Valley-ridge Inflection Points Research Articles

Chiral-at-Metal Racemization Unraveled for MX2 (a-chel)2 by means of a Computational Analysis of MoO2 (acnac)2.

Octahedral chiral-at-metal complexes MX2 (a-chel)2 (a-chel=asymmetric chelate) can rearrange their ligands by four mechanisms known as the Bailar (B), Ray-Dutt (RD), Conte-Hippler (CH), and Dhimba-Muller-Lammertsma (DML) twists. Racemization involves their interconnections, which were computed for MoO2 (acnac)2 (acnac=β-ketoiminate) using density functional theory at ωB97x-D with the 6-31G(d,p) and 6-311G(2d,p) basis sets and LANL2DZ for molybdenum. Racemizing the cis(NN) isomer, being the global energy minimum with trans oriented imine groups, is a three step process (DML-CH-DML) that requires 17.4 kcal/mol, while all three cis isomers (cis(NN), cis(NO), and cis(OO)) interconvert at ≤17.9 kcal/mol. The B and RD twists are energetically not competitive and neither are the trans isomers. The interconnection of all enantiomeric minima and transition structures is summarized in a graph that also visualizes valley ridge inflection points for two of the three CH twists. Geometrical features of the minima and twists are given. Lastly, the influence of N-substitution on the favored racemization pathway is evaluated. The present comprehensive study serves as a template for designing chiral-at-metal MX2 (a-chel)2 catalysts that may retain their chiral integrity.

Symmetric Post-Transition State Bifurcation Reactions with Berry Pseudomagnetic Fields

We investigate how the Berry force (i.e., the pseudomagnetic force operating on nuclei as induced by electronic degeneracy and spin-orbit coupling (SOC)) might modify a post-transition state bifurcation (PTSB) reaction path and affect product selectivity for situations when multiple products share the same transition state. To estimate the magnitude of this effect, Langevin dynamics are performed on a model system with a valley-ridge inflection (VRI) point in the presence of a magnetic field (that mimics the Berry curvature). We also develop an analytic model for such selectivity that depends on key parameters such as the surface topology, the magnitude of the Berry force, and the nuclear friction. Within this dynamical model, static electronic structure calculations (at the level of generalized Hartree-Fock with spin-orbit coupling (GHF+SOC) theory) suggest that electronic spin induced Berry force effects may indeed lead to noticeable changes in methoxy radical isomerization.

The Influence of a Parameter that Controls the Asymmetry of a Potential Energy Surface with an Entrance Channel and Two Potential Wells

In this paper we study an asymmetric valley-ridge inflection point (VRI) potential, whose energy surface (PES) features two sequential index-1 saddles (the upper and the lower), with one saddle having higher energy than the other and two potential wells separated by the lower index-1 saddle. We show how the depth and the flatness of our potential changes as we modify the parameter that controls the asymmetry as well as how the branching ratio (ratio of the trajectories that enter each well) is changing as we modify the same parameter and its correlation with the area of the lobes as they have formulated by the stable and unstable manifolds that have been extracted from the gradient of the LD scalar fields.

The dynamical significance of valley-ridge inflection points

In this paper we demonstrate that valley-ridge inflection (VRI) points of a potential energy surface (PES) have a dynamical influence on the fate of trajectories of the underlying Hamiltonian system. These points have attracted the attention of chemists in the past decades when studying selectivity problems in organic chemical reactions whose energy landscape exhibits a post-transition-state bifurcation in the region between two sequential saddles without an intervening energy minimum. To address the dynamical significance of valley-ridge inflection points, we construct a symmetric potential energy function that allows us to move the location of the VRI point while keeping the locations and energies of the critical points fixed. In this setup, we carry out a parametric study of the dynamical behavior of ensembles of trajectories in terms of the energy of the chemical system and the position of the VRI point. Our analysis reveals that the location of the VRI point controls the fraction of trajectories that recross the high energy saddle region of the PES without entering either of the potential wells that are separated by the low energy saddle.

