C. D. Gelatt

There is a deep and useful connection between statistical mechanics (the behavior of systems with many degrees of freedom in thermal equilibrium at a finite temperature) and multivariate or combinatorial optimization (finding the minimum of a given function depending on many parameters). A detailed analogy with annealing in solids provides a framework for optimization of the properties of very large and complex systems. This connection to statistical mechanics exposes new information and provides an unfamiliar perspective on traditional optimization problems and methods.

We present a conceptual model and calculational procedure for the study of the electronic structure of metallic compounds. The model consists of spherical atoms compressed into finite volumes appropriate to the solid. The model involves no adjustable or experimentally derived parameters. All contributions to the total energy (other than the Madelung energy) are obtained from independent compressed-atom calculations. Interatomic interactions enter the calculations through the electronic configuration (the distribution of the valence charge among $s$, $p$, $d$, etc., states) and boundary conditions which give the atomic valence levels a finite width. These environmental constraints, which specify the state of the compressed atoms, are obtained from energy-band calculations. For the latter we introduce a new method, which we call the augmented-spherical-wave (ASW) method to stress its conceptual similarity to Slater's augmented-plane-wave (APW) method. The ASW method is a direct descendant of the linear-muffin-tin-orbitals technique introduced by Andersen; when applied to pure metals, it yields results which closely approximate those of the much more elaborate Korringa-Kohn-Rostoker calculations of Moruzzi, Williams, and Janak. The combined ASW compressed-atom procedure is tested on (i) the empty lattice, (ii) the pure metals Na, Al, Cu, and Mo, and (iii) the ordered stoichiometric compounds NaCl, NiAl, and CuZn. Finally, we demonstrate the utility of the procedure by using it to study the anomalous tendency of Ni and Pd (as compared to their Periodic Table neighbors Co, Cu, Rh, and Ag) to form hydride phases. We have calculated the total energies of the six pure metals and their monohydrides. The total energy differences exhibit the anomaly and an analysis of quantities internal to the calculation reveals its origin.

We present a theory of the chemical bond in compounds consisting of both transition metals and nontransition metals. Chemical trends in the bonding properties are established by directly comparing the total energies of a large number of such compounds with the total energies of their constituents. These chemical trends are analyzed in terms of the $s$-, $p$-, and $d$-like state densities of the compounds and the constituents. Rather different types of bonding are shown to result when the atomic $s$ and $p$ levels of the nontransition metal lie above, below, and near the energy of the transition-metal $d$ level. The heat of compound formation is shown to result from a competition between two simple physical effects: (1) the weakening of the transition-metal bonds by the lattice dilatation required for the accommodation of the nontransition metal, and (2) the increased bonding which results from the occupation of the bonding members of the hybrid states formed from the interaction between the transition-metal $d$ states and the $s\ensuremath{-}p$ states on the nontransition metal. Our theoretical values for the heats of formation of these compounds are generally similar to those given by Miedema's empirical formula. Distinctive aspects of the variation of the heat of formation with the number of valence electrons reveal, however, that the microscopic picture on which the empirical formula is based is quite different from that given by our self-consistent energy-band theory.

Calculations of the electronic structure of transition-metal hydrides are applied to the cohesive energy of $3d$ and $4d$ monohydrides, and the single-particle lifetime of states in nonstoichiometric Cu and Pd hydrides. A simple formula is presented which delineates the principal contributions to the cohesive energy of the hydrides: (i) the formation of a metal-hydrogen bonding level derived of states of the pure metal band structure which have $s$ symmetry about the site of the added proton, (ii) a slight increase in binding of the metal $d$ bands due to the added attractive potential, and (iii) the addition of an extra electron to the metal electron sea. The calculations, corrected for Coulomb repulsion at the hydrogen sites, qualitatively reproduce the experimental trends of the heats of formation of the transition-metal hydrides. The single-particle lifetime calculations are in quantitative agreement with Dingle-temperature measurements and they correctly predict the existence of essentially undamped states on the hole sheets of the $\ensuremath{\alpha}$-phase PdH Fermi surface.