The Nudged Elastic Band Method and its Implementation in VASP S. Sinthika

Goals Determine minimum-energy structures of complex systems. • Calculate activation energies. • Predict the mechanisms for chemical/physical processes.

Simple Potential Energy Curve Equilibrium Bond length Higher dimensional cases?

Stationary point- (forces are zero) Stable configurations Local and Global minima

Reaction Path Using Optimization techniques: can locate local minima (reactant and product). Evolution of system or reaction path

Minimum energy path A path is an MEP only if the forces defined by any image along the path are oriented directly along the path Image Transition State MEP

Elastic Band Methods To determine transition states and MEP: Construct a chain of images that connects two known potential energy minima. ‘Elastic band’ anchored on end points (minima) Allow the intermediate states to relax : minimizing a function (‘objective function’)

Minimize the function: Coupled harmonic spring Potential Energy 1) States move toward PE minima 2) Spring tends to keep the images equidistant with straight path to minimize the energy “Corner cutting” and “sliding down” problems may occur  NEB

Difficulties Corner cutting : component of spring force which is perpendicular to the path tends to pull images off the MEP (at large K). Sliding down : component of the true force in the direction of the path : slides to potential minima (at small K).

Nudged elastic band method Project parallel component of spring force and perpendicular component of physical force (‘nudging’). F NEB[r(i)] = FV⊥[r(i)] + FS∥ [r(i)] Components of Physical force: FV∥ [r(i)] = - (𝞩i V[r(i)].τ)τ FV⊥[r(i)] = -𝞩i V[r(i)] + (𝞩i V[r(i)].τ)τ

Components of Spring force: Can relax the elastic band towards MEP using iterative algorithms like steepest descent. The images move in a direction perpendicular to the path.

Set up calculation in VASP Directory : INCAR, KPOINTS, and POTCAR. Set of subdirectories (numbered 00,01,02...) must be created, and each subdirectory must contain one POSCAR file. The tag : IMAGES= number of images   forces VASP to run the elastic band method. 4) The number of nodes must be dividable by the number of images  5) Positions for the end points (fixed) : 00/POSCAR and XX/POSCAR (XX=number of images+1)

Set up calculation in VASP All output goes to the subdirectories (except 00 and XX) Set spring constants using the tag: SPRING=-5 Negative value sets up a Nudge Elastic Band calculation. SPRING= 0 : Images are optimized but only allowed to move perpendicular to the hyper-tangent (normal vector between two neighbouring images)

Tips & Tricks for setting up a calculation recommended to use the Quasi Newton algorithm (IBRION =1)or Damped MD (IBRION =3). make sure that the sum of all positions is the same for each cell. the number of CPUs to be used has to be an integer multiple of IMAGES if convergence is not reached within NSW steps, the calculation can be continued by a continuation run, just like for a standard ionic relaxation. recommended to keep the number of images small. If the region close to the transition state is to be refined, one can do another NEB-calculation, using the ionic configurations of the IMAGES adjacent to the transition state as the new initial and final states for the follow-up run.

CI-NEB The highest energy image is driven up to the saddle point. This image does not feel the spring forces along the band. True force at this image along the tangent is inverted. 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.

Interpolate images Decide how many images Append POSCAR1 and POSCAR2 to POSCAR1_POSCAR2. POSCAR should not contain element names. Make directories 00, 01…NN Use interpolatePOSCAR (awk script) syntax: interpolatePOSCAR POSCAR1_POSCAR2 POSCARs created in each directory (Modify code to required no. of images)

POSCAR 02 POSCAR 00 POSCAR 01 POSCAR 03 POSCAR 04 POSCAR 05

NEB – Reaction Path

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