Jahn-Teller distortion, octahedra rotations and orbital ordering in perovskites: KScF 3 as a model system.

The KScF 3 perovskite has been used as a model for investigating the relative importance of the Jahn-Teller (JT) lift of degeneracy, the ScF 6 octahedra rotation (OR), and the quadrupole-quadrupole interaction linked to different occupancy of the Sc t 2 g subshell in various sites of the unit cell (orbital ordering, OO). The group-subgroup sequence P m 3 ¯ m , P 4 m m m , P 4 m b m , and P n m a , supplemented by C m m m and I 4 m c m , has been explored by using an all electron Gaussian type basis set, hybrid functionals, and the CRYSTAL17 code. The JT lift of degeneracy provides a stabilization about 5 times larger than the sum of the OO and OR effects. The energy gained in the transition from P 4 m m m to P 4 m b m , consisting in a rotation of the octahedra around the c axis, is 1077 µ E h . From P 4 m b m to P n m a , additional rotations around the a and b axes are possible, and the d Sc electron can occupy a different t 2 g orbital, with a total energy reduction of 2318 µ E h . The rotation of the octahedra reduces the strength of superexchange: in going from P 4 m m m to P n m a the G-AFM stabilization with respect to FM shrinks by a factor 4.

Band gap, Jahn-Teller deformation, octahedra rotation in transition metal perovskites LaTiO 3 .

The LaTiO 3 perovskite (where Ti is in a d1 state) is investigated by using an all electron Gaussian basis and many functionals, ranging from pure GGA (PBE), to hybrids (full range, B3LYP and PBE0, and range separated, HSE06) to Hartree Fock. Recently, Varignon et al. (Phys. Rev. Res 1, 033131, 2019), showed that, when GGA+U or HSE06 are used, a metallic solution and fractional occupancy of the t 2 g subshell are obtained. Here, it is shown that when a full range hybrid functional is used, an integer occupancy is obtained, as suggested by the Jahn-Teller theorem. When the exact exchange percentage varies from 0 to 100, the system is insulating when it exceeds 20. By reducing progressively the symmetry from cubic down to orthorhombic, the relative importance of the Jahn-Teller deformation and of the rotation of the octahedra is explored.

Calculation of Anharmonic IR and Raman Intensities for Periodic Systems from DFT Calculations: Implementation and Validation.

An extension of the CRYSTAL program is presented allowing for calculations of anharmonic infrared (IR) intensities and Raman activities for periodic systems. This work is a follow-up of two papers devoted to the computation of anharmonic vibrational states of solids from DFT (density functional theory) calculations (Erba et al. J. Chem. Theory Comput. 2019, 15, 3755-3765 and Erba et al. J. Chem. Theory Comput. 2019, 15, 3766-3777). The approach presented here relies on the evaluation of integrals of the dipole moment and polarizability operators over anharmonic wave functions obtained from either VSCF (vibrational self-consistent field) or VCI (vibrational configuration interaction) calculations. With this extension, the program now allows for a more complete characterization of the vibrational spectroscopic features of solids within the density functional theory. In particular, it is able (i) to provide reliable positions and intensities for the most intense spectral features and (ii) to check whether a first overtone or a combination band has a nonvanishing IR intensity or Raman activity. Therefore, it becomes possible to assign the transition(s) corresponding to satellite peak(s) around a fundamental transition or the overtones or combination bands that may be as intense as their corresponding fundamental transitions through the strongest mode-mode couplings, as in so-called Fermi resonances. The present method is assessed on two molecular systems, H2O and H2CO, as well as on two solid state cases, boron hydrides BH4 and their deuterated species BD4 in a crystalline environment of alkali metals (M = Na, K). The solid state cases are particularly insightful as, in the B-H (or B-D) stretching region here considered, they exhibit many spectral features entirely due to anharmonic effects: two out of three in the IR spectrum and four out of six in the Raman spectrum. All IR and Raman active overtones and combination bands experimentally observed are correctly predicted with our approach. The effect of the adopted quantum-chemical model (DFT exchange-correlation functional/basis set) for the electronic structure calculations on the computed spectra is discussed and found to be significant, which suggests some special care is needed for the analysis of subtle spectral features.

