Let’s consider three types of static perturbations
- atomic displacements
with
with
and 
- homogeneous strains
with 
- static electric field
with 
By performing a Taylor expansion of the total energy $E$ in terms of these perturbations we obtain[1]

The derivatives of the energy with respect to an electric field are the polarization, with respect to atomic displacements are the forces, with respect to changes in the lattice vectors are the stress tensor.



This leads to the following ‘clamped-ion’ or ‘frozen-ion’ definitions:






To compare with experimental results, however, the static response properties should take into account the ionic relaxation. This follows from the Taylor expansion above by looking at the ionic positions where the energy is minimal:

The physical ‘relaxed-ion’ tensors are

The second term on the right-hand side of each of these equations is called the ionic contributions to the dielectric susceptibility, elastic tensor, and piezoelectric tensor.
The ionic contributions to the dielectric tensor are:

The ionic contributions to the elastic tensor

The ionic contributions to the piezoelectric tensor

References