On the use of symmetry in configurational analysis for the simulation of disordered solids

The starting point for a quantum mechanical investigation of disordered systems usually implies calculations on a limited subset of configurations, generated by defining either the composition of interest or a set of compositions ranging from one end member to another, within an appropriate supercell of the primitive cell of the pure compound. The way in which symmetry can be used in the identification of symmetry independent configurations (SICs) is discussed here. First, Pólya's enumeration theory is adopted to determine the number of SICs, in the case of both varying and fixed composition, for colors numbering two or higher. Then, De Bruijn's generalization is presented, which allows analysis of the case where the colors are symmetry related, e.g. spin up and down in magnetic systems. In spite of their efficiency in counting SICs, neither Pólya's nor De Bruijn's theory helps in solving the difficult problem of identifying the complete list of SICs. Representative SICs are obtained by adopting an orderly generation approach, based on lexicographic ordering, which offers the advantage of avoiding the (computationally expensive) analysis and storage of all the possible configurations. When the number of colors increases, this strategy can be combined with the surjective resolution principle, which permits the efficient generation of SICs of a problem in |R| colors starting from the ones obtained for the (|R| − 1)-colors case. The whole scheme is documented by means of three examples: the abstract case of a square with C4v symmetry and the real cases of the garnet and olivine mineral families

Symmetry and random sampling of symmetry independent configurations for the simulation of disordered solids

A symmetry-adapted algorithm producing uniformly at random the set of symmetry independent configurations (SICs) in disordered crystalline systems or solid solutions is presented here. Starting from Pólya's formula, the role of the conjugacy classes of the symmetry group in uniform random sampling is shown. SICs can be obtained for all the possible compositions or for a chosen one, and symmetry constraints can be applied. The approach yields the multiplicity of the SICs and allows us to operate configurational statistics in the reduced space of the SICs. The present low-memory demanding implementation is briefly sketched. The probability of finding a given SIC or a subset of SICs is discussed as a function of the number of draws and their precise estimate is given. The method is illustrated by application to a binary series of carbonates and to the binary spinel solid solution Mg(Al,Fe)2O4

The Superexchange mechanism in crystalline compounds. The case of KMF3(M=Mn, Fe, Co, Ni) perovskites.

International audienceAbstract The ferromagnetic and antiferromagnetic wavefunctions of four KMF 3 (M= Mn, Fe, Co and Ni) perovskites have been obtained quantum-mechanically with the CRYSTAL code, by using the Hartree-Fock (HF) Hamiltonian and three flavours of DFT (PBE, B3LYP and PBE0) and an all-electron Gaussian type basis set. In the Fe and Co cases, with d6and d7occupation, the Jahn-Teller distortion of the cubic cell is as large as 0.12 Å. Various features of the superexchange interaction energies (SIE), namely additivity, dependence on the M-M distance, on the MFM̂ angle, and on the adopted functional, are explored. The contribution to SIE by the Coulomb, exchange and kinetic energy terms is analyzed. It is shown that, when using density functionals, SIE clearly correlates with the amount of exact (Hartree-Fock) exchange in the functional. The effect of SIE on the equilibrium geometry and volume of the unit cell is discussed, and it is shown that the key quantity is the spin polarization of the (closed shell) F ions along the M-F-M path. The effect of this magnetic pressure is evaluated quantitatively for the first time

The <i>d</i> Orbital Multi Pattern Occupancy in a Partially Filled d Shell: The KFeF₃ Perovskite as a Test Case

The occupancy of the d shell in KFeF3 is t2g4eg2, with five α and one β electrons. The Jahn–Teller lift of degeneracy in the t2g sub-shell produces a tetragonal relaxation of the unit cell (4.09 vs. 4.22 Å, B3LYP result) not observed experimentally. In order to understand the origin of this apparent contradiction, we explored, with a 2 × 2 × 2 supercell (40 atoms per cell), all possible local structures in which contiguous Fe atoms have a different occupancy of the t2g orbitals with the minority spin electron. A total of 6561 configurations (with occupancies from (8,0,0) to (3,2,2) of the 3 t2g orbitals of the 8 Fe atoms) have been explored, with energies in many cases lower (by up to 1550 μEh per 2 Fe atoms) than the one of the fully ordered case, both for the ferromagnetic and the anti-ferromagnetic solutions. The results confirm that the orientation of the β d electron of Fe influences the electrostatics (more efficient relative orientation of the Fe quadrupoles of the d shell) of the system, but not the magnetic interactions. Three hybrid functionals, B3LYP, PBE0, and HSE06, provide very similar results

Ab initio Hartree-Fock investigation of the structural, electronic, and magnetic properties of Mn 3 O 4

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Longitudinal and transverse hyperpolarizabilities of carbon nanotubes: a computational investigation through the coupled-perturbed Hartree–Fock/Kohn–Sham scheme

