Reiher Markus, Wolf Alexander
Lehrstuhl für Theoretische Chemie, Universität Bonn, Wegelerstrasse 12, D-53115, Germany.
J Chem Phys. 2004 Aug 1;121(5):2037-47. doi: 10.1063/1.1768160.
Exact decoupling of positive- and negative-energy states in relativistic quantum chemistry is discussed in the framework of unitary transformation techniques. The obscure situation that each scheme of decoupling transformations relies on different, but very special parametrizations of the employed unitary matrices is critically analyzed. By applying the most general power series ansatz for the parametrization of the unitary matrices it is shown that all transformation protocols for decoupling the Dirac Hamiltonian have necessarily to start with an initial free-particle Foldy-Wouthuysen step. The purely numerical iteration scheme applying X-operator techniques to the Barysz-Sadlej-Snijders (BSS) Hamiltonian is compared to the analytical schemes of the Foldy-Wouthuysen (FW) and Douglas-Kroll-Hess (DKH) approaches. Relying on an illegal 1/c expansion of the Dirac Hamiltonian around the nonrelativistic limit, any higher-order FW transformation is in principle ill defined and doomed to fail, irrespective of the specific features of the external potential. It is shown that the DKH method is the only valid analytic unitary transformation scheme for the Dirac Hamiltonian. Its exact infinite-order version can be realized purely numerically by the BSS scheme, which is only able to yield matrix representations of the decoupled Hamiltonian but no analytic expressions for this operator. It is explained why a straightforward numerical iterative extension of the DKH procedure to arbitrary order employing matrix representations is not feasible within standard one-component electronic structure programs. A more sophisticated ansatz based on a symbolical evaluation of the DKH operators via a suitable parser routine is needed instead and introduced in Part II of this work.
在酉变换技术框架下讨论了相对论量子化学中正能量态和负能量态的精确解耦。对每种解耦变换方案都依赖于所采用酉矩阵的不同但非常特殊的参数化这一模糊情况进行了批判性分析。通过对酉矩阵参数化应用最一般的幂级数假设，表明所有用于解耦狄拉克哈密顿量的变换协议都必然要从初始的自由粒子福尔德 - 伍森步骤开始。将应用X算符技术于巴里什 - 萨德莱 - 斯尼德斯（BSS）哈密顿量的纯数值迭代方案与福尔德 - 伍森（FW）和道格拉斯 - 克罗尔 - 赫斯（DKH）方法的解析方案进行了比较。由于在非相对论极限附近对狄拉克哈密顿量进行了非法的1/c展开，任何高阶FW变换原则上都是定义不明确的且注定要失败，无论外部势的具体特征如何。结果表明，DKH方法是狄拉克哈密顿量唯一有效的解析酉变换方案。其精确的无穷阶版本可以通过BSS方案纯数值地实现，该方案只能产生解耦哈密顿量的矩阵表示，而不能给出该算符的解析表达式。解释了为什么在标准的单组分电子结构程序中，将DKH过程直接数值迭代扩展到任意阶并使用矩阵表示是不可行的。相反，需要一种基于通过合适的解析器例程对DKH算符进行符号评估的更复杂假设，并在本工作的第二部分中引入。
