Lennard-Jones potential

Numerical analysis · SimulationFinite element · Boundary element Lattice Boltzmann · Riemann solverDissipative particle dynamicsSmoothed particle hydrodynamicsThe Lennard-Jones potential (also termed the L-J potential, 6-12 potential, or 12-6 potential) is a mathematically simple model that approximates the interaction between a pair of neutral atoms or molecules. A form of this interatomic potential was first proposed in 1924 by John Lennard-Jones. The most common expressions of the L-J potential are V LJ ( r c ) = V LJ ( 2.5 σ ) = 4 ε [ ( σ 2.5 σ ) 12 − ( σ 2.5 σ ) 6 ] ≈ − 0.0163 ε , {displaystyle displaystyle V_{ ext{LJ}}(r_{ ext{c}})=V_{ ext{LJ}}(2.5sigma )=4varepsilon leftapprox -0.0163varepsilon ,} V LJ ( r ) = 4 ε [ ( σ r ) 12 − ( σ r ) 6 ] . {displaystyle displaystyle V_{ ext{LJ}}(r)=4varepsilon left.} V LJ trunc ( r ) := { V LJ ( r ) − V LJ ( r c ) for  r ≤ r c 0 for  r > r c . {displaystyle displaystyle V_{{ ext{LJ}}_{ ext{trunc}}}(r):={egin{cases}V_{ ext{LJ}}(r)-V_{ ext{LJ}}(r_{ ext{c}})&{ ext{for }}rleq r_{ ext{c}}\0&{ ext{for }}r>r_{ ext{c}}.end{cases}}} The Lennard-Jones potential (also termed the L-J potential, 6-12 potential, or 12-6 potential) is a mathematically simple model that approximates the interaction between a pair of neutral atoms or molecules. A form of this interatomic potential was first proposed in 1924 by John Lennard-Jones. The most common expressions of the L-J potential are where ε is the depth of the potential well, σ is the finite distance at which the inter-particle potential is zero, r is the distance between the particles, and rm is the distance at which the potential reaches its minimum. At rm, the potential function has the value −ε. The distances are related as rm = 21/6σ ≈ 1.122σ. These parameters can be fitted to reproduce experimental data or accurate quantum chemistry calculations. Due to its computational simplicity, the Lennard-Jones potential is used extensively in computer simulations even though more accurate potentials exist. The r−12 term, which is the repulsive term, describes Pauli repulsion at short ranges due to overlapping electron orbitals, and the r−6 term, which is the attractive long-range term, describes attraction at long ranges (van der Waals force, or dispersion force). Differentiating the L-J potential with respect to r gives an expression for the net inter-molecular force between 2 molecules. This inter-molecular force may be attractive or repulsive, depending on the value of r. When r is very small, the molecules repel each other. Whereas the functional form of the attractive term has a clear physical justification, the repulsive term has no theoretical justification. It is used because it approximates the Pauli repulsion well and is more convenient due to the relative computing efficiency of calculating r12 as the square of r6. The L-J potential is a relatively good approximation. Due to its simplicity, it is often used to describe the properties of gases and to model dispersion and overlap interactions in molecular models. It is especially accurate for noble gas atoms and is a good approximation at long and short distances for neutral atoms and molecules. The lowest-energy arrangement of an infinite number of atoms described by a Lennard-Jones potential is a hexagonal close-packing. On raising temperature, the lowest-free-energy arrangement becomes cubic close packing, and then liquid. Under pressure, the lowest-energy structure switches between cubic and hexagonal close packing. Real materials include body-centered cubic structures also. The Lennard-Jones (12,6) potential was improved by the Buckingham potential (exp-6) later proposed by Richard Buckingham, incorporating an extra parameter and the repulsive part is replaced by an exponential function: Other more recent methods, such as the Stockmayer potential, describe the interaction of molecules more accurately. Quantum chemistry methods, Møller–Plesset perturbation theory, coupled cluster method, or full configuration interaction can give extremely accurate results, but require large computing cost.