Madelung Constants

Madelung constants are simple numbers which depend on the type of structure investigated. They are needed for the calculation (using the Born-Haber cycle) of lattice energies and enthalpies of formation of ionic compounds. Each Madelung constant is the sum of partial Madelung constants which represent the contributions of the individual ions to the total lattice energy. The partial Madelung constants depend on the ionic charge and, clearly though not stringently, on the coordination number. On the other hand, each Madelung constant can be represented by a sum of Madelung constants for related simple primitive AB structures. Surprisingly, these Madelung constants are numerically interrelated in a simple manner, and are related to the partial Madelung constants of interstitial sites. – Madelung constants of parameter-dependent structures (e.g. of the rutile or anatase type) and their variations with the structure-determining quantities are of particular interest. Madelung constants also yield information in the case of complex compounds and – surprisingly – of non-metal compounds (e.g. XeF₂, XeF₄, XeF₂·XeF₄).     

Born-Haber-Fajans cycle generalized: linear energy relation between molecules, crystals, and metals

Classical procedures to calculate ion-based lattice potential energies (U(POT)) assume formal integral charges on the structural units; consequently, poor results are anticipated when significant covalency is present. To generalize the procedures beyond strictly ionic solids, a method is needed for calculating (i) physically reasonable partial charges, delta, and (ii) well-defined and consistent asymptotic reference energies corresponding to the separated structural components. The problem is here treated for groups 1 and 11 monohalides and monohydrides, and for the alkali metal elements (with their metallic bonds), by using the valence-state atoms-in-molecules (VSAM) model of von Szentpály et al. (J. Phys. Chem. A 2001, 105, 9467). In this model, the Born-Haber-Fajans reference energy, U(POT), of free ions, M(+) and Y(-), is replaced by the energy of charged dissociation products, M(delta)(+) and Y(delta)(-), of equalized electronegativity. The partial atomic charge is obtained via the iso-electronegativity principle, and the asymptotic energy reference of separated free ions is lowered by the "ion demotion energy", IDE = -(1)/(2)(1 - delta(VS))(I(VS,M) - A(VS,Y)), where delta(VS) is the valence-state partial charge and (I(VS,M) - A(VS,Y)) is the difference between the valence-state ionization potential and electron affinity of the M and Y atoms producing the charged species. A very close linear relation (R = 0.994) is found between the molecular valence-state dissociation energy, D(VS), of the VSAM model, and our valence-state-based lattice potential energy, U(VS) = U(POT) - (1)/(2)(1 - delta(VS))(I(VS,M) - A(VS,Y)) = 1.230D(VS) + 86.4 kJ mol(-)(1). Predictions are given for the lattice energy of AuF, the coinage metal monohydrides, and the molecular dissociation energy, D(e), of AuI. The coinage metals (Cu, Ag, and Au) do not fit into this linear regression because d orbitals are strongly involved in their metallic bonding, while s orbitals dominate their homonuclear molecular bonding.     

Volume-based thermodynamics: estimations for 2:2 salts

The lattice energy of an ionic crystal, U(POT), can be expressed as a linear function of the inverse cube root of its formula unit volume (i.e., Vm(-1/3)); thus, U(POT) approximately 2I(alpha/Vm(1/3) + beta), where alpha and beta are fitted constants and I is the readily calculated ionic strength factor of the lattice. The standard entropy, S, is a linear function of Vm itself: S approximately kVm + c, with fitted constants k and c. The constants alpha and beta have previously been evaluated for salts with charge ratios of 1:1, 1:2, and 2:1 and for the general case q:p, while values of k and c applicable to ionic solids generally have earlier been reported. In this paper, we obtain alpha and beta, k and c, specifically for 2:2 salts (by studying the ionic oxides, sulfates, and carbonates), finding that U(POT)[MX 2:2]/(kJ mol(-1)) approximately 8(119/Vm(1/3) + 60) and S degree [MX 2:2]/(J K(-1) mol(-1)) approximately 1382V(m) + 16.     

