Freely rotating chain

A freely rotating chain has fixed bond length and fixed angle between successive forward bond vectors, with independently uniform torsion angles. It neglects interactions between distant chain segments. The fixed angle introduces short-range orientational correlations even though the torsional rotations are unrestricted. Its characteristic ratio of a freely rotating chain converts those correlations into an end-to-end size; its equivalent long-chain link is the Kuhn length of a freely rotating chain.

With independently uniform torsion, consecutive forward tangent correlations multiply by $\cos \vartheta$. After $n$ bonds the correlation is $(\cos \vartheta {)}^n$. Matching it to $e^{-nb/{\ell }_p}$ gives the displayed decay length for positive correlation. This discrete orientational definition need not equal half the equivalent Kuhn length exactly.

For $N$ bonds of length $a$, fixed forward angle $\theta$ and uniform independent torsions, conditional averaging gives $⟨{\mathbf{a}}_i\cdot {\mathbf{a}}_j⟩=a^2{\alpha }^{|j-i|}$. Consequently $⟨R^2⟩=N{a}^2+2a^2{\sum }_{k=1}^{N-1}(N-k){\alpha }^k$. Dividing by the independent-step result $N{a}^2$ and summing the geometric series gives the displayed $C_N$, for $\alpha \ne 1$. For fixed $|\alpha |<1$, $C_N\to C_{\infty }=(1+\alpha )/(1-\alpha )$.

The angle is measured between forward vectors; a chemical internal bond angle is supplementary. At $\alpha =1$, the finite chain is a rigid rod with $C_N=N$, so taking the infinite-chain limit before the rigid limit would be invalid. The Kuhn length of a freely rotating chain is asymptotically $a{C}_{\infty }$.