1-PDM低稀疏性似乎与表示方式无关，甚至在将1-PDM投影到实空间网格后仍然存在，这导致得出结论：这种“稀疏性诅咒”仅仅是基组的人工产物，出乎意料的是，它在更大的基组中变得更糟，似乎与一个明确的基组极限概念相矛盾。 我们表明，这是由于反变基函数的局部性较低，如逆重叠矩阵$\mathbf{S}^{-1}$所量化，其稀疏性明显低于其协变对偶。 通过引入一个无限非相互作用氦原子链的模型系统，我们能够量化指数衰减率与基组的扩散性和局部不完整性成正比，这意味着小而扩散的基组受影响最大。 最后，我们提出了一种解决方案，即结合互补辅助基组（CABS）单修正和紧凑的低l量子数基组，显示出在非共价相互作用中的良好前景。 显示英文摘要

摘要： Diffuse atomic orbital basis sets have proven to be essential to obtain accurate interaction energies, especially in regard to non-covalent interactions. However, they also have a detrimental impact on the sparsity of the one-particle density matrix (1-PDM), to a degree stronger than the spatial extent of the basis functions alone could explain. This is despite the fact that the matrix elements of the 1-PDM of insulators (systems with significant HOMO-LUMO gaps) are expected to decay exponentially with increasing real-space distance from the diagonal and the asymptotic decay rate is expected to have a well-defined basis set limit. The observed low sparsity of the 1-PDM appears to be independent of representation and even persists after projecting the 1-PDM onto a real-space grid, leading to the conclusion that this "curse of sparsity" is solely a basis set artifact, which, counterintuitively, becomes worse for larger basis sets, seemingly contradicting the notion of a well-defined basis set limit. We show that this is a consequence of the low locality of the contra-variant basis functions as quantified by the inverse overlap matrix $\mathbf{S}^{-1}$ being significantly less sparse than its covariant dual. Introducing the model system of an infinite non-interacting chain of helium atoms, we are able to quantify the exponential decay rate to be proportional to the diffuseness as well as local incompleteness of the basis set, meaning small and diffuse basis sets are affected the most. Finally, we propose one solution to the conundrum in the form of the complementary auxiliary basis set (CABS) singles correction in combination with compact, low l-quantum-number basis sets, showing promising results for non-covalent interactions.