# Transferability of Molecular Potentials for 2D Molybdenum Disulphide

## Abstract

**:**

## 1. Introduction

## 2. Computational Methodology

#### 2.1. Ab Initio Calculations

**M**[1/2,0,0]-

**K**[1/3,1/3,0]-$\mathsf{\Gamma}$[0,0,0], and for 1T’-MoS${}_{2}$: $\mathsf{\Gamma}$[0,0,0]-

**Z**[0,1/2,0]-

**C**[1/2,1/2,0]-

**Y**[1/2,0,0]-$\mathsf{\Gamma}$[0,0,0]) [37] of the structures examined were then utilised to identify their dynamical stability [34,38], complementary to the elastic stability.

#### 2.2. Molecular Calculations

#### Molecular Potentials

**SW2013**[13]: the Stillinger–Weber (SW) potential fitted to an experimentally obtained phonon spectrum along the $\mathsf{\Gamma}$-M direction for bulk 2H-MoS${}_{2}$.**SW2015**[14]: the Stillinger–Weber (SW) potential derived from the valence force-field model.**SW2016**[46]: the Stillinger–Weber (SW) potential fitted to lattice parameters, distance between two chalcogen atoms and elastic constants for SL 1H-MoS${}_{2}$ obtained from DFT calculations.**REBO**[48]: the reactive many-body potential (REBO) fitted to structure and energetics of Mo molecules, three-dimensional Mo crystals, two-dimensional Mo structures, small S molecules and binary Mo-S crystal structures.**SNAP**[49]: the machine-learning-based spectral neighbour analysis potential (SNAP) fitted to total energies and interatomic forces in SL 1H-MoS${}_{2}$ obtained from first-principles density functional theory (DFT) calculations.**ReaxFF**[50]: the reactive force-field (ReaxFF) parameters fitted to a training set of geometries, energies, and charges derived from DFT calculations for both clusters and periodic Mo${}_{x}$S${}_{y}$ systems.

## 3. Results

#### 3.1. Structural and Mechanical Properties

#### 3.2. Phonon Spectra

## 4. Conclusions

- The transferability of analysed molecular potentials leaves much to be desired.
- Three potentials: SW2016, SW2017 and REBO demonstrate the best quantitative performance.
- None of the above three potentials correctly reproduces the dynamical stability of all SL MoS${}_{2}$ phases.
- Only the REBO potential distinguishes three different 2D molybdenum disulphide allotropes.
- Two potentials, ReaxFF and SNAP, demonstrate significantly lower quantitative efficiency.
- It seems that the low transferability of the analysed potentials is a result of the improper fitting of their parameters.
- To increase the transferability of potentials, the number of configurations to be taken into account in the parameter optimisation process should be significantly increased.

