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, Local variational study of 2d lattice energies and application to Lennard-Jones type interactions, Nonlinearity, 31(9) (2018), 973-4005
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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.