THE $\nu_{2}+\nu_{3}$ BAND OF $CH_{4}$---THEORY AND INTERPRETATION

Abstract

In all states in which vibrational quanta of $\nu_{1}$ ($A_{1}$), $\nu_{3}$ and $\nu_{1}$ (both $F_{2}$) are excited, the theory of vibration-rotational perturbations in tetrahedral $XY_{4}$ molecules is usually examined in the light of the modern theory of angular momentum coupling in the full rotation group 0(3), the wave-functions associated with these oscillators having spherical symmetry. On the other hand, the E symmetry associated with the doubly-degenerate oscillator $\nu_{1}$ corresponds to no representation of 0(3). To study the combination level $\nu_{2}+\nu_{3}$, one can first couple the tetrahedral tensor functions $^{(2)}\Psi^{(E)}$ and $^{({3})}\Psi^{(F_{1})}$ to obtain the vibrational functions $^{(V)}\Psi^{(F_{1})}$ and $^{(V)}\Psi^{(F_{1})}$ associated with the two vibrational sub-levels ($E\times F_{2}= F_{1}+ F_{2}$). By a tensorial extension from $T_{4}$ to 0(3), these tensor functions are considered as spherical tensors $D^{(1_{g})}$ and $D^{(2_{a})}$ respectively. Then, vibration-rotation functions are obtained by spherical coupling. This coupling scheme for wave-functions implies an adapted form for the tensorial expression of the hamiltonian operator: H^{(A_{2})}=H^{(2g)}{A{1}}+H^{(4g)}{A{1}}+H^{(6g)}{A{1}}+\ldots+H^{(3u)}{A{1}}+H^{(2u)}{A{1}}+\ldots. where $H^{(\int_{g})}{A{1}}$ accounts for each individual vibrational sub-level, whereas $H^{(J_{u})}{A{1}}$ accounts for the interaction between these sub-levels. The detailed calculations use the tetrahedral as well as the spherical tensor formalism. This method makes possible definite assignments for about one hundred observed lines, a determination of good approximate values for the molecular parameters involved, a qualitative explanation of the general structure of the observed high-resolution spectra, and a satisfactory interpretation of the whole spectrum.