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Is there a simple physical argument why lattice structures involving 3 and 4 fold coordinations as in graphite and diamond are not stable in radially symmteric pair potentials?

I read in the bond order potential molecular dynamics literature as motivation that simple Lennard-Jones potentials do not imply diamond or graphite structures because directional dependence would be needed for these systems.

But I can not see a simple reason and I could not find more details or simulations.

YoussefMabrouk
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A succinct way to see why purely radial (isotropic) potentials do not stabilize diamond or graphite is that an isotropic pair potential rewards maximizing the number of neighbors at the preferred bond distance, pushing atoms into close‐packed structures. Diamond and graphite, on the other hand, each have relatively low coordination numbers (4 and 3, respectively) because of their strong, directional sp³/sp² bonds.

In an isotropic potential, there is no energetic advantage to having such low coordination; you would simply pack as many neighbors around each atom as possible. By contrast, directional bonding (e.g. covalent overlap of specific orbitals) is what stabilizes those lower‐coordination, “open” lattices of diamond and graphite. Without that angular dependence, you will not favor a diamond or graphite structure over the higher‐coordination alternatives.

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In principle, there is no hard rule that says that isotropic pair potentials cannot give rise to low-coordinated crystal structures. In fact, there are several papers where the authors have carefully designed interaction potentials to stabilize such structures. Two examples are:

  • Dimensionality and Design of Isotropic Interactions that Stabilize Honeycomb, Square, Simple Cubic, and Diamond Lattices, A. Jain, J. R. Errington, T. M. Truskett, Phys. Rev. X 4, 031049 (2014). Arxiv: https://arxiv.org/pdf/1408.6776

  • Synthetic Diamond and Wurtzite Structures Self-Assemble with Isotropic Pair Interactions, M.C. Rechtsman, F.H. Stillinger, and S. Torquato, Phys. Rev. E 75, 031403 (2007). Arxiv: https://arxiv.org/pdf/0709.3807

However, this clearly does take some careful design of the interactions, and an arbitrary isotropic interaction that is not specifically designed for this purpose is in practice much more likely to give rise to close-packed structures.

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