\(\renewcommand{\AA}{\text{Å}}\)

# pair_style rann command

## Syntax

    pair_style rann
    pair_coeff file Type1_element Type2_element Type3_element...

## Examples

    pair_style rann
    pair_coeff * * Mg.rann Mg
    pair_coeff * * MgAlalloy.rann Mg Mg Al Mg

## Description

Pair style rann computes pairwise interactions for a variety of materials using rapid atomistic neural network (RANN) potentials (Dickel , Nitol). Neural network potentials work by first generating a series of symmetry functions i.e. structural fingerprints from the neighbor list and then using these values as the input layer of a neural network. There is a single output neuron in the final layer which is the energy. Atomic forces are found by analytical derivatives computed via back-propagation. For alloy systems, each element has a unique network.

## Potential file syntax

The RANN potential is defined by a single text file which contains all the fitting parameters for the alloy system. The potential file also defines the active fingerprints, network architecture, activation functions, etc. The potential file is divided into several sections which are identified by one of the following keywords:

- atomtypes

- mass

- fingerprintsperelement

- fingerprints

- fingerprintconstants

- screening (optional)

- networklayers

- layersize

- weight

- bias

- activationfunctions

- calibrationparameters (ignored)

The ‘#’ character is treated as a comment marker, similar to LAMMPS input scripts. Sections are not required to follow a rigid ordering, but do require previous definition of prerequisite information. E.g., fingerprintconstants for a particular fingerprint must follow the fingerprints definition; layersize for a particular layer must follow the declaration of network layers.

atomtypes are defined as follows using element keywords separated by spaces.

    atomtypes:
    Fe Mg Al etc.

mass must be specified for each element keyword as follows:

    mass:Mg:
    24.305
    mass:Fe:
    55.847
    mass:Al:
    26.982

fingerprintsperelement specifies how many fingerprints are active for computing the energy of a given atom. This number must be specified for each element keyword. Active elements for each fingerprint depend upon the type of the central atom and the neighboring atoms. Pairwise fingerprints may be defined for a Mg atom based exclusively on its Al neighbors, for example. Bond fingerprints may use two neighbor lists of different element types. In computing fingerprintsperelement from all defined fingerprints, only the fingerprints defined for atoms of a particular element should be considered, regardless of the elements used in its neighbor list. In the following code, for example, some fingerprints may compute pairwise fingerprints summing contributions about Fe atoms based on a neighbor list of exclusively Al atoms, but if there are no fingerprints summing contributions of all neighbors about a central Al atom, then fingerprintsperelement of Al is zero:

    fingerprintsperelement:Mg:
    5
    fingerprintsperelement:Fe:
    2
    fingerprintsperelement:Al:
    0

fingerprints specifies the active fingerprints for a certain element combination. Pair fingerprints are specified for two elements, while bond fingerprints are specified for three elements. Only one fingerprints header should be used for an individual combination of elements. The ordering of the fingerprints in the network input layer is determined by the order of element combinations specified by subsequent fingerprints lines, and the order of the fingerprints defined for each element combination. Multiple fingerprints of the same style or different ones may be specified. If the same style and element combination is used for multiple fingerprints, they should have different id numbers. The first element specifies the atoms for which this fingerprint is computed while the other(s) specify which atoms to use in the neighbor lists for the computation. Switching the second and third element type in bond fingerprints has no effect on the computation:

    fingerprints:Mg_Mg:
    radial_0 radialscreened_0 radial_1
    fingerprints:Mg_Al_Fe:
    bond_0 bondspin_0
    fingerprints:Mg_Al:
    radial_0 radialscreened_0

The following fingerprint styles are currently defined. See the formulation section below for their definitions:

