Ground-state electronic angular momentum $J=1/2$ couples to nuclear spin $I=3/2$, giving $F=1,2$. For contact hyperfine interaction $A\mathbf{I}\cdot \mathbf{J}/{\hslash }^2$, their zero-field energies are $-5A/4$ and $3A/4$. The splitting is $2A$. Electronic weak-field contributions give $g_{F=1}\simeq -1/2$ and $g_{F=2}\simeq +1/2$; the small nuclear contribution refines these numbers.

The rotational state populations of a spin-free polar linear rigid rotor are proportional to $(2J+1)e^{-{\theta }_{\mathrm{rot}}J(J+1)/T}$. Differentiating their logarithm with respect to continuous $J$ gives the displayed maximum when it is nonnegative. The actual maximum must be selected among nonnegative integers by comparing adjacent populations. This population maximum generally differs from a maximum in measured spectral line intensity.

The angular momentum associated with end-over-end molecular rotation. In an open-shell diatomic molecule it is useful to distinguish $\mathbf{N}$ from the total $\mathbf{J}=\mathbf{N}+\mathbf{S}$, which also contains the electronic spin angular momentum.

For a unit quaternion $q=(w,v)$ with $v\in {\mathbb{R}}^3$, the Euler-Rodrigues formula gives the rotationwhere $[v{]}_{\times }x=v\times x$. It follows by expanding quaternion conjugation $x\mapsto qx{q}^{-1}$ for a purely imaginary quaternion $x$.

Writing $w=\cos (\theta /2)$ and $v=n\sin (\theta /2)$ gives rotation through angle $\theta$ about the unit axis $n$. Both $q$ and $-q$ give the same rotation matrix, displaying the double cover by unit quaternions. The formula is useful for combining spatial rotations without singular angle coordinates.

In rapid rotation, both rotating-frame modes are nearly circular and counter-rotate relative to the axes at frequency near $|\Omega |$. Transforming back to the laboratory leaves a slow nearly harmonic oscillation and small rapid micromotion. For positive rotation the slow ellipse also has higher-order prograde precession at rate ${\omega }^4/(8{\Omega }^3)$. These follow by expanding $\sqrt{{\Omega }^2\pm {\omega }^2}$ and combining the two laboratory circular components.^[ref]

A Fibonacci tree is a recursively defined binary tree whose two subtrees have consecutive smaller orders. One convention sets $T_0$ and $T_1$ to a single vertex and defines $T_n$ for $n\ge 2$ by attaching $T_{n-1}$ and $T_{n-2}$ to a new root. Then its vertex count satisfies $N_n=1+N_{n-1}+N_{n-2}$, so $N_n=2F_{n+1}-1$ for Fibonacci numbers $F_0=0,F_1=1$. The counts begin $1,1,3,5,9$. Initial trees vary across applications, so the convention must be stated; the common feature is the consecutive-order recursive branching.

For rolling without slipping with contact radius $r$, substitute $\omega =v/r$ into translational plus rotational kinetic energy:The coefficient acts as an effective inertia for a translation coordinate. It is not additional gravitational mass. A quadratic potential energy $K{x}^2/2$ then gives small-oscillation frequency ${\omega }_0=\sqrt{K/M_{\mathrm{eff}}}$ when this coefficient is constant.