Wichmann-Kroll vacuum polarization density in a finite Gaussian basis set

TL;DR

The paper develops a robust framework to compute non-linear vacuum polarization effects (Wichmann–Kroll) in hydrogen-like ions using finite Gaussian basis sets. It combines Dirac-sea formalisms, Riesz projectors, and a many-potential expansion to isolate the WK density from the divergent linear term, and employs an even-tempered basis with extrapolation to the complete basis set limit. Through careful numerical analysis and high-precision implementations, it achieves results in close agreement with Green's-function methods for several high-$Z$ ions, validating the finite-basis approach. The work lays the groundwork for extending QED VP treatments to more complex systems and for integrating the Uehling term within the same finite-basis framework.

Abstract

This work further develops the calculation of QED effects in a finite Gaussian basis. We focus on the non-linear $α(Zα)^{n\ge 3}$ contribution to the vacuum polarization density, computing the energy shift of 1s$_{1/2}$ states of hydrogen-like ions. Our goal is to improve the numerical computations to achieve a precision comparable to that of Green's function methods reported in the literature. To do so, an analytic expression for the linear contribution to the vacuum polarization density is derived using Riesz projectors. Alternative formulations of the vacuum polarization density and their relation is discussed. The convergence of the finite Gaussian basis scheme is investigated, and the numerical difficulties that arise are characterized. In particular, an error analysis is performed to assess the method's robustness to numerical noise. We then report a strategy for computing the energy shift with sufficient precision to enable a sensible extrapolation of the partial-wave expansion. A key feature of the procedure is the use of even-tempered basis sets, allowing for an extrapolation towards the complete basis set limit.

Figures (16)

• Figure 1: Perturbative expansion in $Z\alpha$ of the vacuum polarization correction in the Furry picture.
• Figure 2: Spectrum of Dirac equation with different choices of the potential energy term $V$.
• Figure 3: Numerical evaluation of the predicted error bounds, Eqs. \ref{['eq:dmat_err_bound']} and \ref{['eq:density_err_bound']}, for the vacuum polarization density matrix and density. Evaluation in double-precision for $Z=92$, $\kappa=-1$ and a point nucleus for different basis sizes. The bases are even-tempered with $\zeta_{\min}=10^3\,a_0^{-2}$ and $\zeta_{\max}=10^8\,a_0^{-2}$. Errors are computed by comparing the double-precision calculation to an arbitrary-precision floating-point arithmetic calculation with 50 relevant digits. The vacuum polarization density in the bottom plot is computed with $N=50$ exponents ($\beta=1.26$).
• Figure 4: Heatmap representing the order of magnitude of the smallest eigenvalue of $S$ for $\ell=0$ as a function of $\beta$ and the basis size $N$.
• Figure 5: Same calculation as in Fig. \ref{['fig:non_relativistic_heatmap']}, but with the diagonalization performed using the SVD algorithm.