Many physical properties depend upon a system response to some form of perturbation. Examples include polarisabilities, phonons, Raman intensities and infra-red absorption cross-sections to name but a few. Density functional perturbation theory (DFPT) is a particularly powerful and flexible theoretical technique that allows calculation of such properties within the density functional framework, thereby facilitating an understanding of the microscopic quantum mechanical mechanisms behind such processes, as well as providing a rigorous testing ground for theoretical developments. System responses to external perturbations may be calculated using DFT with the addition of some perturbing potential; however, as such methods involve obtaining the system response through a series of single-point energy calculations carried out at varying strengths of the external perturbation, it can be said that they are somewhat crude and aesthetically unappealing. More fundamentally, such techniques are sometimes restricted in application: for example, they cannot readily be used to calculate the response of crystalline systems to electric field perturbations, and cannot, without large computational effort, calculate phonon responses at arbitrary wavevector; points which will be expounded upon in this chapter. We now consider the application of perturbation theory to DFT, and use this formalism to derive equations allowing the calculation of phonon and electric field responses within crystalline materials.
The two main formalisms of DFPT are due to Baroni  and Gonze ; although the two may be shown to be equivalent, there are differences in the implementation that may result in one method being preferable to another. The Baroni formalism is centred upon obtaining a series of equations that may be solved self-consistently using Green's function methods; the Gonze formalism is based rather upon a perturbative expansion of the Kohn-Sham energy functional, leading to a variational problem for even orders of expansion akin to the zeroth order problem.