Spin Currents via the Gauge Principle for Meta-Generalized Gradient Exchange-Correlation Functionals.

The prominence of density functional theory in the field of electronic structure computation stems from its ability to usefully balance accuracy and computational effort. At the base of this ability is a functional of the electron density: the exchange-correlation energy. This functional satisfies known exact conditions that guide the derivation of approximations. The strongly constrained and appropriately normed (SCAN) approximation stands out as a successful, modern, example. In this Letter, we demonstrate how the SU(2) gauge invariance of the exchange-correlation functional in spin current density functional theory allows us to add an explicit dependence on spin currents in the SCAN functional (here called JSCAN)-and similar meta-generalized-gradient functional approximations-solely invoking first principles. In passing, a spin-current dependent generalization of the electron localization function (here called JELF) is also derived. The extended forms are implemented in a developer's version of the crystal23 program. Applications on molecules and materials confirm the practical relevance of the extensions.

Efficient calculation of derivatives of integrals in a basis of non-separable Gaussians.

A computational procedure is developed for the efficient calculation of derivatives of integrals over non-separable Gaussian-type basis functions, used for the evaluation of gradients of the total energy in quantum-mechanical simulations. The approach, based on symbolic computation with computer algebra systems and automated generation of optimized subroutines, takes full advantage of sparsity and is here applied to first energy derivatives with respect to nuclear displacements and lattice parameters of molecules and materials. The implementation in the Crystal code is presented, and the considerably improved computational efficiency over the previous implementation is illustrated. For this purpose, three different tasks involving the use of analytical forces are considered: (i) geometry optimization; (ii) harmonic frequency calculation; and (iii) elastic tensor calculation. Three test case materials are selected as representatives of different classes: (i) a metallic 2D model of the Cu(111) surface; (ii) a wide-gap semiconductor ZnO crystal, with a wurtzite-type structure; and (iii) a porous metal-organic crystal, namely the ZIF-8 zinc-imidazolate framework. Finally, it is argued that the present symbolic approach is particularly amenable to generalizations, and its potential application to other derivatives is sketched.

Mechanisms for Pressure-Induced Isostructural Phase Transitions in EuO.

We study pressure-induced isostructural electronic phase transitions in the prototypical mixed valence and strongly correlated material EuO using the global-hybrid density functional theory. The simultaneous presence in the valence of highly localized d- and f-type bands and itinerant s- and p-type states, as well as the half-filled f-type orbital shell with seven unpaired electrons on each Eu atom, have made the description of the electronic features of this system a challenge. The electronic band structure, density of states, and atomic oxidation states of EuO are analyzed in the 0-50 GPa pressure range. An insulator-to-metal transition at about 12 GPa of pressure was identified. The second isostructural transition at approximately 30-35 GPa, previously believed to be driven by an oxidation from Eu(II) to Eu(III), is shown instead to be associated with a change in the occupation of the Eu d orbitals, as can be determined from the analysis of the corresponding atomic orbital populations. The Eu d band is confined by the surrounding oxygens and split by the crystal field, which results in orbitals of e_{g} symmetry (i.e., d_{x^{2}-y^{2}} and d_{2z^{2}-x^{2}-y^{2}}, pointing along the Eu-O direction) being abruptly depopulated at the transition as a means to alleviate electron-electron repulsion in the highly compressed structures.

Spin-orbit coupling from a two-component self-consistent approach. II. Non-collinear density functional theories.

We revise formal and numerical aspects of collinear and non-collinear density functional theories in the context of a two-component self-consistent treatment of spin-orbit coupling. Theoretical and numerical analyses of the non-collinear approaches confirm their ability to yield the proper collinear limit and provide rotational invariance of the total energy for functionals in the local-density or generalized-gradient approximations (GGAs). Calculations on simple molecules corroborate the formal considerations and highlight the importance of an effective screening algorithm to provide the sufficient level of numerical stability required for a rotationally invariant implementation of non-collinear GGA functionals. The illustrative calculations provide a first numerical comparison of both previously proposed non-collinear formulations for GGA functionals. The proposed screening procedure allows us to effectively deal with points of small magnetization, which would otherwise be problematic for the evaluation of the exchange-correlation energy and/or potential for non-collinear GGA functionals. Both previously suggested formulations for the non-collinear GGA are confirmed to be adequate for total energy calculations, provided that the screening is achieved on a sufficiently fine grid. All methods are implemented in the Crystal program.

