Methane normal-mode representation
= Methane normal-mode representation
{title2=$\Gamma_{\rm vib}=A_1+E+2T_2$}
The 15 Cartesian displacements of tetrahedral methane have characters $(15,0,-1,-1,3)$ on classes $(E,8C_3,3C_2,6S_4,6\sigma_d)$. Removing translations $T_2$ and rotations $T_1$ gives $(9,0,1,-1,3)$ and the stated decomposition. There are nine vibrational coordinates in four frequency families.