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     

Remarks on the updated Hessian matrix methods

Optimizing a function with respect to a set of variables using the quasi‐Newton–Raphson method implies updating the Hessian matrix at each iteration. The Broyden–Fletcher–Goldfarb–Shanno update formula is used for minimization and the Murtagh–Sargent–Powell update formula for optimization of first‐order saddle points. Two new formulae are proposed to update the Hessian matrix. One of these formulae is derived using exponential weights and should be used to locate first‐order saddle points. The second formula is a modification of the TS–Broyden–Fletcher–Goldfarb–Shanno update and could used for both minimum and first‐order saddle point optimizations. These two update Hessian matrix formulae present a performance that is the same and in many cases better that the Broyden–Fletcher–Goldfarb–Shanno and Murtagh–Sargent–Powell formulae. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem 94: 324–332, 2003     

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.     

Parallel algorithm for efficient calculation of second derivatives of conformational energy function in internal coordinates

A parallel algorithm for efficient calculation of the second derivatives (Hessian) of the conformational energy in internal coordinates is proposed. This parallel algorithm is based on the master/slave model. A master processor distributes the calculations of components of the Hessian to one or more slave processors that, after finishing their calculations, send the results to the master processor that assembles all the components of the Hessian. Our previously developed molecular analysis system for conformational energy optimization, normal mode analysis, and Monte Carlo simulation for internal coordinates is extended to use this parallel algorithm for Hessian calculation on a massively parallel computer. The implementation of our algorithm uses the message passing interface and works effectively on both distributed-memory parallel computers and shared-memory parallel computers. We applied this system to the Newton–Raphson energy optimization of the structures of glutaminyl transfer RNA (Gln-tRNA) with 74 nucleotides and glutaminyl-tRNA synthetase (GlnRS) with 540 residues to analyze the performance of our system. The parallel speedups for the Hessian calculation were 6.8 for Gln-tRNA with 24 processors and 11.2 for GlnRS with 54 processors. The parallel speedups for the Newton–Raphson optimization were 6.3 for Gln-tRNA with 30 processors and 12.0 for GlnRS with 62 processors. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1716–1723, 1998     

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     

Third-order methods for molecular geometry optimizations

Third-order optimization methods that require the evaluation of the gradient and initial estimates for the second and third derivatives are described. Update algorithms for the Hessian and the third-derivative tensor are outlined. The direct inversion in the iterative subspace scheme is extended to third order and is combined with the third-order update procedures. For geometry optimization, an approximate third-derivative tensor is constructed from simple empirical formulas. Examples of application to Hartree–Fock geometry optimization problems are given. © 1993 John Wiley & Sons, Inc.     

Global optimization using bad derivatives: Derivative-free method for molecular energy minimization

A general method designed to isolate the global minimum of a multidimensional objective function with multiple minima is presented. The algorithm exploits an integral “coarse-graining” transformation of the objective function, U, into a smoothed function with few minima. When the coarse-graining is defined over a cubic neighborhood of length scale ϵ, the exact gradient of the smoothed function, 𝒰_ϵ, is a simple three-point finite difference of U. When ϵ is very large, the gradient of 𝒰_ϵ appears to be a “bad derivative” of U. Because the gradient of 𝒰_ϵ is a simple function of U, minimization on the smoothed surface requires no explicit calculation or differentiation of 𝒰_ϵ. The minimization method is “derivative-free” and may be applied to optimization problems involving functions that are not smooth or differentiable. Generalization to functions in high-dimensional space is straightforward. In the context of molecular conformational optimization, the method may be used to minimize the potential energy or, preferably, to maximize the Boltzmann probability function. The algorithm is applied to conformational optimization of a model potential, Lennard–Jones atomic clusters, and a tetrapeptide. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1445–1455, 1998     

Technique for incorporating the density functional Hessian into the geometry optimization of biomolecules, solvated molecules, and large floppy molecules

Traditional geometry optimization methods require the gradient of the potential surface, together with a Hessian which is often approximated. Approximation of the Hessian causes difficulties for large, floppy molecules, increasing the number of steps required to reach the minimum. In this article, the costly evaluation of the exact Hessian is avoided by expanding the density functional to second order in both the nuclear and electronic variables, and then searching for the minimum of the quadratic functional. The quadratic search involves the simultaneous determination of both the geometry step and the associated change in the electron density matrix. Trial calculations on Taxol indicate that the cost of the quadratic search is comparable to the cost of the density functional energy plus gradient. While this procedure circumvents the bottleneck coupled-perturbed step in the evaluation of the full Hessian, the second derivatives of the electron-repulsion integrals are still required for atomic-orbital-based calculations, and they are presently more expensive than the energy plus gradient. Hence, we anticipate that the quadratic optimizer will initially find application in fields in which existing optimizers breakdown or are inefficient, particularly biochemistry and solvation chemistry.     

