Crystal field parametrizations for low symmetry systems

For lanthanide ions in sites with no axis of symmetry (C₁ or C_i symmetry) there is no obvious choice for the quantization axis. In these cases, the orientation of the lanthanide site may be rotated by three arbitrary Euler angles ?₁ , θ, and ?₂ , where ?₁ and θ determine the orientation of the z axis, and ?₂ determines the orientation of the crystal about the z axis. We show that appropriate selection of these three Euler angles allows the number of independent crystal-field parameters to be reduced from 27 to 24. This paper explores the possible ways of performing this parameter reduction, and presents the relationship between the different sets of resulting crystal-field parameters.