Characteristic ratio of a freely rotating chain ($C_N=\frac{1+\alpha }{1-\alpha }-\frac{2\alpha (1-{\alpha }^N)}{N(1-\alpha {)}^2},\alpha =\cos \theta$)

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 $a{C}_{\infty }$.