The modified embedded atom method (MEAM) is an empirical extension of EAM by Baskes, which

includes angular forces. As in the EAM, there are pairwise repulsions and an embedding function. In the EAM, the pi is interpreted as a linear supposition of species-dependent spherically averaged atomic elec­tron densities (here designated by f (r)); in MEAM pj is augmented by angular terms. The spherically sym­metric partial electron density p(0) is the same as the electron density in the EAM:

r(0)= X fm(ri)

j

where the sum is over all atoms j, not including the atom at the specific site of interest i. The angular contributions to the density are similar to spherical harmonics: they are given by similar formulas weighted by the x, y, and z components of the distances between atoms (labeled by a, b, g):

2

(r(1))2=E

a

(r(2))2 —

a, b

(r(3))2=E

a. p.g

The f(l) are so-called ‘atomic electron densities,’ which decrease with distance from the site of interest, and the a, b, and g summations are each over the three coordinate directions (x, y z). The functional forms for the partial electron densities were chosen to be trans­lationally and rotationally invariant and are equal to zero for crystals with inversion symmetry about all atomic sites. Although the terms are related to powers of the cosine of the angle between groups of three atoms, there is no explicit evaluation of angles, and all the information required to evaluate the MEAM is available in standard MD codes. Typically, atomic electron densities are assumed to decrease exponen­tially, that is, f(l) (R) — exp [—b(l) (R/re — 1)] where the decay lengths (re and b(l)) are constants.

While there is no derivation of the MEAM from electronic structure, it also introduced the physically reasonable idea ofmany-body screening, which is miss­ing in pair-functional forms such as EAM. Thus, f(Rj) is reduced by a screening factor determined by the other atoms k forming three-body triplets with i and j: primar­ily those lying between j and j. This eliminates the need for an explicit cut-off in the ranges of V (r) and f(l) (r).

For close-packed materials, the improvement of MEAM over standard EAM is marginal; the angular terms come out to be small. For sp-bonded materials, a large three-body term can stabilize tetrahedrally coor­dinated structures, but since the physics arises from preferred 109° angles rather than preferred fourfold coordination, it suffers problems similar to Stillinger- Weber type potentials (discussed below). Very high angular components enable one to fit the complex phases of lanthanides and actinides. It is tempting to attribute this to the correct capture of the f-electron physics, although the additional functional freedom may play a role in enabling fits to low symmetry structures.