Estimating the hessian for gradient-type geometry optimizations

Optimization methods that use gradients require initial estimates of the Hessian or second derivative matrix; the more accurate the estimate, the more rapid the convergence. For geometry optimization, an approximate Hessian or force constant matrix is constructed from a simple valence force field that takes into account the inherent connectivity and flexibility of the molecule. Empirical rules are used to estimate the diagonal force constants for a set of redundant internal coordinates consisting of all stretches, bends, torsions and out-of-plane deformations involving bonded atoms. The force constants are transformed from the redundant internal coordinates to Cartesian coordinates, and then from Cartesian coordinates to the non-redundant internal coordinates used in the specification of the geometry and optimization. This method is especially suitable for cyclic molecules. Problems associated with the choice of internal coordinates for geometry optimization are also discussed.Fellow of the Alfred P. Sloan Foundation, 1981–83     

Molecular tailoring approach for geometry optimization of large molecules: energy evaluation and parallelization strategies

A linear-scaling scheme for estimating the electronic energy, gradients, and Hessian of a large molecule at ab initio level of theory based on fragment set cardinality is presented. With this proposition, a general, cardinality-guided molecular tailoring approach (CG-MTA) for ab initio geometry optimization of large molecules is implemented. The method employs energy gradients extracted from fragment wave functions, enabling computations otherwise impractical on PC hardware. Further, the method is readily amenable to large scale coarse-grain parallelization with minimal communication among nodes, resulting in a near-linear speedup. CG-MTA is applied for density-functional-theory-based geometry optimization of a variety of molecules including alpha-tocopherol, taxol, gamma-cyclodextrin, and two conformations of polyglycine. In the tests performed, energy and gradient estimates obtained from CG-MTA during optimization runs show an excellent agreement with those obtained from actual computation. Accuracy of the Hessian obtained employing CG-MTA provides good hope for the application of Hessian-based geometry optimization to large molecules.     

Using bonding to guide transition state optimization

Optimization of a transition state typically requires both a good initial guess of the molecular structure and one or more computationally demanding Hessian calculations to converge reliably. Often, the transition state being optimized corresponds to the barrier in a chemical reaction where bonds are being broken and formed. Utilizing the geometries and bonding information for reactants and products, an algorithm is outlined to reliably interpolate an initial guess for the transition state geometry. Additionally, the change in bonding is also used to increase the reliability of transition state optimizations that utilize approximate and updated Hessian information. These methods are described and compared against standard transition state optimization methods. © 2015 Wiley Periodicals, 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.     

Intra‐ and intermolecular interactions in crystals of polar molecules. A study by the mixed quantum mechanical/molecular mechanical SCMP‐NDDO method

Stabilization energies of crystals of polar molecules were calculated with the recently developed NDDO‐SCMP method that determines the wave function of a subunit embedded in the symmetrical environment constituted by the copies of the subunit. The total stabilization energies were decomposed into four components. The deformation energy is the difference between the energy of the molecule in the geometries adopted in the crystal on the one hand, and in vacuo, on the other hand. Further energy components are derived from the molecular geometry found in the crystal phase. The electrostatic component is the interaction energy of the molecule with the crystal field, corresponding to the charge distribution obtained in vacuo. The polarization component is the energy lowering resulted in the self‐consistent optimization of the wave function in the crystal field. The rest of the stabilization energy is attributed to the dispersion–repulsion component, and is calculated from an empirical potential function. The major novelty of this decomposition scheme is the introduction of the deformation energy. It requires the optimization of the structural parameters, including the molecular geometry, the intermolecular coordinates, and the cell parameters of the crystal. The optimization is performed using the recently implemented forces in the SCMP‐NDDO method, and this new feature is discussed in detail. The calculation of the deformation energy is particularly important to obtain stabilization energies for crystals in which the molecular geometry differs considerably from that corresponding to the energy minimum of the isolated molecule. As an example, crystals of diastereoisomeric salts are investigated. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1679–1690, 2001     

