3.4. ReaxFF: Training set for cobalt

The cobalt training set documented here was used in the parametrization of the Cobalt force field, Co.ff, published in “Development of a Transferable Reactive Force Field for Cobalt” by M. R. LaBrosse et al. Within the publication the choice of training set entries is discussed from a scientific standpoint.

This page only discusses the training set, not the parametrization.

Contents:

The job_collection.yaml and training_set.yaml files can be found in the directory $AMSHOME/scripting/scm/params/examples/import_old_ReaxFF/Co/converted_to_params, where $AMSHOME is the AMS installation directory.

In the ParAMS GUI, select File → Open and browse to the job_collection.yaml file.

3.4.1. Weighting of individual entries

To estimate the quality of a given set of ReaxFF parameters with respect to the training data a sum of squared errors loss function (or objective function, or error function) is defined:

\[\textrm{Error} = \sum_{i=1}^{n} w_i {\Big[ \frac{x_{i,ref} - x_{i,ReaxFF} }{\sigma_{i}} \Big]}^2\]

where the sum runs over all training set entries. Each difference between the reference property (xi,ref) and the ReaxFF value (xi,ReaxFF) is weighted individually via the weight wi and σi.

For ReaxFF parametrization up to AMS2021, only the σ value was used. In ParAMS, both w and σ are used. You can modify one or both of them, depending on your preference. For more details, see Sigma vs. weight: What is the difference?.

Note

The value of this loss function is the quantity that is minimized by the optimization algorithms used for the force field fitting. It is therefore important to choose the weightings such that there is no unwanted bias towards one or the other entry.

In total the Co training set contains 144 entries all of which are energies. The training data is distributed as follows:

/scm-uploads/doc.2023/params/_images/Co-objective-function-trainingset.png

The contribution of each entry to the overall error function can to some extent be visualized using the weightings of the error function.

/scm-uploads/doc.2023/params/_images/Co-objective-function-weightings-trainingset.png

For example, the most stable phases hcp and fcc are given much higher weights during the optimization (25.4% and 36.0%) than the less favorable diamond phase (0.5%). In practice one would use the breakdown of the error function (Loss contribution) to finetune the weights of the loss function, which is probably how the above weightings were set too.

3.4.2. General energies, Cluster models and the Co₂ dimer

The energy differences between optimized cubic phases are included in the training data

/scm-uploads/doc.2023/params/_images/Co-general-energies-trainings.png 
Weight   Sigma    Reference_value     Expression
6.300    1.255    -101.300 kcal/mol   +energy("hcp_opt")/2-energy("Co_1_atom")/1
3937.683 1.255    -0.520   kcal/mol   +energy("hcp_opt")/2-energy("fcc_opt")/4
157.507  1.255    -2.210   kcal/mol   +energy("hcp_opt")/2-energy("bcc_opt")/2
1.575    1.255    -17.000  kcal/mol   +energy("hcp_opt")/2-energy("sc_opt")/1
0.394    1.255    -28.600  kcal/mol   +energy("hcp_opt")/2-energy("diam_opt")/8

In addition cohesive energies for a set of small clusters of sizes 2, 3, 4, 5, 6, 8, and 13 atoms are included

/scm-uploads/doc.2023/params/_images/CO-Clusters-training.png 
Weight   Sigma    Reference_value     Expression
0.394    1.255    -29.790  kcal/mol   +energy("Co_2_atom")/2-energy("Co_1_atom")/1
0.394    1.255    -38.800  kcal/mol   +energy("Co_3_atom")/3-energy("Co_1_atom")/1
0.394    1.255    -46.940  kcal/mol   +energy("Co_4_atom")/4-energy("Co_1_atom")/1
0.394    1.255    -56.910  kcal/mol   +energy("Co_5_atom")/5-energy("Co_1_atom")/1
0.394    1.255    -64.580  kcal/mol   +energy("Co_6_atom")/6-energy("Co_1_atom")/1
0.394    1.255    -65.680  kcal/mol   +energy("Co_8_atom")/8-energy("Co_1_atom")/1
0.394    1.255    -71.060  kcal/mol   +energy("Co_13_atom")/13-energy("Co_1_atom")/1

