Anharmonic coupling coefficients

Evidently, we would arrive at the same result if the problem were considered in the Cartesian system of coordinates, with accordingly different values of anharmonic coupling coefficients ... 

To invoke the perturbation theory for a small anharmonic coupling coefficient, we use the Wick theorem for the coupling of the creation and annihilation operators of low-frequency modes in expression (A3.19). Retaining the terms of the orders y and y2, we are led to the following expressions for the shift AQ and the width 2T of the high-frequency vibration spectral line 184... 

Figure 18. The coefficient of thermal conductivity calculated for myoglobin from T = 20 K to 320 K is plotted as a solid curve in (a). The dashed curve gives the thermal conductivity if no anharmonic coupling of normal modes is included in the calculation, (b) Thermal diffusivity calculated for myoglobin is plotted from 20 K to 320 K with (solid) and without (dashed) contributions of anharmonicity.

In myoglobin, we find that the anharmonic contribution significantly enhances thermal conduction over that in the harmonic limit, by more than a factor of 2 at 300 K. Moreover, the thermal conductivity rises with temperature for temperatures beyond 300 K as a result of anharmonicity, whereas it appears to saturate around 100 K if we neglect the contribution of anharmonic coupling of vibrational modes. The value for the thermal conductivity of myoglobin at 300 K is about half the value for water. The value for the thermal diffusivity that we calculate for myoglobin is the same as the value for water. Thermal transport coefficients for other proteins will be presented elsewhere. 

The harmonic contribution to the thermal conductivity is Ki,(r), which we calculated in the previous section. The main purpose of Kjni,(r) is to allow for localized modes, which in a protein such as myoglobin correspond to vibrational modes above 150 cm", to contribute to thermal flow via anharmonic coupling to other vibrational modes. The contribution to the diffusion coefficient due to anharmonic coupling can be thought of as == where l/x is the transfer rate between modes... 

We have derived an expression for the specific heat of the anharmonic linear chain which is valid at high temperatures (5.65) due to anharmonicity, Cj (T) is proportional to T at high temperatures. The same result also applies for the specific heat c (T) of three-dimensional crystals [5.31,32]. As in (5.65), the coefficient of proportionality is determined by the third and fourth-order anharmonic coupling constants. In addition, the generalization of the expression (5.67) for Cj - Cj to cubic crystals is straightforward, namely,... 

The high-frequency mode is coupled through the anharmonicity coefficient y with the low-frequency mode which is a resonant one due to harmonic interaction with the surface reservoir. For simplicity, the last term is written in the Gaitler-London approximation (compare with more general Eq. (4.1.8) where this restriction is absent). The required GF of the high-frequency mode can be obtained from the... 

Due to the rotational structure as well as the so-called hot hand absorptions (Sec. 2.5.3), the contour of a rovibrational band depends on the temperature. Today it is possible to determine molecular constants such as moments of inertia, Coriolis coupling constants, centrifugal distortion constants, and anharmonicity coefficients by FTIR as precisely as possible in order to calculate the intensity and shape of an absorption band. In such a simulation process the temperature may be used as a parameter. The results can be compared to the experimental spectra and the temperature may be deduced by fitting the calculated to the observed bands. This is possible with IR as well as with Raman bands. A review of curve fitting procedures and their limitations has been given by Maddams (1980). 

Adiabatic Treatment of Torsional Anharmonicity and Mode Coupling in Molecular Partition Functions and Statistical Rate Coefficients Application to Hydrogen Peroxide... 

For radical-radical reactions, the full mode coupling and anharmonicity effects for the relative and overall rotational motions must be explicitly accounted for. We have derived a direct variable reaction coordinate transition state theory approach that appears to 3deld accurate rate coefficients for a number of alkyl radical reactions.This approach is analogous to that embodied in Eq. (4.10) for the long-range transition state, but includes variational optimizations of the form of the reaction coordinate and does not make the large orbital moment of inertia assumption. A detailed description of this approach was provided in some of our recent articles. 

The coefficients C cover (besides intramode anharmonic contributions) mode-mode coupling in the potential energy... 

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