Techniques for geometry optimization: A comparison of cartesian and natural internal coordinates

A comparison is made between geometry optimization in Cartesian coordinates, using an appropriate initial Hessian, and natural internal coordinates. Results on 33 different molecules covering a wide range of symmetries and structural types demonstrate that both coordinate systems are of comparable efficiency. There is a marked tendency for natural internals to converge to global minima whereas Cartesian optimizations converge to the local minimum closest to the starting geometry. Because they can now be generated automatically from input Cartesians, natural internals are to be preferred over Z-matrix coordinates. General optimization strategies using internal coordinates and/or Cartesians are discussed for both unconstrained and constrained optimization. © John Wiley & Sons, Inc.     

Size‐guided multi‐seed heuristic method for geometry optimization of clusters: Application to benzene clusters

Since searching for the global minimum on the potential energy surface of a cluster is very difficult, many geometry optimization methods have been proposed, in which initial geometries are randomly generated and subsequently improved with different algorithms. In this study, a size‐guided multi‐seed heuristic method is developed and applied to benzene clusters. It produces initial configurations of the cluster with n molecules from the lowest‐energy configurations of the cluster with n − 1 molecules (seeds). The initial geometries are further optimized with the geometrical perturbations previously used for molecular clusters. These steps are repeated until the size n satisfies a predefined one. The method locates putative global minima of benzene clusters with up to 65 molecules. The performance of the method is discussed using the computational cost, rates to locate the global minima, and energies of initial geometries. © 2018 Wiley Periodicals, Inc.     

Fast geometry optimizationin self-cosistent reaction field computations on solvated molecules

The free energy gradient or Hessian of a molecule interacting with a liquid represented by a dielectric continuum is derived in the self-consistent reaction field formalism. An ellipsoidal approximation of the cavity is proposed with an algorithm to automatically define the ellipsoid from the nuclear coordinates of the atoms. With this approximation, geometry optimization of the solvated molecule becomes very fast. This method has been implemented in some standard ab initio or semiempirical computational codes. As a first test of the method, full geometry optimization of formamide in a high dielectric constant medium reveals that the CPU time needed for one optimization cycle is less than 3% longer for a solvated species than for the corresponding free molecule.     

A quantum chemical method for rapid optimization of protein structures

A quantum chemical method for rapid optimization of protein structures is proposed. In this method, a protein structure is treated as an assembly of amino acid units, and the geometry optimization of each unit is performed with taking the effect of its surrounding environment into account. The optimized geometry of a whole protein is obtained by repeated application of such a local optimization procedure over the entire part of the protein. Here, we implemented this method in the MOPAC program and performed geometry optimization for three different sizes of proteins. Consequently, these results demonstrate that the total energies of the proteins are much efficiently minimized compared with the use of conventional optimization methods, including the MOZYME algorithm (a representative linear-scaling method) with the BFGS routine. The proposed method is superior to the conventional methods in both CPU time and memory requirements.     

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     

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     

TheRate: Program for ab initio direct dynamics calculations of thermal and vibrational-state-selected rate constants

We introduce TheRate (THEoretical RATEs), a complete application program with a graphical user interface (GUI) for calculating rate constants from first principles. It is based on canonical variational transition-state theory (CVT) augmented by multidimensional semiclassical zero and small curvature tunneling approximations. Conventional transition-state theory (TST) with one-dimensional Wigner or Eckart tunneling corrections is also available. Potential energy information needed for the rate calculations are obtained from ab initio molecular orbital and/or density functional electronic structure theory. Vibrational-state-selected rate constants may be calculated using a diabetic model. TheRate also introduces several technical advancements, namely the focusing technique and energy interpolation procedure. The focusing technique minimizes the number of Hessian calculations required by distributing more Hessian grid points in regions that are critical to the CVT and tunneling calculations and fewer Hessian grid points elsewhere. The energy interpolation procedure allows the use of a computationally less demanding electronic structure theory such as DFT to calculate the Hessians and geometries, while the energetics can be improved by performing a small number of single-point energy calculations along the MEP at a more accurate level of theory. The CH₄+H↔CH₃+H₂ reaction is used as a model to demonstrate usage of the program, and the convergence of the rate constants with respect to the number of electronic structure calculations. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1039–1052, 1998