mIt is shown how the regularized two-component while many basis functions are required. If one wants to use a variational technique, one has to make sure that no spurious solutions appear. This can be done using so-called kinetically balanced basis sets, but, then, one needs an even larger basis for the small component than for the large component of the Dirac spinor. An attractive alternative is to transform the Dirac Hamiltonian into two-component form. We refer to the discussion by Kutzelnigg 111 for a detailed exposition of the various approaches and the difficulties that they give rise to in the form of divergent terms and singularities at the nuclei.The source of the difficulties is that the expansions that are being used implicitly or explicitly rely on an expansion in ( E -V )/2mc2, which is invalid for particles in a Coulomb potential, where there will always be a region of space (close to the

Seven different types of Slater type basis sets for the elements H (Z = 1) up to E118 (Z = 118), ranging from a double zeta valence quality up to a quadruple zeta valence quality, are tested in their performance in neutral atomic and diatomic oxide calculations. The exponents of the Slater type functions are optimized for the use in (scalar relativistic) zeroth-order regular approximated (ZORA) equations. Atomic tests reveal that, on average, the absolute basis set error of 0.03 kcal/mol in the density functional calculation of the valence spinor energies of the neutral atoms with the largest all electron basis set of quadruple zeta quality is lower than the average absolute difference of 0.16 kcal/mol in these valence spinor energies if one compares the results of ZORA equation with those of the fully relativistic Dirac equation. This average absolute basis set error increases to about 1 kcal/mol for the all electron basis sets of triple zeta valence quality, and to approximately 4 kcal/mol for the all electron basis sets of double zeta quality. The molecular tests reveal that, on average, the calculated atomization energies of 118 neutral diatomic oxides MO, where the nuclear charge Z of M ranges from Z = 1-118, with the all electron basis sets of triple zeta quality with two polarization functions added are within 1-2 kcal/mol of the benchmark results with the much larger all electron basis sets, which are of quadruple zeta valence quality with four polarization functions added. The accuracy is reduced to about 4-5 kcal/mol if only one polarization function is used in the triple zeta basis sets, and further reduced to approximately 20 kcal/mol if the all electron basis sets of double zeta quality are used. The inclusion of g-type STOs to the large benchmark basis sets had an effect of less than 1 kcal/mol in the calculation of the atomization energies of the group 2 and group 14 diatomic oxides. The basis sets that are optimized for calculations using the frozen core approximation (frozen core basis sets) have a restricted basis set in the core region compared to the all electron basis sets. On average, the use of these frozen core basis sets give atomic basis set errors that are approximately twice as large as the corresponding all electron basis set errors and molecular atomization energies that are close to the corresponding all electron results. Only if spin-orbit coupling is included in the frozen core calculations larger errors are found, especially for the heavier elements, due to the additional approximation that is made that the basis functions are orthogonalized on scalar relativistic core orbitals.

In this paper we will discuss relativistic total energies using the zeroth order regular approximation (ZORA). A simple scaling of the ZORA one-electron Hamiltonian is shown to yield energies for the hydrogenlike atom that are exactly equal to the Dirac energies. The regular approximation is not gauge invariant in each order, but the scaled ZORA energy can be shown to be exactly gauge invariant for hydrogenic ions. It is practically gauge invariant for many-electron systems and proves superior to the (unscaled) first order regular approximation for atomic ionization energies. The regular approximation, if scaled, can therefore be applied already in zeroth order to molecular bond energies. Scalar relativistic density functional all-electron and frozen core calculations on diatomics, consisting of copper, silver, and gold and their hydrides are presented. We used exchange-correlation energy functionals commonly used in nonrelativistic calculations; both in the local-density approximation (LDA) and including density-gradient (‘‘nonlocal’’) corrections (NLDA). At the NLDA level the calculated dissociation energies are all within 0.2 eV from experiment, with an average of 0.1 eV. All-electron calculations for Au2 and AuH gave results within 0.05 eV of the frozen core calculations.

Analytical expressions are derived for the evaluation of energy gradients in the zeroth order regular approximation ͑ZORA͒ to the Dirac equation. The electrostatic shift approximation is used to avoid gauge dependence problems. Comparison is made to the quasirelativistic Pauli method, the limitations of which are highlighted. The structures and first metal-carbonyl bond dissociation energies for the transition metal complexes W͑CO͒ 6 , Os͑CO͒ 5 , and Pt͑CO͒ 4 are calculated, and basis set effects are investigated.

In this paper we will calculate the effect of spin-orbit coupling on properties of closed shell molecules, using the zero-order regular approximation to the Dirac equation. Results are obtained using density functionals including density gradient corrections. Close agreement with experiment is obtained for the calculated molecular properties of a number of heavy element diatomic molecules.