Reactivity boundaries to separate the fate of a chemical reaction associated with an index-two saddle

Abstract

Reactivity boundaries that divide the destination and the origin of trajectories are of crucial importance to reveal the mechanism of reactions. We investigate whether such reactivity boundaries can be extracted for higher index saddles in terms of a nonlinear canonical transformation successful for index-one saddles by using a model system with an index-two saddle. It is found that the true reactivity boundaries do not coincide with those extracted by the transformation taking into account a nonlinearity in the region of the saddle even for small perturbations, and the discrepancy is more pronounced for the less repulsive direction of the index-two saddle system. The present result indicates an importance of the global properties of the phase space to identify the reactivity boundaries, relevant to the question of what reactant and product are in phase space, for saddles with index more than one. Saddle points and the dynamics in their vicinities play crucial roles in chemical reactions. A saddle point on a multidimensional potential energy surface is defined as a stationary point at which the Hessian matrix does not have zero eigenvalues and, at least, one of the eigenvalues is negative. Saddle points are classified by the number of the negative eigenvalues, and a saddle that has n negative eigenvalues is called an index-n saddle. An index-one saddle on a potential surface has especially long been considered to make a bottleneck for reactions [1][2][3], the sole unstable direction corresponding to the "reaction coordinate." This is because index-one saddles are considered to be the lowest energy stationary point connecting two potential minima, of which one corresponds to the reactant and the other to the product, and the system must traverse the index-one saddle from the reactant to the product [4][5][6][7].To estimate reaction rate constants across the saddles, transition state theory was proposed [1][2][3], by envisaging the existence of a nonrecrossing dividing surface [i.e., transition state (TS)] in the region of an index-one saddle. Recent studies of nonlinear dynamics in the vicinity of index-one saddles have revealed the firm theoretical ground for the robust existence of the no-return TS in the phase space [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] (see also Refs. [25,26], and references therein). and systems with rovibrational couplings [18,19] and showed the robust existence of reaction boundaries even while a no-return TS ceases to exist [13].For complex molecular systems, the potential energy surface becomes more complicated, and transitions from a potential basin to another involve not only index-one saddles but also higher index saddles [28][29][30]. For example, it was shown in a computer simulation of an inert gas cluster containing seven atoms that transitions from a solid-like phase to a liquid-like phase occur mostly through index-two saddles rather than through index-one saddle with the increase of kinetic temperature [29]. This indicates that the more rugged a ...

Roaming: A Phase Space Perspective

“…In this way, we were able to define a global DS together with the associated mechanism DS. Such global DSs have been studied in the context of bifurcations of NHIMs and associated DSs (61-63; see also 116,117). These concepts have yet to be applied to the case of roaming.…”

Reactivity boundaries for chemical reactions associated with higher-index and multiple saddles

“…When only one saddle point plays a dominant role in determining the occurrence of the reaction, normal form theory readily extracts the seed of reactivity boundaries in an analytical way. In contrast, there is still no practical method applicable to general cases where more than one saddle point are involved in the reaction process [73].…”

“…For the former assumption, Toda [72] addressed that linear resonance between two reactive modes may introduce a breakdown of the reactivity boundary. As for the latter assumption, Nagahata et al [73] reported recently that the reactivity boundary extracted by use of the normal form does not necessarily give the barrier separating the reactivity in the original coordinate space for higher-index saddles.…”

“…In contrast to normal mode approximation or normal form theory locally expanded in the vicinity of a single saddle point, for our purposes, the analytic formula of the underlying reaction coordinate is hard to derive and the invariant manifold locally extracted in the vicinity of a single point or a collection of multiple saddle points with different indices might not necessarily provide the boundary to divide the fates of the reactions originated from the well region far from the saddles [73].…”

Reaction coordinates for elucidating reaction dynamics with anharmonic couplings

“…Actually, it was shown in the study of reaction dynamics through a rank-two saddle point that the reactivity boundary identified as x 1 50 correctly indicates the future sign of q 1 but the sign of q 1 is not identical to the sign of q 1 that corresponds to the correct reactants and products. [17] All these issues show the importance of understanding the reaction dynamics in the global space from the reactants region to the products region. It should be noted here that efforts are continuously being made to increase the convergence radius of the normal form transformation.…”