From Poincaré Maps to Lagrangian Descriptors: The Case of the Valley Ridge Inflection Point Potential

In this paper we compare the method of Lagrangian descriptors with the classical method of Poincare maps for revealing the phase space structure of two degree-of-freedom Hamiltonian systems. The comparison is carried out by considering the dynamics of a two degree-of-freedom system having a valley ridge inflection point (VRI) potential energy surface. VRI potential energy surfaces have four critical points: a high energy saddle and a lower energy saddle separating two wells. In between the two saddle points is a valley ridge inflection point that is the point where the potential energy surface geometry changes from a valley to a ridge. The region between the two saddles forms a reaction channel and the dynamical issue of interest is how trajectories cross the high energy saddle, evolve towards the lower energy saddle, and select a particular well to enter. Lagrangian descriptors and Poincare maps are compared for their ability to determine the phase space structures that govern this dynamical process.

Tuning the branching ratio in a symmetric potential energy surface with a post-transition state bifurcation using external time dependence

Chemical selectivity, as quantified by a branching ratio, is a phenomenon relevant for many organic chemical reactions. It may be exhibited on a potential energy surface (PES) that features a valley-ridge inflection point (VRI) in the region between two sequential index-1 saddles, with one saddle having higher energy than the other. Reaction occurs when a trajectory crosses the region of the higher energy saddle (the “entrance channel”) and approaches the lower energy saddle. On both sides of the lower energy saddle, there are two wells and the question we address is that given an initial ensemble of trajectories, what is the relative fraction of trajectories that enter each well. For a symmetric PES this fraction is 1:1. We consider a symmetric PES subject to a time-periodic forcing characterized by an amplitude, frequency, and phase. In this letter we analyze how the branching ratio depends on these three parameters.

The phase space mechanism for selectivity in a symmetric potential energy surface with a post-transition-state bifurcation

Chemical selectivity is a phenomenon displayed by potential energy surfaces (PES) that is relevant for many organic chemical reactions whose PES feature a valley-ridge inflection point in the region between two sequential index-1 saddles. In this letter we describe the underlying dynamical phase space mechanism that qualitatively determines the product distributions resulting from bifurcating reaction pathways. We show that selectivity is a consequence of the heteroclinic and homoclinic connections established between the invariant manifolds of the families of unstable periodic orbits (UPOs) present in the system. The geometry of the homoclinic and heteroclininc connections is determined using the technique of Lagrangian descriptors, a trajectory-based scalar technique with the capability of unveiling the geometrical template of phase space structures that characterizes transport.

A model for a driven Frenkel–Kontorova chain

We study a Frenkel–Kontorova (FK) model of a finite chain with free-end boundary conditions. The model has two competing potentials. Newton trajectories are an ideal tool to understand the circumstances under a driving of an FK chain by external forces. To reach the insights we calculate some stationary structures for a chain with 23 particles. We search the lowest energy saddle points for a complete minimum energy path of the chain for a movement over the full period of the on-site potential, a sliding. If an additional tilting is set, then one is interested in barrier breakdown points (BBPs) on the potential energy surface for a critical tilting force named the static frictional force. In symmetric cases, such BBPs are often valley-ridge inflection points of the potential energy surface. We explain the theory and demonstrate it with an example. We propose a model for a DC drive, as well as an AC drive, of the chain using special directional vectors of the external force.

Designing Reactions with Post-Transition-State Bifurcations: Asynchronous Nitrene Insertions into C–C σ Bonds

Summary Discoveries of chemical reactions with potential energy surfaces with post-transition-state bifurcations are increasing in frequency. Although such potential energy surface features are often discovered accidentally, we demonstrate here that they can be designed rationally. We describe a process for designing such reactions and apply it to the design of a nitrene addition/alkyl-shift reaction with a post-transition state bifurcation. We examine substituent effects on this reaction and propose a dynamics-based model for selectivity control.

