Computational methods are commonly used by materials scientists to make predictions about materials. These methods can achieve in hours what would take days or weeks to accomplish in a lab. However, there are limits to what computational methods can do and how accurate the predictions are.A limiting factor for computational materials science is the size of the search space. The space of potential materials is infinite. Selecting specific systems of elements on a fixed lattice to study reduces the number of possible arrangements of atoms in the lattice to a finite number. However, this number can still be very large. Additionally this list of arrangements will contain duplicates, i.e., two different atomic arrangements could be equivalent by a rotation or translation of the lattice. Using symmetry to eliminate the duplicates saves time and resources. In order to ensure that the final list of unique structures will fit into computer memory it is also useful to know how many unique arrangements there are before actually finding them. For this reason the Pòlya enumeration algorithm was created to determine the number of unique arrangements before enumerating them. A new atomic enumeration algorithm has also been implemented in the enumlib package. This new algorithm has been optimized to find the symmetrically unique arrangements for systems with large amounts of configurational freedom, such as high-entropy alloys, which have been too computationally expensive for other algorithms.A popular computational method in materials science is Density Functional Theory (DFT). DFT codes perform first principles calculations by calculating the electron energy using numerical integrals. It is well known that the accuracy of the integrals depends heavily on the number of sample points, k-points, used. We have conducted a detailed study of how k-point sampling methods effect the accuracy of DFT calculations. This study shows that the most efficient k-point grids are those that have the fewest symmetrically distinct k-points, we call these general regular (GR) grids. GR grids are, however, difficult to generate, requiring a search across many possible grids. In order to make GR grids more accessible to the DFT community we have implemented an algorithm that can search k-point grids for the grid that has the fewest symmetry reduction in a matter of seconds.

For centuries, scientists have dreamed of creating materials by design. Rather than discovery by accident, bespoke materials could be tailored to fulfill specific technological needs. Quantum theory and computational methods are essentially equal to the task, and computational power is the new bottleneck. Machine learning has the potential to solve that problem by approximating material behavior at multiple length scales. A full end-to-end solution must allow us to approximate the quantum mechanics, microstructure and engineering tasks well enough to be predictive in the real world. In this dissertation, I present algorithms and methodology to address some of these problems at various length scales. In the realm of enumeration, systems with many degrees of freedom such as high-entropy alloys may contain prohibitively many unique possibilities so that enumerating all of them would exhaust available compute memory. One possible way to address this problem is to know in advance how many possibilities there are so that the user can reduce their search space by restricting the occupation of certain lattice sites. Although tools to calculate this number were available, none performed well for very large systems and none could easily be integrated into low-level languages for use in existing scientific codes. I present an algorithm to solve these problems. Testing the robustness of machine-learned models is an essential component in any materials discovery or optimization application. While it is customary to perform a small number of system-specific tests to validate an approach, this may be insufficient in many cases. In particular, for Cluster Expansion models, the expansion may not converge quickly enough to be useful and reliable. Although the method has been used for decades, a rigorous investigation across many systems to determine when CE "breaks" was still lacking. This dissertation includes this investigation along with heuristics that use only a small training database to predict whether a model is worth pursuing in detail. To be useful, computational materials discovery must lead to experimental validation. However, experiments are difficult due to sample purity, environmental effects and a host of other considerations. In many cases, it is difficult to connect theory to experiment because computation is deterministic. By combining advanced group theory with machine learning, we created a new tool that bridges the gap between experiment and theory so that experimental and computed phase diagrams can be harmonized. Grain boundaries in real materials control many important material properties such as corrosion, thermal conductivity, and creep. Because of their high dimensionality, learning the underlying physics to optimizing grain boundaries is extremely complex. By leveraging a mathematically rigorous representation for local atomic environments, machine learning becomes a powerful tool to approximate properties for grain boundaries. But it also goes beyond predicting properties by highlighting those atomic environments that are most important for influencing the boundary properties. This provides an immense dimensionality reduction that empowers grain boundary scientists to know where to look for deeper physical insights.

The perovskite family of materials is extremely large and provides a template for designing materials for different purposes. Among them, hybrid organic-inorganic perovskites (HOIPs) are very interesting and have been recently identified as possible next generation light harvesting materials because they combine low manufacturing cost and relatively high power conversion efficiencies (PCEs). In addition, some other applications like light emitting devices are also highly studied. This thesis starts with an introduction to the solar cell technologies that could use HOIPs. In Chapter 2, previously published results on the structural, electronic, optical and mechanical properties of HOIPs are reviewed in order to understand the background and latest developments in this field. Chapter 3 discusses the computational and experimental methods used in the following chapters. Then Chapter 4 describes the discovery of several hybrid double perovskites, with the formula (MA)$_2$M$^I$M$^{III}$X$_6$ (MA = methylammonium, CH$_3$NH$_3$, M$^I$ = K, Ag and Tl, M$^{III}$ = Bi, Y and Gd, X = Cl and Br). Chapter 5 presents studies on the variable presure and temperature response of formamidinium lead halides FAPbBr$_3$ (FA = formamidinium, CH(NH$_2$)$_2$) as well as the mechanical properties of FAPbBr$_3$ and FAPbI$_3$, followed by a computational study connecting the mechanical properties of halide perovskites ABX$_3$ (A = K, Rb, Cs, Fr and MA, X = Cl, Br and I) to their electronic transport properties. Chapter 6 describes a study on the phase stability, transformation and electronic properties of low-dimensional hybrid perovskites containing the guanidinium cation Gua$_x$PbI$_{x+2}$ (x = 1, 2 and 3, Gua = guanidinium, C(NH$_2$)$_3$). The conclusions and possible future work are summarized in Chapter 7. These results provide theoreticians and experimentalists with insight into the design and synthesis of novel, highly efficient, stable and environmentally friendly materials for solar cell applications as well as for other purposes in the future.

Materials play an important role in society. Historically, and even at present, materials have been discovered by trial and error, and many of the most useful materials have been discovered by chance. The high-throughput approach aims to remove (as much as possible) chance and guesswork at the experimental level by filtering out materials candidates that are not predicted to exist. Many successes have been recorded. In the high-throughput approach to materials discovery, machined-learned models of materials are created from databases of theoretical materials. These databases are the result of millions of density-functional-theory (DFT) simulations. The size and accuracy of the data in the databases (and, consequently, the predictions of machined-learned models) are most affected by the band energy calculation; most of the computation of a DFT simulation is computing the band energy in the self-consistency cycle, and most of the error in the simulation comes from band energy error. The band energy is obtained from a two-part multidimensional numerical integral over the Brillouin or irreducible Brillouin zone. A quadratic approximation and integration algorithm for computing the band energy in 2D and 3D is described. Analytic and semi-analytic integration of quadratic polynomials over simplices improves the accuracy and efficiency of the calculation. A method is proposed for estimating the error bounds of the quadratic approximation that does not require additional eigenvalues. Error propagation of approximation errors leads to an adaptive refinement approach that is driven by band energy error. Because adaptive meshes have little symmetry, integration is performed within the irreducible Brillouin zone, and a general algorithm for computing the irreducible Brillouin zone is described. The efficiency of quadratic integration is tested on realistic empirical pseudopotentials. When compared to current integration methods, uniform quadratic integration over the irreducible Brillouin zone sometimes results in fewer k-points for a given accuracy. Adaptive refinement fails to improve integration performance because band energy error bounds are inaccurate, especially at accidental crossings at the Fermi level.