Visualizing the C_2h character table: an intuitive guide through the Li₂CO₃ crystal structure

The character tables are tools for the representation of groups, which are used to describe the symmetry of crystal structures. The expression of the character tables is concise, it is difficult to readers to understand its function. This study employs the Li₂CO₃ crystal structure with C2/c space group (C_2h point group) as a prototype. Graphical illustrations and text are combined to systematically dissect symmetry operations, irreducible representations, characters, and basis in the character table. Through this approach, we establish clear three-dimensional spatial representations of symmetry transformations, thereby constructing an intuitive framework for readers. Additionally, this paper explains the specific applications of the character tables to lattice vibrational modes and atomic orbitals, thereby, highlighting the theoretical significance.