Mechanochemistry on the Müller–Brown surface by Newton trajectories

AbstractChemical processes which suffer the application of mechanical force are theoretically described by effective potential energy surface (PES). We worked out (W. Quapp, J. M. Bofill, Theor. Chem. Acc. 2016, 135, 113) that the changes due to the force for the minimums and for the saddle points can be described by Newton trajectories (NT) of the original PES. If the force is so high that the saddle point disappears into a shoulder then the mechanochemical action is fulfilled: the pulling force breaks down the reaction barrier. The point is named barrier breakdown point. Different families of NTs form corridors on the original PES which describe qualitative different actions of the force. The border regions of such corridors are governed by the valley‐ridge inflection points (VRI) of the surface. Here, we discuss all this on the basis of the well‐known Müller–Brown surface, and we describe a new kind of NT‐corridor.

A Computational Study on the Addition of HONO to Alkynes toward the Synthesis of Isoxazoles; a Bifurcation, Pseudopericyclic Pathways and a Barrierless Reaction on the Potential Energy Surface.

Homopropargyl alcohols react with t-BuONO to form acyloximes which can be oxidatively cyclized to yield ioxazoles. The mechanism for the initial reaction of HONO with alkynes to form acyloximes (e.g., 13c) has been explored at the B3LYP/6-31G(d,p) + ZPVE level of theory. The observed chemoselectivity and regioselectivity are explained via an acid-catalyzed mechanism. Furthermore, the potential energy surface revealed numerous surprising features. The addition of HONO (8) to protonated 1-phenylpropyne (18) is calculated to follow a reaction pathway involving sequential transition states (TS6 and TS8), for which reaction dynamics likely play a role. This reaction pathway can bypass the expected addition product 21 as well as transition state TS8, directly forming the rearranged product 23. Nevertheless, TS8 is key to understanding the potential energy surface; there is a low barrier for the pseudopericylic [1,3]-NO shift, calculated to be only 8.4 kcal/mol above 21. This places TS8 well below TS6, making the valley-ridge inflection point (VRI or bifurcation) and direct formation of 23 possible. The final tautomerization step to the acyloxime can be considered to be a [1,5]-proton shift. However, the rearrangement in the case of 17h to 13c is calculated to be barrierless, arguably because the pathway is pseudopericyclic and exothermic.

Nontotally symmetric trifurcation of an SN 2 reaction pathway.

A new type of reaction pathway which involves a nontotally symmetric trifurcation was found and investigated for a typical SN 2-type reaction, NC(-) + CH3 X → NC-CH3 + X(-) (X = F, Cl). A nontotally symmetric valley-ridge inflection (VRI) point was located along the C3 v reaction path. For X = F, the minimum energy path (MEP) starting from the transition state (TS) leads to a second-order saddle point with C3v symmetry, which connects three product minima of Cs symmetry. For X = Cl, four product minima have been observed, of which three belong to Cs symmetry and one to C3v symmetry. The branching path from the VRI point to the lower symmetry minima was determined by a linear interpolation technique. The branching mechanism is discussed based on the reaction path curvature and net atomic charges, and the possibility of a nonotally symmetric n-furcation is discussed.

A Comprehensive Analysis in Terms of Molecule-Intrinsic Quasi-Atomic Orbitals. IV. Bond Breaking and Bond Forming along the Dissociative Reaction Path of Dioxetane.

The quantitative analysis of molecular density matrices in terms of oriented quasi-atomic orbitals (QUAOs) is shown to yield detailed conceptual insight into the dissociation of dioxetane on the basis of ab initio wave functions. The QUAOs persist and can be followed throughout the reaction path. The kinetic bond orders and the orbital populations of the QUAOs quantitatively reveal the changes of the bonding interactions along the reaction path. At the transition state the OO bond is broken, and the molecule becomes a biradical. After the transition state the reaction path bifurcates. The minimum energy path gently descends from the transition state via a valley-ridge inflection point to a second saddle point, from which two new minimum energy paths lead to two equivalent formaldehyde dimers. The CC bond breaks, and the π-bonds of the formaldehyde fragments form in close vicinity of the second saddle point. The changes of the interactions in this region are elucidated by the analysis of the rearrangements of the QUAOs.

