Valley-ridge Inflection Research Articles

Analysis of the Valley-Ridge inflection points through the partitioning technique of the Hessian eigenvalue equation

The Valley-Ridge inflection (VRI) points are related to the branching of a reaction valley or reaction channel. These points are a special class of points of the potential energy surface (PES). They are also special points of the Valley-Ridge borderline of the PES. The nature of the VRI points and their differences with respect to the other points of the Valley-Ridge borderline is analyzed using the Lowdin’s partitioning technique applied to the eigenvalue equation of the Hessian matrix. Eigenvalues and eigenvectors of the Hessian are better imaginable than the former used adjoint matrix.

Search of manifolds of nonsymmetric Valley-Ridge inflection points on the potential energy surface of HCN

The potential energy surface (PES) of a chemical reaction contains Valley-Ridge inflection points (VRI) if—an often occurring phenomenon—the reaction path branches. In this paper, we introduce a new direct search method to detect these VRI points. It is based on an inductive execution of Gauss–Newton steps. For the first time, we were able to find not only singular nonsymmetric VRI points, but also whole VRI manifolds on the PES of HCN.

Trifurcation of the reaction pathway

A concept of trifurcation of a reaction pathway is introduced to analyze the case where a downhill path from the first-order saddle point accompanies three branches via the valley–ridge inflection region, leading to three different product minima on the potential energy surface. We provide a detailed analysis on the reaction path for an electron transfer reaction, HCHO−+CH3Cl→OH2C–CH3…Cl−, as an illustrative example of the trifurcating reaction path.

Search for conical intersection points (CI) by Newton trajectories

Recently, valley–ridge inflection points on the potential energy surface for the ring opening of the cyclopropyl radical have been determined using Newton trajectories (NTs) [W. Quapp, J.M. Bofill, A. Aguilar-Mogas, Theor. Chem. Acc. 129 (2011) 803]. This Letter is the report about the utilization of NTs for the search for conical intersection (CI) points. These points play a main role in the understanding of intersections of different electronic surfaces which open the door for photochemical reactions. We explain the reason why Newton trajectories can find CI points, and report a CI seam on the CASSCF (3,3) surface of the allyl radical ring closure.

Topography of cyclopropyl radical ring opening to allyl radical on the CASSCF(3,3) surface: valley-ridge inflection points by Newton trajectories

Valley-ridge inflection points (VRI) on the potential energy surface for the ring opening of the cyclopropyl radical to the allyl radical are determined using the tool of Newton trajectories (NTs) (Quapp and Schmidt in Theor Chem Acc 128:47, 2011). The UHF surface is treated in a former paper (Quapp et al. in Theor Chem Acc 129:803, 2011). This paper is the extension to the more expensive CASSCF(3,3) surface. We compare the results on the UHF surface with the more appropriate calculation: there are quantitative as well as qualitative changes, of course. But many fundamental relations are the same on both surfaces. However, we could detect new pathways on the CASSCF(3,3) surface which highlight the bifurcation problem of this radical. VRIs play a role in the understanding of bifurcating reactions. The region where the bifurcation takes place is governed by a VRI point. Because the transition state of the ring opening is not symmetric, the steepest descent (SD) from the transition state is not along a symmetry axis either, and in this case the SD usually fails a downhill VRI point. The SD from the transition state of the ring opening goes to the disrotatory minimum of the allyl radical. In contrast, we find some pathways which end at the conrotatory minimum, and which go along so-called non steepest descent paths, at least in parts. The region of interest (around the SP of the ring opening) is crossed by electronic intersection seams. Conical intersection points on the seam can be detected by NTs. We use the possibility to explore parts of the intersection seam of the lower CAS surface and we determine connected VRI points being the corner stones of the possible ring opening channels in the disrotatory and the conrotatory case.

The Variational Structure of Gradient Extremals.

The gradient extremals can be taken as a representation of reaction paths. We prove that this type of curve possesses a variational nature. We report the conditions such that these curves have the character of a minimal curve. Finally we discuss the relations between the points of these curves being turning points with respect to other special points of the potential energy surface, like the valley-ridge inflection points.

A significant role of the totally symmetric valley-ridge inflection point in the bifurcating reaction pathway

Appearance of the valley-ridge inflection (VRI) point on the intrinsic reaction path (IRP) introduces geometrical instability of the reaction coordinate and sometimes leads to two different product minima on the potential energy surface (PES). A significant role of the totally symmetric VRI point on the IRP is discussed from the viewpoint of branching of the reaction pathway. As illustrative examples, ab initio calculations were performed to determine the IRP for XCHO− + CH3Cl (X = H, CH3) at the Møller–Plesset second-order perturbation theory (MP2) level with 6-31+G(d) basis sets and geometric features of the PES around the IRP have been analyzed.

