  We present efficient angle-dependent low-energy Hamiltonians to describe the
properties of the twisted bilayer graphene (tBLG) heterostructure, based on
{\it ab initio} calculations of mechanical relxation and electronic structure.
The angle-dependent relaxed atomic geometry is determined by continuum
elasticity theory, which induces both in-plane and out-of-plane deformations in
the stacked graphene layers. The electronic properties corresponding to the
deformed geometry are derived from a Wannier transformation to local
interactions obtained from Density Functional Theory calculations. With these
{\it ab initio} tight-binding Hamiltonians of the relaxed heterostructure, the
low-energy effective theories are derived from the projections near Dirac cones
at K valleys. For twist angles ranging from 0.7$^\circ$ to 4$^\circ$, we
extract both the intra-layer pseudo-gauge fields and the inter-layer coupling
terms in the low-energy Hamiltonians, which extend the conventional low-energy
continuum models. We further include the momentum dependent inter-layer
scattering terms which give rise to the particle-hole asymmetric features of
the electronic structure. Our model Hamiltonians can serve as a starting point
for formulating physically meaningful, accurate interacting electron theories.

Mesoscale and Nanoscale Physics