Bifurcation of reaction pathways: the set of valley ridge inflection points of a simple three-dimensional potential energy surface

This paper serves for the better understanding of the branching phenomenon of reaction paths of potential energy hypersurfaces in more than two dimensions. We apply the recently proposed reduced gradient following (RGF) method for the analysis of potential energy hypersurfaces having valley-ridge inflection (VRI) points. VRI points indicate the region of possible reaction path bifurcation. The relation between RGF and the so-called global Newton search for stationary points (Branin method) is shown. Using a 3D polynomial test surface, a whole 1D manifold of VRI points is obtained. Its relation to RGF curves, steepest descent and gradient extremals is discussed as well as the relation of the VRI manifold to bifurcation points of these curves. Received: 8 July 1998 / Accepted: 24 August 1998 / Published online: 23 November 1998     

Improved RGF method to find saddle points

The predictor-corrector method for following a reduced gradient (RGF) to determine saddle points [Quapp, W. et al., J Comput Chem 1998, 19, 1087] is further accelerated by a modification allowing an implied corrector step per predictor but almost without additional costs. The stability and robustness of the RGF method are improved, and the new version in addition reduces the number of gradient and Hessian calculations.     

Determination of energy minima and saddle points using multireference configuration interaction methods in combination with reduced gradient following: the S(0) surface of H(2)CO and the T(1) and T(2) surfaces of acetylene

The implementation of the reduced gradient following (RGF) method into the COLUMBUS quantum-chemical program system is reported using the newly developed analytic MR-CISD/AQCC gradient feature. By this combination a very useful tool has been developed for general searches of stationary points on ground- and excited-state energy surfaces. This procedure is applied to the S(0) surface of H(2)CO and the T(1) and T(2) surfaces of acetylene. For H(2)CO we investigated three minima (formaldehyde, s-trans, and s-cis hydroxycarbene) and five saddle points. For the T(1) and T(2) states of acetylene the cis- and trans-minima and the planar and nonplanar saddle points were computed.     

Following the streambed reaction on potential-energy surfaces: a new robust method

A simple procedure with low computational efforts is proposed to follow the reaction path of the potential-energy hypersurface (PES) starting from minima or saddle points. The method uses a modification of the so-called “following the reduced gradient” [Quapp W, Hirsch M, Imig O, Heidrich D (1998) J Comput Chem 19:1087]. The original method connects points where the gradient has a constant direction. In the present article the procedure is replaced by taking iterative varying directions of the gradient controlled by the last tangent of the searched curve. The resulting minimum energy path is that valley floor gradient extremal (GE) which belongs to the smallest (absolute) eigenvalue of the Hessian and, hence, that GE which usually leads along the streambed of a chemical reaction. The new method avoids third derivatives of the PES and obtains the GE of least ascent by second-order calculations only. Nevertheless, we are able to follow the streambed GE uphill or downhill. We can connect a minimum with its saddles if the streambed leads up to a saddle, or we find a turning point or a bifurcation point. The effectiveness and the characteristic properties of the new algorithm are demonstrated by using polynomial test surfaces, an ab initio PES of H₂O, and the analytic potentials of Lennard-Jones (LJ) clusters. By tracing the streambeds we located previously identified saddle points for LJ _N with N=3, 7, 8, and 55. Saddles for LJ _N with N=15, 20, and 30 as presented here are new results. Received: 8 March 2000 / Accepted: 17 July 2000 / Published online: 24 October 2000     

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.     

