For a diatomic molecule with a center of symmetry, such as H 2 , N 2 , C O 2 , or H 2 C 2 (i.e. D ∞ ∞ h point group), rotation of a molecule by π π radian about an axis perpendicular to the molecule axis and going through the center of mass will interchange pairs of equivalent atoms. The spin–statistics theorem of quantum mechanics requires that the total molecular wavefunction be either symmetric or antisymmetric with respect to this rotation depending upon whether an even or odd number of pairs of fermion nuclear pairs are exchanged. A given electronic & vibrational wavefunction will either be symmetric or antisymmetric with respect to this rotation. The rotational wavefunction with quantum number J will have a sign change of ( − − 1 ) J . The nuclear spins states can be separated into those that are symmetric or antisymmetric with respect to the nuclear permutations produced by the rotation. For the case of a symmetric diatomic with nuclear spin quantum number I for each nucleus, there are ( I + 1 ) ( 2 I + 1 ) symmetric spin functions and I ( 2 I + 1 ) are antisymmetric functions for a total number of nuclear functions g ns = ( 2 I + 1 ) 2 . Nuclei with an even nuclear mass number are bosons and have integer nuclear spin quantum number, I. Nuclei with odd mass number are fermions and had half integer I. For the case of H 2, rotation exchanges a single pair of fermions and so the overall wavefunction must be antisymmetric under the half rotation. The vibration-electronic function is symmetric and so the rotation-vibration-electronic will be even or odd depending upon whether J is an even or odd integer. Since the total wavefunction must be odd, the even J levels can only use the antisymmetric functions (only one for I = 1/2) while the odd J levels can use the symmetric functions (three for I = 1/2). For D2, I = 1 and thus there are six symmetric functions, which go with the even J levels to produce an overall symmetric wavefunction, and three antisymmetric functions that must go with odd J rotational levels to produce an overall even function. The number of nuclear spin functions that are compatible with a given rotation-vibration-electronic state is called the nuclear spin statistical weight of the level, often represented as g J . Averaging over both even and odd J levels, the mean statistical weight is ( 1 / 2 ) ( 2 I + 1 ) 2 , which is one half the value of g ns expected ignoring the quantum statistical restrictions. In the high temperature limit, it is traditional to correct for the missing nuclear spin states by dividing the rotational partition function by a factor σ σ = 2 with σ σ known as the rotational symmetry number which is 2 for linear molecules with a center of symmetry and 1 for linear molecules without.