Douglas-Kroll-Transformed Spin-Orbit Operators

Douglas-Kroll-Transformed Spin-Orbit Operators 

In chapter 16, we showed how approximate relativistic spin-free Hamiltonians could be derived using unitary ttansformations. For applications to molecular systems the most frequently used model is the Douglas-Kroll-Hess model. Having successfully carried out a spin-free Douglas-Kroll-Hess calculation, we may be faced with the challenge of trying to account for the spin-orbit effects. In order to do this, we have to identify the terms that have been neglected in order to arrive at a spin-free formalism. 

The one-electton part of the second-order Douglas-KroU transformed Hamiltonian is given in (16.61) and can be expressed in two-component form for the positive-energy states as 

The first parenthesis contains the first-order term from the free-particle Foldy-Wouthuysen transformation, and the second parenthesis contains the second-order Douglas-Kroll term. Both A and Ep are spin-free, so the spin-dependence comes from the transformations involving 1 2, which was defined as 

Note that VVi is linear in 11.2, and thus both terms contain spin-dependence. In practice, we would want to include the spin-orbit operators from both terms even though the second-order term is the spin-orbit interaction behaves as 1/r and therefore 

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Douglas-Kroll-Transformed Spin-Orbit Operators... 

Models related to spin-forbidden reactions are discussed in this chapter. Coupling between two surfaces of different spin and symmetry is given by various levels of approximation for spin-orbit operators from the reduction of relativistic quantum mechanics. Well-established methods such as the Breit-Pauli Hamiltonian exist, but new relativistic methods such as the Douglas-Kroll Hamiltonian and other new transformation schemes are also being investigated and implemented today. 

Here we see a modified kinetic energy term that is cut off near the nuclei, a spin-free relativistic correction, and a spin-orbit term, both of which are regularized and behave as 1/r for small r. We may compare this with the regularization of the free-particle Foldy-Wouthuysen or Douglas-Kroll transformed Hamiltonian of section 16.3. The regularization clearly corrects the overestimation of relativistic effects that plagues the Pauli Hamiltonian. There is another consequence of the small r behavior. Since the relativistic correction operator behaves like 1 jr, it ought to be possible to use T zoRA variationally—and in fact we may demonstrate that there is a variational lower bound. 

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