  Gamow shell model (GSM) is usually performed within the Woods-Saxon (WS)
basis in which the WS parameters need to be determined by fitting experimental
single-particle energies including their resonance widths. In the multi-shell
case, such a fit is difficult due to the lack of experimental data of
cross-shell single-particle energies and widths. In this paper, we develop an
{\it ab-initio} GSM by introducing the Gamow Hartree-Fock (GHF) basis that is
obtained using the same interaction as the one used in the construction of the
shell-model Hamiltonian. GSM makes use of the complex-momentum Berggren
representation, then including resonance and continuum components. Hence, GSM
gives a good description of weakly bound and unbound nuclei. Starting from
chiral effective field theory and employing many-body perturbation theory
(MBPT) (called nondegenerate $\hat Q$-box folded-diagram renormalization) in
the GHF basis, a multi-shell Hamiltonian ({\it sd-pf} shells in this work) can
be constructed. The single-particle energies and their resonance widths can
also been obtained using MBPT. We investigated $^{23-28}$O and $^{23-31}$F
isotopes, for which multi-shell calculations are necessary. Calculations show
that continuum effects and the inclusion of the {\it pf} shell are important
elements to understand the structure of nuclei close to and beyond driplines.
