The modeling of magnetic materials at finite temperatures is an ongoing challenge in the field of theoretical physics. This field has strongly benefited from the development of computational methods, which allow to predict material’s properties and explain physical effects on the atomic scale, and are now employed to direct the design of new materials. However, simulations need to be as accurate as possible to give reliable insights into solid-state phenomena, which means that, most desirably, all competing effects occurring in a system at realistic conditions should be included. This task is particularly difficult in the modeling of magnetic materials from first principles, due to the quantum nature of magnetism and its interplay with other phenomena related to the atomic degrees of freedom. The aim of this thesis is therefore to develop methods that enable the inclusion of magnetic effects in finite temperature simulations based on density functional theory (DFT), while considering on the same footing vibrational and structural degrees of freedom,with a particular focus on the high-temperature paramagnetic phase. The type of couplings investigated in this thesis can be separated in two big categories: interplay between magnetism and structure, and between magnetism and vibrations.

Regarding the former category, I have tried to shine some light on the effect of the paramagnetic state on atomic positions in a crystal in the presence of defects or for complicated systems, as opposed to the ordered magnetic state. To model the high-temperature paramagnetic phase of magnetic materials, the disordered local moment (DLM) approach is employed in the whole work. In this framework, I have developed a method to perform local lattice relaxations in the disordered magnetic state, which consists of a step-wise partial relaxation of the atomic positions, while changing the configuration of the magnetic moments at each step of the procedure. This method has been tested on point defects in paramagnetic bcc Fe, namely the single vacancy and, separately, the C interstitial in octahedral position, and on Fe_1-xCr_x alloys, finding non-negligible effects on formation energies. In addition, the feasibility of investigating extended defects like dislocations in the paramagnetic state with this method has also been proven by studying the screw dislocation in bcc Fe. The DLM-relaxation method has then been used to investigate intrinsic and extrinsic defects in CrN, an antiferromagnetic semiconductor studied for thermoelectric applications, found in the paramagnetic state at operating temperature, and a newly synthesized compound, Fe₃CO₇, which features a complicated crystal structure and unusual electronic properties, with possible important implications for the chemistry of Earth’s mantle.

The other focus of this thesis is the coupling between magnetism and lattice vibrations. As a pre-step to perform fully coupled atomistic spin dynamics-ab initio molecular dynamics (ASD-AIMD) simulations, I have first investigated the effect of vibrations on the so called longitudinal spin fluctuations, a mechanism occurring at finite temperatures and important for itinerant electron magnetic systems. I have developed a framework to investigate the dependence of the local moment’s energy landscapes on the instantaneous positions of the atoms, testing it on Fe at different temperature and pressure conditions. This study has laid the foundation to apply machine learning techniques to the prediction of the energy landscapes during an ASD-AIMD simulation. Finally, I have investigated the phase stability of Fe at ambient pressure from the theoretical Curie temperature up to its melting point with ASD-AIMD. This task is carried out by applying a pool of thermodynamic techniques to calculate free energy differences, and therefore I have defined a strategy to discern the thermodynamic equilibrium structure in magnetic materials in the high temperature paramagnetic phase based on first principles dynamical simulations. The methodologies developed and applied in this work constitute an improvement towards the simulation of magnetic materials accounting for the coupling of all effects, and the hope is to bridge a gap between theory and experiments.

Accurate simulation of magnetic materials using computational methods is essential for under-standing their fundamental behavior and enabling their use in technological applications. In this work, I use first-principles calculations to investigate systems with magnetic properties and to develop new methods for predicting the behavior of these materials. The systems studied are characterized by magnetic moments that are localized near the atomic sites. The paramagnetic state, at which these magnetic moments are disordered, and the magnetic order-disorder transition are of specific interest in this work.

To better capture finite-temperature magnetic behavior, a machine learning (ML) model is developed to predict the magnitudes of the magnetic moments at finite temperatures. This enables the inclusion of longitudinal spin fluctuations in coupled spin-lattice dynamics simulations, which would otherwise be computationally prohibitive. The ML model is applied to Fe at both the magnetic transition temperature, 1043 K, and at a pressure and temperature comparable to the conditions of the Earth’s inner core.

Evidently, the magnetic order-disorder transition temperature of ferromagnetic materials, known as the Curie temperature, is a fundamental property, since these materials lose their macroscopic magnetization above this point. Predicting this temperature is therefore crucial for the discovery and design of new magnetic materials. An approach is proposed which is based on the energy difference between magnetically ordered and disordered states, obtained from density functional theory (DFT) calculations. This method offers a balance between accuracy and computational efficiency, allowing its application to a wide variety of systems and making it suitable for high-throughput screening. The approach is fitted to and benchmarked against several known ferro- and ferrimagnetic materials and further evaluated on a particularly challenging class of systems: substitutionally disordered alloys. Finally, this approach enables a high-throughput exploration of Fe-, Mn-, and Co-containing systems to identify promising candidates for magnetic applications.

In addition, the debated role of constraining fields in DFT calculations for constrained non-collinear magnetism is investigated. The study shows that these fields can be used to propagate the transverse dynamics of magnetic moments, thereby providing a theoretical foundation for their use in adiabatic spin dynamics simulations.