  Complex quantum coupling phenomena of halide perovskites are examined through
ab-initio calculations and exact diagonalization of model Hamiltonians to
formulate a set of fundamental guiding rules to engineer the bandgap through
strain. The bandgap tuning in halides is crucial for photovoltaic applications
and for establishing non-trivial electronic states. Using CsSnI$_3$ as the
prototype material, we show that in the cubic phase, the bandgap reduces
irrespective of the nature of strain. However, for the tetragonal phase, it
reduces with tensile strain and increases with compressive strain, while the
reverse is the case for the orthorhombic phase. The reduction can give rise to
negative bandgap in the cubic and tetragonal phases leading to normal to
topological insulator phase transition. Also, these halides tend to form a
stability plateau in a space spanned by strain and octahedral rotation. In this
plateau, with negligible cost to the total energy, the bandgap can be varied in
a range of 1eV. Furthermore, we present a descriptor model for the perovskite
to simulate their bandgap with strain and rotation. Analysis of band topology
through model Hamiltonians led to the conceptualization of topological
influencers that provide a quantitative measure of the contribution of each
chemical bonding towards establishing a normal or topological insulator phase.
On the technical aspect, we show that a four orbital based basis set
(Sn-$\{s,p\}$ for CsSnI$_3$) is sufficient to construct the model Hamiltonian
which can explain the electronic structure of each polymorph of halide
perovskites.
