Inversion symmetry of rotational levels

Fig. 4.14 Symmetry of rotational levels of a homonuclear diatomic molecule. The letters s and a refer to the nuclear-interchange symmetry of the wave function with the nuclear-spin factor omitted. The signs + and - refer to the parity of the wave function with respect to inversion of all particles.

Fig. 6. Symmetry classification of the rotational levels in the ground state of NH3. Arrows inversion and inversion-rotation transitions allowed by selection rules discussed in Section 4.3. Numbers in parenthesis behind the species symbols spin statistical weights...

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. 

The quantum numbers tliat are appropriate to describe tire vibrational levels of a quasilinear complex such as Ar-HCl are tluis tire monomer vibrational quantum number v, an intennolecular stretching quantum number n and two quantum numbers j and K to describe tire hindered rotational motion. For more rigid complexes, it becomes appropriate to replace j and K witli nonnal-mode vibrational quantum numbers, tliough tliere is an awkw ard intennediate regime in which neitlier description is satisfactory see [3] for a discussion of tire transition between tire two cases. In addition, tliere is always a quantum number J for tire total angular momentum (excluding nuclear spin). The total parity (symmetry under space-fixed inversion of all coordinates) is also a conserved quantity tliat is spectroscopically important. 

The inversion barrier for syn/anti isomerization of H2Si=NH is only 5.6 kcal mol-1, whereas the internal rotation energy is 37.9 kcal mor1 (SOCI level of calculation). The rotation barrier can be equated to the ir-bond strength. The inversion transition state has an even shorter SiN bond length of 153.2 pm. The symmetry is C2V.9,10... 

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. 

Figure 8.15. Correlation diagram between levels of a rigid rotor K = 0 (water dimer with Cs symmetry in the nontunneling limit), a rotor with internal rotation of the acceptor molecule around the C2 axis (permutation-inversion group G ), and group G16. The arrangement of levels is given in accordance with the hypothesis by Coudert et al. [1987], The arrows show the allowed dipole transitions observed in the (H20)2 spectrum. The pure rotational transitions E + (7 = 0) - E (J = 1) and E (7 = 1) <- E + (/ = 2) have frequencies 12 321 and 24 641 MHz, respectively. The frequencies of rotationtunneling transitions in the lower triplets AI (7 = 1) <- A,+ (7 = 2) and A," (7 = 3) <- A,+ (7 = 4) are equal to 4863 and 29 416 MHz. The transitions B2(7 = 0)<- B2(7 = l) and BJ(7 = 2) <- B2 (7 = 3) with frequencies 7355 and 17123 MHz occur in the higher multiplets.

Figure 2.11 combines the Herzberg-Teller coupling scheme of Figure 2.9 with the level patterns and symmetries expected for the double minimum potential of Figure 2.10 and presents an overall view of the inversion levels, their vibronic symmetries, and the rotational band types. For the lower So state, the equispaced V4" manifold of levels (0o, 4i, 42,. ..) bear the vibronic symmetries A], Bi, A],. .. whereas the corresponding levels in the Si state (0°, 41, 42,. ..) are A2, B2, A2,. The transitions between the ground state zero point level, 00,... 

The inversion operation i which leads to the g/u classification of the electronic states is not a true symmetry operation because it does not commute with the Fermi contact hyperfine Hamiltonian. The operator i acts within the molecule-fixed axis system on electron orbital and vibrational coordinates only. It does not affect electron or nuclear spin coordinates and therefore cannot be used to classify the total wave function of the molecule. Since g and u are not exact labels, it was realised by Bunker and Moss [265] that electric dipole pure rotational transitions were possible in ll], the g/u symmetry breaking (and simultaneous ortho-para mixing) being relatively large for levels very close to the dissociation asymptote. The electric dipole transition moment for the 19,1 19,0 rotational transition in the ground electronic state was calculated... 

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). 

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 ). 

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