= Solution
Use the <Christoffel symbol> formula with its first index raised:
$$
\Gamma^i{}_{jk}=\frac12g^{i\ell}(\partial_jg_{\ell k}+\partial_kg_{\ell j}-\partial_\ell g_{jk}).
$$
The diagonal <metric tensor> has $g_{rr}=e^{2\chi}$ and $g_{\theta\theta}=r^2e^{2\chi}$, with inverse components $g^{rr}=e^{-2\chi}$ and $g^{\theta\theta}=r^{-2}e^{-2\chi}$. Since $\partial_r\chi=\chi'$ and all angular derivatives vanish,
$$
\boxed{\Gamma^r{}_{rr}=\chi',\qquad
\Gamma^r{}_{\theta\theta}=-r-r^2\chi',\qquad
\Gamma^\theta{}_{\theta r}=\Gamma^\theta{}_{r\theta}=\frac1r+\chi'.}
$$
For example, $\Gamma^r{}_{\theta\theta}=-\tfrac12e^{-2\chi}\partial_r(r^2e^{2\chi})$. The other components vanish. The formula supplied in the paper gives the first-kind coefficients $\Gamma_{ijk}$; multiplication by the inverse <metric tensor> is needed for the requested second-kind coefficients. These <polar coordinates> are understood away from $r=0$.
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