By A. Baldereschi, R. Resta (auth.), J. T. Devreese, V. E. Van Doren, P. E. Van Camp (eds.)
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Extra info for AB Initio Calculation of Phonon Spectra
39) after a few moments will give a good approximation to X. The main advantage of this method is that we only need valence states. In the calculations discussed in the next paragraph two approximations to Eq. (39) have been used. 10 is kept. 10' which depends upon wavevector and band-index, is replaced by a constant. This constant is calculated explicitly [21-23]. e. Eq. (20)). Approximately 113 plane waves are used in the Hamiltonian. For the integration over the Brillouin zone in Eq. (38) an 8-division mesh is used.
However, it was only recently (1979) that the fir~t real ab-initio (parameterless) calculation of phonon frequencies of a semiconductor appeared in the literature. The major cause for the time delay between the theory and an actual calculation is the matrix property of the polarizability. Whereas in a simple metal the offdiagonal elements are somehow small and usually neglected, they are very essential and have to be taken into account in a semiconductor or insulator. The early dielectric screening calculations struggled with apparent instabilities of the crystal against shear.
Dens~ty 1y n-+ (r') o~. ~n (24) , We can define an output potential ~n t ou using (9) and to linear order we find from (22) and (24) using the definition (20). from nn out ~:ut(r) = Jd 3 r 1d 3 r 2 Vt(~'~l) X (~1'~2) o~~n(~2)' (25) or in matrix notation, using eq. (23c), = (l+~' -1 v xc c )(l-~' xc X* )(l-s)~ n.