  We derive electronic tight-binding Hamiltonians for strained graphene,
hexagonal boron nitride and transition metal dichalcogenides based on Wannier
transformation of {\it ab initio} density functional theory calculations. Our
microscopic models include strain effects to leading order that respect the
hexagonal crystal symmetry and local crystal configuration, and are beyond the
central force approximation which assumes only pair-wise distance dependence.
Based on these models, we also derive and analyze the effective low-energy
Hamiltonians. Our {\it ab initio} approaches complement the symmetry group
representation construction for such effective low-energy Hamiltonians and
provide the values of the coefficients for each symmetry-allowed term. These
models are relevant for the design of electronic device applications, since
they provide the framework for describing the coupling of electrons to other
degrees of freedom including phonons, spin and the electromagnetic field. The
models can also serve as the basis for exploring the physics of many-body
systems of interesting quantum phases.
