Two-dimensional, finite-difference method of solving the Dirac equation for diatomic molecules revisited

Abstract

Two-dimensional, fully numerical, finite-difference approach to the second-order Dirac equation for one-electron diatomics is revisited. Instead of using the transformation of variables to get rid of the singularities of σ-type spinors the Gaussian nuclear charge distribution model is employed. The approach is tested on the several lowest σ, π, δ and φ states of H, Kr${}^{35+}$ and Th${}^{89+}$ systems with the reference data of 12-digit accuracy provided by the GRASP${}^2$ and QRHF atomic codes.

The method is also tested on H${}_2^{+}$, Kr${}_2^{71+}$ and Th${}_2^{179+}$ systems. The energies of $1{\sigma }_{1/2\mathrm{g}}$ and $1{\sigma }_{1/2\mathrm{u}}$ states of H${}_2^{+}$ are converged to 9 significant figures and the relativistic correction to 6. In the case of the Th${}_2^{179+}$ system the calculations for the $1{\sigma }_{1/2\mathrm{g}}$ state were performed for a range of internuclear separations $2/90\le R\le 4.0$ a.u. and compared with the eXact 2-Component (X2C) and 4-component results. The agreement to within 1 milihartree or better is observed for all the separations greater than 0.2 a.u. Alas, the $1{\sigma }_{1/2\mathrm{g}}$ orbital energy of the system with $R=2/90$ a.u., could only be calculated with 0.1% accuracy. This problem also seems to haunt Sundholm's implementation of the method and we indicate the reasons behind the low accuracy at extremely short internuclear distances.

GRAPHICAL ABSTRACT

Keywords:

• Dirac equation
• diatomic molecules
• Gaussian nuclear charge distribution
• prolate spheroidal coordinates
• 8th-order discretisation
• successive overrelaxation

Acknowledgments

We would like to thank Jacek Karwowski for his comments on the draft version of the manuscript. We are also very much indebted to the referee for the host of valuable remarks that helped improve the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Notes

1 The following values of atomic masses (in amu) were used in the DIRAC2D and Molcas programs: H – 1.0078250350, Kr – 83.911507, Rn – 222.017571, Th – 232.0380508 (in the DIRAC program the mass of the hydrogen nucleus is set to 1.007825).

2 Note that the term $\frac{1}{4c^2}{R}^2({\xi }^2-{\eta }^2)(ϵ-V{)}^2$ is missing in Equation 11a,b of the paper by Sundholm et al. [13].

3 The calculations were realised via the complete active space self-consistent field method (CASSCF) [49,50]

4 According to Bağcı and Hoggan [51] the relativistic energy of 1s${}_{1/2}$ state of the hydrogen atom with the point nucleus can be calculated via the Rayleigh-Ritz method as $-0.5000066565965473$ hartree and – according to this paper – is equal to the binding energy that can be obtained from the analytical formula (as given by Eg. 62 of that paper), when the velocity of light equal to $137.035999139$ a.u. is used. However, when the formula is used with this very velocity of light it leads to a slightly different value of the energy, namely, $-0.500006656596547032$ hartree. This value can be obtained by means of the bc utility implementing an arbitrary precision calculator.