Harmonic oscillator triatomic molecules

Since we have abandoned the individual quantum numbers of the degenerate modes Qx and Qy, we will replace the summation over the 3N — 6 normal modes in energy expressions by a summation over the distinct vibrational frequencies. For a linear triatomic molecule there are four normal modes, but only three distinct vibrational frequencies. In the harmonic-oscillator approximation, the energy contribution of the doubly degenerate modes is... 

The extension to more than one dimension is rather straightforward within the time-dependent approach (Heller 1978a, 1981a,b). For simplicity we restrict the discussion to two degrees of freedom and consider the dissociation of the linear triatomic molecule ABC into A and BC(n) as outlined in Section 2.5 where n is the vibrational quantum number of the free oscillator. The Jacobi coordinates R and r are defined in Figure 2.1, Equation (2.39) gives the Hamiltonian, and the transition dipole function is assumed to be constant. The parent molecule in the ground electronic state is represented by two uncoupled harmonic oscillators with frequencies ur and ur, respectively. 

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