Gradient Extremal Research Articles

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.

One-dimensional models of internal rotation in CX 3NO molecules (X = H, D, F)

Various non-empirical methods for estimating the parameters of one-dimensional internal rotation potentials and energies of torsional transitions were compared for the CX 3NO molecules (X = H, D, F) in the ground ( S 0) and lowest excited singlet ( S 1) electronic states. The potential energy surfaces were studied by the ab initio MR-AQCC/cc-pVTZ, MR-AQCC/cc-pVTZ(- f), MP2/6-311G(2 d), and MP2/6-311G( d, p) methods. The one-dimensional internal rotation problem was solved using the following models: (1) geometry optimization at a given internal rotation coordinate; (2) intrinsic reaction path; (3) gradient extremal; and (4) use of only the data on potential energy surface stationary points. Special attention was paid to the problem of calculation of kinematic coefficient. In all cases, the calculated torsional energies for CX 3NO molecules (X = H, D, F) are in agreement with experiment. The results from different methods for constructing torsional cross-sections of the potential energy surface are virtually equivalent and differ insignificantly from the results of calculations within the framework of the simplest model, hence, estimates of the barrier to internal rotation are of most importance. It was found that a change in the zero-point energy could give a correction to the internal rotation potential as large as 15% of the potential barrier. However, in the case under consideration the calculations in the harmonic approximation taking into account this correction do not improve the agreement between the calculated torsional transitions and the experimental data.

The reaction pathway of a potential energy surface as curve with induced tangent

The reaction path is an important concept of theoretical chemistry. We employ the definitions of the Newton trajectory (NT) and of the gradient extremal (GE). An NT follows a curve where the gradient is always a pointer to a fixed direction. A GE is a curve where the gradient of the potential energy surfaces (PES) is eigenvector of the Hessian. Examples are given with the Müller–Brown potential and further model PESs. An induced tangent opens the possibility to follow the curve by a predictor–corrector method. We show that the two kinds of curves can be treated as curves with induced tangent, however, the gradient descent, or intrinsic reaction coordinate not.

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.

Reaction pathways and projection operators: application to string methods.

The reaction path is an important concept of theoretical chemistry. We discuss the definition with the help of diverse projection operators for the intrinsic reaction coordinate (IRC), for the following of the reduced gradient (RGF) or Newton trajectory (NT), and for the Gradient Extremal (GE). We describe different numerical schemes for the definitions in the context of string methods. It comes out that Newton trajectories are the best ansatz for a string method.

Exploring the potential energy surface of the ethyl cation by new procedures

The MP2/6-31G∗∗ potential energy hypersurface (PES) of the chemically interesting molecule ion, C2H5+, is analyzed by new procedures proposed by us recently. The reaction paths (RPs) start from the minima or saddle points by following the so-called reduced gradient procedure and/or its modification, the tangent search concept (TASC). The minimum energy paths (MEP) obtained by TASC are fine approximations of those gradient extremals, which follow the valley floor (or ridge) along the smallest (absolute) eigenvalue of the Hessian. It is the valley floor gradient extremal which is also termed the streambed of the PES. Tracing the streambed uphill we locate saddle points of the H-scrambling in C2H5+. The potential energy surface of this cation is used to systematically explore the properties of RP definitions with particular focus to RP bifurcations. The intrinsic reaction coordinate (IRC), which is mostly used as MEP in chemistry, is included for the purpose of comparison.

Searching minima of an N-dimensional surface: A robust valley following method

A procedure is proposed to follow the “minimum path” of a hypersurface starting anywhere in the catchment region of the corresponding minimum. The method uses a modification of the so-called “following the reduced gradient” [1]. The original method connects points where the gradient has a constant direction. In the present letter, this is replaced by the successive directions of the tangent of the searched curve. The resulting pathway is that valley floor gradient extremal which belongs to the smallest (absolute) eigenvalue of the Hessian. The new method avoids third derivatives of the objective function. The effectiveness of the algorithm is demonstrated by using a polynomial test, the notorious Rosenbrock function in two, 20, and in 100 dimensions.

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 LJN with N=3, 7, 8, and 55. Saddles for LJN with N=15, 20, and 30 as presented here are new results.

Gradient extremal bifurcation and turning points: An application to the H2CO potential energy surface

Gradient extremals are curves on potential energy surfaces connecting points where the derivative of the gradient norm, subject to the constraint that the energy is constant, is zero. The bifurcation of such curves is analyzed, and exemplified by tracing out some of the gradient extremals on the HF/STO-3G surface for H2CO. It is shown that gradient extremal following provides a semisystematic way of locating stationary points.

Gradient fields of potential energy surfaces

While contour plots provide conceptual pictures of potential energy surfaces and exhibit their critical points, quantitative determinations of critical points and reaction paths as well as dynamical calculations require a knowledge of the gradient fields. The corresponding orthogonal trajectory maps are more complex than contour maps, but they provide additional insights. They are found to contain certain frequently occurring structural elements and these patterns are here examined. It is shown that many of them result from local confluences of orthogonal trajectory bundles with gradient extremals. The analysis leads to the distinction between eight different kinds of such gradient extremal channels. The most important ones are the streambeds and ridges, the former being typical conceptual prototypes of reaction channels, the latter being prototypes of reaction barriers. Gradient extremal channels emanate from second order critical points in the directions of all normal modes, but they do not necessarily follow along the entire reaction path of any one reaction. They can also exist unrelated to critical points. The conclusions are exemplified on a number of model potential energy surfaces.

Gradient extremals and steepest descent lines on potential energy surfaces

Relationships between steepest descent lines and gradient extremals on potential surfaces are elucidated. It is shown that gradient extremals are the curves which connect those points where the steepest descent lines have zero curvature. This condition gives rise to a direct method for the global determination of gradient extremals which is illustrated on the Müller–Brown surface. Furthermore, explicit expressions are obtained for the derivatives of the steepest-descent-line curvatures and, from them, for the gradient extremal tangents. With the help of these formulas, a new gradient extremal following algorithm is formulated.

Following gradient extremal paths

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.

How to find a saddle point

AbstractThis 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.