text: Negative thermal expansion is usually observed in non-close-packed systems with directional interactions (e.g. ice, graphene, etc.) and complex compounds (e.g. , , beta-quartz, some zeolites, etc.). However, in a paper, it was shown that negative thermal expansion (NTE) is also realized in single-component close-packed lattices with pair central force interactions. The following sufficient condition for potential giving rise to NTE behavior is proposed for the interatomic potential, , at the equilibrium distance :
  where  is shorthand for the third derivative of the interatomic potential at the equilibrium point:
  This condition is (i) necessary and sufficient in 1D and (ii) sufficient, but not necessary in 2D and 3D. An approximate necessary and sufficient condition is derived in a paper
  where  is the space dimensionality. Thus in 2D and 3D negative thermal expansion in close-packed systems with pair interactions is realized even when the third derivative of the potential is zero or even negative. Note that one-dimensional and multidimensional cases are qualitatively different. In 1D thermal expansion is caused by anharmonicity of interatomic potential only. Therefore, the sign of thermal expansion coefficient is determined by the sign of the third derivative of the potential. In multidimensional case the geometrical nonlinearity is also present, i.e. lattice vibrations are nonlinear even in the case of harmonic interatomic potential. This nonlinearity contributes to thermal expansion. Therefore, in multidimensional case both  and  are present in the condition for negative thermal expansion.