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:

* Weighting of individual entries

* General energies, Cluster models and the Co₂ dimer

* Description of crystalline phases

* Description of Co-surfaces

* Adatoms

* Vacancies and defects

* Elastic strain moduli

The "job_collection.yaml" and "training_set.yaml" files can be found
in the directory "$AMSHOME/scripting/scm/params/examples/import_old_R
eaxFF/Co/converted_to_params", where "$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 (x_i,ref)  and the ReaxFF value
(x_i,ReaxFF) is weighted individually via the weight w_i and σ_i.

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:

[image]

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

[image]

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

[image]

   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

[image]

   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

[image]

   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**

[image]

   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**

[image]

   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.

[image]

   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.

[image]

   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.

[image]

   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.

[image]

   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).

[image]

   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:

[image]

   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**.
