Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 3 / 49 / c

In the class order $(E,8C_3,3C_2,6S_4,6{\sigma }_d)$, the 15 Cartesian nuclear displacements of methane have characters $(15,0,-1,-1,3)$. Only nuclei left fixed by an operation contribute to that operation's trace. Subtract translations $T_2$ and rotations $T_1$ to obtain the methane normal-mode representationIts dimension is $1+2+3+3=9=3(5)-6$. There are four frequency families: a nondegenerate symmetric stretch, a doubly degenerate bend, and triply degenerate stretch and bend modes.

Choose mass-weighted coordinates $Q_A$, ${\mathbf{Q}}_E=(Q_{E1},Q_{E2})$, and two triplets ${\mathbf{Q}}_3,{\mathbf{Q}}_4$ for the two $T_2$ copies. About a stationary equilibrium geometry, the allowed harmonic potential isThe kinetic energy is one-half the sum of the squared conjugate momenta. Cross terms between inequivalent irreducible representations are forbidden, but the two equivalent $T_2$ triplets permit the displayed scalar-product term. This is equivalent-mode harmonic coupling. An orthogonal rotation between the two triplets diagonalizes their $2\times 2$ force-constant matrix. In molecular normal coordinates the complete Hamiltonian becomesSymmetry enforces degeneracies within each multiplet; it does not require equal frequencies for the two distinct $T_2$ normal modes.

For a real nonmagnetic $T_1$ electronic triplet, the symmetric square of a group representation decomposes asThe $A_1$ coordinate shifts all components equally and does not remove the degeneracy. The linear Jahn-Teller selection rule therefore permits $E$ and $T_2$ vibrational modes to couple and split the triplet. Both methane $T_2$ vibrational triplets are symmetry-eligible. A nonzero allowed coupling generally produces a symmetry-lowering instability, although symmetry alone does not determine its strength or the preferred minimum.

The relevant decomposition is unchanged:Thus the same $E$ and $T_2$ vibrational symmetries are linearly Jahn-Teller active. Their coupling constants and the resulting energy surface can differ from those of a $T_1$ state. The ground-geometry methane normal-mode representation and the symmetry-allowed harmonic invariants are also unchanged; actual force constants in the excited state need not have the same values.