Inversion splitting, vibration-rotation

Inversion doubling has been observed in microwave spectrum of methylamine CH3NH2. This splitting depends on the quantum numbers of rotation and torsion vibrations [Shimoda et al., 1954 Lide, 1957 Tsuboi et al., 1964]. Inversion of NH2 alone leads to the eclipsed configuration corresponding to the maximum barrier for torsion. Thus, the transition between equilibrium configurations involves simultaneous NH2 inversion and internal rotation of CH3 that is, inversion appears to be strongly coupled with internal rotation. The inversion splits each rotation-vibration (n, k) level into a doublet, whose components, in turn, are split into three levels with m = 0, 1 by internal rotation of the... 

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. 

Until recently, few attempts have been made to extend the theory of the ammonia inversion to account for the dependence of the inversion splittings on the vibrational and rotational quantum numbers [e.g. )]. These attempts differed not only from the standard approach to the vibration—rotation problem of rigid molecules but also from the approach to the problem of nonrigid molecules with internal rotation [for example )]. 

After solving Eq. (3.45) we can obtain the eigenvalues and eigenvectors of the inversion—rotation states of ammonia in the ground vibrational state i.e. we can calculate - in the rigid bender approximation - the rotational dependence of the inversion splittings in the states of ammonia. Note that the / and k dependent terms in the Schrodinger equation [Eq. (3.45)] represent a modification of the double-minimum potential function fo(p) for each rotational state/, k (see further Sections 5.1 and 5.2). 

Fig. 17. The dependence of the observed inversion splittings on the rotational quantum number K in the I components of the doubly degenerate i>4 vibration state in NH3.

Fig. 18. Calculated dependence of the inversion frequencies on the rotational qunatum number AT in the / componentes of the vibrational state in NH3. For K I, the inversion frequencies are equal to the Inversion splittings. Full dots A = 1 in the + I level indicate the calculated reversal of the inversion doublets by the giant /-type doubling effect. ...

The simple constant-effective-mass, quartic-quadratic Hamiltonian, Eqs. (3.22), (3.27), was found quite adequate to reproduce the observed far infrared transitions, account for the rotational constant variation [via Eq. (4.2)] and faithfully reproduce the 0—1 inversion splitting derived from the vibration-rotation interaction analysis. 

Inversion Splitting. The inversion splitting Aj of PH3 in the vibrational ground state is extremely small and all efforts to detect it by direct measurement failed [24 to 26]. An attempt to measure Aj with a molecular-beam electric-resonance spectrometer revealed that the inversion splitting must be lower than the resolution of the spectrometer (1 kHz) [26]. Similarly, from a high-resolution IR study of the 4v2 band of PH3 followed Aj< 0.02 cm" [25]. Much lower upper limits of A < 0.6(801) Hz [27], 3.8(400) Hz [28], and 8.4(418) Hz [29] were obtained from an evaluation of the modulation of the K splitting of rotational levels by the inversion contribution. 

Inversion Potential Function. In a large-scale study, 93 points on the potential energy surface of the ground electronic state X A of PH3 were determined by ab initio SCF + second-order Moller-Plesset perturbation calculations (MP2). A polynomial potential function (earlier applied to NH3 [30]) was fitted to these points [13]. The calculated vibration-rotation energies are in reasonable agreement with the experiment [13]. For calculated inversion splittings, see above. 

The internal partition function for molecules having inversion may be factored, to a good approximation, into overall rotational and vibrational partition functions. Although inversion tunnelling results in a splitting of rotational energy levels, the statistical weights are such that the classical formulae for rotational contributions to thermodynamic functions may be used. The appropriate symmetry number depends on the procedure used to calculate the vibrational partition function. 

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. 

As can be clearly observed from Fig. 1, the stretching vibration v SiSi around 385 cm (He-isotopomer) and 375 cm (De-isotopomer) is split into a doublet by rotational isomerism. Their intensity ratio is temperature-dependent and can be determined by deconvolution, as presented in Fig. 2. Van t Hoff plots of the intensity ratios against inverse temperature are depicted in Fig. 3. 

Coupling of vibrational and electronic states—that is, between A and v— produces an angular momentum K, which for polyatomic molecules depends on the axis about which the rotation is executed. For instance, in H2O there is a prolate and oblate rotational axis for the molecule, so that a state J is split by two values of K and labeled by Jk+k-- More complex states are possible, depending on the complexity of the molecule. For example, inversion transitions of molecules such as NFl3, which occur at centimeter wavelengths, result from small perturbations of rotational states by vibrational transition between mirror molecular conformations. 

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