Cohesive Energy of Simple Metals

The cohesive energy of a solid is the energy required to disassemble it into atoms. 

This is in fact the bonding energy. The units of the cohesive energy are Ry/atom or eV/atom or kj/mol or kcal/mol (1 eV/atom = 96.35 kj/mol = 23.05 kcal/mol). 

A most important test of the theory is to compare their data with the bonding energy of real metals. This comparison has been performed. It turned out that the cohesive energy of simple metals could be explained by the theory described in Sections 5.2-5.4. Simple metals are those without d electrons in the electron shells. 

The Wigner-Seitz model of a spherical unit cell is used for calculation. This is a sphere of radius tsUo with one nucleus at the center. Each sphere has overall neutrality, since one-electron charge at the center is canceled by the positive charge inside the volume of the sphere. In this model the spheres exert no electrical forces on each other. Of course this is only an approximate model, since the unit cells are not truly spheres - spheres cannot be packed together to cover aU volume. However, the error made by the approximation is remarkably small. 

For Z 1 the Wigner-Seitz radius rws = The total electrostatic ener- 

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We consider a homogenous gas that consists of free electrons in Chapter 5. Notions of exchange energy and correlation energy of electrons are introduced. The theory enables one to calculate some macroscopic properties of simple metals, which have the ns external electronic shell. The calculated cohesive energy of simple metals turns out to fit the experimental values satisfactorily. 

Hafner (1985) applies to binary alloys formed by s and p elements the same approach already used by Heine Weaire (1970) in the evaluation of the cohesion energy of simple metals (sp-bonded metals). 

In the particular example chosen, the maximum attractive energy is found when the metal atomic orbitals are half-filled. Upon a further increase of the electron density per metal atom, antibonding surface fragment orbitals become occupied and the interaction energy will decrease. Interestingly, in the case of the simple cubic lattice also, the cohesive energy of the metal has a maximum when the electron occupancy per metal atom exceeds half of its capacity, because in the metal antibonding metal orbitals become occupied as well. 

Unfortunately there are no simple theories to predict the cohesive energies of the metals like the coulomb attraction in ionic crystals. More sophisticated quantum mechanical theories using pseudopotential or other modeling techniques are generally required. There are some interesting correlations, however. 

Strongly for the ionic crystals, yet the bulk modulus for the alkali halides varies as d. The cl trend for the bulk modulus will show up in the study of simple metals, and in terms of the pseudopotentials that will be used in the study of simple metals, d" -dependence takes on a particularly fundamental role. In Problem 15-3, the simple metal theory is used to give a good account of the bulk modulus in C, Si, and Gc. It should be noted also that the simple metal theory docs not give a good account of cohesive energy itself there is much cancellation between terms for that property, and there are important contributions (for example, that do not vary as... 

Potassium is a typical simple metal. The model of the electronic gas (Chapter 5, Table 5.2) describes the cohesive energy of the bulk potassium well. The completely delocalized s electrons ensure the interatomic bond in potassium. 

As with the simple metals, a theoretical treatment of the cohesive energies of the transition metals requires a more sophisticated quantum mechanical treatment. A few trends can be observed, however. Unlike the simple metals, the strongest bonds occur... 

In Fig. 7 the results of the model for the cohesive energy are given, and compared with the experimental values and with the results of band calculations. The agreement is satisfactory (at least of the same order as for similar models for d-transition metals). For americium, the simple model yields too low a value, and one needs spin-polarized full band calculations (dashed curve in Fig. 7) to have agreement with the experimental value. 

The cohesive energy (energy of atomization) of the simple metals, in cV. Values of some semiconductors and scmimctals are also included. 

Muller DA (1998) Simple model for relating EELS and XAS spectra of metals to changes in cohesive energy. Phys Rev B 58 5989... 

Assuming that the cohesive energy is due to the delocalization energy of partly filled bands, we can conclude that Equations (5.20), (5.22) and (5.23) are valid. We can also assume that the coulombic energy a is the same as for the free metal atoms. This is what is done in the simple Huckel theory used for other covalent bonding. Then we can write... 

In the present context, this example was intended to serve as a reminder of how one formulates a simple model for the quantum mechanics of electrons in metals and, also, how the Pauli principle leads to an explicit algorithm for the filling up of these energy levels in the case of multielectron systems. In addition, we have seen how this model allows for the explicit determination (in a model sense) of the cohesive energy and bulk modulus of metals. 

PW91 appears to improve the properties of both simple[43,110] and transition metals[lll]. For example, LSD tends to underestimate bond lengths of molecules and solids, including metals. Even in the alkali metals Li and Na, which are often compared with the uniform electron gas, this error is about 4%. PW91 expands the lattice, producing lattice constants in better agreement with experiment, as seen in Table 8. GGA s also improve cohesive energies, just as they improve atomization en-... 

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