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Molecular Physics
An International Journal at the Interface Between Chemistry and Physics
Volume 120, 2022 - Issue 19-20: Special Issue of Molecular Physics in Memory of Lutosław Wolniewicz
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Special Issue of Molecular Physics in Memory of Lutosław Wolniewicz

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

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Article: e2092563 | Received 24 Jan 2022, Accepted 15 Jun 2022, Published online: 29 Jun 2022
 

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, Kr35+ and Th89+ systems with the reference data of 12-digit accuracy provided by the GRASP2 and QRHF atomic codes.

The method is also tested on H2+, Kr271+ and Th2179+ systems. The energies of 1σ1/2g and 1σ1/2u states of H2+ are converged to 9 significant figures and the relativistic correction to 6. In the case of the Th2179+ system the calculations for the 1σ1/2g state were performed for a range of internuclear separations 2/90R4.0 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 1σ1/2g 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

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 14c2R2(ξ2η2)(ϵV)2 is missing in Equation 11a,b of the paper by Sundholm et al. [Citation13].

3 The calculations were realised via the complete active space self-consistent field method (CASSCF) [Citation49,Citation50]

4 According to Bağcı and Hoggan [Citation51] the relativistic energy of 1s1/2 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.

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