Exploration of cyclopropyl radical ring opening to allyl radical by Newton trajectories: importance of valley-ridge inflection points to understand the topography

Valley-ridge inflection points (VRI) on the potential energy surface for the ring opening of the cyclopropyl radical to allyl radical are determined using the tool of Newton trajectories (Quapp and Schmidt in Theor Chem Acc 128:47, 2011). VRIs play a role in the understanding of bifurcating reactions. The region where the bifurcation takes place is usually governed by a VRI point. For this important ring opening, the knowledge of the VRI point after the transition state was demanded some years ago (Mann and Hase in J Am Chem Soc 124:3208, 2002). Because the transition state is not symmetric, also the steepest descent from the transition state is not along a symmetry axis, and in such cases the steepest descent can fail the VRI point. That is the case here, indeed, though the pathway of the steepest descent goes near to a VRI point downhill. However, an electronic intersection seam disturbs the relations. The exploration of the notorious curvilinear potential energy surface of this ring opening has delivered some further VRI points, which are reported. They give a frame for possible ring-opening channels including the conrotatory case.

Do we fully understand what controls chemical selectivity?

Reaction rates and product selectivity of kinetically controlled reactions are not always sufficiently described by standard RRKM or TST theory. Reactions taking place on potential energy surfaces featuring a valley ridge inflection point belong to this class of reactions. Though various research groups could show that reaction path bifurcations are far from being an exception in organic reactions the underlying principles that govern product distributions of those bifurcating reaction pathways are yet not fully understood. This Perspective has the intention to provide an overview of how far our understanding and the development of the theoretical foundation have progressed.

An empirical, variational method of approach to unsymmetric valley-ridge inflection points

Valley-ridge inflection points (VRIs) emerge on a potential energy surface of a chemical reaction if the reaction pathway bifurcates. The valley of the reaction path branches into two valleys, and a ridge in between. It can happen in uphill or in downhill direction. Newton trajectories (NT) are curves for the description of the reaction path. They are curves where at every point the gradient of the potential energy surface points into the same direction. Singular Newton trajectories are a special case: they bifurcate at VRI points. To find a singular Newton trajectory is quasi equivalent with the determination of the corresponding VRI point where this NT bifurcates. Often the bifurcation of the reaction path is governed by a symmetry of the problem. Then the symmetry axis is usually the first branch of the singular NT, and so its determination is easy. In case of an unsymmetric branching, however, such a guiding line is missing. We name the place of such a bifurcation a skew VRI. We propose a variational calculation of the singular NT through the VRI of interest by an empirical, iterative method. Before, the variational theory of possible reaction pathways is developed and applied to the intrinsic reaction coordinate (IRC), as well as to NTs. We have to employ the theory of NTs with its many facets, we use especially the Branin equation. The developed method is applied to the calculation of VRI points on the potential energy surface of HCN and to a VRI point of alanine dipeptide being adjacent to the C5 minimum.

Quantum Chemical Study of Trimolecular Reaction Mechanism between Nitric Oxide and Oxygen in the Gas Phase

Singlet and triplet potential energy surfaces of the reaction between molecular oxygen and two nitric oxide(II) molecules were studied by quantum chemical methods (coupled cluster, CASSCF, and density functional theory: B3LYP, TPSS, VSXC, BP86, PBE, B2-PLYP, B2K-PLYP). Elementary steps involving various N2O4 isomers (cyclic, cis-cis-, cis-trans-, trans-trans-ONOONO, cis- and trans-ONONO2, O2NNO2) were considered, as well as weakly bound molecular clusters preceding formation of O2NNO2, and Coupe-type quasi-aromatic hexagonal ring intermediate NO2.O2N. We found that activation energy strongly depends on the conformation of ONOONO peroxide, which is formed barrierlessly. The best agreements with experimental values were achieved by the B3LYP functional with aug-pc3 basis set. The lowest transition state (TS) energies correspond to the following reaction channel: 2NO + O2 (0 kJ/mol) --> cis-cis-ONOONO (-45 kJ/mol) --> TS1 --> NO2.O2N (-90 kJ/mol) --> TS2 --> cis-ONONO2 (-133 kJ/mol)--> TS3 --> trans-ONONO2 (-144 kJ/mol) --> TS4 --> O2NNO2 (-193 kJ/mol). A valley ridge inflection (VRI) point is located on the minimum energy path (MEP) connecting NO2.O2N and cis-ONONO2. The energy landscape between NO2.O2N and CC-TS2 can be classified as a downhill valley-pitchfork VRI bifurcation according to a recent classification of bifurcation events [Quapp, W. J. Mol. Struct. 2004, 95, 695-696]. The first and second transition states correspond to barrier heights of 10.6 and 37.0 kJ/mol, respectively. These values lead to the negative temperature dependence of the rate constant. The apparent activation enthalpy of the overall reaction was calculated to be Delta(r)H(0) = -4.5 kJ/mol, in perfect agreement with the experimental value.

