  It is well-known that any Lennard-Jones type potential energy must a have
periodic ground state given by a triangular lattice in dimension 2. In this
paper, we describe a computer-assisted method that rigorously shows such global
minimality result among $2$-dimensional lattices once the exponents of the
potential have been fixed. The method is applied to the widely used classical
$(12,6)$ Lennard-Jones potential, which is the main result of this work.
Furthermore, a new bound on the inverse density (i.e. the co-volume) for which
the triangular lattice is minimal is derived, improving those found in [L.
B\'etermin and P. Zhang, Commun. Contemp. Math., 17 (2015), 1450049] and [L.
B\'etermin, SIAM J. Math. Anal., 48 (2016), 3236-3269]. The same results are
also shown to hold for other exponents as additional examples and a new
conjecture implying the global optimality of a triangular lattice for any
parameters is stated.
