second, matrix elements of the spin-orbit ( ĤSO ) and Zeeman ( ĤZ ) Hamiltonians are evaluated in the basis of these non-relativistic states. As spin-orbit operator ĤSO , we use the one-electron operator derived from the Breit-Pauli spin-orbit Hamiltonian, within the spin-orbit mean-field approximation of the two electron part, as implemented by Pokhilko, Krylov, and co-workers. Magnetic sublevels are then obtained by diagonalizing the perturbed Hamiltonian, i.e., Ĥ = Ĥ0 + ĤSO + ĤZ , where Ĥ0 is the Born-Oppenheimer Hamiltonian. This procedure also enables quantification of the spin-reversal barrier in single-molecule and single-atom magnets using the computed magnetic sublevels (see Figure in Section II C). Magnetization and susceptibility are then derived from the resulting partition function using Boltzmann statistics.