Hamiltonian derivatives are linear in the spin operators, see Eqs. ( ) and ( ). Therefore, we can evaluate the quantities ⟨⟨S i S j ⟩⟩ (the spin dyadic tensor) and ⟨⟨S i S j S k S l ⟩⟩ (which can be called the "spin tetradic" tensor) and from these obtain the three needed tensors. See the Appendix for the "response theory" notation used here. Note that our definition of the spin dyadic differs by a factor of -β from the spin dyadic ⟨S i S j ⟩ of Vaara and coworkers (also used by us previously ), which is equal to the Z-matrix introduced by Autschbach and coworkers: 38