Lattice energies of apatites and the estimation of DeltaH f degrees (PO 4 3-, g)

Experimentally based lattice energies are calculated for the apatite family of double salts M(5)(PO(4))(3)X, where M is a divalent metal cation (Ca, Sr, Ba) and X is hydroxide or a halide. These values are also shown to be estimable, generally to within 4%, using the recently derived Glasser-Jenkins equation, U(POT) = AI(2I/V(m))(1/3), where A = 121.39 kJ mol(-)(1). The apatites exhibiting greater covalent character (e.g., M = Pb, Cd, etc.) are less well reproduced but are within 8% of the experimentally based value. The lattice energy for ionic apatites (having identical lattice ionic strengths, I) takes the particularly simple form U(POT)/kJ mol(-)(1) = 26680/(V(m)/nm(3))(1/3), reproducing cycle values of U(POT) well when V(m) is estimated by ion volume summation and employing a volume for the PO(4)(3)(-) ion (not previously quantified with an associated error) of 0.063 +/- 0.003 nm(3). A value for the enthalpy of formation of the gaseous phosphate ion, DeltaH(f)( ) degrees (PO(4)(3)(-), g), is absent from current thermochemical tabulations. Examination of solution and solid state thermochemical cycles for apatites, however, leads us to a remarkably consistent value of 321.8 +/- 1.2 kJ mol(-)(1). Experimental and estimated lattice energies were used along with other thermodynamic data to determine enthalpies, entropies, and free energies of dissolution for apatites of uncertain stabilities. These dissolution values are compared with the corresponding values for stable apatites and are used to rationalize the relative instability of certain derivatives.     

Lattice potential energy estimation for complex ionic salts from density measurements

This paper is one of a series exploring simple approaches for the estimation of lattice energy of ionic materials, avoiding elaborate computation. The readily accessible, frequently reported, and easily measurable (requiring only small quantities of inorganic material) property of density, rho(m), is related, as a rectilinear function of the form (rho(m)/M(m))(1/3), to the lattice energy U(POT) of ionic materials, where M(m) is the chemical formula mass. Dependence on the cube root is particularly advantageous because this considerably lowers the effects of any experimental errors in the density measurement used. The relationship that is developed arises from the dependence (previously reported in Jenkins, H. D. B.; Roobottom, H. K.; Passmore, J.; Glasser, L. Inorg. Chem. 1999, 38, 3609) of lattice energy on the inverse cube root of the molar volume. These latest equations have the form U(POT)/kJ mol(-1) = gamma(rho(m)/M(m))(1/3) + delta, where for the simpler salts (i.e., U(POT)/kJ mol(-1) < 5000 kJ mol(-1)), gamma and delta are coefficients dependent upon the stoichiometry of the inorganic material, and for materials for which U(POT)/kJ mol(-1) > 5000, gamma/kJ mol(-1) cm = 10(-7) AI(2IN(A))(1/3) and delta/kJ mol(-1) = 0 where A is the general electrostatic conversion factor (A = 121.4 kJ mol(-1)), I is the ionic strength = 1/2 the sum of n(i)z(i)(2), and N(A) is Avogadro's constant.     

Study of Surface Cell Madelung Constant and Surface Free Energy of Nanosized Crystal Grain

1 INTRODUCTION 2. 1 Madelung constant of crystal Surface energy of crystal grain has crucial influ- The Madelung constant, which is used to calculate ence on the electrical and mechanical performances lattice energy and so on[1], is of central importance in of material, especially for material making up of na- the theory of ionic crystal and property of crystal nosized crystal grains because all outstanding per- structure. There is no special difficulty in calculating formances of the mat…     

Ionic hydrates,M(p)X(q).nH(2)O: lattice energy and standard enthalpy of formation estimation

This paper is one of a series (see: Inorg. Chem. 1999, 38, 3609; J. Am. Chem. Soc. 2000, 122, 632; Inorg. Chem. 2002, 41, 2364) exploring simple approaches for the estimation of lattice energies of ionic materials, avoiding elaborate computation. Knowledge of lattice energy can lead, via thermochemical cycles, to the evaluation of the underlying thermodynamics involving the preparation and subsequent reactions of inorganic materials. A simple and easy to use equation for the estimation of the lattice energy of hydrate salts, U(POT)(M(p)X(q).nH(2)O) (and therefore for solvated salts, M(p)X(q).nS, in general), using either the density or volume of the hydrate, or of another hydrate, or of the parent anhydrous salt or the volumes of the individual ions, is derived from first principles. The equation effectively determines the hydrate lattice energy, U(POT)(M(p)X(q).nH(2)O), from a knowledge of the (estimated) lattice energy, U(POT)(M(p)X(q)), of the parent salt by the addition of ntheta(U) where theta(U)(H(2)O)/kJ mol(-1) = 54.3 and n is the number of water molecules. The average volume of the water molecule of hydration, V(m)(H(2)O)/nm(3) = 0.0245, has been determined from data on a large series of hydrates by plotting hydrate/parent salt volume differences against n. The enthalpy of incorporation of a gaseous water molecule into the structure of an ionic hydrate, [Delta(f)H degrees (M(p)X(q).nH(2)O,s) - Delta(f)H degrees (M(p)X(q),s) - nDelta(f)H degrees (H(2)O,g)], is shown to be a constant, -56.8 kJ (mol of H(2)O)(-1). The physical implications with regard to incorporation of the water into various types of solid-state structures are considered. Examples are given of the use of the derived hydrate lattice energy equation. Standard enthalpies of formation of a number of hydrates are thereby predicted.     

