# Modified embedded atom method¶

The modified embedded atom method (MEAM) potential scheme was developed as a generalization of the embedded atom method [Bas87]. In contrast to the latter, MEAM potentials include angular dependent interactions, which enter via the electron density term. As a result these potentials can describe directional bonding, which is most apparent in covalent materials such as silicon and diamond but also in e.g., body-centered cubic (BCC) metals [Bas92].

While initially simple analytic forms were adopted and the (angular) interactions were restricted to first-nearest neigbors [Bas87], [Bas92], the MEAM form can be easily extended to include a larger number of neighbors [LeeBasKim01] and support general functional forms (e.g., splines [LenSadAlo00]). According to the classification of Carlsson [Car90] MEAM potentials fall in the category of cluster functionals, which also includes e.g., analytic bond order potential.

The general form for the total energy is (also compare the embedded atom method (EAM) format)

$E = \sum_{ij} V(r_{ij}) + \sum_i F(\rho_i)$

where the angular dependence enters via the electron density term

$\rho_i = \sum_j \rho(r_{ij}) + \sum_{jk} f(r_{ij}) f(r_{ik}) g\left(\cos(\theta_{ijk})\right).$

The second term in the summation constitutes they key difference compared to the EAM format.

As in the case of EAM potentials various functional forms have been proposed for $$V$$, $$F$$, $$\rho$$, $$f$$, and $$g$$. In the present implementation it is possible to compose functional forms using a math parser. This allows one not only to replicate any of the original forms but to define practically arbitrary functional forms. This is demonstrated by the construction of a MEAM potential with user defined functions.

The following code block illustrates the definition of a rather simple functional form. The two main subelements of the <meam> block are <mapping> and <functions>. The <functions> block comprises the definitions of the various functions and parameters as described in detail in the section on the specification of functional forms. Each function defined has to be assigned an id using the id attribute. The id is used in the <mapping> block to attach the functions to certain types of interaction (pair potential V, electron density rho along with the supporting functions f and g) or atom type (embedding function F).

<meam id="Al" species-a="*" species-b="*">
<export-functions>Si_meam_out</export-functions>

<mapping>
<pair-interaction species-a='*' species-b='*' function='V' />
<electron-density species-a='*' species-b='*' function='rho' />
<f-function species-a='*' species-b='*' function='f' />
<g-function species-a='*' species-b='*' species-c='*' function='g' />
<embedding-energy species='*' function='F' />
</mapping>

<functions>
<spline id='V'>
<derivative-left>-42.66967</derivative-left>
<cutoff>4.5</cutoff>
<nodes>
<node x='1.500000000'  y=' 6.92994'   enabled='true' />
<node x='1.833333333'  y='-0.43995'   enabled='true' />
<node x='2.166666667'  y='-1.70123'   enabled='true' />
<node x='2.500000000'  y='-1.62473'   enabled='true' />
<node x='2.833333333'  y='-0.99696'   enabled='true' />
<node x='3.166666667'  y='-0.27391'   enabled='true' />
<node x='3.500000000'  y='-0.02499'   enabled='true' />
<node x='3.833333333'  y='-0.01784'   enabled='true' />
<node x='4.166666667'  y='-0.00961'   enabled='true' />
<node x='4.500000000'  y=' 0.0    '   enabled='true' />
</nodes>
</spline>
...
</functions>
</meam>


Elements and attributes

• <mapping>: This block defines the mapping of the functions defined in the <functions> block onto different atom types and pairs of atom types.
• <pair-interaction>: Assign a pair interaction using the attributes species-a and species-b to specify the atom types involved and the function attribute to specify the id of the function. The function id has to match exactly one of the functions defined in the <functions> block.
• <electron-density>: Assign an electron density function using the attributes species-a and species-b to specify the atom types involved and the function attribute to specify the id of the function. The function id has to match exactly one of the functions defined in the <functions> block.
• <f-function>: Assign a supporting pairwise function that enters the angular dependent part of the electron density calculation using the attributes species-a and species-b to specify the atom types involved and the function attribute to specify the id of the function. The function id has to match exactly one of the functions defined in the <functions> block.
• <g-function>: Assign a supporting angular function that enters the angular dependent part of the electron density calculation using the attributes species-a, species-b, and species-c to specify the atom types involved and the function attribute to specify the id of the function. The function id has to match exactly one of the functions defined in the <functions> block.
• <embedding-energy>: Assign an embedding function using the attribute species to specify the atom type involved and the function attribute to specify the id of the function. The function id has to match exactly one of the functions defined in the <functions> block.
• <functions>: This block comprises the definitions of the various functions and parameters. The definition of a function is described :ref:here <function_definition>. Each function defined has to be assigned an id using the id attribute. The id is used in the <mapping> block to attach a function to a certain type of interaction (pair potential and electron density) or atom type (embedding function).
• <export-functions> (optional): Name of file, to which potential parameters are being written in Lammps format [Default: no file is written].

Further information

More detailed information can be found in [Bas87], [Bas92], [LeeBasKim01], [LenSadAlo00].