  We have applied the Gamow shell model (GSM) in the coupled-channel
representation to study the astrophysical $S$-factor for the proton radiative
capture reaction of $^6$Li. Reaction channels are built by coupling the proton
wave function expanded in different partial waves with the GSM wave functions
of the ground state ($1^+$) and the excited states ($3^+_1$, $0^+_1$ and
$2^+_1$) of $^6$Li. All relevant $E1$, $M1$, and $E2$ transitions from the
initial continuum states in $^7$Be to the final bound states (${3/2}^-_1$ and
${1/2}^-_1$) states are included. It is found that the calculated total
astrophysical $S$ factor for this reaction agrees well with the experimental
data.
