Multi-Lennard-Jones Potential¶

The Multi-Lennard-Jones calculator implements the classic Lennard-Jones potential with support for multiple chemical species and custom mixing rules.

Theory¶

Potential Form¶

The Lennard-Jones potential between atoms \(i\) and \(j\) is:

\[ u_{ij}(r) = 4\epsilon_{ij} \left[\left(\frac{\sigma_{ij}}{r}\right)^{12} - \left(\frac{\sigma_{ij}}{r}\right)^6\right] \]

Where:

\(\sigma_{ij}\): Distance at which potential is zero
\(\epsilon_{ij}\): Depth of potential well
\(r\): Interatomic distance

Force¶

\[ \mathbf{F}_{ij} = 24\epsilon_{ij} \left[2\left(\frac{\sigma_{ij}}{r}\right)^{12} - \left(\frac{\sigma_{ij}}{r}\right)^6\right] \frac{\mathbf{r}_{ij}}{r^2} \]

Usage¶

Single Species¶

from a2c_ase.potentials.mlj import MultiLennardJones
from ase import Atoms

# Create atoms
atoms = Atoms('Ar10', positions=..., cell=..., pbc=True)

# Create calculator
calculator = MultiLennardJones(
    epsilon=1.0,    # eV
    sigma=3.4,      # Å
    rc=10.0        # Cutoff (Å)
)

atoms.calc = calculator
energy = atoms.get_potential_energy()

Multiple Species¶

# Dictionary specification
calculator = MultiLennardJones(
    epsilon={"Fe": 0.5, "B": 0.3},
    sigma={"Fe": 2.5, "B": 1.8},
    rc=10.0
)

Parameters¶

Parameter	Type	Default	Description
`epsilon`	float/dict	1.0	Well depth (eV)
`sigma`	float/dict	1.0	Zero-crossing distance (Å)
`rc`	float	None	Cutoff distance (auto: 3σ)
`ro`	float	None	Smooth cutoff onset (auto: 0.66rc)
`smooth`	bool	False	Use smooth cutoff
`mixing_rule`	str	"lorentz_berthelot"	Mixing rule
`cross_interactions`	dict	None	Explicit cross-interactions

Mixing Rules¶

Lorentz-Berthelot (Default)¶

\[ \sigma_{ij} = \frac{\sigma_i + \sigma_j}{2} \]

\[ \epsilon_{ij} = \sqrt{\epsilon_i \epsilon_j} \]

calculator = MultiLennardJones(
    epsilon={"A": 1.0, "B": 1.5},
    sigma={"A": 1.0, "B": 0.8},
    mixing_rule="lorentz_berthelot"
)

Geometric¶

\[ \sigma_{ij} = \sqrt{\sigma_i \sigma_j} \]

\[ \epsilon_{ij} = \sqrt{\epsilon_i \epsilon_j} \]

calculator = MultiLennardJones(
    epsilon={"A": 1.0, "B": 1.5},
    sigma={"A": 1.0, "B": 0.8},
    mixing_rule="geometric"
)

Custom Cross-Interactions¶

Override mixing rules for specific pairs:

calculator = MultiLennardJones(
    epsilon={"A": 1.0, "B": 1.5},
    sigma={"A": 1.0, "B": 0.88},
    mixing_rule="lorentz_berthelot",
    cross_interactions={
        ("A", "B"): {"sigma": 0.8, "epsilon": 1.5}
    }
)

Cutoff Methods¶

Shifted Cutoff (Default)¶

Simple shift to make energy continuous:

\[ u_{\text{shifted}}(r) = u(r) - u(r_c) \]

calculator = MultiLennardJones(
    epsilon=1.0,
    sigma=3.4,
    rc=10.0,
    smooth=False  # Default
)

Smooth Cutoff¶

Smoothly goes to zero between ro and rc:

\[ u_{\text{smooth}}(r) = u(r) \times S(r) \]

where \(S(r)\) is a switching function:

\[ S(r) = \begin{cases} 1 & r < r_o \\ \frac{(r_c - r)^2(r_c + 2r - 3r_o)}{(r_c - r_o)^3} & r_o \leq r < r_c \\ 0 & r \geq r_c \end{cases} \]

calculator = MultiLennardJones(
    epsilon=1.0,
    sigma=3.4,
    rc=10.0,
    ro=6.6,      # Onset of cutoff
    smooth=True
)

Examples¶

Argon (Noble Gas)¶

from ase.lattice.cubic import FaceCubicFactory
from a2c_ase.potentials.mlj import MultiLennardJones

# Argon parameters (from literature)
calc = MultiLennardJones(
    epsilon=0.0103,  # eV (0.0103 eV ≈ 120 K)
    sigma=3.405,     # Å
    rc=10.0
)

# Create FCC lattice
atoms = FaceCubicFactory()(symbol='Ar', size=(3,3,3), latticeconstant=5.26)
atoms.calc = calc

energy = atoms.get_potential_energy()
print(f"Energy per atom: {energy/len(atoms):.4f} eV")

Kob-Andersen Binary Mixture¶

Classic binary LJ mixture:

calc = MultiLennardJones(
    epsilon={"A": 1.0, "B": 0.5},
    sigma={"A": 1.0, "B": 0.88},
    mixing_rule="lorentz_berthelot",
    cross_interactions={
        ("A", "B"): {"sigma": 0.8, "epsilon": 1.5}
    },
    rc=3.0,
    smooth=True
)