3 gen terms and rescaled these by the Ge polarizability for the anion-cation terms. The Gaussian parameters too were added after the fit and then the comparison with the first-principles data was run again to check that its quality remained unchanged. The parameters d ij determine the range at which the overlap of the charge densities affects the induced dipoles, the parameters c ij determine the strength of the ion re- sponse to this effect. It is important to notice that anion- anion damping terms have been taken into account, con- trary to what was done in $^{15}$. The addition of anion-anion damping terms was found to greatly improve the ability to match the first-principles data. In Figure 1 we report the agreement between the DFT calculated forces and dipoles and those predicted by the fitted potential, for a set of representative ions; the ab- scissa shows the x component of the forces and dipoles, while the ordinate is simply the ion number. The quality of the representation is quite good and comparable with the one obtained in $^{22}$; this can be regarded as a very good result, considering the simplicity of the model (both ion shape deformation effects and quadrupoles are not taken into account in the present model $^{13,14}$). The parameters obtained for the DIPPIM potential are summarized in Table I. The parameters in the interaction potential are deter- mined by matching the dipoles and forces on the ions calculated from first-principles on condensed phase ionic configurations $^{13,14}$. Starting from the empirical pair po- tential in $^{7}$, we obtained atomic configurations for GeO$_{2}$ by running short MD simulations on small cells (150 ions); a total of two liquid configurations were obtained for GeO$_{2}$; for each of these, the Hellman-Feynman forces acting on individual ions of the simulation cell were cal- culated using planewave-DFT code CPMD $^{18}$. In all the calculations we used norm-conserving pseudopotentials and planewave energy cut-offs of 1360 eV ; all calcula- tions were performed using the generalized gradient ap- proximation (GGA) according to the Perdew, Burke and Ernzernhof (PBE) scheme. For the calculation of first- principles dipoles, the Kohn-Sham orbitals are localized via a Wannier transformation to construct maximally lo- calized Wannier functions (MLWF) and the dipoles de- termined from the positions of the centres of the Wannier functions associated with each ion $^{19}$. B. The DFT reference calculations C. Fitting procedure FIG. 1: Agreement between the DFT calculated dipoles (top panel) and forces (bottom panel) and those predicted by a fitted DIPPIM potential model for a set of 50 representative ions. In each panel the ordinate gives the ion number and the abscissa gives the DFT values of the x component of the forces and dipoles as points and the fitted values as the continuous line. The potential parameters are optimized by fitting the forces and dipoles, obtained with the DIPPIM poten- tials for the reference configurations, to the respective results from the DFT calculations; the 2 configurations provide a total of about 1800 data points to fit, compris- ing three Cartesian force components of each individual ion and three components for the dipoles. While most of the potential parameters are left as free parameters in the fits, there are some exceptions. The O 2 − polar- izability, for instance, was fixed to α$_{O}$ 2 − = 11 a.u. , i.e. the value Salmon et al obtained experimentally in $^{5}$. This value for the oxide polarizability is also compatible with the range of values obtained for this quantity in several magmatic melts from first-principles calculations $^{20}$. One problem in DFT calculations is the uncontrolled repre- sentation of the dispersion interaction. Although disper- sion only contributes a tiny fraction to the total energy, it has a considerable influence on phase transition pres- sures and on the material density and stress tensor. For this reason, we decided not to include the dispersion pa- rameters in the fits as discussed in 13 and to add them afterwards. We used the parameters in 21 for the oxy- We performed MD simulations in the NV T ensem- ble through the use of the Nos´ e-Hoover chain thermostat method $^{23,24}$. The simulation cell contained 288 oxide and 144 germanium ions, and its volume was set to 6856 ˚ A 3 in order to match the experimental density at 300 K, which is 3.66 g.cm − $^{3}$. We used a time step of 1 fs to in- tegrate the equations of motion and the minimization of the polarization energy was carried out with a conjugate gradient method. The system was studied in both the liquid and amorphous phases. III. SIMULATION DETAILS