# The Stillinger-Weber potential

## 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 $$h_{ijk}=\epsilon\lambda \exp \left[ \frac{\gamma}{r_{ij}/\sigma-a} + \frac{\gamma}{r_{ik}/\sigma-a} \right] \left(\cos \theta_{ijk} - h \right)^2.$$

## Parameters

 Parameter Units $A$ eV $B$ dimensionless $\epsilon$ dimensionless $\sigma$ A $a$ dimensionless $\lambda$ eV $\gamma$ dimensionless $h$ dimensionless

## Potential file

### Single-element system

• For one-element systems, the potential file reads
   sw_1985
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
   sw_1985_2
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 $\epsilon$ in the potential file here and $A$ and $\lambda$ actually stand for $\epsilon A$ and $\epsilon\lambda$.

### Three-element system

• If there are three atom types, the potential file reads:
   sw_1985_3
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.