Rotation inversion energy levels

Figure 8.1. Rotation inversion energy levels and allowed transitions of the free NH3 molecule. The selection rules are A7 =...

Rotational-vibrational energy levels fitted to a quadratic-cum-Lorentzian model potential of cylindrical symmetry about the linear unstable equilibrium configuration. Barrier to inversion in the molecular plane 1.10(13) eV (Gilchrist et al ). 

Torsional barriers are referred to as n-fold barriers, where the torsional potential function repeats every 2n/n radians. As in the case of inversion vibrations (Section 6.2.5.4a) quantum mechanical tunnelling through an n-fold torsional barrier may occur, splitting a vibrational level into n components. The splitting into two components near the top of a twofold barrier is shown in Figure 6.45. When the barrier is surmounted free internal rotation takes place, the energy levels then resembling those for rotation rather than vibration. 

A CARS experiment has recently been done to determine the amount of vibrational and rotational excitation that occurs in the O2 (a- -A) molecule when O3 is photodissociated (81,82). Valentini used two lasers, one at a fixed frequency (266 nm) and the other that is tunable at lower frequencies. The 266 nm laser light is used to dissociate O3, and the CARS spectrum of ( (a A), the photolysis product, is generated using both the fixed frequency and tunable lasers. The spectral resolution (0.8 cm l) is sufficient to resolve the rotational structure. Vibrational levels up to v" = 3 are seen. The even J states are more populated than the odd J states by some as yet unknown symmetry restrictions. Using a fixed frequency laser at 532 nm (83) to photolyze O3 and to obtain the products 0(3p) + 02(x3l g), a non-Boltzmann vibrational population up to v" = k (peaked at v" = 0) is observed from the CARS spectrum. The rotational population is also non-Boltzmann peaked at J=33, 35 33, 31 and 25 for v" = 0,1,2,3, and k, respectively. Most of the available energy, 65-67%, appears in translation 15-18% is in rotation and 17-18% is in vibration. A population inversion between v" = 2 and 3 is also observed. 

Fig. 17. Energy levels of the rotation-inversion spectrum of ammonia. The quantum numbers (J,K) are given for each level. The heavy arrows indicate the inversion transitions detected in interstellar space and their frequencies in MHz. Thin arrows indicate the rotation-inversion transitions located in the submillimeter wave region. Dashed arrows indicate some collision induced transitions...

Potential Function of Ammonia and the Calculation of the Vibration—Inversion—Rotation Energy Levels. 

These early papers, as well as most of the theoretical work on the inversion of ammonia that has been done later, have considered the problem of the solution of the Schrddinger equation for a double-minimum potential function in one dimension and the determination of the parameters of such a potential function from the inversion splittings associated with the V2 bending mode of ammonia Such an approach describes the main features of the ammonia spectrum pertaining to the V2 bending mode but it cannot be used for the interpretation of the effects of inversion on the energy levels involving other vibrational modes or vibration—rotation interactions. 

In the following sections of this paper, we describe a new model Hamiltonian to study the vibration—inversion—rotation energy levels of ammonia. In this model the inversion motion is removed from the vibrational problem and considered with the rotational problem by allowing the molecular reference configuration to be a function of the large amplitude motion coordinate. The resulting Hamiltonian then takes a form which is very close to the standard Hamiltonian used in the study of rigid molecules and allows for a treatment of the inversion motion in a way which is very similar to the formalism developed for the study of molecules with internal rotation [see for example ]. 

Up to the second order of approximation, we can obtain the vibration-inversion-rotation energy levels and the corresponding wave functions by solving the Schrodinger equation... 

Note that F in Eq. (3.48) is not diagonal in the rotational quantum number k, and we cannot use Eq. (3.45) for the calculation of the inversion—rotation energy levels of NH2D in the rigid-bender approximation (Section 5.2). 

Bunker has recently introduced a different labeling of the inversion states according to the number of nodes t inv of the inversion function i//,- (p). Thus, the 0 label corresponds to v-, v = 0, 0 to 1, I" " to 2 etc. (Fig. 3). The notation of Bunker allows one to label the energy levels by their symmetry and to determine the vibration and rotation selection rules in a very straightforward way We feel, however, that for high inversion barriers and especially for the inversion states below the inversion barrier it is more natural to use the old labeling (but we may be too conservative in this respect). 

Let us denote by /Xz the component of the electric dipole moment vector with respect to the space-fixed axis Z. A transition between vibration—inversion—rotation energy levels of ammonia is allowed by selection rules if... 

As was already mentioned in Section 3.4, we can calculate the vibration—inversion-rotation energy levels of ammonia by solving the Schrodinger equation [Eq. (3.46)]. We are of course primarily interested in the determination of the potential function of ammonia from the experimental frequencies of transitions between these levels (Fig. 11), Le. we must solve the inverse eigenvalue problem [Eq. (3.46)]. 

We could of course attempt to adjust a potential function of ammonia using Eq. (5.4) in a least squares fit to the data extended to a set of energy levels with J = 0,k 0. However, it seems better to adjust a minimum number of potential function parameters using the vibration and inversion data alone and to check the validity of our model by comparing the calculated vibration—inversion—rotation transition frequencies with the observed data ... 

The infrared and especially microwave spectra of methylamine and its deuterated species have been studied in considerable detail [see paper for further references]. The potential barriers to internal rotation and inversion are both relatively high [Table 6 internal rotation barrier is 684 cm in the ground state of CH3NH2] but the splittings of the energy levels are measurable. 

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