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     

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.     

A gradient extremal walking algorithm

Gradient extremals define stream beds connecting stationary points on molecular potential energy surfaces. Using this concept we have developed an algorithm to determine transition states. We initiate walks at equilibrium geometries and follow the gradient extremals until a stationary point is reached. As an illustration we have investigated the mechanism for exchange of protons on carbon in methylenimine (H₂C=NH) using a multi-reference self-consistent-field wave function.     

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     

Reaction rates in a theory of mechanochemical pathways

If one applies mechanical stress to a molecule in a defined direction then one generates a new, effective potential energy surface (PES). Changes for minima and saddle points (SP) by the stress are described by Newton trajectories on the original PES (Quapp and Bofill, Theor. Chem. Acc. 2016, 135, 113). The barrier of a reaction fully breaks down for the maximal value of the norm of the gradient of the PES along a pulling Newton trajectory. This point is named barrier breakdown point (BBP). Depending on the pulling direction, different reaction pathways can be enforced. If the exit SP of the chosen pulling direction is not the lowest SP of the reactant valley, on the original PES, then the SPs must change their role anywhere: in this case the curve of the log(rate) over the pulling force of a forward reaction can show a deviation from the normal concave curvature. We discuss simple, two‐dimensional examples for this model to understand more deeply the mechanochemistry of molecular systems under a mechanical stress. © 2016 Wiley Periodicals, Inc.     

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.     

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     

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     

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     

Variational nature, integration, and properties of Newton reaction path

The distinguished coordinate path and the reduced gradient following path or its equivalent formulation, the Newton trajectory, are analyzed and unified using the theory of calculus of variations. It is shown that their minimum character is related to the fact that the curve is located in a valley region. In this case, we say that the Newton trajectory is a reaction path with the category of minimum energy path. In addition to these findings a Runge-Kutta-Fehlberg algorithm to integrate these curves is also proposed.     

The effects of the column pressure drop on retention and efficiency in packed and open tubular supercritical fluid chromatography

The effects of the pressure drop across the column on retention and efficiency in SFC have been studied. Numerical methods are described which enable the prediction of hold-up time and pressure drop in both packed and open tubular columns. Predictions of both hold-up time and pressure drop are in good agreement with experimental data. The density gradient along the column can be calculated using the numerical methods and a procedure is described which enables the calculation of the overall capacity factors of the solutes from the density profile in the column. Significant variations of the capacity factor are observed along the column. The effect of the density gradient along the column on local diffusivity and dispersion is studied. The column efficiency in systems with significant pressure drops is affected by changes in: the linear velocity of the mobile phase; the diffusion coefficients; and the capacity factors of the solutes along the column. The overall efficiency of the chromatographic system can be calculated if, as is the case for open tubular columns, adequate plate height equations are available.     

A growing string method for the reaction pathway defined by a Newton trajectory

The reaction path is an important concept of theoretical chemistry. We use a projection operator for the following of the Newton trajectory (NT) along the reaction valley of the potential energy surface. We describe the numerical scheme for the string method, adapting the proposal of a growing string (GS) by [Peters et al.,J. Chem. Phys. 120, 7877 (2004)]. The combination of the Newton projector and the growing string idea is an improvement of both methods, and a great saving of the number of iterations needed to find the pathway over the saddle point. This combination GS-NT is at the best of our knowledge new. We employ two different corrector methods: first, the use of projected gradient steps, and second a conjugated gradient method, the CG+ method of Liu, Nocedal, and Waltz, generalized by projectors. The executed examples are Lennard-Jones clusters, LJ(7) and LJ(22), and an N-methyl-alanyl-acetamide (alanine dipeptide) rearrangement between the minima C7(ax) and C5. For the latter, the growing string calculation is interfaced with the GASSIAN03 quantum chemical software package.     

Algorithmic tools in the study of semiempirical potential surfaces

The most efficient optimization methods implemented in the semiempirical package AMPAC are presented. They concern the minimization of the energy or of the gradient norm by either pseudo-Newton or quadratic procedures, the search for transition states, and the intrinsic reaction coordinate in conjunction with variational transition-state theories. Nonlocal methods such as simulated annealing are also introduced. © 1992 John Wiley & Sons, Inc.     

Geometric theory of trigger waves — A dynamical system approach

We propose a geometric model for wave propagation in excitable media. Our model is based on the Fermat principle and it resembles that of Wiener and Rosenblueth. The model applies to the propagation of excitations, such as chemical and biological wave fronts, grass fire, etc. Starting from the Fermat principle, some consequences of the assumptions are derived analytically. It is proved that the model describes a dynamical system, and that the wave propagates along ignition lines (extremals). The theory is applied to the special cases of tube reactor and annular reactor. The asymptotic shape of the wave fronts is derived for these cases: they are straight lines perpendicular to the tube, and involutes of the central obstacle, respectively.     

