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Perturbation-Theoretical Model Calculation of the Lattice Mechanics of Ionic Crystals in the Point-Dipole Approximation
Abstract
A perturbation‐theoretic model calculation is developed in the point‐dipole approximation for the study of ionic crystals from a knowledge of the Hartree‐Fock wave functions of the free ions that constitute the solid. The salient features of the present calculation are that it a) provides a method for calculating the different lattice static and dynamic properties of a crystal directly from the free ion wave functions without using any crystal property; b) clearly gives an empirical justification for the major assumption of the phenomenological shell model; c) develops a new method for evaluating the Coulomb overlap interaction; d) suggests a modified short‐range polarisation mechanism; and lastly e) shows how far actually the free ion wave functions are realistic as a starting point. The method is applied to calculate a number of lattice mechanical properties of the two ionic crystals namely, NaCl and KCl. Despite the simplicity of the approach the agreement, with observation for both the crystals is quite impressive. The source of the remaining discrepancy is discussed.
The present investigation for the alteration of the free-ion wave functions, when the ions are put in a crystal, the effect of the crystal environment is simulated by the Watson potential. The developed perturbation theoretical model for ionic solids by Basu and sengupta has been used for calculation of the different lattice-dynamical properties for the low-polarizability ionic crystal. The feature of used method it requered only input data necessary for the study of (KF) crystal properties are the Hartree-Fock wave functions of the constituent ions. Certain properties like, the dielectric properties and some phonons in symmetry directions, depend on the excited states of the crystal, are rather sensitive to this effect which varies from crystal. A unified study of the phase transition, cohesion, elastic constant, dielectric and vibrational properties of the (KF) crystal is given without using any adjustable free parameter, our approach and overall agreement obtained is well satisfactory. Finally the limitations and reliability of the present model for estimating the effect of the surroundings are critically discussed.
Starting from the Hartree?Fock wave functions of free ions different lattice mechanical properties of a low?polarisability ionic solid, namely, the RbF crystal are calculated within the framework of the perturbation?theoretical?model approach of ionic solids developed by Basu and Sengupta. In a recent investigation it is shown that the dipolar polarisation in an ionic solid may be additively decomposed into three parts, two short range polarisation effects and a long range one apart from the displacement polarisation. From an analysis of the Szigeti relations it is found that both the short range polarisation effects in the low polarisability ionic crystals are small compared to those in high polarisability crystals. This fact leads to a major simplification in calculation for this group of solids. The overall agreement obtained broadly justifies the approximations. Since no crystal property is used to obtain the parameters the discrepancies clearly indicate the regions where the omitted interactions are important. Finally a critical analysis of the results is presented and the difficulties of the existing phenomenological models for these solids are also discussed.
The specific form of the change in overlap interaction between ions due to the presence of dipole polarizability implied by the shell model lacks any microscopic justification, while all other terms of the model have a sound quantum-mechanical basis. Recently we have shown that within the framework of the Heitler-London approach this modified overlap term of the shell model is microscopically justifiable. This calculation provides a method of evaluating the parameters of the model directly from the wave functions of the ions. Since the shell model has been successful in describing the different lattice-mechanical properties, it seems instructive to investigate how far it can reproduce the collective dynamics of the electrons in insulators. Starting from the Hartree-Fock wave function of the ions the calculation is presented for the dispersion relations of the collective oscillation of the shells in the shell model by relaxing the usual adiabatic condition for the two crystals: namely, KCl and NaCl. It is interesting to note that the q⃗→0 frequencies of these collective oscillations of shells of these two crystals together with those of the other fourteen crystals obtained phenomenologically are found to compare satisfactorily with the measured plasma frequencies.
A simplified version of the perturbation-theoretical-model approach to ionic solids developed by Basu and Sengupta has been employed for calculating the different lattice-mechanical properties of the two low-polarizability ionic crystals. Although this is not a microscopic calculation, the interesting feature of the method is that the only input data necessary for predicting the crystal properties are the Hartree-Fock wave functions of the constituent ions. In addition, the present investigation for the first time makes an attempt to study the effect of the alteration of the free-ion wave functions, when the ions are put in a crystal, on the lattice-dynamical properties. The effect of the crystal environment is thought to be simulated by the Watson potential. The calculated properties with and without the potential give a rough estimate of the order of magnitude of this effect. It is found that certain properties, namely, the dielectric properties and some phonons in the symmetry directions that depend on the excited states of the crystal, are rather sensitive to this effect which varies from crystal to crystal. Apart from this, a unified treatment of the cohesion, the phase transition, and the elastic, the dielectric, and the vibrational properties of the NaF and RbF crystals is presented without any adjustable free parameter. In view of the simplicity of the approach, the overall agreement obtained is satisfactory. Finally the reliability and the limitations of the present method of estimating the effect of the surroundings are critically discussed.
