Linear response formalism for the Douglas-Kroll-Hess approach to the Dirac-Kohn-Sham problem: First- and second-order nuclear displacement derivatives of the energy

Starting from a matrix representation of the modified Dirac equation, we offer an alternative approach to the Douglas-Kroll-Hess (DKH) protocol, specifically for the popular decoupling of electronic and positronic degrees of freedom to second-order in the potential. Then we use that approach for a convenient analysis of response properties. We discuss details of the linear and quadratic response of the DKH Hamiltonian to a geometric distortion encountered in calculations of first- and second-order derivatives of the energy. We present an implementation of analytic force constants in a quasi-relativistic (scalar) model constructed by the commonly used DKH transformation of second order in the nuclear potential only. We demonstrate the performance of the method by a comparison to the effective core potential method, using two isomers of the cluster Ir₄ and aqua complexes of the uranyl dication, UO₂²⁺(H₂O)_n (n = 3,... 6) as examples.