Embedding of the saddle point of index two on the PES of the ring opening of cyclobutene

Abstract

The ring opening of cyclobutene is characterized by a competition of the two different pathways: a usual pathway over a saddle of index one (SP 1 ) along the conrotatory behavior of the end groups, as well as a "forbidden" pathway over a saddle point of index two (SP 2 ) along the disrotatory behavior of the end CH 2 groups. We use the system of ordinary differential equations for the method of the gentlest ascent dynamics (GAD) to determine saddle points of the potential energy surface (PES) of the ring opening of cyclobutene to cis-butadiene. We apply generalized GAD formulas for the search of a saddle point of index two. To understand the relation of the different regions of the PES (around minimums, around SPs of index one or two) we also calculate valley-ridge inflection (VRI) points on the PES using Newton trajectories (NT). VRIs and the corresponding singular NTs subdivide the regions of "attraction" of the different SPs. We calculate the connections of the SP 2 (in its different symmetry versions) with different SPs of index one of the PES by different "reaction pathways." We compare the possibilities of the tool of the GAD curves for the exploration of PESs with these of NT. The barrier of the disrotatory SP 2 is somewhat higher than the barrier of the conrotatory SP 1 , however, pathways across the slope to the SP 2 open additional reaction valleys. IntroductionMathematics has had a profound impact on science, providing a means to understand the world around us in unprecedented ways.[1] With the advantage of the computer age, the subject of modelization has hugely grown in importance. In particular, over the last decades significant advances in our understanding of the world of chemistry have culminated in the development of a very large set of concepts. Some of these concepts are the potential energy surface (PES) [2][3][4] and the chemical reaction path (RP) [5] being the basic concepts for the theories of chemical dynamics and chemical transformations. The PES is a continuous function of the coordinates of the nuclei over an IR N , thus it is an N-dimensional hypersurface. It should have continuous derivatives of at least second order for our treatments. If Cartesian coordinates are used, then N53 n, however, we may also use N53 n26 nonredundant, internal coordinates for an n-atomic molecule. The PES can be seen as formally divided in catchments associated with local minima.[2] For such a picture, one has in mind steepest descent (SD) curves. The first order saddle points or transition states (TSs) are located at the deepest points of the boundary of the basins. TSs and minima correspond to stationary points of the PES. Two adjacent minima of the PES can be connected through a TS via a continuous curve in the N-dimensional coordinate space, describing the coordinates of the nuclei. The curve characterizes a RP. The concept of the RP has found a widespread use in descriptions of reaction mechanisms, which are based on geometrical rearrangements of the atoms comprising a given chemical system. [6] Mor...

A contribution to a theory of mechanochemical pathways by means of Newton trajectories

“…The search of VRI points by NTs is possible [68,69]. It is especially reported on the PES of alanine dipeptide [70], on the PES of the cyclopropyl radical [71], as well as on the PES associated to the ring opening of the cyclobutene [72].…”

Reaction rates in a theory of mechanochemical pathways

“…Thus, when we study the pulling problem by NTs, we have to include these higher index SPs, compare Ref. .…”

Toward a theory of mechanochemistry: Simple models from the very beginnings

“…[43] Thermal ring-opening is only allowed via a conrotatory pathway, yielding a different isomer of the ring-opened product (E,Z or E,E) for each starting isomer (cis-or trans-, respectively), the disrotatory pathway is forbidden, for a computational study see for example, Ref. [122] Nevertheless, when the reaction is under an external mechanical force, the ring-opening always yields the E,E isomer as a product, irrespective of the starting material. The generic rationalization of this phenomenon is just exposed in section 8.…”