On the Padé method for sequence acceleration and calculation of Madelung constants

An iterative variant of Padé approximants (PA) is presented and employed to accelerate convergence of sequences. Pilot calculations on partial lattice sums for NaCl and CsCl crystals have been explicitly shown to offer accurate estimates of their Madelung constants.     

Computation of the Madelung constant for hypercubic crystal structures in any dimension

A new method of computing the Madelung constants for hypercubic crystal structures in any dimension \(n\ge 2\) is given. It is shown for \(n\ge 3\) that the Madelung constant may be obtained in a simple, efficient and unambiguous way as the Hadamard finite part of the integral representation of the potential within the crystal which is divergent at any point charge location. Such a regularization method fails in the bidimensional case due to the logarithmic nature of singularities for the potential. In that case, a specific approach is proposed taking in account the scale invariance of the Poisson equation and the existence of a finite horizon for each point charge in the plane. Since a closed-form exact solution for the 2D electrostatic potential may be derived, one shows that the Madelung constant may be defined via an appropriate limit calculation as the mean value of potential energies of charges composing the unit cell.     

Construction of a generalized gradient approximation by restoring the density-gradient expansion and enforcing a tight Lieb-Oxford bound

Recently, a generalized gradient approximation (GGA) to the density functional, called PBEsol, was optimized (one parameter) against the jellium-surface exchange-correlation energies, and this, in conjunction with changing another parameter to restore the first-principles gradient expansion for exchange, was sufficient to yield accurate lattice constants of solids. Here, we construct a new GGA that has no empirical parameters, that satisfies one more exact constraint than PBEsol, and that performs 20% better for the lattice constants of 18 previously studied solids, although it does not improve on PBEsol for molecular atomization energies (a property that neither functional was designed for). The new GGA is exact through second order, and it is called the second-order generalized gradient approximation (SOGGA). The SOGGA functional also differs from other GGAs in that it enforces a tighter Lieb-Oxford bound. SOGGA and other functionals are compared to a diverse set of lattice constants, bond distances, and energetic quantities for solids and molecules (this includes the first test of the M06-L meta-GGA for solid-state properties). We find that classifying density functionals in terms of the magnitude mu of the second-order coefficient of the density gradient expansion of the exchange functional not only correlates their behavior for predicting lattice constants of solids versus their behavior for predicting small-molecule atomization energies, as pointed out by Perdew and co-workers [Phys. Rev. Lett. 100, 134606 (2008); Perdew ibid. 80, 891 (1998)], but also correlates their behavior for cohesive energies of solids, reaction barriers heights, and nonhydrogenic bond distances in small molecules.     

Why are [P(C6H5)4](+)N3- and [As(C6H5)4](+)N3- ionic salts and Sb(C6H5)4N3 and Bi(C6H5)4N3 covalent solids? A theoretical study provides an unexpected answer

A recent crystallographic study has shown that, in the solid state, P(C(6)H(5))(4)N(3) and As(C(6)H(5))(4)N(3) have ionic [M(C(6)H(5))(4)](+)N(3)(-)-type structures, whereas Sb(C(6)H(5))(4)N(3) exists as a pentacoordinated covalent solid. Using the results from density functional theory, lattice energy (VBT) calculations, sublimation energy estimates, and Born-Fajans-Haber cycles, it is shown that the maximum coordination numbers of the central atom M, the lattice energies of the ionic solids, and the sublimation energies of the covalent solids have no or little influence on the nature of the solids. Unexpectedly, the main factor determining whether the covalent or ionic structures are energetically favored is the first ionization potential of [M(C(6)H(5))(4)]. The calculations show that at ambient temperature the ionic structure is favored for P(C(6)H(5))(4)N(3) and the covalent structures are favored for Sb(C(6)H(5))(4)N(3) and Bi(C(6)H(5))(4)N(3), while As(C(6)H(5))(4)N(3) presents a borderline case.     

