gas clusters, and yield both structural and optical properties of clusters exhibiting isomerization and phase transitions (Tsoo et al., 1990, 1992; Estrin et al., 1992). Simulated annealing has been applied to the Na₄ cluster using a generalized valence bond electronic wavefunction, but only a 500-femtosecond test run has been reported so far (Hartke and Carter, 1992). Ab initio quantum electronic structure methods for simulations would benefit if mathematical scientists improved the computer algorithms or advanced current theory.

Atomic-scale theories have proved to be very valuable in elucidating the properties of all sorts of materials. Electronic structure calculations have been essential in understanding electronic, optical, and transport properties, including elastic constants, conductivities, magnetic properties, and many more. Molecular dynamics is an explanatory mechanism and contributes to determining structural and mechanical properties of complex materials. It should be borne in mind, however, that atomic-scale theories are limited to length scales of a maximum of a few hundred angstroms. Most properties of complex materials, however, are determined by their collective microstructure (grain boundaries, precipitates, inclusions, microvoids, and so on) where the length scale is measured in microns and beyond (Baskes et al., 1992; Pantelides, 1992). Similarly, time scales of atomic simulations are measured in picoseconds or nanoseconds, whereas many diffusive and slip phenomena that underlie plasticity and other processes occur over time scales of microseconds, milliseconds, and beyond. It is therefore necessary to develop theories relating results obtained on atomic scales to desired material properties on more macroscopic length and time scales. A wide-open frontier for new mathematical sciences research is that of building connecting links between materials science theories developed for different length scales.