Rule of the mean

An additional useful measure of how well an even-power expansion represents the quantum corrections of order n can be obtained by examining the deviations from the rules of the mean (2, 3, 28) of order n. In a molecule containing two or more equivalent atoms, an even-power expansion of the form of Equation 62, for a pair which diflEer by one isotopic substitution at one of the equivalent positions, is equal to the... 

This is the first rule of the mean. It holds if the coeflScients c are independent of the isotopic substitutions, as they are for the Bernoulli series. The same condition is also satisfied for the Jacobi expansions, when a quantity common to a given molecular species, such as v max of the lightest isotopic molecule, is used for evaluating the modulating coefficients. For special combinations of isotopic pairs the rule holds to higher orders and... 

Using a quantity such as v, ax of HoO for evaluating the modulating coefficients for both isotopic pairs, the one-term Jacobi expansion predicts the quantum correction to be zero, thus satisfying the first rule of the mean. The first contribution to the quantum correction arises from n = 2 in the expansion. To describe the bending vibrations adequately, however, we need at least n = 3. In Table XVI, quantum corrections predicted by expansion formulae are compared with the exact quantum correction for the disproportionation among the isotopic water molecules. No entry is made for the Bernoulli series at 300°K. because the series does not exist at this temperature. 

Table XVII shows similar comparisons of the deviations from the first rule of the mean for an 0-disproportionation between carbon dioxide species. Here, the normal modes have been divided into 2 linear modes and 2 non-linear modes. The approximations for the non-linear modes are much better than those for the linear modes, simply because v max for the former is only a quarter of v max for the latter. Calculations (not shown in the Table) made with the Chebyshev (L = 5) yielded In K s of 2.550 X 10 and 2.086 X 10 for n = 2 and 3, respectively, compared with the exact value of 3.546 X 10 at 300°K. Rule of the Mean tests are especially useful for examining the suitability of a polynomial to a particular molecule, because they permit an examination of the correspondence between individual terms of the expansion and motions of groups of atoms in the molecule.

Table XVI. Deviation from the First Rule of the Mean of Water ...

The W2A2E6u term in Eq. (26) allows one to predict the sign and magnitude of the deviation from the rule of the mean In the harmonic approximation. From this term we can derive the theorem in an isotopic disproportionation reaction to form asymmetric molecules from symmetrical ones, the symmetry nundier corrected equilibrium constant is always less than unity. Only the term 2([j - ia )ia.a2 in 26u leads to deviations from the rule of the mean. Quantitative calculation of the deviations from the rule of the mean by Eq. (26) will require at least the term j = 3. The corrections are proportional to i.a, . The largest deviations are expected for the equilibrium ... 

Isotope effect between the HH, HD, DH, and DD isotopomers was used as an important tool to determine the mechanism of the double-proton transfer. For concerted degenerate double-proton transfers in the absence of tunneling, the rule of the geometrical mean (RGM) should hold in good approximation, which states that /chh/ hd = /cdh/ dd-Tunneling may lead to a breakdown of this rule but the relation /chh > hd = dh > dd should remain valid. In the absence of secondary isotope effects the relation /chh HD = DH = 2 /cdd sliould liold for a stepwise pathway, even if tunneling is involved. 

The truncated part of the integral can be obtained by numerical integration (e.g. by means of the trapezium rule) of the function rCp(r) between times 0 and T. The mean residence time MRT is an important pharmacokinetic parameter, especially when a substantial fraction of the drug is excreted or metabolized during its first pass through an organ, such as the liver. 

A number of authors have suggested various mixing rules, according to which the quantity a could be calculated for a measured electrolyte in a mixture, starting from the known individual parameters of the single electrolytes and the known composition of the solution. However, none of the proposed mixing relationships has found broad application. Thus, the question about the dependence of the mean activity coefficients of the individual electrolytes on the relative contents of the various electrolytic components was solved in a different way. 

