  Theoretical and experimental investigation of the electronic and magnetic
structure of transition metal atoms on an insulating interface with a metallic
substrate at low temperatures is quite challenging. In this paper, we show a
density functional theory plus Hubbard $U$ based protocol to study an Mn adatom
on three symmetrically allowed absorption sites, namely on O, hollow (between
two Mg and two O) and on Mg on MgO(001). We added a thick enough(bulk-like)
metallic Ag slab beneath MgO for faithful replication of scanning tunneling
microscopy (STM) experiments. Our study reveals that to determine the stable
most binding site, we need to obtain a Hubbard $U$ value from the density
functional theory(DFT) calculations, and our results show agreement with STM
experiments. Our calculated Hubbard $U$ values for the three adatom sites are
different. When Mn sits on O, it retains 2.4 $\mu_B$ spin moment, close to its
atomic spin moment. However, when Mn sits on other adatom sites, the spin
moment decreases. Using the atom projected density of states, we find Mn on O
atom shows very narrow crystal field splitting among \textit{d} orbitals,
whereas on other adatom sites Mn \textit{d} orbitals show significant
splitting. We calculate charge and spin densities and show vertical and
horizontal propagation of charge and spin density varies widely between sites.
Mn on Mg top shows an unusual feature, that Mn pushes Mg below, to make a
coplanar of Mn geometry with the four oxygen atoms. In addition, we explained
the reason for spin leaking down to the Ag layers. Mn on MgO/Ag does not show
any spin-flip behavior in the STM, an unusual phenomenon, and we used our first
principles-based approach to explain this unique observation.
