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The orbital ordering (OO) resulting from the partial occupancy of the t 2 g d subshell of the transition metals in KBF 3 (B = Sc, Ti, Ffe, Co) perovskites, and the many possible patterns arising from the coupling between the B sites, have been investigated at the quantum mechanical level ( all electron Gaussian type basis set, B3LYP hybrid functional) in a 40 atoms supercell. The numerous patterns are distributed into 162 classes of equivalent configurations. For each fluoroperovskite, one representative per class has been calculated. The four compounds behave similarly: an identical dependence of the energy and volume (or cell parameters) on the OO pattern; the spanned energy interval is small (1 to 2 mE h per formula unit), suggesting that most of the configurations are occupied at room and even at low temperature. A linear model, taking into account the relative orbital order in contiguous sites, reproduces the energy order in the full set for each compound, suggesting that it could be used for studying OO in larger supercells., (© 2024 Wiley Periodicals LLC.)

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., (© 2023 Wiley Periodicals LLC.)

The ferromagnetic and antiferromagnetic wave functions of the KMnF 3 perovskite have been evaluated quantum-mechanically by using an all electron approach and, for comparison, pseudopotentials on the transition metal and the fluorine ions. It is shown that the different number of α and β electrons in the d shell of Mn perturbs the inner shells, with shifts between the α and β eigenvalues that can be as large as 6 eV for the 3s level, and is far from negligible also for the 2s and 2p states. The valence electrons of F are polarized by the majority spin electrons of Mn, and in turn, spin polarize their 1s electrons. When a pseudopotential is used, such a spin polarization of the core functions of Mn and F can obviously not take place. The importance of such a spin polarization can be appreciated by comparing (i) the spin density at the Mn and F nuclear position, and then the Fermi contact constant, a crucial quantity for the hyperfine coupling, and (ii) the ferromagnetic-antiferromagnetic energy difference, when obtained with an all electron or a pseudopotential scheme, and exploring how the latter varies with pressure. This difference is as large as 50% of the all electron datum, and is mainly due to the rigid treatment of the F ion core. The effect of five different functionals on the core spin polarization is documented., (© 2022 Wiley Periodicals LLC.)

When different proposals exist (or can reasonably be formulated) for the size of the unit cell (in terms of number of atoms) and space group of crystalline compounds, a strategy for exploring with simulation methods the various cases and for investigating their relative stability must be defined. The optimization schemes of periodic quantum mechanical codes work in fact at fixed space group and number of atoms per unit cell, so that only the fractional coordinates of the atoms and the lattice parameters are optimized. A strategy is here presented, based on four standard tools, used synergistically and in sequence: (1) the optimization of inner coordinates and unit cell parameters; (2) the calculation of the vibrational frequencies not only at Γ , but also at a set of k → points (in the example presented here they are eight, generated by a shrinking factor 2), looking for possible negative wavenumbers. The latter correspond to maxima, rather than minima, along the coordinate described by the corresponding normal mode; (3) the exploration of the total energy along the mode with negative wavenumber, looking for the minimum of the curve; (4) the identification of the new space group corresponding to the reduced symmetry resulting from the previous step. The strategy is illustrated with reference to the KMnF 3 perovskite compound, for which many space groups are proposed in the literature, ranging from cubic Pm 3 ¯ m to tetragonal P 4 m bm or I 4 m cm and orthorhombic (Pnma and Cmcm) down to monoclinic (P2 1 /m). The corresponding primitive cells contain 5, 10, and 20 atoms in the various cases, and the point symmetry reduces from 48 to 4 operators. In nature, the KMnF 3 phase transitions also include the magnetic phases. For simplicity, here we limit the analysis to the ones that take place between ferromagnetic phases, as they are sufficiently rich for illustrating the proposed strategy. As the total energy differences involved can be as small as, say, 10-50 μHartree, a high numerical accuracy at each one of the steps mentioned above is required. The present calculations, performed with the CRYSTAL code, by using an all electron basis set and the Hartree-Fock and B3LYP functionals, document such an accuracy. The energy difference between the tetragonal I 4 m cm and cubic Pm 3 ¯ m phases is 225 μHartree, with a volume reduction of 0.58 Å 3 ; the differences between the orthorhombic and tetragonal phases are an order of magnitude smaller, being 23 μHartree and 0.06 Å 3 for total energy and cell volume, respectively., (© 2021 Wiley Periodicals LLC.)

The Infrared (IR) and Raman spectra of various interstitial carbon defects in silicon are computed at the quantum mechanical level by using an all electron Gaussian type basis set, the hybrid B3LYP functional and the supercell approach, as implemented in the CRYSTAL code (Dovesi et al. J. Chem. Phys. 2020, 152, 204111). The list includes two 〈100〉 split interstitial I XY defects, namely I CC and I CSi , a couple of related defects that we indicate as I X I Y , the so called C i C s 0 in its A and B form, as well as SiC i Si and C s C i C s , in which the interstitial carbon atom is twofold coordinated. The second undergoes a large relaxation, and the final configuration is close to I CC C s . Geometries, relative stabilities, electronic, and vibrational properties are analysed. All these defects show characteristic features in their IR spectrum (above 730 cm - 1 ), whereas the Raman spectrum is dominated, in most of the cases, by the pristine silicon peak at 530 cm -1 , that hides the defect peaks., (© 2021 Wiley Periodicals LLC.)

