# Notes¶

The modified embedded-atom-method (MEAM) potential extends the conventional embedded-atom-method by including directional bonding [Bas92]. This allows the application to both metallic and covalent systems, as well as mixed systems.

The energy function for the Modified Embedded Atom Method as presented e.g. in [GWS03] is given as the sum of some pair term and some embedding term as follows:

$E= \sum_i \left\{ F_i(\bar{\rho}_i) + \frac{1}{2} \sum_{j\ne i} \Phi_{ij}(r_{ij})\right\}$

The embedding function is of the form

$F_i(\bar{\rho}_i) = A_i E_i^0\bar{\rho}_i \log \bar{\rho}_i$

for $$\bar{\rho}_i > 0$$ and for $$\bar{\rho}_i \le 0$$

$F_i(\bar{\rho}_i) = 0$

if embedding_negative is 0 or

$F_i(\bar{\rho}_i) = -A_i E_i^0 \bar{\rho}_i$

if it is 1, where $$A_i$$ and $$E_i^0$$ are element dependent parameters. The background electron density $$\bar{\rho}_i$$ is given by:

$\bar{\rho}_i = \frac{\bar{\rho}_i^{(0)}}{\rho_i^0} G_i (\Gamma_i)$

where

$\Gamma_i = \sum_{k=1}^3 t_i^{(k)}\left(\frac{\bar{\rho}_i^{(k)}}{\bar{\rho}_i^{(0)}}\right)^2$

and $$G_i$$ is one of

(1)$\begin{split}G_i(\Gamma) = \begin{cases} \sqrt{1+\Gamma} & gamma=0\\ \exp(\Gamma / 2) & gamma=1\\ \text{sign}(1+\Gamma) \sqrt{|1+\Gamma|} & gamma=2\\ \frac{2}{1+\exp(-\Gamma)} &gamma=3\\ \sqrt{1+\Gamma} & gamma=4 \end{cases}\end{split}$

With the element dependent constant $$\rho_i^0$$, the number of nearest neighbors in the reference structure $$Z_{i0}$$ and $$\Gamma_i^\text{ref}$$ $$\Gamma$$ evaluated in the reference structure, the composition dependent density scaling can be chosen to be either

(2)$\rho_i^0 = \rho_{i0}Z_{i0}G_i(\Gamma_i^\text{ref})$

or

(3)$\rho_i^0 = \rho_{i0} Z_{i0}$

Note that for gamma=0 and gamma=2 $$G_i(\Gamma_i^\text{ref})$$ is always set to 1. The partial electron densities are defined as

$\begin{split}\begin{eqnarray*} \bar{\rho}_i^{(0)} &=& \sum_{j\ne i} \rho_j^{a(0)}(r_{ij})S_{ij} \\ (\bar{\rho}_i^{(1)})^2 &=& \sum_{\alpha = 1}^3 \left[\sum_{j \ne i} \rho_j^{a(1)} \frac{r_{ij\alpha}}{r_{ij}} S_{ij} \right]^2 \\ (\bar{\rho}_i^{(2)})^2 &=& \sum_{\alpha =1}^3 \sum_{\beta =1}^3 \left[\sum_{j\ne i} \rho_j^{a(2)} \frac{r_{ij\alpha}r_{ij\beta}}{r_{ij}^2} S_{ij} \right]^2 - \frac{1}{3} \left[ \sum_{j\ne i} \rho_j^{a(2)}S_{ij}\right]^2\\ (\bar{\rho}_i^{(3)})^2 &=& \sum_{\alpha =1}^3 \sum_{\beta =1}^3 \sum_{\gamma =1}^3 \left[\sum_{j\ne i} \rho_j^{a(3)} \frac{r_{ij\alpha}r_{ij\beta}r_{ij\gamma}}{r_{ij}^3} S_{ij}\right]^2 - \frac{3}{5}\sum_{\alpha =1}^3\left[\sum_{j \ne i} \rho_j^{a(3)} \frac{r_{ij\alpha}}{r_{ij}} S_{ij}\right]^2 \end{eqnarray*}\end{split}$

with the atomic electron densities given by

$\rho_i^{a(k)}(r_{ij}) = \rho_{i0} \exp \left[-\beta_i^{(k)} \left(\frac{r_{ij}}{r_i^0} - 1 \right)\right]$

Finally, the average weighting factors are given by

(4)$\begin{split}t_i^{(k)} = \begin{cases} \frac{1}{\bar{\rho}_i^{(0)}} \sum_{j\ne i} t_{0,j}^{(k)} \rho_j^{a(0)} S_{ij} & \text{wf-mixing=0} \\ \frac{ \sum_{j\ne i} t_{0,j}^{(k)} \rho_j^{a(0)} S_{ij}} { \sum_{j\ne i} \left(t_{0,j}^{(k)}\right)^2 \rho_j^{a(0)} S_{ij}}& \text{wf-mixing=1} \\ t_{0,i}^{(k)} & \text{wf-mixing=2} \end{cases}\end{split}$

with element dependent parameters $$t_{0,i}^{(k)}$$.

