# What's the Theory Behind Nudged Elastic Band?

I'm interested in doing some calculations to model the interstitial addition of nitrogen into a structure and I'm thinking of using Nudged Elastic Band to do it.

From what I have read you start with two scenarios: an initial position, and the end result you hope to achieve, then the algorithm finds you a low energy path between the two.

From this I expect I will be able to calculate the minimum energy required by the nitrogen to infiltrate the structure.

Is this possible with nudged elastic band? What's the fundamental idea driving the method? Are there any good books or papers on the subject? What software would you recommend for this?

• I think the question is better with the additional context, but you can roll it back to the smaller version if you prefer.
– Tyberius
Commented Nov 4, 2021 at 1:12
• Commented Nov 5, 2021 at 11:46
• Thanks very much @Tyberius! Commented Nov 6, 2021 at 14:27

The Nudged Elastic Band (NEB) method is a way to find the Minimum Energy Path (MEP) between a set of reactants and products. Some variants, like the Climbing Image (CI-NEB), also specifically find the transition state along this path. So if your goal is to determine the reaction pathway, and possibly activation energy, for nitrogen infiltration into a solid, this method could be able to help with that.

The idea behind the method is to consider a set of "images" or interpolated structures between the reactants and products. Rather than treating these as individual structure, we instead think of creating a large band between reactants and products, where adjacent images are linked using harmonic spring potential. We then optimize the structure of the band, considering just these spring forces along the band and the actual structural forces along the directions perpendicular to the band. In equation form, the force on each image $$i$$ is given as

$$\mathbf{F}_i=\mathbf{F}^S_{i}|_{\parallel}-\mathbf{\nabla}E(\mathbf{R}_i)|_{\perp}\tag{1}\label{1}$$ $$\mathbf{F}^S_{i}|_{\parallel}=k(|\mathbf{R}_{i+1}-\mathbf{R}_i|-|\mathbf{R}_{i}-\mathbf{R}_{i-1}|)\boldsymbol{\tau}_i \tag{2}\label{2}$$ $$\mathbf{\nabla}E(\mathbf{R}_i)|_{\perp}= \mathbf{\nabla}E(\mathbf{R}_i)-\mathbf{\nabla}E(\mathbf{R}_{i})|_{\parallel}=\mathbf{\nabla}E(\mathbf{R}_i)-\mathbf{\nabla}E(\mathbf{R}_i)\cdot\boldsymbol{\tau}_i\boldsymbol{\tau}_i \tag{3}\label{3}$$

where $$S$$ denotes the spring force and $$\tau$$ is the tangent along the band (the definition requires some extra details to avoid kinks in the band, see the papers in the linked Henkleman group page above). The spring forces keep each of the images apart while the perpendicular forces move the band closer to the MEP.

The CI variant doesn't take too much additional effort: after a few regular iterations, find the highest energy image, and replace its forces with

$$F_{i,\text{max}}=-\mathbf{\nabla}E(\mathbf{R}_{i,\text{max}})|_{\perp}+\mathbf{\nabla}E(\mathbf{R}_{i,\text{max}})|_{\parallel}=-\mathbf{\nabla}E(\mathbf{R}_{i,\text{max}})+2\mathbf{\nabla}E(\mathbf{R}_{i,\text{max}})|_{\parallel}\tag{4}\label{4}$$

So we remove the spring force and invert the structural forces along the band, making this image climb up in energy along the band until it reaches a saddle point (maximum along the band, minimum for all other coordinates).

If you are comfortable doing VASP calculations, NEB is implemented in this program and it would probably be well suited for this sort of computation on a solid. Alternatively, if you have a preferred electronic structure program, you can interface it with ASE, which can externally run NEB by using the electronic structure program as a calculator.