View source for Two-body DeepPot-SE

{{Infobox descriptor
| name = Two-body DeepPot-SE (full)
| key = se_e2_a
| alias = se_a
| version = {{DeePMD-kit version|v0.8.2}}
| compression = {{ok}}
| ref = <ref name="zhang2018">Linfeng Zhang, Jiequn Han, Han Wang, Wissam A. Saidi, Roberto Car, and E. Weinan. 2018. End-to-end symmetry preserving inter-atomic potential energy model for finite and extended systems. In Proceedings of the 32nd International Conference on Neural Information Processing Systems (NIPS'18). Curran Associates Inc., Red Hook, NY, USA, 4441–4451.</ref>
}}
{{Infobox descriptor
| name = Two-body DeepPot-SE (radial-only)
| key = se_e2_r
| alias = se_r
| version = {{DeePMD-kit version|v1.0.0}}
| compression = {{not ok}}
}}

'''Two-body DeepPot-SE''' is a [[descriptor]] of the [[Deep Potential Smooth Edition]] (DeepPot-SE) constructed from two-body information of atomic configurations.<ref name="zhang2018"/> It has two versions: '''se_e2_a''' with full information and '''se_e2_r''' with radius-only information.

== Theory ==
The two-body DeepPot-SE descriptor is given by

<math>
\mathcal{D}^i = (\mathcal{G}^i)^T \mathcal{R}^i (\mathcal{R}^i)^T \mathcal{G}^i_<
</math>

where <math>\mathcal{R}^i \in \mathbb{R}^{N \times \{1,4\}}</math> is the coordinate matrix, and each row of <math>\mathcal{R}^i</math> can be constructed as one of the following forms

<math>
    (\mathcal{R}^i)_j =
    \begin{cases}
    \{
    \begin{array}{cccc}
    s(r_{ji}) & \frac{s(r_{ji})x_{ji}}{r_{ji}} & \frac{s(r_{ji})y_{ji}}{r_{ji}} & \frac{s(r_{ji})z_{ji}}{r_{ji}} 
    \end{array}
    \}, &\text{full information}  \\
    \{
    \begin{array}{c}
    s(r_{ji})
    \end{array}
    \}, &\text{radial-only information} 
    \end{cases}
    \label{eq:rij}
</math>

where <math>\mathbf{R}_{ji}=\mathbf{R}_j-\mathbf{R}_i = (x_{ji}, y_{ji}, z_{ji})</math> is the relative coordinate and <math>r_{ji}=\lVert \mathbf{R}_{ji} \lVert</math> is its norm. The [[switching function]] <math>s(r)</math> is defined as:

<math>
    s(r)=
    \begin{cases}
    \frac{1}{r}, & r<r_s \\
    \frac{1}{r} \{ {(\frac{r - r_s}{ r_c - r_s})}^3 (-6 {(\frac{r - r_s}{ r_c - r_s})}^2 +15 \frac{r - r_s}{ r_c - r_s} -10) +1 \}, & r_s \leq r<r_c \\
    0, & r \geq r_c
    \end{cases}
</math>

Each row of the embedding matrix <math>\mathcal{G}^i \in \mathbb{R}^{N \times M}</math> consists of outputs of a [[neural network]] function <math>\mathcal{N}</math> of <math>s(r_{ji})</math>:

<math>
    (\mathcal{G}^i)_j = \mathcal{N}(s(r_{ji}))
    \label{eq:G2}
</math>

where <math>\mathcal{G}^i_< \in \mathbb{R}^{N \times M_<}</math> takes first <math>M_<</math> columns of <math>\mathcal{G}^i</math>.

Both <math>M</math> and <math>M_<</math> are pre-defined parameters.

== References ==
{{reflist}}

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{{DeePMD-kit}}

[[Category:Descriptors]]