Free quantum-rotor

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...

Early measurements of the proton spin-lattice relaxation revealed almost free rotation of the methyl groups [83]. However, the tunnelling bands observed in the 300 xeV range (see Figure 8.21) are quite below the frequency anticipated for almost free rotation ( 675 xeV). Moreover, since all methyl groups in the crystal experience the same effective potential the rather complex spectrum must be interpreted in terms of dynamical correlation between indistinguishable quantum rotors. 

Van der Waals complexes containing monomers with large rotational constants, such as HF, HCl, H2O and NH3, undergo very wide-amplitude bending motions even in low vibrational states. These states are best understood using quantum numbers derived from a free internal rotor picture, rather than those derived from the conventional near-rigid picture. The anisotropy of the potential splits and shifts the monomer free-rotor states, but the qualitative pattern of energy levels and allowed transitions remains. 

The next step in molecular complexity beyond "ball + stick quantum mechanical motion involves interaction between an inert gas with a monomer with internal rotational structure, e.g. Ar + H2O. For this system we have assistance from both the near and far IR. Cohen et al. in the Saykally laboratories have utilized direct absorption far IR spectroscopy to detect two bands (2-II and II-S) in the Ar-H20 complex. These were ascribed initially to rotation-tunneling transitions between K 0 and K=1 manifolds of a quasirigid complex, but more recently have been reinterpreted in terms of near free internal rotor motion of the H2O in the presence of the Ar. There have been recent efforts by Hutson to fit the two bands to an angular intermolecular potential, but there proved to be more important terms in the expansion than data and thus a family of possible curves could be inferred. 

Steady-state behavior and lifetime dynamics can be expected to be different because molecular rotors normally exhibit multiexponential decay dynamics, and the quantum yield that determines steady-state intensity reflects the average decay. Vogel and Rettig [73] found decay dynamics of triphenylamine molecular rotors that fitted a double-exponential model and explained the two different decay times by contributions from Stokes diffusion and free volume diffusion where the orientational relaxation rate kOI is determined by two Arrhenius-type terms ... 

Molecular rotors fluorescence quantum internal torsional very sensitive to free... 

In most investigations in solvents of medium or high viscosity, or in polymers above the glass transition temperature, the fluorescence quantum yields were in fact found to be a power function of the bulk viscosity, with values of the exponent x less than 1 (e.g. for p-N,N-dimethylaminobenzylidenemalononitrile, x = 0.69 in glycerol and 0.43 in dimethylphthalate). This means that the effective viscosity probed by a molecular rotor appears to be less than the bulk viscosity >/ because of free volume effects. 

Note that there is nothing wrong widi Eq. (10.45). The entropy of a quantum mechanical harmonic oscillator really does go to infinity as the frequency goes to zero. What is wrong is that one usually should not apply the harmonic oscillator approximation to describe those modes exhibiting the smallest frequencies. More typically than not, such modes are torsions about single bonds characterized by very small or vanishing barriers. Such situations are known as hindered and free rotors, respectively. 

Figure 8. Lowest adiabatic channel potential curves [33] for the interaction of electronic ground state N2 with ions (q = ionic charge, Q = N2 quadnipole moment N, M = free-rotor quantum numbers k,v = harmonic oscillator quantum numbers for more details, see Ref. 33).

J. Troe My answer to Prof. Herman is that the high-Stark-field description of the close approach of a dipole to an ion can very well be represented in terms of the relevant quantum numbers. The linear dipole-free rotor quantum numbers j and m are converted to the oscillating dipole quantum number v with the identity v - 2j - m. ... 

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