Trajectory, steepest descent

The simplest chemical reactions correspond to overcoming a single reaction barrier on the way from reactants to products through a saddle point along the intrinsic reaction coordinate (IRC). The IRC corresponds to the steepest descent trajectory (in the mass-weighted coordinates) from the saddle point to configurations of reactants and products. 

Steepest Descent Path The steepest descent trajectory (of lowering the electronic energy as a function of configuration of the nuclei) that connects a first-order saddle point with two adjacent energy minima corresponding to the stable configurations of the reactants and products. 

Away from the transition state, the IRC is formally defined by a steepest descent trajectory in mass-weighted Cartesian coordinates [112],... 

This expression highlights the similarities between the second-order steepest-descent trajectory Eq. (46) and the Levenberg-Marquardt trajectory. In particular, we notice that both trajectories end at the minimum of the second-order surface for t = oo. For small steps, we expand around t = 0 and obtain... 

In the light of the path-integral representation, the density matrix p Q-,Q-,p) may be semi-classically represented as oc exp[ —Si(Q )], where Si(Q ) is the Eucledian action on the -periodic trajectory that starts and ends at the point Q and visits the potential minimum Q = 0 for r = 0. The one-dimensional tunneling rate, in turn, is proportional to exp[ —S2(Q-)], where S2 is the action in the barrier for the closed straight trajectory which goes along the line with constant Q. The integral in (4.32) may be evaluated by the method of steepest descents, which leads to an optimum value of Q- = Q. This amounts to minimization of the total action Si -i- S2 over the positions of the bend point Q. ... 

The functional B[(2(r)] actually depends only on the velocity dQ/dr at the moment when the non-adiabaticity region is crossed. If we take the path integral by the method of steepest descents, considering that the prefactor B[(2(r)] is much more weakly dependent on the realization of the path than Sad[Q(A]> we shall obtain the instanton trajectory for the adiabatic potential V a, then B[(2(t)] will have to be calculated for that trajectory. Since the instanton trajectory crosses the dividing surface twice, we finally have... 

The Rearrangement of 1,2,6-Heptatriene. The experimental facts and basic mechanistic idea behind this reaction were outlined in section 1.2. The molecular dynamics study began with a single CASSCF(8,8)/6-31G(d) trajectory started from TSl (see Fig. 21.2) with no kinetic energy (not even ZPE) in any of the real-frequency normal modes. The purpose of such an unphysical trajectory calculation is to see what is the steepest descent path down from the transition state... 

Trajectories were initiated at either the TS 63 or a random sampling of structures about this TS. Most trajectories reach a product, either 62 or 64 within 30 fs, and all finished within 85 fs. These short trajectories suggest little opportunity for IVR to occur. Most of the trajectories, 72 out of 101, end up at 62, the product that is directly connected to the TS. The other 29 trajectories terminate at 64. Some of these trajectories were allowed to go on for another 100 fs but none exited the area about 64. Given sufficient time, these trajectories would cross over the barrier about 65 and make the final product 62. The upshot is that the diradical is produced even though there is no TS that takes reactant into it. The ridge provides the opportunity for trajectories to fall into two possible energy sinks. Most follow the steepest descent downhill toward 62, but some will divert into the neighboring minimum, the diradical 64. 

Beck and Berry - have performed a similar analysis on all of the clusters studied in the iV-dependence work of Beck et al. Steepest-descent quenches were performed at regular intervals along trajectories at three energies two... 

In this expression, ° is the energy at the bottom of the basin a. Q- are the stationary points satisfying V" (q ) = 0, which have arisen from the steepest descent evaluation of the numerator of Eq. (55). The Hessian V"(Qi) is taken only within the space of the boundary da. Furthermore, only those stationary points are supposed to appear in Eq. (56) whose N — 1 eigenvalues are all positive. Thus, the stationary points taken into account of this estimate are actually the first-rank saddles, and it is assumed that the boundary da can be topologically built up by connecting such first-rank saddles by ridge lines [actually (N — 1)-dimensional manifolds], (Note that these particular roles of the first-rank saddle is just a matter of the steepest decent estimate. Indeed trajectories actually pass many different places, too. See the next subsubsection and Fig. 15.)... 

SDEL Trajectories Approach the Steepest Descent Path... 

After presenting the algorithm, we show that the proposed numerical protocol interpolates (as a function of the approximation quality) from a steepest descent path (a poor approximation to the dynamics) to an exact Newtonian trajectory. Finally, we present an efficient algorithm to compute relative rates that can be used with our formulation to compute experimental observables,... 

IV. SDEL TRAJECTORIES APPROACH THE STEEPEST DESCENT PATH... 

The question we address in the present section is the asymptotic behavior of a trajectory when the step becomes quite large. We are unable to answer that question in a useful way for the SDET algorithm however, an intriguing result is obtained for SDEL. In the present section we will show a connection between SDEL trajectories and the usual definition of the reaction coordinate, the steepest descent path (SDP). 

The R state correlates to product, X-A/ Y (A = L3C), since it contains a c )x-crlF bond-pair that becomes the X-A bond, and at the same time the occupancy of the a y orbital causes the cleavage of the C-Y bond to release the Y anion. Furthermore, the R state contains information about the stereochemical pathway. Since the bond-pair involves a x ay overlap, due to the nodal properties of the a y orbital the bond-pair will be optimized when the X is coupled to the substrate in a collinear X-A-Y fashion. Thus, the steepest descent of the R state, and the lowest crossing point will occur along a backside trajectory of the nucleophile toward the substrate. 

Figure 3. Another expanded view of the PES of Fig. 1, sketching the projection of a classical trajectory onto the PES (solid line) and a similar projection of the path of steepest descent (dashed line).

The algorithm produces an interpolation between the steepest descent path (SDP) and a true classical trajectory. Hence, even trajectories with low-resolution can be useful in reaction path studies. 

We define the center of the tube with the help of (1) a guiding trajectory, (2) a steepest descent path (SDP) calculation, (3) and a flexible choice of collective variables. 

Here, we consider how we can approximately describe the size dependence of the potential energy in Fig. 23.3. As discussed above, molecular coordinates at the imaginary time t = 8/2 are distributed according to the exact ground state wavefunction, I o(/ )p. Each structure at t = /2 along the VPIMD trajectory is mapped onto a nearest local minimum structure in the configuration space. This can be realized by the steepest descent minimization technique ... 

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