Data

The production of the data sets used in the training of the neural networks followed the strategies of Huran et al. [476] and Artrith et al. [212]. We start by the exploration of the chemical environment, and the selection of allowed crystal structures for each chemical composition using the MHM (see section 2.1.1). All the calculations were performed at the level of DFT with the PBE approximation for the exchange and correlation functional, as implemented in vasp code [477, 478]. More information on the calculations can be found in the appendix A. The output of this undertaking provided two sets of structures: one corresponding to the local minima of the PES, and the other to different steps of short MD simulations. Furthermore, we applied a series of geometrical distortions to the local-minima structure, namely volume-conserving orthorhombic and monoclinic strains (see fig. 4.3), and scaling of the lattice constants by up to ±10%. Lastly, we complemented the data sets of Si and Ge with two-dimensional minima structures obtained following the strategy of Borlido et al. [479, 480].

Figure 4.3: Cubic Si structure (middle panel) deformed by a volume-conserving monoclinic (left panel) and orthorhombic (right panel) strain.

4.4. EXAMPLE FORCE-FIELDS 57 Note that, contrary to parameterizations that include physical constrains, such as those for DFTB, and that for this reason might require a small number of training structures [476], machine learning algorithms are expected to identify the underlying patterns and physical relations only from the data. As such they normally require large training sets and, in fact, the accuracy of the model usually increases with increasing sizes of the training sets. To give an idea of the size of the data sets we will use Si as an example. Our data set for Si includes 26360 structures, out of which 131 correspond to minima, 54 to 2D minima, 7142 to distorted, and 19033 to MD structures. These structures were then divided into two sets:

70% for the training set and 30% for the test set. Table 4.1 displays the number corresponding to each type of structures in the training set for Si and for the other materials, that were divided in the same fashion. A quick glance at the table reveals that our data sets contain a rather low number of minima (both 3D and 2D), precisely the type of structures we are more interested in describing. For this reason, we increased the weight of these structures.

Moreover, to insure a better description of the regions close to dynamical minima, we included a dimensionless weight factor uσ in the objectives:

uσ = 0.2

0.2 + ¯F_σ² . (4.26)

where, ¯Fσ corresponds to the average norm of forces acting on the atoms in theσ^th structure.

The maximum of this function occurs for ¯Fσ = 0 and decreases monotonically with the increase of the forces.

On the other hand, we are mostly interested in energy differences and not in the absolute value of the total energy (which is meaningless for solids). An example of such difference is the formation energy. This quantity is defined as the difference between the energy of a structure and that of the ground-state of its elementary substances. Hence, we also found useful to increase the weight of the ground-state structures in the training set in order to improve the accuracy of their description with the neural network force-fields.

Table 4.2 exhibits the ranges of the energies, forces, and stresses in our data sets. These values will be important to understand the meaning of the errors calculated during the validation. The column for the norm of the forces reveals that no force in our training set exceeds a magnitude of 2.0 eV/˚A. This is the case due to the filter we applied to clean the data when constructing the data sets. We removed duplicates and neglected structures with very high forces.

The distribution of the target properties in our data sets can be visualized, for example, in figs. 4.4 and 4.5 (for Cu and Si, respectively). It is evident that both data sets were constructed in a similar fashion. In fact, the differences between the distributions for Si and for Cu, come solely from the number of minima structures found for each of them. We note again that the structures identified as distorted come from distortions of these minima.

A comparison between the energy distributions reveals that the few lowest energy minima found for Cu are closer in energy than those found for Si. This is the reason why the energy distribution for Cu looks cut in fig. 4.4. Obviously, formation energies and energies follow the same distribution.

The forces distribution shows that our construction of the data sets focused on structures around the minima of the PES. Nevertheless, they also provide a rather complete description

Formation Energy Forces Forces Stress Stress

Energy Component Norm

eV/atom eV/atom eV/˚A eV/˚A kBar eV/˚A³

Si [0.0,7.8] [-5.4,2.3] [-5.2,4.9] [0.0,2.0] [-1218,6827] [-0.8,4.2]

Ge [0.0,6.0] [-4.6,1.4] [-3.9,4.4] [0.0,2.0] [ -549,3372] [-0.3,2.1]

SiGe [0.0,7.0] [-5.4,1.7] [-4.8,4.1] [0.0,2.0] [-1167,4633] [-0.7,2.9]

Cu [0.0,3.0] [-4.1,-1.1] [-3.3,3.3] [0.0,2.0] [ -925,3098] [-0.6,1.9]

Au [0.0,3.3] [-3.3,0.0] [-4.2,4.2] [0.0,2.0] [ -754,2726] [-0.5,1.7]

Table 4.2: Range for the formation energy, energy, forces (norm and component) and stresses in our data sets (training and test).

of the forces space between 0 and 2 eV/˚A. Both distributions look almost symmetrical around the maximum of 0.75 eV/˚A (after neglecting the 0 eV/˚A bar).

Lastly, the distribution of the stresses, for both Si and Cu, is the combination of 2 almost symmetric distributions, one around a maximum at 0 kBar and another around a maximum at −200 kBar. Most of the structures belong to the former, and the latter occurs since we performed some MHM runs at a pressure of 20 GPa. We included in the sets a few examples of high pressure structures (above 50 GPa) to increase the performance of the fit when the atoms are close together, during MD simulations or when the structures are compressed.

Furthermore, in fig. 4.6 we present the distributions for Cu separated by type of structure.

As most of our structures come from MD simulations it is not surprising that the distributions for MD structures (panels g, h, and i) and the total distributions in our data sets look very similar. It is obvious that we mainly rely on this type of structures to obtain a fairly extensive description of all regions of the PES.

Regarding the minima structures (panels a, b, and c), we note that we do not have much control over their selection, they correspond to the structures found by the MHM. We can only increase the temperature to try to find higher energy structures or the pressure of the system to find structures subjected to some strain.

While the MD structures provide a rather broad set of forces, the distorted structures improve the description of the stresses in a broader region of the space (as observed in the panel f) corresponding to the stresses of the distorted structures). Meanwhile, the forces provided by the distorted set concern mainly the region in the proximity of the minima of the PES (see panel e).

Finally, we would like to mention that the distributions for the sets of the other materials resemble those depicted above for Cu and Si.

We anticipate that the force-fields, constructed from data sets generated in this manner, will provide an accurate description of structures close to dynamical equilibrium. Meanwhile, the inclusion of the MD and distorted structures assures the correct description of structures under different conditions of temperature and pressure, and with relatively large forces. Due to the Behler and Parrinelo approach and the extrapolation capabilities of neural networks, we do not expect force-fields created from these data sets to properly describe single atoms, molecules, or clusters. However, we believe them able to describe supercells that resemble locally the structures contained in the data sets, since the cut-off radius centered around

4.4. EXAMPLE FORCE-FIELDS 59

−4.00 −3.75 −3.50 −3.25 −3.00 Energy (eV/atom)

0 200 400

Numberofstructures

0.0 0.5 1.0

Formation Energy (eV/atom) 0

200 400 600

Numberofstructures

0.0 0.5 1.0 1.5 2.0

Average norm of forces (eV/˚A) 0

100 200 300

Numberofstructures

−200 0 200

Pressure (kBar) 0

200 400 600

Numberofstructures

Figure 4.4: Distribution of the energies (top left panel), formation energies (top right panel), forces (bottom left panel), and stresses (bottom right panel) in our data set for Cu.

each atom encompasses the larger contributions to the energy.