Born effective charges

The Born effective charges represent a useful means of characterising the response of a potential model to perturbations, particularly those that create an electric field. Increasingly values are becoming available from ab initio techniques for solids through the application of linear response methods. Hence, this quantity can provide a useful comparison between the results of formal-charge shell model calculations and more accurate first principles methods.

One of the first materials for which the Born effective charges were determined using planewave techniques is $\alpha $-quartz. In Table 2.3 the values obtained from the shell model of Sanders et al [120] are compared with those yielded by a planewave calculation using the Local Density Approximation and norm-conserving pseudopotentials [121].

Table 2.3: Born effective charges (in a.u.) calculated according to the shell model of Sanders et al [120] and from first principles techniques [121]. Values are shown for the asymmetric unit atoms with the approximate positions of the Si atom at (0.46,0,0) and the O atom at (0.41,0.27,0.11) in space group 154, with the origin set to (0,0,1/3).
    Si     O  
  3.122 0.0 0.0 -1.406 0.368 0.252
Shell model 0.0 3.530 0.292 0.364 -1.920 -0.517
  0.0 -0.171 3.422 0.176 -0.568 -1.711
  3.016 0.0 0.0 -1.326 0.429 0.222
LDA 0.0 3.633 0.282 0.480 -1.999 -0.718
  0.0 -0.324 3.453 0.298 -0.679 -1.726

The comparison of the Born effective charge tensors demonstrates that the oxygen shell model is surprisingly successful at reproducing the quantum mechanical data, especially in comparison to rigid ion models, which would yield a diagonal matrix with all components equivalent. Furthermore, the polarisability of the shell leads to the ions behaving as partially charged species with realistic magnitudes. Consequently, this explains why the seemingly unreasonable use of a formal charge for $Si^{4+}$actually works extremely well in practice. Similar observations have been previously made for perovskite materials [122].