Strong diffuse scattering characterised by blobs ('spots') of diffuse intensity well separated from neighbouring Bragg peaks and by diffuse streaks perpendicular to the (100)* directions have been observed in zone axis electron diffraction patterns of KCl. An ab initio lattice dynamical investigation based on the shell model has been performed and the intensity of thermal diffuse scattering calculated from the dispersion relations obtained. The calculated results indicate that the rather strong and characteristic diffuse intensity distribution of KCl can, at least qualitatively, be well understood in terms of the thermal excitation of low-energy phonon modes.
A unified study of the different lattice mechanical properties of the three fluorites CaF2, SrF2, and BaF2 is presented, using a semi-microscopic model and a single set of parameters. The cohesive energy, elastic constants, dielectric constants, and phonon dispersion curves are calculated. The temperature dependence of the phonon frequencies is estimated from a simple quasi-harmonic approach which indicated softening of a 〈100〉 zone boundary phonon mode which may lead to the diffuse transition to the superionic state.
Verschiedene gittermechnische Eigenschaften der drei Fluorite CaF2, SrF2 und BaF2 werden anhand eines halbmikroskopischen Modells unter Verwendung eines einzigen Parametersatzes untersucht. Kohäsionsenergie, elastische Konstanten, dielektrische Konstanten und Phononendispersionskurven werden berechnet. Die Temperaturabhängigkeit der Phononenfrequenzen wird in quasiharmonischer Näherung abgeschätzt, dabei wird die Erweichung einer 〈100〉-Zonengrenzenphononenmode festgestellt, was zu dem diffusen Übergang zum superionischen Zustand führen kann.
Spherical charge deformation within the ionic description is responsible for a considerable improvement over rigid-ion elastic constants and bulk moduli, but some of the remaining discrepancies cannot in principle be eliminated within a spherical-ion model. In particular, the effect of spherical breathing on the optical vibrational frequencies is not dramatic and further improvement (e.g., reduction of LO-TO splitting) will require the inclusion of nonspherical charge relaxation (polarizability).
Phonon dispersion relations in NaCl at 80°K have been determined in the crystallographic [100], [101], and [111] symmetry directions using inelastic neutron scattering. Some measurements have also been made at 300°K, and at (1.0, 0.5, 0) in reciprocal space. The results are compared with theoretical calculations: those of Karo and Hardy for their "deformed-dipole" model, those of Schröder and Nüsslein for a "breathing-shell" model, and those of Caldwell and Klein for a shell model. Satisfactory agreement is obtained in many places, but significant discrepancies still remain, particularly close to the zone boundaries. Comparison is also made with earlier x-ray work on NaCl. The work was performed on a new crystal spectrometer at the R2 reactor at Studsvik, which is equipped with a two-crystal monochromator.
Phonon dispersion relations in KCl at 80°K have been determined using inelastic neutron scattering. The measurements have been performed mainly in the symmetry directions [100], [110], and [111], but also at some other points in the reciprocal space. More than one third of the phonons in all branches were also studied at 300°K in order to obtain and estimate of the temperature dependence of their frequencies. A comparison is made with dispersion curves calculated by Nüsslein and Schröder for their “breathing shell” variant of the shell model for ionic crystals. The agreement for the dispersion curves is quite satisfactory in the most cases.Die Dispersionskurven der Gitterschwingungen in KCl bei 80°K wurden durch inelastische Neutronenstreuung gemessen. Die Messungen wurden meistens in den kristallographischen Symmetrierichtungen [100], [110] und [111], aber zum Teil auch in anderen Punkten im reziproken Raum durchgeführt. Mehr als ein Drittel der Phononen sind auch bei 300°K untersucht worden, um die Temperaturabhngigkeit ihrer Frequenzen zu erhalten. Die Ergebnisse werden mit dem „breathing shell”-Modell von Nüsslein und Schröder verglichen. Die Übereinstimmung ist in den meisten Fllen sehr befriedigend.