A scan of the bond stretch for the Co₂ dimer is included as well

/scm-uploads/doc.2023/params/_images/CO-dimer-training.png 
Weight   Sigma    Reference_value     Expression
0.063    1.255    -22.750  kcal/mol   +energy("dimer_sep_1")/2-energy("Co_1_atom")/1
0.063    1.255    -27.360  kcal/mol   +energy("dimer_sep_2")/2-energy("Co_1_atom")/1
0.063    1.255    -29.430  kcal/mol   +energy("dimer_sep_3")/2-energy("Co_1_atom")/1
0.063    1.255    -29.790  kcal/mol   +energy("dimer_sep_4")/2-energy("Co_1_atom")/1
0.063    1.255    -29.750  kcal/mol   +energy("dimer_sep_5")/2-energy("Co_1_atom")/1
0.063    1.255    -29.520  kcal/mol   +energy("dimer_sep_6")/2-energy("Co_1_atom")/1
0.063    1.255    -28.890  kcal/mol   +energy("dimer_sep_7")/2-energy("Co_1_atom")/1
0.063    1.255    -27.270  kcal/mol   +energy("dimer_sep_8")/2-energy("Co_1_atom")/1
0.063    1.255    -25.410  kcal/mol   +energy("dimer_sep_9")/2-energy("Co_1_atom")/1
0.063    1.255    -23.420  kcal/mol   +energy("dimer_sep_10")/2-energy("Co_1_atom")/1
0.063    1.255    -22.170  kcal/mol   +energy("dimer_sep_11")/2-energy("Co_1_atom")/1

Not surprisingly the dimer entries of the bond stretch are single point calculations. See how to import your own bond scans into ParAMS.

3.4.3. Description of crystalline phases

The training set includes equation of state (EOS) curves (energy-volume curves) for the following crystalline phases

  • hcp

  • fcc

  • bcc

  • sc

  • diamond

The curves were generated by performing a complete relaxation with a fixed cell volume. The according energies are defined per-atom and set relative to the structure with the lowest energy. For example, the EOS for the hcp and diamond phases are defined as follows:

hcp phase

/scm-uploads/doc.2023/params/_images/EOS-hcp-training.png 
Weight   Sigma    Reference_value     Expression
9.844    1.255    -13.420  kcal/mol   +energy("EOS_hcp_6")/2-energy("EOS_hcp_1")/2
39.377   1.255    -1.860   kcal/mol   +energy("EOS_hcp_6")/2-energy("EOS_hcp_2")/2
157.507  1.255    -1.040   kcal/mol   +energy("EOS_hcp_6")/2-energy("EOS_hcp_3")/2
157.507  1.255    -0.460   kcal/mol   +energy("EOS_hcp_6")/2-energy("EOS_hcp_4")/2
157.507  1.255    -0.120   kcal/mol   +energy("EOS_hcp_6")/2-energy("EOS_hcp_5")/2
157.507  1.255    -0.010   kcal/mol   +energy("EOS_hcp_6")/2-energy("EOS_hcp_6")/2
157.507  1.255    -0.080   kcal/mol   +energy("EOS_hcp_6")/2-energy("EOS_hcp_7")/2
157.507  1.255    -0.340   kcal/mol   +energy("EOS_hcp_6")/2-energy("EOS_hcp_8")/2
157.507  1.255    -0.770   kcal/mol   +energy("EOS_hcp_6")/2-energy("EOS_hcp_9")/2
157.507  1.255    -1.340   kcal/mol   +energy("EOS_hcp_6")/2-energy("EOS_hcp_10")/2
39.377   1.255    -2.000   kcal/mol   +energy("EOS_hcp_6")/2-energy("EOS_hcp_11")/2
9.844    1.255    -6.790   kcal/mol   +energy("EOS_hcp_6")/2-energy("EOS_hcp_12")/2

