  Our understanding of collective spin fluctuation in materials relies largely
on Heisenberg-type spin Hamiltonians. Implicit in these spin models is the
atomic moment picture that in transverse spin dynamics the magnetization around
an atom undergoes precessional motion as a rigid moment, which has been
challenged by emerging theoretical and experimental advances. To assess the
validity of the atomic moment picture in spin dynamics, however, necessitates
magnon wavefunctions from \textit{ab initio} methods without \textit{a priori}
spin models. To this end, we develop an efficient model-free {\it ab initio}
method for computing magnon spectrum and wavefunctions. Niu-Kleinman's
adiabatic spin-wave dynamics is reformulated using linear perturbation theory
into a generalized eigenvalue problem, which can be solved to produce magnon
spectrum and wavefunctions without assuming atomic moments. We have implemented
this method in the framework of density functional perturbation theory (DFPT).
A dynamical extension of Niu-Kleinman equation of motion is proposed to improve
inaccurate predicted magnon energies due to imperfect adiabaticity at higher
energies. Based on so-obtained {\it ab initio} magnon wavefunctions, we find
the atomic moment picture to be valid in typical ferromagnets and
antiferromagnets, but fails in the molecular orbital crystal Na$_2$IrO$_3$. Our
results suggest that the usual spin Hamiltonian approach should be taken with a
grain of salt, and possible experimental ramification on the issue is
discussed.