Can we understand the branching of reaction valleys for more than two degrees of freedom?

The model of the chemical reaction path is fundamental in Chemistry. We usually understand it as the pathway along a valley of the potential energy surface (PES). However, often a valley bifurcation occurs. This is controlled by a valley-ridge-inflection point (VRI). Up to now, 2-dimensional (2D) figures of a PES govern our understanding. But for more degrees of freedom, this might be misleading. Here, we explain the matter over a 3D configuration space; the PES is then a 3D hypersurface in an \(R^4\). In this case a visualization is possible. We still can project curves on the PES down into the \(R^3\) of the configuration space. A method for the calculation of Newton trajectories (NT) is applied, because NTs bifurcate at VRI points. The example used is a simple mathematical test function. It ends with a threefold combination manifold of three 1D VRI lines. The intersection of the VRI lines forms a super-VRI point. The corresponding singular NT at the super-VRI point has eight branches.

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

The ring opening of cyclobutene is characterized by a competition of the two different pathways: a usual pathway over a saddle of index one (SP1) along the conrotatory behavior of the end groups, as well as a “forbidden” pathway over a saddle point of index two (SP2) along the disrotatory behavior of the end CH2 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 SP2 (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 SP2 is somewhat higher than the barrier of the conrotatory SP1, however, pathways across the slope to the SP2 open additional reaction valleys. © 2015 Wiley Periodicals, Inc.

Reaction Path Bifurcation in an Electrocyclic Reaction: Ring-Opening of the Cyclopropyl Radical.

Following previous work [J. Chem. Phys. 2013, 139, 154108] on a simple model of a reaction with a post-transition state valley ridge inflection point, we study the chemically important example of the electrocyclic cyclopropyl radical ring-opening reaction using direct dynamics and a reduced dimensional potential energy surface. The overall reaction requires con- or disrotation of the methylenes, but the initial stage of the ring-opening involves substantial internal rotation of only one methylene. The reaction path bifurcation is then associated with the relative sense of rotation of the second methylene. Clear deviations of reactive trajectories from the disrotatory intrinsic reaction coordinate (IRC) for the ring-opening are observed and the dynamical mechanism is discussed. Several features observed in the model system are found to be preserved in the more complex and higher dimensional ring-opening reaction. Most notable is the sensitivity of the reaction mechanism to the shape of the potential manifested as a Newtonian kinetic isotope effect upon deuterium substitution of one of the methylene hydrogens. Dependence of the product yield on frictional dissipation representing external environmental effects is also presented. The dynamics of the post-transition state cyclopropyl radical ring-opening are discussed in detail, and the use of low dimensional models as tools to analyze complicated organic reaction mechanisms is assessed in the context of this reaction.

Quantum chemical study of intramolecular rearrangements of the cyclooctatetraene complexes C8H8X (X = Mg, Si, S)

Intramolecular rearrangements of the complexes of cyclooctatetraene with magnesium, silicon, and sulfur have been studied by DFT quantum chemical calculations using B3LYP functional and 6-311++G(d,p) and 6-311++G(3df,3pd) basis sets. All complexes are stabilized as nonplanar stereochemically nonrigid structures liable to fast intramolecular circular rearrangements. Valley-ridge inflection points have been detected on the potential energy surfaces of the C8H8Mg and C8H8Si systems, and the reaction path starting from a saddle point and terminating at a minimum passes through a point corresponding to a hilltop, twodimensional for the magnesium atom and four-dimensional for the silicon atom.

