Beijing National Laboratory for Molecular Sciences, Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, and Center for Computational Science and Engineering, Peking University, Beijing 100871, People's Republic of China.
J Chem Phys. 2012 Oct 21;137(15):154114. doi: 10.1063/1.4758987.
The separation of the spin-free and spin-dependent terms of a given relativistic Hamiltonian is usually facilitated by the Dirac identity. However, this is no longer possible for the recently developed exact two-component relativistic Hamiltonians derived from the matrix representation of the Dirac equation in a kinetically balanced basis. This stems from the fact that the decoupling matrix does not have an explicit form. To resolve this formal difficulty, we first define the spin-dependent term as the difference between a two-component Hamiltonian corresponding to the full Dirac equation and its one-component counterpart corresponding to the spin-free Dirac equation. The series expansion of the spin-dependent term is then developed in two different ways. One is in the spirit of the Douglas-Kroll-Hess (DKH) transformation and the other is based on the perturbative expansion of a two-component Hamiltonian of fixed structure, either the two-step Barysz-Sadlej-Snijders (BSS) or the one-step exact two-component (X2C) form. The algorithms for constructing arbitrary order terms are proposed for both schemes and their convergence patterns are assessed numerically. Truncating the expansions to finite orders leads naturally to a sequence of novel spin-dependent Hamiltonians. In particular, the order-by-order distinctions among the DKH, BSS, and X2C approaches can nicely be revealed. The well-known Pauli, zeroth-order regular approximation, and DKH1 spin-dependent Hamiltonians can also be recovered naturally by appropriately approximating the decoupling and renormalization matrices. On the practical side, the sf-X2C+so-DKH3 Hamiltonian, together with appropriately constructed generally contracted basis sets, is most promising for accounting for relativistic effects in two steps, first spin-free and then spin-dependent, with the latter applied either perturbatively or variationally.
给定的相对论哈密顿量的自旋无关项和自旋相关项的分离通常通过狄拉克恒等式来促进。然而，对于最近从动力学平衡基中的狄拉克方程的矩阵表示导出的精确双分量相对论哈密顿量，这不再可能。这源于去耦矩阵没有显式形式的事实。为了解决这个形式上的困难，我们首先将自旋相关项定义为对应于完整狄拉克方程的双分量哈密顿量与其对应于无自旋狄拉克方程的单分量对应物之间的差。然后以两种不同的方式展开自旋相关项的级数展开式。一种是在道格拉斯-克罗尔-赫斯（DKH）变换的精神下，另一种是基于固定结构的双分量哈密顿量的微扰展开，无论是两步 Barysz-Sadlej-Snijders（BSS）还是一步精确双分量（X2C）形式。为这两种方案提出了构造任意阶项的算法，并对其收敛模式进行了数值评估。将展开式截断到有限阶自然会导致一系列新的自旋相关哈密顿量。特别是，DKH、BSS 和 X2C 方法之间的逐阶区别可以很好地揭示出来。通过适当近似去耦和重整化矩阵，还可以自然地恢复著名的 Pauli、零阶正则逼近和 DKH1 自旋相关哈密顿量。在实际方面，sf-X2C+so-DKH3 哈密顿量，以及适当构造的广义收缩基集，最有希望分两步（先无自旋，后自旋相关）考虑相对论效应，后者可以通过微扰或变分方式应用。
