Curvature coupling elements

The curvature of the reaction path at a nonstationary point can be calculated from only the gradient and the force constants via Eq (8). Then from Eq (13), the curvature coupling elements are given by... 

The curvature coupling elements are thus simply off-diagonal matrix elements of the unprojected force constant matrix in the basis of eigenvectors of the projected force constant matrix. The classical notion that a trajectory will overshoot the path and climb the wall if the path curves on the way down the hill is a reflection of this curvature coupling. Climbing the wall in a transverse direction is tantamount to exchanging energy between the reaction path and the transverse vibration. 

To describe energy transfer along the reaction path, curvature vector K(s), curvature coupling elements B, s(s) arid mode-mode coupling elements B, v(s) have to be calculated [53,54], of which only the former will be discussed here. The... 

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) ... 

The SCP-IOS approximation has also been used to describe the effects of the curvature coupling elements on tunneling probabilities in chemical reactions. For example, the probability of tunneling through a simple barrier is given within the SCP-IOS model by... 

The structures of the reactants, the transition states, and the products are optimized and the intrinsic reaction coordinates (IRC) connecting them are traced by the analytical gradient method at the 6-31G /RHF (for Rl) and MP2 (M011er-Plesset) (for R2) levels. The vibrational frequency analyses [19] along IRC are carried out to obtain the curvature of IRC and the coupling elements among the vibrational modes perpendicular to IRC. 

A qualitatively appealing feature of the RPH is that the kinematic coupling between motion along the reaction coordinate and the vibrational modes is made explicit in the matrix elements. From the early work of Polanyi and co-workers on collinear triatomic reactions, we know that from the position of the saddle point relative to the corner in the reaction path, qualitative conclusions can be drawn about the effect of the energy distribution in the reactants on the rate and about the distribution of energy in the products [20, 22]. The Bj appear to contain quite analogous and quite detailed information about curvature effects for polyatomic systems, identifying which modes should couple to (and... 

A third type of fixed temperature element uses bimetallic strips or disks. The two metals forming the strip or disk each expand at different rates with temperature increases. This causes the strip to bend or the disk to snap to a different curvature. The movement couples to electrical circuits. 

Thus, the fifth term in eq. (5.9) arises from the expectation value of the linear term of the intermolecular potential function (which is assumed to be expanded up to second order in the oscillator coordinates) and is referred to as the V-T (vibration-translation) coupling term. The V-T term mainly depends on the linear force fk, the first term of which contains the coupling term BkF which couples the reaction path to the perpendicular vibrational mode A , and a matrix element hk s) of the second quantized creation and anhilation operators. BkF is a measure of the curvature of the reaction path and determines the amount of coupling to a given vibrational degree of freedom. 

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