How good is a Broyden–Fletcher–Goldfarb–Shanno-like update Hessian formula to locate transition structures? Specific reformulation of Broyden–Fletcher–Goldfarb–Shanno for optimizing saddle points

Based on a study of the Broyden–Fletcher–Goldfarb–Shanno (BFGS) update Hessian formula to locate minima on a hypersurface potential energy, we present an updated Hessian formula to locate and optimize saddle points of any order that in some sense preserves the initial structure of the BFGS update formula. The performance and efficiency of this new formula is compared with the previous updated Hessian formulae such as the Powell and MSP ones. We conclude that the proposed update is quite competitive but no more efficient than the normal updates normally used in any optimization of saddle points. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 349–362, 1998     

A survey of optimization procedures for stable structures and transition states

We examine a variety of methods for obtaining the stable geometry of molecules and the transition states of simple systems and summarize some of our findings. We find the most efficient methods for optimizing structure to be those based on calculated gradients and estimated second derivative (Hessian) matrices, the later obtained either from the Broyden–Fletcher–Goldfarb–Shanno (BFGS ) quasi-Newton update method or from approximations to the coupled perturbed Hartree–Fock method. For uncovering transition states we find particularly useful a variety of the augmented Hessian theory used to uncover regions of the potential energy hypersurface with one and only one negative eigenvalue of the Hessian matrix characterizing the catchment region of the transition state. Once this region is found we minimize the norm of the gradient vector to catch the nearest extreme point of the surface. Examples of these procedures are given.     

Use of symmetric rank-one Hessian update in molecular geometry optimization

The use of the symmetric rank-one Hessian update and the Broyden–Fletcher–Goldfarb–Shano (BFGS) update formula are considered in an ab initio molecular geometry optimization algorithm. It is noted that the symmetric rank-one Hessian update has an advantage when compared with the BFGS update formula and this advantage must be more evident in the optimization of molecular geometry, because the total energy surface is a near-quadratic function with a small nonlinearity close to a minimum point. The results obtained in geometry optimization of a test group of molecules support this proposal and show that the use of the symmetric rank-one Hessian update formula permits reduction of the number of energy and gradient evaluations needed to locate a minimum on the energy surface. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1877–1886, 1998     

Computational strategies for the optimization of equilibrium geometries and transition-state structures at the semiempirical level

Several improvements have been made to the gradient algorithms commonly used to optimize equilibrium and transition-state geometries at the semiempirical level. A gradient algorithm derived from a combination of a variable metric method (Davidon–Fletcher–Powell/Broyden–Fletcher–Goldfarb–Shanno) and Pulay's direct inversion in the iterative subspace method for geometry optimization (GDIIS) is compared with the variable metric method combined with an accurate linear search algorithm. The latter method is used routinely in the standard semiempirical program packages, MNDO, MOPAC, and AMPAC. The combined variable metric and GDIIS algorithm is also compared with GDIIS which uses a static metric. The performance of these algorithms is examined for a wide range of systems with respect to both choice of coordinate system (for cyclic molecules) and guess for the initial Hessian. The results show that the GDIIS method is up to ca. 40% more efficient than the variable metric combined with accurate line search algorithm: however, the exact savings vary depending on the coordinate system and initial Hessian. For noncyclic systems, variable-metric GDIIS is usually equal or superior to static-metric GDIIS, and consistently performs ca. 30% more efficiently than the variable metric combined with accurate line search algorithm. For the optimization of cyclic molecules, an improved estimate of the initial Hessian has increased the efficiency by at least a factor of two. Greater efficiencies (usually >40%) are also obtained when static-metric GDIIS is used to refine the geometry after the initial application of a transition-state search based on the variable metric combined with line search algorithm. On the basis of these results, we recommend several changes to the algorithms as currently implemented in the standard semiempirical program packages.     

The search for stationary points on a quantum mechanical/molecular mechanical potential-energy surface

 We propose a methodology to locate stationary points on a quantum mechanical/molecular mechanical potential-energy surface. This algorithm is based on a suitable approximation of an initial full Hessian matrix, either a modified Broyden–Fletcher–Goldfarg–Shanno or a Powell update formula for the location of, respectively, a minimum or a transition state, and the so-called rational function optimization. The latter avoids the Hessian matrix inversion required by a quasi-Newton–Raphson method. Some examples are presented and analyzed. Received: 16 July 2001 / Accepted: 9 October 2001 / Published online: 9 January 2002     

Geometric molecular similarity from volume-based distance minimization: Application to saxitoxin and tetrodotoxin

The geometric molecular dissimilarity between two molecules is defined as the difference between the volume of their union minus the volume of their intersection. This dissimilarity has the mathematical properties of a distance. This distance is minimized under all rotations and translations using a discrete Broyden, Fletcher, Goldfarb & Shanno (B.F.G.S.) algorithm. The optimal geometric superimposition of saxitoxin and tetrodotoxin is discussed. © 1995 by John Wiley & Sons, Inc.     

Another way to implement the Powell formula for updating Hessian matrices related to transition structures

A way to update the Hessian matrix according to the Powell formula is given. With this formula one does not need to store the full Hessian matrix at any iteration. A method to find transition structures, which is a combination of the quasiNewton–Raphson augmented Hessian algorithm with the proposed Powell update scheme, is also given. The diagonalization of the augmented Hessian matrix is carried out by Lanczoslike methods. In this way, during all the optimization process, one avoids to store full matrices.     

