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 Π Π ‴ ( a ) a > − − ( d − − 1 ) Π Π ″ ( a ) , where d 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 Π Π ″ and Π Π ‴ are present in the condition for negative thermal expansion.