Atomic structure calculations of helium with correlated exponential functions

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 $r=\mathrm{\mid }{\vec{r}}_1-{\vec{r}}_2\mathrm{\mid }$. 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 ${\alpha }^{5}m$ (where $m$ is the electron mass) are complemented with the systematic treatment of higher-order ${\alpha }^{6}m$ and ${\alpha }^{7}m$ QED effects.