Finite-Basis-Set Approach to the Two-Center Heteronuclear \\ Dirac Problem

The rigorous two-center approach based on the dual-kinetically balanced finite-basis-set expansion is applied to one-electron, heteronuclear diatomic Bi-Au, U-Pb, and Cf-U quasimolecules. The obtained $1\sigma$ ground-state energies are compared with previous calculations, when possible. Upon analysis of three different placements of the coordinate system's origin in the monopole approximation of the two-center potential: (1) in the middle, between the nuclei, (2) in the center of the heavy nucleus, and (3) in the center of the light nucleus, a substantial difference between the results is found. The leading contributions of one-electron quantum electrodynamics (self-energy and vacuum polarization) are evaluated within the monopole approximation as well.

Previously, we considered the one-and two-electron homonuclear quasimolecules of xenon, lead, and uranium in both the rigorous two-center approach and the monopole approximation within the dual-kinetically balanced finite-basis-set approach [36,37]. We showed that the obtained solution is in good agreement with other independent calculations of the energy spectra. In Ref. [37], it was shown that an analysis of different placements of the coordinate system's origin (c.s.o.) could provide an estimation of the non-monopole correction to contributions that are not presently available for rigorous two-center evaluation.
In the present work, we extend our approach to the case of one-electron heteronuclear quasimolecules: Bi-Au, U-Pb, and Cf-U. The ground-state energy is evaluated in a wide range of internuclear distances, up to 1000 fm, in both two-center and monopole potentials. 2 (fine-structure constant), are used throughout the paper.

II. METHOD
We start with the Born-Oppenheimer approximation, in which the electron is described by the two-center Dirac equation, where r and R A,B are the position vectors of the electron and the nuclei, respectively; V A,B nucl (r) are the spherically symmetric binding potentials generated by the nuclei; p is the momentum operator; α and β are the standard 4 × 4 Dirac matrices. The distance between the nuclei In this work, we use the Fermi model of the nuclear-charge distribution. The corresponding explicit formulas are well-known and can be found, e.g., in Ref. [38].
The two-center (TC) potential is axially symmetric with respect to the internuclear axis.
In the spherical coordinate system (r, θ, ϕ) with the polar angle θ measured from this axis, the potential can be expanded into the following series: The first term in this series, V 0 (r), corresponds to the widely used monopole approximation (MA). Within this approximation, the initial axially symmetric problem is reduced to the spherically symmetric one. Numerous methods developed for the atomic problem can be applied to solve the corresponding Dirac equation. We use the dual-kinetically balanced finite-basis-set approach for both the TC and MA potentials; see Refs. [36,37,39,40] for more details.
The spherical coordinates are used with three different placements of the c.s.o., namely: (1) in the middle between the nuclei, (2)  In addition to the Dirac energies, we also evaluate the leading QED corrections selfenergy and vacuum polarization. These terms are only treated within the monopole approximation; that is, the MA(1) potential is used in this case. The computations follow the procedures discussed, e.g., in Refs. [41][42][43][44][45][46]. They are based on the expansion of the electron propagator in powers of the binding potential in order to isolate ultraviolet divergences and perform renormalization.

III. RESULTS
In Figure 1 Furthermore, we note an almost constant difference between the MA(2) and the MA (3) results within the presented range.
There are two main sources for the total uncertainty of the obtained results: (1) the numerical error of the computational scheme, which is determined by the quality of the finite basis set employed in the practical calculations (basis-set error), and (2) the error associated with the uncertainties of the nuclear model and the root-mean-square radii (nuclear error).
The nuclear error provides the main contribution to the total uncertainty at the small internuclear distances and rapidly decreases towards the larger D. Meanwhile, the basis-set error is rather small, and its value, e.g., for the U-Pb quasimolecule, does not exceed 10 eV in the entire studied range. Therefore, the total uncertainty at the small D (up to 300 fm)  (3) potentials. E * 1σ corresponds to the data from Ref. [35] .
In Figure 2, we compare the ground-state energies for the U-Pb quasimolecule evaluated using the TC approach with the available data [35]. The difference ∆E = E Ref. [35] 1σ − E T C 1σ is plotted. All the values are in good agreement, except for the one with D = 50 fm. For this internuclear distance, we estimate our total numerical error to be ±30 eV, which is three times smaller than the corresponding uncertainty presented in Ref. [35]. The reasons for this deviation are unclear to us.
The numerical data, including the self-energy and vacuum polarization contributions of all the quasimolecules under consideration, can be found in Table I. We note that the difference between TC and MA(1) for the binding energies is significantly larger, more than an order of magnitude in most cases, than the total QED correction. Thus, an evaluation of the Dirac energy within the rigorous two-center approach is crucial for the accurate determination of the electronic spectra.   In this work, the ground-state energies of the Bi-Au, U-Pb, and Cf-U quasimolecules at different internuclear distances, up to 1000 fm, were evaluated within the rigorous two-center approach. The monopole approximation was also considered using three different placements of the coordinate system's origin: (1) in the middle between the nuclei, (2) in the center of the heavy nucleus, and (3) in the center of the light nucleus. The results obtained within the two-center approach were found to be in good agreement with previous independent calculations for the Bi-Au and U-Pb quasimolecules. The leading QED contributions, selfenergy and vacuum polarization, were also evaluated within the monopole approximation.
Accurate theoretical predictions of the quasimolecular spectra require further development of the presented methods, including rigorous two-center evaluation of the QED contributions.

ACKNOWLEDGMENTS
The one-electron energy calculations were funded by the Russian Science Foundation