  We study the nature of spin excitations of individual transition metal atoms
(Ti, V, Cr, Mn, Fe, Co and Ni) deposited on a Cu$_2$N/Cu(100) surface using
both spin-polarized density functional theory (DFT) and exact diagonalization
of an Anderson model derived from DFT. We use DFT to compare the structural,
electronic and magnetic properties of different transition metal adatoms on the
surface. We find that the average occupation of the transition metal d shell,
main contributor to the magnetic moment, is not quantized, in contrast with the
quantized spin in the model Hamiltonians that successfully describe spin
excitations in this system. In order to reconcile these two pictures, we build
a multi-orbital Anderson Hamiltonian for the d shell of the transition metal
hybridized with the p orbitals of the adjacent Nitrogen atoms, by means of
maximally localized Wannier function representation of the DFT Hamiltonian. The
exact solutions of this model have quantized total spin, without quantized
charge at the d shell. We propose that the quantized spin of the models
actually belongs to many-body states with two different charge configurations
in the d shell, hybridized with the p orbital of the adjacent Nitrogen atoms.
This scenario implies that the measured spin excitations are not fully
localized at the transition metal.
