Curvature coupling coefficient

Figure 19a. Characterization of the reaction path curvature k(s) (thick solid line) in terms of normal mode-curvature coupling coefficients Bn,s(s) (dashed lines). The curve k(s) has been shifted by 0.5 units to more positive values to facilitate the distinction between k(s) and Bji s(s). For a definition of the internal coordinates, compare with Figure 17. The position of the transition state corresponds to s = 0 amu /2 Bohr and is indicated by a vertical line.

Analysis of the RP curvature k(s) helps to identify those path regions with strong curvature and a coupling between translational and transverse vibrational modes. For this purpose, the curvature is investigated in terms of normal mode-curvature coupling coefficients and adiabatic internal mode-curvature coupling amplitudes At.,. 

The curvature coupling elements, (.v), which represent coefficients of the expansion of the curvature vector in terms of generalized normal modes l (i), are defined by Eq. (94) ... 

Fig. 14.7. A scheme the Coriolis coupling coefficient (B12) and the curvature coefficients (Bij and B2s) related to the normal modes 1 and 2 and reaction coordinate Diagonalization of the two Hessians calculated at points = and s = S2 gives two corresponding normal mode eigenvectors L (si) and L2 ( si) as well as Li ( 2) and La 2)- At points si and S2, we also calculate the versors w (si) and w (si) that are tangent to the IRC (curved linej. Hie calculated vectors inserted into the formulas give the approximations to Bij, B2j and B12-...

The in-1 vibrational frequencies, C0 (s), are obtained from normal-mode analyses at points along the reaction path via diagonalization of a projected force constant matrix that removes the translational, rotational, and reaction coordinate motions. The B coefficients are defined in terms of the normal mode coefficients, with those in the denominator of the last term determining the reaction path curvature, while those in the numerator are related to the non-adiabatic coupling of different vibrational states. A generalization to non-zero total angular momentum is available [59]. 

Here the A terms describe curvature distortions of the smectic planes, the B terms the distortions of the director when the smectic planes are unperturbed, and the C terms the coupling between these two types of distortions. All the coefficients are approximately of the same order of magnitude as the nematic elastic constants. A term of the type B(du/dzY may also be included to allow for the compression of the layers, but we shall neglect it in the present discussion. 

The curvature of the drainpipe is hidden in constants their definition wiU be given later in this chapter. Coefficient Bkie(s) teUs us how normal modes k and k are eoupled together, while Bksis) is responsible for a similar coupling between reaction path jcirc( )... 

Eor ferrocene sites at the end of long alkanethiols self-organized at gold electrodes and diluted with unsubstituted thiols with the redox moiety in contact with the electrolyte (Fig. 4a), Chidsey has reported [34] curved Tafel plots (Fig. 4b), which could be fitted by equations derived from Marcus theory with values of k = 0.85 eV and Z = 6.73 x 10 s"l eV" for a reaction rate of A = 2.5 s at in Fig. 4(b). Similar curvature in Tafel plots has been reported by Faulkner and coworkers [35] for adsorbed osmium complexes at ultramicro-electrodes (UME). The temperature dependence of the rate coefficient could also be fitted from Marcus equation and electron states in the metal and coupling factors given by quantum mechanics. 

As stated by Remark 6.4, the wall thickness of the considered beams is small in comparison to the cross-sectional dimensions, being reflected in r (s) and rs(s), and to the radius of curvature R s). These geometric orders of magnitude also enter the constitutive description of the laminated beam wall in Eq. (6.4b). The plate stiffness coefficients 613(5) and 633(5) and coupling stiffnesses 831(5) and 633(5) essentially depend upon the difference of cubed, respectively squared, laminae positions in the thickness direction, while the membrane stiffness kss s) is a function of the laminae thicknesses. To comply with Remark 6.4, it is necessary to revise the formulation of the warping... 

The remaining beam stiffness coefficients depend on the shell stiffnesses that result from coupling on the laminae level. This concerns the coupling between shear and extension, Ai3(s), between extension and lengthwise curvature, 13(5) and 631(5), as well as between lengthwise curvature and twist, 613(5). So, the beam stiffness coefficients responsible for the coupling of extension with shear and torsion read ... 

The vectors a, b and c are defined with respect to a SmC layer, as shown in Fig. 10(a). The coefficients A describe curvature of the planes, B nonuniform rotations of the optic axis about z in a flat structure, and the C coefficients are coupling terms between A and B, and are shown schematically in Fig. 10(b). All of the temperature dependences of these coefficients may be included in the tilt angle. 

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