An expression for the electronic polarizability of ions in ideal alkali halides is obtained. For this purpose a second order correction of the perturbation theory to the energy of the crystal, placed in an external electric field, is used. As a perturbation the interaction of ions in the crystal and the interaction of the crystal with the external macroscopic electric field are taken simultaneously. The additivity rule for the electronic polarizability of the ideal crystal and for the polarization energy of the non-ideal crystal is confirmed. The difference between the value of the electronic polarizability of the ion in a crystal and that of the free ion is qualitatively correct.
Es wird ein Ausdruck für die elektronische Polarisierbarkeit von Ionen in idealen Alkalihalogeniden entwickelt. Zu diesem Zweck wird eine Korrektur zweiter Ordnung in der Störungstheorie für die Energie eines Kristalls, der sich in einem äußeren elektrischen Feld befindet, durchgeführt. Als Störung wird gleichzeitig die Wechselwirkung der Ionen im Kristall und die Wechselwirkung des Kristalls mit dem äußeren, makroskopischen, elektrischen Feld angenommen. Das Additivitätsgesetz für die elektronische Polarisierbarkeit eines idealen Kristalls und die Polarisationsenergie eines nichtidealen Kristalls wird bestätigt. Die Differenz zwischen dem Wert der elektronischen Polarisierbarkeit eines Ions im Kristall und dem eines freien Ions ist qualitativ richtig.
Theoretical and experimental studies were made of the lattice dynamics ; of alkali halides. A theory of the lattice dynamics of ionic crystals is given ; based on replacement of a polarizable ion by a mcdel in which a rigid shell of ; electrons (taken to have zero mass) can move with respect to the massive ionic ; core. The dipolar approximation then makes the model exactly equivalent to a ; Born-von Karman crystal in which there are two "atoms" of differing charge at ; each lattice point, one of the "atoms" having zero mass. The model was ; specialized to the case of an alkali halide in which only one atom is ; polarizable, and computations of dispersion curves were carried out for Nal. The ; dispersion nu (q) relation of the lattice vibrations in the symmetric STA001!, ; STA110!, and STA111! directions of Nal at 110 deg K were determined by the ; methods of neutron spectrometry. The transverse acoustic, longitudinal ; acoustic, and transverse optic branches were determined completely with a ; probable error of about 3%. -ine dispersion relation for the longitudinal optic ; (LO) branch was determined for the STA001! directions with less accuracy. The ; agreement between the experimental results and the calculations based on the ; shell model, while not complete, is quite satisfactory. The neutron groups ; corresponding to phonons of the LO branch were anomalously energy broadened, ; especially for phonons of long wavelength, suggesting a remarkably short lifetime ; for the phonons of this branch. (auth);
The present work is an attempt to unify the lattice statics and dynamics of an alkali halide crystal through a phenomenological approach. A particular variant of the shell model, namely the deformable shell model has been chosen for the purpose. The different lattice static and dynamic properties of the NaF crystal have been calculated on the model, the same set of parameters being used throughout. The results of the calculation are compared with experiment and the possible source of improvement is suggested. In view of the limited number of parameters used and the drastic approximation introduced in the calculation of certain properties, the overall agreement seems to be quite encouraging and gives some confidence in this approach of considering an alkali halide crystal in all its entirety.
A microscopic theory of lattice dynamics of ionic crystals is developed in terms of non-orthogonal highly localized wave functions. It is shown that this approach is in close correspondence to some shell models and leads to a microscopic understanding of the model parameters. The phonon dispersion of LiD has been calculated without using any adjustable parameters and it agrees with experiment to within 10% on the average.
It is shown that a deformation-dipole model more general than that of Hardy follows rigorously from Zeyher's microscopic theory of lattice dynamics of ionic crystals. In addition to the self-energy correction term due to the electric dipoles in Hardy's model, Zeyher's theory gives a similar term due to the deformation dipoles. The general shell model corresponds to a special form of the new deformation-dipole model.
The ion-pair interaction potentials obtained by using the modle presented in our previous paper are applied to some ionic crystals. The calculated lattice properties of alkali-halide and alkaline-earth-dihalide crystals agree quite well with experimental data. The polymorphic transitions of alkali-halide crystals at high pressures are also successfully described by the calculations.
the mechanism of charge transfer is incorporated into the shell model in an effort to better describe the lattice dynamics of crystals of the zincblende structure. The long-wavelength aspects of the resulting new model are treated in detail. A preliminary application is made to GaAs.
