  We present a simple and general coherence model for multiorbital tunnel
ionization of molecules, which we incorporate into our previously developed
density matrix approach for sequential double ionization [Yuen and Lin, Phys.
Rev. A 106, 023120 (2022)]. The influence of this coherence is investigated
through simulations of single ionization and sequential double ionization of
N$_2$ and O$_2$ using few-cycle near-infrared laser pulses. In the case of
single ionization, our results reveal the crucial role played by this coherence
in generating population inversion in N$_2^+$, suggesting a potential mechanism
for air lasing. Regarding sequential double ionization, we observe only minor
changes in the kinetic energy release spectra when the coherence is included,
while noticeable differences in the angle-dependent dication yield for both
N$_2$ and O$_2$ are found. Based on these findings, we recommend the inclusion
of multiorbital tunnel ionization coherence in models for single ionization of
general molecules, while suggesting that it can be safely neglected in the case
of sequential double ionization.
