  According to standard stellar evolution, lithium abundance is believed to be
a useful indicator of the stellar age. However, many evolved stars like red
giants show huge fluctuations around expected theoretical abundances that are
not yet fully understood. The better knowledge of nuclear reactions that
contribute to the creation and destruction of lithium can help to solve this
puzzle. In this work we apply the Gamow shell model (GSM) formulated in the
coupled-channel representation (GSM-CC) to investigate the mirror radiative
capture reactions $^6$Li$(p,\gamma)$$^7$Be and $^6$Li$(n,\gamma)$$^7$Li. The
cross-sections are calculated using a translationally invariant Hamiltonian
with the finite-range interaction which is adjusted to reproduce spectra,
binding energies and one-nucleon separation energies in $^{6-7}$Li, $^7$Be. All
relevant $E1$, $M1$, and $E2$ transitions from the initial continuum states to
the final bound states $J={3/2}_1^-$ and $J={1/2}^-$ of $^7$Li and $^7$Be are
included. We demonstrate that the $s$-wave radiative capture of proton
(neutron) to the first excited state $J^{\pi}=1/2_1^+$ of $^7$Be ($^7$Li) is
crucial and increases the total astrophysical $S$-factor by about 40 \%.
