The Symmetry of a Molecule Picks a Character: Nuclear spin statistics, eta quotients and Dirichlet L-functions

Summary: The rotational levels of a molecule with identical nuclei carry statistical weights that chemists have tabulated since Wilson (1935). We show that, written in the variable n = 2J + 1, these weights are finite combinations of Dirichlet characters, and that the rotation group alone decides which characters occur. A rotation by 2π/m acts on a level of angular momentum J by the Weyl character sin(nπ/m)/sin(π/m); its values lie in {0, ±1} for all odd n exactly when m ∈ {2, 3, 4}, and then they are χ₋₄(n), χ₋₃(n) and χ₋₈(n), the odd quadratic characters of ℚ(i), ℚ(√−3) and ℚ(√−2). Consequences: (i) the rotational partition functions of methane and SF₆ are exact finite sums of a classical term and eta quotients of weight 3/2, for instance Z(CH₄) = e^(σ/4) [ (4/3) A(σ) + (8/3) η(3τ)⁵/η(6τ)² + η(τ)³ ] with τ = iσ/π, σ = Θ_R/T; McDowell's high-temperature hierarchy e^(−π²/(m²σ)) (1987) follows from one modular transformation; (ii) the Mellin transform of the χ₋₃-term is 4^s Γ(s) (1 + 2^(1−2s)) L(2s − 1, χ₋₃), and the vanishing of odd L-functions at negative odd integers is the arithmetic reason why the textbook symmetry-number rule is exact up to exponentially small terms; (iii) at the self-dual temperatures the character terms take closed values with Γ(1/4), for example η(3τ)⁵/η(6τ)² = η(i)⁵/η(2i)² = Γ(1/4)³ 2^(−9/4) π^(−9/4) at T = 3Θ_R/π; (iv) in ammonia the inversion tunnelling switches on the hydrogen character, Z ⊃ −2 (1 − e^(−Δ/kT)) e^(σ/4) η(iσ/π)³, while χ₋₃ is absent from the exact form; (v) for ¹²C₆₀ the five-fold axes give the quartic characters modulo 5. Finally, for every finite rotation group whose pole orders divide |G|/2 (the tetrahedral, octahedral and icosahedral groups and D_n with n even), the allowed J for spinless nuclei form a symmetric numerical semigroup whose largest gap is |G|/2 − 1, one less than the number of edges of the polyhedron. Two primers explain the physics for number theorists and the number theory for physicists; labels separate proved, cited, classical, numerical and open statements. The companion code (Python scripts with mpmath, the three pre-registered tests with their registrations and SHA-256 files, a runner that checks every step against its expected outcome, reference logs, the figure script and a reference checker) is included as P14_companion_code.zip; the figures used by the LaTeX source are in P14_figures.zip. - nuclear spin statistics - rotational partition function - spherical top - methane - sulfur hexafluoride - ammonia - fullerene C60 - benzene - Dirichlet characters - Dirichlet L-functions - eta quotients - modular forms of weight 3/2 - Weyl character formula - Mellin transform - symmetry number - Chowla–Selberg formula - numerical semigroups - Molien series - pre-registration - 11F20 (Dedekind eta function, Dedekind sums) - 11M06 (ζ(s) and L(s, χ)) - 20C35 (applications of group representations to physics) - 81V55 (molecular physics) - 80A10 (classical and relativistic thermodynamics) - 11D07 (the Frobenius problem)