## 4 Slater-Koster Parameters from Fitting to Ab Initio Calculations  
Finding the optimal set of Slater-Koster parameters for the TB model considered here is a difficult task. Two main requirements must be satisfied: a good reproduction of the structure of the relevant electronic bands, and faithful representation of the orbital contribution along such bands. The last condition is especially relevant because, for example, different kinds of strains do not affect all the hoppings in the same manner. Therefore, capturing the proper orbital contribution is essential when using the TB model for calculations of physical properties of strained membranes. The same happens when one considers the effect of vacancies, adatoms, etc. In this work, we have obtained the Slater-Koster parameters for each compound from a minimization procedure that has the possibility to consider a band/momentum resolved weight that allows us to resolve more accurately particular \(\mathbf{k}\) regions of selected bands (e.g., edges of the valence and conduction bands)(The calculation of tight-binding band structure and fitting to DFT have been performed using the MathQ package, developed by [PERSON]. (Source: [[http://www.icmm.csic.es/sanjose/MathQ/MathQ.html](http://www.icmm.csic.es/sanjose/MathQ/MathQ.html)]([http://www.icmm.csic.es/sanjose/MathQ/MathQ.html](http://www.icmm.csic.es/sanjose/MathQ/MathQ.html)))). Furthermore, we can apply constrictions for the orbital contribution at specific band regions, taking as a reference the information from the DFT wave-functions. The sets of Slater-Koster parameters that we have obtained for the four compounds are given in Table 2. During the fitting, we have used only the \(6\times 6\) block of the Hamiltonian because, as explained above, it accounts for the valence and conduction bands. Therefore, we obtain as output all the Slater-Koster parameters but one, \(\Delta_{1}\), which is the crystal field corresponding to \(d_{2}=d_{xz,yz}\) orbitals, whose contribution is absent in the \(6\times 6\) block. What we have done to estimate the value of \(\Delta_{1}\) is to impose that the edges of the TB and DFT bands coincide for the lower energy band of the \(5\times 5\) block at the K point of the Brillouin zone. The band structure calculated with the \(6\times 6\) block of this model is plotted in Figure 3, as compared to DFT calculations. In Figures 4 and 5 and Figures 11-13, we compare the orbital contribution of the TB model, using the Slater-Koster parameters of Table 2, to the corresponding orbital contribution as obtained from DFT. We show the results for the most relevant orbitals (\(d_{0}=d_{3z^{2}-r^{2}}\), \(d_{2}=d_{xy},d_{x^{2}-y^{2}}\), \(p_{xy}=p_{x},p_{y}\) and \(p_{z}\)), and we can conclude that the TB model not only presents an acceptable fit to the band structure, but, importantly, the wave-functions also reproduce the DFT orbital contribution at the most important points of the band structure. Table 3 contains the main orbital contribution of each compound at the most relevant edges of the band structure, namely valence and conduction bands at K point, and valence band at \(\Gamma\) point of the Brillouin zone. We notice that the main restriction of the TB model considered here is that it only includes up to next-nearest-neighbor hopping terms, and this is why the fit to the DFT bands cannot be perfect. More sophisticated methods such as DFT based tight-binding Hamiltonians represented in the basis of maximally localized Wannier functions can lead to better agreements, at the cost of inclusion of longer-range hopping terms [52]. For the case presented here, and due to the automatised fitting procedure, we can conclude that the set of parameters presented in Table 2 must be close to the ideal solution.