1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
|
.. module:: ase.calculators.tersoff
:synopsis: Tersoff Interatomic Potential
Tersoff Calculator
==================
The Tersoff potential is a three-body empirical potential that can model covalently
bonded solids like silicon or carbon. This implementation provides a native ASE
calculator that follows the `LAMMPS-style Tersoff
<https://docs.lammps.org/pair_tersoff.html>`_ parameterization.
.. note::
The performance of the routines for this calculator have not been optimized for
speed nor benchmarked with the LAMMPS implementation.
Theory
------
The many-body interaction is based on bond-order concept that includes both two-body and
three-body terms. The total energy is given by:
.. math::
\begin{split}
E & = \frac{1}{2} \sum_i \sum_{j \neq i} V_{ij} \\
V_{ij} & = f_C(r_{ij}) \left[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \right]
\end{split}
where the repulsive and attractive pair terms are:
.. math::
\begin{split}
f_R(r) & = A \exp (-\lambda_1 r) \\
f_A(r) & = -B \exp (-\lambda_2 r)
\end{split}
The cutoff function :math:`f_C(r)` ensures smooth decay of interactions:
.. math::
f_C(r) = \begin{cases}
1 & r < R - D \\
\frac{1}{2} - \frac{1}{2} \sin \left( \frac{\pi}{2} \frac{r-R}{D} \right) & R-D < r < R + D \\
0 & r > R + D
\end{cases}
The bond order term :math:`b_{ij}` captures the short-range local atomic environment:
.. math::
\begin{split}
b_{ij} & = \left( 1 + \beta^n {\zeta_{ij}}^n \right)^{-\frac{1}{2n}} \\
\zeta_{ij} & = \sum_{k \neq i,j} f_C(r_{ik}) g(\theta_{ijk})
\exp \left[ {\lambda_3}^m (r_{ij} - r_{ik})^m \right]
\end{split}
where :math:`\theta_{ijk}` is the angle between bonds :math:`ij` and :math:`ik`. The
angular function :math:`g(\theta)` is:
.. math::
g(\theta) = \gamma \left( 1 + \frac{c^2}{d^2} -
\frac{c^2}{d^2 + (h - \cos \theta)^2} \right)
where :math:`h = \cos \theta_0` defines the preferred angle :math:`\theta_0`.
The parameters :math:`A`, :math:`B`, :math:`\lambda_1`, :math:`\lambda_2`,
:math:`\lambda_3`, :math:`\beta`, :math:`n`, :math:`c`, :math:`d`, :math:`h`,
:math:`\gamma`, :math:`m`, :math:`R`, and :math:`D` define the potential for each
interaction type.
For a complete description of the functional forms and parameters, see the
:mod:`ase.calculators.tersoff` module documentation.
Parameters
----------
The Tersoff potential is defined by 14 parameters for each three-body interaction:
========= ===================================
Parameter Description
========= ===================================
A Repulsive pair potential prefactor
B Attractive pair potential prefactor
lambda1 Decay length for repulsive term
lambda2 Decay length for attractive term
lambda3 Decay length for angular term
beta Angular strength parameter
n Angular exponent
c Angular coefficient
d Angular parameter
h Cosine of angle parameter
gamma Angular scaling
m Bond order exponent
R Cutoff distance
D Cutoff width
========= ===================================
See :class:`ase.calculators.tersoff.TersoffParameters` for details.
Examples
--------
Silicon Diamond Structure
~~~~~~~~~~~~~~~~~~~~~~~~~
Using calculator on silicon crystal in the diamond structure:
.. code-block:: python
from ase.build import bulk
from ase.calculators.tersoff import Tersoff, TersoffParameters
# Create silicon diamond structure
si = bulk("Si", "diamond", a=5.43)
# Define Tersoff parameters for Si
si_params = {
("Si", "Si", "Si"): TersoffParameters(
A=3264.7,
B=95.373,
lambda1=3.2394,
lambda2=1.3258,
lambda3=1.3258,
beta=0.33675,
gamma=1.00,
m=3.00,
n=22.956,
c=4.8381,
d=2.0417,
h=0.0000,
R=3.00,
D=0.20,
)
}
# Set up calculator
calc = Tersoff(si_params)
si.calc = calc
# Calculate properties
energy = si.get_potential_energy()
forces = si.get_forces()
stress = si.get_stress()
Parameter Updates
~~~~~~~~~~~~~~~~~
The calculator parameters can be updated after initialization:
.. code-block:: python
# Update single parameters
calc.set_parameters(("Si", "Si", "Si"), R=2.9, D=0.25)
# Or replace entire parameter set
new_params = TersoffParameters(...)
calc.set_parameters(("Si", "Si", "Si"), params=new_params)
Interface to LAMMPS Files
~~~~~~~~~~~~~~~~~~~~~~~~~
.. warning::
This interface has only been tested with `Si.tersoff
<https://github.com/lammps/lammps/blob/ea1607f1d8a70941cc675f18c4255c25c95c459f/potentials/Si.tersoff>`_
and `SiC.tersoff
<https://github.com/lammps/lammps/blob/ea1607f1d8a70941cc675f18c4255c25c95c459f/potentials/SiC.tersoff>`_
LAMMPS files so it is not necessarily guaranteed to parse all LAMMPS Tersoff files. Therefore, it is
recommended to format the LAMMPS Tersoff file using similar to the *Si.tersoff* or
*SiC.tersoff* files.
Read parameters from a `LAMMPS-style Tersoff file
<https://docs.lammps.org/pair_tersoff.html>`_:
.. code-block:: python
# Initialize from LAMMPS file
calc = Tersoff.from_lammps("SiC.tersoff")
Tersoff Calculator Class
++++++++++++++++++++++++
.. autoclass:: ase.calculators.tersoff.TersoffParameters
:class-doc-from: class
.. autoclass:: ase.calculators.tersoff.Tersoff
:members: from_lammps, read_lammps_format, set_parameters
References
----------
1. J. Tersoff, "New empirical approach for the structure and energy of covalent
systems", *Phys. Rev. B* **37**, 6991 (1988)
2. J. Tersoff, "Modeling solid-state chemistry: Interatomic potentials for
multicomponent systems", *Phys. Rev. B* **39**, 5566(R) (1989)
3. S. J. Plimpton, A. Kohlmeyer, A. P. Thompson, et al., "LAMMPS: Large-scale
Atomic/Molecular Massively Parallel Simulator". Zenodo, Aug. 02, 2023.
`doi:10.5281/zenodo.10806852 <https://doi.org/10.5281/zenodo.10806852>`_.
|