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

G6-TMD | Group 6 transition metal dichalcogenide |

SL MoS${}_{2}$ | single-layer molybdenum disulphide |

MS | molecular statics |

DFT | density functional theory |

DFPT | density functional perturbation theory |

PP-PW | pseudopotential, plane-wave |

XC | exchange-correlation |

LDA | local density approximation |

GGA | generalized gradient approximation |

PBE | Perdew–Burke–Ernzerhof |

## Appendix A

`# CIF file 1H-MoS2`

`# This file was generated by FINDSYM`

`# Harold T. Stokes, Branton J. Campbell, Dorian M. Hatch`

`# Brigham Young University, Provo, Utah, USA`

`data_findsym-output`

`_audit_creation_method FINDSYM`

`_symmetry_space_group_name_H-M "P -6 m 2"`

`_symmetry_Int_Tables_number 187`

`_cell_length_a 3.16544`

`_cell_length_b 3.16544`

`_cell_length_c 16.00000`

`_cell_angle_alpha 90.00000`

`_cell_angle_beta 90.00000`

`_cell_angle_gamma 120.00000`

`loop_`

`_space_group_symop_id`

`_space_group_symop_operation_xyz`

`1 x,y,z`

`2 -y,x-y,z`

`3 -x+y,-x,z`

`4 x,x-y,-z`

`5 -x+y,y,-z`

`6 -y,-x,-z`

`7 -x+y,-x,-z`

`8 x,y,-z`

`9 -y,x-y,-z`

`10 -x+y,y,z`

`11 -y,-x,z`

`12 x,x-y,z`

`loop_`

`_atom_site_label`

`_atom_site_type_symbol`

`_atom_site_symmetry_multiplicity`

`_atom_site_Wyckoff_label`

`_atom_site_fract_x`

`_atom_site_fract_y`

`_atom_site_fract_z`

`_atom_site_occupancy`

`Mo1 Mo 1 d 0.33333 0.66667 0.50000 1.00000`

`S1 S 2 i 0.66667 0.33333 0.40249 1.00000`

`# CIF file 1T-MoS2`

`# This file was generated by FINDSYM`

`# Harold T. Stokes, Branton J. Campbell, Dorian M. Hatch`

`# Brigham Young University, Provo, Utah, USA`

`data_findsym-output`

`_audit_creation_method FINDSYM`

`_symmetry_space_group_name_H-M "P -3 2/m 1"`

`_symmetry_Int_Tables_number 164`

`_cell_length_a 3.19358`

`_cell_length_b 3.19358`

`_cell_length_c 16.00000`

`_cell_angle_alpha 90.00000`

`_cell_angle_beta 90.00000`

`_cell_angle_gamma 120.00000`

`loop_`

`_space_group_symop_id`

`_space_group_symop_operation_xyz`

`1 x,y,z`

`2 -y,x-y,z`

`3 -x+y,-x,z`

`4 x-y,-y,-z`

`5 y,x,-z`

`6 -x,-x+y,-z`

`7 -x,-y,-z`

`8 y,-x+y,-z`

`9 x-y,x,-z`

`10 -x+y,y,z`

`11 -y,-x,z`

`12 x,x-y,z`

`loop_`

`_atom_site_label`

`_atom_site_type_symbol`

`_atom_site_symmetry_multiplicity`

`_atom_site_Wyckoff_label`

`_atom_site_fract_x`

`_atom_site_fract_y`

`_atom_site_fract_z`

`_atom_site_occupancy`

`Mo1 Mo 1 b 0.00000 0.00000 0.50000 1.00000`

`S1 S 2 d 0.33333 0.66667 0.40182 1.00000`

`# CIF file 1T’-MoS2`

`# This file was generated by FINDSYM`

`# Harold T. Stokes, Branton J. Campbell, Dorian M. Hatch`

`# Brigham Young University, Provo, Utah, USA`

`data_findsym-output`

`_audit_creation_method FINDSYM`

`_symmetry_space_group_name_H-M "P 1 21/m 1"`

`_symmetry_Int_Tables_number 11`

`_cell_length_a 5.75123`

`_cell_length_b 3.17711`

`_cell_length_c 16.00000`

`_cell_angle_alpha 90.00000`

`_cell_angle_beta 90.00000`

`_cell_angle_gamma 90.00000`

`loop_`

`_space_group_symop_id`

`_space_group_symop_operation_xyz`

`1 x,y,z`

`2 -x,y+1/2,-z`

`3 -x,-y,-z`

`4 x,-y+1/2,z`

`loop_`

`_atom_site_label`

`_atom_site_type_symbol`

`_atom_site_symmetry_multiplicity`

`_atom_site_Wyckoff_label`

`_atom_site_fract_x`

`_atom_site_fract_y`

`_atom_site_fract_z`

`_atom_site_occupancy`

`Mo1 Mo 2 e 0.70568 0.25000 0.49753 1.00000`

`S1 S 2 e 0.41977 0.25000 0.60514 1.00000`

`S2 S 2 e 0.07725 0.25000 0.41677 1.00000`

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**Figure 4.**Phonon dispersion of SL MoS${}_{2}$ from DFT (

