Harmonic functions, periodic structures

Unslaving of the harmonics The width of the sidebands grows either when the control parameter B increases or if the curvature, as a function of the wavenumber, at the maximum of the linear critical frequency becomes smaller, for instance as the ratio DaiDh decreases. Then overtones of the critical modes, that were otherwise slaved, progressively rejoin the active set of modes and their resonant interactions with the fundamental modes must be taken into account. This effect also grows more important as the space dimension of the pattern increases. Indeed for a periodic structure in one dimension the first harmonic to become excited lies at k = 2kc, whereas for patterns of hexagonal symmetry that are periodic in two dimensions the first overtone is at A = /3 kc- Further, for 3D patterns they appear... 

We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

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