Multidimensional Exploration of Valley−Ridge Inflection Points on Potential-Energy Surfaces

The ene reaction between singlet oxygen ((1)O(2)) and simple alkenes has been reported as the first experimentally supported example of a potential-energy surface (PES) featuring a valley-ridge inflection that contributes to the selectivity of product formation (J. Am. Chem. Soc. 2003, 125, 1319-1328). That PES, based on gas-phase ab initio calculations and experimental kinetic isotope effects, has shaped the current (1)O(2) ene mechanism by advocating a concerted "two-step no-intermediate" mechanism. Our current investigation of the ene reaction between (1)O(2) and tetramethylethylene in water, DMSO, and cyclohexane using novel 3-dimensional potentials of mean force (3-D PMF) calculations coupled to QM/MM simulations predicts an alternative free-energy surface that supports a traditional stepwise mechanism interpretation featuring a symmetric charge-separated perepoxide intermediate. Solvent polarity dictates the stability of the intermediate and controls the activation barrier for ene product formation. Transformation of the higher order condensed-phase 3-D PMF potential-energy surface, computed following three simultaneous reaction coordinates, into a downgraded 2-D surface results in the "two-step no-intermediate" mechanism. CCSD(T) and MP4(SDQ) free-energy maps in DMSO, constructed from a 3-D grid of B3LYP geometries using the CPCM solvent model, reproduce the QM/MM results and confirm that when solvent is taken into account the gas-phase bifurcation reaction pathway converts into a stepwise mechanism. This manuscript provides new insight into the biologically and synthetically important (1)O(2) ene reaction and highlights a new multidimensional approach for constructing potential-energy surfaces.

A theoretical study of the effects of the charge state and size of gold clusters on the adsorption and dissociation of H2

The adsorption and dissociation of H(2) on the neutral and charged gold clusters Au(n) (m)(m=0,+/-1; n=1-6) is investigated using the density functional theory PW91 functional. H(2) interacts very weakly with Au(n) (-1), whereas the interaction with Au(n) (+1) is relatively strong. The binding energies on neutral clusters are between those on the cationic and anionic systems. The binding energy decreases monotonically with the size increase of the cationic clusters while it goes up first and then goes down on the neutral systems with the maximum value of 0.78 eV at Au(3). Au cations show no propensity for the dissociation barrier reduction and are thermodynamically unfavorable for the dissociation. For the first time we find that H(2) dissociation involves valley-ridge inflection points on some clusters. Our results indicate that H(2) dissociates facilely at low temperatures on both neutral and cationic Au(4) and Au(5). The phenomenon that H(2) dissociation was not observed experimentally is not due to the higher dissociation barrier and weak binding of H(2). We also show that the coordination number of the Au atom may not play a determining role in H(2) dissociation.

Applications of analytic and geometry concepts of the theory of Calculus of Variations to the Intrinsic Reaction Coordinate model¶

A mathematical analysis of several algorithms, for the integration of the differential equation associated to the Intrinsic Reaction Coordinate path, is performed. This analysis first shows that the Intrinsic Reaction Coordinate path can be derived from a variational problem, so that it has the properties of an extremal curve. Then, one may borrow the mathematical methods for the integration of extremal curves, to formulate new algorithms for the integration of the Intrinsic Reaction Coordinate path. One may use also this theoretical framework, to recast recently formulated algorithms based on direct minimization of an arbitrary curve, such as the Nudged Elastic Band Method or String Method. In this view a new algorithm is proposed. Finally, the theory of broken extremals is used to analyse an Intrinsic Reaction Coordinate path possessing a valley ridge inflection point. ¶Dedicated in honor of Professor Peter Pulay on his 65th birthday.

Can the MaxFlux algorithm describe bifurcating paths?

Temperature-dependent reaction paths have been calculated with the MaxFlux algorithm for a bifurcating region of the potential energy surface of the blocked alanine dipeptide. The resulting paths agree with an ensemble of trajectories generated with Transition Path sampling, which has also been used to determine product distributions. The Valley Ridge Inflection point has been shown to lie close to the point of maximum curvature of the MaxFlux paths.

