  The interplay between structural phase and electronic correlations has been
an intriguing topic of research. An prominent example is the pressure-induced
uncollapsed to collapsed tetragonal phase transition observed in
CaFe$_2$As$_2$, which is accompanied with the emergence of superconductivity in
the collapsed phase. Recently, a very similar structural phase transition was
discovered in LaFe$_2$As$_2$, but in contrast to CaFe$_2$As$_2$,
superconductivity was only observed in the uncollapsed phase, not the collapsed
phase. Previous studies have attributed this puzzling observation to the
differences in the two materials' band coherence, orbital occupation, and Fermi
surface topology. Here, we present a comparative study of LaFe$_2$As$_2$ and
CaFe$_2$As$_2$ using the DFT+DMFT method. Surprisingly, we find that although
La appears to have a valence higher than Ca, the doped one electron actually
primarily resides on the La site. This leads to almost the same total Fe-$3d$
occupancy and electronic correlation strength as well as similar Lifshiftz
transition in the Fermi surface topology for the two materials. In addition, we
show that the two materials in both structural phases belong to the category of
Hund's metals. Our results indicate that the electronic structures of
LaFe$_2$As$_2$ and CaFe$_2$As$_2$ are not too different, which further suggest
that superconductivity might also be induced in the collapsed phase of
LaFe$_2$As$_2$ under similar non-hydrostatic conditions as for CaFe$_2$As$_2$.