**a**) 1H, (

**b**) 1T and (

**c**) 1T’. High symmetry points: $\mathsf{\Gamma}$[0,0,0],

**M**[1/2,0,0],

**K**[1/3,1/3,0],

**Z**[0,1/2,0],

**C**[1/2,1/2,0],

**Y**[1/2,0,0].

**Figure 5.**Phonon dispersion of SL MoS${}_{2}$ from SW2017 potential (

**a**) 1H, (

**b**) 1T and (

**c**) 1T’. Black lines represent SW2017 results, red lines represent DFT results.

**Figure 6.**Phonon dispersion of SL MoS${}_{2}$ from SW2016 potential (

**a**) 1H, (

**b**) 1T and (

**c**) 1T′. Black lines represent SW2016 results, red lines represent DFT results.

**Figure 7.**Phonon dispersion of SL MoS${}_{2}$ from reactive many-body (REBO) potential (

**a**) 1H, (

**b**) 1T and (

**c**) 1T′. Black lines represent REBO results, red lines represent DFT results.

**Table 1.**Structural and mechanical properties of SL MoS${}_{2}$ phases from density functional theory (DFT) calculations: lattice parameters a,b (Å), average cohesive energy ${E}_{c}$ (eV/atom), average bond length d (Å), average height h (Å), 2D elastic constants ${C}_{ij}$ (N/m) and 2D Kelvin moduli ${K}_{i}$ (N/m).

Polymorph | 1H | 1T | 1T′ | ||||||
---|---|---|---|---|---|---|---|---|---|

Source | Present | Exp. | DFT | Present | Exp. | DFT | Present | Exp. | DFT |

a | 3.165 | 3.157 ^{a} | 3.183 ^{b} | 3.194 | 3.179 ^{b} | 5.751 | 5.717 ^{b} | ||

b | 3.165 | 3.157 ^{a} | 3.183 ^{b} | 3.194 | 3.176 ^{b} | 3.177 | 3.179 ^{b} | ||

$-{E}_{c}$ | 5.64 | 5.35 ^{a} | 5.52 | 5.56 | |||||

${d}_{Mo-S}$ | 2.403 | 2.38 ^{a} | 2.43 ^{a} | 2.422 | 2.430 ^{c} | 2.415 ^{‡} | |||

${h}_{S-S}$ | 3.120 | 3.116 ^{a} | 3.11 ^{a} | 3.142 | 3.184 ^{c} | 3.364 | |||

${C}_{11}$ | 126.5 | 127.2 ^{d} | 84.1 | 103.8 ^{d} | 68.1 | 94.0 ^{d} | |||

${C}_{22}$ | 126.5 | 127.2 ^{d} | 84.1 | 103.8 ^{d} | 78.9 | 119.2 ^{d} | |||

${C}_{12}$ | 28.5 | 25.8 ^{d} | 5.0 | −2.5 ^{d} | 18.2 | 17.2 ^{d} | |||

${C}_{44}$ | 49.0 | 51.0 ^{d} | 39.6 | 52.8 ^{d} | 43.2 | 37.5 ^{d} | |||

${K}_{I}$ | 155.0 | 89.1 | 90.9 | ||||||

${K}_{II}$ | 98.0 | 79.1 | 56.1 | ||||||

${K}_{III}$ | 98.0 | 79.1 | 86.4 |

**Table 2.**Structural and mechanical properties of SL 1H-MoS${}_{2}$ from molecular calculations: lattice parameters a,b (Å), average cohesive energy ${E}_{c}$ (eV/atom), average bond length d (Å), average height h (Å), 2D elastic constants ${C}_{ij}$ (N/m), 2D Kelvin moduli ${K}_{i}$ (N/m), mean absolute percentage error (MAPE) (%).

