This paper examines a number of aspects of evaluating the reaction path Hamiltonian (RPH) of Miller, Handy, and Adams. The reaction path is represented as a Taylor series expansion of mass weighted Cartesian coordinates as a function of arc length. The second (path tangent) and third (path curvature) coefficients in the Taylor series are important in the RPH. General analytical formulas for all the coefficients as explicit functions of energy derivatives are derived. If the Taylor series is expanded about the saddle point, special limiting formulas for the coefficients are required. These are obtained using L′Hospital’s rule. In a local quadratic approximation (LQA) third and higher energy derivatives are ignored. Within this approximation all but the first two coefficients in the Taylor series expansion of the path are zero when the expansion point is the saddle point. At nonstationary points on the path the first three Taylor series coefficients are evaluated exactly within the LQA while the others have nonzero approximate values. The resulting LQA Taylor series can be summed exactly. This leads to a new method of stepping along the reaction path which is superior to the traditional Euler method and should be used whenever second energy derivatives are available. Extensions of this method which include third energy derivative information are also presented. Exact analytical formulas for the RPH coupling parameters are derived. These include

simplified formulas for the projection matrix and its derivative. At nonstationary points, the couplings of the transverse vibrations to the path depend only on first and second energy derivatives and hence are exactly calculated in the LQA. The remaining RPH parameters depend on third energy derivatives as well but have nonzero approximate values in the LQA. At the saddle point, all of the RPH parameters depend on third energy derivatives and they are zero when third derivatives are ignored. In general, when the complete set of RPH parameters are calculated, the same energy derivative information is required at the saddle point as at nonstationary points, namely the gradient, the force constants, and the components of the third derivatives along the path tangent. It is demonstrated that severe errors can occur when the RPH parameters are calculated at a point near the saddle point lying on the eigenvector corresponding to the negative eigenvalue of the force constant matrix at the saddle point. These errors occur even when the exact formulas are used and are due to slight deviations of this eigenvector from the exact reaction path. A remedy is described.

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This follows by induction.$T1$ is proportional to third derivatives. Therefore, so is $ν0(1)$ by Eq. (17). $T2$ is proportional to $ν0(1)$ and third and higher derivatives. Therefore, so is $ν0(2)$ by Eq. (17), etc.
13.
These devices (Ref. 8) appear to work quite well. One should note, however, the following formal objection to their use. Of all the solutions to Eq. (2) only one terminates at the saddle point. This is the reaction path. It cannot be denned or calculated without referencing the saddle point. When one has left the path one attempts to return to it using only locally available information in which reference to the saddle point has been lost. Therefore, one cannot always expect these devices to work. The problem of determining the path from local information only is discussed in Ref. 14.
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Given third derivative information, one can devise various alternative methods of approximating the reaction path, none of which we have yet investigated. In generalized normal coordinates, for example, the Taylor series including the second and diagonal third derivatives can be summed exactly and the remaining third derivatives included by partial summing to high order.
17.
This, however, will never occur in practice for many‐dimensional problems, since it requires that somewhere between the saddle point and products the path tangent must simultaneously satisfy Eq. (2) and the gradient extremal condition (that the path tangent is an eigenvector of F) (Ref. 14). While a gradient extremal path might cross the reaction path in a two‐dimensional problem it is improbable in many dimensions.
18.
At the saddle point, the rotations, translations, and path tangent are already eigenvectors of F, so that projecting them out is unnecessary. Although K and F share a set of eigenvectors at the saddle point, they are not identical. The negative eigenvalue of F corresponding to the path tangent becomes a zero eigenvalue of K.
19.
This reflects the fact that unless the center of mass is at the origin, a rotation about the origin contains a translational component.
20.
If desired, vectors $vi$ which correspond to translations along and rotations about the principle inertial axes can be obtained from the $6×6$ column eigenvector matrix U and the diagonal matrix of eigenvalues ε of S. The result is $v = bUε−1/2.$
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