Numerical electronic structure calculations for atoms. II. Generalized variable transformation and relativistic calculations

Abstract

ABSTRACT:The pairs of radial functions P i and Q i , which are part of the four-component single-particle spinors in the relativistic description of the electronic structure of bound states of atoms, are usually determined as solutions of eigenvalue problems. The latter constitute two-point boundary value problems which involve coupled pairs of first-order ordinary radial differential equations. To introduce a suitable notation, the theory involved in relativistic electronic structure calculations for atoms is briefly reviewed, including a general treatment of arbitrary transformations for the radial variable. Such variable transformations must be specified then, either explicitly (by a transformation function) or implicitly (by the solution method itself), for any actual calculation, though initially the variable transformation is almost completely independent of the numerical method to be applied. We consider suitable transformation functions out of which an actual choice can be made within wide limits. In our approach, the resulting transformed radial differential equations are discretized then with finite-difference methods. Since the accuracy and efficiency of these methods is increased when a constant step size h between contiguous points (in the transformed variable) is used, it is only here where the careful choice of the variable transformation is important. The resulting system of linear equations is solved for the radial functions with standard linear algebra methods rather than "shooting" methods. The present work extends the general study of variable transformations, given recently for nonrelativistic electronic structure calculations in Part I, to the relativistic case. Important results following from the present work are (i) a discretization scheme for first-order ordinary differential equations similar to the ANDRAE, REIHER, AND HINZE well-known standard Numerov scheme used within the nonrelativistic framework, (ii) a consistent numerical algorithm with an overall truncation error of order h 4 , (iii) a method for handling effective potentials behaving singularly like r −1 at the origin (as encountered, e.g., when the atomic nucleus is represented by a pointlike charge density distribution), and, in connection with this last point, (iv) the avoidance of nonanalytic short-range behavior of the solutions to be obtained from the differential equations.

On the definition of local spin in relativistic and nonrelativistic quantum chemistry

“…These radial functions of the 4-spinor are obtained on a logarithmic grid consisting of 3500 inner grid points distributed over an r-interval of [0,10] bohr using a grid parameter of b = 0.000004. 48 Regarding the accuracy achieved in the numerical Dirac calculations we note that the ground state energies agree very well with the analytically known eigenvalue. In both atomic structure programs, a point-like nucleus with singular Coulombic nuclear attraction potential ÀZ/r was employed.…”

Mössbauer spectroscopy for heavy elements: a relativistic benchmark study of mercury

“…In the relativistic case, it is already common to introduce an extended-nucleus model [11][12][13][14] because this facilitates the use of Gaussian type orbitals (GTOs). GTOs have zero slope at the nucleus, which is consistent with the exact solutions for extended-nuclear models [15][16][17][18] in both non-relativistic and relativistic theory. This feature thus makes GTOs a natural expansion set for relativistic orbitals for which one may rely on well-established basis sets augmented by steep functions (as e.g., demonstrated recently for contact densities at iron nuclei [18]).…”

The Shell Structure of Atoms

“…Atomic four-component Dirac−Hartree−Fock and MCSCF calculations were carried out fully numerically . While all angular degrees of freedom are treated analytically, the two radial functions P i ( r ) = P n i κ i ( r ) and Q i ( r ) = Q n i κ i ( r ) of the 4-spinor are represented on an equidistant, logarithmic grid of points in the new variable s , which is calculated from the original radial variable r (see refs and for details on this type of radial grid). The Laplacian of the spherically averaged electron density was calculated numerically employing a three-point finite difference formula.…”