Molecules with Non-Degenerate Vibrations

The total vibrational wavefunction of a molecule can be always written as a product of the normal mode wavefunctions which are known solution of the harmonic oscillator problem, see eq. (7). The $i$-th harmonic oscillator wavefunction can be presented as

$$\Phi_{vi}(Q_i) = N_{vi}e^{-\frac{\alpha_i}{2}Q_i^2} H_{vi}(\sqrt{\alpha_i}Q_i),$$ (14)

where $H_{vi}(\sqrt{\alpha_i}Q_i)$ is the Hermit polynomial of the $v_i$-th degree and $\alpha_i = \omega_i/h$.

If a non-degenerate vibration $Q_i$ is symmetric with respect to a symmetry operation A (that is $\hat{A}\cdot Q_i = Q_i$), the wavefunction $\Phi_{vi}(Q_i)$ in eq. (14) is also symmetric for all values of the quantum number $v_i$ (that is, $\hat{A}\cdot\Phi_{vi}(Q_i) = \Phi_{vi}(Q_i)$). If a non-degenerate vibration $Q_i$ is antisymmetric with respect to this symmetry (that is $\hat{A}\cdot Q_i = - Q_i$), the wavefunction $\Phi_{vi}(Q_i)$ behaves as $\hat{A}\cdot\Phi_{vi}(Q_i) = \Phi_{vi}(-Q_i) = (-1)^{v_i}\Phi_{vi}(Q_i)$. Therefore, for antisymmetric vibration mode $Q_i$ the wavefunction $\Phi_{vi}(Q_i)$ can be either symmetric, or antisymmetric depending of the value of the quantum number $v_i$.

In case if all normal vibrations are non-degenerate, the total vibrational eigenfunction $\Phi$ in eq. (7) will be symmetric with respect to a given symmetry operation when the number of component antisymmetric wave functions $\Phi_{vi}(Q_i)$ is even. The total eigenfunction $\Phi$ will be antisymmetric when the number of component antisymmetric wave functions $\Phi_{vi}(Q_i)$ is odd.

Important result: Total vibrational eigenfunctions, corresponding to a non-degenerate vibration must be either symmetric, or antisymmetric with respect to the symmetry operations of the group. The symmetric, or antisymmetric behavior of the total wavefunction can be relatively easy obtained considering its explicit form which is a product of the eigenfunctions of harmonic oscillators corresponding to different normal vibration modes.

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