  We provide a formal derivation of a reduced model for twisted bilayer
graphene (TBG) from Density Functional Theory. Our derivation is based on a
variational approximation of the TBG Kohn-Sham Hamiltonian and asymptotic limit
techniques. In contrast with other approaches, it does not require the
introduction of an intermediate tight-binding model. The so-obtained model is
similar to that of the Bistritzer-MacDonald (BM) model but contains additional
terms. Its parameters can be easily computed from Kohn-Sham calculations on
single-layer graphene and untwisted bilayer graphene with different stackings.
It allows one in particular to estimate the parameters $w_{\rm AA}$ and $w_{\rm
AB}$ of the BM model from first-principles. The resulting numerical values,
namely $w_{\rm AA}= w_{\rm AB} \simeq 126$ meV for the experimental interlayer
mean distance are in good agreement with the empirical values $w_{\rm AA}=
w_{\rm AB}=110$ meV obtained by fitting to experimental data. We also show that
if the BM parameters are set to $w_{\rm AA}= w_{\rm AB} \simeq 126$ meV, the BM
model is an accurate approximation of our reduced model.
