Introduction:

The finite basis set approach is widely accepted as the standard for many atomic and molecular calculations. It has been successfully used in quantum electrodynamic calculations involving electron self-energy loops. However, it has not yet been applied to vacuum polarization loops for a long time. In [Salman and Saue (PRA, 2023)], the Gaussian finite basis set method was used to calculate the main partial contribution |κ| = 1 of the many-potential vacuum polarization charge density. In our work, [Ivanov et al., PRA, 2024], we repeated this technique and extended the calculations up to |κ| = 5. This allowed us to find energy shift corrections that can be compared to existing results [Persson et al., PRA 1993].

Methods:

The presented calculations were obtained using the finite basis set approach. In this method, the solution to the differential equation — the Dirac equation in our case — is presented as a linear combination of basis functions. The coefficients in this combination were found via the variational principle. The obtained solution can accurately present the Green's function in the vacuum polarization loop.

Results and Discussion:

We present our recent progress in vacuum polarization calculations in hydrogen-like ions using finite basis sets. Our calculations used larger basis sets than those used in [Ivanov et al., PRA, 2024]. This allowed us to find the energy correction with higher precision, proving the convergence of the method.

Conclusions:

Our results demonstrate the potential of applying the finite basis set method to vacuum polarization calculations. Our numerical results approach those reachable by standard methods. Promising applications include calculating the vacuum polarization correction to Zeeman and hyperfine splittings (see, for example, [Beier, Physics Reports, 2000]).