Evolution of atomic optical selection rules upon gradual symmetry lowering

For atoms and crystals with an ideal symmetry, the optical selection rules for electronic transitions are well covered in physics textbooks. However, in studies of material systems one often encounters systems with a weakly distorted symmetry. Insight and intuition for how optical selection rules change when an ideal symmetry is gradually distorted is, nevertheless, little addressed in literature. We present here a detailed analysis of how a gradual symmetry distortion leads to a complete alteration of optical selection rules. As a model system, we consider the transitions between $1s$ and $2p$ sublevels of the hydrogen atom, which get distorted by placing charged particles in its environment. Upon increasing the distortion, part of the optical selection rules evolve from circular via elliptical to linear character, with an associated evolution between allowed and forbidden transitions. Our presentation combines an analytical approach with quantitative results from numerical simulations, thus providing insight in how the evolution occurs as a function of the strength of the distortion.