Dielectric and spectroscopic measurements have been made on the alkali halides and several thallium and silver halides, at 2 and 290 K, in order to determine, for each, values for the parameters which characterize the dispersion of the dielectric constant due to lattice vibrations. These have then been used to assess critically the current theoretical treatments and interpretations of the electronic structures and interionic forces in ionic crystals. The measurements were of audio and radio-frequency dielectric constants, of far infrared transmission spectra and of refractive indices through the visible spectrum.
The deformable-shell model developed by Basu and Sengupta and later substantiated by a potential form by Sarkar and Sengupta has found wide application in describing the different static properties of ionic crystals of both NaCl and CsCl structures. But so far a complete calculation of dynamical properties of ionic crystals has been reported for only one crystal. Further, from a critical comparison of the different lattice-dynamical models which effectively introduce many-body interactions between the ions, we have found that there are certain differences between them, some of which are quite fundamental in nature. Moreover, of the current phenomenological models, the deformable-shell model alone is capable of reasonably treating both the static and the dynamic properties of the crystals. Hence it is important to know the results of the calculation according to different models. In this work we present the lattice-dynamic calculation on the following five crystals, NaCl, NaBr, KI, KCl, and KBr according to the deformable-shell model. In order to obtain the parameters, the well-known macroscopic quantities have been used and no least-square-fitting procedure has been adoped. The parameters obtained from the theory have been used to calculate the phonon dispersion relation in both the symmetry and the off-symmetry directions (where experimental results are available) and the variation of the Debye temperature from the frequency spectra for these crystals. We have consistently used the polarizable negative-ion model for all of them. The results thus obtained agree well with experiment. Other theoretical-model results are also discussed in detail.
Starting from the Hartree-Fock wave functions of the free ions a method is suggested to make a comprehensive calculation of the lattice static and dynamic properties of alkali halides. This is achieved through the intermediary of a model, the parameters of which are extracted from a first-principles calculation. It may be mentioned here that until now no calculation of lattice dynamics of alkali halides, without using any experimental input, has been reported. In the present calculation for the KCl crystal neither any measured property of the solid is used nor any parameter is varied arbitrarily to fit experiment. The calculated properties are found to agree quite satisfactorily with the measured ones.
In this report we present a simple statistical-model derivation of the three-body interaction envisaged in the deformable-shell model developed by Basu and Sengupta. Next the model is applied to a unified study of the lattice statics and dynamics of the CsBr crystal, the same set of parameters being used throughout. Among the alkali halide crystals, the cesium group of halides are distinguished from the rest by two characteristic problems of the their own—one related to the stability of the static lattice structure and the other concerning the correlation of the dielectric properties and the dispersion of phonons. The results of the present calculation show that this simple model gives a good description of the lattice mechanics of the CsBr crystal in all its totality and the source of improvement for the remaining discrepancy is suggested.
A microscopic calculation is performed of the phonon spectrum of the alkali halide crystals KCl and KBr in the approximation where the dielectric matrix is assumed to be of separable form. The calculations involve two adjustable microscopic parameters. Reasonable agreement with experiment is obtained. A detailed discussion is also given of the microscopic treatment of effective charges and overlap interactions in ionic crystals. It is shown how qualitative conclusions regarding the nature of the overlap forces may be drawn from a simple consideration of the nature of the valence- and conduction-band orbitals.
A simple model is presented for calculating the forces between closed‐shell atoms and molecules in the regions both of the attractive well and of the repulsive wall at shorter distances. Account is taken of both the overlap of the separate atomic densities and of electron correlation. Applications to pairs of rare gas atoms and to alkali halide molecules demonstrate quantitative agreement with empirically determined intermolecular potentials for these systems over the whole range of separations inside and including the potential minimum.
