ML LMAX2: Difference between revisions

Latest revision as of 08:11, 9 May 2023

ML_LMAX2 = [integer]

Default: ML_LMAX2	= 3	for ML_LAFILT2 = .TRUE.
	= 6	else

Description: This tag specifies the maximum angular momentum quantum number $L_\max$ of spherical harmonics used to expand atomic distributions within the machine learning force field method.

To construct an atomic environment descriptor the atomic probability density

${\displaystyle \rho_{i}\left(\mathbf{r}\right) = \sum\limits_{j=1}^{N_{\mathrm{a}}} f_{\mathrm{cut}}\left(r_{ij}\right) g\left(\mathbf{r}-\mathbf{r}_{ij}\right), }$

where $g\left(\mathbf{r}\right)$ is an approximation of the delta function, is expanded in terms of radial basis functions $\chi_{nl}(r)$ and spherical harmonics $Y_{lm}\left(\hat{\mathbf{r}}\right)$ (see this section for more details):

${\displaystyle \rho_{i} \left( \mathbf{r} \right) = \sum\limits_{l=1}^{L_{\mathrm{max}}} \sum\limits_{m=-l}^{l} \sum\limits_{n=1}^{N^{l}_{ \mathrm{R}}} c_{nlm}^{i}\chi_{nl} \left( r \right) Y_{lm} \left( \hat{\mathbf{r}} \right), }$

The tag ML_LMAX2 specifies the maximum angular momentum quantum number $L_\max$ of spherical harmonics used in this expansion.

Mind: This tag is only relevant for the angular descriptor

\rho^{(3)}_i(r)

. The corresponding number of expansion coefficients

p_{n\nu l}^{i}

scales linearly with

L_\max

(however the calculation of the spherical harmonics scales quadratically) and is also depending on ML_MRB2 and the number of elements present.

By default an angular filtering (see ML_LAFILT2 and ML_IAFILT2) with a filtering parameter of ML_AFILT2=0.002 is used where a value of ML_LMAX2=3 is perfectly safe for most applications. If you need to improve the accuracy, you might also try to increase it to ML_LMAX2=4. If the angular filter is switched off, we recommend to use a value of ML_LMAX2=6 (this is then the default value).

@@ Line 1: / Line 1: @@
-{{TAGDEF|ML_FF_LMAX2_MB|[integer]|6}}
+{{DISPLAYTITLE:ML_LMAX2}}
+{{TAGDEF|ML_LMAX2|[integer]}}
+{{DEF|ML_LMAX2|3|for {{TAG|ML_LAFILT2}} {{=}} .TRUE.|6|else}}
-Description:
+Description: This tag specifies the maximum angular momentum quantum number <math>L_\max</math> of spherical harmonics used to expand atomic distributions within the machine learning force field method.
 ----
+To construct an atomic environment descriptor the atomic probability density
-== Related Tags and Sections ==
+<math>
+\rho_{i}\left(\mathbf{r}\right) = \sum\limits_{j=1}^{N_{\mathrm{a}}} f_{\mathrm{cut}}\left(r_{ij}\right) g\left(\mathbf{r}-\mathbf{r}_{ij}\right),
+</math>
-{{sc|ML_FF_LMAX2_MB|Examples|Examples that use this tag}}
+where <math>g\left(\mathbf{r}\right)</math> is an approximation of the delta function, is expanded in terms of radial basis functions <math>\chi_{nl}(r)</math> and spherical harmonics <math>Y_{lm}\left(\hat{\mathbf{r}}\right)</math> (see [[Machine learning force field: Theory#Basis set expansion|this section]] for more details):
+<math>
+\rho_{i} \left( \mathbf{r} \right) = \sum\limits_{l=1}^{L_{\mathrm{max}}} \sum\limits_{m=-l}^{l} \sum\limits_{n=1}^{N^{l}_{ \mathrm{R}}} c_{nlm}^{i}\chi_{nl} \left( r \right) Y_{lm} \left( \hat{\mathbf{r}} \right),
+</math>
+The tag {{TAG|ML_LMAX2}} specifies the maximum angular momentum quantum number <math>L_\max</math> of spherical harmonics used in this expansion.
+{{NB|mind|This tag is only relevant for the angular descriptor <math>\rho^{(3)}_i(r)</math>. The corresponding number of expansion coefficients <math>p_{n\nu l}^{i}</math> scales '''linearly''' with <math>L_\max</math> (however the calculation of the spherical harmonics scales quadratically) and is also depending on {{TAG|ML_MRB2}} and the number of elements present.}}
+By default an angular filtering (see {{TAG|ML_LAFILT2}} and {{TAG|ML_IAFILT2}}) with a filtering parameter of {{TAG|ML_AFILT2}}=0.002 is used where a value of {{TAG|ML_LMAX2}}=3 is perfectly safe for most applications. If you need to improve the accuracy, you might also try to increase it to {{TAG|ML_LMAX2}}=4. If the angular filter is switched off, we recommend to use a value of {{TAG|ML_LMAX2}}=6 (this is then the default value).
+== Related tags and articles ==
+{{TAG|ML_LMLFF}}, {{TAG|ML_MRB1}}, {{TAG|ML_MRB2}}, {{TAG|ML_LAFILT2}}, {{TAG|ML_IAFILT2}}, {{TAG|ML_AFILT2}}
+{{sc|ML_LMAX2|Examples|Examples that use this tag}}
 ----
-[[Category:INCAR]][[Category:Machine Learning]][[Category:Machine Learned Force Fields]]
+[[Category:INCAR tag]][[Category:Machine-learned force fields]]