2. Calculation of the equilibrium geometry and the wavefunction of ionic crystals


The ab initio Perturbed Ion (aiPI) (Luaña and Pueyo 1990, Luaña et al 1993) method has been used to determine the theoretical equilibrium geometry for every crystal analyzed in this work, and then for obtaining the crystal wavefunction at this geometry. Theoretical geometries have been used because the experimental geometries are not available for most of the studied compounds.

The aiPI method solves the Hartree-Fock equations of a solid in a localized Fock space by partitioning the crystal wavefunction into local, weakly overlapping, group functions, each containing a number of electrons but a single nucleus. The method brings together (Martín Pendás et al. 1992) ideas from the Electron Separability Theory (McWeeny 1992, Huzinaga et al. 1973) and from the Theory of Localizing Potentials (Adams 1961, 62, Gilbert 1964) to produce a very efficient treatment for ionic materials.

The aiPI method has been successfully applied to the description of a variety of electronic, energetic and structural properties of ionic crystals, including halides (Luaña and Pueyo 1990, Recio et al. 1993, Martín Pendás et al 1994, Martín Pendás et al 1994b ) simple and complex oxides (Luaña et al 1990b, Franco et al 1994, Franco et al 1996, Andrés and Beltrán 1994) and sulfides (Recio et al 1993).

The best available Slater Type function (STO) basis sets have been used for all ions (Clementi and Roetti 1974). The unrelaxed hard Coulomb hole formalism (uCHF) has been used to estimate the correlation energy when obtaining the equilibrium geometry (Chakravorti and Clementi 1989), but the crystal wavefunction has been obtained at the Hartree-Fock level, not being modified by the correlation estimation procedure. All coulomb interactions have been integrated exactly. Interionic exchange and projection operators, on the other hand, have been accounted for neighbors up to 25 bohr away.

The perovskite structure [Cubic, Pm3m, A at (1/2,1/2,1/2), M at (0,0,0), and X at (1/2,0,0), See Fig. 2.1] fixes all geometrical parameters but one: the cell side length a. We have optimized a for all the 120 compounds using a simplex method.

Fig. 2.1 Crystallographic (balls and sticks) and Wigner-Seitz (transparent box) unit cells of the cubic perovskites. Dark grey balls represents M ions, green balls X ions, and blue balls are A ions. Small red balls are placed at the Wyckoff's 3c position and will play a very significant role in the topology. properties. The figure was designed with pairpot3 and rendered with POV-Ray.

The equilibrium properties of a sample of the 120 compounds, those perovskites for which reliable experimental geometry of the cubic phase exists, are collected in Table 1.1. Our theoretical geometries are found to agree within 1.1% with these experimental values. The largest discrepancies are found on the Cs crystals, and can be related to the degradation of the hard Coulomb hole functional parameters for the heaviest elements. Previous analyses on alkali and alkaline earth halides indicate that a more realistic estimation of the errors of the uCHF aiPI calculations relative to the experimental equilibrium properties is: 2-4%, 3-6% and 10-15% for the lattice parameters, binding energies and isothermal bulk moduli, respectively.

Table 1.1. Equilibrium properties of a sample of the 120 perovskites.
a: lattice parameter, B: bulk moduli, and Elatt: lattice energy.
Crystal a (A) Elatt (kcal/mol) B (GPa) Exp. a (A) Exp. ref.
NaMgF3 3.836 -951.97 87.999 3.833 Zhao et al. 1993
KMgF3 3.943 -934.77 82.692 3.989 Massa and Babel 1988
RbCaF3 4.483 -825.07 47.275 4.431 Bulou 1980
CsCaF3 4.459 -830.34 44.147 4.505 Ridou et al. 1984
NaZnF3 3.791 -984.30 89.764 3.884 Lutger and Babel 1992
KZnF3 3.897 -965.80 85.485 4.035 Ridou et al. 1984
RbZnF3 3.900 -970.97 86.921 4.110 Massa and Babel 1988
KMgCl3 4.998 -737.96 33.849 4.998 Seifert and Vebach 1985
CsCaCl3 5.563 -673.35 23.386 5.396 Seifert and Vebach 1985

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