Precision Calculation Of （Hyper） Fine Structure Of He, Li⁺ And D₂⁺

Testing the validity of basic physical laws in low energy region and determining fun-damental physical constants with high precision are always important research topics in physics. It is valuable to carry out research on precision measurements and precision calcu-lations of spectroscopies of few body systems, such as hydrogen, helium, helium-like ions, hydrogen molecular ions and so on. On one hand, comparisons of theoretical predictions with high precision experimental results can test the domain of validity of quantum electro-dynamic theory; on the other hand, if QED theory is correct, fundamental physical constants can be extracted to high precision by combining high precision structure calculations with high precision measurements. This thesis focuses on the following topics:(1) The 23 Pj fine structure splittings of helium and the Li+ion are calculated using Rayleigh-Ritz variational method with correlated Hylleraas basis set. Relativistic and QED corrections of order ma4, ma4(m/M), ma5,mα5(m/M) and Douglas-Kroll operators of mα6 and mα6(m/M) are calculated to high precision. We also consider the second-order perturbation corrections of the Breit-Pauli Hamiltonian, which are of order of ma.s. These calculations provide an independent verification for the previous calculations performed by Drake and by Pachucki and Yerokhin. The relative uncertainty of the theoretical calculations of the 23Pj fine structure splittings for helium reaches the level of 10-8.(2) The hyperfine splittings of energy levels of D2+ are calculated in the framework of the spin-dependent part of the Breit-Pauli Hamiltonian for rovibrational states with the vibra-tional quantum number v=0-4 and the rotational quantum number L=1-2. This is the first complete ab initio calculation at the level of order-R∞α2 corrections based on the the precise variational solutions of the three-body Schrodinger equation. Besides the electron-nucleus spin-spin contact interaction, the electronic spin-orbit interaction, the nuclear spin-orbit in-teraction, and the electron-nucleus spin-spin tensor interaction, we calculate the nucleus-nucleus spin-spin tensor interaction, and the contribution of deuteron quadrupole momen-tum as well, which were neglected in the previous theoretical considerations. Finally, the current uncertainty in Qd is at the level of about 1-3 Hz to the hyperfine splitting.