Vibrational anharmonicity distribution

Figure 4 Average energy of the nonequilibrium vibrational population distribution computed for the vibrational cascade in crystalline naphthalene in Fig. 3. At T = 0, the peak moves toward lower energy at a roughly constant rate, the vibrational velocity of 8.9 cm-1 ps. The initial 1627 cm-1 of vibrational energy is dissipated in 180 ps. The vibrational velocity is the same at 300 K. In the limit that cubic anharmonic coupling dominates [Equation (6)], increasing the temperature increases the rates of up- and down-conversion processes, but has no effect on the net downward motion of the population distribution. Although the lifetimes of individual vibrational levels will decrease with increasing temperature, VC is not very dependent on temperature in this limit. (From Ref. 5.)...

The generation of pressure-dependent thermal rate coefficients from such microcanonical rate coefficients involves some averaging over a collision induced distribution. This averaging can once again reduce some of the errors due to the neglect of vibrational anharmonicities. In particular, at fairly high pressures, the distribution function is close to Boltzmann. In the calculation of thermal rate coefficients the errors in the distribution function then cancel with those in the microcanonical rate coefficients, just as in the high pressure limit. 

Note added in proof In view of the failure of the harmonic oscillator model to account for the observed rate of activation in unimolecular dissociation reactions (the dissociation lag problem) these calculations have been repeated for a Morse anharmonic oscillator with transition between nearest and next-nearest neighbor levels [S. K. Kim, /. Chem. Phys. (to be published)]. The numerical evaluation of the analytical results obtained by Kim has not yet been carried out. From the results obtained by us and our co-workers [Barley, Montroll, Rubin, and Shuler, /. Chem. Phys. in press)] on the relaxation of vibrational nonequilibrium distributions of a system of Morse anharmonic oscillators it seems clear, however, that the anharmonic oscillator model with weak interactions (i.e., adiabatic perturbation type matrix elements) does not constitute much of an improvement on the harmonic oscillator model in giving the observed rates of activation. The answer to tliis problem would seem to lie in a recalculation of the collisional matrix elements for translational-vibrational energy exchange which takes account of the strong interactions in highly energetic collisions which can lead to direct dissociation. 

The assumption of harmonic vibrations and a Gaussian distribution of neighbors is not always valid. Anharmonic vibrations can lead to an incorrect determination of distance, with an apparent mean distance that is shorter than the real value. Measurements should preferably be carried out at low temperatures, and ideally at a range of temperatures, to check for anharmonicity. Model compounds should be measured at the same temperature as the unknown system. It is possible to obtain the real, non-Gaussian, distribution of neighbors from EXAFS, but a model for the distribution is needed and inevitably more parameters are introduced. 

In general, all observed intemuclear distances are vibrationally averaged parameters. Due to anharmonicity, the average values will change from one vibrational state to the next and, in a molecular ensemble distributed over several states, they are temperature dependent. All these aspects dictate the need to make statistical definitions of various conceivable, different averages, or structure types. In addition, since the two main tools for quantitative structure determination in the vapor phase—gas electron diffraction and microwave spectroscopy—interact with molecular ensembles in different ways, certain operational definitions are also needed for a precise understanding of experimental structures. 

The relevant question regarding secondary IEs on acidity is the extent to which IEs affect the electronic distribution. How can an inductive effect be reconciled with the Born-Oppenheimer approximation Although the potential-energy function and the electronic wave function are independent of nuclear mass, an anharmonic potential leads to different vibrational wave functions for different masses. Averaging over the ground-state wave function leads to different positions for the nuclei and thus averaged electron densities that vary with isotope. This certainly leads to NMR isotope shifts (IEs on chemical shifts), because nuclear shielding is sensitive to electron density.16... 

Calculations of vibrational frequencies are never accurate enough to verify that the secondary IE arises entirely from zero-point energies. Therefore although they do confirm a role for zero-point energies, which was never at issue, they cannot exclude the possibility of an additional inductive effect arising from changes of the average electron distribution in an anharmonic potential. The question then is whether it is necessary to invoke anharmonicity to account for a part of these secondary IEs. 

Since experimental results were available for the high-frequency (" -2080 cm-1) diatomic CN in water (as opposed to CH3C1) (17), with an estimated T value of some 25 ps, an MD study was undertaken by Rey and Hynes (29) to clarify the role of Coulomb forces for VET in this accesible case. The charge distribution of CN in the solvent was modeled by a negative charge on N and a finite dipole located on the C site (30). The equilibrium solvent structure about this ion involved greater solvation number on the N end compared to the C end, a result consistent with some small cluster calculations (31). Since the frequency shift from the vacuum and the anharmonicity in the CN bond are both relatively small (29), the static vibrational aspects of the ion are evidently fairly clean. ... 

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