An all-electron study of the low-lying excited states and optical constants of Al2O3 in the range 5-80 eV.

This paper reports calculated energies and electronic structures of O(2p), O(2s) and Al(2p) excited states in bulk [Formula: see text]-Al2O3, at the [Formula: see text] and [Formula: see text] surfaces and in the presence of O vacancy defects, obtained from all-electron HF, B3LYP, GGA and LDA calculations based on a recently described direct [Formula: see text]-SCF approach (Mackrodt et al 2018 J. Phys.: Condens. Matter 30 495901). The closely related frequency-dependent optical constants derived from B3LYP calculations within the CPHF/DF framework are also reported, where both sets of results are shown to compare favourably with the experimental spectra. The differences between the directly calculated excited state energies, which in [Formula: see text]-Al2O3 are equal to the leading excitation edges, based on the four functionals, are substantially less than the differences between the corresponding (ground state) band gaps, as reported previously for AFII NiO (Mackrodt et al 2018 J. Phys.: Condens. Matter 30 495901). For the B3LYP functional, these energies are 8.7 eV, 12.5 eV and 73.7 eV for the O(2p), O(2s) and Al(2p) excitations respectively. The O(2p) edge is predicted to be degenerate, with distinct excitations from O(2p) states that are parallel to and perpendicular to the c-axis, in agreement with the reported spectra (Tomiki et al 1993 J. Phys. Soc. Japan 62 573). Detailed analyses of the charge and spin distributions in the four bulk excited states indicate that these are essentially charge-transfer excitonic, with acceptor sites at the nearest neighbour positions. Despite the close proximity of the O([Formula: see text]) and O(2p[Formula: see text]) excited state energies, the charge and spin distributions are predicted to be quite different.

Large-Scale Condensed Matter DFT Simulations: Performance and Capabilities of the CRYSTAL Code.

Nowadays, the efficient exploitation of high-performance computing resources is crucial to extend the applicability of first-principles theoretical methods to the description of large, progressively more realistic molecular and condensed matter systems. This can be achieved only by devising effective parallelization strategies for the most time-consuming steps of a calculation, which requires some effort given the usual complexity of quantum-mechanical algorithms, particularly so if parallelization is to be extended to all properties and not just to the basic functionalities of the code. In this Article, the performance and capabilities of the massively parallel version of the Crystal17 package for first-principles calculations on solids are discussed. In particular, we present: (i) recent developments allowing for a further improvement of the code scalability (up to 32 768 cores); (ii) a quantitative analysis of the scaling and memory requirements of the code when running calculations with several thousands (up to about 14 000) of atoms per cell; (iii) a documentation of the high numerical size consistency of the code; and (iv) an overview of recent ab initio studies of several physical properties (structural, energetic, electronic, vibrational, spectroscopic, thermodynamic, elastic, piezoelectric, topological) of large systems investigated with the code.

The V + I defects in diamond: An ab initio investigation of the electronic structure, of the Raman and IR spectra, and of their possible recombination.

