Wolfgang Quapp

Searching for saddle points of potential energy surfaces by following a reduced gradient

The old coordinate driving procedure to find transition structures in chemical systems is revisited. The well‐known gradient criterion, ∇E(x)=0, which defines the stationary points of the potential energy surface (PES), is reduced by one equation corresponding to one search direction. In this manner, abstract curves can be defined connecting stationary points of the PES. Starting at a given minimum, one follows a well‐selected coordinate to reach the saddle of interest. Usually, but not necessarily, this coordinate will be related to the reaction progress. The method, called reduced gradient following (RGF), locally has an explicit analytical definition. We present a predictor–corrector method for tracing such curves. RGF uses the gradient and the Hessian matrix or updates of the latter at every curve point. For the purpose of testing a whole surface, the six‐dimensional PES of formaldehyde, H2CO, was explored by RGF using the restricted Hartree–Fock (RHF) method and the STO‐3G basis set. Forty‐nine minima and saddle points of different indices were found. At least seven stationary points representing bonded structures were detected in addition to those located using another search algorithm on the same level of theory. Further examples are the localization of the saddle for the HCN⇌CNH isomerization (used for steplength tests) and for the ring closure of azidoazomethine to 1H‐tetrazole. The results show that following the reduced gradient may represent a serious alternative to other methods used to locate saddle points in quantum chemistry. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1087–1100, 1998

Analysis of the concept of minimum energy path on the potential energy surface of chemically reacting systems

Some confusion regarding the properties of minimum energy paths is evident in the literature. We show that a way of steepest descent on a potential surface can be defined independently upon the choice of the coordinate systems. The result is applied to mass-weighted coordinates and their use is critically reviewed. Fukuis IRC appears to be a special case of the steepest descent path starting from a saddle point. The impossibility to define a general ascent path is illustrated and the relations of IRC to real trajectories are discussed.

Gradient extremals and valley floor bifurcations on potential energy surfaces

Gradient extremals are curves in configuration space denned by the condition that the gradient of the potential energy is an eigenvector of the Hessian matrix. Solutions of a corresponding equation go along a valley floor or along a crest of a ridge, if the norm of the gradient is a minimum, and along a cirque or a cliff or a flank of one of the two if the gradient norm is a maximum. Properties of gradient extremals are discussed for simple 2D model surfaces including the problem of valley bifurcations.

REDUCED GRADIENT METHODS AND THEIR RELATION TO REACTION PATHS

The reaction path is an important concept in theoretical chemistry. We discuss different definitions, their merits as well as their drawbacks: IRC (steepest descent from saddle), reduced gradient following (RGF), gradient extremals, and some others. Many properties and problems are explained by two-dimensional figures. This paper is both a review and a pointer to future research. The branching points of RGF curves are valley-ridge inflection (VRI) points of the potential energy surface. These points may serve as indicators for bifurcations of the reaction path. The VRI points are calculated with the help of Branins method. All the important features of the potential energy surface are independent of the coordinate system. Besides the theoretical definitions, we also discuss the numerical use of the methods.

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.

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.

A gradient-only algorithm for tracing a reaction path uphill to the saddle of a potential energy surface

We propose a procedure to follow the weakest ascent along a valley using gradient-only calculations. The device allows one to search for saddles starting somewhere near a minimum provided that the saddle is connected with the minimum by a valley. We define a local criterion for a minimum energy path by comparison of gradients. The algorithm is a predictor-corrector method using two parameters: step length and tolerance. There is no need for a guess of the saddle point region. We calculate valley pathways on 2D test examples, on the HCN surface, and on a 12D potential of an argon 4-cluster.

Determination of energy minima and saddle points using multireference configuration interaction methods in combination with reduced gradient following: The S0 surface of H2CO and the T1 and T2 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 S0 surface of H2CO and the T1 and T2 surfaces of acetylene. For H2CO we investigated three minima (formaldehyde, s‐trans, and s‐cis hydroxycarbene) and five saddle points. For the T1 and T2 states of acetylene the cis‐ and trans‐minima and the planar and nonplanar saddle points were computed.

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 mechanism of a barrierless reaction: hidden transition state and hidden intermediates in the reaction of methylene with ethene

The chelotropic addition reaction of singlet methylene to ethene yielding cyclopropane (reaction 1) was investigated with the help of the Unified Reaction Valley approach (URVA) using different levels of theory (B3LYP, MP2, MP4, CCSD(T), G3) and two basis sets (6-31G(d,p), 6-311++G(3df,3pd)). At all levels of theory, reaction (1) proceeds without barrier and transition state (TS). Nevertheless, reaction (1) possesses a distinct mechanism comprising four different reaction phases: (i) a van der Waals phase, in which the stereochemistry of the reaction is decided; (ii) an electrophilic attack phase, in which charge is transferred from ethene to methylene to establish a weak bonding interaction between the reaction partners typical of those encountered in TSs of CC bond forming reactions; (iii) a nucleophilic attack phase, in which charge transfer between methylene and ethene is reverted and a trimethylene biradical structure is formed; (iv) a ring closure phase, in which the trimethylene structure closes to the three-membered ring. The URVA analysis identifies a hidden TS and two hidden intermediates at the transitions from one phase to the next. If methylene is replaced by difluorocarbene (reaction 2) or germylene (reaction 3), the 4-phase mechanism is retained, however the hidden TS and one of the hidden intermediates are converted into a real TS and a real intermediate thus establishing 2-step mechanisms with strongly different energy profiles along the reaction path.