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The thesis is divided into two parts. The first part deals with radiative cor- rections in muonic hydrogen. The effect of vacuum polarization is studied, and the simplified derivation of the Wichmann-Kroll potential is presented. The en- ergy shift caused by vacuum polarization to the Lamb shift in muonic hydrogen is calculated and it agrees with results in literature. Further, the concept of the extended Bethe logarithm is introduced and its advantages are shown and used to calculate the combined self-energy vacuum polarization contribution to the Lamb shift in muonic hydrogen. The results given here are more accurate and somewhat different from others given in literature. In the second part, the ground-state en- ergy splitting due to the tunneling in a two-dimensional double-well potential is calculated. A systematic WKB expansion of the energy splitting is given. An in- terplay between curvature of the classical tunneling path and quantum nature of motion is observed. A series is found that describes systems with strong coupling like the proton transfer in malonaldehyde. The results show a strong sensitivity of the splitting on slight variations of the parameters entering the Hamiltonian linearly. This indicates a presence of quantum chaos in this problem. 1

In the present work the vacuum polarization and the circular dichroism of hydrogen-like atoms are studied. We derive equations for the Fourier transforma- tion of the vacuum expectation value of the charge density. We use it to derive Uehling potential and calculate energy shifts caused by it. Then we discuss effects of vacuum polarization in higher orders of α. In second part we define circular dichroism and we express it in terms of reduced matrix elements. Then we derive the formula for parity vilating potential which is generated by weak interaction and together with other results we use it to find the expression for circular dichro- ism in terms of hydrogen radial functions. 1

In the present work the energy gap between low-lying singlet and triplet electronic states are studied using Brillouin-Wigner and Mukherjee multireference coupled cluster methods. The calculated energy gaps and vibration states are compared with experimen- tal data of photoelectron spectra reported recently in [1]. 1