The double defect in diamond, vacancy (V) plus 〈100〉 self-split-interstitial (V+I), is investigated at the ab initio quantum mechanical level, by considering the vicinal case VI1 (V is one of the first neighbors of one of the two C atoms constituting the I defect) and the two possible "second neighbors" cases, VI2D, VI2S, in which a carbon atom is a first neighbor of both V and I. The case in which the two defects are at a larger distance is simulated by considering the two isolated defects separately (VI∞). A 6-21G local Gaussian-type basis set and the B3LYP hybrid functional are used for most of the calculations; richer basis sets and other functionals (a global hybrid as PBE0, a range-separated hybrid as HSE06, LDA, PBE, and Hartree-Fock) have also been used for comparison. With this computational approach we evaluate the energy difference between the various spin states, the location of the corresponding bands in the energy gap of pristine diamond, as well as the defect formation energy of the four defects. The path for the recombination of V and I is explored for the vicinal case, by using the distinguished reaction coordinate strategy. A barrier as high as 0.75 eV is found with B3LYP between VI1 and the perfect diamond recombined structure; when other hybrids are used, as PBE0 or HSE06, the barrier increases up to 1.01 eV (pure density functional theory produces lower barriers: 0.62 and 0.67 for PBE and LDA, respectively). Such a barrier is lower than the one estimated in a very indirect way through experimental data, ranging from 1.3 to 1.7 eV. It confirms however the evidence of the extremely low recombination rate also at high temperature. The Raman (and IR) spectra of the various defects are generated, which permit one to unambiguously attribute to these defects (thanks also to the graphical animation of the modes) many of the peaks observed in damaged diamond above the dominant peak of perfect bulk. For the residual non-attributed peaks, more complicated aggregations of defects should be explored.

Piezo-optic tensor of crystals from quantum-mechanical calculations.

An automated computational strategy is devised for the ab initio determination of the full fourth-rank piezo-optic tensor of crystals belonging to any space group of symmetry. Elastic stiffness and compliance constants are obtained as numerical first derivatives of analytical energy gradients with respect to the strain and photo-elastic constants as numerical derivatives of analytical dielectric tensor components, which are in turn computed through a Coupled-Perturbed-Hartree-Fock/Kohn-Sham approach, with respect to the strain. Both point and translation symmetries are exploited at all steps of the calculation, within the framework of periodic boundary conditions. The scheme is applied to the determination of the full set of ten symmetry-independent piezo-optic constants of calcium tungstate CaWO4, which have recently been experimentally reconstructed. Present calculations unambiguously determine the absolute sign (positive) of the π61 constant, confirm the reliability of 6 out of 10 experimentally determined constants and provide new, more accurate values for the remaining 4 constants.

Anharmonic thermal oscillations of the electron momentum distribution in lithium fluoride.

Anharmonic thermal effects on the electron momentum distribution of a lithium fluoride single crystal are experimentally measured through high-resolution Compton scattering and theoretically modeled with ab initio simulations, beyond the harmonic approximation to the lattice potential, explicitly accounting for thermal expansion. Directional Compton profiles are measured at two different temperatures, 10 and 300 K, with a high momentum space resolution (0.10 a.u. in full width at half maximum), using synchrotron radiation. The effect of temperature on measured directional Compton profiles is clearly revealed by oscillations extending almost up to |p|=4 a.u., which perfectly match those predicted from quantum-mechanical simulations. The wave-function-based Hartree-Fock method and three classes of the Kohn-Sham density functional theory (local-density, generalized-gradient, and hybrid approximations) are adopted. The lattice thermal expansion, as described with the quasiharmonic approach, is found to entirely account for the effect of temperature on the electron momentum density within the experimental accuracy.

Structural and elastic anisotropy of crystals at high pressures and temperatures from quantum mechanical methods: The case of Mg2SiO4 forsterite.

We report accurate ab initio theoretical predictions of the elastic, seismic, and structural anisotropy of the orthorhombic Mg2SiO4 forsterite crystal at high pressures (up to 20 GPa) and temperatures (up to its melting point, 2163 K), which constitute earth's upper mantle conditions. Single-crystal elastic stiffness constants are evaluated up to 20 GPa and their first- and second-order pressure derivatives reported. Christoffel's equation is solved at several pressures: directional seismic wave velocities and related properties (azimuthal and polarization seismic anisotropies) discussed. Thermal structural and average elastic properties, as computed within the quasi-harmonic approximation of the lattice potential, are predicted at high pressures and temperatures: directional thermal expansion coefficients, first- and second-order pressure derivatives of the isothermal bulk modulus, and P-V-T equation-of-state. The effect on computed properties of five different functionals, belonging to three different classes of approximations, of the density functional theory is explicitly investigated.