Lattice dynamics has some claim to being the oldest branch of solid state physics. Planck's Theory of Radiation and the Theory of Specific Heat was published by Einstein in 1907, On Vibrations in Space Lattices by Born and von Karman in 1912, and On the Theory of Specific Heat by Debye in the same year. Other early papers on lattice dynamics included those by Debye and by Waller on the effect of temperature op the scattering of x-rays by a crystal. In the 1920's Peierls made fundamental contributions to the theory of thermal conductivity and of electrical conductivity involving lattice dynamics, and in the 1930's Blackman brought to an end the acceptance of the complete validity of the Debye theory of specific heat and also contributed the first papers on anharmonic effects in the absorption of infrared radiation by ionic crystals. The work we have mentioned used the formal theory of lattice dynamics to account for the thermal and other properties of crystals. It was not until 1940 that Kellermann made the first moderately successful attempt to predict the dynamical properties of a crystal—sodium chloride—from something approaching first principles, and it was in the late 1950's that the development of the technique of neutron inelastic scattering made it possible for the first time t o acquire detailed knowledge of the spectrum of lattice vibrations of a crystal. The development of this experimental technique, pioneered by Brockhouse, was possibly the main impetus for the greatly increased activity in the subject at the present time. The formal theory of lattice dynamics in the harmonic approximation is now on a firm foundation, largely provided by the work of Born and his collaborators, and was recently reviewed by Maradudin, Montroll and Weiss.1 Much recent work has been concerned with anharmonic effects and their influence on the thermal, dielectric, and radiation-scattering properties of crystals, and on the theory of the dynamics of imperfect crystals. The subject has become comparatively wide-ranging. In this review we shall concentrate on work where the ultimate objective is to understand the dynamical properties of a crystal in terms of the interaction between electrons and nuclei or, more realistically, between ‘ion cores’ interacting directly and through the valence electrons. This objective has been most nearly achieved for the alkali metals because of their relatively simple electronic structure, but we largely exclude metals from consideration since the subject, Lattice Dynamics of Metals, has recently been reviewed by Joshi and Rajagopal.2 Kellermann's theory3 of the dynamics of sodium chloride involved interactions between point charges representing the ions, Born having already shown4 that the cohesive energy of alkali halides could be understood on this basis. The theory of the cohesive energy of ionic crystals has recently been reviewed by Tosi.5 We begin at this point in the development of the theory for nonmetallic crystals since many of the concepts in more recent work can be expressed in terms of those which apply to the simple point ion model of an ionic crystal. The dielectric and dynamical properties of ionic crystals are so intimately connected that we also briefly discuss the former.
A theory for the change in polarisabilities of ions in crystal environment from their free ion values is presented. The contribution to this change is shown to come from the pure third order perturbation and second order exchange terms in the perturbation energy expression for the system. These contributions are separately estimated. The main contribution is shown to arise from third order perturbation effect. The inadequacies of the earlier treatments by Ruffa [1] and by Ledovskaya [2] are pointed out. Calculations for the change in polarisability due to third order perturbation effects are carried out for cations only. For anions some difficulties in the application of the perturbation theory are discussed. The results are then compared with the empirical values of Tessmann, Kahn, and Shockley [3] and also with those of Ruffa.
Es wird eine Theorie für die Polarisierbarkeitsiinderung eines Ions in einer Kristallumgebung vom Wert des freien Ions angegeben. Es wird gezeigt, daß die Beiträge zu dieser Änderung von reinen Störungen dritter Ordnung oder Austauschtermen zweiter Ordnung im Störungsenergieausdruck des Systems herrühren. Diese Beiträge werden getrennt berechnet. Es wird gezeigt, daß der früptbeitrag von einem Storungseffekt dritter Ordnung stammt. Es wird ferner gezeigt, daß die früheren Behandlungen von Ruffa [1] und von Ledovskaya [2] inadäquat sind. Rechnungen für die Polarisierbarkeitsünderungen infolge des Störungseffekts dritter Ordnung werden für fiir Kationen durchgeführt. Für Anionen werden einige Schwierigkeiten bei der Anwendung der Störungstheorie diskutiert. Die Ergebnisse werden dann mit den empirischen Werten von Tessmann, Kahn und Shockley [3] sowie mit denen von Ruffa verglichen.
The approach is based on the adiabatic principle and the harmonic approximation. The aim is to find those quantum mechanical approximations which are necessary to derive expressions of the same structural form as the classical shell model equations of the lattice dynamics of ionic crystals. To this purpose the long range interaction of the ionic dipoles is treated in first order perturbation theory, whereas for the short range part of the adiabatic potential a more restricted approximation is used. By this means equations of motions are obtained of the form of the equations of the classical shell models. But in contrast to the phenomenological models by quantum mechanical arguments two different coordinates are introduced for the description of a T1u-motion of an electronic shell.Es werden diejenigen quantenmechanischen Näherungen untersucht, die nötig sind, um Bewegungsgleichungen von der Form der Gleichungen der klassischen Schalenmodelle abzuleiten. Hierzu werden das adiabatische Prinzip und die harmonische Näherung benutzt. Die weitreichende Wechselwirkung der Dipole der Ionen wird mit Störungstheorie in erster Näherung behandelt, während für die kurzreichweitigen Beiträge des adiabatischen Potentials eine spezielle Näherung Verwendung findet. Damit werden Bewegungsgleichungen von gleicher Form wie die Gleichungen der klassischen Schalenmodelle erhalten. Aber im Gegensatz zu den phänomenologischen Modellen führen quantenmechanische Überlegungen zu zwei verschiedenen Koordinaten zur Beschreibung der T1u-Bewegung einer Elektronenschale.
