One-electron energy spectra of heavy highly charged quasimolecules

The generalized dual-kinetic-balance approach for axially symmetric systems is employed to solve the two-center Dirac problem. The spectra of one-electron homonuclear quasimolecules are calculated and compared with the previous calculations. The analysis of the monopole approximation with two different choices of the origin is performed. Special attention is paid to the lead and xenon dimers, Pb$^{82+}$-Pb$^{82+}$-e$^{-}$ and Xe$^{54+}$-Xe$^{54+}$-e$^{-}$, where the energies of the ground and several excited $\sigma$-states are presented in the wide range of internuclear distances. The developed method provides the quasicomplete finite basis set and allows for construction of the perturbation theory, including within the bound-state QED.


II. METHOD
In heavy atomic systems the parameter αZ (α is the fine-structure constant and Z is the nuclear charge), which measures the coupling of electrons with nuclei, is not small. Therefore, the calculations for these systems should be done within the fully relativistic approach, i.e., to all orders in αZ. With this in mind, we start with the Dirac equation for the two-center potential, Here r and R 1,2 are the coordinates of the electron and nuclei, respectively, V A,B nucl (r) is the nuclear potential at the distance r generated by nucleus with the charge Z A,B , α and β are the standard 4 × 4 Dirac matrices: where σ is a vector of the Pauli matrices.
In the following we consider the identical nuclei, i.e. Z A = Z B , with the Fermi model of the nuclear charge distribution: where ρ 0 is the normalization constant, a is skin thickness constant and c is the half-density radius, for more details see, e.g., Ref. [41].
The solution of Eq. (1) is obtained within the dual-kinetic-balance (DKB) approach, which allows one to solve the problem of "spurious" states. Originally, this approach was implemented for spherically symmetric systems, like atoms, [38], using the finite basis set constructed from the B-splines [42,43]. Later, authors of Ref. [39] generalized it to the case of axially symmetric systems (A-DKB) they considered atom in the external homogeneous field. This situation was also considered within this method in Refs. [44][45][46] to evaluate the higher-order contributions to the Zeeman splitting in highly charged ions. In Ref. [40] we have adapted the A-DKB method to diatomic systems, which also possess axial symmetry.
Below we provide a brief description of the calculation scheme.
The system under consideration is rotationally invariant with respect to the z-axis directed along the internuclear vector D = R 2 − R 1 . Therefore, the z-projection of the total angular momentum with the quantum number m J is conserved and the electronic wave function can be written as, The (r, θ)-components of the wave function are represented using the finite-basis-set expansion: where B ir (r) are Legendre polynomials of the argument 2θ/π − 1, and e u 4 u=1 are the standard four-component basis vectors: The Λ-matrix imposes the dual-kinetic-balance conditions on the basis set. With the given form of Φ and the finite basis set one can find the corresponding Hamiltonian matrix H ij . The eigenvalues and eigenfunctions are found by diagonalization of H ij . As a result, we obtain quasicomplete finite set of wave functions and electronic energies for the two-center Dirac equation. Ground and several lowest excited states are reproduced with high accuracy while the higher-excited states effectively represent the infinite remainder of the spectrum. The negative-energy continuum is also represented by the finite number of the negative energy eigenvalues. This quasicomplete spectrum can be used to construct the Green function, which is needed for the perturbation theory calculations.

III. RESULTS
Relativistic calculations of the binding energies of heavy one-electron quasimolecules were presented, in particular, in Refs. [20,28,31,33,35,36], see also references therein. Ref. [36] provides nearly the most accurate up-to-date values for the very broad range of Z and taking into account the finite nuclear size. So, we use just these data for comparison, see Table I, where the ground-state energies are presented for Z = 1 . . . 100 at the so-called "chemical" distances, D = 2/Z a.u. We observe that the results are in good agreement, the relative deviation varies from 2 × 10 −6 for hydrogen to 5 × 10 −5 for Z = 100. This deviation is consistent with our own estimation of the numerical uncertainty, which is evaluated by inspecting the convergence of the results with respect to the size of the basis set. In this calculation up to N r = 320 B-splines and N θ = 54 Legendre polynomials are used, for heavy nuclei this number of basis functions ensures the uncertainty, which is comparable to or smaller than the uncertainty of the finite nuclear size effect at all internuclear distances from 0 to 2/Z a.u.
Next, we present the obtained one-electron spectra of the Pb 82+ -Pb 82+ -e − and Xe 54+ -Xe 54+ -e − quasimolecules in the wide range of the internuclear distances from few tens of fermi up to the "chemical" distances. In the present figures only σ-states (m J = ± 1 2 ) are shown. The precise quantum numbers are m J and parity, g (gerade) or u (ungerade). In addition, we determine the quantum numbers of the "merged atom", i.e. the state of the system with internuclear distance D → 0, and put it to the left of molecular term symbol, e.g., the ground state is 1s 1/2 σ g .
In Figure 1, the energies of the ground (n = 1) and first 9 (n = 2 . . . 10) excited states of Pb 82+ -Pb 82+ -e − system as the functions of the internuclear distance are shown. Here, n has no connection with atomic principal quantum number, it simply enumerates the σ-states.
To visually compare the data obtained with the ones by Soff et al. we zoom the second plot   in Fig. 1 to match the scale of the corresponding figure from Ref. [15]. Although we cannot compare the numerical results, the plots for all the states under consideration appear to be in very good agreement all the states are identified correctly, all the crossings and avoided crossings appear at the same internuclear distances. The similar results for xenon, i.e., the energies of the ground (n = 1) and first 9 (n = 2 . . . 10) excited states of Xe 54+ -Xe 54+ -e − system are shown in Figure 3. Tables II and III,    uncertainty serves as a non-trivial self-test of the method, since the basis-set expansion (6) is essentially different for the two cases. In fact, due to the lower symmetry of the second c.s., the uncertainty of the TC(2) values is much larger and completely determines the difference between TC(1) and TC (2). The differences between the TC(1) and MA(1) results Table II: Ground-state binding energy E 1σg (in eV) of the Pb 82+ -Pb 82+ -e − quasimolecule for the two-center potential (TC) and for the monopole-approximation potential (MA), with coordinate system origin at the center of mass of the nuclei (1) and at the one of the nuclear centers (2). are not yet available for the TC calculations, e.g., the two-photonexchange and QED corrections [40].

IV. DISCUSSION AND CONCLUSION
In this work, the two-center Dirac equation is solved within the dual-kinetic-balance method [38,39]. The energies of the ground and several excited σ-states in such heavy diatomic systems as Pb 82+ -Pb 82+ -e − and Xe 54+ -Xe 54+ -e − are plotted as a function of the The notations are the same as in Table II.