  Recently, signatures of superconductivity with critical temperature from 20
to 30 K have been reported in pressured trilayer nickelate La$_4$Ni$_3$O$_{10}$
through a pressure-induced structure transition. Here we explore the evolution
of electronic structures and electronic correlations in different phases of
La$_4$Ni$_3$O$_{10}$ under corresponding pressure regions, by using density
functional theory (DFT) combined with dynamical mean-field theory (DMFT).
Similar to bilayer superconductor La$_3$Ni$_2$O$_{7}$, the electronic bands in
superconducting La$_4$Ni$_3$O$_{10}$ are dominated by Ni-3$d_{x^2-y^2}$ and
3$d_{z^2}$ orbits near the Fermi level, in contrast, the inner Ni-O plane in
La$_4$Ni$_3$O$_{10}$ generates a doublet hole-pocket Fermi surfaces around the
Brillouin-zone corner, meanwhile one branch of the Ni-$3d_{z^2}$ bands is
pushed very close above the Fermi level, which can induce an electron pocket
through small electron doping. The DFT+DMFT simulations suggest that the
electronic correlations only give minor modification to the Fermi surfaces,
meanwhile the Ni-$3d_{z^2}$ and 3$d_{x^2-y^2}$ states on outer Ni-O layers have
considerable greater mass enhancements than on the inner layer. The
sensitiveness of electronic structure under doping and unique layer dependence
of correlation suggest a distinct superconducting mechanism with respect to
bilayer La$_3$Ni$_2$O$_{7}$. Based on the DFT and DFT+DMFT simulations, we
eventually derive a trilayer effective tight-binding model, which can produce
rather precise electronic bands and Fermi surfaces, hence can serve as an
appropriate model to further study the superconducting mechanism and paring
symmetry in trilayer La$_4$Ni$_3$O$_{10}$.