diamond phase

/scm-uploads/doc.2023/params/_images/EOS-diam-training.png 
Weight   Sigma    Reference_value     Expression
0.025    1.255    -14.800  kcal/mol   +energy("EOS_diam_5")/8-energy("EOS_diam_1")/8
0.025    1.255    -5.590   kcal/mol   +energy("EOS_diam_5")/8-energy("EOS_diam_2")/8
0.025    1.255    -1.630   kcal/mol   +energy("EOS_diam_5")/8-energy("EOS_diam_3")/8
0.025    1.255    -0.350   kcal/mol   +energy("EOS_diam_5")/8-energy("EOS_diam_4")/8
0.025    1.255    -0.010   kcal/mol   +energy("EOS_diam_5")/8-energy("EOS_diam_5")/8
0.025    1.255    -0.430   kcal/mol   +energy("EOS_diam_5")/8-energy("EOS_diam_6")/8
0.025    1.255    -1.510   kcal/mol   +energy("EOS_diam_5")/8-energy("EOS_diam_7")/8
0.025    1.255    -3.100   kcal/mol   +energy("EOS_diam_5")/8-energy("EOS_diam_8")/8
0.025    1.255    -5.040   kcal/mol   +energy("EOS_diam_5")/8-energy("EOS_diam_9")/8
0.025    1.255    -7.260   kcal/mol   +energy("EOS_diam_5")/8-energy("EOS_diam_10")/8
0.025    1.255    -12.350  kcal/mol   +energy("EOS_diam_5")/8-energy("EOS_diam_11")/8
0.025    1.255    -17.950  kcal/mol   +energy("EOS_diam_5")/8-energy("EOS_diam_12")/8

3.4.4. Description of Co-surfaces

The training set contains several surfaces for which the training values are modified surface formation energies defined as the per atom energy of the surface relative to the energy per atom of the bulk phase (optimized hcp).

For the cubic surfaces (fcc,bcc,sc) both low-Miller and high-Miller surfaces are included. The (0001) surface has been added for the hcp phase only.

/scm-uploads/doc.2023/params/_images/Co-surfaces-training.png 
Weight   Sigma    Reference_value     Expression
0.394    1.255    9.670    kcal/mol   +energy("Surface_bcc100")/28-energy("hcp_opt")/2
0.394    1.255    15.970   kcal/mol   +energy("Surface_bcc110")/11-energy("hcp_opt")/2
0.394    1.255    7.790    kcal/mol   +energy("Surface_bcc310")/30-energy("hcp_opt")/2
157.507  1.255    6.690    kcal/mol   +energy("Surface_fcc100")/14-energy("hcp_opt")/2
0.025    1.255    25.200   kcal/mol   +energy("Surface_fcc110")/7-energy("hcp_opt")/2
0.394    1.255    6.840    kcal/mol   +energy("Surface_fcc510")/37-energy("hcp_opt")/2
630.029  1.255    4.710    kcal/mol   +energy("Surface_hcp1000")/7-energy("hcp_opt")/2
0.005    1.255    19.720   kcal/mol   +energy("Surface_sc100")/7-energy("hcp_opt")/2
0.005    1.255    35.550   kcal/mol   +energy("Surface_sc110")/7-energy("hcp_opt")/2
0.005    1.255    23.140   kcal/mol   +energy("Surface_sc111")/14-energy("hcp_opt")/2
0.005    1.255    20.950   kcal/mol   +energy("Surface_sc510")/32-energy("hcp_opt")/2
0.000    1.255    10.000   kcal/mol   +energy("Surface_bcc510")/54-energy("hcp_opt")/2
0.003    1.255    48.060   kcal/mol   +energy("Surface_bcc111")/14-energy("hcp_opt")/2
0.025    1.255    0.810    kcal/mol   +energy("Surface_fcc310")/21-energy("hcp_opt")/2

Tip

Creating various surfaces and bulk materials is easy with the GUI.

