= Kuhn length of a freely rotating chain
{c}
{title2=$l_K\simeq b(1+c)/(1-c),\quad c=\cos\alpha$}
For fixed bond length $b$, fixed forward bond angle $\alpha$, and independently uniform torsion angles, the bond correlations are $\langle\boldsymbol b_i\cdot\boldsymbol b_{i+s}\rangle=b^2c^s$. Thus for $N$ bonds
$$
\langle R^2\rangle=b^2\left[N+2\sum_{s=1}^{N-1}(N-s)c^s\right].
$$
For fixed $|c|<1$ and a sufficiently long chain, divide by contour length $Nb$ and sum the <geometric series> to obtain the title formula. Positive forward persistence raises the <Kuhn length>, and negative correlation lowers it relative to $b$. A chemical internal angle is supplementary to the forward-vector angle. The long-chain formula cannot be used uniformly as $c\to1$ at fixed $N$: the exact chain obeys $R\leq Nb$ and hence $l_K\leq Nb$, with equality in the rigid-rod limit.
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