Douglas-Kroll-Hess Hamiltonians

The example of neon, where relativistic contributions account for as much as a0.5% of 711, shows that relativistic effects can turn out to be larger for high-order NLO properties and need to be included if aiming at high accuracy. Some efforts to implement linear and nonlinear response functions for two- and four-component methods and to account for relativity in response calculations using relativistic direct perturbation theory or the Douglas-Kroll-Hess Hamiltonian have started recently [131, 205, 206]. But presently, only few numerical investigations are available and it is unclear when it will become important to include relativistic effects for the frequency dispersion. 

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/+. 

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