We distinguish two extreme classes of perturbation problems depending on the signs of second-order response properties. The first class refers to a positive value of the same for any state, and is overwhelmingly more probable. The other category offers all-but-one negative values, or at least some negative values for highly excited states. The classes are seen to differ in reproducing results of finite-dimensional matrix Hamiltonian perturbations, allowing the emergence of a type of sum rule. A few analytical findings are employed for direct demonstration. The outcomes provide notable restrictions on second order response properties of quantum states. © 2014 Springer International Publishing Switzerland.