Motion, coupled

For this reaction the transition state drawn in Equation 11.82 involving simultaneous proton and hydride transfers has been proposed to explain the observed isotope effects. 

---

QCMD describes a coupling of the fast motions of a quantum particle to the slow motions of a classical particle. In order to classify the types of coupled motion we eventually have to deal with, we first analyze the case of an extremely heavy classical particle, i.e., the limit M —> oo or, better, m/M 0. In this adiabatic limit , the classical motion is so slow in comparison with the quantal motion that it cannot induce an excitation of the quantum system. That means, that the populations 6k t) = of the... 

Beeause there are no terms in this equation that couple motion in the x and y directions (e.g., no terms of the form x yb or 3/3x 3/3y or x3/3y), separation of variables can be used to write / as a product /(x,y)=A(x)B(y). Substitution of this form into the Schrodinger equation, followed by collecting together all x-dependent and all y-dependent terms, gives ... 

At present, the Brownian motions of isolated rigid macromolecules are quite well understood. The challenge now is to understand the Brownian deformations of nonrigid macromolecules and to ascertain the time scales on which the coupled motions of their subunits relax various experimental signals. 

Table 1 Experimental studies that led to the coupled motion and tunneling model... 

In the following year, Cleland and his coworkers reported further and more emphatic examples of the phenomenon of exaltation of the a-secondary isotope effects in enzymic hydride-transfer reactions. The cases shown in Table 1 for their studies of yeast alcohol dehydrogenase and horse-liver alcohol dehydrogenase would have been expected on traditional grounds to show kinetic isotope effects between 1.00 and 1.13 but in fact values of 1.38 and 1.50 were found. Even more impressively, the oxidation of formate by NAD was expected to exhibit an isotope effect between 1.00 and 1/1.13 = 0.89 - an inverse isotope effect because NAD" was being converted to NADH. The observed value was 1.22, normal rather than inverse. Again the model of coupled motion, with a citation to Kurz and Frieden, was invoked to interpret the findings. 

In 1983, Huskey and Schowen tested the coupled-motion hypothesis and showed it to be inadequate in its purest form to account for the results. If, however, tunneling along the reaction coordinate were included along with coupled motion, then not only was the exaltation of the secondary isotope effects explained but also several other unusual feamres of the data as well. Fig. 4 shows the model used and the results. The calculated equilibrium isotope effect for the NCMH model (the models employed are defined in Fig. 4) was 1.069 (this value fails to agree with the measured value of 1.13 because of the general simplicity of the model and particularly defects in the force field). If the coupled-motion hypothesis were correct, then sufficient coupling, as measured by the secondary/primary reaction-coordinate amplimde ratio should generate secondary isotope effects that... 

The overall conclusion drawn by Huskey and Schowen was that a combination of coupled motion and tunneling through a relatively sharp barrier was required to explain the exaltation of secondary isotope effects. They also noted that this combination predicts that a reduction of exaltation in the secondary effect will occur if the transferring hydrogen is changed from protium to deuterium for point A in Fig. 4, the secondary effect is reduced by a factor of 1.09. Experimentally, reduction factors of 1.03 to 1.14 had been reported. For points B, C, and D on the diagram, all of which lack a combination of coupled motion and tunneling, no such reductions in the secondary isotope effect were calculated. 

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. 

For further important work on this and related concepts, see Rucker, J. and Kliman, J.P. (1999). Computational study of tunneling and coupled motion in alcohol dehydrogenase-catalyzed reactions Implication for measured hydrogen and carbon isotope effects. J. Am. Chem. Soc. 121, 1997 -2006, and Kohen, A. and Jensen, J.H. (2002). Boundary conditions for the Swain-Schaad relationship as a criterion for hydrogen tunneling. J. Am. Chem. Soc. April 17, 124(15), 3858-3864. 

