Potentials

In order to compute the energy or the forces acting on a particle, two informations are needed : a potential energy function, and a computation algorithm. The potential function is a description of the variations of the potential with the particles positions, and a potential computation is a way to compute the values of this potential function. The following image shows the all the potentials functions types currently available in Jumos.

We can see these potentials are classified as four main categories: pair potentials, bond potentials, angles potentials (between 3 atoms) and dihedral potentials (between 4 atoms).

The only implemented pair potentials are short-range potentials. Short-range pair potentials go to zero faster than the \(1/r^3\) function, and long-range pair potentials go to zero at the same speed or more slowly than \(1/r^3\). A typical example of long-range pair potential is the Coulomb potential between charged particles.

Potential functions

Short-range pair potential

Short-range pair potential are subtypes of PairPotential. They only depends on the distance between the two particles they are acting on. They should have two main properties:

• They should go to zero when the distance goes to infinity;
• They should go to zero faster than the \(1/r^3\) function.

Lennard-Jones potential

A Lennard-Jones potential is defined by the following expression:

\[V(r) = 4\epsilon \left( \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^6 \right)\]

LennardJones(epsilon, sigma)

Creates a Lennard-Jones potential with \(\sigma = \text{sigma}\), and \(\epsilon = \text{epsilon}\). sigma should be in angstroms, and epsilon in \(kJ/mol\).

Typical values for Argon are: \(\sigma = 3.35\ A, \epsilon = 0.96\ kJ/mol\)

Null potential

This potential is a potential equal to zero everywhere. It can be used to define “interactions” between non interacting particles.

NullPotential()

Creates a null potential instance.

Bonded potential

Bonded potentials acts between two particles, but does no go to zero with an infinite distance. A contrario, they goes to infinity as the two particles goes apart of the equilibirum distance.

Harmonic

An harmonic potential have the following expression:

\[V(r) = \frac12 k (\vec r - \vec r_0)^2 - D_0\]

\(D_0\) is the depth of the potential well.

Harmonic(k, r0, depth=0.0)

Creates an harmonic potential with a spring constant of k (in \(kJ.mol^{-1}.A^{-2}\)), an equilibrium distance \(r_0\) (in angstroms); and a well’s depth of \(D_0\) (in \(kJ/mol\)).

Potential computation

As stated at the begiging of this section, we need two informations to compute interactions between particles: a potential function, and a potential computation. The potential compuatation algorithms come in four flavors:

• The DirectComputation is only a small wrapper on the top of the potential functions, and directly calls the potential function methods for energy and force evaluations.
• The CutoffComputation is used for short range potentials. All interactions at a longer distance than the cutoff distance are set to zero. The default cutoff is \(12\ A\), and this can be changed by passing a cutoff keyword argument to the add_interaction function. With this computation, the energy is shifted so that their is a continuity in the energy at the cutoff distance.
• The TableComputation use table lookup to extrapolate the potential energy and the forces at a given point. This saves computation time at the cost of accuracy. This algorithm is parametrized by an integer, the size of the undelying array. Increases in this size will result in more accuracy, at the cost of more memory usage. The default size is 2000 points — which corresponds to roughly 15kb. TableComputation has also a maximum distance for computations, rmax. For any bigger distances, the TableComputation will returns a null energy and null forces. So TableComputation can only be used if you are sure that the particles will never be at a greater distance than rmax.
• The LongRangeComputation is not implemented yet.

Which computation for which potential ?

Not all computation algorithms are suitable for all potential functions. The usable associations are in the table below.

Function	DirectComputation	CutoffComputation	TableComputation
ShortRangePotential
BondedPotential
AnglePotential
DihedralPotential