# How can I scan the whole PES of a bulk structure?

I am not very familiar with scanning a PES for a bulk structure, and I'd be very appreciative for some suggestions.

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• Welcome to our community! – Camps Jun 30 at 17:59
• I think that as the bulk is periodic, you cannot scan the PES. All the atom positions are "forced" due to symmetry operations. – Camps Jun 30 at 17:59
• Welcome to the site! If you haven't yet, take the tour and visit the help center. You will probably need a bit more detail to get a good answer. For example 1. what sort of compound are you looking at? 2. what modeling software are you trying to use? 3. are you just trying to map the PES or are you interested in some specific features (e.g. a particular transition state, the location of minima)?. These details will make it easier for someone to address your particular problem and will make the question more useful to future users. – Tyberius Jun 30 at 17:59
• It is worth to limit what area of PES you want to scan: the whole PES would include all kind of geometries, even gas or liquid. What you are generally looking for is PES around equilibrium, or along a sub space, eg reaction path. All of these are extremely high dimensional spaces, so even there you generally use some approximation how to reduce dimensions. – Greg Jul 1 at 6:46

The potential energy surface (PES) is a 3N-dimensional function for a bulk system containing N atoms (in reality 3N-3 to account for the trivial translational degrees of freedom). For a bulk structure, N typically represents the number of atoms in a simualtion cell with periodic boundary conditions, which is of the order of $$10^2$$-$$10^3$$, so the function is of very high dimension. The number $$10^2$$-$$10^3$$ comes from a typical DFT calculation; if you use cheaper methods (e.g. force fields) then that number can be larger, or if the method is more expensive (e.g. quantum chemistry) then the number is somewhat smaller. In any case, you are sampling a high dimensional function, which is an extremely hard problem.

So what can be done to scan this potential energy surface? These are a few options:

1. Phonon calculations. A (meta)stable bulk structure sits at a local minimum of the PES. A phonon calculation determines the Hessian around this minimum, that is, the curvature of the PES around the minimum. Phonon calculations are relatively straight-forward to perform (say at the DFT level), and typically provide very useful information about the low-energy part of the PES. This is useful for calculating thermodynamic properties of a crystal at relatively low temperatures (when atoms do not move very far away from the minimum). One can explore a little further away by including terms beyond the second derivative in the expansion about the minimum (referred to as anharmonic terms), but these still essentially give a local view of the PES around the minimum.
2. Transition states. Considering two structures associated with two nearby minima in the PES, it is possible to locate the minimum energy path between the two. This path goes through a saddle point of the PES, which is called a transition state. A well-known technique to find this is the nudged elastic band method, which requires knowledge of the two minima and calculates the saddle point. Another method that can allow one to "jump" from one minimum to another nearby minimum is molecular dynamics. These methods still provide a local view of the PES, but they explore it in a wider region than a phonon calculation would.
3. Structure prediction. If you want a more uniform sampling of the PES, then you are entering the realm of structure prediction. These methods try to explore as many minima as possible of the potential energy surface. The basic idea is to generate structures at random (corresponding to arbitrary points in the PES) and then relax them to the local minimum. Repeating this many times allows you to explore basins around different minima of the PES. There are different methods that allow you to do this, which differ by how they generate the "next" structure (e.g. stochastic, genetic, particle swarm algorithms). These methods provide a global view of the potential energy surface, but lack in detail, so, for example, would have to be complemented with phonon calculations to explore the important low-energy region of a material at low temperatures.

David Wales has a book on some of this, you can find more details in his website.

• +1. Another good one from ProfM. As a molecular person, I would say the PES is 3N-6 dimensional or 3N-5 for linear molecules. So for the diatomic molecule H$_2$, we have 3N-5 = 3(2)-5 = 1 dimension, which is the internuclear distance R, between the two H atoms. Since the question is about bulk structures maybe your PES is typically 3N dimensional. Sorry if my comment is very naive! – Nike Dattani Jun 30 at 22:56
• Thank you @NikeDattani, you are absolutely correct that some degrees of freedom are trivial. In bulk systems, there are 3 trivial degrees of freedom, corresponding to translations of the system in the three spatial dimensions. I have updated the response to clarify this. – ProfM Jul 1 at 4:59