On how differently the quasi-harmonic approximation works for two isostructural crystals: thermal properties of periclase and lime.

Harmonic and quasi-harmonic thermal properties of two isostructural simple oxides (periclase, MgO, and lime, CaO) are computed with ab initio periodic simulations based on the density-functional-theory (DFT). The more polarizable character of calcium with respect to magnesium cations is found to dramatically affect the validity domain of the quasi-harmonic approximation that, for thermal structural properties (such as temperature dependence of volume, V(T), bulk modulus, K(T), and thermal expansion coefficient, α(T)), reduces from [0 K-1000 K] for MgO to just [0 K-100 K] for CaO. On the contrary, thermodynamic properties (such as entropy, S(T), and constant-volume specific heat, CV(T)) are described reliably at least up to 2000 K and quasi-harmonic constant-pressure specific heat, CP(T), up to about 1000 K in both cases. The effect of the adopted approximation to the exchange-correlation functional of the DFT is here explicitly investigated by considering five different expressions of three different classes (local-density approximation, generalized-gradient approximation, and hybrids). Computed harmonic thermodynamic properties are found to be almost independent of the adopted functional, whereas quasi-harmonic structural properties are more affected by the choice of the functional, with differences that increase as the system becomes softer.

In silico infrared and Raman spectroscopy under pressure: the case of CaSnO3 perovskite.

The CaSnO3 perovskite is investigated under geochemical pressure, up to 25 GPa, by means of periodic ab initio calculations performed at B3LYP level with local Gaussian-type orbital basis sets. Structural, elastic, and spectroscopic (phonon wave-numbers, infrared and Raman intensities) properties are fully characterized and discussed. The evolution of the Raman spectrum of CaSnO3 under pressure is reported to remarkably agree with a recent experimental determination [J. Kung, Y. J. Lin, and C. M. Lin, J. Chem. Phys. 135, 224507 (2011)] as regards both wave-number shifts and intensity changes. All phonon modes are symmetry-labeled and bands assigned. The single-crystal total spectrum is symmetry-decomposed into the six directional spectra related to the components of the polarizability tensor. The infrared spectrum at increasing pressure is reported for the first time and its main features discussed. All calculations are performed using the Crystal14 program, taking advantage of the new implementation of analytical infrared and Raman intensities for crystalline materials.

Pressure effect on elastic anisotropy of crystals from ab initio simulations: the case of silicate garnets.

A general methodology has been devised and implemented into the solid-state ab initio quantum-mechanical Crystal program for studying the evolution under geophysical pressure of the elastic anisotropy of crystalline materials. This scheme, which fully exploits both translational and point symmetry of the crystal, is developed within the formal frame of one-electron Hamiltonians and atom-centered basis functions. Six silicate garnet end-members, among the most important rock-forming minerals of the Earth's mantle, are considered, whose elastic anisotropy is fully characterized under high hydrostatic compressions, up to 60 GPa. The pressure dependence of azimuthal anisotropy and shear-wave birefringence of seismic wave velocities for these minerals are accurately simulated and compared with available single-crystal measurements.

High pressure elastic properties of minerals from ab initio simulations: the case of pyrope, grossular and andradite silicate garnets.

A computational strategy is devised for the accurate ab initio simulation of elastic properties of crystalline materials under pressure. The proposed scheme, based on the evaluation of the analytical stress tensor and on the automated computation of pressure-dependent elastic stiffness constants, is implemented in the CRYSTAL solid state quantum-chemical program. Elastic constants and related properties (bulk, shear and Young moduli, directional seismic wave velocities, elastic anisotropy index, Poisson's ratio, etc.) can be computed for crystals of any space group of symmetry. We apply such a technique to the study of high-pressure elastic properties of three silicate garnet end-members (namely, pyrope, grossular, and andradite) which are of great geophysical interest, being among the most important rock-forming minerals. The reliability of this theoretical approach is proved by comparing with available experimental measurements. The description of high-pressure properties provided by several equations of state is also critically discussed.