Comment “On the quadratic reaction path evaluated in a reduced potential energy surface model and the problem to locate transition states” [by J. M. Anglada,E. Besalú, J. M. Bofill,and R. Crehuet,J Comput Chem 2001, 22, 4, 387–406]

The difference is explained between steepest ascent and following a reduced gradient (distinguished coordinate method) for the location of saddle points. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 537–540, 2001     

An approach to reaction path branching using valley–ridge inflection points of potential-energy surfaces

Valley–ridge inflection (VRI) points of a potential-energy surface (PES) may have a strong relation to the occurrence of bifurcations along reaction pathways of molecular rearrangements. We discuss two different definitions of VRI points in the literature. The calculation of symmetric VRI points has already been reported [W. Quapp et al. (1998) Theor. Chem. Acc. 100: 285–299]. Here, we in addition calculate special asymmetric VRI points which are placed on gradient extremals (GE). Following a GE opens the possibility to find the VRI point on it. An application is presented to search for asymmetric VRI points near the isomerization valley of the PES of the HCN molecule. A new method for GE-following is based on a mathematical connection between the following of a reduced gradient and the calculation of GEs. The tangent search method to follow a GE to the smallest eigenvalue [W. Quapp et al. (2000) Theor. Chem. Acc. 105: 145–155] is extended to follow also GEs to higher eigenvalues in order to find a VRI point. The new method needs gradient and second derivatives of the PES only.     

Stationary points on the H2CO potential energy surface: dependence on theoretical level

A total of 36 stationary points have been located on the H₂CO potential energy surface by means of gradient extremal following. These 36 points are believed to represent all the important stationary points on this surface. There is no indication that the structure of the surface becomes less complicated as the size of the basis set is enlarged at the Hartree-Fock level of theory, but many of the second- and third-order saddle points disappear when electron correlation is introduced. Of the ten first-order saddle points (transition structures) located, the majority have reaction paths entering the associated minima in a side-on approach, i.e. these cannot be located by uphill walking from the minimum. Received: 5 February 1998 / Accepted: 21 May 1998 / Published online: 29 July 1998     

Newton Trajectories in the Curvilinear Metric of Internal Coordinates

The reaction path is an important concept of theoretical chemistry. We discuss the definition with the help of the following of the reduced gradient (RGF) [see Quapp et al., Theoret. Chem. Acc. 100 (1998) 285], also named the Newton trajectory. All the important features of the potential energy surface are definable independently of the coordinate system. We demonstrate it for the Newton trajectory. We design a numerical scheme for the RGF method including the intrinsic curvilinear metric of internal coordinates. For the path following we extend the previous method to the use of a generalized singular value decomposition (SVD). An example of the HCN isomerization pathway is discussed.     

Geometrical properties of the potential energy of the soft-sphere binary mixture

We report a detailed study of the stationary points (zero-force points) of the potential energy surface (PES) of a model structural glassformer. We compare stationary points found with two different algorithms (eigenvector following and square gradient minimization), and show that the mapping between instantaneous configuration and stationary points defined by those algorithms is as different as to strongly influence the instability index K versus temperature plot, which relevance in analyzing the liquid dynamics is thus questioned. On the other hand, the plot of K versus energy is much less sensitive to the algorithm employed, showing that the energy is the good variable to discuss geometric properties of the PES. We find new evidence of a geometric transition between a minima-dominated phase and a saddle-point-dominated one. We analyze the distances between instantaneous configurations and stationary points, and find that above the glass transition, the system is closer to saddle points than to minima.     

Quantenchemische untersuchungen zum mechanismus der elektrophilen substitution. I. Zur potentialhyperfläche des systems benzol/h+

Results of semiempirical calculations (CNDO/2-FK and MINDO/2 methods) for the σ-π complex problem on protonated benzene are given and compared with previous ones. The semiempirical methods were chosen according to the agreement of their results with new theoretical energy data (E_HF + E_korrel) concerning the classical–nonclassical problem of protonated ethylene. By these methods the corresponding part of the energy surface of the benzene/H⁺ system is simulated. The stationary points of this surface are found by a gradient method with complete optimization of the geometry. On the basis of this method we determined the energy profile of a reaction coordinate between the classical (σ-complex) and nonclassical (π-complex) cation. The so called strong π-complex is a saddle point between two σ-complex minima and can be interpreted as transition state of 1,2-proton shifts. Hypotheses for possible minimum energy paths of electrophilic attacks in the given region of the surface are discussed.     