- radial

- radialscreened

- radialspin

- radialscreenedspin

- bond

- bondscreened

- bondspin

- bondscreenedspin

fingerprintconstants specifies the meta-parameters for a defined fingerprint. For all radial styles, re, rc, alpha, dr, o, and n must be specified. re should usually be the stable interatomic distance, rc is the cutoff radius, dr is the cutoff smoothing distance, o is the lowest radial power term (which may be negative), and n is the highest power term. The total length of the fingerprint vector is (n-o+1). alpha is a list of decay parameters used for exponential decay of radial contributions. It may be set proportionally to the bulk modulus similarly to MEAM potentials, but other values may provided better fitting in special cases. Bond style fingerprints require specification of re, rc, alphak, dr, k, and m. Here m is the power of the bond cosines and k is the number of decay parameters. Cosine powers go from 0 to m-1 and are each computed for all values of alphak. Thus the total length of the fingerprint vector is m*k.

    fingerprintconstants:Mg_Mg:radialscreened_0:re:
    3.193592
    fingerprintconstants:Mg_Mg:radialscreened_0:rc:
    6.000000
    fingerprintconstants:Mg_Mg:radialscreened_0:alpha:
    5.520000 5.520000 5.520000 5.520000 5.520000
    fingerprintconstants:Mg_Mg:radialscreened_0:dr:
    2.806408
    fingerprintconstants:Mg_Mg:radialscreened_0:o:
    -1
    fingerprintconstants:Mg_Mg:radialscreened_0:n:
    3

screening specifies the Cmax and Cmin values used in the screening fingerprints. Contributions form neighbors to the fingerprint are omitted if they are blocked by a closer neighbor, and reduced if they are partially blocked. Larger values of Cmin correspond to neighbors being blocked more easily. Cmax cannot be greater than 3, and Cmin cannot be greater than Cmax or less than zero. Screening may be omitted in which case the default values Cmax = 2.8, Cmin = 0.8 are used. Since screening is a bond computation, it is specified separately for each combination of three elements in which the latter two may be interchanged with no effect.

    screening:Mg_Mg_Mg:Cmax:
    2.700000
    screening:Mg_Mg_Mg:Cmin:
    0.400000

networklayers species the size of the neural network for each atom. It counts both the input and output layer and so is 2 + <hidden layers>.

    networklayers:Mg:
    3

layersize specifies the length of each layer, including the input layer and output layer. The input layer is layer 0. The size of the input layer size must match the summed length of all the fingerprints for that element, and the output layer size must be 1:

    layersize:Mg:0:
    14
    layersize:Mg:1:
    20
    layersize:Mg:2:
    1

weight specifies the weight for a given element and layer. Weight cannot be specified for the output layer. The weight of layer i is a m x n matrix where m is the layer size of i and n is the layer size of i+1:

    weight:Mg:0:
    w11 w12 w13 ...
    w21 w22 w23 ...
    ...

bias specifies the bias for a given element and layer. Bias cannot be specified for the output layer. The bias of layer i is a nx1 vector where n is the layer size of i+1:

    bias:Mg:0:
    b1
    b2
    b3
    ...

activationfunctions specifies the activation function for a given element and layer. Activation functions cannot be specified for the output layer:

    activationfunctions:Mg:0:
    sigI
    activationfunctions:Mg:1:
    linear

The following activation styles are currently specified. See the formulation section below for their definitions.

- sigI

- linear

calibrationparameters specifies a number of parameters used to calibrate the potential. These are ignored by LAMMPS.

## Formulation

In the RANN formulation, the total energy of a system of atoms is given by:

\[\begin{split}E = \sum_{\alpha} E^{\alpha}\\\\ E^{\alpha} = {}^{N}\!A^{\alpha}\\\\ {}^{n+1}\!A_i^{\alpha} = {}^{n}\!F\left({}^{n}\!W_{ij}{\;}^{n}\!A_j^{\alpha}+{}^{n}\!B_i\right)\\\\ {}^{0}\!A_i^{\alpha} = \left[\begin{array}{c} {}^1\!S\!f^\alpha\\ {}^2\!S\!f^\alpha \\...\\\end{array}\right]\end{split}\]

Here \(E^\alpha\) is the energy of atom \(\alpha\), \({}^n\!F()\), \({}^n\!W_{ij}\) and \({}^n\!B_i\) are the activation function, weight matrix and bias vector of the n-th layer respectively. The inputs to the first layer are a collection of structural fingerprints which are collected and reshaped into a single long vector. The individual fingerprints may be defined in any order and have various shapes and sizes. Multiple fingerprints of the same type and varying parameters may also be defined in the input layer.

