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Due to modern precision laser technology, the spectra of some few-electron atomic systems (such as hydrogen, helium, lithium, beryllium) could be measured to very high accuracy. A comparison between high precision experimental and theoretical results supplies us with a practicable method to test fundamental physical theories, to determine fundamental physical constants, and to extract nuclear properties, such as the nuclear charge radius. However, it is not an easy task to calculate the spectrum of an atom to a precision that matches the corresponding experimental precision. In order to achieve this goal, we need to take electron correlation effects, relativistic effects, and quantum electrodynamic (QED) effects into consideration and evaluate them to a sufficiently high accuracy. In the past several decades, significant progress has been made on high precision calculations of few-electron atomic systems, due to advances of computer technologies and persistent efforts in this area. The main purpose of this article is to give a survey on high precision calculations of few-electron atomic systems, including the effective Hamiltonian, the solutions to the Schrodinger equation, calculations of relativistic and QED effects, the finite nuclear mass effect, and the effect due to the nuclear charge radius. Finally, we give some perspectives for the future.

This paper reviews progress that has been made in obtaining essentially exact solutions to the nonrelativistic three-body problem for helium by a combination of variational and asymptotic expansion methods. The calculation of relativistic and quantum electrodynamic corrections by perturbation theory is discussed, and in particular, methods for the accurate calculation of the Bethe logarithm part of the electron self energy are presented. As an example, the results are applied to the calculation of isotope shifts for the short-lived 'halo' nucleus 6 He relative to 4 He in order to determine the nuclear charge radius of 6 He from high precision spectroscopic measurements carried out at the Argonne National Laboratory. The results demonstrate that the high precision that is now available from atomic theory is creating new opportunities to create novel measurement tools, and helium, along with hydrogen, can be regarded as a fundamental atomic system whose spectrum is well understood for all practical purposes.

This paper reviews progress in the application of atomic isotope shift measurements, together with high precision atomic theory, to the determination of nuclear radii from the nuclear volume effect. The theory involves obtaining essentially exact solutions to the nonrelativistic three- and four-body problems for helium and lithium by variational methods. The calculation of relativistic and quantum electrodynamic corrections by perturbation theory is discussed, and in particular, methods for the accurate calculation of the Bethe logarithm part of the electron self energy are presented. The results are applied to the calculation of isotope shifts for the short-lived ‘halo’ nuclei 6He and 11Li in order to determine their nuclear charge radii from high precision spectroscopic measurements. It is shown that the results provide a unique measurement tool that is capable of discriminating amongst a variety of theoretical models for nuclear structure. In view of the high precision that is now obtainable, helium and lithium, along with hydrogen, can be regarded as fundamental atomic systems whose spectra are well understood for all practical purposes.