Improved hybrid functional for solids: the HSEsol functional

We introduce the hybrid functional HSEsol. It is based on PBEsol, a revised Perdew-Burke-Ernzerhof functional, designed to yield accurate equilibrium properties for solids and their surfaces. We present lattice constants, bulk moduli, atomization energies, heats of formation, and band gaps for extended systems, as well as atomization energies for the molecular G2-1 test set. Compared to HSE, significant improvements are found for lattice constants and atomization energies of solids, but atomization energies of molecules are slightly worse than for HSE. Additionally, we present zero-point anharmonic expansion corrections to the lattice constants and bulk moduli, evaluated from ab initio phonon calculations.     

Madelung energy of the donor-acceptor crystal (BEDT-TTF)2I3

Using the Ewald method for different degrees of ionic character, we have calculated the Madelung energy of the Β phase of the donor-acceptor molecular crystal (BEDT-TTF)₂I₃ for pressures 1 bar and 9.5 kbar.     

On the relationship between the chemical bond energy and the electron energy levels of strongly ionic compounds

The analysis of experimental data suggests that the energy levels of ions in a crystal shift rigidly by an amount equal to the Madelung energy (electrostatic bond energy) maintaining the same separation as in the free ion. This leads to the conclusion that the electrostatic bonding energies of ions in a crystal is taken up partly by the orbital electrons and partly by the nuclear charge.     

Static and dynamic ionization levels of transition metal-doped zinc chalogenides

Transition metal (TM) impurities in semiconductors have a considerable effect on the electronic properties and on the lattice vibrations. The unfilled d shell permits the impurity atoms to exist in a variety of charge states. In this work, the static donor and acceptor ionization energies of ZnX:M, with X = S, Se, Te and M:Sc, Ti, V, Fe, Co, Ni are obtained from first principles total energy calculations and compared with experimental results in the literature where they exist. From these results, many of the TM-doped zinc chalogenides have an amphoteric behavior. To analyze the rule of the deep gap levels in both the radiative and non-radiative processes, the dynamic ionization energies are obtained as a function of the inward and outward M–X displacements. In many cases, the changes in the mass and the force constants resulting from the substitution of an impurity center for a lattice atom are small. When the charge or the environment of the impurity changes, the electron population tend to remain compensated. As consequence, the changes in the lattice vibrational modes are small.     

Importance of Madelung potential in quantum chemical modeling of ionic surfaces

The importance of the inclusion of the Madelung potential in cluster models of ionic surfaces is the subject of this work. We have determined Hartree-Fock all electron wave functions for a series of MgO clusters with and without a large array of surrounding point charges (PC) chosen to reproduce the Madelung potential. The phenomena investigated include: the reactivity of surface oxygens toward CO₂, atomic hydrogen, and H⁺; the geometry and adsorption energy of water and the vibrational shift of CO adsorbed at Mg²⁺ sites; the electronic structure and the hyperfine coupling constants of oxygen vacancies, the paramagnetic F_s⁺ centers. While some clusters give results which are virtually independent of the presence of the PCs, other clusters, typically of small size, give physically incorrect results when the PCs are not included. The embedding of the cluster in PCs guarantees a similar response for clusters of different size, at variance with the bare clusters, where the long range coulombic interactions are not included. © 1997 by John Wiley & Sons, Inc.     

Interpolation determination of the lattice energy of ionic crystals within the framework of stereoatomic model

A new method based on graph theory was suggested for interpolation calculation of the lattice energy U of ionic crystals. The method is based on revealing matrix correlation between the ionic radii and U values for MX compounds, where M is a metal and X is halogen, hydrogen, or chalcogen. A new formula was obtained for calculating the lattice energy solely from the ionic radii, without introduction of abitrary factors. The mean error of determining U for alkali metal halides is 0.49%. The lattice energies were calculated for a large group of inorganic substances. The accuracy of the interpolation calculation of the lattice energy of ionic crystals depends on the degree of ionicity of the bond: With an increase in the covalent contribution, the error increases.     

