Modified Embedded Atom Method with Bond Order (MEAM-BO) implementation in LAMMPS

This package implements the MEAM-BO force field as a LAMMPS pair style based on the MEAM/C (https://lammps.sandia.gov/doc/pair_meamc.html, 5Jun2019 version) package where bond order calculations are added.

Implementation by Sungkwang Mun, Ricolindo L Carino, and Michael Baskes
[email protected], [email protected], [email protected]
Center for Advanced Vehicular Systems
Mississippi State University, MS 39762

Contributors:
Andrew Bowman, US Army Corps of Engineers - ERDC
Doyl Dickel, Mississippi State University
Mark Horstemeyer, Liberty University
Sasan Nouranian, University of Mississippi
Steven Gwaltney, Mississippi State University

The challenge in the current MEAM formalism is a lack of capability to distinguish bond order in different materials, such as acetylene (bond order: 3), ethylene (2), graphene (4/3) and benzene (3/2). To solve this issue, additional energy terms are added to the original MEAM energy as follows.

where E_MEAM is the energy from the original MEAM formalism, f_Ibond is a function that gives fractional energy with respect to the bond I, E_bond is an increment of energy due to the bond saturation, and R_I is the bond length between two atoms. It should be noted that E₂ deals with the bond order equals or less than two such as graphene and benzene. See Ref. [1] for the detail.

1. Copy User-MEAMBO folder to the LAMMPS src/ folder.
2. Type "make yes-user-meambo" in the command line to enable this package.
3. Rest of the procedures are the same as specified in the following page. https://lammps.sandia.gov/doc/Build_make.html

In your LAMMPS input script, specify

  pair_style meambo
  pair_coeff * * parameter_1 el1 ... parameter_2 el1 ... bond_order_parameter

to enable the use of this implementation. The parameter files used with pair_style meam/c are usable by pair_style meambo; however, the following keyword=value lines, for example (CH binary system), should be added to parameter_2 file for the bond order calculation (as well as van der Waals interaction ):

ntypBC=1           // number of elements that have unsaturated bonds
rcutBC(1)=3.3      // cutoff radius for 1st element for bond order calculation 
rcutBC(2)=2.4      // cutoff radius for 2nd element for bond order calculation 
vdW_form = 1       // selection parameter for form of vdW function
                   // 0: no vdW effect, 1: 9-6 Lenard Jones equation
evdW_96LJ(1)=0.435 // epsilon (energy in eV) parameter for 1st element in 9-6 LJ
evdW_96LJ(2)=0     // epsilon (energy in eV) parameter for 2nd element in 9-6 LJ 
svdW_96LJ(1)=3.01  // sigma (distance in Angstrom) parameters for 1st element in 9-6 LJ 
svdW_96LJ(2)=0     // sigma (distance in Angstrom) parameters for 2nd element in 9-6 LJ

Replacing the bond order file name with NULL such as the following

pair_coeff * * parameter_1 el1 ... parameter_2 el1 ... NULL

will execute the original MEAM calculation without the bond order calculation so that the results will be exactly the same as those of meam/c.

The bond parameter file for hydrocarbon is shown below.

# elt_i1   elt_i2   bond    Z(0)   Z(1)   beta0    beta1    beta2   beta3   p0     p1     p2     p3
# double bond: e2a_0(1:3), e2a_1(1:3), e2a_2(1:3), re2a(1:3)
# triple bond: e3(1:3), re3

'C'     'C'      2       3     0     5       1       4.5     1.2     1       1       1       1.2     
-1.296    -4.618    -14.21   
-0.632    -4.462    21.841   
0.299     -5.452    55.653   
1.3396    -0.0959   0.0364    

'C'     'C'      3       2     0     5       5       4.5     0.46    1       1       1       1       
-3.647          -3.03           -13.003         1.203           

The first line consists of bond information applicable to double or triple bonds. elt_i1 and elt_i2 are element names of unsaturated bonds that must match with the element names in parameter_1 file. bond 2 or 3 to represent a double or a triple bond, respectively. Z(0) and Z(1) are the number of neighbors required for the Z0 and Z1 counting factor function to satisfy the geometrical condition, e.g. Z(0) for triple bond (acetylene) is 2 (two neighbors). beta0/1/3 and p0/1/3 are parameters for the Gaussian-like delta function. beta2 and p2 are parameters to correct the rotational barriers.

The parameters in the next lines are determined by the value of bond. If 2 (double bond), then additional 12 parameters must follow. If 3 (triple bond), then additional 4 parameters must follow. e3 values are triple bond parameters to fit the difference of energy between MEAM and first principle (FP) calculation as a function of distance. re3 is an experimental distance at the equilibrium for a reference structure for a triple bond, e.g. acetylene for hydrocarbon. e2a_0/1/2 and re2a are auxiliary parameters that will result in e2 and re2 values similar to the triple bond case.

1. New reference structures are added
   a. ch4: methane-like structure only for binary system
   b. dia3: diamond structure with primary 1NN and secondary 3NN interaction
   c. tri: H2O-like structure that has an angle
   e. zig: zigzag structure with a uniform angle
   d. lin: linear structure (180 degree angle)
2. tri, zig, and lin reference structures require angle information (in degree) in the parameter_2 file such as the following.

theta = 109.5

1. Performance optimization
2. Generalized vdW interaction function will be added, i.e. DFT-D3 function (See Ref. [2] for detail)
3. The equation related to nI term (local density) will be slightly modified to treat different kinds of unsaturated bonds such as C-O and N-O. However, the results will not be changed.

[1] Mun, S.; Bowman, A. L.; Nouranian, S.; Gwaltney, S. R.; Baskes, M. I.; Horstemeyer, M. F. Interatomic Potential for Hydrocarbons on the Basis of the Modified Embedded-Atom Method with Bond Order (MEAM-BO). J. Phys. Chem. A 2017. https://doi.org/10.1021/acs.jpca.6b11343.
[2] Dickel, D.; Gwaltney, S. R.; Mun, S.; Baskes, M. I.; Horstemeyer, M. F. A Dispersion-Corrected Modified Embedded-Atom Method Bond Order Interatomic Potential for Sulfur. The Journal of Physical Chemistry A 2018. https://doi.org/10.1021/acs.jpca.8b07410.