Low-mode conformational search elucidated: Application to C39H80 and flexible docking of 9-deazaguanine inhibitors into PNP

We previously described a new conformational search method, termed low-mode search (LMOD), and discussed its utility for conformational searches performed on cycloalkanes and a cyclic penta-peptide. 1 In this report, we discuss a rigorous implementation of mode following (c-LMOD) for conformational searching, and we demonstrate that for a conformational search involving cycloheptadecane, this rigorous implementation is capable of finding all of the previously known structures. To the best of our knowledge, this is the first computational proof that mode following can be used for conformational searches conducted on a complex molecular system. We show, however, that, as expected, it is generally inefficient to perform a conformational search in this manner. Nonetheless, c-LMOD has been shown to be an excellent method for conducting conformational analyses involving conformational interconversions, where the location of saddle points is important. We also describe refinement to our original LMOD procedure (l-LMOD) and discuss its utility for a difficult conformational search problem, namely locating the global minimum energy conformation of C₃₉H₈₀. For this search, l-LMOD combined with limited torsional Monte Carlo movement was able to locate the lowest energy structures yet reported, and significantly outperformed a pure torsional Monte Carlo and a genetic algorithm-based search. Furthermore, we also demonstrate the utility of l-LMOD combined with random translation/rotation of a ligand for the extremely difficult problem of docking flexible ligands into flexible protein binding sites on a system that includes 9-deaza-guanine-based inhibitors docked into the flexible biding site of PNP. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1671–1684, 1999     

Anab initio quantum-chemical study of proton addition to F-, Ch3-, and CF3-substituted ethylene derivatives

Protonated forms of the molecules of ethylene derivatives with the general formula C₂X₂Y₂ (X=Y=H) (1), F (2), CH₃ (3) CH₃ (4); X=F, Y=H:cis-(5)trans- (6)) were calculated by theab initio MP2/6-31 G^* method with full geometry optimization. The minima and saddle points located on the potential energy surface (PES) of the protonated ethylene molecule correspond to the stationary states and transition states of proton migration, respectively. The stationary states are characterized by a nonclassical geometry of carbocations similar to that of π-complexes, whereas the transition states have a classical structure. Unlike1, the carbocations of molecules2–6 have the classical structure. The saddle points on the PES of the ethylene derivatives correspond to the structures of the π-complex type, which are the transition states of proton migration between the C atoms of the ethylene bond. The barrier to rotation about, the C−C bond depends essentially on the substituent nature. Published inIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 8, pp. 1333–1337, August, 2000.     

How to find a saddle point

This paper explains a method for finding saddle points on a multidimensional surface and shows how it may be used to define saddle-point seeking curves that have properties similar to well-known orthogonal trajectories. It is shown that a gradient extremal is a special case of one of these curves, and its chemical significance as the path, defined by local criteria, which starts from a stable structure and leads to a transition state, is discussed briefly in relation to the intrinsic reaction coordinate. It is emphasized that this theory gives a natural method for locating points that have Hessians of similar structure to those of transition states.     

Ab initio conformational analysis of cyclooctane molecule

The potential energy surface (PES) for the cyclooctane molecule was comprehensively investigated at the Hartree–Fock (HF) level of theory employing the 3–21G, 6–31G, and 6–31G* basis sets. Six distinct true minimum energy structures (named B, BB, BC, CROWN, TBC, and TCC₁), characterized through harmonic frequency analysis, were located on the multidimensional PES. Two transition state structures were also located on the PES for the cyclooctane molecule. Electron correlation effects were accounted for using the Møller–Plesset second-order perturbation theory (MP2) approach. The predicted global minimum energy structure on the ab initio PES for the cyclooctane molecule is the BC conformer. A gas phase electron diffraction study at 300 K suggested a conformational mixture while an NMR study in solution at 161.5 K predicted the BC conformer as the predominant form. The equilibrium constants reported in the present study, which were evaluated from the ab initio calculated total Gibbs free energy change values, were in good agreement with both experimental investigations. The ab initio results showed that the low temperature condition significantly favored the BC conformer while above room temperature both BC and CROWN structures can coexist. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 524–534, 1998     

