Geometry optimization, including searching for transition states, accounts for most of the CPU time spent in quantum chemistry, computational surface science, and solid-state physics, and also plays an important role in simulations employing classical force fields. We have implemented a geometry optimizer, called DL-FIND, to be included in atomistic simulation codes. It can optimize structures in Cartesian coordinates, redundant internal coordinates, hybrid-delocalized internal coordinates, and also functions of more variables independent of atomic structures. The implementation of the optimization algorithms is independent of the coordinate transformation used. Steepest descent, conjugate gradient, quasi-Newton, and L-BFGS algorithms as well as damped molecular dynamics are available as minimization methods. The partitioned rational function optimization algorithm, a modified version of the dimer method and the nudged elastic band approach provide capabilities for transition-state search. Penalty function, gradient projection, and Lagrange-Newton methods are implemented for conical intersection optimizations. Various stochastic search methods, including a genetic algorithm, are available for global or local minimization and can be run as parallel algorithms. The code is released under the open-source GNU LGPL license. Some selected applications of DL-FIND are surveyed.

The discrete path sampling method was used to investigate the folding of a three-stranded antiparallel beta-sheet peptide, Beta3s, described by an empirical potential and implicit solvent model. After application of a coarse-graining scheme that groups together sets of minima in local equilibrium, the calculated folding time was in reasonable agreement with other simulations and consistent with the experimental upper bound. The folding mechanism exhibited by the most significant discrete paths involves early formation of the C-terminal hairpin followed by docking of the N-terminal strand.

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We report a new algorithm for constructing pathways between local minima that involve a large number of intervening transition states on the potential energy surface. A significant improvement in efficiency has been achieved by changing the strategy for choosing successive pairs of local minima that serve as endpoints for the next search. We employ Dijkstra's algorithm [E. W. Dijkstra, Numer. Math. 1, 269 (1959)] to identify the "shortest" path corresponding to missing connections within an evolving database of local minima and the transition states that connect them. The metric employed to determine the shortest missing connection is a function of the minimized Euclidean distance. We present applications to the formation of buckminsterfullerene and to the folding of various biomolecules: the B1 domain of protein G, tryptophan zippers, and the villin headpiece subdomain. The corresponding pathways contain up to 163 transition states and will be used in future discrete path sampling calculations.

The results of basin-hopping global optimization simulations are presented for four small, alpha-helical proteins described by a coarse-grained potential. A step-taking scheme that incorporates the local conformational preferences extracted from a large number of high-resolution protein structures is compared with an unbiased scheme. In addition, the discrete path sampling method is used to investigate the folding of one of the proteins, namely, the villin headpiece subdomain. Folding times from kinetic Monte Carlo simulations and iterative calculations based on a Markovian first-step analysis for the resulting stationary-point database are in good mutual agreement, but differ significantly from the experimental values, probably because the native state is not the global free energy minimum for the potential employed.

A quasi-continuous interpolation (QCI) scheme is introduced for characterizing physically realistic initial pathways from which to initiate transition state searches and construct kinetic transition networks. Applications are presented for peptides, proteins, and a morphological transformation in an atomic cluster. The first step in each case involves end point alignment, and we describe the use of a shortest augmenting path algorithm for optimizing permutational isomers. The QCI procedure then employs an interpolating potential, which preserves the covalent bonding framework for the biomolecules and includes repulsive terms between unconstrained atoms. This potential is used to identify an interpolating path by minimizing contributions from a connected set of images, including terms corresponding to minima in the interatomic distances between them. This procedure detects unphysical geometries in the line segments between images. The most difficult cases, where linear interpolation would involve chain crossings, are treated by growing the structure an atom at a time using the interpolating potential. To test the QCI procedure, we carry through a series of benchmark calculations where the initial interpolation is coupled to explicit transition state searches to produce complete pathways between specified local minima.

Refinement of databases of connected stationary points to describe global kinetics is discussed for the GB1 hairpin peptide modelled by an empirical potential and an implicit solvent model. Two approaches to the removal of artificial kinetic frustration caused by undersampling are separately applied to an initial database of stationary points. We consider both additional sampling between minima close in energy but separated by high barriers, and the removal of stationary points that do not contribute significantly to the calculated rate constants for the initial database. Results from these two approaches are found to be consistent: the transition networks produced in both cases exhibit structure-seeking properties because most of the initial frustration is removed. Excluding stationary points from the initial database that do not appear on kinetically relevant paths proves to be much less computationally expensive than subsequently finding better connections for them. After application of a coarse-graining scheme that groups together sets of minima separated by low barriers, the calculated folding time is consistent with expectations for beta-hairpins modelled using implicit solvent. The folding mechanism corresponding to the most significant kinetic paths involves early compaction, followed by formation of the turn and then completion of the hydrophobic core.

Initially in a collision between antihydrogen and hydrogen the electron is bound to the proton and the positron to the antiproton. Clearly, if the proton and the antiproton coincide, they cannot bind the light particles. In this letter, an upper bound value of 0.8 a 0 is obtained for the critical value of the internuclear distance below which the electron and the positron cease to be bound to the nuclei as they can attain a lower energy by separating from the nuclei and forming positronium.A small number of antihydrogen (AH) atoms were prepared at CERN and Fermilab in 1996. These atoms were travelling at speeds close to that of light. However, an antiproton decelerator is under construction at CERN that should soon make it possible to trap antihydrogen at very low temperatures (<1 K) in an inhomogeneous magnetic field. This will enable experimentalists to study the properties of antihydrogen in its rest frame. Two experiments (ATHENA and ATRAP) are planned to carry out such studies, starting in 1999. For a review of the background to research on antihydrogen, see Charlton et al (1994) andHolzscheiter et al (1997).These developments can be expected to result in increasing interest in antihydrogen and its properties on the part of theoreticians. One area requiring attention is collisions that cause loss of trapped AH. These losses are mainly due to AH collisions with H 2 and He at energies up to room temperature. Accurate cross sections for these collision processes would make it possible for experimentalists to choose the simplest method for maximizing the AH lifetime (Charlton 1996).As a first step towards carrying out such calculations, we have started work on the interaction between AH and H.