The Calculation of Lattice Energies

A brief account will be given in this Section of the various methods that have been employed to calculate the lattice energies of ionic crystals, starting with a discussion of the extended classical calculation. Each term in this calculation will be discussed separately. 

This term arises froin the electrostatic interaction of the point charges with which the ions have been replaced in the theoretical consideration of the problem. The interaction energy of two point charges ZiC and 22c, distance ri2 apart is Z Z2e /ri2. Similarly, the total electrostatic energy Um of n such charges of magnitude zt (i = 1,2,3,. . . , n) is 

Here Um is the energy per mole z+e and z e are the absolute values of the charges on the positive and negative ions I is one of the characteristic crystal dimensions jVa is the Avogadro number and Mi is the Madelung constant, a pure number characteristic of the crystal structure and independant of the dimensions of the lattice. 

Born (16) developed a method of computing the Madelung series of more general applicability. The unit cell was divided into subcells by means pf systems of equidistant planes parallel to the sides of the cell so that each ion in the unit cell was located at one of the sublattice points (Pi/s, Pa/s, Pa/s) where Pi, P2, and Pg are integers which may assume the values 0, 1, 2, 3,. .., s-1. For a cubic crystal the Madelung 

The superscript prime indicates that the cases G = t2 = t are to be excluded. The Born Grundpotential is a special case of the generalized zeta-function developed by Epstein (SS, S4) and Emersleben SO, SI) has evaluated the zeta-function and hence the Grundpotential Tl(xyz) for a cubic lattice for all cases from 11(000) to in steps of %2- 

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An alternative (and probably more precise) method for evaluating defect energies is based on the calculation of lattice energy potentials. 

The calculation of lattice energies (and other Coulomb s law energies) is complicated somewhat by the fact that in SI the permittivity (dielectric constant) of a vacuum is no longer defined as one but has an experimentally determined value. Furthermore, for reasons we need not explore at present. Coulomb s law is stated in the form ... 

The hydridic model is also naive, yet it provides a useful rationalization of internuclear distances in hydrides (as does the atomic model), and it permits estimation of their lattice energies. The latter is true only because the calculation of lattice energies by the method of Bom et al. (11) works reasonably well for solids which are not very ionic. 

D. The Calculation of Lattice Energies from Hydration Enthalpies. . 186... 

An entirely different approach to the calculation of lattice energies was initiated by Hylleraas (59) who applied a general quantum mechanical treatment to the calculation of the lattice energy of lithium hydride. He used one-electron wave functions of the hydrogenic type with nuclear screening so that the entire computation could be performed analytically. The wave functions were... 

The calculation of lattice energy has been refined by including terms arising from the van der Waals (London) forces and from the zero-point energy. The former is important only if both ions are readily polarizable, as may be seen from the following figures ... 

The calculation of lattice energies of ionic compounds is very important since, in general, there is no direct way to measure them experimentally, although they can be obtained from certain experimental data using the Born-Haber cycle which is discussed immediately below. For example, the heat of vaporization of NaCl does not give the lattice energy because up to the highest temperatures at which accurate measurements can be made the gas phase consists of NaCl molecules (or ion pairs), and it has so far proved impossible to get an accurate estimate of the heat of dissociation of NaCl(g) into Na+(g) and Cl (g) since NaCl(g) normally dissociates into atoms. 

There is a second theoretical approach to the calculation of lattice energies. Anatolii Kapustinsldi su ested that in the absence of specific knowledge about the crystal structure of a compound, the lattice energy could be estimated using Equation (8.11) ... 

The physical interpretation of the necessary parameters is difficult or impossible, if one considers how difficult it must be to reconcile the simple expressions 4.38-4.40 with the intricate electronic effects that have been sketched in the preceding sections. A recommendable attitude [46] is then to consider the atom-atom model as a useful numerical machinery for the calculation of lattice energies, without attaching too much physical significance to it. Thus, terms are attractive terms but cannot be considered to represent dispersion or polarization, or exponential terms are repulsive terms but do not necessarily represent Pauli repulsion, and even the terms are of a coulomb-law type but, as discussed in former sections of this chapter, are but a poor representation of the true coulombic potential energy. 

This discussion highlights some of the many lurking variables that affect the calculation of lattice energies. Calculations can give results that differ by up to 2-5 kJ moU for a number of often-hidden reasons while this is demonstrated here for atom-atom... 

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