Following gradient extremal paths

Summary For any point on a gradient extremal path, the gradient is an eigenvector of the hessian. Two new methods for following the gradient extremal path are presented. The first greatly reduces the number of second derivative calculations needed by using a modified updating scheme for the hessian. The second method follows the gradient extremal using only the gradient, avoiding the hessian evaluation entirely. The latter algorithm makes it possible to use gradient extremals to explore energy surfaces at higher levels of theory for which analytical hessians are not available.Dedicated to Prof. Klaus Ruedenberg     

Paths to which the nudged elastic band converges

A recent letter to the editor (Quapp and Bofill, J Comput Chem 2010, 31, 2526) claims that the nudged elastic band (NEB) method can converge toward gradient extremal paths and not to steepest descent paths, as has been assumed. Here, we show that the NEB does in fact converge to steepest descent paths and that the observed tendency for the NEB to approach gradient extremal paths was a consequence of implementation errors. We also note that while the NEB finds steepest descent paths, these are not necessarily minimum energy paths in the sense of being a set of points which are minima in the potential energy surface perpendicular to the path. An example is given where segments of steepest descent paths follow potential energy ridges.     

Reaction Pathways and Convexity of the Potential Energy Surface: Application of Newton Trajectories

The reaction path is an important concept of theoretical chemistry. We employ the definitions of the intrinsic reaction coordinate (IRC), the gradient extremal (GE), and the Newton trajectory (NT). The usual imagination in chemistry is that a minimum energy path is in a convex region of the potential energy surface. We describe different schemes of convexity to handle the situation. It comes out that NTs are the best ansatz for the problem: NTs, which monotonically increase (or monotonically decrease), are automatically strictly pseudo-convex throughout, and they go throughout along a valley between minimum and saddle point.     

A comment to the nudged elastic band method

The minimum energy path (MEP) is an important reaction path concept of theoretical chemistry, and the nudged elastic band (NEB) method with its many facets is a central method to determine the MEP. We demonstrate in this comment that the NEB does not have to lead to a steepest descent pathway (as always assumed). In contrast, as long as it is used without spring forces, it can lead to a gradient extremal. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Tunnelling and reaction path curvature effects in the isomerization of the methoxy radical

The reaction path hamiltonian is used to investigate the isomerisation CH₃O CH₂OH; the reaction path, the frequencies along it and the coupling coefficients describing reaction path curvature were calculated by ab-initio methods. Correlation effects were included by configuration interaction using a double-zeta-plus-polarisation basis set. Including both tunnelling and curvature gives temperature dependent rate constants in broad agreement with experiment, whereas previous results were in error by several orders of magnitude.     

The geometric minimum action method for computing minimum energy paths

An algorithm is proposed to calculate the minimum energy path (MEP). The algorithm is based on a variational formulation in which the MEP is characterized as the curve minimizing a certain functional. The algorithm performs this minimization using a preconditioned steepest-descent scheme with a reparametrization step to enforce a constraint on the curve parametrization.     

HNCO+OH->H2O+NCO的反应机理

采用从头算分子轨道法 (UHF/6 31G 水平 ,并用MP4加以相关能校正 )研究了HNCO OHH2 O NCO反应机理 .同时用Morokuma数值法获得了反应途径即内禀反应坐标 (IRC) .沿着IRC ,运用反应途径哈密顿理论 ,获得反应途径动态学信息 .在此基础上 ,根据过渡态理论和相应隧道效应校正 ,计算了在不同温度下的反应速率常数 ,得到了和实验相一致的结果 .计算结果表明 ,此反应是一步直接型的抽提H反应 .     

Potential energy surface exploration with equilibrial paths. Part I: Theory

The equilibrial path concept is further developed. Special attention is spent the symmetry conservation along equilibrial paths and symmetry-breaking. Symmetry-breaking can occur only at singular points. The simple singular points of an equilibrial path are valley–ridge inflection points. In contrast to the intrinsic reaction paths and the gradient extremal paths, the equilibrial paths enable to describe the branching of reaction channels.     

A new look at the reduced-gradient-following path

 The mathematical structure of the reduced-gradient-following (RGF) path introduced by Quapp et al. (1988 J. Comput. Chem. 19:1087) is reviewed and analyzed. We report two new algorithms to evaluate the RGF path. The RGF path is also compared mathematically and computationally with the gradient extremals path. An example of the evaluation of the RGF path is also reported. Received: 21 May 2001 / Accepted: 27 September 2001 / Published online: 9 January 2002     

Newton leaves on potential energy surfaces

The reaction path (RP) is an important concept of theoretical chemistry. We generalize the definition of the Newton trajectory (NT), as an RP, to Newton leaves in a higher dimensional subspace of the configuration space. Our standpoint is that of Bofill and Anglada [(2001) Theor. Chem. Acc. 105:436], who used a reduced potential energy surface for finding an RP. An NT follows an RP curve where the gradient is always a pointer to a fixed direction. More generally, a Newton leaf is a subspace of coordinates where the gradient can move in a subspace of directions. We report some known mathematical properties of Newton leaves. We explain the construction of Newton leaves with the example of a 3D test surface in ⁴ [W.Quapp et al. (1998) Theor. Chem. Acc. 100:285], because three coordinate dimensions are the smallest number of dimensions one needs at least to understand a Newton leaf in contrast to the known NTs.Acknowledgement The work was made possible through financial support of the Deutsche Forschungsgemeinschaft. The authors thank D. Heidrich for stimulating discussions.     