Peculiar structure of the potential energy surfaces of typical electrocyclic reactions in the areas of the symmetry-forbidden reaction paths

The mechanisms of the electrocyclic ring opening reactions of cyсlobutene to trans-l,3-butadiene and 1,3-cyclohexadiene to 1,3,5-hexatriene were computationally studied with the use of B3LYP/6-311++G(d,p), MP2/6-311G++(3df,2p), CCSD/6-311++G(df,p), CCSD/6-311G(df,p) and CCSD(T)/6-311++G(df,p)//CCSD methods. Special attention was paid to the study of the potential energy surfaces (PESs) of both systems in the domains pertaining to the disrotatory movement of the vicinal methylene groups. No symmetry-forbidden disrotatory reaction channel was found within the PES of the cyclobutene–1,3-butadiene system. Instead of that, disrotatory rotation of two methylene groups of cyсlobutene results in elimination of a molecule of dihydrogen giving rise to the formation of a low-stable π-complex H2·cyсlobutadiene. Similar type π-complex H2·benzene was also located on the 1,3-cyclohexadiene–1,3,5-hexatriene PES along the disrotatory reaction coordinate. The topologies of the PESs of both systems analyzed on the basis of the patterns of the gradient lines are characterized by an existence on both of them of valley-ridge inflection (VRI) points corresponding to the bifurcation of the disrotatory reaction paths leading to the formation of the π-complexes formed upon elimination of a dihydrogen molecule from cyclobutene and 1,3-cyclohexadiene.

Nonstatistical dynamics on potentials exhibiting reaction path bifurcations and valley-ridge inflection points

We study reaction dynamics on a model potential energy surface exhibiting post-transition state bifurcation in the vicinity of a valley ridge inflection (VRI) point. We compute fractional yields of products reached after the VRI region is traversed, both with and without dissipation. It is found that apparently minor variations in the potential lead to significant changes in the reaction dynamics. Moreover, when dissipative effects are incorporated, the product ratio depends in a complicated and highly non-monotonic fashion on the dissipation parameter. Dynamics in the vicinity of the VRI point itself play essentially no role in determining the product ratio, except in the highly dissipative regime.

Depletion of atmospheric ozone by nitrogen dioxide: a bifurcated reaction pathway

Theoretical investigation on the reaction of NO2 with O3 in the context of ozone depletion is reported in this study. The progress of this reaction in single or concrete steps is an open and highly challenging question for both experimentalists and theoreticians. The reaction is studied through DFT methods, M06-2X, BH and HLYP and MPW1K with 6-311++G(2df,2p) basis set and also at UCCSD and CASPT2 methods with 6-31G(d,p) basis set. The single-point energies of the reactive species are calculated at CASSCF, UCCSD(T), UQCISD(T) and MRCI methods with 6-311++G(2df,2p) basis set. The reaction proceeds along an intermediate through a rate-limiting first transition state. This intermediate is followed by a second transition state, and the intrinsic reaction coordinate (IRC) calculations from this transition state show that the reaction path follows a product channel and another transition state. Two sequential transition states are calculated and the potential energy surface bifurcation is observed through a valley-ridge inflection point. The IRC calculations from third transition state lead to second product channel in which NO and O2 are formed. The energy barrier calculated for the formation of NO3 and O2 in the first product channel is 33.96 kcal/mol, and an energy barrier of 34.07 kcal/mol is calculated for the formation of NO + 2O2 product channel. The rate constant calculated at 298 K for the two product channels is 2.31 × 10−7 cm3 molecule−1 s−1. These results show that the formation of the two product channels is equally probable. The photodissociation wavelength of the reactive species involved in the reaction system is calculated using equation of motion coupled cluster (EOMCC) method. The calculated photodissociation wavelength shows that a significant amount of NO2 photofragment enters into the stratosphere and potentially involves in the depletion of ozone. The photodissociation wavelength of the reactive species calculated using EOMCC method is in good agreement with the available experimental data.