Saddle second-order

Orbital-based methods can be used to compute transition structures. When a negative frequency is computed, it indicates that the geometry of the molecule corresponds to a maximum of potential energy with respect to the positions of the nuclei. The transition state of a reaction is characterized by having one negative frequency. Structures with two negative frequencies are called second-order saddle points. These structures have little relevance to chemistry since it is extremely unlikely that the molecule will be found with that structure. 

Or, more precisely, a firsl-order saddle paint, where the order indicates the number of dimensions in which the saddle point is a maximum. A second-order saddle point would be a maximum in two dimensions and a minimum in all others. Transition structures are first-order saddle points. 

A more balanced description requires MCSCF based methods where the orbitals are optimized for each particular state, or optimized for a suitable average of the desired states (state averaged MCSCF). It should be noted that such excited state MCSCF solutions correspond to saddle points in the parameter space for the wave function, and second-order optimization techniques are therefore almost mandatory. In order to obtain accurate excitation energies it is normally necessarily to also include dynamical Correlation, for example by using the CASPT2 method. 

Saddle point. 170 Salt effects. 206-214 Scavenging (see Reactions, trapping) Second-order kinetics. 18-22, 24 in one component, 18-19 in two components (mixed), 19-22 Selectivity. 112 Sensitivity analysis. 118 Sensitivity factor, 239-240 Sequential reactions (see Consecutive reactions)... 

One of the fundamental assumptions of TST states that the reaction rate is determined by the dynamics in a small neighborhood of the saddle point, which we place at q = 0. It is then a reasonable approximation to expand the potential U(q) in a Taylor series around the saddle and retain only the lowest-order terms. Because VqU(q = 0) =0 at the saddle point itself, these are of second order and lead to the Hamiltonian... 

As before, we make the fundamental assumption of TST that the reaction is determined by the dynamics in a small neighborhood of the saddle, and we accordingly expand the Hamiltonian around the saddle point to lowest order. For the system Hamiltonian, we obtain the second-order Hamiltonian of Eq. (2), which takes the form of Eq. (7) in the complexified normal-mode coordinates, Eq. (6). In the external Hamiltonian, we can disregard terms that are independent of p and q because they have no influence on the dynamics. The leading time-dependent terms will then be of the first order. Using complexified coordinates, we obtain the approximate Hamiltonian... 

As explained above, the QM/MM-FE method requires the calculation of the MEP. The MEP for a potential energy surface is the steepest descent path that connects a first order saddle point (transition state) with two minima (reactant and product). Several methods have been recently adapted by our lab to calculate MEPs in enzymes. These methods include coordinate driving (CD) [13,19], nudged elastic band (NEB) [20-25], a second order parallel path optimizer method [25, 26], a procedure that combines these last two methods in order to improve computational efficiency [27],... 

G and Huzinaga basis sets used for K, Rb and Cs, 6-31+G and 6-31+G on C used for C3H5. 6-31+G. Number of imaginary frequencies is given in parentheses (1) a transition state (2) a second-order saddle point. 

The Kuhn-Tucker necessary conditions are satisfied at any local minimum or maximum and at saddle points. If (x, A, u ) is a Kuhn-Tucker point for the problem (8.25)-(8.26), and the second-order sufficiency conditions are satisfied at that point, optimality is guaranteed. The second order optimality conditions involve the matrix of second partial derivatives with respect to x (the Hessian matrix of the... 

The integrals in the numerator and denominator can be estimated by using the saddle point method again. By expanding g (x) around the maximum contribution at x , we get, up to second order. 

An equivalent linear structure of CoP-CO was found to be a second-order saddle point (N-x = 2) when the 6-31G(d) basis set... 

The MEP for inversion corresponds to 6 = 0 and is characterized by the barrier height VWhen C/2V, > 1, apart from this MEP, there is a path that includes two segments described by Eq. (8.42) and a second-order saddle point. The barrier along this path is greater than V, and equal to U,(l + 2V0/C). The transverse frequency along the straight-line MEP for inversion has a minimum at the saddle point q = 0, 0 = 0 consequently, the vibrationally adiabatic barrier is lower than the static one. 

When the response surface has an extreme, then all coefficients of a canonic equation have the same signs and the center of the figure is close to the center of experiment. A saddle-type surface has a canonic equation where all coefficients have different signs. In a crest-type surface some canonic equation coefficients are insignificant and the center of the figure is far away from the center of experiment. To obtain a surface approximated by a second-order model for two factors, it is possible to get four kinds of contour curves-graphs of constant values ... 

Why are chemists but rarely interested in finding and characterizing second-order and higher saddle points (hilltops) ... 

What kind(s) of stationary points do you think a second-order saddle point connects ... 

---