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Mathematical Research in Materials Science: Opportunities and Perspectives
In analyzing and predicting the phases of a material and the nature of transitions between these phases, an electronic structure calculation (Callaway, 1991) is an important tool. Typically, electronic structure calculations are used to develop models of interatomic forces and to predict the zero-temperature configuration, that is, the crystal structure, of a large number of atoms and molecules. In these areas, algorithms are needed that can handle complicated materials efficiently and effectively.
Improved electronic structure methods and approximations are needed to treat systems with a large number of atoms or molecules. This is particularly true for such complicated structures as polymers, alloys, ceramics, and materials in which defects such as surfaces and dislocations are present. Improving the calculations of alloy properties will probably require the development of radically new methods because disorder is generally present in these materials. Determining microscopic forces between defects is an example of an especially fundamental and challenging calculation.
Improving many electronic structure methods requires going beyond the local density approximation to density functional theory. Density functional methods are based on a theorem that says the ground-state electronic energy is a universal functional of the electronic density, but that theorem provides no information on the nature of the functional. Mathematical analysis expresses the functional in pieces that are exactly known and lumps the unknown parts into a piece that is approximated, known as the local density approximation. This latter piece contains the dominant part of the electronic correlation energy, which is intrinsically a quantum many-body effect. Accurate computation of the electron structure of many materials requires improved approximations of this energy. How to do this has been the object of research for a number of years. Can bounds or inequalities be established for this part of the energy? and How must the assumptions be changed for different classes of systems? are natural questions to ask.
A general issue in electronic structure calculations is the effective utilization of parallel computers. The use of parallel computers for electronic structure calculations has been developing slowly because of the size of the computational codes, uncertainty in possible computational gains, and the difficulty in adjusting well-developed procedures to novel computational environments. Most parallel computing in this area has involved the joint efforts of a materials theorist, an applied mathematician, and a computer scientist.
Ab initio electronic structure calculations for clusters, which are common activities of the quantum chemist, are an example of a situation where strong coupling appears between treating the electron correlation energy properly and using parallel computers. The major difficulty with these methods, as with their counterparts used in condensed matter physics (mentioned in Chapter 2), is the need to correlate many electrons. Knowing how to effectively exploit parallel computers for such intensive calculations would be beneficial.
In recent years, electronic structure calculations and the study of quantum many-body phenomena have been attempted by Monte Carlo methods (Doll and Gubernatis, 1990; De Raedt and von der Linden, 1992). The quantum mechanics imposes significant constraints on the nature and utility of the algorithms used. In almost all cases, these algorithms possess what is called the sign problem, in which the transition probabilities needed in the Monte