Karsten, W.E., Hwang, C.C. and Cook, P.F. (1999). Alpha-secondary tritium kinetic isotope effects indicate hydrogen tunneling and coupled motion occur in the oxidation of L-malate by NAD-malic enzyme. Biochemistry 38, 4398-4402... 

Klinman, J.P. (1991). Hydrogen tunneling and coupled motion in enzyme reactions. In Enzyme Mechanism from Isotope Effects, Cook, P.F. (ed.), pp. 127-148. CRC Press, Boca Raton... 

Kinetic complexity definition, 43 Klinman s approach, 46 Kinetic isotope effects, 28 for 2,4,6-collidine, 31 a-secondary, 35 and coupled motion, 35, 40 in enzyme-catalyzed reactions, 35 as indicators of quantum tunneling, 70 in multistep enzymatic reactions, 44-45 normal temperature dependence, 37 Northrop notation, 45 Northrop s method of calculation, 55 rule of geometric mean, 36 secondary effects and transition state, 37 semiclassical treatment for hydrogen transfer,... 

An important difference between the BO and non-BO internal Hamiltonians is that the former describes only the motion of electrons in the stationary field of nuclei positioned in fixed points in space (represented by point charges) while the latter describes the coupled motion of both nuclei and electrons. In the conventional molecular BO calculations, one typically uses atom-centered basis functions (in most calculations one-electron atomic orbitals) to expand the electronic wave function. The fermionic nature of the electrons dictates that such a function has to be antisymmetric with respect to the permutation of the labels of the electrons. In some high-precision BO calculations the wave function is expanded in terms of basis functions that explicitly depend on the interelectronic distances (so-called explicitly correlated functions). Such... 

When this model is applied to the interpretation of spin relaxation of polymers in solution the extent of cooperation motion can he measured hy a parameter R = (W /Wb) / which is found to take on values from 1 to 50. If a bond is the smallest moving unit, then R, called the "range", corresponds approximately to the number of bonds Involved in cooperative or coupled motion ( ). Both Wg and are strongly temperature dependent and vary non-monoton-Ically with temperature which appears to complicate this simple identification of R. 

In enzyme catalysis, 249, 373-397 [bovine serum amine oxidase, 249, 393-394 coupled motion and, 249, 386-388 demonstration, 249, 374-386 (breakdown of rule of geometric mean in,... 

For (-)-cyclohexanone-2,6-chair conformations are equivalent, with the axial and equatorial CD bonds chirally oriented. The CD stretching VCD, positive at —2200 cm and negative between 2130-2170 cm , is consistent with that predicted for coupled motion of the two CD bonds. In this case, the splitting between the modes is due to different CD bond force constants for the axial and equatorial positions. 

In a molecular dynamic simulation147 of bulk atomic diffusion by a vacancy mechanism, two atoms may occasionally jump together as a pair. The temperature of the simulation is close to the melting point of the crystal. In FTM studies of single atom and atomic cluster diffusion, the temperature is only about one tenth the melting point of the substrate. All cluster diffusion, except that in the (1 x 1) to (1 x 2) surface reconstruction of Pt and Ir (110) surfaces already discussed in Section 4.1.2(b), are consistent with mechanisms based on jumps of individual atoms.148,149 In fact, jumps of individual atoms in the coupled motion of adatoms in the adjacent channel of the W (112) surface can be directly seen in the FTM if the temperature of the tip is raised to near 270 K.150... 

An observation of motion of single atoms and single atomic clusters with STEM was reported by Isaacson et al,192 They observed atomic jumps of single uranium atoms on a very thin carbon film of —15 A thickness or less. Coupled motion of two to three atoms could also be seen. As the temperature of the thin film could not be controlled, no Arrhenius plot could be obtained. Instead, the Debye frequency , kTIh, was used to calculate the activation energy of surface diffusion, as is also sometimes done in field ion microscopy. That the atomic jumps were not induced by electron bombardment was checked by observing the atomic hopping frequencies as a function of the electron beam intensity. 

---