Two-component Douglas-Kroll Hamiltonians

For most chemical applications, one is not interested in negative energy solutions of a four-component Dirac-type Hamiltonian. In addition, the computational expense of treating four-component complex-valued wave functions often limited such calculations to benchmark studies of atoms and small molecules. Therefore, much effort was put into developing and implementing approximate quantum chemistry methods which explicitly treat only the electron degrees of freedom, namely two- and one-component relativistic formulations [2]. This analysis also holds for a relativistic DFT approach and the solutions of the corresponding DKS equation. 

Solutions with positive and negative energies of a one-particle Dirac equation of a molecular system are represented by states where either electronic or positronic contributions of four-component wave functions dominate. With chemical systems in mind, electronic and positronic components are also referred to as large and small components, respectively. However, small components cannot simply be neglected or projected out to arrive at a simpler two-component description because, in an intrinsic fashion they also contribute in a fully relativistic description of a chemical system. Thus, a projection step, in which positronic components are discarded, can only be applied after a suitable decoupling of electron and positron degrees of freedom. Then the effects of the small components are implicitly accounted for. 

The Foldy-Wouthuysen (FW) transformation [67] offers a decoupling, which in principle is exact, but it is impractical and leads to a singular expansion in 1/c in the important case of a Coulomb potential [68]. Douglas and Kroll (DK) suggested an alternative decoupling procedure based on a series of appro- 

This first-order DK transformation can be represented by a unitary matrix 

the two-component second-order operators 2 0, 2 represent the 

---