Rule of the geometric mean

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 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. 

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... 

These studies had therefore found the tunneling phenomenon, with coupled motion, as the explanation for failures of these systems to conform to the expectations that the kinetic secondary isotope effects would be bounded by unity and the equilibrium effect and that the primary and secondary effects would obey the Rule of the Geometric Mean (Chart 3), as well as being consistent with the unusual temperature dependences for isotope effects that were predicted by Bell for cases involving tunneling. 

Chart 7. Huskey s rules for violations of the rule of the geometric mean and the Swain-Schaad relationships... 

Geometric Mean (Chart 3), Rp = Rs = 1. If the Rule of the Geometric Mean is invalid, e.g., as a result of strong vibrational coupling between the primary and secondary centers, Rp and Rs may not be unity. 

The observation of systematic deviations from the Rule of the Geometric Mean between primary and a-secondary isotope effects. The magnitude of both effects should be greater (more normal) when the other position is occupied by a lighter isotope than when it is occupied by a heavier isotope. 

All these conclusions are somewhat dependent on the assumption that the other possible complications of the analysis mentioned in Section IV are absent. The success of the analysis in the cases studied so far (Williams and Kreevoy, 1968) is some justification of the value l = 0-69 and the conclusions about the approximate cancellation of effects due to the imperfect validity of the rule of the geometric mean. The limitations of these somewhat optimistic inferences can only be settled by further, more accurate, experiments. 

Equation (145) gave the best fit to the data for the assumption of Y = 0-96, i.e. quite a small transfer effect. However, this calculation by Gold and Waterman (1968a) disregarded the deviation from the rule of the geometric mean. In the case of hydrogen ion-catalysed reactions we have remarked that a correction for this effect is hard to evaluate because of its unknown size for the hydrogen ion. In the li ... 

Again, the calculation in Table 4 with and without consideration of the breakdown of the rule of the geometric mean (i.e. by equations (57) and (148)) should place the importance of this factor in perspective. The case considered here is one in which this effect can be properly computed and is also one in which it exerts a particularly large influence because the ratio H/ D is large. Even so the difference between the predictions of equations (57) and (148) is small compared with the accuracy normally attainable in all but the most exact kinetic studies. 

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