We have recently developed a non-empirical Debye-like model for the inclusion of thermal effects in the equation of state (EOS) of solids. This model allows the calculation of many thermodynamical properties from the E-V relationship. We report the results of a theoretical investigation that explores the EOS of two ionic solids: MgF2 and Al2O3. The interionic interactions are modelized using either the ab initio Perturbed Ion (aiPI) method or the electron gas formalism along with aiPI electronic wavefunctions, which are allowed to relax with crystal strains. Our EOS results are in overall good agreement with experimental data. Other thermodynamic properties behave in the same way, although Gruneisen constant and related quantities have significant errors. This may be caused by numerical inaccuracies on the high order derivatives needed for its calculation.

Thermodynamical properties of solids from microscopic theory : applications to mgf2 and al2o3

Implementation of the projector augmented-wave method in the ABINIT code: Application to the study of iron under pressure

Critic2: A program for real-space analysis of quantum chemical interactions in solids

The structural and thermodynamic properties of MgF2 have been investigated in a wide range of pressures (0−80 GPa) and temperatures (0−850 K) by coupling quantum-mechanical ab initio perturbed ion ...

Quantum-Mechanical Study of Thermodynamic and Bonding Properties of MgF2

We pose and solve the analogue of Slepian's time-frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the sphere or, alternatively, of strictly spacelimited functions that are optimally concentrated in the spherical harmonic domain. Such a basis of simultaneously spatially and spectrally concentrated functions should be a useful data analysis and representation tool in a variety of geophysical and planetary applications, as well as in medical imaging, computer science, cosmology, and numerical analysis. The spherical Slepian functions can be found by solving either an algebraic eigenvalue problem in the spectral domain or a Fredholm integral equation in the spatial domain. The associated eigenvalues are a measure of the spatiospectral concentration. When the concentration region is an axisymmetric polar cap, the spatiospectral projection operator commutes with a Sturm--Liouville operator; this enables the eigenfunctions to be computed extremely accurately and efficiently, even when their area-bandwidth product, or Shannon number, is large. In the asymptotic limit of a small spatial region and a large spherical harmonic bandwidth, the spherical concentration problem reduces to its planar equivalent, which exhibits self-similarity when the Shannon number is kept invariant.

https://www.jstor.org/stable/pdfplus/20453835.pdf

Spatiospectral Concentration on a Sphere

Rotation matrices (or Wigner D functions) are the matrix representations of the rotation operators in the basis of spherical harmonics. They are the key entities in the generation of symmetry-adapted functions by means of projection operators. Although their expression in terms of ordinary (complex) spherical harmonics and Euler rotation angles is well known, an alternative representation using real spherical harmonics is desirable. The aim of this contribution is to obtain a general algorithm to compute the representation matrix of any point-group symmetry operation in the basis of the real spherical harmonics, paying attention to the use of recurrence relationships that allow the treatment of functions with high angular momenta.

Evaluation of the rotation matrices in the basis of real spherical harmonics

Quantum mechanical cluster calculations of ionic materials : the ab initio perturbed ion (version 7) program

Combined first-principles pairwise simulations and quantum-mechanical ab initio perturbed ion (AIPI) calculations have been extensively performed to determine the static equations of state (EOS) of the cubic (fluorite-type) and orthorhombic (\ensuremath{\alpha}-${\mathrm{PbCl}}_{2}$-type) polymorphs of ${\mathrm{CaF}}_{2}$. This theoretical investigation covers the range of pressures experimentally available (0\char21{}45 GPa). The elastic behavior and the equilibrium crystal parameters have been accurately determined by means of efficient numerical procedures involving a Richardson-iterated, finite-difference formula for the derivatives and the combination of downhill simplex and modified Powell methods for the multidimensional optimizations. For the bulk modulus and the effective elastic constants of the cubic phase, the simulations and AIPI calculations give increasing functions of pressure with small negative curvatures. Besides, the zero-pressure AIPI computations agree with the values of Catlow et al. [J. Phys. C 11, 3197 (1978)] for the bulk modulus and the elastic constants. The computed EOS also reproduces quantitatively the most recent experimental p-V data for the cubic phase. For the orthorhombic phase, we optimize nine crystal parameters for each value of pressure. This set provides a full structural characterization of this phase, as well as a global description of the p-V relationship that is consistent with the synchrotron-radiation x-ray-diffraction study of Gerward et al. [J. Appl. Cryst. 25, 578 (1992)]. Our simulation techniques are able to detect a first-order phase transition from the low-pressure fluorite-type to the high-pressure \ensuremath{\alpha}-${\mathrm{PbCl}}_{2}$-type polymorph. The computed thermodynamic transition pressure lies below the experimental values, as it should for this kind of structural transformation exhibiting large pressure hysteresis.

Static simulations of CaF2 polymorphs.

Core projection effects in near-ab-initio valence calculations. II: Ground state geometry of octahedral chromium (I, II, III, and IV) hexafluorides

We present a general computational scheme to extend the spinodal equation of state [Garcia Baonza et al., Phys. Rev. B 51, 28 (1995)] to the interpretation of the cell parameters response to hydrostatic pressure in orthogonal lattices. As an important example, we analyze the pressure (formula presented)–volume (formula presented)–temperature (T) data of the rutile phase of (formula presented) We show that results of ab initio perturbed ion calculations and very recent x-ray-diffraction experiments of isothermal compression on this system closely follow the spinodal conduct. The computational scheme permits the incorporation of temperature effects in the static calculation as well as in the room-temperature experimental data. Overall, we find highly consistent results and good theory-experiment agreement for a significant series of observables, including structural parameters, (formula presented) diagram, bulk modulus, linear compressibilities, and thermal-expansion coefficient. The observed discrepancies in the pressure first derivative of the bulk modulus can be traced back to the difference between the theoretical and the experimental spinodal pressure. © 2003 The American Physical Society.

/pdf/spinodal-equation-of-state-for-rutile-tio-2-5cl7jpvbdf.pdf

M. Bermejo

Papers

Evaluation of the rotation matrices in the basis of real spherical harmonics

Quantum mechanical cluster calculations of ionic materials : the ab initio perturbed ion (version 7) program

Static simulations of CaF2 polymorphs.

Core projection effects in near-ab-initio valence calculations. II: Ground state geometry of octahedral chromium (I, II, III, and IV) hexafluorides

Spinodal equation of state for rutile TiO 2