Since we neglected Hamiltonian derivatives of higher than first order, we can, in analogy to the derivative theory of the ground state energy, write the derivatives of the Helmholtz free energy as temperature-dependent generalizations of zero-frequency response functions, i.e., The advantage of this notation is that it is more general than derivatives of the free energy, also allowing operators that are not derivatives of the Hamiltonian. From the sum-overstates expressions, it becomes clear that these "response functions" are linear in each of the operators. For example, in the spin Hamiltonian approximation, ∂H/∂B 0,i = 1 2 j g ij S j , i.e., the derivative of the Hamiltonian is a linear combination of spin operators. Because of linearity of the "response function" in each operator, that means that we can for example write ∂ 2 F ∂B 0,i ∂B 0,j = 1 4