Application of a dynamic method of minimisation in the study of reaction surfaces

This paper describes the application of Snyman's dynamic minimisation method to a fitted potential surface of H₃. Comparisons are made with conventional algorithms. A method is described to extend Snyman's method so that it will find only a particular kind of stationary point. It is emphasized that this method enables saddle points to be found without having to resort to approaches based on trial and error.     

CASPT2 study of the potential energy surface of the HSO2 system

The importance of the HSO(2) system in atmospheric and combustion chemistry has motivated several works dedicated to the study of associated structures and chemical reactions. Nevertheless controversy still exists in connection with the reaction SH + O(2)→ H + SO(2) and also related to the role of the HSOO isomers in the potential energy surface (PES). Here we report high-level ab initio calculation for the electronic ground state of the HSO(2) system. Energetic, geometric, and frequency properties for the major stationary states of the PES are reported at the same level of calculations: CASPT2/aug-cc-pV(T+d)Z. This study introduces three new stationary points (two saddle points and one minimum). These structures allow the connection of the skewed HSOO(s) and the HSO(2) minima defining new reaction paths for SH + O(2) → H + SO(2) and SH + O(2) → OH + SO. In addition, the location of the HSOO isomers in the reaction pathways have been clarified.     

On the quadratic reaction path evaluated in a reduced potential energy surface model and the problem to locate transition states*

Knowledge of the location of saddle points is crucial to the study the chemical reactivity. Using a path following method defined in a reduced potential energy surface, and starting at either the reactant or product region, we propose an algorithm that locates the corresponding saddle point. The reduced potential energy surface is defined by the set of molecular geometry parameters, namely bond distances, bond angles, and dihedral angles that undergo the largest change for the reaction under consideration; the rest of the coordinates are forced to have a null gradient. Consequently, the proposed method can be seen as a new formulation of the distinguished coordinate method. The method is based on a quadratic model; consequently, it only requires the calculation of the energy and the gradient. The Hessian matrix is normally updated except in the first step and the steps where the resulting updated Hessian matrix is not adequate. Some examples are presented and analyzed. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 387–406, 2001     

Simulated annealing to locate various stationary points in semiempirical methods

In this work, an algorithm was developed to study the potential energy surfaces in the coordinate spaces of molecules by a nonlocal way, in contrast to classic energy minimizers as the BFGS or the DFP method. This algorithm, based on the specificities of semiempirical methods, mixes simulated annealing and local searches to reduce computation costs. By this technique, the global energy minimum can be localized. Moreover, local minima that are close in energy to the global minimum are also obtained. If the search is not only for minima but for all stationary points (minima, saddle points…), then the energy is replaced by the gradient norm, which reaches its minimum values at stationary points. The annealing process is stopped before having accurately reached the global minimum and generates a list of geometries whose energies (respectively, whose gradients) are optimized by local minimizers. This list of geometries is shortened from the nearly equivalent geometries by a dynamic single-clustering analysis. The energy/gradient local minimizers act on the clustered list to produce a set of minima/stationary points. A targeted search of these points and reduction of the costs are reached by the way of several penalty functions. They eliminate—without energy calculation—most of the points generated by random walks on the potential energy surface. These penalty functions (on the total moment of inertia or on interatomic distances) are specific to the class of problem studied. They account for the nonrupture of either specific chemical bonds or rings in cyclic molecules, they assure that molecular systems are kept bonded, and they avoid the collapsing of atoms. © 1992 John Wiley & Sons, Inc.     

Mechanochemistry on the Müller–Brown surface by Newton trajectories

Chemical 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.