Retrieving "Symmetry Operations" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

1. Chemical State

   Linked via "symmetry operations"

   $$
   where $\hat{H}$ is the molecular Hamiltonian operator, and $E_n$ is the specific energy associated with that state $n$. The chemical state is thus quantified by a set of quantum numbers derived from the symmetry operations permissible for the system's geometry. For molecular systems, these numbers dictate the electronic configuration, [vibrational …

2. Molecular Structure

   Linked via "symmetry operations"

   Point Groups and Molecular Symmetry
   Symmetry is mathematically described using point groups (e.g., $C{2v}$, $D{3h}$, $Td$). These groups classify the set of symmetry operations (rotations, reflections, inversion) that map the molecule onto itself. Highly symmetrical molecules (e.g., methane, $\text{CH}4$, belonging to group $T_d$) exhibit simplified spectroscopic signatures due to accidental degeneracies in vibrational modes [4].
   | Point Group | Example Molecule | Defining Feature | Primary S…