  Asteroseismology is a powerful tool to infer the evolutionary status and
chemical stratification of white dwarf (WD) stars, and to explore the physical
processes that lead to their formation. This is particularly true for the
variable H-rich atmosphere (DA) WDs, known as DAV or ZZ Ceti stars. We present
a new grid of DA WD models that take into account the last advances in the
modeling and input physics of both the progenitor and the WD stars, thus
avoiding and improving several shortcomings present in the set of DA WD models
employed in the asteroseismological analyses of ZZ Ceti stars we carried out in
our previous works. These new models are derived from a self-consistent way
with the changes in the internal chemical distribution that result from the
mixing of all the core chemical components induced by mean molecular-weight
inversions, from $^{22}$Ne diffusion, Coulomb sedimentation, and from residual
nuclearburning. In addition, the expected nuclear-burning history and mixing
events along the progenitor evolution are accounted for, in particular the
occurrence of third dredge-up, which determines the properties of the core and
envelope of post-AGB and white dwarf stars, as well as the WD initial-final
mass relation. The range of H envelopes of our new ZZ star models extends from
log(M_H/M_*)= -4 to -5, to log(M_H/M_*)= -13.5. This allows, for the first
time, to consider seismological solutions for ZZ Ceti stars with extremely thin
H envelopes. Our new H-burning post-AGB models predict chemical profiles of O
and C near the stellar centre of ZZ Ceti stars substantially different from
those we used in our previous works. We find that the pulsation periods of $g$
modes and the mode-trapping properties of the new models differ significantly
from those characterizing the ZZ Ceti models of our previous works,
particularly for long periods.
