Exact solutions of regular approximate relativistic wave equations for hydrogen-like atoms

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Abstract
Apart from relativistic effects originating from high kinetic energy of an electron in a flat potential, which are treated in first order by the Pauli Hamiltonian, there are relativistic effects even for low‐energy electrons if they move in a strong Coulomb potential. The latter effects can be accurately treated already in the zeroth order of an expansion of the Foldy–Wouthuysen transformation, if the expansion is carefully chosen to be nondivergent for r→0 even for Coulomb potentials, as shown by Van Lenthe et al. [J. Chem. Phys. 99, 4597 (1993)] (cf. also Heully et al. [J. Phys. B 19, 2799 (1986)] and Chang et al. [Phys. Scr. 34, 394 (1986)]). In the present paper, it is shown that the solutions of the zeroth order of this two‐component regular approximate (ZORA) equation for hydrogen‐like atoms are simply scaled solutions of the large component of the Diracwave function for this problem. The eigenvalues are related in a similar way. As a consequence, it is proven that under some restrictions, the ZORA Hamiltonian is bounded from below for Coulomb‐like potentials. Also, an exact result for the first order regular approximate Hamiltonian is given. The method can also be used to obtain exact results for regular approximations of scalar relativistic equations, like the Klein–Gordon equation. The balance between relativistic effects originating from the Coulombic singularity in the potential (typically core penetrating s and p valence electrons in atoms and molecules) and from high kinetic energy (important for high‐energy electrons in a flat potential and also for core‐avoiding high angular momentum (d, f, and g states in atoms) are discussed.

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