  A study of correlation effects in twisted bilayer graphene is presented using
the extended coupled cluster method at consistent mean-field and beyond
mean-field truncations. In order to describe the phase transitions in this
strongly correlated system, equations and a suitable implementation is
developed for the method. Combining modern tensor contraction techniques with
singular value decomposition, the correlation effects are successfully
described in a qualitative manner, with the majority of both short-range and
long-range Coulomb interactions included. The superconducting gap is found to
peak at an angle $\theta_c = 1.00^\circ$ with a BCS value for the critical
temperature of $T_\text{c}^\text{BCS} = 0.8$K matching closely with experiment.
Additionally, a homogeneous (s-wave) gap is found. While the simulations are
somewhat limited by the truncations used, nevertheless, a strong candidate for
the mechanism behind superconductive phases in twisted bilayer graphene is
presented.