Hessian‐free low‐mode conformational search for large‐scale protein loop optimization: application to c‐jun N‐terminal kinase JNK3

A Hessian‐free low‐mode search algorithm has been developed for large‐scale conformational searching. The new method is termed LLMOD, and it utilizes the ARPACK package to compute low‐mode eigenvectors of a Hessian matrix that is only referenced implicitly, through its product with a series of vectors. The Hessian × vector product is calculated utilizing a finite difference formula based on gradients. LLMOD is the first conformational search method that can be applied to fully flexible, unconstrained protein structures for complex loop optimization problems. LLMOD has been tested on a particularly difficult model system, c‐jun N‐terminal kinase JNK3. We demonstrate that LLMOD was able to correct a P38/ERK2/HCL‐based homology model that grossly misplaced the crucial glycine‐rich loop in the ATP‐binding site. © 2000 John Wiley & Sons, Inc. J Comput Chem 22: 21–30, 2001     

Finding transition states using reduced potential-energy surfaces

 We propose a method to locate saddle points that is based on the interplay between the driving coordinate and the restricted quasi-Newton algorithm. The method locates the transition state using a reduced potential-energy surface. The reduced potential-energy surface is characterized by the set of driving coordinates. The proposed algorithm starts at a point on the surface that is slightly perturbed from either reactant or product and, in principle, converges to the transition state. Finally we give a special type of update Hessian matrix formula that should be applied in optimizations carried out on reduced potential-energy surfaces. Received: 29 September 2000 / Accepted: 3 January 2001 / Published online: 3 April 2001     

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     

Comparison of methods for finding saddle points without knowledge of the final states

Within the harmonic approximation to transition state theory, the biggest challenge involved in finding the mechanism or rate of transitions is the location of the relevant saddle points on the multidimensional potential energy surface. The saddle point search is particularly challenging when the final state of the transition is not specified. In this article we report on a comparison of several methods for locating saddle points under these conditions and compare, in particular, the well-established rational function optimization (RFO) methods using either exact or approximate Hessians with the more recently proposed minimum mode following methods where only the minimum eigenvalue mode is found, either by the dimer or the Lanczos method. A test problem involving transitions in a seven-atom Pt island on a Pt(111) surface using a simple Morse pairwise potential function is used and the number of degrees of freedom varied by varying the number of movable atoms. In the full system, 175 atoms can move so 525 degrees of freedom need to be optimized to find the saddle points. For testing purposes, we have also restricted the number of movable atoms to 7 and 1. Our results indicate that if attempting to make a map of all relevant saddle points for a large system (as would be necessary when simulating the long time scale evolution of a thermal system) the minimum mode following methods are preferred. The minimum mode following methods are also more efficient when searching for the lowest saddle points in a large system, and if the force can be obtained cheaply. However, if only the lowest saddle points are sought and the calculation of the force is expensive but a good approximation for the Hessian at the starting position of the search can be obtained at low cost, then the RFO approaches employing an approximate Hessian represent the preferred choice. For small and medium sized systems where the force is expensive to calculate, the RFO approaches employing an approximate Hessian is also the more efficient, but when the force and Hessian can be obtained cheaply and only the lowest saddle points are sought the RFO approach using an exact Hessian is the better choice. These conclusions have been reached based on a comparison of the total computational effort needed to find the saddle points and the number of saddle points found for each of the methods. The RFO methods do not perform very well with respect to the latter aspect, but starting the searches further away from the initial minimum or using the hybrid RFO version presented here improves this behavior considerably in most cases.     

On the restricted step method coupled with the augmented Hessian for the search of stationary points of any continuous function

In any optimization using the augmented Hessian technique, the step is not restricted to any length. Since the restriction of the step at each iteration is very important in order to achieve good convergence, we present a coupled method such that the augmented Hessian automatically gives both the adequate length of the step and the correct Hessian structure. The method is showed for the minima and saddle points of any order. © 1997 John Wiley & Sons, Inc.     

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     

A conceptual realization of stationary points using the Hessian matrix

In this study we use surface-fitting equations to generate the energy and its first derivative in terms of two torsional angles in the methanediol model, using GAUSSIAN 88 at the 4-31G level of approximation. The Hessian matrix is further used to locate the stationary points, and a Gaussian fit of the absolute values of the sum of the eigenvalues of the Hessian is used in order to generate a surface in which all the stationary points are identified.     

Computation of stationary points via a homotopy method

A homotopy method is presented that locates both minimizers and saddle points of energy functions in an efficient manner. In contrast to other methods, it makes possible the exploration of large parts of potential energy surfaces. Along a homotopy path stationary points of odd and even order occur alternately. A path tracing procedure requiring only gradients and at most one evaluation of the Hessian matrix is given. Test results on a model potential and three MINDO/3 potentials are reported. Received: 6 May 1996 / Accepted: 2 April 1998 / Published online: 23 June 1998     

Spinor optimization for a relativistic spin-dependent CASSCF program

Detailed formulae for the implementation of the multi-configuration SCF spinor optimization in a basis of Kramers pair 2-spinors – i.e. exploiting time-reversal symmetry – are presented. Full expressions for the spinor gradient and spinor Hessian elements are given in abstract form as well as within the usual CASSCF subspace division. As far as possible, the resulting terms are grouped to relativistic inactive and active Fock matrices, which have been introduced previously. Approximations for the Hessian are introduced so as to initialize it in an inverse Hessian update algorithm for a diagonal first approximation within the standard quasi-Newton-Raphson procedure. The effects of double group symmetry arising from spin dependence on Fock matrices and therefore gradient and Hessian are discussed and a group scheme for the implementation is proposed. Received: 14 January 1997 / Accepted: 3 February 1997