# Nudged Elastic Band

#### 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 feel 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 could develop very different spacing on either side of the saddle point. That said, it is generally safe to use the NEB with the climbing images turned on at all times.

#### 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 its lower energy neighbor. An intuitive way of thinking about this tangent is that each image is hanging from its 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 double ended search method. It is important to remember that every atom in the initial state maps to every atom in the final. The method treats each atom as a distinguishable atom. It is quite common for a user to overlook this aspect when setting up a calculation. The NEB will attempt to calculate a barrier, however there is a good chance that the resultant saddle will not be the most energeticly favorable saddle. This 7 atom Lennard-Jones island illustrates how energetically degenerate configurations can are connected via different energy saddle points. In the same vein of thinking the NEB only minimizes to the local MEP. There can be multiple saddles that connect two states. For example there are two degenerate saddle points which connect the cis and trans isomers of 2-butene.

#### The NEB in VASP and TSSE

The NEB method has been incorperated into the DFT code VASP (Vienna Ab-initio Simulation Package). Our local collection of information, source and scripts are located here. In addition it has also be implemented in our python code tsase (transition states for the atomic simulation environment)

### References

D. Sheppard, R. Terrell, and G. Henkelman, Optimization methods for finding minimum energy paths, J. Chem. Phys. 128, 134106 (2008).

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).

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).