The latest release of the Crystal program for solid-state quantum-mechanical ab initio simulations is presented. The program adopts atom-centered Gaussian-type functions as a basis set, which makes it possible to perform all-electron as well as pseudopotential calculations. Systems of any periodicity can be treated at the same level of accuracy (from 0D molecules, clusters and nanocrystals, to 1D polymers, helices, nanorods, and nanotubes, to 2D monolayers and slab models for surfaces, to actual 3D bulk crystals), without any artificial repetition along nonperiodic directions for 0–2D systems. Density functional theory calculations can be performed with a variety of functionals belonging to several classes: local-density (LDA), generalized-gradient (GGA), meta-GGA, global hybrid, range-separated hybrid, and self-consistent system-specific hybrid. In particular, hybrid functionals can be used at a modest computational cost, comparable to that of pure LDA and GGA formulations, because of the efficient implementation of exact nonlocal Fock exchange. Both translational and point-symmetry features are fully exploited at all steps of the calculation, thus drastically reducing the corresponding computational cost. The various properties computed encompass electronic structure (including magnetic spin-polarized open-shell systems, electron density analysis), geometry (including full or constrained optimization, transition-state search), vibrational properties (frequencies, infrared and Raman intensities, phonon density of states), thermal properties (quasi-harmonic approximation), linear and nonlinear optical properties (static and dynamic [hyper]polarizabilities), strain properties (elasticity, piezoelectricity, photoelasticity), electron transport properties (Boltzmann, transport across nanojunctions), as well as X-ray and inelastic neutron spectra. The program is distributed in serial, parallel, and massively parallel versions. In this paper, the original developments that have been devised and implemented in the last 4 years (since the distribution of the previous public version, Crystal14, occurred in December 2013) are described.

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Quantum-mechanical condensed matter simulations with CRYSTAL

Consistent basis sets of triple-zeta valence with polarization quality for main group elements and transition metals from row one to three have been derived for periodic quantum-chemical solid-state calculations with the crystalline-orbital program CRYSTAL. They are based on the def2-TZVP basis sets developed for molecules by the Ahlrichs group. Orbital exponents and contraction coefficients have been modified and reoptimized, to provide robust and stable self-consistant field (SCF) convergence for a wide range of different compounds. We compare results on crystal structures, cohesive energies, and solid-state reaction enthalpies with the modified basis sets, denoted as pob-TZVP, with selected standard basis sets available from the CRYSTAL basis set database. The average deviation of calculated lattice parameters obtained with a selected density functional, the hybrid method PW1PW, from experimental reference is smaller with pob-TZVP than with standard basis sets, in particular for metallic systems. The effects of basis set expansion by diffuse and polarization functions were investigated for selected systems.

Consistent Gaussian basis sets of triple-zeta valence with polarization quality for solid-state calculations.

We present a detailed ab initio investigation of the stability, the structural, electronic, and magnetic properties of the (0001) surfaces of hematite $({\mathrm{Fe}}_{2}{\mathrm{O}}_{3})$ and chromia or eskolaite $({\mathrm{Cr}}_{2}{\mathrm{O}}_{3})$. Strong electron correlation effects not included in a density-functional description are described by a Hubbard-type on-site Coulomb repulsion (the $\mathrm{DFT}+U$ approach). For bulk chromia we find, complementing our recent work on hematite [Rollmann et al., Phys. Rev. B 69, 165107 (2004)] that the inclusion of correlation effects leads to an improved description of the structural, electronic, and magnetic properties. In particular, the increased exchange splitting of the $d$ band changes the character of the insulating gap from a pure $d\text{\ensuremath{-}}d$ Mott-Hubbard type to intermediate between $d\text{\ensuremath{-}}d$ and charge-transfer insulator. For both oxides, the strong correlation effects have a dramatic influence on the surface stability: oxygen-terminated surfaces are strongly disfavored because of the increased energetic cost of stabilizing a higher oxidation state of the transition metal close to the surface. The stability of metal-terminated surfaces even under oxidizing conditions agrees with the most recent STM and LEED data. For ${\mathrm{Cr}}_{2}{\mathrm{O}}_{3}(0001)$ where detailed experimental information on the surface structure is available, quantitative agreement of the calculated surface relaxations is achieved. Detailed results on the surface electronic structure (valence-band spectra and core-level shifts) and the surface magnetic properties are presented.

Ab initio study of the (0001) surfaces of hematite and chromia: Influence of strong electronic correlations

https://www.crystal.unito.it/supplement/ref_geom/ref_geom.pdf

Hartree–Fock geometry optimisation of periodic systems with the Crystal code

Many recent articles have suggested that density functional theory (DFT) with the generalized-gradient approximation does not provide the correct electronic structure for the oxides of titanium. The current opinion is that a Hubbard U correction improves the DFT results. There is no generally accepted method for deciding what the value of the U parameter should be, and we propose that, if one intends to study catalysis, U should be chosen to fit the reaction energy for the oxidation of Ti2O3 to TiO2. We show that the value of U derived in this manner provides additional improvements in the description of the electronic structure.

Choice of U for DFT+U Calculations for Titanium Oxides

Electronic, magnetic and crystal structure of Cr2O3 by theoretical methods

The ground-state electronic and magnetic properties of rhombohedral ${\mathrm{Ti}}_{2}$${\mathrm{O}}_{3}$ and ${\mathrm{V}}_{2}$${\mathrm{O}}_{3}$ have been investigated by ab initio all-electron periodic Hartree-Fock calculations. Both unrestricted open-shell and restricted closed-shell methods have been employed, with basis sets of atomic orbitals represented by contracted Gaussian-type functions. The ${\mathrm{t}}_{2\mathrm{g}}$ degeneracy of d electrons is removed by the rhombohedral field, giving rise to orbital ordering between ${\mathrm{a}}_{1\mathrm{g}}$ and ${\mathrm{e}}_{\mathrm{g}}^{\mathrm{\ensuremath{\pi}}}$ levels. The self-consistent-field variational solutions are spin-polarized insulating states with single (${\mathrm{a}}_{1\mathrm{g}}$) and double (${\mathrm{e}}_{\mathrm{g}}^{\mathrm{\ensuremath{\pi}}}$) electron occupations for ${\mathrm{Ti}}_{2}$${\mathrm{O}}_{3}$ and ${\mathrm{V}}_{2}$${\mathrm{O}}_{3}$, respectively. Conducting or semiconducting states, with different relative energies of ${\mathrm{a}}_{1\mathrm{g}}$ and ${\mathrm{e}}_{\mathrm{g}}^{\mathrm{\ensuremath{\pi}}}$ bands, have also been obtained by changing the c/a ratio of the hexagonal unit cell. The charge transfer into ${\mathrm{e}}_{\mathrm{g}}^{\mathrm{\ensuremath{\sigma}}}$ levels is discussed and compared to the ${\mathrm{Cr}}_{2}$${\mathrm{O}}_{3}$ and ${\mathrm{Fe}}_{2}$${\mathrm{O}}_{3}$ behavior.

G. Sandrone

Papers

Electronic, magnetic and crystal structure of Cr2O3 by theoretical methods

Periodic unrestricted Hartree-Fock study of corundumlike Ti 2 O 3 and V 2 O 3