Density functional theory studies on the dissociation energies of metallic salts: relationship between lattice and dissociation energies

The formation and physicochemical properties of polymer electrolytes strongly depend on the lattice energy of metal salts. An indirect but efficient way to estimate the lattice energy through the relationship between the heterolytic bond dissociation and lattice energies is proposed in this work. The heterolytic bond dissociation energies for alkali metal compounds were calculated theoretically using the Density Functional Theory (DFT) of B3LYP level with 6‐311+G(d,p) and 6‐311+G(2df,p) basis sets. For transition metal compounds, the same method was employed except for using the effective core potential (ECP) of LANL2DZ and SDD on transition metals for 6‐311+G(d,p) and 6‐311+G(2df,p) calculations, respectively. The dissociation energies calculated by 6‐311+G(2df,p) basis set combined with SDD basis set were better correlated with the experimental values with average error of ca. ±1.0% than those by 6‐311+G* combined with the LANL2DZ basis set. The relationship between dissociation and lattice energies was found to be fairly linear (r>0.98). Thus, this method can be used to estimate the lattice energy of an unknown ionic compound with reasonably high accuracy. We also found that the dissociation energies of transition metal salts were relatively larger than those of alkaline metal salts for comparable ionic radii. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 827–834, 2001     

Influence of madelung (interatomic Coulomb) energy on Wolfsberg–Helmholz calculations

A model is presented for the estimation of ionicities in molecules and complex ions. The model uses the minimization of total energy by the method of differential ionization energies. The effect of Madelung corrections to the energies is considered, and the model is refined by evaluating the covalent-bond energies. Wolfsberg–Helmholz calculations have been applied to the same type of model, also incorporating Madelung corrections. The Madelung corrections make the metal ionization energy curves less steep, and the ligand ionization energies are nearly invariant with charge. This creates a situation which has previously been artificially imposed by selecting the ligand ionization energies to give desirable terms in the Wolfsberg–Helmholz secular determinant. The effect of Madelung energy is shown to be the primary influence in describing the ionicity and total energy of a chromophore; covalent bonding effects are shown to be secondary when the ligands and the central atom have fairly different electronegativities.     

Why are the Homoleptic Diyl Complexes M(InR)4 with M = Ni and Pt Stable Compounds while only Ni(CO)4 but not Pt(CO)4 can become Isolated? A Theoretical Study of M(EMe)44 and M(CO)4 (M = Ni,Pd, Pt; E = B,Al, Ga,In, Tl)

The equilibrium geometries and first bond dissociation energies of the homoleptic complexes M(EMe)₄ and M(CO)₄ with M = Ni, Pd, Pt and E = B, Al, Ga, In, Tl have been calculated at the gradient corrected DFT level using the BP86 functionals. The electronic structure of the metal‐ligand bonds has been examined with the topologial analysis of the electron density distribution. The nature of the bonding is revealed by partitioning the metal‐ligand interaction energies into contributions by electrostatic attraction, covalent bonding and Pauli repulsion. The calculated data show that the M‐CO and M‐EMe bonding is very similar. However, the M‐EMe bonds of the lighter elements E are much stronger than the M‐CO bonds. The bond energies of the latter are as low or even lower than the M‐TlMe bonds. The main reason why Pd(CO)₄ and Pt(CO)₄ are unstable at room temperature in a condensed phase can be traced back to the already rather weak bond energy of the Ni‐CO bond. The Pd‐L bond energies of the complexes with L = CO and L = EMe are always 10 — 20 kcal/mol lower than the Ni‐L bond energies. The calculated bond energy of Ni(CO)₄ is only D_o = 27 kcal/mol. Thus, the bond energy of Pd(CO)₄ is only D_o = 12 kcal/mol. The first bond dissociation energy of Pt(CO)₄ is low because the relaxation energy of the Pt(CO)₃ fragment is rather high. The low bond energies of the M‐CO bonds are mainly caused by the relatively weak electrostatic attraction and by the comparatively large Pauli repulsion. The σ and π contributions to the covalent M‐CO interactions have about the same strength. The π bonding in the M‐EMe bonds is less than in the M‐CO bonds but it remains an important part of the bond energy. The trends of the electrostatic and covalent contributions to the bond energies and the σ and π bonding in the metal‐ligand bonds are discussed.