The infrared (IR) and Raman spectra of eight substitutional carbon defects in silicon are computed at the quantum mechanical level by using a periodic supercell approach based on hybrid functionals, an all electron Gaussian type basis set and the CRYSTAL code. The single substitutional C s case and its combination with a vacancy (C s V and C s SiV) are considered first. The progressive saturation of the four bonds of a Si atom with C is then examined. The last set of defects consists of a chain of adjacent carbon atoms C s i , with i = 1-3. The simple substitutional case, C s , is the common first member of the three sets. All these defects show important, very characteristic features in their IR spectrum. One or two C related peaks dominate the spectra: at 596 cm -1 for C s (and C s SiV, the second neighbor vacancy is not shifting the C s peak), at 705 and 716 cm -1 for C s V, at 537 cm -1 for C s 2 and C s 3 (with additional peaks at 522, 655 and 689 for the latter only), at 607 and 624 cm -1 , 601 and 643 cm -1 , and 629 cm -1 for SiC s 2 , SiC s 3 , and SiC s 4 , respectively. Comparison with experiment allows to attribute many observed peaks to one of the C substitutional defects. Observed peaks above 720 cm -1 must be attributed to interstitial C or more complicated defects., (© 2020 Wiley Periodicals, Inc.)

Two alternative approaches for the quantum-mechanical calculation of the nuclear-relaxation term of elastic and piezoelectric tensors of crystalline materials are illustrated and their computational aspects discussed: (i) a numerical approach based on the geometry optimization of atomic positions at strained lattice configurations and (ii) a quasi-analytical approach based on the evaluation of the force- and displacement-response internal-strain tensors as combined with the interatomic force-constant matrix. The two schemes are compared both as regards their computational accuracy and performance. The latter approach, not being affected by the many numerical parameters and procedures of a typical quasi-Newton geometry optimizer, constitutes a more reliable and robust mean to the evaluation of such properties, at a reduced computational cost for most crystalline systems. © 2016 Wiley Periodicals, Inc., (© 2016 Wiley Periodicals, Inc.)

It is shown that ab initio simulation can be used as a powerful complementary tool in the interpretation of the experimental reflectance spectra R(ν) of crystalline compounds. Experimental frequencies and intensities are obtained from a best fit of R(ν) with a set of damped harmonic oscillators, whose number and initial position in frequency can dramatically influence the final results, as the parameters are strongly correlated. Computed ab initio values for frequencies and intensities are accurate enough to represent an excellent starting point for the best fit process. Moreover, at variance with respect to experiment, simulation permits to identify all the symmetry allowed modes, also when characterized by low intensity or when close to a very intense peak. Overall, simulation-aided analysis of experimental spectra prevents from classifying combination modes as fundamental modes and permits to discard artifacts due to superposition of bands, background, and noise. Finally, it allows to (almost) completely characterize the set of fundamental modes., (Copyright © 2013 Wiley Periodicals, Inc.)

A fully ab initio technique is discussed for the determination of dynamical X-ray structure factors (XSFs) of crystalline materials, which is based on a standard Debye-Waller (DW) harmonic lattice dynamical approach with all-electron atom-centered basis sets, periodic boundary conditions, and one-electron Hamiltonians. This technique requires an accurate description of the lattice dynamics and the electron charge distribution of the system. The main theoretical parameters involved and final accuracy of the technique are discussed with respect to the experimental determinations of the XSFs at 298 K of crystalline silicon. An overall agreement factor of 0.47% between the ab initio predicted values and the experimental determinations is found. The best theoretical determination of the anisotropic displacement parameter, of silicon is here 60.55 × 10(-4) Å(2), corresponding to a DW factor B = 0.4781 Å(2)., (Copyright © 2012 Wiley Periodicals, Inc.)

Fully ab initio treatment of complex solid systems needs computational software which is able to efficiently take advantage of the growing power of high performance computing (HPC) architectures. Recent improvements in CRYSTAL, a periodic ab initio code that uses a Gaussian basis set, allows treatment of very large unit cells for crystalline systems on HPC architectures with high parallel efficiency in terms of running time and memory requirements. The latter is a crucial point, due to the trend toward architectures relying on a very high number of cores with associated relatively low memory availability. An exhaustive performance analysis shows that density functional calculations, based on a hybrid functional, of low-symmetry systems containing up to 100,000 atomic orbitals and 8000 atoms are feasible on the most advanced HPC architectures available to European researchers today, using thousands of processors., (Copyright © 2012 Wiley Periodicals, Inc.)

