7 of the relative velocity of iα along the i → iα bond, i.e. Similar expressions may be written down for the veloc- ities of the other symmetry coordinates $^{30}$. The corre- sponding DOS is then obtained by a Fourier transform of the corresponding velocity autocorrelation function: cross-term strongly reduces the intensity of the two low frequency bands relative to that which would be obtained from the charge fluctuations alone, whereas for the high- frequency band the cancellation is much weaker. For the high frequency band the relationship of the charge-charge and cross-terms is different to that found previously for SiO$_{2}$ and BeF$_{2}$ $^{26,29}$. In the latter cases, the charge- charge and cross-terms had the same sign, so that the net band intensity was slightly larger than that predicted by the charge-charge term alone. This difference in the behaviour of the calculated spectra might arise from the inclusion of an anion-anion damping term in the polar- ization part of the interaction potential here, which was not the case for the previous studies. v ‖ iα = ( v iα − v $_{i}$) · ( r iα − r $_{i}$) . (12) DOS$_{A}$( ν ) = ℜ ∫ ∞ 0 e 2 $^{πıνt}$〈 v A i ( t ) · v A i (0) 〉 d t. (13) FIG. 6: Decomposition of the total IR absorption spectrum in its three component parts. FIG. 7: The vibrational densities of states, for the DIP- PIM, of both type of ions (top panel), and their projections in terms of the A (symmetric stretch), F 2 (asymmetric stretch), E (bend), and F 2 (bend) modes of the Ge0 4 tetrahedra (bot- tom panel). We can now attempt to associate these IR frequencies with the vibrational modes of the system. The upper part of figure 7 shows the vibrational density of states (VDOS) of each ion. Some of these features can be associated to the vibrations of the GeO$_{4}$ tetrahedra by comparing them with the tetrahedral-VDOS as obtained from the correla- tion functions of the velocities of various local tetrahedral symmetry coordinates. The latter would correspond to the velocities of GeO$_{4}$ normal modes, if the tetrahedra were isolated ( i.e. not mechanically linked in a network). For a tetrahedral molecule, vibrational normal coordi- nates ν$_{1}$ ( A$_{1}$ , symmetric stretch), ν$_{2}$ ( E , bend), ν$_{3}$ ( F$_{2}$ , asymmetric stretch) and ν$_{4}$ ( F$_{2}$ , bend) are expected $^{30}$. We may obtain velocities associated with the symmetry coordinates of each GeO$_{4}$ unit. For example, for the sym- metric stretching motion Were the network to vibrate as a collection of isolated tetrahedra, the spectra of each tetrahedral-VDOS would exhibit a single peak corresponding to the characteris- tic frequency of the corresponding normal mode of vi- bration. The four tetrahedral-VDOS are shown on the where i labels the Ge of a tetrahedral complex, and iα the four oxide anions at the vertices. v ‖ iα is the projection v A i = ∑ iα =1 − 4 v ‖ iα (11)