Douglas-Kroll-Hess Theory

Chapter 11 introduced the basic principles for elimination-of-the-small-component protocols and noted that the Foldy Wouthuysen scheme applied to one-electron operators including scalar potentials yield ill-defined 1 /c-expansions of the desired block-diagonal Hamiltonian. In contrast, the Douglas Kroll-Hess transformation represents a unique and valid decoupling protocol for such Hamiltonians and is therefore investigated in detail in this chapter. 

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Accounting for Relativistic Effects within the Douglas-Kroll-Hess Theory... 

Of course, what has just been stated for the one-electron Dirac Hamiltonian is also valid for the general one-electron operator in Eq. (11.1). However, the coupling of upper and lower components of the spinor is solely brought about by the off-diagonal ctr p operators of the free-partide Dirac one-electron Hamiltonian and kinetic energy operator, respectively. We shall later see that the occurrence of any sort of potential V will pose some difficulties when it comes to the determination of an explicit form of the unitary transformation U. A universal solution to this problem will be provided in chapter 12 in form of Douglas-Kroll-Hess theory. 

This Hamiltonian can then be used variationally in quantum chemical calculations, since because of its derivation no negative energy states can occur. It should be anticipated that this Hamiltonian is conceptually equivalent to the infinite-order Douglas-Kroll-Hess Hamiltonian to be discussed in section 12.3, because both schemes do not apply any expansion in 1/c. Also the expressions for Ep and Ap are strictly evaluated in closed form within both approaches. However, whereas Douglas-Kroll-Hess theory yields analytic exressions for each order in V, the infinite-order two-component method summarizes all powers of V in the final matrix representation of/+. 

M. Reiher. Relativistic Douglas-Kroll-Hess Theory. WIREs Comp. Mol. Sci, 2 (2012) 139-149. 

A. Wolf, M. Reiher. Exact decoupling of the Dirac Hamiltonian. IV. Automated evaluation of molecular properties within the Douglas-Kroll-Hess theory up to arbitrary order. /. Chem. Phys., 124 (2006)... 

M. Reiher, Douglas-Kroll-Hess Theory a relativistic electrons-only theory for chemistry, Theor. Chem. Acc., 2006, 116, 241-252. 

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