Representative helicoidal conformations of polyglycine infinite chains have been investigated by using periodic boundary conditions, the B3LYP hybrid functional, and large basis sets, by means of the CRYSTAL code. The exploitation of the helix roto-translational symmetry permits to optimize at a relatively low cost the structure of systems whose unit cell contains more than 300 atoms, much larger than the one investigated till now. In the present calculations, the helix symmetry is exploited at three levels. First, for the automatic generation of the structure. Second, for the calculation of the one- and two-electron integrals that enter into the Fock matrix definition. Only the irreducible wedge of the Fock matrix is computed. Finally, for the diagonalization of the Fock matrix, where each irreducible representation is separately treated. The efficiency and accuracy of the computational scheme are documented, by considering cells containing up to 47 glycine residues. Results are compared with previous calculations and available experimental data. The role of hydrogen bonding in stabilizing polyglycine conformers is also addressed., (Copyright 2010 Wiley Periodicals, Inc.)

Nanotubes can be characterized by a very high point symmetry, comparable or even larger than the one of the most symmetric crystalline systems (cubic, 48 point symmetry operators). For example, N = 2n rototranslation symmetry operators connect the atoms of the (n,0) nanotubes. This symmetry is fully exploited in the CRYSTAL code. As a result, ab initio quantum mechanical large basis set calculations of carbon nanotubes containing more than 150 atoms in the unit cell become very cheap, because the irreducible part of the unit cell reduces to two atoms only. The nanotube symmetry is exploited at three levels in the present implementation. First, for the automatic generation of the nanotube structure (and then of the input file for the SCF calculation) starting from a two-dimensional structure (in the specific case, graphene). Second, the nanotube symmetry is used for the calculation of the mono- and bi-electronic integrals that enter into the Fock (Kohn-Sham) matrix definition. Only the irreducible wedge of the Fock matrix is computed, with a saving factor close to N. Finally, the symmetry is exploited for the diagonalization, where each irreducible representation is separately treated. When M atomic orbitals per carbon atom are used, the diagonalization computing time is close to Nt, where t is the time required for the diagonalization of each 2M x 2M matrix. The efficiency and accuracy of the computational scheme is documented., ((c) 2009 Wiley Periodicals, Inc.)

The Coupled Perturbed Hartree-Fock (CPHF) scheme has been implemented in the CRYSTAL06 program, that uses a gaussian type basis set, for systems periodic in 1D (polymers), 2D (slabs), 3D (crystals) and, as a limiting case, 0D (molecules), which enables comparison with molecular codes. CPHF is applied to the calculation of the polarizability alpha of LiF in different aggregation states: finite and infinite chains, slabs, and cubic crystal. Correctness of the computational scheme for the various dimensionalities and its numerical efficiency are confirmed by the correct trend of alpha: alpha for a finite linear chain containing N LiF units with large N tends to the value for the infinite chain, N parallel chains give the slab value when N is sufficiently large, and N superimposed slabs tend to the bulk value. CPHF results compare well with those obtained with a saw-tooth potential approach, previously implemented in CRYSTAL. High numerical accuracy can easily be achieved at relatively low cost, with the same kind of dependence on the computational parameters as for the SCF cycle. Overall, the cost of one component of the dielectric tensor is roughly the same as for the SCF cycle, and it is dominated by the calculation of two-electron four-center integrals., ((c) 2008 Wiley Periodicals, Inc.)

The central-zone vibrational spectrum of alpha-quartz (SiO2) is calculated by building the Hessian matrix numerically from the analytical gradients of the energy with respect to the atomic coordinates. The nonanalytical part is obtained with a finite field supercell approach for the high-frequency dielectric constant and a Wannier function scheme for the evaluation of Born charges. The results obtained with four different Hamiltonians, namely Hartree-Fock, DFT in its local (LDA) and nonlocal gradient corrected (PBE) approximation, and hybrid B3LYP, are discussed, showing that B3LYP performs far better than LDA and PBE, which in turn provide better results than HF, as the mean absolute difference from experimental frequencies is 6, 18, 21, and 44 cm(-1), respectively, when a split valence basis set containing two sets of polarization functions is used. For the LDA results, comparison is possible with previous calculations based on the Density Functional Perturbation Theory and usage of a plane-wave basis set. The effects associated with the use of basis sets of increasing size are also investigated. It turns out that a split valence plus a single set of d polarization functions provides frequencies that differ from the ones obtained with a double set of d functions and a set of f functions on all atoms by on average less than 5 cm(-1)., (Copyright 2004 Wiley Periodicals, Inc.)

The finite field approach has been implemented in the periodic ab initio CRYSTAL program and been used for calculating the dielectric constants of crystalline LiF and MgO (FCC structure) and BeO (wurtzite structure). To maintain the periodicity along the applied field direction, a "sawtooth" potential is used in conjunction with a supercell scheme. Supercells four to five times longer than the primitive cell in the direction of the applied field provide well-converged results. The influence of the computational parameters is discussed. An alternative scheme has also been implemented, for inner check, that consists of applying a static electric field to a slab of increasing thickness in the direction orthogonal to the surface; the dielectric response at the center of the slab is shown to converge rapidly to the bulk value evaluated with the sawtooth field. The method is accurate and permits the determination of nonlinear corrections to the dielectric constant. When used in conjunction with the local density approximation (LDA) scheme, it provides for the dielectric constant of the three above-mentioned compounds values close to those recently obtained with a time-dependent density functional theory approach., (Copyright 2003 Wiley Periodicals, Inc.)