Saddle energy

A, B and C being the Coulomb energy functions, a, 0, and y the resonance energy functions over the distances XY, YZ and XZ. Hence, if these functions are known (a usual approximation is to simply subtract 10 % of the total energies for the Coulomb interaction), the saddle energy may be calculated for certain distances as being the activation energy of the reaction... 

At energies slightly above the saddle energy, there exists a single unstable classical periodic orbit. This periodic orbit corresponds in general to symmetric stretching motion (or an equivalent mode in XYZ-type molecules). The Lyapunov exponent of this periodic orbit tends to the one of the equilibrium point as the threshold energy is reached from above. 

Just above the saddle energy, the quantization can be performed by the usual perturbation theory applied to scattering systems as described by Miller and Seideman [24], This equilibrium point quantization uses Dunham expansions of the form (2.8) with imaginary coefficients. This method is valid for relatively low-lying resonances above the saddle, up to the point where anhar-monicities become so important that the Dunham expansion is no longer applicable (see the discussion in Section II.B). 

The resonances are then obtained by searching for the complex zeros of the zeta functions (4.12) in the complex surface of the energy. Assuming that the action is approximately linear, S(E,J) = T(E - Ei), while the stability eigenvalues are approximately constant near the saddle energy E, the quantization condition (4.12) gives the resonances [10]... 

This shows, e.g., that the flux scales with E for energies close to the saddle energy. The key advantage of the normal form coordinates resulting from the normal form algorithm is that it allows one to include the nonlinear corrections to (22) to any desired order. 

The types of critical points can be labelled by the number of less than zero. Specifically, the critical points are labelled by M. where is the number of which are negative i.e. a local minimum critical point would be labelled by Mq, a local maximum by and the saddle points by (M, M2). Each critical point has a characteristic line shape. For example, the critical point has a joint density of state which behaves as = constant x — ttiiifor co > coq and zero otherwise, where coq corresponds to thcAfQ critical point energy. At... 

Figure Al.6.26. Stereoscopic view of ground- and excited-state potential energy surfaces for a model collinear ABC system with the masses of HHD. The ground-state surface has a minimum, corresponding to the stable ABC molecule. This minimum is separated by saddle points from two distmct exit chaimels, one leading to AB + C the other to A + BC. The object is to use optical excitation and stimulated emission between the two surfaces to steer the wavepacket selectively out of one of the exit chaimels (reprinted from [54]).

Ionova I V and Carter E A 1993 Ridge method for finding saddle points on potential energy surfaces J. Chem. Phys. 98 6377... 

Muller K and Brown L D 1979 Location of saddle points and minimum energy paths by a constrained simplex optimization procedure Theor. Chim. Acta 53 75... 

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... 

Fig. 5. The left hand side figure shows a contour plot of the potential energy landscape due to V4 with equipotential lines of the energies E = 1.5, 2, 3 (solid lines) and E = 7,8,12 (dashed lines). There are minima at the four points ( 1, 1) (named A to D), a local maximum at (0, 0), and saddle-points in between the minima. The right hand figure illustrates a solution of the corresponding Hamiltonian system with total energy E = 4.5 (positions qi and qs versus time t).

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. 

To fin d a first order saddle poiri t (i.e., a trail sition structure), a m ax-imiim must be found in on e (and on/y on e) direction and minima in all other directions, with the Hessian (the matrix of second energy derivatives with respect to the geometrical parameters) bein g varied. So, a tran sition structu re is ch aracterized by th e poin t wh ere all th e first derivatives of en ergy with respect to variation of geometrical parameters are zero (as for geometry optimization) and the second derivative matrix, the Hessian, has one and only one negative eigenvalue. 

The lowest-energy path from one minimum to another passes through a saddle point. 

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