Ab-initio phonon force constants

Two examples of such chains are given in Fig. 5.3.1. The atomic planes are connected by interplanar force constants k and, for symmetric choices of propagation direction, the equations of motion of the linear chain lead to a simple secular equation 2 x 2, whatever the range of the forces different sets k are then needed for different choices of polarization and of propagation direction. For choices less symmetric than k [100] or k  

Different sets of force-constants are obtained for longitudinal and transverse vibrations for a different propagation direction we have to start from a different unit cell - such as the one in Fig. 

2b for [111] - and determine another set of force constants, which is then pertinent to a different linear chain of planes -such as the one shown for [111] in Fig. 5.3.1b. 

The above picture of the linear chain is not new. The new ingredient is the faculty to determine the planar force constants ab initio, independently of any phenomenological model for the interactions. By switching from interatomic to interplanar force constants we have achieved two goals 1) The ab initio evaluation of k requires supercells which are large only in one dimension -and wRich thus become feasible. 2) Moreover, the k fall off with distance faster than the corresponding ( k V ). This property will be particularly useful in polar crystal where the k include all electrostatic interactions. 

We first illustrate the method on the example of Ge - which is sufficiently simple to demonstrate the basic steps - before addressing a more complicated system, a polar crystal. We follow closely the path used in Ref. 39. In the supercells shown in Figs. 5.3.2a,b, we choose the origin of coordinates so that the atom at (000) is in the plane positive directions [100] or [HI] are such that atom at a/4(lll) is in the plane +1, atoms a/4(-l,l,-l) and a/4(-l,-l,l) in the plane -1. The symmetry of the structure 

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