Method | DFT | SW2013 | SW2015 | SW2016 | SW2017 | REBO | SNAP | ReaxFF |
---|---|---|---|---|---|---|---|---|

a | 3.165 | 3.062 | 3.117 | 3.174 | 3.196 | 3.168 | 3.139 | 3.186 |

b | 3.165 | 3.062 | 3.117 | 3.174 | 3.196 | 3.168 | 3.139 | 3.186 |

$-{E}_{c}$ | 5.64 | 3.00 | 0.62 | 1.84 | 5.11 | 7.16 | 2.28 | 5.05 |

${d}_{Mo-S}$ | 2.403 | 2.399 | 2.382 | 2.515 | 2.441 | 2.445 | 2.392 | 2.431 |

${h}_{S-S}$ | 3.120 | 4.223 | 4.257 | 4.032 | 3.194 | 3.242 | 3.124 | 3.183 |

${C}_{11}$ | 126.5 | 103.9 | 45.8 | 90.0 | 118.9 | 154.4 | 140.3 | 237.3 |

${C}_{22}$ | 126.5 | 103.9 | 45.8 | 90.0 | 118.9 | 154.4 | 140.3 | 262.4 |

${C}_{12}$ | 28.5 | 33.4 | 8.0 | 30.1 | 40.9 | 45.8 | 35.7 | 121.2 |

${C}_{44}$ | 49.0 | 35.2 | 18.9 | 30.0 | 39.0 | 54.3 | 52.3 | 71.2 |

${K}_{I}$ | 155.0 | 137.3 | 53.8 | 120.1 | 159.8 | 200.2 | 176.0 | 370.4 |

${K}_{II}$ | 98.0 | 70.5 | 37.8 | 59.9 | 78.0 | 108.6 | 104.6 | 129.3 |

${K}_{III}$ | 98.0 | 70.4 | 37.8 | 60.0 | 78.0 | 108.6 | 104.6 | 142.4 |

MAPE${}_{1\mathrm{H}}$ | 19.797 | 48.204 | 25.342 | 11.263 | 16.602 | 11.886 | 66.398 |

**Table 3.**Structural and mechanical properties of SL 1T-MoS${}_{2}$ from molecular calculations: lattice parameters a,b (Å), average cohesive energy ${E}_{c}$ (eV/atom), average bond lengths d (Å), average height h (Å), 2D elastic constants ${C}_{ij}$ (N/m), 2D Kelvin moduli ${K}_{i}$ (N/m), mean absolute percentage error (MAPE) (%).

Method | DFT | SW2013 | SW2015 | SW2016 | SW2017 | REBO | SNAP | ReaxFF |
---|---|---|---|---|---|---|---|---|

a | 3.194 | 3.062 * | 3.117 * | 3.174 * | 3.307 | 3.194 | 3.072 | 3.162 |

b | 3.194 | 3.062 * | 3.117 * | 3.174 * | 3.307 | 3.194 | 3.072 | 3.162 |

$-{E}_{c}$ | 5.52 | 3.00 | 0.62 | 1.84 | 4.96 | 7.05 | 2.31 | 4.84 |

${d}_{Mo-S}$ | 2.422 | 2.399 | 2.382 | 2.515 | 2.42 | 2.445 | 2.476 | 2.433 |

${h}_{S-S}$ | 3.142 | 4.223 | 4.257 | 4.032 | 2.973 | 3.211 | 3.454 | 3.203 |

${C}_{11}$ | 84.1 | 103.9 | 45.8 | 91.7 | 121.8 | 118.2 | 437.1 | 173.3 |

${C}_{22}$ | 84.1 | 103.9 | 45.8 | 91.7 | 121.8 | 118.2 | 437.1 | 32.1 |

${C}_{12}$ | 5.0 | 33.4 | 8.0 | 28.4 | 28.6 | 32.4 | 6.1 | 83.8 |

${C}_{44}$ | 39.6 | 35.2 | 18.9 | 31.7 | 46.6 | 42.9 | 215.5 | 9.4 |

${K}_{I}$ | 89.1 | 137.3 | 53.8 | 120.1 | 150.4 | 150.6 | 443.2 | 147.8 |

${K}_{II}$ | 79.1 | 70.5 | 37.8 | 63.3 | 93.2 | 85.8 | 431.0 | 57.6 |

${K}_{III}$ | 79.2 | 70.4 | 37.8 | 63.4 | 93.2 | 85.8 | 431.0 | 18.8 |

MAPE${}_{1\mathrm{T}}$ | 65.962 | 39.849 | 56.735 | 58.860 | 62.843 | 222.509 | 167.192 |

^{*}Input 1T converges to 1H.

