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MEAM

Description

The meam_force operator computes the Modified Embedded-Atom Model potential with many-body screening, for a single species. The energy of atom \(i\) reads

\[ E_i = F\left(\bar\rho_i\right) + \frac{1}{2}\sum_{j \neq i} S_{ij}\,\phi(r_{ij}) \]

Partial electron densities. Each neighbor \(j\) contributes atomic densities \(\rho^{(l)}(r) = e^{-\beta_l\left(r/r_0 - 1\right)}\), \(l = 0 \dots 3\), weighted by the screening factor \(S_{ij}\). They are combined into a spherical density \(\rho_i^{(0)}\) and angular densities \(\rho_i^{(1)}\), \(\rho_i^{(2)}\), \(\rho_i^{(3)}\). The background density is

\[ \bar\rho_i = \rho_i^{(0)}\,\frac{2}{1 + e^{-\Gamma_i}}, \qquad \Gamma_i = \sum_{l=1}^{3} t_l\left(\frac{\rho_i^{(l)}}{\rho_i^{(0)}}\right)^2 \]

Embedding function.

\[ F(\bar\rho) = A\,E_c\,\frac{\bar\rho}{Z}\,\ln\frac{\bar\rho}{Z} \]

where \(E_c\) is the cohesive energy and \(Z\) the coordination of the reference structure. The reference-structure density, used to build the pair term, combines the partial densities with the weights \(s_l\,t_l\).

Tip

Both Ecoh and E0 enter the energy. In the examples shipped with exaStamp they are set to the same value.

Pair term. The pair interaction follows from a Rose-type equation of state, \(E^u(r) = -E_0\left(1 + a^* + \delta\,a^{*3} r_0/r\right)e^{-a^*}\) with \(a^* = \alpha\left(r/r_0 - 1\right)\).

Screening. The screening factor \(S_{ij}\) of a pair is the product of the screening by every common neighbor \(k\), computed from the ellipse parameter \(C_{ikj}\) and the bounds \(C_{\min}\), \(C_{\max}\) (full screening below \(C_{\min}\), none above \(C_{\max}\)). A polynomial smoothing brings the interactions to zero between \(r_c - r_p\) and \(r_c\).

Ghost atoms

By default (ghost: true) the MEAM terms are also computed on ghost atoms, so no extra communication is needed, at the price of a ghost layer of twice the MEAM cutoff (ghost_dist_max is raised automatically). With ghost: false, the ghost layer is one cutoff wide, but the force contributions written on ghost atoms must be sent back to their owners: add includes: [ config_update_symmetric_forces.msp ] to the input file.

Parameter Units Description
rmax distance Maximum distance for the density contributions
rmin distance Minimum distance for the density contributions
Ecoh energy Cohesive energy
E0 energy Energy scale of the pair term and of the embedding function
A — Embedding amplitude
r0 distance Nearest-neighbor distance of the reference structure
alpha — Exponent of the equation of state
delta — Cubic coefficient of the equation of state
beta0 … beta3 — Decay rates of the partial densities
t0 … t3 — Weights of the partial densities
s0 … s3 — Reference-structure weights of the partial densities
Cmin, Cmax — Screening bounds
Z — Number of first neighbors in the reference structure
rc distance Cutoff of the MEAM interactions
rp distance Width of the smoothing interval \([r_c - r_p, r_c]\)

All parameters are required.

Syntax
meam_force:
  rcut: <float>
  ghost: <bool>
  parameters: { rmax: <float>, rmin: <float>, Ecoh: <float>, E0: <float>, A: <float>, r0: <float>,
                alpha: <float>, delta: <float>, beta0: <float>, beta1: <float>, beta2: <float>, beta3: <float>,
                t0: <float>, t1: <float>, t2: <float>, t3: <float>, s0: <float>, s1: <float>, s2: <float>, s3: <float>,
                Cmin: <float>, Cmax: <float>, Z: <float>, rc: <float>, rp: <float> }
Parameters
rcut:        float, required       # Base cutoff radius. The actual cutoff may be larger, to include the screening neighbors.
ghost:       bool, default true    # true: compute the MEAM terms on ghost atoms too (ghost layer of 2 x cutoff, no extra communication). false: ghost layer of 1 x cutoff, ghost forces must be sent back (include config_update_symmetric_forces.msp).
parameters:  map, required         # MEAM parameters, see the table above.

Usage example

Tin

meam_force:
  rcut: 3.69 ang
  parameters:
    rmax: 3.69 ang
    rmin: 0.0
    Ecoh: 4.93474273599999995322e-19 J
    E0: 4.93474273599999995322e-19 J
    A: 1.000E+00
    r0: 3.44 ang
    alpha: 6.200E+00
    delta: 0.000E+00
    beta0: 6.200E+00
    beta1: 6.000E+00
    beta2: 6.000E+00
    beta3: 6.000E+00
    t0: 1.000E+00
    t1: 4.500E+00
    t2: 6.500E+00
    t3: -0.183E+00
    s0: 1.440E+02
    s1: 0.000E+00
    s2: 0.000E+00
    s3: 0.000E+00
    Cmin: 0.800E+00
    Cmax: 2.800E+00
    Z: 1.200E+01
    rc: 4.0 ang
    rp: 0.1 ang

compute_force: meam_force

Complete inputs are in exaStamp/data/regression_new/potentials/meam/.