The energy in potentials of the Modified Embedded Atom type behaves like the EAM type and consists of two parts, a pair potential term specified by the function $\Phi\left(r\right)$ representing the electrostatic core-core repulsion, and a cohesive term specified by the function $F\left(n\right)$ representing the energy the ion core gets when it is “embedded” in the “Electron Sea”. This Embedding Energy is a function of the local electron density, which in turn is constructed as a superposition of contributions from neighboring atoms. The “electron” density depends on the electron transfer, $\rho\left(r\right)$, and the 3-body terms $f\left(r\right)$ and $g\left(\cos\theta\right)$.
The embedding function $F_i\left(n\right)$ and the 3-body angular function $g_i\left(\cos\theta\right)$ both depend on the type of the embedded atom, the transfer function $\rho_j\left(r\right)$ depends on the type of the donating atom, and the pair potential $\Phi_{ij}\left(r\right)$ and the 3-body radial function $f_{ij}\left(r\right)$ both depend on the types $i$ and $j$ of both atoms involved. All potential functions are given in a single file, whose format is described here.
The Modified Embedded Atom Method (MEAM) was adapted for use in potfit by Jeremy Nicklas. It is currently offered in the following Molecular Dynamics codes: (untested and yet-to-be-released version) IMD implemented by Jeremy Nicklas, ohmms implemented by Jeongnim Kim, and a private version of LAMMPS implemented by Richard Hennig.
The Modified Embedded Atom Method was suggested by Baskes 1) 2). A simplified version of MEAM that employs cubic splines instead of predetermined analytic functions was developed later by Lenosky 3). The explicit 3-body term may improve upon the description of materials with highly-directional bonds while only being about 3-5 times slower than EAM.
Total energy form for MEAM potential:
$$E_{\text{total}}=\frac{1}{2}\sum_{i,j}^{N}\Phi_{ij}\left(r_{ij}\right)+\sum_i^N F_i\left(n_i\right)$$ $$n_i=\sum_{j\neq i}^N\rho_j\left(r_{ij}\right)+ \frac{1}{2}\sum_{j,k\neq i}^N f_{ij}\left(r_{ij}\right)f_{ik}\left(r_{ik}\right)g_{i}\left(\cos\theta_{jik}\right)$$
The functions $\Phi_{ij}\left(r_{ij}\right)$, $\rho_j\left(r_{ij}\right)$, and $F_i\left(n_i\right)$ are taken from the EAM model. The explicit 3-body term is included in the local “electron” density through the dependence on $f_{ij}\left(r_{ij}\right)$ and $g_{i}\left(\cos\theta_{jik}\right)$, where $\theta_{jik}$ is the angle between atoms $j$, $i$, and $k$ centered on atom $i$.
The invariance properties of MEAM are:
$$\rho_j\left(r\right)\rightarrow \alpha\rho_j\left(r\right)$$ $$g_i\left(\cos\theta\right)\rightarrow \alpha g_i\left(\cos\theta\right)$$ $$F_i\left(n\right)\rightarrow F_i\left(\tfrac{n}{\alpha}\right)$$
and
$$f_{ij}\left(r\right)\rightarrow \beta f_{ij}\left(r\right)$$ $$g_i\left(\cos\theta\right)\rightarrow \tfrac{1}{\beta^2} g_i\left(\cos\theta\right)$$
To describe a system of $N$ atom types you need $N(N+4)$ potentials.
| # atom types | $\Phi_{ij}$ | $\rho_j$ | $F_i$ | $f_{ij}$ | $g_i$ | Total # potentials |
|---|---|---|---|---|---|---|
| $N$ | $N(N+1)/2$ | $N$ | $N$ | $N(N+1)/2$ | $N$ | $N(N+4)$ |
| 1 | 1 | 1 | 1 | 1 | 1 | 5 |
| 2 | 3 | 2 | 2 | 3 | 2 | 12 |
| 3 | 6 | 3 | 3 | 6 | 3 | 21 |
| 4 | 10 | 4 | 4 | 10 | 4 | 32 |
The potential table is assumed to be symmetric, i.e. the potential for the atom types 1-0 is the same as the potential 0-1.
The order of the potentials in the potential file for $N$ atom types is:
$\Phi_{00}, \ldots, \Phi_{0N}, \Phi_{11}, \ldots, \Phi_{1N}, \ldots, \Phi_{NN}$
$\rho_0, \ldots, \rho_N$
$F_0, \ldots, F_N$
$f_{00}, \ldots, f_{0N}, f_{11}, \ldots, f_{1N}, \ldots, f_{NN}$
$g_0, \ldots, g_N$
Tabulated MEAM potentials require the embedding function $F_i$ to be defined at a density of $1.0$. This is necessary to fix the gauge degrees of freedom.