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     

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.     

甲亚胺1,2—脱氢过程中化学键变化的量子拓扑研究

在IRC解析的基础上,用电子密度的量子拓扑分析方法研究了甲亚胺1,2-脱氢过程中化学键的变化.反应途径中各点化学键的拓扑性质可以清楚地反映出化学键断裂及生成过程,计算结果进一步证明该反应为协同非同步反应.本工作为研究化学反应过程提供了一种新的方法。     

Reaction path dynamics and theoretical rate constants for the SiH3Cl+H→SiH2Cl+H2 reaction by ab initio direct dynamics method

Ab initio direct dynamics method has been used to study the title reaction. Electronic structure information including geometries, gradients and force constants (Hessians) are calculated at the UQCISD/6-311+G^** level. Energies along the minimum energy path are improved by a series of single-point G2//QCISD calculations. The changes of the geometries, vibratioanal frequencies, potential energies and total curvature along the reaction path are discussed. The rate constants in the temperature range 200–3000 K are calculated by canonical variational transition state theory with small-curvature tunneling correction (CVT/SCT) method. The results show that the variational effect is small and in the lower temperature range, the small curvature tunneling effect is important for the reaction.     

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     

Gradient extremals

Gradient extremals on N-dimensional energy hypersurfaces V=V(x ₁ x _n ) are curves defined by the condition that the gradient V is an eigenvector of the hessian matrix V. For variations which are restricted to any (N–1) dimensional hypersurface V(x ₁ x _N ) = V ₀= constant, the absolute value of the gradient V is an extremum at those points where a gradient extremal intersects this surface. In many, though not all, cases gradient extremals go along the bottom of a valley or along the crest of a ridge. The properties of gradient extremals are discussed through a detailed differential analysis and illustrated by an explicit example. Multidimensional generalizations of gradient extremals are defined and discussed.Operated for the U.S. Department of Energy by Iowa State University under Contract No. W-7405-ENG-82. This work was supported by the Office of Basic Energy Sciences     

Monte Carlo approach to the decay rate of a metastable system with an arbitrarily shaped barrier

A path integral Monte Carlo method based on the fast-Fourier transform technique combined with the important sampling method is proposed to calculate the decay rate of a metastable quantum system with an arbitrary shape of a potential barrier. The contribution of all fluctuation actions is included which can be used to check the accuracy of the usual steepest-descent approximation, namely, the perturbation expansion of potential. The analytical approximation is found to produce the decay rate of a particle in a cubic potential being about 20% larger than the Monte Carlo data at the crossover temperature. This disagreement increases with increasing complexity of the potential shape. We also demonstrate via Langevin simulation that the postsaddle potential influences strongly upon the classical escape rate.     

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.     

适用于HCCI燃烧的汽油替代燃料化学动力学简化模型

建立了一个适用于由正庚烷、异辛烷、甲苯和二异丁烯组成的汽油替代燃料均质压燃着火(HCCI)燃烧过程的简化机理模型, 包含103 种组分199 个反应. 二异丁烯主要通过燃料的脱氧反应消耗掉, 生成三种同分异构体, JC₈H₁₅-A、JC₈H₁₅-B和JC₈H₁₅-D; 燃料的分解反应也是二异丁烯的另外一条主要消耗路径, 生成两种重要的C₄产物, TC₄H₉和IC₄H₇. 这些产物是CH₂O的主要来源. 甲苯掺比燃料(TRF)机理主要是基于Andrae 等建立的TRF半详细机理, 甲苯和二异丁烯子机理是通过路径分析和敏感性分析得到. 简化机理能够很好地模拟激波管里的着火延迟和HCCI发动机实验, 由此可知, 本文提出的简化机理用来模拟HCCI燃烧是可靠的.     

Location of saddle points and minimum energy paths by a constrained simplex optimization procedure

Two methods are proposed, one for the location of saddle points and one for the calculation of steepest-descent paths on multidimensional surfaces. Both methods are based on a constrained simplex optimization technique that avoids the evaluation of gradients or second derivative matrices. Three chemical reactions of increasing structural complexity are studied within the PRDDO SCF approximation. Predicted properties of reaction hypersurfaces are in good overall agreement with those determined by gradient minimization and gradient following algorithms in connection with various ab initio SCF methods. Computational efforts required by the new procedures are discussed.