## 2 Theoretical Background
### How to Speed Up Hybrid Functional Calculations  
First-principles calculations based on DFT have become an indispensable tool for contemporary computational materials science investigations. Recent advances in methods development and the wide-spread availability and efficient use of high-performance computing facilities have paved the way towards _predictive_ computational materials investigations. The achievable accuracy of current investigations is mostly governed by the choice of the exchange and correlation functional. While (semi)local functionals allow for very fast calculations, they mostly fail to accurately determine band gaps and optical properties of semiconducting materials. So-called hybrid functionals, where a particular fraction of Hartree-Fock exact exchange is mixed into an underlying (semi)local functional, capture the electronic and optical properties of semiconducting materials in much better agreement with experimental results, however, this accuracy has to be paid for by an increase in computational resources. While hybrid functional calculations have become standard for the calculation of structural, electronic, and optical properties of bulk materials [22, 23, 24, 25, 26, 27, 28, 29], their widespread application for problems that require a large number of calculations has been hindered by the computational demands. One way to solve this particular problem relies on a two-step process: firstly, the structural relaxations are performed using a computationally cheap (semi)local functional to obtain the ground state structure, and secondly, only for this ground state structure, perform a single-shot hybrid functional calculation to obtain more accurate electronic and optical properties. One obvious flaw with this approach is that the hybrid functional calculations are performed at geometries that are not the ground state for the applied hybrid functional. A recent investigation dealt with the performance and accuracy of different exchange and correlation functionals for the structural, electronic, and optical properties of CZTSe and Ag\({}_{2}\)ZnSnSe\({}_{4}\)[22]. Focusing only on CZTSe as of interest for the present work, it has been shown that the structural properties calculated by means of the SCAN functional [30] provided the best agreement with experimental results, however, the electronic and optical properties were severely underestimated. On the other hand, hybrid functional calculations employing the HSE06 functional [31] showed only slightly less good agreement concerning the structural properties, but improved massively for the electronic and optical properties when compared to experimental results. The results for the electronic properties calculated by a single-shot HSE06 calculation for the SCAN optimised ground state structure is slightly less good compared to a full hybrid functional investigation, but still improved over a SCAN functional investigation alone. Here, we employ a recently devised method to speed up hybrid functional calculations [19]. All the structural relaxations are typically performed by relaxing the internal coordinates of a given unit cell for different fixed unit cell volumes around the experimentally known ground state volume. The resulting total energy curves can then be fitted by an equation of state, e.g., [PERSON]'s [32,33],  
\[E(V)=E(V_{0})+\frac{B_{0}V}{B_{0}^{{}^{\prime}}}\left[\frac{(V_{0}/V)^{B_{0}^ {{}^{\prime}}}}{B_{0}^{{}^{\prime}}-1}+1\right]-\frac{V_{0}B_{0}}{B_{0}^{{}^{ \prime}}-1}\,, \tag{1}\]that besides the ground state volume \(V_{0}\) provides additional access to the bulk modulus \(B_{0}\) and its pressure derivative \(B_{0}^{{}^{\prime}}\) for the materials under investigation. The internal coordinates of the identified ground state volume undergo a final structural relaxation, and provide the starting point for subsequent calculations, e.g., electronic band structures or dielectric functions for the optical properties.