Faculty of Chemistry, Warsaw University of Technology, Noakowskiego 3, 00-664 Warsaw, Poland.
J Chem Phys. 2012 Jan 21;136(3):034104. doi: 10.1063/1.3674163.
We present an analytical approach to treat higher order derivatives of Hartree-Fock (HF) and Kohn-Sham (KS) density functional theory energy in the Born-Oppenheimer approximation with respect to the nuclear charge distribution (so-called alchemical derivatives). Modified coupled perturbed self-consistent field theory is used to calculate molecular systems response to the applied perturbation. Working equations for the second and the third derivatives of HF/KS energy are derived. Similarly, analytical forms of the first and second derivatives of orbital energies are reported. The second derivative of Kohn-Sham energy and up to the third derivative of Hartree-Fock energy with respect to the nuclear charge distribution were calculated. Some issues of practical calculations, in particular the dependence of the basis set and Becke weighting functions on the perturbation, are considered. For selected series of isoelectronic molecules values of available alchemical derivatives were computed and Taylor series expansion was used to predict energies of the "surrounding" molecules. Predicted values of energies are in unexpectedly good agreement with the ones computed using HF/KS methods. Presented method allows one to predict orbital energies with the error less than 1% or even smaller for valence orbitals.
我们提出了一种分析方法，用于在 Born-Oppenheimer 近似下处理 Hartree-Fock (HF) 和 Kohn-Sham (KS) 密度泛函理论能量的高阶导数，这些导数与核电荷分布有关（即所谓的化学导数）。修正的耦合微扰自洽场理论用于计算分子系统对施加的微扰的响应。推导出 HF/KS 能量的二阶和三阶导数的工作方程。同样，报道了轨道能量的一阶和二阶导数的解析形式。计算了 Kohn-Sham 能量的二阶导数和核电荷分布的 Hartree-Fock 能量的三阶导数。考虑了实际计算中的一些问题，特别是基组和 Becke 加权函数对微扰的依赖性。对于一系列等电子分子，计算了可用的化学导数，并使用泰勒级数展开来预测“周围”分子的能量。预测的能量值与使用 HF/KS 方法计算的值出人意料地吻合。所提出的方法允许人们以小于 1%的误差甚至更小的误差来预测价轨道的能量。