57Fe Mössbauer isomer shift of pure iron and iron oxides at high pressure-An experimental and theoretical study.

The 57Fe isomer shift (IS) of pure iron has been measured up to 100 GPa using synchrotron Mössbauer spectroscopy in the time domain. Apart from the expected discontinuity due to the α → ε structural and spin transitions, the IS decreases monotonically with increasing pressure. The absolute shifts were reproduced without semi-empirical calibrations by periodic density functional calculations employing extensive localized basis sets with several common density functionals. However, the best numerical agreement is obtained with the B1WC hybrid functional. Extension of the calculations to 350 GPa, a pressure corresponding to the Earth's inner core, predicted the IS range of 0.00 to -0.85 mm/s, covering the span from Fe(0) to Fe(VI) compounds measured at ambient pressure. The calculations also reproduced the pressure trend from polymorphs of prototypical iron oxide minerals, FeO and Fe2O3. Analysis of the electronic structure shows a strong donation of electrons from oxygen to iron at high pressure. The assignment of formal oxidation to the Fe atom becomes ambiguous under this condition.

Topology of the Electron Density and of Its Laplacian from Periodic LCAO Calculations on f-Electron Materials: The Case of Cesium Uranyl Chloride.

The chemistry of f-electrons in lanthanide and actinide materials is yet to be fully rationalized. Quantum-mechanical simulations can provide useful complementary insight to that obtained from experiments. The quantum theory of atoms in molecules and crystals (QTAIMAC), through thorough topological analysis of the electron density (often complemented by that of its Laplacian) constitutes a general and robust theoretical framework to analyze chemical bonding features from a computed wave function. Here, we present the extension of the Topond module (previously limited to work in terms of s-, p- and d-type basis functions only) of the Crystal program to f- and g-type basis functions within the linear combination of atomic orbitals (LCAO) approach. This allows for an effective QTAIMAC analysis of chemical bonding of lanthanide and actinide materials. The new implemented algorithms are applied to the analysis of the spatial distribution of the electron density and its Laplacian of the cesium uranyl chloride, Cs2UO2Cl4, crystal. Discrepancies between the present theoretical description of chemical bonding and that obtained from a previously reconstructed electron density by experimental X-ray diffraction are illustrated and discussed.

Spin-orbit coupling from a two-component self-consistent approach. II. Non-collinear density functional theories.

We revise formal and numerical aspects of collinear and noncollinear 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 noncollinear magnetization is formally well-defined within the local density approximation, and therefore results in a numerically stable theory, this is not the case within the generalized gradient approximation (GGA). Previously reported formulations of noncollinear 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 or antiparallel 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 noncollinear 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 noncollinear DFT.

Spin-orbit coupling from a two-component self-consistent approach. I. Generalized Hartree-Fock theory.

Formal and computational aspects are discussed for a self-consistent treatment of spin-orbit coupling within the two-component generalization 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 ßß diagonal spin blocks, one has also to deal with the αß and ßα 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 us to treat local magnetic torque as well as noncollinear 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 allows possible convergence to the ground-state solution in all of the discussed test cases.

Benzoquinone-Bridged Heterocyclic Zwitterions as Building Blocks for Molecular Semiconductors and Metals.

In pursuit of closed-shell building blocks for single-component organic semiconductors and metals, we have prepared benzoquino-bis-1,2,3-thiaselenazole QS, a heterocyclic selenium-based zwitterion with a small gap (λmax = 729 nm) between its highest occupied and lowest unoccupied molecular orbitals. In the solid state, QS exists in two crystalline phases and one nanocrystalline phase. The structures of the crystalline phases (space groups R3 c and P21/ c) have been determined by high-resolution powder X-ray diffraction methods at ambient and elevated pressures (0-15 GPa), and their crystal packing patterns have been compared with that of the related all-sulfur zwitterion benzoquino-bis-1,2,3-dithiazole QT (space group Cmc21). Structural differences between the S- and Se-based materials are interpreted in terms of local intermolecular S/Se···N'/O' secondary bonding interactions, the strength of which varies with the nature of the chalcogen (S vs Se). While the perfectly two-dimensional "brick-wall" packing pattern associated with the Cmc21 phase of QT is not found for QS, all three phases of QS are nonetheless small band gap semiconductors, with σRT ranging from 10-5 S cm-1 for the P21/ c phase to 10-3 S cm-1 for the R3 c phase. The bandwidths of the valence and conduction bands increase with applied pressure, leading to an increase in conductivity and a decrease in thermal activation energy Eact. For the R3 c phase, band gap closure to yield an organic molecular metal with a σRT of ∼102 S cm-1 occurs at 6 GPa. Band gaps estimated from density functional theory band structure calculations on the ambient- and high-pressure crystal structures of QT and QS correlate well with those obtained experimentally.

