RIKEN Center for Computational Science, 7-1-26 Minatojima-minami, Cyuo-ku, Kobe, Hyogo 650-0047, Japan.
J Chem Phys. 2023 Jan 28;158(4):044103. doi: 10.1063/5.0131926.
We extended the conventional Douglas-Kroll (DK) and infinite order two-component (IOTC) methods to a technique applicable to Fock matrices, called extended DK (EDK) and extended IOTC (EIOTC), respectively. First, we defined a strategy to divide the Dirac-Fock operator into zero- and first-order terms. We then demonstrated that the first-order extended DK transformation, which is the Foldy-Wouthuysen transformation for the zero-order term, as well as the second- and third-order EDK and EIOTC, could be well defined. The EDK- and EIOTC-transformed Fock matrix, kinetic energy operator, nuclear attraction operator, and density matrix were derived. These equations were numerically evaluated, and it was found that these methods were accurate. In particular, EIOTC was consistent with the four-component approach. Four-component and extended two-component calculations are more expensive than non-relativistic calculations due to small-component-type two-electron integrals. We developed a new approximation formula, RIS-V, for small-component-type two-electron integrals, including the spin-orbit interaction between electrons. These results suggest that the RIS-V formula effectively accelerates the four-component and extended two-component methods.
我们将传统的 Douglas-Kroll (DK) 和无限阶双组份 (IOTC) 方法扩展到适用于 Fock 矩阵的技术，分别称为扩展 DK (EDK) 和扩展 IOTC (EIOTC)。首先，我们定义了一种策略，将 Dirac-Fock 算子分为零阶和一阶项。然后，我们证明了一阶扩展 DK 变换（即零阶项的 Foldy-Wouthuysen 变换）以及二阶和三阶 EDK 和 EIOTC 可以很好地定义。推导出了 EDK 和 EIOTC 变换后的 Fock 矩阵、动能算子、核吸引算子和密度矩阵。这些方程进行了数值评估，发现这些方法是准确的。特别是，EIOTC 与四分量方法一致。由于小分量型双电子积分，四分量和扩展双分量计算比非相对论计算更昂贵。我们开发了一种新的小分量型双电子积分的近似公式 RIS-V，包括电子之间的自旋轨道相互作用。这些结果表明，RIS-V 公式有效地加速了四分量和扩展双分量方法。
