The Stillinger-Weber potential

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Potential form

Here we consider the original Stillinger-Weber potential as proposed by Stillinger and Weber in 1985. The total potential energy consists of a two-body part and a three-body part. The site potential is $$ U_i = \frac{1}{2} V_2(r_{ij}) + \frac{1}{2}\sum_{j\neq i}\sum_{k\neq i,j} h_{ijk}, $$ where the two-body part is $$ V_2(r_{ij}) = \epsilon A \left[ B \left( \frac{\sigma }{ r_{ij} } \right)^{4}-1 \right] \exp\left( \frac{1}{ r_{ij} / \sigma - a} \right) $$ and the three-body part is $$ h_{ijk}=\epsilon\lambda \exp \left[ \frac{\gamma}{r_{ij}/\sigma-a} + \frac{\gamma}{r_{ik}/\sigma-a} \right] \left(\cos \theta_{ijk} - h \right)^2. $$


Parameter Units
[math]A[/math] eV
[math]B[/math] dimensionless
[math]\epsilon[/math] dimensionless
[math]\sigma[/math] A
[math]a[/math] dimensionless
[math]\lambda[/math] eV
[math]\gamma[/math] dimensionless
[math]h[/math] dimensionless

Potential file

Single-element system

  • For one-element systems, the potential file reads
   epsilon lambda A B a gamma sigma h

Double-element system

  • For two-element systems, the two-body parameters can generally have 3 different values and the three-body parameters can generally have 8 different values. The potential file reads
   A_00 B_00 a_00 sigma_00 gamma_00
   A_01 B_01 a_01 sigma_01 gamma_01
   A_11 B_11 a_11 sigma_11 gamma_11
   lambda_000 cos0_000
   lambda_001 cos0_001
   lambda_010 cos0_010
   lambda_011 cos0_011
   lambda_100 cos0_100
   lambda_101 cos0_101
   lambda_110 cos0_110
   lambda_111 cos0_111
  • Note that we have removed the redundant parameter [math]\epsilon[/math] in the potential file here and [math]A[/math] and [math]\lambda[/math] actually stand for [math]\epsilon A[/math] and [math]\epsilon\lambda[/math].

Three-element system

  • If there are three atom types, the potential file reads:
   A_00 B_00 a_00 sigma_00 gamma_00
   A_01 B_01 a_01 sigma_01 gamma_01
   A_02 B_02 a_02 sigma_02 gamma_02
   A_10 B_10 a_10 sigma_10 gamma_10
   A_11 B_11 a_11 sigma_11 gamma_11
   A_12 B_12 a_12 sigma_12 gamma_12
   A_20 B_20 a_20 sigma_20 gamma_20
   A_21 B_21 a_21 sigma_21 gamma_21
   A_22 B_22 a_22 sigma_22 gamma_22
   lambda_000 cos0_000
   lambda_001 cos0_001
   lambda_002 cos0_002
   lambda_010 cos0_010
   lambda_011 cos0_011
   lambda_012 cos0_012
   lambda_020 cos0_020
   lambda_021 cos0_021
   lambda_022 cos0_022
   lambda_100 cos0_100
   lambda_101 cos0_101
   lambda_102 cos0_102
   lambda_110 cos0_110
   lambda_111 cos0_111
   lambda_112 cos0_112
   lambda_120 cos0_120
   lambda_121 cos0_121
   lambda_122 cos0_122
   lambda_200 cos0_200
   lambda_201 cos0_201
   lambda_202 cos0_202
   lambda_210 cos0_210
   lambda_211 cos0_211
   lambda_212 cos0_212
   lambda_220 cos0_220
   lambda_221 cos0_221
   lambda_222 cos0_222
  • Do we need to consider more than three atom types? I don't think so. So stop here.