Hannes Jónsson

A climbing image nudged elastic band method for finding saddle points and minimum energy paths

A modification of the nudged elastic band method for finding minimum energy paths is presented. One of the images is made to climb up along the elastic band to converge rigorously on the highest saddle point. Also, variable spring constants are used to increase the density of images near the top of the energy barrier to get an improved estimate of the reaction coordinate near the saddle point. Applications to CH4 dissociative adsorption on Ir~111! and H2 on Si~100! using plane wave based density functional theory are presented.

Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points

An improved way of estimating the local tangent in the nudged elastic band method for finding minimum energy paths is presented. In systems where the force along the minimum energy path is large compared to the restoring force perpendicular to the path and when many images of the system are included in the elastic band, kinks can develop and prevent the band from converging to the minimum energy path. We show how the kinks arise and present an improved way of estimating the local tangent which solves the problem. The task of finding an accurate energy and configuration for the saddle point is also discussed and examples given where a complementary method, the dimer method, is used to efficiently converge to the saddle point. Both methods only require the first derivative of the energy and can, therefore, easily be applied in plane wave based density-functional theory calculations. Examples are given from studies of the exchange diffusion mechanism in a Si crystal, Al addimer formation on the Al(100) surfa...

A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives

The problem of determining which activated (and slow) transitions can occur from a given initial state at a finite temperature is addressed. In the harmonic approximation to transition state theory this problem reduces to finding the set of low lying saddle points at the boundary of the potential energy basin associated with the initial state, as well as the relevant vibrational frequencies. Also, when full transition state theory calculations are carried out, it can be useful to know the location of the saddle points on the potential energy surface. A method for finding saddle points without knowledge of the final state of the transition is described. The method only makes use of first derivatives of the potential energy and is, therefore, applicable in situations where second derivatives are too costly or too tedious to evaluate, for example, in plane wave based density functional theory calculations. It is also designed to scale efficiently with the dimensionality of the system and can be applied to ve...

Reversible work transition state theory: application to dissociative adsorption of hydrogen

Abstract A practical method for finding free energy barriers for transitions in high-dimensional classical and quantum systems is presented and used to calculate the dissociative sticking probability of H 2 on a metal surface within the transition state theory. The reversible work involved in shifting the system confined to a hyperplane from the reactant region towards products is evaluated directly. Quantum mechanical degrees of freedom are included by using Feynman path integrals with the hyperplane constraint applied to the centroid of the cyclic paths. An optimal dividing surface for the rate estimated by the transition state theory is identified naturally in the course of the reversible work evaluation. The free energy barrier is determined relative to the reactant state directly so that an estimate of the transition rate can be obtained without requiring a solvable reference model for the transition state. The method has been applied to calculations of the sticking probability of a thermalized hydrogen gas on a Cu(110) surface. The two hydrogen atoms and eight surface Cu atoms were included quantum mechanically and over two hundred atoms in the Cu crystal where included classically. The activation energy for adsorption and desorption was determined and found to be significantly lowered by tunneling at low temperature. The calculated values agree quite well with experimental estimates for adsorption and desorption. Dynamical corrections to the classical transition state theory rate estimate were evaluated and found to be small.

Systematic analysis of local atomic structure combined with 3D computer graphics

Abstract The implementation of a method for systematic analysis of local atomic structure in combination with 3D computer graphics is described. The method, Common Neighbor Analysis, is a decomposition of the radial distribution function according to the local environment of the pairs of atoms and can provide direct interpretation of various features of the radial distribution function in terms of atomic structure. It can also be used to identify atoms in particular environment, such as FCC, HCP, BCC or icosahedral. We describe an application of this program to a study of crystal nucleation in a molten Cu slab. While the majority of atoms in the resulting crystals are classified as being FCC, stacking faults are observed and can be traced back to the near-critical nuclei.

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.

Long time scale kinetic Monte Carlo simulations without lattice approximation and predefined event table

We present a method for carrying out long time scale dynamics simulations within the harmonic transition state theory approximation. For each state of the system, characterized by a local minimum on the potential energy surface, multiple searches for saddle points are carried out using random initial directions. The dimer method is used for the saddle point searches and the rate for each transition mechanism is estimated using harmonic transition state theory. Transitions are selected and the clock advanced according to the kinetic Monte Carlo algorithm. Unlike traditional applications of kinetic Monte Carlo, the atoms are not assumed to sit on lattice sites and a list of all possible transitions need not be specified beforehand. Rather, the relevant transitions are found on the fly during the simulation. A multiple time scale simulation of Al(100) crystal growth is presented where the deposition event, occurring on the time scale of picoseconds, is simulated by ordinary classical dynamics, but the time i...

Density functional theory calculations for the hydrogen evolution reaction in an electrochemical double layer on the Pt(111) electrode

We present results of density functional theory calculations on a Pt(111) slab with a bilayer of water, solvated protons in the water layer, and excess electrons in the metal surface. In this way we model the electrochemical double layer at a platinum electrode. By varying the number of protons/electrons in the double layer we investigate the system as a function of the electrode potential. We study the elementary processes involved in the hydrogen evolution reaction, 2(H(+) + e(-)) --> H(2), and determine the activation energy and predominant reaction mechanism as a function of electrode potential. We confirm by explicit calculations the notion that the variation of the activation barrier with potential can be viewed as a manifestation of the Brønsted-Evans-Polanyi-type relationship between activation energy and reaction energy found throughout surface chemistry.

Methods for Finding Saddle Points and Minimum Energy Paths

The problem of finding minimum energy paths and, in particular, saddle points on high dimensional potential energy surfaces is discussed. Several different methods are reviewed and their efficiency compared on a test problem involving conformational transitions in an island of adatoms on a crystal surface. The focus is entirely on methods that only require the potential energy and its first derivative with respect to the atom coordinates. Such methods can be applied, for example, in plane wave based Density Functional Theory calculations, and the computational effort typically scales well with system size. When the final state of the transition is known, both the initial and final coordinates of the atoms can be used as boundary conditions in the search. Methods of this type include the Nudged Elastic Band, Ridge, Conjugate Peak Refinement, Drag method and the method of Dewar, Healy and Stewart. When only the initial state is known, the problem is more challenging and the search for the saddle point represents also a search for the optimal transition mechanism. We discuss a recently proposed method that can be used in such cases, the Dimer method.

Molecular multipole moments of water molecules in ice Ih

We have used an induction model including dipole, dipole–quadrupole, quadrupole–quadrupole polarizability and first hyperpolarizability as well as fixed octopole and hexadecapole moments to study the electric field in ice. The self-consistent induction calculations gave an average total dipole moment of 3.09 D, a 67% increase over the dipole moment of an isolated water molecule. A previous, more approximate induction model study by Coulson and Eisenberg [Proc. R. Soc. Lond. A 291, 445 (1966)] suggested a significantly smaller average value of 2.6 D. This value has been used extensively in recent years as a reference point in the development of various polarizable interaction potentials for water as well as for assessment of the convergence of water cluster properties to those of bulk. The reason for this difference is not due to approximations made in the computational scheme of Coulson and Eisenberg but rather due to the use of less accurate values for the molecular multipoles in these earlier calculations.