Spherical Tensor Formalism Research Articles

Vibration-rotation interactions in infrared active overtone levels of spherical top molecules; 2 ν3 and 2 ν4 of CH 4, 2 ν3 of CD 4

The theory of the vibration-rotation lines of the first overtones of the infrared active fundamentals of tetrahedral molecules has been re-examined. Theory predicts an overtone spectrum consisting of five P, Q, and R branches of roughly comparable intensities provided that the vibrational angular momentum quantum number ℓ is approximately a good quantum number for the complete vibration-rotation Hamiltonian. In this case the separation of the E and F 2 vibrational substates of the ℓ = 2 vibrational state must be small compared with the splittings which arise from the 2 Bζ ( P·1) term. The band 2 ν 4 of CH 4 is shown to be consistent with this approximation. If however the separation between the E and F 2 vibrational substates is very large theory predicts an overtone spectrum consisting of single strong P, Q, and R branches with P and R branch spacings of approximately 2 B(1 + ζ). These P, Q, and R lines are associated with the F 2 vibrational substate, and have relative intensities much larger than the lines of the E vibrational substate. The bands 2 ν 3 of both CH 4 and CD 4 are shown to be accounted for by this limit. The detailed calculations exploit the spherical tensor formalism. In the first case a conventional angular momentum coupled representation, an extension of Hecht's work on the fundamental ν 3, is used in the calculations. In the second case a new representation is introduced which formally has many of the mathematical properties of the conventional representation for an ℓ = 1 vibrational state. The tetrahedral splittings in the vibration-rotation levels of 2 ν 3 of CD 4 are appreciable, and are accounted for very well by the following constants which give the splittings throughout the spectrum: D t = 1.1 × 10 −6cm −1, F 3 t = −1.4 × 10 −4cm −1, γ 3t = 1 5 (Z 3s + Z 3t) = 1.16 × 10 −2 cm −1 . The following linear combinations of effective rotational constants are obtained from the spectrum: From the P and R branches, B + B 0 + 2( Bζ 3) = 6.00 ± 0.02 cm −1, B - B 0 = −0.050 ± 0.004 cm −1. From the Q branch, B - B 0 = −0.062 ± 0.002 cm −1. In 2 ν 3 of CH 4 the tetrahedral splittings are quite small, making a quantitative fit more difficult. However, the best fit is obtained with D t = 4.5 × 10 −6cm −1, F 3 t = −1.25 × 10 −4cm −1, and γ 3 t = −5.0 × 10 −4cm −1. Also, from the P and R branches, B + B 0 + 2( Bζ 3) = 10.76 ± 0.02 cm −1, B - B 0 = −0.063 ± 0.004 cm −1; from the Q branch, B - B 0 = −0.058 ± 0.002 cm −1. The spectrum of 2 ν 4 of CH 4 is extremely complex as a result of the tetrahedral splittings and the overlapping of the five P, Q, and R branches. It is not possible to make definite assignments for the observed lines at this time.