  For fast and accurate calculations of band gaps of solids, we present an {\it
ab initio} method that extends the density functional theory plus on-site
Hubbard interaction (DFT+$U$) to include inter-site Hubbard interaction ($V$).
This formalism is appropriate for considering various interactions such as a
local Coulomb repulsion, covalent hybridizations, and their coexistence in
solids. To achieve self-consistent evaluations of $U$ and $V$, we adapt a
recently proposed Agapito-Curtarolo-Buongiorno Nardelli pseudohybrid functional
for DFT$+U$ to implement a density functional of $V$ and obtain band gaps of
diverse bulk materials as accurate as those from $GW$ or hybrid functionals
methods with a standard DFT computational cost. Moreover, we also show that
computed band gaps of few layers black phosphorous and Si(111)-($2\times1$)
surface agree with experiments very well, thus meriting the new method for
large-scale as well as high throughput calculations with higher accuracy.