The pair term $$\Phi_{ij}$$ is defined as

(5)$\begin{split}\begin{eqnarray} \Phi_{ij}(r_{ij}) &=& \bar{\Phi}_{ij}(r_{ij}) S_{ij}\\ \bar{\Phi}_{ij}(r_{ij}) &=& \frac{1}{Z_{ij0}} [2E_i^u(r_{ij}) - F_i(\hat{\rho}_i (r_{ij})) - F_j(\hat{\rho}_j(r_{ij})] \end{eqnarray}\end{split}$

where $$Z_{ij0}$$ is the number of nearest neighbors in the reference structure, $$F_i(\hat{\rho}_{ij})$$ the embedding function evaluated in the reference structure and

(6)$\begin{split}E_i^u(r_{ij}) = \begin{cases} -E_{ij}^c \left(1+a_{ij}^{*} + a_{ij}^{(3)} {a_{ij}^{*}}^3 \frac{r^0_{ij}}{r_{ij}}\right) \exp(-a_{ij}^*) & \text{erose=0} \\ -E_{ij}^c \left(1+a_{ij}^* + \left(-\text{attrac}+\frac{\text{repuls}}{r_{ij}}\right){a_{ij}^*}^3\right)\exp(-a_{ij}^*) & \text{erose=1}\\ -E_{ij}^c\left(1+a_{ij}^* + a_{ij}^{(3)}{a_{ij}^*}^3\right)\exp(-a_{ij}^*) & \text{erose=2} \end{cases}\end{split}$

with

$a_{ij}^*(r_{ij}) = \alpha_{ij} \left(\frac{r_{ij}}{r^0_{ij}} - 1\right)$

and

$\begin{split}a_{ij}^{(3)} = \begin{cases} \text{repuls} & a_{ij}^* < 0\\ \text{attrac} & a_{ij}^* \ge 0 \end{cases}\end{split}$

Some reference structures use a more complex form of (5). For small $$r_{ij}$$, this pair term is blended with a ZBL potential by default.newline Last, the screening function $$S_{ij}$$ is defined by

$\begin{split}\begin{eqnarray*} S_{ij}&=&\bar{S}_{ij} f_c\left(\frac{r_c - r_{ij}}{\Delta r}\right)\\ \bar{S}_{ij} &=& \prod_{k\ne i,j} S_{ikj}\\ S_{ikj} &=& f_c\left(\frac{C_{ikj}-C_{\text{min},ikj}}{C_{\text{max},ikj} -C_{\text{min},ikj}}\right)\\ C_{ikj}&=& 1+2\frac{r_{ij}^2 r_{ik}^2 + r_{ij}^2r_{jk}^2-r_{ij}^4}{r_{ij}^4-(r_{ik}^2-r_{jk}^2)^2}\\ f_c(x)&=&\begin{cases} 1 & x \ge 1 \\ [1-(1-x)^4]^2 & 0<x<1\\ 0 & x \le 0 \end{cases} \end{eqnarray*}\end{split}$

where $$r_c$$ is the global cutoff radius, $$\Delta r$$ is the length of a smoothing region for $$r$$ near $$r_c$$ and $$C_\text{min}$$ and $$C_\text{max}$$ are element-triple dependent parameters.

Note that the cutoff radius for this term needs to be larger than $$r_c$$, depending on $$C_{\text{max}}$$, because even triples with some particle distance $$> r_c$$ may have a screening effect.

The parameter lattice defines the reference structure for the given particle type. The parameters alpha, beta_k, re, Ec, scaling_factor, weighting_factor_k, rho, gamma, attrac and repuls correspond to the variables $$\alpha$$, $$\beta^{(k)}$$, $$r^0$$, $$E^c$$, $$A$$, $$t_0^{(k)}$$, $$\rho_{0}$$, attrac and repuls from the previous section respectively. nn2 enables the second nearest neighbor formulation as described in cite{lee2000second}, zbl enables ZBL blending for small distances if set to 1.

re and Ec correspond to $$r^0$$ and $$E^c$$. Note that all element_data entries must be given before the first element_pair entry.newline If augment_1st is set to 1, we set

$t_{0,i}^{(1)} = t_{0,i}^{(1)} + \frac{3}{5} t_{0,i}^{(3)}$

for all $$i$$. This is usually only needed for older parameter sets, modern ones already include this correction. The default density_scaling uses (2), setting it to 1 uses (3) instead.

The latticeType parameter must be one of the following strings:

• dia = diamond
• fcc = face centered cubic
• bcc = body centered cubic
• dim = dimer
• b1 = rock salt
• hcp = hexagonal close-packed
• c11 = MoSi2 structure
• l12 = Cu3Au structure
• b2 = CsCl structure

 [Bas92] M. I. Baskes. Modified embedded-atom potentials for cubic materials and impurities. Phys. Rev. B, 46:2727–2742, Aug 1992. URL: http://link.aps.org/doi/10.1103/PhysRevB.46.2727, doi:10.1103/PhysRevB.46.2727.
 [GWS03] PM Gullet, G Wagner, and A Slepoy. Numerical tools for atomistic simulations. SANDIA Report, 8782:2003, 2003.