Curvature vector

Another entity that we shall need belongs to the realm of intrinsic geometry geodesic curvature. Consider a surface x, a point P on x and a curve on x passing through P. The curvature vector of at P joins P to the centre of curvature of This curvature vector may be decomposed into mutually orthogonal components. These components are given by projection of the... 

Figure 27.8 Potential energy contours for two harmonic vibrational modes, which are orthogonal to the reaction coordinate, for the Cl + CH, reaction at s = -0.49 Oo on the reaction coordinate. The straight line is the direction u, of the reaction-path curvature vector and the symbols are turning points for zero-point harmonic motion along g, (square), gj (triangle), and u, (circle).

In Eq. (54), and are known as the tangent and curvature vectors. The tangent vector is given by... 

Gonzalez and Schlegel [209] have also developed a series of third- and fourth-order methods. All of their higher order approaches use implicit integrators and are extensions of the GS2 algorithm. One of these, a fourth-order method, uses the tangent and curvature vectors at the initial and final points of each step. 

MEP can cause large errors in the tangent and curvature vectors. 

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) ... 

Once generalized adiabatic modes anR(s) have been defined, the normal modes and curvature vector can be analyzed utilizing the CNM approach of Section 7 [20,21]. For this purpose, the amplitude An,s is defined [22]... 

In the small-curvature tunneling approximation, k(T) requires, in addition to some of the information detailed above, the curvature components Cm(,s) of the curvature of the reaction path, where each curvature component measures the projection of the curvature vector on a particular generalized normal mode direction m. Calculation of Kl-CT(r) or kPOMT(7 requires, in addition, values of the Bom-Oppenheimer potential V in the reaction swath, typically at points where it cannot be computed from the available harmonic expansion around the MEP. 

Expressions for the curvature vector and curvature of a plane curve corresponding to different representations are ... 

The curvature vector indicates how sharply the reaction path bends, and points toward the inside of the bend. The magnitude of the curvature, k, is equal to the inverse of the radius of curvature, /c = i) = 1/, i.e., large curvature corresponds to a tight bend with a small radius. At the transition structure (TS), both the numerator and the denominator of equation (3) are zero however, the curvature can be found by applying rHospital s rule ... 

The LQA, CLQA, and GS methods yield the exact tangent and curvature vectors along the path in the limit of infinitesimal step size, whereas the first-order methods reproduce only the tangent. The GS and CLQA methods also give the correct curvature vector at the transition structure, but the LQA method does not. 

The first two coefficients in expansion (16) correspond to the normalized reaction path vector (path tangent) t(s) and the reaction path curvature vector k(s) (see Figure 3) ... 

Figure 3 Schematic representation of reaction path vector t(.y) and curvature vector k(s) at a point y of a curved reaction path Xs(s)...

With the help of the generalized adiabatic modes a (s), both normal modes l is) and curvature vector k(s) can be analy.sed utilizing appropriately defined amplitudes A. (/ s) and Ak.Ak,s) ... 

The dynamic RPH is related to the RSH (Section 2.7.3) in so far as a reaction surface is used in the form of a flexible reaction plane defined by the reaction path vector t s) and the curvature vector (j) (see Figure 6). Since the curvature vector orients itself at each new path point in configuration space, the reaction plane always adjusts to the direction of strongest RP curvature. Accordingly, use of the reaction plane does not require a preselection of LAM coordinates and the calculation of a minimum energy reaction surface. In addition, the rotation of the reaction plane provides a measure for determining the sequence of interactions between translational motion and transverse LAMs. All LAMs and all deviations from the RP because of sharp curvature ate coveted without increasing the dimensions of the reaction surface. Of course, in those... 

Figure 6 Schematic iiiustration of a RP with two reaction planes defined by reaction path vector and curvature vector at path positions I l and. s 2...

---