Bom-von Karman zone

In addition to the investigation of the components of the hyperpolarizability tensors associated with directions perpendicular to the chain axis, Schmidt and Springborg [178 proposed a method based on Wannier functions to determine the longitudinal components. Their method is related to the one of Kune and Resta [115] since the number of k points is finite and the dipole moment operator (longitudinal component) is rewritten to display the same periodicity as the Bom von Karman zone. Consequently, the accuracy of the longitudinal hyperpolarizabilities per unit ceU will increase with the number of k points. 

Fig. 1. Schematic representation of the external field for (a) a finite system with the true field, (b) the approximation that the field has the periodicity of the lattice, and (c) that it has the periodicity of the Bom von Karman zone.

M = 2K). Then, all Bloch functions have the periodicity of the Bom von Karman zone, i.e., of the lengfli 2K a, with a being the length of one unit cell. With n being the band index, any function with this periodicity can be expanded in flie Bloch functions. 

We shall approximate the scalar potential of the electrostatic field by the sawtooth curve of Fig. 5 that has the periodicity of the Bom von Karman zone, i.e., of the length of one unit cell times the number of k points that is used in a calculation. We have here assumed that the potential takes both positive and negative values. By doing so, the average potential from the field vanishes and we have therefore an optimal starting point for eliminating effects that are linear in the number of k points of the calculation (i.e., in the length of the Bom von Karman zone). 

Fig. 5. The left part shows the sawtooth curve z that is periodic (and linear) with the periodicity of the Bom von Karman zone. It is decomposed into the lattice-periodic part of the middle part, and the piecewise constant part shown in the right part of the figure, z = Zi+Z2-...

We have applied this approach on a linear chain of carbon atoms with alternating bond lengths of 2.7 and 2.5 a.u. In Fig. 6 we show the band structures for this system without any external field. The calculations were done using seven k points in half-part of the Brillouin zone, giving a Bom von Karman zone of 12 unit cells, each with two carbon atoms. Moreover, we included the 20 energetically lowest bands that all are shown in the figure (notice, however, that tt and 8 bands are pairwise degenerate) and, for the sake of simplicity, we applied a local-density approximation within density-functional Aeory. 

Fig. 8. Changes in the number of electrons for the different bands relative to the numbers for the undistorted system (24 for the first four bands, and 0 for the remaining). The curves are (in order of decreasing amplitude) for field strengths of e-E = 0.0002, 0.0005, and 0.0002 hartree for (the most oscillating curve) the case of the field with the symmetry of the Bom von Karman zone, and (the other curves) for the case that the field has the lattice periodicity.

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