numerically. But many deep problems remain unsolved. A prime illustration of the unsettled state of this field is that we do not yet have a satisfactory way of characterizing the intrinsically nonequilibrium, amorphous—i.e., glassy—states of matter.

Quantum Mechanical Calculations

The modern quantum theory of the structure of materials has its origins in the calculation of the cohesive energy of metals by E.Wigner and F.Seitz in the 1930s. With the advent of large computers during the past two decades, such calculations have achieved quantitative predictive capabilities when applied to regular (or very nearly regular) crystalline solids. Recent developments open the possibility of similar accuracy in describing irregular configurations such as crystalline deformations near defects, surfaces, or grain boundaries. It is even possible that the new methods will allow studies of metastable or strongly disordered states of matter.

In order to predict the structure of a solid, in principle, it is first necessary to calculate the total energy of the underlying many-body system of interacting electrons and nuclei for an arbitrary configuration of these constituents, and then to find the specific configuration that minimizes this energy. A typical computation of the kind that has been tested carefully during the past decade might proceed by, first, fixing the positions of the ion cores, then using what are known as self-consistent “density-functional” and “pseudopotential” methods to find the electronic ground-state energy in this configuration, including ion-ion interactions to compute the total energy, and finally comparing this energy with that of other configurations in order to determine the equilibrium state of the system as a whole. In recent applications of this technique, total energy differences between alternative crystalline structures have been obtained accurately to within a few tenths of an electron volt per atom, structural parameters to within tenths of angstroms, and bulk moduli and phonon frequencies to within a few percent. Note, however, that the method described above pertains only to zero-temperature ground states of regular crystalline arrays of atoms and not to irregular configurations or to alternative phases that might occur at higher temperatures.

Very recently, new methods for performing ab initio total energy calculations have been suggested that provide a novel way of carrying out the above procedure and that also can deal with arbitrary configurations of fairly large arrays of atoms—50 to 100 atoms in a supercell geometry using currently available computers. The basic idea is to minimize the total energy of the system by allowing both the electronic and the ionic degrees of freedom to relax toward equilibrium simultaneously. The useful computational scheme is known as “simulated annealing,” a recent development in mathematical optimization theory that has been borrowed



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