Saddle point coordinates

It should be noted that in the cases where y"j[,q ) > 0, the centroid variable becomes irrelevant to the quantum activated dynamics as defined by (A3.8.Id) and the instanton approach [37] to evaluate based on the steepest descent approximation to the path integral becomes the approach one may take. Alternatively, one may seek a more generalized saddle point coordinate about which to evaluate A3.8.14. This approach has also been used to provide a unified solution for the thennal rate constant in systems influenced by non-adiabatic effects, i.e. to bridge the adiabatic and non-adiabatic (Golden Rule) limits of such reactions. 

This formula, however, tacitly supposes that the instanton period depends monotonically on its amplitude so that the zero-amplitude vibrations in the upside-down barrier possess the smallest possible period 2nla>. This is obvious for sufficiently nonpathological one-dimensional potentials, but in two dimensions this is not necessarily the case. Benderskii et al. [1993] have found that there are certain cases of strongly bent two-dimensional PES when the instanton period has a minimum at a finite amplitude. Therefore, the cross-over temperature, formally defined as the lowest temperature at which the instanton still exists, turns out to be higher than that predicted by (4.7). At 7 > Tc the trivial solution Q= Q Q is the saddle-point coordinate) emerges instead of instanton, the action equals S = pV (where F " is the barrier height at the saddle point) and the Arrhenius dependence k oc exp( — F ") holds. 

For convenience of notation we accept from here on, that each frequency of the problem co has a dimensionless counterpart denoted by a capital Greek letter, so that co,- = coofl,. The model (4.28) may be thought of as a particle in a one-dimensional cubic parabola potential coupled to the q vibration. The saddle-point coordinates, defined by dVjdQ = dVjdq = 0, are... 

Fig. 5. The ground state energies of a Z 30 hydrogen-like atom in the Gaussian basis obtained from the Dirac (D) andfrom the L y-Leblond (L) equations as functions of a and. The saddle point coordinates equal to (282,168) in the Dirac case and (271,164) in the Levy-Leblondcase. The cross-sections of...

Fig. 4. The same as in figures 1 and 3, but the values ofL and S are not equal to the ones of the exact solutions. In the case of the Dirac equation L = S = and in the case of the Levy-Leblond equation L = S y have been taken. The saddle point coordinates (Uq, Po are equal to (114,99) in the Dirac case and (66,84) in the Levy-Leblond case. The cross-sections of the energy surface by the planes a = P (broad solid line), and a = a q (broken line) are plotted versus P, while the cross-sections by theplane P = Pq (broken line with dots) and by the surface P = Pmax( 0 (thin solid line) areplotted versus a, in D2 (Dirac) and in L2 (Levy-Leblond). The scale o/P is shown in the horizontal axes. The scale of a has been chosen so that the curve for which P = Po and the one for which a = ttg match at the saddle point. The scale of a may be obtained by adding ag - Pg to the values o/P displayed in the axes of D2 and L2.

The barrier on the surface in figure A3,7,1 is actually a saddle point the potential is a maximum along the reaction coordinate but a minimum along the direction perpendicular to the reaction coordinate. The classical transition state is defined by a slice tlirough the top of tire barrier perpendicular to the reaction coordinate. 

Techniques have been developed within the CASSCF method to characterize the critical points on the excited-state PES. Analytic first and second derivatives mean that minima and saddle points can be located using traditional energy optimization procedures. More importantly, intersections can also be located using constrained minimization [42,43]. Of particular interest for the mechanism of a reaction is the minimum energy path (MEP), defined as the line followed by a classical particle with zero kinetic energy [44-46]. Such paths can be calculated using intrinsic reaction coordinate (IRC) techniques... 

Using the coordinates of special geometries, minima, and saddle points, together with the nearby values of potential energy, you can calculate spectroscopic properties and macroscopic therm ody-riatriic and kinetic parameters, sncfi as enthalpies, entropies, and thermal rate constants. HyperChem can provide the geometries and energy values for many of these ealeulatiori s. 

The reaction coordinate is one specific path along the complete potential energy surface associated with the nuclear positions. It is possible to do a series of calculations representing a grid of points on the potential energy surface. The saddle point can then be found by inspection or more accurately by using mathematical techniques to interpolate between the grid points. 

HyperChem can calculate transition structures with either semi-empirical quantum mechanics methods or the ab initio quantum mechanics method. A transition state search finds the maximum energy along a reaction coordinate on a potential energy surface. It locates the first-order saddle point that is, the structure with only one imaginary frequency, having one negative eigenvalue. 

Fig. 17. Contour plots for a

Although intrinsic reaction coordinates like minima, maxima, and saddle points comprise geometrical or mathematical features of energy surfaces, considerable care must be exercised not to attribute chemical or physical significance to them. Real molecules have more than infinitesimal kinetic energy, and will not follow the intrinsic reaction path. Nevertheless, the intrinsic reaction coordinate provides a convenient description of the progress of a reaction, and also plays a central role in the calculation of reaction rates by variational state theory and reaction path Hamiltonians. 

An intrinsic reaction coordinate (IRC) is concerned with travel along the reaction path it can be defined by the path taken by a classical particle sliding from a saddle point down to a minimum. 

---