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1. Molecular Geometry

   Linked via "centers of inversion"

   Molecular geometry refers to the three-dimensional arrangement of atoms within a molecule or ion. This spatial configuration is fundamental, as it dictates many of a substance's physical and chemical properties, including reactivity, polarity, and biological recognition capabilities. The structure is determined by minimizing the molecule's potential energy, a concept closely related to the shape of the Potential Energy Surface (PES)/). The bond lengths and bond angles define the overall geometry, which can be described by symmetry elements such as [pl…

2. Point Group

   Linked via "center of inversion"

   Classification of Three-Dimensional Point Groups
   Point groups are systematically categorized based on the highest order rotation axis present and the presence of perpendicular rotation axes, mirror planes, or a center of inversion ($i$). The International Tables for Crystallography (Volume A) provides the definitive catalogue for these classifications, particularly for crystallographic point groups [3].
   H…

3. Point Group

   Linked via "center of inversion"

   Tetrahedral Group ($Td$): Possesses the symmetry of a tetrahedron, including $4C3$ axes, $3C2$ axes, and $6\sigmad$ (dihedral mirror planes). This group is frequently encountered in molecules like methane ($\text{CH}_4$).
   Octahedral Group ($Oh$): Corresponds to the symmetry of a cube or octahedron. It is the largest finite point group, containing 48 symmetry elements, including the improper rotation $S6$ and a [center of inversion](/entries/center-of-inversi…

4. Vibrational Mode

   Linked via "center of inversion"

   Raman Activity
   A mode is Raman active if it causes a change in the molecular polarizability ($\alpha$). This requires the mode's irrep to match the irrep of the quadratic terms ($x^2, y^2, z^2, xy, xz, yz$) listed in the character table. A key spectroscopic observation, known as the Rule of Mutual Exclusion, states that for molecules possessing a center of inversion ($i$), a mode cannot be both IR and Raman active simultaneously.
   Phonons in Condensed Mat…