Lennard-Jones Potential

A Lennard-Jones potential is a radial pair interaction of the form

(1)

where , , , and . Mathematical energy minimization asks for configurations of distinct points minimizing , where is the Euclidean norm. The classical exponents are and (Blanc and Lewin 2015). Dividing the energy by reduces the potential to the case with a rescaled .

In the normalization , Luo (2026) classified every global minimizer for exactly four points in the plane. For , the infimum is zero and no minimizer exists. For , the minimizing configuration is unique up to isometry and relabeling. Define

(2)

The minimizer is a square when , and otherwise a rhombus with acute angle strictly between and . For a rhombus of acute angle , write . Its minimizing side length satisfies

(3)

For the classical exponents , the minimizing angle is , where is the unique root in of

(4)

With , its side length is approximately 0.9980123196944918. This classification is for a finite four-point configuration, not for an infinite lattice.