  Motivated by the recent experimental determination of the three-dimensional
Fermi surface of overdoped La-based cuprate superconductors [Horio et al.,
Phys. Rev. Lett. 2018, 121, 077004], we revisit the tight-binding
parameterization of their conduction band. We construct a minimal tight-binding
model entailing eight orbitals, two of them involving apical oxygen ions.
Parameter optimization allows to almost perfectly reproduce the
three-dimensional conduction band as obtained from density functional theory
(DFT). We discuss how each parameter entering this multiband model influences
it, and show that the peculiar form of its dispersion severely constraints the
parameter values. We then evidence that standard perturbative derivation of an
effective one-band model is poorly converging because of the comparatively
small value of the charge transfer gap. Yet, this allows us to unravel the
microscopical origin of the in-plane and out-of-plane hopping amplitudes. An
alternative approach to the computation of the tight-binding parameters of the
effective model is presented and worked out. It results that the agreement with
DFT is preserved provided longer-ranged hopping amplitudes are retained. A
comparison with existing models is also performed. Finally, the Fermi surface,
showing staggered pieces alternating in size and shape, is compared to
experiment, with the density of states also being calculated.