New valley ridge inflection point associated to the bifurcation of a valley on potential energy surfaces

AbstractThere are currently two definitions of valley ridge inflection point on an adiabatic potential energy surface (PES), one introduced by Basilevsky and the other by Valtazanos and Ruedenberg. Here, we investigate the validity of both definitions on bidimensional surfaces without symmetry constraints, with neither producing completely satisfactory results. Hence, we propose new conditions which must obey a valley bifurcation point (VBP) locally and demonstrate that such conditions are necessary when a valley splits into two, finishing at two energy minima. Finally, we localize such VBPs on the bidimensional surfaces and check the certainty of our previous deductions. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

Better Understanding of the Ring-Cleavage Process of Cyanocyclopropyl Anionic Derivatives. A Theoretical Study Based on the Electron Localization Function

[reaction: see text] Theoretical calculations at the B3LYP/6-31+G(d), MP2/6-31+G(d), and G3(MP2) levels have been carried out to understand the alternative reaction pathways (the cyclopropyl ring cleavage (RC) and the retrocycloaddition reaction (rCA)) of a constrained tricyanocyclopropyl anionic derivative. The more energetically favorable path is found to be the RC process, a formally "forbidden" rearrangement (Leiviers, M.; Tam, I.; Groves, K.; Leung, D.; Xie, Y.; Breslow, R. Org. Lett. 2003, 5, 19, 3407) yielding an allylic anion system via a concerted transition structure, in agreement with experimental outcomes. rCA is more energetically favorable along a two-stage mechanism, via an intermediate, than a synchronous concerted process. By using isodesmic reactions, we have found that B3LYP presents limitations when it calculates carbon-carbon bond-breaking processes along the present rCA reaction. A detailed analysis of the nature of the topology of the reactive potential energy surface for the RC process points out the presence of a valley-ridge inflection point in the uphill part. An explanation for the low-energy barrier associated with RC is furnished on the analysis of the evolution of the twisting (dis-/conrotatory) motions of cyano substituents in the cyclopropyl ring as well as on the number and type of electron pairs provided by the electron localization function (ELF).

Wave packets in a bifurcating region of an energy landscape: Diels-Alder dimerization of cyclopentadiene

Quantum dynamics in a valley ridge inflection (VRI) point region is analyzed in the case of the Diels-Alder endo-dimerization of cyclopentadiene pointed out recently by [Caramella et al., J. Am. Chem. Soc. 124, 1130 (2002)]. The VRI point is located along the reaction path connecting the bispericyclic symmetrical transition structure put in evidence by Caramella et al. and the transition state of the Cope rearrangement. Dynamics is carried out by using constrained Hamiltonian methodology. The active coordinates are the first formed C-C bond length and the difference between the two other C-C bond lengths which achieve the dimerization as 4+2 or 2+4 adducts. A two-dimensional (2D) minimum-energy surface have been computed at the Becke 3 Lee-Yong-Parr6-31G* level. The energy landscape can be classified as an uphill ridge-pitchfork VRI bifurcation according to a recent classification of bifurcation events [W. Quapp, J. Mol. Struct. 695-696, 95 (2004)]. Dynamics does not describe the thermal reaction but concerns wave packets which could be prepared by pulse reagents, i.e., by coherent control. We analyze how the shape and initial location on the ground potential-energy surface are linked to the synchronous or asynchronous mechanism of the final step after the first transition state. We use a one-dimensional model of optimum control theory to check the feasibility of such a coherent preparation. The wave-packet evolution in the VRI domain is well explained by semiclassical predictions even with the negative curvature of the unstable ridge. Finally, a crude model of dissipation has been introduced to test the stability of the 2D predictions.

Variational transition state theory as a tool to determine kinetic selectivity in reactions involving a valley-ridge inflection point.

Variational transition state theory has been used to calculate the kinetic isotope effects affecting product ratios in the reaction between (1)O(2) and d(6)-tetramethylethylene. The minimum energy path on the potential energy surface for this process reaches a valley-ridge inflection point and then bifurcates leading to the two final products. Using canonical variational transition state theory, two distinct dynamical bottlenecks were located corresponding to the H- and the D-abstraction, respectively. The calculated KIE at 263 K turns out to be 1.126. Analogously, a H/T KIE of 1.17 at the same temperature has been found for the reaction of (1)O(2) with the tritiated derivative of tetramethylethylene.

Reaction channels of the potential energy surface: application of Newton trajectories

The reaction path is an important concept of theoretical chemistry. We employ the definition of the Newton trajectory (NT). An NT follows a curve where the gradient is always a pointer to a fixed direction. Usually, a whole family of NTs connects two adjacent stationary points of an index difference of one. We will name such a family a reaction channel. The border between two reaction channels is formed by singular NTs which cross valley-ridge inflection (VRI) points. Examples are given with the Müller–Brown potential, and the potential energy surfaces of water and formaldehyde.