梯度接触角表面的构建与应用

梯度接触角是梯度表面张力的反映,固体表面的润湿性由表面化学组成和表面微观形貌共同决定。通过表面化学组成和表面微观形貌的梯度化,可制备接触角变化范围不同的梯度接触角表面。本文综述了梯度接触角表面在液滴移动、微流体流动和生物吸附等领域中的应用。梯度接触角表面具有的不平衡杨氏力是促进液滴移动的主要原因,而表面所产生的接触角滞后则阻碍液滴移动;在生物学领域,梯度接触角表面会造成蛋白质和细胞选择性吸附或黏附。最后,简要探讨了梯度接触角表面存在的问题和发展方向。     

Quantum chemistry, Sobolev spaces and SCF

Extended wavefunctions, including the wavefunction gradient, and the norm induced Sobolev spaces are presented as a mathematical structure well adapted to the approximate quantum-chemical formalism, customarily used to handle the Schrödinger equation. A useful application, related to the solution of SCF Euler equations in matrix form, is also analysed.     

Dynamic ion-exchange chromatography with post-column reaction detection for the determination of the rare earth elements in monazites

Summary Separation and determination of lanthanum, cerium, praseodymium, neodymium and samarium in monazites have been achieved by dynamic ion-exchange chromatography. The ore samples are decomposed by sulfuric acid and the rare earths are separated in a group as oxalates. The rare earth elements are then separated from each other on a column of bonded phase silica by gradient elution with 0.05 to 0.5 M lactic acid (pH 3.5) in the presence of 0.01 M sodium 1-octanesulfonate. Post-column reaction with Arsenazo III is used for detection and quantification of the individual rare earth elements. Results are quoted for lanthanum, cerium, praseodymium, neodymium and samarium in monazites. Detection limit is 1 μg ml⁻¹ with a S/N ratio of 3. The separation is complete within 27 min valley to valley resolution. Precision of better than 1% can usually be obtained.     

Peak compression in reversed-phase gradient elution

Previous reports suggest that peak widths in linear gradient elution are consistently larger than predicted by theory; however, if gradient compression is ignored, experiment and theory are in reasonable agreement. This suggests that gradient compression might represent an incorrect or poorly understood concept. In the present study, an experimental program was carried out to better understand the role of gradient compression and the reason for past differences between experiment and theory. It is concluded that the concept of gradient compression is correct.     

Diffusioosmosis of electrolyte solutions in a fine capillary slit

The steady diffusioosmotic flows of an electrolyte solution along a charged plane wall and in a capillary channel between two identical parallel charged plates generated by an imposed tangential concentration gradient are theoretically investigated. The plane walls may have either a constant surface potential or a constant surface charge density. The electrical double layers adjacent to the charged walls may have an arbitrary thickness and their electrostatic potential distributions are determined by the Poisson-Boltzmann equation. Solving a modified Navier-Stokes equation with the constraint of no net electric current arising from the cocurrent diffusion, electric migration, and diffusioosmotic convection of the electrolyte ions, the macroscopic electric field and the fluid velocity along the tangential direction induced by the imposed electrolyte concentration gradient are obtained semianalytically as a function of the lateral position in a self-consistent way. The direction of the diffusioosmotic flow relative to the concentration gradient is determined by the combination of the zeta potential (or surface charge density) of the wall, the properties of the electrolyte solution, and other relevant factors. For a given concentration gradient of an electrolyte along a plane wall, the magnitude of fluid velocity at a position in general increases with an increase in its electrokinetic distance from the wall, but there are exceptions. The effect of the lateral distribution of the induced tangential electric field and the relaxation effect in the double layer on the diffusioosmotic flow are found to be very significant.     

Analytical equilibrium gradient methods

Analytical equilibrium gradient methods are non-linear separation methods in which the separation mechanism involves a force gradient along the separation channel. These methods can be classified into two categories: those in which the gradient is a field gradient applied along the separation channel (i.e., field gradient), and those in which the channel is subjected to a constant field with a gradient formed in some other property (i.e., constant field). Standard deviation of peak width, resolution and peak capacity are important parameters in characterizing equilibrium gradient methods, and general expressions can be obtained from considering both the point of force acting on the analyte and the basic flux equation. Several successful examples, such as density gradient sedimentation, isoelectric focusing and electromobility focusing are discussed. Based on equilibrium gradient methods in the field gradient category, a method to dynamically improve peak capacity is described. An example of such an approach is given using electromobility focusing.