3.4.5. Adatoms

Since the migration of Cobalt atoms on various surfaces is essential for the forming of energetically favorable surfaces, the training set contains a variety of adatom structures (top, bridge, hollow sites) on a variety of surfaces (fcc, bcc, sc). All DFT references were optimized, to ensure that the adatom is located in a local minimum.

/scm-uploads/doc.2023/params/_images/Co-surfaces-adatom-training.png 
Weight   Sigma    Reference_value     Expression
0.394    1.255    9.830    kcal/mol   +energy("Surf_adatom_bcc100")/29-energy("hcp_opt")/2
0.394    1.255    8.200    kcal/mol   +energy("Surf_adatom_bcc110")/49-energy("hcp_opt")/2
0.394    1.255    7.360    kcal/mol   +energy("Surf_adatom_fcc100")/57-energy("hcp_opt")/2
0.394    1.255    8.190    kcal/mol   +energy("Surf_adatom_fcc110")/37-energy("hcp_opt")/2
0.394    1.255    6.650    kcal/mol   +energy("Surf_adatom_fcc111")/25-energy("hcp_opt")/2
0.394    1.255    7.380    kcal/mol   +energy("Surf_adatom_fcc310")/43-energy("hcp_opt")/2
0.394    1.255    8.220    kcal/mol   +energy("Surf_adatom_fcc510")/63-energy("hcp_opt")/2
0.063    1.255    21.740   kcal/mol   +energy("Surf_adatom_sc110")/29-energy("hcp_opt")/2

3.4.6. Vacancies and defects

To consider vacancies and defects in bulk cobalt the training set contains

  • bulk fcc cobalt with missing Co atoms (vacancies)

  • amorphous bulk Co structures

  • stacking fault defects

The training set contains formation energies of 1−6 coalesced vacancies in fcc cobalt. As discussed in the paper, the training data shows that it is most energetically favorable to have two vacancies as nearest neighbors.

/scm-uploads/doc.2023/params/_images/Co-vacancies-training.png 
Weight   Sigma    Reference_value     Expression
1.575    1.255    -154.090 kcal/mol   +energy("Vac_0v")/1-energy("Vac_1v")/1-energy("Co_1_atom")/1
0.098    1.255    -305.260 kcal/mol   +energy("Vac_0v")/1-energy("Vac_2v1")/1-energy("Co_1_atom")/0.5
0.098    1.255    -309.220 kcal/mol   +energy("Vac_0v")/1-energy("Vac_2v2")/1-energy("Co_1_atom")/0.5
0.098    1.255    -307.460 kcal/mol   +energy("Vac_0v")/1-energy("Vac_2v3")/1-energy("Co_1_atom")/0.5
0.098    1.255    -308.340 kcal/mol   +energy("Vac_0v")/1-energy("Vac_2v4")/1-energy("Co_1_atom")/0.5
0.098    1.255    -307.880 kcal/mol   +energy("Vac_0v")/1-energy("Vac_2v5")/1-energy("Co_1_atom")/0.5
0.098    1.255    -308.270 kcal/mol   +energy("Vac_0v")/1-energy("Vac_2v6")/1-energy("Co_1_atom")/0.5
0.098    1.255    -308.330 kcal/mol   +energy("Vac_0v")/1-energy("Vac_2v7")/1-energy("Co_1_atom")/0.5
0.098    1.255    -308.000 kcal/mol   +energy("Vac_0v")/1-energy("Vac_2v8")/1-energy("Co_1_atom")/0.5
0.098    1.255    -308.370 kcal/mol   +energy("Vac_0v")/1-energy("Vac_2v9")/1-energy("Co_1_atom")/0.5
0.098    1.255    -452.510 kcal/mol   +energy("Vac_0v")/1-energy("Vac_3v")/1-energy("Co_1_atom")/0.33333333
0.098    1.255    -598.220 kcal/mol   +energy("Vac_0v")/1-energy("Vac_4v")/1-energy("Co_1_atom")/0.25
0.098    1.255    -738.330 kcal/mol   +energy("Vac_0v")/1-energy("Vac_5v")/1-energy("Co_1_atom")/0.2
0.098    1.255    -865.980 kcal/mol   +energy("Vac_0v")/1-energy("Vac_6v")/1-energy("Co_1_atom")/0.16666667

