  In this work, we have performed large-scale shell-model calculations for the
first-forbidden $\beta ^{-}$ decay of $^{207}$Hg into the one-proton-hole
nucleus $^{207}$Tl corresponding to the recently available experimental data
from ISOLDE-CERN [T. A. Berry et al., Phys. Rev. C 101, 054311 (2020)]. We have
used the one-particle one-hole ($1p$-$1h$) truncation for both protons and
neutrons simultaneously across the doubly-shell closure at $^{208}$Pb in the
final states of $^{207}$Tl. In our calculations, we have also considered the
effect of mesonic enhancement $\epsilon _{\mathrm{mec}}=2.01\pm 0.05$ in the
rank-0 for the axial-charge matrix element $\gamma _{5}$. Here, we have
calculated the $\log ft$ values from the ground-state of $^{207}$Hg to the
several excited states of $^{207}$Tl and obtained a good agreement between the
calculated and the experimental data. In the experimental data spin and parity
for some states are not yet confirmed, thus based on the shell-model results
for the $\log ft$ values we have given the prediction for these states. This is
the first theoretical calculation for the $\log ft$ values for these
transitions.
