We derive the most general parametrization of the unitary matrices in the Douglas-Kroll ͑DK͒ transformation sequence for relativistic electronic structure calculations. It is applied for a detailed analysis of the generalized DK transformation up to fifth order in the external potential. While DKH2-DKH4 are independent of the parametrization of the unitary matrices, DKH5 turns out to be dependent on the third expansion coefficient of the innermost unitary transformation which is carried out after the initial free-particle Foldy-Wouthuysen transformation. The freedom in the choice of this expansion coefficient vanishes consistently if the optimum unitary transformation is sought for. Since the standard protocol of the DK method is the application of unitary transformations to the one-electron Dirac operator, we analyze the DKH procedure up to fifth order for hydrogenlike atoms. We find remarkable accuracy of the higher-order DK corrections as compared to the exact Dirac ground state energy. In the case of many-electron atomic systems, we investigate the order of magnitude of the higher-order corrections in the light of the neglect of the DK transformation of the two-electron terms of the many-particle Hamiltonian. A careful analysis of the silver and gold atoms demonstrates that both the fourth-and fifth-order one-electron DK transformation yield a smaller contribution to the total electronic energy than the DK transformation of the two-electron terms. In order to improve significantly on the third-order correction DKH3, it is thus mandatory to include the DK transformation of the two-electron terms as well as the spin-dependent terms before proceeding to higher orders in the transformation of the one-electron terms. However, an analysis of the ionization energies of these atoms indicates that already DKH3 yields a highly accurate treatment of the scalar-relativistic effects on properties.

In order to achieve exact decoupling of the Dirac Hamiltonian within a unitary transformation scheme, we have discussed in part I of this series that either a purely numerical iterative technique (the Barysz-Sadlej-Snijders method) or a stepwise analytic approach (the Douglas-Kroll-Hess method) are possible. For the evaluation of Douglas-Kroll-Hess Hamiltonians up to a pre-defined order it was shown that a symbolic scheme has to be employed. In this work, an algorithm for this analytic derivation of Douglas-Kroll-Hess Hamiltonians up to any arbitrary order in the external potential is presented. We discuss how an estimate for the necessary order for exact decoupling (within machine precision) for a given system can be determined from the convergence behavior of the Douglas-Kroll-Hess expansion prior to a quantum chemical calculation. Once this maximum order has been accomplished, the spectrum of the positive-energy part of the decoupled Hamiltonian, e.g., for electronic bound states, cannot be distinguished from the corresponding part of the spectrum of the Dirac operator. An efficient scalar-relativistic implementation of the symbolic operations for the evaluation of the positive-energy part of the block-diagonal Hamiltonian is presented, and its accuracy is tested for ground-state energies of one-electron ions over the whole periodic table. Furthermore, the first many-electron calculations employing sixth up to fourteenth order DKH Hamiltonians are presented.

Exact decoupling of positive- and negative-energy states in relativistic quantum chemistry is discussed in the framework of unitary transformation techniques. The obscure situation that each scheme of decoupling transformations relies on different, but very special parametrizations of the employed unitary matrices is critically analyzed. By applying the most general power series ansatz for the parametrization of the unitary matrices it is shown that all transformation protocols for decoupling the Dirac Hamiltonian have necessarily to start with an initial free-particle Foldy-Wouthuysen step. The purely numerical iteration scheme applying X-operator techniques to the Barysz-Sadlej-Snijders (BSS) Hamiltonian is compared to the analytical schemes of the Foldy-Wouthuysen (FW) and Douglas-Kroll-Hess (DKH) approaches. Relying on an illegal 1/c expansion of the Dirac Hamiltonian around the nonrelativistic limit, any higher-order FW transformation is in principle ill defined and doomed to fail, irrespective of the specific features of the external potential. It is shown that the DKH method is the only valid analytic unitary transformation scheme for the Dirac Hamiltonian. Its exact infinite-order version can be realized purely numerically by the BSS scheme, which is only able to yield matrix representations of the decoupled Hamiltonian but no analytic expressions for this operator. It is explained why a straightforward numerical iterative extension of the DKH procedure to arbitrary order employing matrix representations is not feasible within standard one-component electronic structure programs. A more sophisticated ansatz based on a symbolical evaluation of the DKH operators via a suitable parser routine is needed instead and introduced in Part II of this work.

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In this paper, the calculation of electric-field-like properties based on higher-order Douglas-Kroll-Hess (DKH) transformations is discussed. The electric-field gradient calculated within the Hartree-Fock self-consistent field framework is used as a representative property. The properties are expressed as an analytic first derivative of the four-component Dirac energy and the nth-order DKH energy, respectively. The differences between a "forward" transformation of the relativistic energy or the "back transformation" of the wave function is discussed in some detail. Detailed test calculations were carried out on the electric-field gradient at the halogen nucleus in the series HX (X=F,Cl,Br,I,At) for which extensive reference data are available. The DKH method is shown to reproduce (spin-free) four-component Dirac-Fock results to an accuracy of better than 99% which is significantly closer than previous DKH studies. The calculations of both the Hamiltonian and the property operator are shown to be essentially converged after the second-order transformation, even for elements as heavy as At. In addition, we have obtained results within the density-functional framework using the DKHZ and zeroth-order regular approximation (ZORA) methods. The latter results included picture-change effects at the scalar relativistic variant of the ZORA-4 level and were shown to be in quantitative agreement with earlier results obtained by van Lenthe and Baerends. The picture-change effects are somewhat smaller for the ZORA method compared to DKH. For heavier elements significant differences in the field gradients predicted by the two methods were found. Based on comparison with four-component Dirac-Kohn-Sham calculations, the DKH results are more accurate. Compared to the spin-free Dirac-Kohn-Sham reference values, the ZORA-4 formalism did not improve the results of the ZORA calculations.