Einsteins single-frequency model

To simplify the analysis, the Einstein single frequency (coe) model is used. The Einstein frequency is given by ... 

A more transparent representation of the temperature dependence can be obtained in simple models. Consider for example an Einstein-type model where the phonon spectrum is represented by a single frequency coa. The rate is loosely written in this case in the form... 

The collective modes of vibration of the crystal introduced in the previous paragraph involve all the atoms, and there is no longer a single vibrational frequency, as was the case in the Einstein model. Different modes of vibration have different frequencies, and in general the number of vibrational modes with frequency between v and v + dv are given by... 

Phonons At least two phonon branches are involved in the observed absorption the acoustic phonons and the optical 46-cm "1 branch. Our model includes a single acoustic branch [with cutoff frequency f2max, and isotropic Debye dispersion hfiac q) = hQmaxq/qmax] and an optical dispersionless branch (Einstein s model, with frequency /20p). 

The two main nuclear modes affecting electronic energies of the donor and acceptor are intramolecular vibrations of the molecular skeleton of the donor-acceptor complex and molecular motions of the solvent. If these two nuclear modes are uncoupled, one can arrive at a set of simple relations between the two spectral moments of absorption and/or emission transitions and the activation parameters of ET. The most transparent representation is achieved when the quantum intramolecular vibrations are represented by a single, effective vibrational mode with the frequency Vy (Einstein model). 

The vibronic envelope ECWD (v) in Eq. [129] can be an arbitrary gas-phase spectral profile. In condensed-phase spectral modeling, one often simplifies the analysis by adopting the approximation of a single effective vibrational mode (Einstein model) with the frequency Vv and the vibrational reorganization energy Xy. The vibronic envelope is then a Poisson distribution of... 

This quite complicated temperature dependence of the macroscopic heat capacity must now be explained by a microscopic model of thermal motion. Neither a single Einstein function, as given in Eq. (5) of Fig. 5.13, nor any of the Debye functions of Fig. 5.14 have any resemblance to the experimental data. It helps in the analysis that the vibration spectrum of crystalline polyethylene is known in detail from calculations using force constants derived from infrared and Raman spectroscopy. Such a spectrum is shown in Fig. 5.18, at the top. Using an Einstein function for each vibration, one can compute the heat capacity by adding the contributions of all the various frequencies. The heat capacity of the crystalline polyethylene shown in Fig. 5.17 can be reproduced above 50 K by these data within experimental error. Below 50 K the experimental data show increasing deviations, an indication that the computation of the low-frequency, skeletal vibrations carmot be carried out correctly at present. ... 

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