Amorphous bulk Co was generated by running ab-initio NVT MD at 2500K of 108 atom fcc lattice. The snapshots from the trajectory are included as single point calculations in the training data.

/scm-uploads/doc.2023/params/_images/Co_amorphous-training.png 
Weight   Sigma    Reference_value     Expression
0.025    1.255    14.130   kcal/mol   +energy("Amorphous_1")/108-energy("hcp_opt")/2
0.025    1.255    10.350   kcal/mol   +energy("Amorphous_2")/108-energy("hcp_opt")/2
0.025    1.255    8.480    kcal/mol   +energy("Amorphous_3")/108-energy("hcp_opt")/2
0.025    1.255    7.270    kcal/mol   +energy("Amorphous_4")/108-energy("hcp_opt")/2
0.098    1.255    8.500    kcal/mol   +energy("Amorphous_5")/108-energy("hcp_opt")/2

Stacking fault energies for the cubic phases were created via a half-lattice offset in [100] direction. The stacking fault energy for the hcp phase is described via a layering transition from hcp(0001) to fcc(111).

/scm-uploads/doc.2023/params/_images/Co-stacking-faults-training.png 
Weight   Sigma    Reference_value     Expression
0.394    1.255    32.720   kcal/mol   +energy("SFE_bcc001")/8-energy("hcp_opt")/2
6.300    1.255    0.320    kcal/mol   +energy("SFE_fcc111")/16-energy("hcp_opt")/2
6.300    1.255    4.940    kcal/mol   +energy("SFE_hcp100")/16-energy("hcp_opt")/2
0.394    1.255    21.280   kcal/mol   +energy("SFE_sc001")/8-energy("hcp_opt")/2

3.4.7. Elastic strain moduli

As discussed in the paper elastic strain moduli are calculated by manipulating the lattice vectors describing the positions of the atoms. The following elastic constants c11, c12, and c44 for bulk Co phases are included in the training data:

/scm-uploads/doc.2023/params/_images/Co_elastic_constants-trainings.png 
Weight   Sigma    Reference_value     Expression
25.201   1.255    3.340    kcal/mol   +energy("Elast_bcc_c11")/1-energy("hcp_opt")/2
25.201   1.255    3.060    kcal/mol   +energy("Elast_bcc_c44")/1-energy("hcp_opt")/2
0.063    1.255    29.890   kcal/mol   +energy("Elast_diam_c11")/2-energy("hcp_opt")/2
0.063    1.255    29.370   kcal/mol   +energy("Elast_diam_c44")/2-energy("hcp_opt")/2
25.201   1.255    2.270    kcal/mol   +energy("Elast_fcc_c11")/1-energy("hcp_opt")/2
25.201   1.255    1.560    kcal/mol   +energy("Elast_fcc_c44")/1-energy("hcp_opt")/2
25.201   1.255    2.080    kcal/mol   +energy("Elast_hcp_c11")/2-energy("hcp_opt")/2
25.201   1.255    0.780    kcal/mol   +energy("Elast_hcp_c44")/2-energy("hcp_opt")/2
0.394    1.255    18.740   kcal/mol   +energy("Elast_sc_c11")/1-energy("hcp_opt")/2
0.394    1.255    16.830   kcal/mol   +energy("Elast_sc_c44")/1-energy("hcp_opt")/2

Tip

When manipulating the lattice in the graphical user interface (Model → Lattice) with the aim of inducing a strain, make sure you check the box Adjust atoms when changing lattice vectors.