Frequency moments

The function Am(s) should describe the low- and high-frequency behavior of AM (s ) to a desired degree. We require that Am(s) reproduces N/, high- and Ni low-frequency moments. Because Am(.v) is determined by an even number of constants an and, one needs to choose Nk I Ni 2N. We refer to the resulting... 

The last step in Urey s derivation is the application of the Redlich-Teller product rule (e.g., Angus et al. 1936 Wilson et al. 1955), which relates the vibrational frequencies, moments of inertia, and molecular masses of isotopically substituted molecules. For CIO,... 

The formulas that we have derived in this chapter to calculate thermochemical data are accurate and easy to apply. This approach can be used to fill in the gaps in species thermochemical data needed in reacting-flow calculations. Their accuracy is limited by the values of the molecular constants used in the calculations, that is, vibrational frequencies, moments of inertia, and the standard-state heat of formation. 

It is a simple but lengthy matter to determine the explicit form of all these frequency moments. From Eqs. (99) and (158) it follows that... 

The integral in the expression (20) is Fourier-transform of function I(t) (20a). If to return this Fourier-transform, i.e. to express I(t) through /(Q), it is easy to find the expressions for the frequency moments... 

U. Balucani and R. Vallauri. Collision induced light scattering at intermediate densities II. The second frequency moment. Molec. Phys., 35 1115-1125 (1479). 

For a very short time interval, one can determine the time evolution of the correlation function by a Taylor expansion. This is an application of the time differentiation theorem already mentioned in connection with Eq. (31). What the theorem provides is a relation between the coefficients in the Taylor expansion and the frequency moments of the... 

Standard manipulations relate the frequency moments of the spectrum to the time derivatives of C(t) at the origin... 

Qualitatively sketch the heat capacity of an ideal gas of diatomic molecules as a function of temperature. Indicate the characteristic temperatures (in terms of vibrational frequency, moment of inertia, and so on) where various degrees of freedom begin to contribute. 

Janoschek and Rossi [140] have calculated energies, harmonic vibration frequencies, moment of inertia, and thermochemical properties, on a set of 32 selected free radicals at the G3MP2B3 level of theory. They compared their calculated values to literature data and show a mean absolute deviation between calculated and experimental enthalpies values of 0.9 kcal moT which was close to the average experimental uncertainty of 0.85 kcal mol. ... 

Mandd with a quite different method. They applied the decoupling (Eq. 10) in the equation of motion for the double-time-retarded commutator of the charge density fluctuation operators, and imposed conservation of frequency moments to all orders in the Hartree-Fock approximation for the... 

Nucleus abundance number (/) ratio" (y) sensitivity frequency" moment" ... 

Because of cost considerations, researchers have tried to combine low level calculations that are used to follow the RP with high level calculations applied just to the stationary points along the RP. Truhlar and co-workers have worked out appropriate correction functions that adjust the potential, frequencies, moments of inertia, etc. obtained with a low level method to the high level results obtained at the stationary points (and some other points). These dual level descriptions have been found to improve the accuracy of reaction rate calculations."... 

One possible approach would focus on the cross section sum rules. Since the stated rules, (18), do not clearly distinguish between the two methods it is possible that higher frequency moments will indicate the source of the differences. Since these higher moments do not correspond to observables, the "experimental" values will have to be provided by Monte Carlo calculations of the corresponding correlation functions. ... 

We define frequency moments (o(n) of the phonon density of states F(a>) as... 

Equation (31) now implies that to leading order in the temperature dependence, the high-temperature heat capacity of harmonic lattice vibrations depends only on the frequency moment co(2). From Eq. (32) we see that the high-temperature vibrational entropy depends only on the moment co(0). A certain combination [see Eq. (34)] of the thermal atomic displacements at high temperatures depends on co(-2). Table 4 summarizes these results, together with the case of very low temperatures, which will be discussed in the following section. 

Table 4 In the Limit of Low or High Temperatures, Some Important Thermodynamic Quantities Depend Only on a Single Frequency Moment (0(n) or Equivalently on a Single Debye Temperature 0d(w)...

We can also define Debye temperatures 0d( ) such that they correctly reproduce the frequency moment (o( ) of a certain density of states F ai). One finds that... 

The phonon spectrum of a solid depends on the interatomic forces as well as on the atomic masses. For an element, the mass dependence is trivial. All phonon frequencies, and hence also all frequency moments co(n), vary with the atomic mass M as In a compound with two or several different atomic masses, the vibrational frequencies depend on the interatomic forces and on the masses in a complex way, with two exceptions. In the low-frequency part of the phonon density of states, which is uniquely given by the sound velocities, the vibrational frequencies vary as p", where p is the mass density of the solid. It follows that 0c (= 0 ) in the limit of low temperatures has interatomic forces and atomic masses separated in the form of two multiplicative factors. The force-constant part is directly related to the elastic coefficients Cy. Hence the low-temperature limit of the Debye temperature gives a certain average over the interatomic forces, as it is reflected in the sound waves. We shall now introduce another average over the interatomic forces, uniquely related to the entire phonon spectrum. 

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