Correlation function reduced

Again, owing to Eq. (M.10), that is to the fact that the average of the Langevin force is zero, this correlation function reduces to... 

In the limit A < 1, called the fast modulation or homogeneous limit, the FID correlation function reduces to... 

Here, ta, and are the correlation time and order parameter for the /th inner motion. When p = 3 and TAi = TA2 ri, the total correlation function reduces to... 

For purely dipolar systems the problem is greatly simplified. The expansions for the correlation functions reduce to... 

The so-called influence parameter c(p(r)) can be related to the direct correlation function of the homogeneous fluid. This equation was first obtained by Yang et al. [4,5] and Bongiomo et al. [6]. In the simple case of the mean-spherical approximation the direct correlation function reduces to -u(r)/kBT, which yields a simple expression for the influence parameter given by ... 

In this report problem of information processing in MIA pulse flaw detectors by means of cross correlation function is considered. Such processing promises to increase the sensitivity and to reduce the noises, mainly the frictional ones. 

Far from the surface, the theory reduces to the PY theory for the bulk pair correlation functions. As we have noted above, the PY theory for bulk pair correlation functions does not provide an adequate description of the thermodynamic properties of the bulk fluid. To eliminate this deficiency, a more sophisticated approximation, e.g., the SSEMSA, should be used. 

By rewriting all of the (5-functions in this manner, the problem of taking the expectation of equation 7.110 has been conveniently reduced to taking expectations of simple products of ct s. Schulman and Seiden go on to show exactly how such expectations can be expressed as functions of second-order correlation functions (or cumulants), <0 02>c = — <02> [schul78] ... 

Fig. 1.15. Translational and angular velocity correlation functions for nitrogen. MD simulation data from [70], T = 122 K, densities are indicated in the figure. Reduced units for time t = (e/cr2), for density p" = p

Fig. 1.16. Angular momentum and kinetic energy correlation functions for compressed nitrogen. MD simulation from [62], T = 300 K. The lines are continuous (800 amagat), close dotted (600 amagat), sparse dotted (400 amagat), dashed (300 amagat) and pairwise dotted (200 amagat). Reduced time units are the same as in Fig. 1.15.

If the second term on the right-hand side of the equation is omitted, the latter is transformed into Eq. (2.76). As the omission is possible only for t < tj, Fourier transformation of the reduced equation holds for co-tj 1 only. Consequently, the equality (2.75) is of asymptotic character, and may not be utilized to find full g(co) or its Fourier-transform Kj(t) at any times. When it was nevertheless used in [117], the rotational correlation function turned out to be alternating in sign. The oscillatory behaviour of Kj(t) occured not only in a compressed gas, but also at normal pressure, when Kj(t) should vanish monotonically, if not exponentially. The origin of these non-physical oscillations is easily... 

A closer look at the original data published by Johnson, Gill, and Pople, 1993, reveals that use of the gradient-corrected LYP correlation functional instead of VWN increases atomization energies for non-hydride species quite significantly, while those containing hydrogen atoms are reduced. This rather systematic trend is portrayed in Table 9-2 for a few example cases. 

Use of the LYP correlation functional apparently reduces the overbinding proportional to the number of hydrogen atoms by about 1 kcal/mol per H from CH to CH4. On the contrary, for the nonhydride diatomics listed, the atomization energies increase by 4-5 kcal/ mol per atom upon substitution of VWN by the LYP functional. For species like C2H2,... 

In the Kohn-Sham Hamiltonian, the SVWN exchange-correlation functional was used. Equation 4.12 was applied to calculate the electron density of folate, dihydrofolate, and NADPH (reduced nicotinamide adenine dinucleotide phosphate) bound to the enzyme— dihydrofolate reductase. For each investigated molecule, the electron density was compared with that of the isolated molecule (i.e., with VcKt = 0). A very strong polarizing effect of the enzyme electric field was seen. The largest deformations of the bound molecule s electron density were localized. The calculations for folate and dihydrofolate helped to rationalize the role of some ionizable groups in the catalytic activity of this enzyme. The results are,... 

D(co)VV (cn)/ii m equal to the expression (l/rr) y y co2). Integrating from frequency zero up to infinite, one gets the empirical formula K(t-x)= (X/ft) y exp(-y t-x ). Here, 1/y represents the memory time of the dissipation and is essentially the inverse of the phonon bandwidth of the heat bath excitations that can be coupled to the oscillator. It reduces to a delta function when y->infinite. The correlation function (t-t), in this model is [133]... 

Much more detailed information about the microscopic structure of water at interfaces is provided by the pair correlation function which gives the joint probability of finding an atom of type/r at a position ri, and an atom of type v at a position T2, relative to the probability one would expect from a uniform (ideal gas) distribution. In a bulk homogeneous liquid, gfn, is a function of the radial distance ri2 = Iri - T2I only, but at the interface one must also specify the location zi, zj of the two atoms relative to the surface. We expect the water pair correlation function to give us information about the water structure near the metal, as influenced both by the interaction potential and the surface corrugation, and to reduce to the bulk correlation Inunction when both zi and Z2 are far enough from the surface. 

The explicit expressions for either the binding constants or the correlation functions are, in general, quite complicated even when the subunits are identical. We present here some selected expressions obtained for special cases. As before, we use the reduced parameters... 

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