  The $s,p-d$ exchange coupling between the spins of band carriers and of
transition metal (TM) dopants ranging from Ti to Cu in ZnO is studied within
the density functional theory. The $+U$ corrections are included to reproduce
the experimental ZnO band gap and the dopant levels. The $p-d$ coupling reveals
unexpectedly complex features. In particular, (i) the $p-d$ coupling constants
$N_0\beta$ vary about 10 times when going from V to Cu, (ii) not only the value
but also the sign of $N_0\beta$ depends on the charge state of the dopant,
(iii) the $p-d$ coupling with the heavy holes and the light holes is not the
same; in the case of Fe, Co and Ni, $N_0\beta$s for the two subbands can differ
twice, and for Cu the opposite sign of the coupling is found for light and
heavy holes. The main features of the $p-d$ coupling are determined by the
$p-d$ hybridization between the $d$(TM) and $p$(O) orbitals. In contrast, the
$s-d$ coupling constant $N_0\alpha$ is almost the same for all TM ions, and
does not depend on the charge state of the dopant. The TM-induced spin
polarization of the $p$(O) orbitals contributes to the $s-d$ coupling,
enhancing $N_0\alpha$.
