Group members
Daniel Sheppard and Rye Terrell
Introduction
This project is a comparison of different saddle point finding methods. All the methods we studied only need the force and energy of the system (and not the Hessian matrix of second derivative). Because of this, the methods scale well with system size, and can be used for relatively large systems using interaction potentialas expensive as density functional theory.
NEB
The Nudged Elastic Band (NEB) method is used to find reaction pathways when both the initial and final states are known. Using this code, the Minimum Energy Path (MEP) for any given chemical process may be calculated, however both the initial and final states must be known. The code works by linearly interpolating a set of images between the known initial and final states (as a "guess" at the MEP), and then minimizes the energy of this string of images. Each "image" corresponds to a specific geometry of the atoms on their way from the initial to the final state, a snapshot along the reaction path. Thus, once the enery of this string of images has been minimized, the true MEP is revealed.
Climbing Image
With this code, it is possible to calculate not only the MEP for the reaction but also the transition state (TS) configuration at the saddle point. The climbing image NEB method has one modification which drives the image with the highest energy up to the saddle point. This image does not see the spring forces along the band. Instead, the true force at this image along the tangent is inverted. In this way, the image tries to maximize it's energy along the band, and minimize in all other directions. When this image converges, it will be at the exact saddle point.
Because the highest image is moved to the saddle point and it does not see the spring forces, the spacing of images on either side of this image will be different. It can be important to do some minimization with the regular NEB method before this flag is turned on, both to have a good estimate of the reaction co-ordinate around the saddle point, and so that the highest image is close to the saddle point. If the maximum image is initially very far from the saddle point, and the climbing image was used from the outset, the path would develop very different spacing on either side of the saddle point.
Improved Tangent
The most natural definition of the tangent along the band is one that leads to problems. This tangent, which is described in detail in references 4 and 5, is defined at an image on the band as being the vector between the two neighboring images of the central image. This is a standard central difference approximation. Unfortunately, this definition can lead to the formation of kinks along the band. If the force along the band becomes large with respect to the curvature perpendicular to the band times the image spacing, these kinks will form. A possible solution is to introduce an artificial angular dependent force which tends to straighten the band if the kinks start to develop. Another solution which is nessecary for using the climbing image is to use a non-central different tangent in which the tangent at an image is defined by the vector between that image and it's lower energy neighbor. An intuitive way of thinking about this tangent is that each image is hanging from it's higher energy neighbor, trying to get in low as energy as possible. Because each image is dependent only on the image above it in energy, the band hangs stabily down from the saddle point.
Common User Pitfalls
The NEB is a two ended search method. It is important to remember that every atom in the initial state maps to every atom in the finial. The method treats each atom as a distinguishable atom. It is quite common for a user to over look this aspect when setting up a job. The NEB will calculate attempt to calculte a barrier, however there is a good chance that any resultant saddle might not be the most energeticly favorable saddle.
The Dimer Method
The dimer method (or more generally a min-mode method) is used to find saddle points on a potential energy surface. It is complimentary to the nudged elastic band method because it does not require a final state. We have used the dimer method in two ways. The first is to evolve a configuration from an initial guess to a saddle point. The second application, which is much more challenging, is to find all low lying saddle points connecting the initial basin to adjacent basins. This problem is of particular interest in rate theory. If the transitions out of a basin are found, their individual rates can be evaluated, and the system can be evolved over long time scales using kinetic Monte Carlo. This involves randomly picking one of the transitions out of a Boltzmann distribution, and moving the system over that saddle point to an adjacent basin.
The Dimer in VASP
The dimer method has been incorperated into the DFT code VASP (Vienna Ab-initio Simulation Package). Our local collection of information, source and scripts is located here.
Benchmarking
See the results of our testing on our benchmarking page.
References
D. Sheppard, R. Terrell, and G. Henkelman,
Optimization methods for finding minimum energy paths,
J. Chem. Phys. 128, 134106 (2008).
R. A. Olsen, G. J. Kroes, G. Henkelman, A. Arnaldsson, and H. Jónsson,
Comparison of methods for finding saddle points without knowledge of the final states,
J. Chem. Phys. 121, 9776-9792 (2004).
G. Henkelman, G. Jóhannesson, and H. Jónsson,
Methods for Finding Saddle Points and Minimum Energy Paths,
in Progress on Theoretical Chemistry and Physics,
Ed. S. D. Schwartz, 269-300 (Kluwer Academic Publishers, 2000).
G. Henkelman, B.P. Uberuaga, and H. Jónsson,
A climbing image nudged elastic band method for finding saddle points and minimum energy paths,
J. Chem. Phys., 113, 9901 (2000).
G. Henkelman and H. Jónsson,
Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points,
J. Chem. Phys., 113, 9978 (2000).
Henkelman and H. Jónsson,
A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives,
J. Chem. Phys., 111, 7010 (1999).
H. Jónsson, G. Mills, K. W. Jacobsen,
Nudged Elastic Band Method for Finding Minimum Energy Paths of Transitions,
in Classical and Quantum Dynamics in Condensed Phase Simulations,
Ed. B. J. Berne, G. Ciccotti and D. F. Coker, 385 (World Scientific, 1998).