Vibrational spectroscopy of hydrogens in diamond: a quantum mechanical treatment.

The electronic and vibrational features of the VHn (n = 1 to 4) family of defects in diamond (hydrogen atoms saturating the dangling bonds of the atoms surrounding a vacancy) are investigated at the quantum mechanical level by using the periodic supercell approach, an all electron Gaussian type basis set, hybrid functionals, and the Crystal code. Most of the results have been collected for supercells containing 64 atoms; however, in order to explore the effect of the defect concentration on both the IR and Raman spectra, supercells containing 216, 512 and 1000 atoms have also been considered in the VH4 case. For each system, all the possible spin states are considered; their relative stability, band structure, charge and spin density distributions are thoroughly described. All the investigated systems present specific IR and Raman spectra, with vibrational spectroscopic features that can in principle be used as fingerprints for their characterization. This is particularly true for the C-H stretching, that ranges between 2500 and 4400 cm-1. The stretching modes are strongly affected by anharmonicity that has been evaluated in this work; it turns out to be extremely sensitive to the H load and spin state of the system, and ranges from -335 cm-1 for VH1 to +85 cm-1 for VH4. All of the investigated defects have very low C-H stretching IR intensity, so that they essentially appear as silent, the exception being VH1. The situation is different for the Raman spectra: the stretching modes of all defects do have similar large intensity; unfortunately here it is the experimental evidence that is lacking.

Characterization of the B-Center Defect in Diamond through the Vibrational Spectrum: A Quantum-Mechanical Approach.

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-1, 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.

Unveiling the Role of Spin Currents on the Giant Rashba Splitting in Single-Layer WSe2.

The Rashba spin splitting in uniaxial, inversion-asymmetric materials has attracted considerable interest for spintronic applications. The most widely used theoretical framework to model such states is Kohn-Sham density functional theory (DFT) in combination with standard (semi)local exchange-correlation density functional approximations (DFAs). However, in the presence of spin-orbit coupling, DFT misses contributions due to modification of the many-body interaction by spin currents J⃗. Inclusion of the latter effects requires a spin current DFT (SCDFT) formulation, which is seldom considered. We investigate the giant Rashba splitting in single-layer WSe2, and we quantify the effect of including spin currents in DFAs of the SCDFT. Crucially, we show that SCDFT allows fully capturing the giant Rashba band splitting in single-layer WSe2, otherwise previously systematically underestimated by standard (semi)local DFAs within the DFT framework. We find the inclusion of J⃗ on the DFA increases the Rashba splitting by about 20%.

Perturbation Theory Treatment of Spin-Orbit Coupling. III: Coupled Perturbed Method for Solids.

A previously proposed noncanonical coupled-perturbed Kohn-Sham density functional theory (KS-DFT)/Hartree-Fock (HF) treatment for spin-orbit coupling is here generalized to infinite periodic systems. The scalar-relativistic periodic KS-DFT/HF solution, obtained with a relativistic effective core potential, is taken as the zeroth-order approximation. Explicit expressions are given for the total energy through third-order, which satisfy the 2N + 1 rule (i.e., requiring only the first-order perturbed wave function for determining the energy through third-order). Expressions for additional second-order corrections to the perturbed wave function (as well as related one-electron properties) are worked out at the uncoupled-perturbed level of theory. The approach is implemented in the Crystal program and validated with calculations of the total energy, electronic band structure, and density variables of spin-current DFT on the tungsten dichalcogenide hexagonal bilayer series (i.e., WSe2, WTe2, WPo2, WLv2), including 6p and 7p elements as a stress test. The computed properties through second- or third-order match well with those from reference two-component self-consistent field (2c-SCF) calculations. For total energies, E(3) was found to consistently improve the agreement against the 2c-SCF reference values. For electronic band structures, visible differences w.r.t. 2c-SCF remained through second-order in only the single-most difficult case of WLv2. As for density variables of spin-current DFT, the perturbed electron density, being vanishing in first-order, is the most challenging for the perturbation theory approach. The visible differences in the electron densities are, however, largest close to the core region of atoms and smaller in the valence region. Perturbed spin-current densities, on the other hand, are well reproduced in all tested cases.

