# Approach for Higher Order Response Properties

There are three (at least that I'm aware of) commonly used approaches to obtain linear response properties (e.g electric polarizability, optical rotation, NMR shielding tensors)

• Sum over states: The properties can formally be written as a sum of matrix elements of the perturbations $$A$$ and $$B$$ over all excited states. In practice, compute enough excited states to converge the property. Tends to converge slowly with the number of states.
• Response functions: The properties can also be written in terms of response relations, which leads us instead to compute the perturbed density with respect to $$A$$ (or $$B$$) and contract it with $$B$$ (or $$A$$) to compute the property.
• Derivatives: These properties are also derivatives of the energy with respect to these perturbations. One can derive analytic formulas or compute numerical derivatives. Analytical formulas are complex and don't (directly) apply to frequency dependent properties. Numerical derivatives require repeated calculations and tuning the step size.

For optical rotation, and I believe most other linear properties, the second approach above has won out as the best way to do the computation in general. But I'm curious if this holds for computing nonlinear properties (e.g. $$n^{\text{th}}$$ hyperpolarizability, the Kerr Effect) as well or whether the cost/benefit analysis of these methods changes.