Zinc oxide nanotubes: an ab initio investigation of their structural, vibrational, elastic, and dielectric properties.

Structural, vibrational, elastic, and dielectric properties of ZnO single-walled nanotubes are investigated theoretically. Calculations are carried out by using a Gaussian basis set and the B3LYP hybrid functional as implemented in the periodic ab initio CRYSTAL code. Nanotubes with increasing radius display asymptotic limits to the infinite monolayer. One soft phonon mode is recognized, whose vibration frequency is shown to be connected to the elastic constant C11 of the monolayer as the 1D → 2D transition is approached. The value of Young's elastic modulus of the nanotubes denotes a remarkable flexibility. Electronic and ionic contributions to the polarizability turn out to be comparable in magnitude. In particular, geometry relaxations at increasing radii show large influence on the transverse vibrational polarizability.

The vibration properties of the (n,0) boron nitride nanotubes from ab initio quantum chemical simulations.

The vibration spectrum of single-walled zigzag boron nitride (BN) nanotubes is simulated with an ab initio periodic quantum chemical method. The trend towards the hexagonal monolayer (h-BN) in the limit of large tube radius R is explored for a variety of properties related to the vibrational spectrum: vibration frequencies, infrared intensities, oscillator strengths, and vibration contributions to the polarizability tensor. The (n,0) family is investigated in the range from n = 6 (24 atoms in the unit cell and tube radius R = 2.5 Å) to n = 60 (240 atoms in the cell and R = 24.0 Å). Simulations are performed using the CRYSTAL program which fully exploits the rich symmetry of this class of one-dimensional periodic systems: 4n symmetry operators for the general (n,0) tube. Three sets of infrared active phonon bands are found in the spectrum. The first one lies in the 0-600 cm(-1) range and goes regularly to zero when R increases; the connection between these normal modes and the elastic and piezoelectric constants of h-BN is discussed. The second (600-800 cm(-1)) and third (1300-1600 cm(-1)) sets tend regularly, but with quite different speed, to the optical modes of the h-BN layer. The vibrational contribution of these modes to the two components (parallel and perpendicular) of the polarizability tensor is also discussed.

Performance of six functionals (LDA, PBE, PBESOL, B3LYP, PBE0, and WC1LYP) in the simulation of vibrational and dielectric properties of crystalline compounds. The case of forsterite Mg2SiO4.

The performance of six different density functionals (LDA, PBE, PBESOL, B3LYP, PBE0, and WC1LYP) in describing the infrared spectrum of forsterite, a crystalline periodic system with orthorhombic unit cell (28 atoms in the primitive cell, Pbmn space group), is investigated by using the periodic ab initio CRYSTAL09 code and an all-electron Gaussian-type basis set. The transverse optical (TO) branches of the 35 IR active modes are evaluated at the equilibrium geometry together with the oscillator strengths and the high-frequency dielectric tensor ϵ(∞) . These quantities are essential to compute the dielectric function ϵ(ν), and then the reflectance spectrum R(ν), which is compared with experiment. It turns out that hybrid functionals perform better than LDA and GGA, in general; that B3LYP overperforms WC1LYP and, in turn, PBE0; that PBESOL is better than PBE; that LDA is the worst performing functional among the six under study.

Periodic ab initio estimates of the dispersive interaction between molecular nitrogen and a monolayer of hexagonal BN.

