The following is the Hubbard contribution to the hamiltonian in the Hubbard-Tight Binding model.
$$H_{hubbard}=U \sum_i n_{i \uparrow}n_{i\downarrow}$$
where $n_{i \sigma}=c_{i\sigma}^\dagger c_{i\sigma}$
And the tight-binding hopping part:
$$H_{TB}=t \sum_{i\sigma} (c_{i\sigma}^\dagger c_{i+1\sigma}+ c_{i+1\sigma}^\dagger c_{i\sigma})$$
The full hamiltonian is then given by:
$$H=H_{TB}+H_{hubbard}$$
If I want to consider a Peierls distortion (which is considering dimerization, breaking the symmetry and now having two different sites A and B within the unit cell) the Tight Binding part is changed to:
$$H_{Peierls}=t_1 \sum_{i\sigma} (a_{i\sigma}^\dagger b_{i\sigma}+ b_{i\sigma}^\dagger a_{i\sigma})+t_2 \sum_{i\sigma} (a_{i+1\sigma}^\dagger b_{i\sigma}+ b_{i\sigma}^\dagger a_{i+1\sigma})$$
In that case, how would the Hubbard hamiltonian change? (To be described in terms of these new operators $a$ and $b$).