The final rule follows from the transformational invariance of the mean squared end-to-end distance Flory exponent). As a hypothesis it is... 

The three most commonly applied external reflectance techniques can be considered in terms of the means employed to overcome the sensitivity problem. Both electrically modulated infrared spectroscopy (EMIRS) and in situ FTIR use potential modulation while polarization modulation infrared reflection absorption spectroscopy (PM-IRRAS) takes advantage of the surface selection rule to enhance surface sensitivity. 

As was discussed above, the surface selection rule arises because of the reduction in magnitude of the mean square electric field strength, < >, of... 

Vibrations in the surface plane, however, will be rather similar to those in the bulk because the coordination in this plane is complete, at least for fee (111) and (100), hep (001) and bcc (110) surfaces. Thus the Debye temperature of a surface is lower than that of the bulk, because the perpendicular lattice vibrations are softer at the surface. A rule of thumb is that the surface Debye temperature varies between about 1/3 and 2/3 of the bulk value (see Table A.2). Also included in this table is the displacement ratio, the ratio of the mean squared displacements of surface and bulk atoms due to the lattice vibrations [1]. 

Pressure design of unlisted components and other piping elements to which the rules in para. IP-3.1 do not apply shall be based on calculations consistent with the design criteria of this Code. These calculations shall be substantiated by one or more of the means stated in (a) through (d) below, considering applicable dynamic, thermal, and cyclic effects in paras. IP-2.1.7 through IP-2.1.8, as well as thermal shock. Calculations and documentation showing compliance with (a), (b), (c), or (d), and (e) shall be available for the owner s approval ... 

VI-1.1.3 Code Interpretations. Code Interpretations provide clarification of the meaning of existing rules in the Code, and are also presented in question and reply format. Interpretations do not introduce new requirements. 

The relationship between the magnitude of secondary deuterium and tritium KIEs and the rule of the geometric mean 223... 

THE RELATIONSHIP BETWEEN THE MAGNITUDE OF SECONDARY DEUTERIUM AND TRITIUM KIEs AND THE RULE OF THE GEOMETRIC MEAN... 

The results of these calculations have implications on the applicability of the rule of the geometric mean, which indicates that the KIE for a doubly labelled species should be the product of the KIEs for the corresponding singly labelled substrates. For instance, the KIE for the doubly labelled [17] should be the product of the secondary deuterium KIE, ]/ ]> associated with the nontransferring hydrogen and the primary deuterium KIE, / , produced by the transferring hydrogen (equation 58)). 

However, the calculations show that (59) does not apply when tunnelling is significant. Therefore, the rule of the geometric mean is not expected to hold if isotopic substitution in one position affects the tunnelling contribution to a KIE at another position. 

A one standard error rule is described in Hastie et al. (Hastie et al. 2001). It is assumed that several values for the measure of the prediction error at each considered model complexity are available (this can be achieved, e.g., by CV or by bootstrap, Sections 4.2.5 and 4.2.6). Mean and standard error (standard deviation of the means, s) for each model complexity are computed, and the most parsimonious model whose mean prediction error is no more than one standard error above the minimum mean prediction error is chosen. Figure 4.4 (right) illustrates this procedure. The points are the mean prediction errors and the arrows indicate mean plus/minus one standard error. 

Using the rule of geometric mean, which states that isotope effects are independent and cumulative we write analogous to Equation 10.12 ... 

Bigeleisen J (1949) The relative velocities of isotopic molecules. J Chem Phys 17 675-678 Bigeleisen J (1955) Statistical mechanics of isotopic systems with small quantum corrections. I. General considerations and the rule of the geometric mean. J Chem Phys 23 2264-2267 Bigeleisen J (1998) Second-order correction to the Bigeleisen-Mayer equation due to the nuclear field shift. Proc National Acad Sci 95 4808-4809... 

Quantum tunneling in enzyme-catalyzed reactions early indications 35 The rule of the geometric mean ( no isotope effects on isotope effects ) The Swain-Schaad relationship 36... 

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