International audienceStatic electronic polarizability α and second hyperpolarizability γ of semiconduting and conducting carbon nanotubes with radius up to 7.5 ˚ A are evaluated using the coupled-perturbed Hartree-Fock/Kohn-Sham scheme, as implemented in the periodic CRYS-TAL14 code, and a split-valence basis set. Two density functionals, namely LDA (pure local) and B3LYP (hybrid), and the Hartree-Fock hamiltonian are compared. A few PBE (gradient corrected) density functional data are also produced for comparison with previous calculations. Convergence of both longitudinal (L) and transverse (T) components is documented. It is shown how the second hyperpolarizability depends critically on the computational conditions, the more so the larger the radius of the nanotube (and thus the smaller the energy gap). The longitudinal component is sensibly affected by the truncation of the exact exchange series (HF and B3LYP), which must include electron-electron interactions at a distance up to 100 A in order to have γ L converged to better than 1%. The transverse γ T component of conducting tubes critically depends on the number of k points in reciprocal space: at least 900 k points are required to converge better than 1% at the LDA level. Coupled-perturbed results are compared to un-coupled values obtained from a sum-overstates (SOS) approach. The difference between the two is particularly important along the transverse direction and when pure DFT functionals are used: the coupled-perturbed correction can shrink the SOS value by several hundreds times. The ratio LDA/HF is roughly constant around 2 for α L ; it ranges between 25 and 60 for γ L. As regards the convergence with the nanotube radius, the R 2 law is confirmed for α L and α T (normalized for the cell parameter) at all levels of theory. For the second hyperpolarizabilities γ L and γ T , a clear R 5 dependence is observed

From anisotropy of dielectric tensors to birefringence: a quantum mechanics approach

International audienceThe way quantum mechanical ab initio computer codes allow to compute, through perturbation theory (the so-called SC-CP, self-consistent coupled-perturbed scheme), many properties resulting from the interaction of the electric field with a crystalline system is illustrated. The polarizability, which leads to the dielectric tensors as well as to the refractive indices and to the birefringence of materials, is the simplest on this list. Higher order tensors, like the first and second hyperpolarizabilities, can be obtained as well with the CRYSTAL code here used. These properties, resulting from the Taylor expansion of the total energy of the solid as a function of the electric field, belong to a large family of phenomena generated by combining in different ways the frequencies of the fields. Second-harmonic generation (SHG), Pockels effect, intensity-dependent refractive index (IDRI), and other quantities now accessible to experiment can be computed at a relatively low cost and with high accuracy

Hydrogen, boron and nitrogen atoms in diamond: a quantum mechanical vibrational analysis

International audienceThe structural, electronic and vibrational properties of two common defects in diamond, CHN and CHB, describing the case in which a carbon C1 atom is substituted by a nitrogen atom, or by a boron atom, breaking a C1–C2 bond, followed by the saturation of the dangling bond of C2 by a hydrogen atom, are investigated at the quantum mechanical level, by using a periodic supercell approach, hybrid DFT functionals and a local Gaussian-type basis set as implemented in the CRYSTAL code. The effect of concentration of the defects has been explored, by considering two supercells containing 64 and 216 atoms (S64 and S216). Formation and hydrogenation energies, geometries, Mulliken charges and the band structure of both defects are reported. The vibrational features of the defects have been investigated, by generating the IR and Raman spectra, and by analyzing graphically and through the isotopic substitution (H → D, 11B → 10B and 14N → 15N) the nature of the most relevant modes related to the defects. The computed C–H stretching mode of CHN, once corrected for anharmonicity (3408 cm−1), falls to wavenumbers very close to the experimental peak observed at 3394 cm−1, which can then be reasonably attributed to this specific defect. The present manuscript is included in a special volume in honor and memory of János Ángyán. Although he did not study in particular the kind of defects discussed in the present manuscript, the many methodological contributions he introduced in computational science have inspired many of the tools we have been using here. One of the present authors, RD, in particular, is grateful to János for the illuminating discussions they had in Paris, Nancy and Torino

Ab initio compressibility of metastable low albite: revealing a lambda-type singularity at pressures of the Earth’s upper mantle

International audienceThe elastic behavior of low albite is investigated ab initio under hydrostatic pressure up to 16 GPa. Our calculations complement and extend previous studies confirming a highly anisotropic character of the feldspar cell compression and, more importantly, revealing a clear change of all structure deformation trends around 8–9 GPa pressure. We correlate this change to the trend of the bulk modulus of low albite as a function of pressure, which we compute in different and independent ways using (1) the Birch–Murnaghan equation of state, (2) the analytical Voigt–Reuss–Hill averaging scheme of calculated elastic constants, and (3) a pressure–volume numerical differentiation procedure. The latter, in particular, uncovers a singularity in the bulk modulus between 8 and 9 GPa pressure which is evocative of a λ-type critical point. We find that the same behavior emerges when comparing with pressure–volume datasets from the experimental literature, where it has been so far overlooked due to the misleading use of a fourth-order Birch–Murnaghan equation of state. Indeed, we show that the equation of state must be extended up to at least the sixth power of the Eulerian strain to approximate the complex elastic behavior of feldspars. The low albite structure softens under increasing pressure between 5 and 8 GPa, as a result of the initiation of auxiliary compression mechanisms—notably, the squeezing of the crankshaft chains along b—and then abruptly resumes a stiffening trend in association with a displacive transformation of the O–O pair interactions. Whether this is an isosymmetric phase transition or a supercritical crossover, it suggests a compatibility with seismological profiles indicating a low wave-velocity anomaly in correspondence of the upper portion of the subducting Pacific plate and the disappearance of such anomaly at greater depths, assuming the alkali feldspar survives as a metastable phase. The data and methodology described here can enable the exploration of important, potentially overlooked features in other minerals, and inspire future high-pressure research in mineral physics