A microscopic theory of lattice dynamics of the ionic crystals is developed in the self-consistent Hartree-Fock approximation. The work is based on a splitting of the dynamical matrix into two parts, a rigid ion and a polarization part. For ionic crystals such a splitting is especially convenient, because here the rigid ion contribution represents the dominant contribution to the total dynamical matrix. In the theory presented, this part results from the classical Coulomb interaction of the cores and electrons among themselves and with each other within the Heitler-London approximation. It is obtained without using perturbation theory, therefore no excited states are involved. For a special choice of the localized wave functions the energy expression of Löwdin [23] and Lundqvist [24] is obtained which describes the perturbation of the electron density during a lattice mode by a rigid displacement of localized electron densities including exchange effects. The polarization part takes into account virtual electronic transitions caused by a weak driving electron-ion potential additively to the rigid ion part. This part is determined by a variational method.
Using the Heitler-London approach a simple expression for the first-order exchange energy between a system of interacting atoms or ions is obtained in the S2 approximation. It is shown that this expression leads to two-body, three-body, and four-body interactions only. A specific expression for the three-body interaction is evaluated using several different approximations. Assumptions which lead to the expression previously obtained by Lundqvist are discussed. It is found that these expressions vanish for neutral particles. Some modification in the assumption is made and an expression for three-body interaction for neutral particles is obtained. Numerical calculations are made for argon, xenon, and helium and the results compared with those obtained by others. The three-body interaction is also applied to study the relative stability of different alkali halide structures. It is found that this three-body potential has a strong preference for CsCl structure.
Mit der Heitler-London-Methode wird ein einfacher Ausdruck für die Austauschenergie erster Ordnung zwischen einem System wechselwirkender Atome oder Ionen in der S2-Näherung erhalten. Es wird gezeigt, daß dieser Ausdruck nur zu Zwei-Teilchen-, Drei-Teilchen- und Vier-Teilchen-Wechselwirkungen führt. Für Drei-Teilchen-Wechselwirkung wird mit verschiedenen Näherungen ein spezifischer Ausdruck abgeleitet. Annahmen, die zu dem kurzlich von Lundquist erhaltenen Ausdruck führen, werden diskutiert. Es wird gefunden, daß diese Ausdrücke für neutrale Teilchen verschwinden. Numerische Berechnungen werden fur Argon, Xenon und Helium dürchgeführt und die Ergebnisse mit denen anderer Autoren verglichen. Die Drei-Teilchen-Wechselwirkung wird auch zur Untersuchung der relativen Stabilität verschiedener Alkalihalogenidstrukturen benutzt. Es wird gefunden, daß dieses Drei-Teilchen-Potential eine starke Bevorzugung der CsCl-Struktur ergibt.
The Heitler-London expression for the cohesive energy of an ionic crystal contains long-range three-centre potentials in the S²-approximation. Their contribution to the dynamical matrix is examined in terms of q-dependent charge tensors and numerically calculated for six alkali halides, using self-consistent free-ion wave functions and a special multipole expansion. Especially numerical values are given for the three-centre contribution to the elastic constants and for the longitudinal and transversal effective charges at q = 0. The parameters of a simple shell model are fitted to macroscopic data with and without the inclusion of the three-centre terms. The corresponding phonon frequencies are compared. It turns out that the usual simple shell model is able to simulate the three-centre contribution to a large extent by a suitable choice of the parameters. Therefore it is found that an explicit treatment of the three-centre terms in the phenomenological models doesn't lead to a remarkable improvement.
Why compute atomic Hartree-Fock functions, an approximation 30 years old, when most of contemporary physics pays relatively minor attention to atomic physics? The simple answer is that there are several unsettled problems and gaps in our knowledge of the electronic structure of the periodic system and that the Hartree-Fock method is presently the most efficient technique available. In the present literature on atomic functions there is no systematic effort comparable to the one presented in these tables, previously reported only briefly in the literature. Indeed, in the past, Hartree-Fock functions have been computed for relatively few cases, and often without sufficient accuracy.
Theoretical Investigation into Some Properties of Ionic Crystals
- Lowdin
Phys. Rev. Letters 35, 174 (1975)
- Lahiri