Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов

an ab initio simulation (solid black line) and a simulation with an effective force field (solid blue line), after [17]. Symbols: Data from two neutron scattering experiments [22].

      (13)

      where p runs over all the eigenmodes, and is the so‐called oscillator strength which is defined as

      (14)

      Here, e _I,k(ω_p) is the part of the eigenvector e(ω_p) that contains the three components of particle I, and the quantity Z_I,jk is the Born effective charge tensor defined as:

      (15)

      where e is the elementary charge, i.e. Z_I,ij is an effective charge that connects the strength of an external electric field ℰ to the force F ^I on particle I.

      These quantities allow to obtain immediately the real and imaginary parts of the dielectric function ɛ(ω) = ɛ₁(ω) + iɛ₂(ω) [24]:

      (16)

      (17)

      Closely related to ɛ(ω) is the absorption spectra α(ω) which is given by

      (18)

All these functions can be measured directly. On one side, they allow to test the predictive power of the simulations and, on the other, they help to interpret the various features found in the experimental spectra. For the infrared spectra of amorphous silica (Figure 5a–c), the theoretical calculation reliably predicts the position of the various peaks although their widths do not match very well the experimental data [18]. For the more complex NBS glass, a broad band is observed below 300 cm⁻¹ where the vDOS is strongly dominated by the motion of Na atoms [6], which explains why the spectra of alkali‐free glass‐formers lack such a low‐frequency band. In contrast, the narrow band found slightly above 400 cm⁻¹ is also present in the spectrum of pure silica where it has been assigned to the bending and rocking motion of oxygen atoms [16]. Therefore, this peak has probably the same origin in NBS as in SiO₂ although it is somewhat broader and shifted to lower frequencies as a result of a more disordered structure.

      

Figure 5 (a)–(c): Infrared spectra for amorphous SiO₂ as obtained from ab initio simulations (solid lines) and from experiments (dotted lines) [18]. (d) and (e): Infrared spectra for a sodium borosilicate glass as obtained from ab initio calculations and compared to experimental data for pure SiO₂ and B₂O₃ glasses, as well as for a sodium borosilicate glass with a similar composition [6].

      At high frequencies one finds two bands. The first one, ranging from 850 to 1200 cm⁻¹, can be assigned to oxygen stretching modes of Si─O bonds. That this band lies at lower frequencies than for silica (sharp peak at around 1070 cm⁻¹) is consistent with earlier results showing that the presence of non‐bridging oxygen atoms shifts the band to lower frequencies [19]. The second band extends between 1200 and 1600 cm⁻¹, and is due to the motions of oxygen atoms belonging to ^[3]B, in agreement with the fact that B₂O₃, which has mainly ^[3]B units, shows a very pronounced peak in this frequency range. Regarding absorption, good agreement is again found between the spectrum of NBS and the ab initio calculations made for a closely related composition (Figure 5d and e). The main deviation is found at high frequency where a peak is absent in the experimental spectrum but present in the simulation, since the latter overestimated the concentration of ^[3]B units because of the high quenching rate [6].

      Finally, we mention that DFT‐based methods have also been proposed in order to compute Raman and hyper‐Raman spectra for periodic solids [25, 26]. These approaches have been used to calculate the corresponding spectra for the main oxide network‐formers, i.e. SiO₂, B₂O₃, or GeO₂, with a good a very good agreement to experimental data [12, 13, 18].

5 Calculations of NMR Spectra

A serious problem faced by NMR experiments on disordered structures is that the observed peaks may overlap so that real‐space interpretations of the spectra are not always straightforward. Theoretical calculations can thus give helpful guidance, but their difficulty is illustrated by the fact that the first NMR schemes have been proposed only 15 years ago. In the glass community, the so‐called GIPAW approach proposed by Pickard and Mauri [27] is certainly the most used one, and implemented in many ab initio packages. However, the theory on how this is done is rather involved, see Ref. [28], and therefore we just illustrate the kind of results obtained with calculations of ²⁹Si Magic Angle Spinning spectra done for binary alkali and alkaline earth silicates (Figure 6).

      Regarding the peak positions, the agreement between the ab initio simulations and the experimental spectra is remarkably good (Figure 6a). For lithium silicates, slight discrepancies are found for the peak intensities but they are likely due to the modest size and high quenching rates of the simulated samples. In fact, one can handle this problem by determining how the concentrations of the various structural units depend on the cooling rate (or on the temperature of the liquid) and then adjusting them to the actual experimental conditions [6].

Figure 6 Application of ab initio simulations to NMR spectroscopy [29]. (a) ^[29]Si magic angle spinning spectra of calcium metasilicate (CS), lithium tetrasilicate (LS4), and sodium tetrasilicate (NS4) glasses. (b) Dependence of the ^[29]Si isotropic chemical shifts on the Si–O–T angle calculated