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An extensive study of gradient approximations to the exchange-correlation and kinetic energy functionals

机译:交换相关和动能泛函的梯度近似的广泛研究

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摘要

We formalize the procedure of functional development, in a general theoretical framework. Expansion in a functional basis set, and fitting via an error functional to a data set, casts functional development as a variational problem to obtain the functional basis-set and data-set limits. Overfitting is avoided by defining the optimum number of parameters. We implement our theory for an investigation of first- and second-order generalized gradient approximations (GGA) to the exchange-correlation and kinetic energy functionals, within an ab initio model. A variety of functional basis sets, including a general finite-element representation, is constructed to represent both one-dimensional and multidimensional GGA enhancement factors. An extensible data set consisting of 429 atomic and diatomic, neutral and cationic species, at stretched and equilibrium geometries, is constructed from Moller–Plesset level exchange-correlation energies, and Hartree–Fock kinetic energies. The range of chemically relevant density and gradient variables is examined. Exhaustive fitting investigations are carried out, to determine the accuracy of the GGA representation of the ab initio models. In the exchange-correlation case we demonstrate that we can reach the functional basis-set and data-set limit, which correspond to a root-mean-square (rms) error of ∼10∼10 mH (6.3 kcal/mol). Changing the functional basis set, higher-order density variables such as the kinetic energy density, multidimensional enhancement factors, and exact exchange yield no significant improvement, and our fits represent an effective solution of the GGA problem for exchange-correlation, at the Møller–Plesset level. In the kinetic energy case, accurate functionals with rms errors of ∼80∼80 mH (50 kcal/mol) are developed. These exhibit a beautifully simple kinetic energy enhancement factor, and are a step towards orbital-free calculations.
机译:我们在一个一般的理论框架中形式化功能开发的过程。扩展功能基础集,并通过误差拟合数据集,将功能开发转换为变量问题,以获得功能基础集和数据集限制。通过定义最佳数量的参数可以避免过拟合。我们从头算模型中,将我们的理论用于一阶和二阶广义梯度近似(GGA)对交换相关和动能函数的研究。构建了包括通用有限元表示形式在内的各种功能基础集,以表示一维和多维GGA增强因子。由Moller–Plesset能级交换相关能和Hartree–Fock动能构成了可扩展的数据集,该数据集包含429个原子和双原子,中性和阳离子物种,处于拉伸状态和平衡状态。检查化学相关的密度和梯度变量的范围。进行了详尽的拟合研究,以确定从头算模型的GGA表示的准确性。在交换相关情况下,我们证明我们可以达到功能基础集和数据集极限,这对应于〜10〜10 mH(6.3 kcal / mol)的均方根(rms)误差。改变功能基础集,高阶密度变量(例如动能密度,多维增强因子和精确交换)没有明显改善,并且我们的拟合代表了Møller处GGA交换相关问题的有效解决方案。 Plesset级。在动能的情况下,开发出了均方根误差为〜80〜80 mH(50 kcal / mol)的精确功能。它们表现出漂亮的简单动能增强因子,是朝​​着无轨道计算迈出的一步。

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