Theory of Sodium, Magnesium, and Aluminum

A theory relating the electronic structure and the properties of metals has been described earlier. It was based upon the self-consistent-field method and a perturbation solution of the energy-band calculation. This theory is now applied explicitly to Na, Mg, and Al. The orthogonalized plane wave (OPW) form factors and the energy-lattice wave-number characteristic were computed by machine and used to compute atomic properties. The correct metallic structure was found to have lowest energy in each case: hcp, hcp, and fcc, respectively. Computed $\frac{c}{a}$ ratios for sodium and magnesium of 1.63 and 1.62, respectively, are close to those observed. The elastic shear constant for axial distortions was computed for the hexagonal phase of each metal; for magnesium the constant is known and the agreement excellent. The vibration spectrum for fcc aluminum is computed and corresponds to errors in the elastic constants of the order of a factor of 2; this sensitivity reflects strong cancellation (which greatly increases from sodium to aluminum) between electrostatic and band-structure contributions to the energy. Calculations of the total binding energy of aluminum and the variation of energy with lattice parameter are quite inaccurate. Inclusion of a free-electron exchange and correlation correction does not significantly improve the results and, in fact, makes the crystal unstable against the formation of lattice distortions. Pressure dependence of the elastic constants was calculated for aluminum and gave discrepancies of a factor of 2. It is concluded that the theory gives a rather good account of changes in energy due to ion rearrangement at constant volume, but not of changes in energy due to changes in volume. A phenomenology is proposed in which the pseudopotential is adjusted to fit the observed vibration spectrum. This phenomenology is applied to aluminum with a single adjustable parameter and the resulting energy-lattice wave-number characteristic given.