**Table 4.**Structural and mechanical properties of SL 1T′-MoS${}_{2}$ from molecular calculations: lattice parameters a,b (Å), average cohesive energy ${E}_{c}$ (eV/atom), average bond lengths d (Å), average height h (Å), 2D elastic constants ${C}_{ij}$ (N/m), 2D Kelvin moduli ${K}_{i}$ (N/m), mean absolute percentage error (MAPE) (%).

Method | DFT | SW2013 | SW2015 | SW2016 | SW2017 | REBO | SNAP | ReaxFF |
---|---|---|---|---|---|---|---|---|

a | 5.751 | 4.944 | 5.757 | 5.263 | 5.728 ^{†} | 5.563 | 5.321 ^{†} | 5.609 |

b | 3.177 | 3.062 | 3.148 | 3.172 | 3.307 ^{†} | 3.245 | 3.072 ^{†} | 3.209 |

$-{E}_{c}$ | 5.56 | 3.02 | 0.55 | 1.87 | 4.96 | 6.93 | 2.31 | 4.83 |

${d}_{Mo-S}$^{‡} | 2.415 | 2.399 | 2.406 | 2.504 | 2.42 | 2.468 | 2.476 | 2.490 |

${h}_{S-S}$ | 3.364 | 4.641 | 5.173 | 4.142 | 2.973 | 3.781 | 3.454 | 3.399 |

${C}_{11}$ | 68.1 | 1.1 | 0.0 | 60.4 | 121.8 | 56.8 | 437.1 | 120.1 |

${C}_{22}$ | 78.9 | 100.5 | 37.6 | 94.6 | 121.8 | 113.0 | 437.1 | 255.7 |

${C}_{12}$ | 18.2 | 1.1 | 0.0 | 20.3 | 28.6 | 23.1 | 6.1 | 68.1 |

${C}_{44}$ | 43.2 | 27.1 | 0.0 | 26.9 | 46.6 | 70.5 | 215.5 | 6.4 |

${K}_{I}$ | 90.9 | 100.5 | 37.6 | 88.4 | 150.4 | 121.3 | 443.2 | 194.3 |

${K}_{II}$ | 56.1 | 1.1 | 0.0 | 66.6 | 93.2 | 48.5 | 431.0 | 181.5 |

${K}_{III}$ | 86.4 | 54.2 | 0.0 | 53.8 | 93.2 | 141.0 | 431.0 | 12.8 |

MAPE${}_{1{\mathrm{T}}^{\prime}}$ | 42.070 | 63.020 | 20.110 | 30.399 | 25.395 | 249.177 | 91.913 | |

∑MAPE | 127.830 | 151.074 | 102.187 | 100.522 | 104.840 | 483.573 | 325.504 |

^{†}Input 1T′ converges to 1H.

^{‡}average first-neighbour bond lengths calculated with cutoff radius = 3.5 and number of histogram bins = 50.

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Maździarz, M.
Transferability of Molecular Potentials for 2D Molybdenum Disulphide. *Materials* **2021**, *14*, 519.
https://doi.org/10.3390/ma14030519

**AMA Style**

Maździarz M.
Transferability of Molecular Potentials for 2D Molybdenum Disulphide. *Materials*. 2021; 14(3):519.
https://doi.org/10.3390/ma14030519

**Chicago/Turabian Style**

Maździarz, Marcin.
2021. "Transferability of Molecular Potentials for 2D Molybdenum Disulphide" *Materials* 14, no. 3: 519.
https://doi.org/10.3390/ma14030519