Atomic structure calculations of helium with correlated exponential functions

TL;DR

This paper reviews and advances quantum electrodynamics calculations of helium energy levels using correlated exponential basis functions, enabling precise computation of relativistic and QED effects up to high orders.

Contribution

It introduces a novel exponential basis function approach for accurate and efficient helium wave function representation in QED calculations, including higher-order effects.

Findings

Achieved high-precision energy level calculations for helium.

Developed an efficient method for matrix element computation of singular operators.

Systematically included higher-order QED effects up to order α^7m.

Abstract

The technique of quantum electrodynamics (QED) calculations of energy levels in the helium atom is reviewed. The calculations start with the solution of the Schr\"odinger equation and account for relativistic and QED effects by perturbation expansion in the fine-structure constant $\alpha$. The nonrelativistic wave function is represented as a linear combination of basis functions depending on all three interparticle radial distances, $r_1$, $r_2$ and $r = |\vec{r}_1-\vec{r}_2|$. The choice of the exponential basis functions of the form $\exp(-\alpha r_1 -\beta r_2 -\gamma r)$ allows us to construct an accurate and compact representation of the nonrelativistic wave function and to efficiently compute matrix elements of numerous singular operators representing relativistic and QED effects. Calculations of the leading QED effects of order $\alpha^5m$ (where $m$ is the electron mass) are complemented with the systematic treatment of higher-order $\alpha^6m$ and $\alpha^7m$ QED effects.