  Machine-learned multi-orbital tight-binding (MMTB) Hamiltonian models have
been developed to describe the electronic characteristics of intermetallic
compounds $\rm Mg_2Si, Mg_2Ge, Mg_2Sn$, and $\rm Mg_2Pb$ subject to strain. The
MMTB models incorporate spin-orbital mediated interactions and they are
calibrated to the electronic band structures calculated via density functional
theory (DFT) by a massively parallelized multi-dimensional Monte-Carlo search
algorithm. The results show that a machine-learned five-band tight-binding
model reproduces the key aspects of the valence band structures in the entire
Brillouin zone. The five-band model reveals that compressive strain localizes
the contribution of the $3s$ orbital of $\rm Mg$ to the conduction bands and
the outer shell $p$ orbitals of $\rm X~(X=Si,Ge,Sn,Pb)$ to the valence bands.
In contrast, tensile strain has a reversed effect as it weakens the
contribution of the $3s$ orbital of $\rm Mg$ and the outer shell $p$ orbitals
of $\rm X$ to the conduction bands and valence bands, respectively. The $\pi$
bonding in the $\rm Mg_2X$ compounds is negligible compared to the $\sigma$
bonding components, which follow the hierarchy
$|\sigma_{sp}|>|\sigma_{pp}|>|\sigma_{ss}|$, and the largest variation against
strain belongs to $\sigma_{pp}$. The five-band model allows for estimating the
strength of spin-orbit coupling (SOC) in $\rm Mg_2X$ and obtaining its
dependence on the atomic number of $\rm X$ and strain. Further, the band
structure calculations demonstrate a significant band gap tuning and band
splitting due to strain. A compressive strain of $-10\%$ can open a band gap at
the $\Gamma$ point in metallic $\rm Mg_2Pb$, whereas a tensile strain of
$+10\%$ closes the semiconducting band gap of $\rm Mg_2Si$. A tensile strain of
$+5\%$ removes the three-fold degeneracy of valence bands at the $\Gamma$ point
in semiconducting $\rm Mg_2Ge$.
