Translation invariance of Lennard-Jones potential #

This section defines the translation invariance of Lennard-Jonnes potential function that describes the intermolecular pair potentials in molecular simulations. The commonly used expression is:
$$ E = 4ε [(\frac{σ}{r})^{12} - (\frac{σ}{r})^6] $$ where:

E is the intermolecular potential between the two atoms or molecules
ε is the well depth and a measure of how strongly the two particles attract each other
σ is the distance at which the intermolecular potential between the two particles is zero
r is the distance of separation between both particles

Here we show that if we translate all the coordinates equally - the molecule is not changing conformations and hence the energy does not change. Using pairwise disntace removes the choice of origin and makes the model translational invariant.

Assumption #

The pairwise distance is given by Euclidean distance or L^2 norm where x and y are the coordinates of the two vectors representing molecular positions.

To-Do #

Proof rotational invariance of the Lennard-Jones function
Generalize proof from properties of system using fundamental theorems of invariant theory

source

theorem LJ_invariance {n : ℕ} (x y t : fin n → ℝ) (r rt a b E Et : ℝ) (a1 : r = has_dist.dist x y) (a2 : rt = has_dist.dist (x + t) (y + t)) (a3 : E = a / r ^ 12 - b / r ^ 6) (a4 : Et = a / rt ^ 12 - b / rt ^ 6) :

E = Et