# Predicting parameters by taking a weighted average of multiple possible structures with energy rather than the structure with least energy

I am currently looking to predict the structure of a material who we know exists under the conditions I am intrusted in(pressure), but the cif file does not exist. Now I have to first make the cif file, for which I need multiple parameters(like the lattice parameters and the octahedral tilts, in order to keep things simple lets say there is one structure parameter S). I created multiple structures with variations in S and optimized all of them with a variable cell, originally I planned of looking at the structure which would yield the lowest energy(so basically I would be looking at the global minima).

But almost all the optimized structures have very similar values of S(~0.1 Angstrom), furthermore almost all the energy values are also quite similar(~10 meV variation). Which makes me think that why are there so many local minimas around one spot.

Now although I do not have S but I have some other parameter(let's say T) and they indicate the structures with lower energy are closer to the true value of T. So would it better to perhaps take a weighted sum of the parameter S with energy. So something like

$$S = \frac{\sum_i S_i E_i}{\sum_i E_i}$$

With this my chances of ending with a wrong structure would reduce. I would have a bit of an average solution but that should have a lesser risk factor of being wrong perhaps? Like if I am not exactly at the global minima.

• I would like to recommend you to get a look at the works of Prof. Artem Oganov. He works with materials under extreme conditions and had developed a method to successfully predict the structures.
– Camps
Commented Nov 25, 2022 at 12:29

If there are really many closely lying minima, this may mean that the material doesn't really have a structure as several of the minima will be populated at room temperature of around 26 meV. One should then look at ensemble averages, instead. For instance, the Boltzmann probability $$p_i$$ of finding the system in the state $$i$$ is given by $$p_i = \exp (-E_i/k_BT)/ \left[\sum_j \exp (-E_j/k_BT)\right]$$ and you might compute observables $$\mathcal{O}$$ as weighted averages of the state-specific values $$\mathcal{O}_i$$: $$\langle \mathcal{O} \rangle = \sum_i p_i \mathcal{O}_i$$