Use extensive or intensive/normalized properties?

Hi there,
probably a short (maybe stupid) question, but I couldn’t find an answer in the ICET documentation or the papers.

The properties that we feed in order to train a cluster expansion, do they need to be provided as extensive properties? For example, if I want to train a CE on the total energy of a system, do I need to provide the total energy as calculated for every single training structure? Or does ICET rely on normalized quantities and I need to normalize the total energy by the number of atoms in the respective training structure first?

I ask because my first feeling was to use total energies or energies of mixing in the way that they scale with the corresponding system size. Double the system = double number of present clusters = double the predicted total energy. But using the total energies as-is, the RMSE value looks really unreasonable, and I am not sure if it is a problem of the fitting or the data itself (because not normalized). Normalizing it looks much better, but then also the numbers are much smaller and the RMSE will be lower anyways…

Could anybody clarify?
Thanks a lot,
Marcel

Hi @Marcel_S,

Not a stupid question at all! The properties you use need to be intensive, meaning energies have to be normalized by the number of atoms in the structure. The reason behind this is that each structure is described by its cluster vector, and the cluster vector is itself normalized by number of atoms – it carries no information about the how large the structure used to calculate it was. For example, think about two structures, one being a primitive structure and the other being that primitive structure repeated three times in periodic directions. They will have the exact same cluster vector, and thereby any property associated with them must be the same too.

Best,
Magnus

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Hi Magnus,
thanks for the fast response, that made everything perfectly clear!
Marcel

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Hi Magnus,
I have some ore follow-up questions. They are partly related on the same topic, but also partly to my other question I had posted in this forum regarding the inclusion of transition states for fitting a CE. Sorry for the lengthy explanation. If it is too much or not clear enough, I am happy to discuss in person.
Let me start giving a bit more context:
The material I investigate is a battery cathode material, which during usage of a battery will change its composition from 100% Li content to 0%. To represent that material with a CE the procedure is in principle clear: Generate a lot of structures with different Li content and Li distribution, relax them with DFT. As the primitive structure we take the structure with 100% Li and we set the chemical species of the Li site to be [“Li”, “X”], where “X” represents the vacancies. There are of course other atoms in the structure, i.e. oxygen and transition metal atoms. Their sublattices are, however, untouched which in the documentation is referred to as inert_species.

First part:
So far, so good. Now, let’s say for simplicity that we wanted to fit a CE that represents the total energy (but the same questions also arise for mixing energies). From the previous answer we know that we need to make this an intensive property by normalizing it because the cluster vector will also be normalized. For a simple A-B alloy this is straightforward to understand because the number of “real” atoms equals the number of lattice sites. But for my case where a) vacancies are present in all phases except the 100% structure and b) where inert_species are present, how should we normalize our property of interest? My feeling tells me we need to normalize it by the actual number of ‘real’ atoms in the structures, meaning the number of Li ions for the current composition as well as the number of inert_species. This is because these atoms actually determine the total energy in DFT calculations. However, after thinking more and more about it, I was not sure if the normalization needs to be changed due to the present “X” species after mapping the relaxed structures to the primitive structures. Could somebody clarify this? Do we need to normalize by the actual number of real atoms (this number changes with different Li content), or rather by the number of available lattice sites (which would result in a constant normalization number independent from Li content)? And do we need to take inert_species into account?
I guess the answer depends on how the clusters are defined. From the documentation and the main ICET paper I understand that as a first step, all possible clusters are determined simply by symmetry/geometry among the active sites of the primitive structure and the given cutoffs. But how is it now for a specific relaxed structure with an arbitrary Li content and distribution? What is the role of the vacancies? Does the cluster vector only contain clusters that are exclusively spanned by Li? Or are there all kind of mixed Li-Li-X, Li-X-X, etc… clusters as well? Are there even clusters of only “X” species?

Second part:
Our final objective here is to also include transition states to be able to do kinetic monte carlo simulations. In our first attempt, we introduced the transition state of Li explicitly. However, because we would only allow one jump at a time, in all our structures only one transition state is occupied at a time. Therefore, all the clusters that contain more than 1 transition state are useless for our purpose. Additionally, introducing the transition states explicitly made the number of clusters increase rapidly. As answered by Paul to my other question, there are optimizers that can deal with such situations and one could also prune the CE afterwards.
Nevertheless, we thought of switching to a different strategy: Instead of explicitly adding the transition states, we rely only on the original Li sub-lattice. The structure of a transition state is now described by introducing a new atom type. Let’s just call it “Ti” to give it a name. For structures with transition states, the initial and the final site of the jump path and occupy them with “Ti”. In terms of composition of the structure, this means that one Li (the jumping one) is deleted and two Ti are added. I would call this a geometrical/mathematical trick. The DFT energy of the transition state calculation we then use together with this modified structure.
But now I have the same questions as above. How is it now the (mixing) energy of this structure? Do we normalize it by the original number of ‘real’ atoms, including/excluding vacancies and inert_species? Or do we need to apply a more complicated normalization as we have now introduced the two “Ti” atoms to represent the transition state?

Any comments on these thoughts are highly appreciated. As mentioned above, if my writing here is hard to understand or if your answers are too much/complicated to type here, I would also be happy to discuss online with one of you. I would then also summarize any insights from such a discussion here afterwards to helpful for anybody else.

Best regards,
Marcel