CRYSTAL23: A Program for Computational Solid State Physics and Chemistry.

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.

Perturbation Theory Treatment of Spin-Orbit Coupling II: A Coupled Perturbed Kohn-Sham Method.

A noncanonical coupled perturbed Kohn-Sham density functional theory (KS-DFT)/Hartree-Fock (HF) treatment of spin-orbit coupling (SOC) is provided. We take the scalar-relativistic KS-DFT/HF solution, obtained with a relativistic effective core potential, as the zeroth-order approximation. Explicit expressions are given for the total energy through the 4th order, which satisfy the 2n + 1 rule. Second-order expressions are provided for orbital energies and density variables of spin-current DFT. Test calculations are carried out on the halogen homonuclear diatomic and hydride molecules, including 6p and 7p elements, as well as open-shell negative ions. The computed properties through second or third order match well with those from reference two-component self-consistent field calculations for total and orbital energies as well as spin-current densities. In only one case (At2-) did a significant deviation occur for the remaining density variables. Our coupled perturbation theory approach provides an efficient way of adding the effect of SOC to a scalar-relativistic single-reference KS-DFT/HF treatment, in particular because it does not require diagonalization in the two-component spinor basis, leading to saving factors on the number of required floating-point operations that may exceed one order of magnitude.

Perturbation Theory Treatment of Spin-Orbit Coupling, Part I: Double Perturbation Theory Based on a Single-Reference Initial Approximation.

We develop a perturbation theory for solving the many-body Dirac equation within a given relativistic effective-core potential approximation. Starting from a scalar-relativistic unrestricted Hartree-Fock (SR UHF) solution, we carry out a double perturbation expansion in terms of spin-orbit coupling (SOC) and the electron fluctuation potential. Computationally convenient energy expressions are derived through fourth order in SOC, second order in the electron fluctuation potential, and a total of third order in the coupling between the two. Illustrative calculations on the halogen series of neutral and singly positive diatomic molecules show that the perturbation expansion is well-converged by taking into account only the leading (nonvanishing) term at each order of the electron fluctuation potential. Our perturbation theory approach provides a computationally attractive alternative to a two-component self-consistent field treatment of SOC. In addition, it includes coupling with the fluctuation potential through third order and can be extended (in principle) to multireference calculations, when necessary for both closed- and open-shell cases, using quasi-degenerate perturbation theory.

Charge Density Analysis of Actinide Compounds from the Quantum Theory of Atoms in Molecules and Crystals.

The nature of chemical bonding in actinide compounds (molecular complexes and materials) remains elusive in many respects. A thorough analysis of their electron charge distribution can prove decisive in elucidating bonding trends and oxidation states along the series. However, the accurate determination and robust analysis of the charge density of actinide compounds pose several challenges from both experimental and theoretical perspectives. Significant advances have recently been made on the experimental reconstruction and topological analysis of the charge density of actinide materials [Gianopoulos et al. IUCrJ, 2019, 6, 895]. Here, we discuss complementary advances on the theoretical side, which allow for the accurate determination of the charge density of actinide materials from quantum-mechanical simulations in the bulk. In particular, the extension of the Topond software implementing Bader's quantum theory of atoms in molecules and crystals (QTAIMAC) to f- and g-type basis functions is introduced, which allows for an effective study of lanthanides and actinides in the bulk and in vacuo, on the same grounds. Chemical bonding of the tetraphenyl phosphate uranium hexafluoride cocrystal [PPh4+][UF6-] is investigated, whose experimental charge density is available for comparison. Crystal packing effects on the charge density and chemical bonding are quantified and discussed. The methodology presented here allows reproducing all subtle features of the topology of the Laplacian of the experimental charge density. Such a remarkable qualitative and quantitative agreement represents a strong mutual validation of both approaches-experimental and computational-for charge density analysis of actinide compounds.

Fundamental Role of Fock Exchange in Relativistic Density Functional Theory.

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.