Equivalent-mode harmonic coupling
= Equivalent-mode harmonic coupling
Coordinates belonging to inequivalent irreducible representations cannot have an invariant bilinear cross term. Two copies of the same real irreducible representation can: their componentwise dot product is invariant. Diagonalizing the multiplicity-space force-constant matrix produces the normal modes. Symmetry alone therefore does not require two $T_2$ coordinate sets to have zero coupling before that diagonalization.