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# ML MRB2

ML_MRB2 = [integer]
Default: ML_MRB2 = ML_MRB1

Description: This tag sets the number ${\displaystyle N_{\text{R}}^{l}}$ (for all ${\displaystyle l}$) of radial basis functions used to expand the angular descriptor within the machine learning force field method.

The angular descriptor is constructed from

${\displaystyle \rho _{i}^{(3)}\left(r,s,\theta \right)=\iint d{\hat {\mathbf {r} }}d{\hat {\mathbf {s} }}\delta \left({\hat {\mathbf {r} }}\cdot {\hat {\mathbf {s} }}-\mathrm {cos} \theta \right)\sum \limits _{j=1}^{N_{a}}\sum \limits _{k\neq j}^{N_{a}}\rho _{ik}\left(r{\hat {\mathbf {r} }}\right)\rho _{ij}\left(s{\hat {\mathbf {s} }}\right),\quad {\text{where}}\quad \rho _{ij}\left(\mathbf {r} \right)=f_{\mathrm {cut} }\left(r_{ij}\right)g\left(\mathbf {r} -\mathbf {r} _{ij}\right)}$

and ${\displaystyle g\left(\mathbf {r} \right)}$ is an approximation of the delta function. In practice, the continuous function above is transformed into a discrete set of numbers ${\displaystyle p_{n\nu l}^{i}}$ by expanding it into a set of radial basis functions ${\displaystyle \chi _{nl}(r)}$ and Legendre polynomials ${\displaystyle P_{l}\left(\mathrm {cos} \theta \right)}$ (see this section for more details):

${\displaystyle \rho _{i}^{(3)}\left(r,s,\theta \right)=\sum \limits _{l=1}^{L_{\mathrm {max} }}\sum \limits _{n=1}^{N_{\mathrm {R} }^{l}}\sum \limits _{\nu =1}^{N_{\mathrm {R} }^{l}}{\sqrt {\frac {2l+1}{2}}}p_{n\nu l}^{i}\chi _{nl}\left(r\right)\chi _{\nu l}\left(s\right)P_{l}\left(\mathrm {cos} \theta \right).}$

The tag ML_MRB2 sets the number ${\displaystyle N_{\text{R}}^{l}}$ of radial basis functions to use in this expansion. The same number is used for all ${\displaystyle l}$.

 Mind: The number of angular descriptor expansion coefficients ${\displaystyle p_{n\nu l}^{i}}$ scales quadratically with ${\displaystyle N_{\text{R}}^{l}}$ set by this tag. It also depends on ML_LMAX2 and the number of elements.