Dispersion coefficients methane

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

The dispersion coefficients for the mixed-symmetry component 7 5 which describes the deviation from Kleinman symmetry are for methane more than an order of magnitude smaller than coefficients of the same order in the frequencies for 7. Their varations with basis sets and wavefunction models are, however, of comparable absolute size and give rise to very large relative changes for the mixed-symmetry dispersion coefficients. 

The CC2 model performes very different for static hyperpolarizabilities and for their dispersion. For methane, CC2 overestimates 70 by a similar amount as it is underestimated by CCS, thus giving no improvement in accuracy relative to the uncorrelated methods CCS and SCF. In contrast to this, the CC2 dispersion coefficients listed in Table 3 are by a factor of 3 - 8 closer to the CCSD values than the respective CCS results. The dispersion coefficients should be sensitive to the lowest dipole-allowed excitation energy, which determines the position of the first pole in the dispersion curve. The substantial improvements in accuracy for the dispersion coefficients are thus consistent with the good performance of CC2 for excitation energies [35,37,50]. 

Table 4 Comparison of various ab initio results and experimental estimates for the dispersion coefficients of the electronic hyperpolarizabilities 7jj and 7 of methane. (All results in atomic units. Results for the dispersion coefficients refer to single point calculations at the equilibrium geometry. Where available, dispersion coefficients for the vibrational average are given in parentheses.)...

The concentration of methane at room temperature at any channel position is described by the impulse response without reaction [Eq. (3.4)] [the dispersion coefficient for a rectangular channel with channel depth b and aspect ratio s is given in Eq. (3.3)] ... 

Adams reported radial dispersion coefficients in the range of 50 to 70 cmVs using methane as a tracer [71]. He found that dispersion increased with gas velocity and decreased with solids suspension density, presumably due to the suppression of turbulent intensity caused by the presence of solids. 

Even the corresponding peak temperatures of the blown bitumens show very small variances in the tests in 10 bar methane and also permit the calculation of statistical means. The resulting coefficients of variation are 3.0 % maximum. This is also true for the colloid components, except for the dispersion medium of the two bitumens 85/40 (sample III) and 85/25 (sample IV). Here again a weight loss caused by distillation even occurs under pressure with the consequence of low values for the activation energy and frequency factor. Only the data of the other three samples was included in the statistics. The average values of the Arrhenius coefficients calculated in this manner and the means of the conversion aie shown in Table 4-93. 

Comparison of the means shows that there is a significant difference in the Arrhenius coefficients for distillation and blown bitumens and their colloid components. The conversions only differ for the dispersion medium whereas the bitumens, the petroleum resins, and the asphaltenes do not differ in the conversions, either in 1 bar argon or in 10 bar methane. 

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