Solution of the Dirac equation using the Rayleigh-Ritz method: Flexible basis coupling large and small components. Results for one-electron systems

Abstract

An algebraic solution of the Dirac equation is reinvestigated. Slater-type spinor orbitals and their corresponding system of differential equations are defined in two- and four-component formalism. They describe the radial function in components of the wave function of the Dirac equation solution to high accuracy. They constitute the matrix elements arising in a generalized eigenvalue equation. These terms are evaluated through prolate spheroidal coordinates. The corresponding integrals are calculated by the numerical global-adaptive method taking into account the Gauss-Kronrod numerical integration extension. Sample calculations are performed using flexible basis sets generated with both signs of the relativistic angular momentum quantum number κ. Applications to one-electron atoms and diatomics are detailed. Variationally optimum values for orbital parameters are obtained at given nuclear separation. Methods discussed in this work are capable of yielding highly accurate relativistic two-center integrals for all ranges of orbital parameters. This work provides an efficient way to overcome the problems that arise in relativistic calculations.

JRAF: A Julia Package for Computation of the Relativistic Molecular Auxiliary Functions

“…The additional computer program code written for the molecular integrals [Eqs. (28−30)] is based on the formulas given in [9,25]. The results show that code presented in this study for the relativistic auxiliary functions based on the analytical method via the series representation of the beta functions allows arbitrary precision floating point calculations.…”

“…The solution two−center nuclear attraction integrals are derived through the single−center potential [9],…”

“…Dropping the restriction on principal quantum number leads to new features [9,25,27] (see also references therein) and wide range of application in physics and chemistry. The analytical method used for computation of molecular auxiliary functions is of course open for improvement.…”

“…The problem in basis spinors derived from the Eqs. (1, 2) such as S −spinors [7], Slater−type spinors orbitals [9] are that they do not pose an addition theorem [10]. The power function r γ in the Eq.…”

Analytical evaluation of relativistic molecular integrals. I. Auxiliary functions

“…The STO basis with non-integer pribcipal quantum numbers provides extra flexibility for closer variational description of trial wavefunction [4,[6][7][8][9][10][11][12][13][14][15] in the linear combination of atomic orbital method [16]. They also lead to use of a Slater-type spinor basis [17,18] in algebraic solution of the four-component Dirac equation [19,20] due to the so-called kinetic balance condition [21][22][23][24][25][26]. The matrix elements arising in a generalized eigenvalue equation are evaluated through prolate spheroidal coordinates and expressed in terms of relativistic molecular auxiliary functions.…”