is the degeneracy of the jth quantum level of an individual particle,
For molecules, under the assumption that total energy levels
and the number of degenerate states are given as products of the single contributions
where "trans", "ns", "rot", "vib" and "e" denotes translational, nuclear spin, rotational and vibrational contributions as well as electron excitation, the molecular partition functions
For a diatomic molecule like CO or HCl, or a linear polyatomic molecule like OCS in its ground vibrational state, the allowed rotational energies in the rigid rotor approximation are
J is the quantum number for total rotational angular momentum and takes all integer values starting at zero, i.e.,
, then replace B by hcB where c is the speed of light in vacuum.
It is also called the classical approximation as this is the result for the canonical partition function for a classical rigid rod.
The mean thermal rotational energy per molecule can now be computed by taking the derivative of
In the high temperature limit approximation, the mean thermal rotational energy of a linear rigid rotor is
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
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
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.
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).
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
Averaging over both even and odd J levels, the mean statistical weight is
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
known as the rotational symmetry number which is 2 for linear molecules with a center of symmetry and 1 for linear molecules without.
A rigid, nonlinear molecule has rotational energy levels determined by three rotational constants, conventionally written
In terms of these constants, the rotational partition function can be written in the high temperature limit as [4]
again known as the rotational symmetry number [5] which in general equals the number ways a molecule can be rotated to overlap itself in an indistinguishable way, i.e. that at most interchanges identical atoms.
Like in the case of the diatomic treated explicitly above, this factor corrects for the fact that only a fraction of the nuclear spin functions can be used for any given molecular level to construct wavefunctions that overall obey the required exchange symmetries.
Another convenient expression for the rotational partition function for symmetric and asymmetric tops is provided by Gordy and Cook:
works for asymmetric, symmetric and spherical top rotors.