Eight types of structural fingerprints are currently defined. In the following, \(\beta\) and \(\gamma\) span the full neighbor list of atom \(\alpha\). \(\delta_i\) are decay meta-parameters, and \(r_e\) is a meta-parameter roughly proportional to the first neighbor distance. \(r_c\) and \(dr\) are the neighbor cutoff distance and cutoff smoothing distance respectively. \(S^{\alpha\beta}\) is the MEAM screening function (Baskes), \(s_i^\alpha\) and \(s_i^\beta\) are the atom spin vectors (Tranchida). \(r^{\alpha\beta}\) is the distance from atom \(\alpha\) to atom \(\beta\), and \(\theta^{\alpha\beta\gamma}\) is the bond angle:

\[cos\left(\theta^{\alpha\beta\gamma}\right)=\frac{\mathbf{r}^{\alpha\beta} \cdot \mathbf{r}^{\alpha\gamma}}{r^{\alpha\beta}r^{\alpha\gamma}}\]

\(S^{\alpha\beta}\) is defined as (Baskes):

\[\begin{split}X^{\gamma\beta} = \left(\frac{r^{\gamma\beta}}{r^{\alpha\beta}}\right)^2\\ \\ X^{\alpha\gamma} = \left(\frac{r^{\alpha\gamma}}{r^{\alpha\beta}}\right)^2\\ \\ C = \frac{2\left(X^{\alpha\gamma}+X^{\gamma\beta}\right)-\left(X^{\alpha\gamma}-X^{\gamma\beta}\right)^2-1}{1-\left(X^{\alpha\gamma}-X^{\gamma\beta}\right)^2}\\ \\ f_c(x) = \left[\begin{array}{l} 1 \; \: x \geq 1\\ \left(1-\left(1-x\right)^4\right)^2 \; \: 0<x<1\\0\; \; x\leq0\end{array}\right.\\ \\ S^{\alpha\beta\gamma} = f_c\left(\frac{C-C_{min}}{C_{max}-C_{min}}\right)\\ \\ S^{\alpha\beta} = \prod_\gamma S^{\alpha\beta\gamma}\\\end{split}\]

The structural fingerprints are computed as follows:

- radial

\[{}^r\!S\!f_i^\alpha = \sum_{\beta} \left(\frac{r^{\alpha\beta}}{r_e}\right)^ie^{-\delta_i \frac{r^{\alpha\beta}}{r_e}}f_c\left(\frac{r_c-r^{\alpha\beta}}{dr}\right)\]

- bond

\[{}^b\!S\!f_{ij}^\alpha = \sum_{\beta}\sum_{\gamma} \left(cos(\theta_{\alpha\beta\gamma})\right)^ie^{-\delta_j \frac{r^{\alpha\beta}}{r_e}}e^{-\delta_j \frac{r^{\alpha\gamma}}{r_e}}f_c\left(\frac{r_c-r^{\alpha\beta}}{dr}\right)f_c\left(\frac{r_c-r^{\alpha\gamma}}{dr}\right)\]

- radialscreened

\[{}^{rsc}\!S\!f_i^\alpha = \sum_{\beta} \left(\frac{r^{\alpha\beta}}{r_e}\right)^ie^{-\delta_i \frac{r^{\alpha\beta}}{r_e}}S^{\alpha\beta}f_c\left(\frac{r_c-r^{\alpha\beta}}{dr}\right)\]

