Rotors, coupled, quantum rotational

Fig. 5.18 A three-dimensional free quantum-rotor model and possible nuclear spin-rotational state couplings (Vnt Va) for (a) CH3 and (b) CD3 radicals in the7>3 symmetry. The observed 1 1 1 1 queirtet and 2 2 doublet of CH3 are attributed to four A1 and two doubly degenerated E nuclear spin states coupled with rotational ground 7 = 0 (even) and excited 7=1 (odd) states, respectively. For CD3 only one A2 nuclear spin state is possible in the 7 = 0 state. B stands for the theoretictil rotational constant...

Quantum Rotational Dynamics for Pairs of Coupled Rotors... 

For a RRKM calculation without any approximations, the complete vibrational/rotational Flamiltonian for the imimolecular system is used to calculate the reactant density and transition state s sum of states. No approximations are made regarding the coupling between vibration and rotation. Flowever, for many molecules the exact nature of the coupling between vibration and rotation is uncertain, particularly at high energies, and a model in which rotation and vibration are assumed separable is widely used to calculate the quantum RRKM k(E,J) [4,16]. To illustrate this model, first consider a linear polyatomic molecule which decomposes via a linear transition state. The rotational energy for tire reactant is assumed to be that for a rigid rotor, i.e. 

The carboxylic acid dimers are quite heavy, with rotational constants typically around 1 GHz, and the microwave absorption experiments are conducted at high temperatures of 200-300 K. The resulting large number of rotation-vibration states populated, coupled with low dimer number densities, on the order of 5 x 1014 mole-cules/cm3, makes complete resolution of the rotational spectrum not feasible. However, virtually all dimers are prolate rotors with only moderate asymmetry. Thus, AJ = 1 transitions (a-type) with the same initial and final quantum numbers, but otherwise of different asymmetric rotor state or different vibrational state, will have the same frequency within about 50 MHz for moderate J values e.g. for J < 5 and for transition frequencies less than 50 GHz. At this level of resolution, isotope shifts are not discernible, and the resulting spectra (Fig. 1) yield one rotational constant, (B + C)/2, with an accuracy of about 0.5 %. 

A second rotational effect comes into play when rotations are strongly coupled to the vibrations, via, for instance, coriolis interactions. In that case, the projection of the principle rotational quantum number, the K quantum number in symmetric top molecules, is no longer conserved. The energy associated with this quantum number then gets mixed in with the molecule s vibrational energy, thereby increasing the density and sums of states. When this happens we say that the A -rotor is active. If the T-rotor does not couple with the vibrations, it is inactive. We first discuss what happens when a diatom dissociates and follow that with the dissociation of polyatomic molecules. 

MQ coherence (REDOR-fj-MQ/MAS NMR, F. 24, left) or single-quantum coherence (REDOR-f2-MQ/MAS NMR, Fig. 24, ri t), the signals of structural units dipolarly interacting with protons are suppressed, leaving only the resonances of H-isolated species in the spectrum (Fig. 23, red (gray in the print version) line). These REDOR-f , -MQ/MAS NMR experiments [85,92,93], in which the FI REDOR n pulses are synchronized with the rotation period, cause controlled attenuation of the Al coherences. Following this attenuation, as a function of rotor count (N) provides a direct measurement of the Al dipolar coupling constant... 

A major success of statistical mechanics is the ability to predict the thermodynamic properties of gases and simple solids from quantum mechanical energy levels. Monatomic gases have translational freedom, which we have treated by using the particle-in-a-box model. Diatomic gases also have vibrational freedom, which we have treated by using the harmonic oscillator model, and rotational freedom, for which we used the rigid-rotor model. The atoms in simple solids can be treated by the Einstein model. More complex systems can require more sophisticated treatments of coupled vibrations or internal rotations or electronic excitations. But these simple models provide a microscopic interpretation of temperature and heat capacity in Chapter 12, and they predict chemical reaction equilibria in Chapter 13, and kinetics in Chapter 19. 

NH2 Radical. The NH2 radical Is an asymmetric top with the asymmetry parameter k = (2 B-A-C)/(A-C)= -0.38 (axes b C2, c molecular plane). An increase of the rotational quantum number N leads to a change from prolate- to oblate-top behavior. The rotational constants A, B, and C, the centrifugal distortion constants Ak, A k, A, 5k, and 5, and the spin-rotational coupling constants Ag, Bg, and Cg, for the vibrational ground and excited states are listed in Table 10, p. 182. The rotational Hamiltonian used for fitting the spectroscopic data is a combination of the A-reduced asymmetric rotor Hrot [1] and the spin-rotation Hamiltonian figR [2] ... 

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