For α ≥ 3/2, the lattice energy Σ |P|^4 e^{-πα|P|^2} over unit-density 2D lattices is uniquely minimized by the hexagonal lattice.
B´ etermin, Minimizing lattice structures for Morse potential energy in two and three dimensions, Journal of Mathematical Physics, 60(10) (2019), 102901
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Minimizing Lattice Energy and Hexagonal Crystallization
For α ≥ 3/2, the lattice energy Σ |P|^4 e^{-πα|P|^2} over unit-density 2D lattices is uniquely minimized by the hexagonal lattice.