The ab initio determination of the leading long-range term of pairwise additive dispersive interactions, based on the independent analysis of the response properties of the interacting objects, is here considered in the case where these are part of a periodic system. The interaction of a nitrogen molecule with a thin film of hexagonal BN has been chosen as a case study for identifying some of the problems involved, and for proposing techniques for their solution. In order to validate the results so obtained, the interaction energy between N(2) and a BN monolayer at different distances has been estimated following a totally different approach, namely by performing post-Hartree-Fock (MP2) supercell calculations using the Crystal+Cryscor suite of programs. The results obtained with the two approaches closely agree over a long range, while the limit of validity of the purely dispersive regime can be clearly assessed.

Chemical and magnetic ordering derived from ab initio simulations: the case of ß'-LiFeO2.

First-principles periodic calculations (with the B3LYP (Becke, three-parameter, Lee-Yang-Parr) hybrid functional, all-electron localized basis functions, and the CRYSTAL code) were coupled to a cluster expansion scheme in order to investigate the monoclinic ß' phase of LiFeO(2), where a partially disordered Fe-Li distribution is observed within a rocksalt-type superstructure. By least-energy optimizing a limited number of ordered configurations, and employing a two-body truncated cluster expansion, the values J(1) = -0.06(2) and J(2) = -0.125(8) eV were obtained for the excess interaction energies J(i) = J(LiFe, i) - (J(LiLi, i) + J(FeFe, i))/2 corresponding to the first and second coordination spheres, respectively; negligible values were computed for third and further coordinations. The ordering phase transformation α−>ß'−>γ was then addressed. Antiferromagnetic versus ferromagnetic ordering was taken into account too, and proved to lower the energy by -0.0577 eV/f.u. The corresponding cluster expansion coefficients J(i) = J(AFM, i) - J(FM, i) are J(1) = 0.007(2) and J(2) = -0.044(5) eV.

Quantum-mechanical ab initio simulation of the Raman and IR spectra of Fe3Al2Si3O12 almandine.

The IR and Raman spectra of the Fe(3)Al(2)Si(3)O(12) almandine garnet were simulated using the periodic CRYSTAL code with an all-electron Gaussian-type basis set and the hybrid B3LYP functional. The wavenumbers of the 25 Raman-active modes (3 of A(1g), 8 of E(g), and 14 of F(2g) symmetry) and 34 F(1u) IR-active modes (17 TO and 17 LO) were computed, as were the IR intensities. Calculated wavenumbers are in excellent agreement with the various sets of experimental results, with the mean absolute difference |Delta| being between 4 and 8 cm(-1). Graphical animation, available on the CRYSTAL Web site, provides a very comprehendible description of the movement of atoms and groups in each vibrational mode. The simulated reflectivity spectrum, obtained using a classical dispersion relation, is in excellent agreement with the measured one.

Ab initio simulation of the IR spectra of pyrope, grossular, and andradite.

IR spectra of pyrope Mg(3)Al(2)Si(3)O(12), grossular Ca(3)Al(2)Si(3)O(12) and andradite Ca(3)Fe(2)Si(3)O(12) garnets were simulated with the periodic ab initio CRYSTAL code by adopting an all-electron Gaussian-type basis set and the B3LYP Hamiltonian. Two sets of 17 F(1u) Transverse Optical (TO) and Longitudinal Optical (LO) frequencies were generated, together with their intensities. Because the generation of LO modes requires knowledge of the high frequency dielectric constant epsilon(infinity) and Born effective charges, they were preliminary evaluated by using a finite field saw-tooth model and well localized Wannier functions, respectively. As a by-product, the static dielectric constant epsilon(0) was also obtained. The agreement of the present calculated wavenumbers (i.e. peak positions) with the available experimental data is excellent, in that the mean absolute difference for the full set of data smaller than 8 cm(-1). Missing peaks in experimental spectra were found to correspond to modes with low calculated intensities. Correspondence between TO and LO modes was established on the basis of the overlap between the eigenvectors of the two sets and similarity of isotopic shifts; as result, the so called LO-TO splitting could be determined. Animation of the normal modes was employed to support the proposed pairing.