- bondscreened

\[{}^{bsc}\!S\!f_{ij}^\alpha = \sum_{\beta}\sum_{\gamma} \left(cos(\theta_{\alpha\beta\gamma})\right)^ie^{-\delta_j \frac{r^{\alpha\beta}}{r_e}}e^{-\delta_j \frac{r^{\alpha\gamma}}{r_e}}S^{\alpha\beta}S^{\alpha\gamma}f_c\left(\frac{r_c-r^{\alpha\beta}}{dr}\right)f_c\left(\frac{r_c-r^{\alpha\gamma}}{dr}\right)\]

- radialspin

\[{}^{rsp}\!S\!f_i^\alpha = \sum_{\beta} \left(\frac{r^{\alpha\beta}}{r_e}\right)^ie^{-\delta_i \frac{r^{\alpha\beta}}{r_e}}\left(\mathbf{s^\alpha \cdot s^\beta}\right)f_c\left(\frac{r_c-r^{\alpha\beta}}{dr}\right)\]

- bondspin

\[{}^{bsp}\!S\!f_{ij}^\alpha = \sum_{\beta}\sum_{\gamma} \left(cos(\theta_{\alpha\beta\gamma})\right)^ie^{-\delta_j \frac{r^{\alpha\beta}}{r_e}}e^{-\delta_j \frac{r^{\alpha\gamma}}{r_e}}\left(\mathbf{s^\alpha \cdot s^\beta}\right)\left(\mathbf{s^\alpha \cdot s^\gamma}\right)f_c\left(\frac{r_c-r^{\alpha\beta}}{dr}\right)f_c\left(\frac{r_c-r^{\alpha\gamma}}{dr}\right)\]

- radialscreenedspin

\[{}^{rscsp}\!S\!f_i^\alpha = \sum_{\beta} \left(\frac{r^{\alpha\beta}}{r_e}\right)^ie^{-\delta_i \frac{r^{\alpha\beta}}{r_e}}S^{\alpha\beta}\left(\mathbf{s^\alpha \cdot s^\beta}\right)f_c\left(\frac{r_c-r^{\alpha\beta}}{dr}\right)\]

- bondscreenedspin

\[{}^{bscsp}\!S\!f_{ij}^\alpha = \sum_{\beta}\sum_{\gamma} \left(cos(\theta_{\alpha\beta\gamma})\right)^ie^{-\delta_j \frac{r^{\alpha\beta}}{r_e}}e^{-\delta_j \frac{r^{\alpha\gamma}}{r_e}}S^{\alpha\beta}S^{\alpha\gamma}\left(\mathbf{s^\alpha \cdot s^\beta}\right)\left(\mathbf{s^\alpha \cdot s^\gamma}\right)f_c\left(\frac{r_c-r^{\alpha\beta}}{dr}\right)f_c\left(\frac{r_c-r^{\alpha\gamma}}{dr}\right)\]

The activation functions are computed as follows:

- sigI

\[F^{sigI}(x) = 0.1x+0.9ln\left(e^x+1\right)\]

- linear

\[F^{linear}(x) = x\]

## Restrictions

Pair style rann is part of the ML-RANN package. It is only enabled if LAMMPS was built with that package. Additionally, if any spin fingerprint styles are used LAMMPS must be built with the SPIN package as well.

Pair style rann does not support computing per-atom stress or using pair_modify nofdotr.

## Defaults

Cmin = 0.8, Cmax = 2.8.

(Baskes) Baskes, Materials Chemistry and Physics, 50(2), 152-158, (1997).

(Dickel) Dickel, Francis, and Barrett, Computational Materials Science 171 (2020): 109157.

(Nitol) Nitol, Dickel, and Barrett, Computational Materials Science 188 (2021): 110207.

(Tranchida) Tranchida, Plimpton, Thibaudeau and Thompson, Journal of Computational Physics, 372, 406-425, (2018).