Two-body DeepPot-SE

Two-body DeepPot-SE (full)

Key	se_e2_a
Alias	se_a
Supported version	v0.8.2
Model compression	Y

Two-body DeepPot-SE (radial-only)

Key	se_e2_r
Alias	se_r
Supported version	v1.0.0
Model compression	N

Two-body DeepPot-SE (se_e2_a or se_e2_r) is a descriptor of the Deep Potential Smooth Edition (DeepPot-SE) constructed from two-body information of atomic configurations.

Theory

The two-body DeepPot-SE descriptor is given by

[math]\displaystyle{ \mathcal{D}^i = (\mathcal{G}^i)^T \mathcal{R}^i (\mathcal{R}^i)^T \mathcal{G}^i_\lt }[/math]

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

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

[math]\displaystyle{ s(r)= \begin{cases} \frac{1}{r}, & r\lt 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\lt r_c \\ 0, & r \geq r_c \end{cases} }[/math]

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

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

where [math]\displaystyle{ \mathcal{G}^i_\lt \in \mathbb{R}^{N \times M_\lt } }[/math] takes first [math]\displaystyle{ M_\lt }[/math] columns of [math]\displaystyle{ \mathcal{G}^i }[/math].

Both [math]\displaystyle{ M }[/math] and [math]\displaystyle{ M_\lt }[/math] are pre-defined parameters.