The Crystal program for quantum-mechanical simulations of materials has been bridging the realm of molecular quantum chemistry to the realm of solid state physics for many years, since its first public version released back in 1988. This peculiarity stems from the use of atom-centered basis functions within a linear combination of atomic orbitals (LCAO) approach and from the corresponding efficiency in the evaluation of the exact Fock exchange series. In particular, this has led to the implementation of a rich variety of hybrid density functional approximations since 1998. Nowadays, it is acknowledged by a broad community of solid state chemists and physicists that the inclusion of a fraction of Fock exchange in the exchange-correlation potential of the density functional theory is key to a better description of many properties of materials (electronic, magnetic, mechanical, spintronic, lattice-dynamical, etc.). Here, the main developments made to the program in the last five years (i.e., since the previous release, Crystal17) are presented and some of their most noteworthy applications reviewed.

Lanthanide sesquioxides are strongly correlated materials characterized by highly localized unpaired electrons in the f band. Theoretical descriptions based on standard density functional theory (DFT) formulations are known to be unable to correctly describe their peculiar electronic and magnetic features. In this study, electronic and magnetic properties of the first four lanthanide sesquioxides in the series are characterized through a reliable description of spin localization as ensured by hybrid functionals of the DFT, which include a fraction of non-local Fock exchange. Because of the high localization of the f electrons, multiple metastable electronic configurations are possible for their ground state depending on the specific partial occupation of the f orbitals: the most stable configuration is here found and characterized for all systems. Magnetic ordering is explicitly investigated, and the higher stability of an antiferromagnetic configuration with respect to the ferromagnetic one is predicted. The critical role of the fraction of exchange on the description of their electronic properties (notably, on spin localization and on the electronic band gap) is addressed. In particular, a recently proposed theoretical approach based on a self-consistent definition -through the material dielectric response -of the optimal fraction of exchange in hybrid functionals is applied to these strongly correlated materials.

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Formal and computational aspects are discussed of a self-consistent treatment of spin-orbit coupling within the two-component generalisation of the Hartree-Fock theory. A molecular implementation into the Crystal program is illustrated, where the standard one-component code (typical of Hartree-Fock and Kohn-Sham spin-unrestricted methodologies) is extended to work in terms of two-component spinors. When passing from a one-to a two-component description, the Fock and density matrices become complex. Furthermore, apart from the αα and ββ diagonal spin blocks, one has also to deal with the αβ and βα off-diagonal spin blocks. These latter blocks require special care as, for open-shell electronic configurations, certain constraints of the one-component code have to be relaxed. This formalism intrinsically allows to treat local magnetic torque as well as non-collinear magnetization and orbital current-density. An original scheme to impose a specified noncollinear magnetization on each atomic center as a starting guess to the self-consistent procedure is presented. This approach turns out to be essential to surpass local minima in the rugged energy landscape and allow possible convergence to the ground-state solution in all of the discussed test cases.

We revise formal and numerical aspects of collinear and non-collinear density functional theory (DFT) in the context of a two-component self-consistent treatment of spin-orbit coupling (SOC). While the extension of the standard one-component theory to a non-collinear magnetization is formally well-defined within the local density approximation (LDA), and therefore results in a numerically stable theory, this is not the case within the generalized gradient approximation (GGA). Previously reported formulations of non-collinear DFT based on GGA exchange-correlation potentials have several limitations: i) they fail at reducing (either formally or numerically) to the proper collinear limit (i.e. when the magnetization is parallel to the z axis everywhere in space); ii) they fail at ensuring a quantitative rotational invariance of the total energy and even a qualitative rotational invariance of the spatial distribution of the magnetization when a SOC operator is included in the Hamiltonian; iii) they are numerically very unstable in regions of small magnetization. All of the above mentioned problems are here shown (both formally and through test examples) to be solved by using instead a new formulation of non-collinear DFT for GGA functionals, which we call the "signed canonical" theory, as combined with an effective screening algorithm for unstable terms of the exchange-correlation potential in regions of small magnetization. All methods are implemented in the Crystal program and tests are performed on simple molecules to compare the different formulations of non-collinear DFT.

We discuss the treatment of spin-orbit coupling (SOC) in time reversal symmetry broken periodic systems for relativistic electronic structure calculations of materials within the generalized noncollinear Kohn-Sham density functional theory (GKS-DFT). We treat SOC self-consistently and express the GKS orbitals in a two-component spinor basis. Crucially, we present a methodology (and its corresponding implementation) for the simultaneous self-consistent treatment of SOC and exact non-local Fock exchange operators. The many advantages of the inclusion of non-local Fock exchange in the self-consistent treatment of SOC, as practically done in hybrid exchange-correlation functionals, are both formally derived and illustrated through numerical examples: i) it imparts a local magnetic torque (i.e. the ability of the two-electron potential to locally rotate the magnetization with respect to a starting guess configuration) that is key to converge to the right solution in non-collinear DFT regardless of the initial guess for the magnetization; ii) because of the local magnetic torque, it improves the rotational invariance of non-collinear formulations of the DFT; iii) it introduces the dependence on specific pieces of the spinors (i.e. those mapped onto otherwise missing spin-blocks of the complex density matrix) into the two-electron potential, which are key to the correct description of the orbital-and spin-current densities and their coupling with the magnetization; iv) when space-inversion symmetry is broken, it allows for the full breaking of timereversal symmetry in momentum space, which would otherwise be constrained by a sum-rule linking the electronic band structure at opposite points in the Brillouin zone (k and -k). The presented methodology is implemented in a developmental version of the public Crystal code. Numerical tests are performed on the model system of an infinite radical chain of Ge2H with both spaceinversion and time-reversal symmetries broken, which allows to highlight all the above-mentioned effects.

The B-center in diamond, which consists of a vacancy whose four first nearest-neighbors are nitrogen atoms, has been investigated at the quantum-mechanical level with an all-electron Gaussian-type basis set, hybrid functionals, and the periodic supercell approach. To simulate various defect concentrations, four cubic supercells have been considered, containing (before the creation of the vacancy) 64, 216, 512, and 1000 atoms, respectively. Whereas the B-center does not affect the Raman spectrum of diamond, several intense peaks appear in the IR spectrum, which should permit us to identify this defect. It turns out that of the seven peaks proposed by Sutherland in 1954, located at 328, 780, 1003, 1171, 1332, 1372, and 1426 cm, and frequently mentioned as fingerprints of the B center, the first one and the last three do not appear in the simulated spectrum at any concentration. The graphical animation of the modes confirms the attribution of the remaining three and also permits investigation of the nature of the full set of modes.

We perform a formal analysis of relativistic density functional theory for the treatment of spin–orbit coupling (SOC), noncollinear magnetization (NCM), and orbital current density (OCD). We identify specific components of the spinors (namely, those mapped onto imaginary diagonal spin-blocks of the density matrix) that arise from the SOC operator and define the OCD. We show that these pieces of the spinors only enter in the bielectronic part of the potential through the exact Fock exchange (FE) operator. The lack of FE therefore leads to a correspondingly incorrect physical description of SOC, NCM, and OCD. This analysis is complemented with an illustrative example, where we show that, while in the absence of FE, the theory fails even at reproducing the expected right-hand relationship between the NCM and OCD, its inclusion provides results that match those from a reference SOC configuration–interaction calculation.