Mathematical Challenges from Theoretical/Computational Chemistry

This chapter highlights some of the most prominent research challenges from theoretical/computational chemistry that appear to be amenable to attack with the help of reasonable advances in the mathematical sciences. Most of the sections have as their s tarting point a challenge facing the chemical sciences, which is described in terms that should be accessible to the nonspecialist. The remaining sections--"Molecular Dynamics Algorithms," "Multivariate Minimization in Computational Chemistry," and "Fast Algebraic Transformation Methods"--contain discussions of topics from the mathematical sciences with specific insight into their relevance to theoretical/computational chemistry. Expositions and references are meant to give the mathematical reader suffici ent insight and direction to be able into subsequently investigate the topic via deeper reading and especially by discussions with colleagues from the chemical sciences. Note that the topics surveyed in Chapter 3 are also sources of continuing research o pportunity, some of which are identified therein.

As an overview of this chapter and a guide for navigating through it, the matrix in Figure 4.1 displays a subjective assessment of the depth of potential cross-fertilization between major challenges from theoretical and computational chemistry and relev ant topics in the mathematical sciences. This matrix is based to some extent on intuition because it is an assessment of future research opportunities, not past results. An "H" in the matrix implies an overlap that appears clearly promising, while an "M " suggests that some synergy between the areas is likely. The absence of an H or an M should not be taken to imply that some clever person will not find an application of that technique to that problem at some point.

	Quantum electronic structure	Molecular mechanics	Condensed- phase simulations	Density functionals	N-represent- ability
Adaptive and Multiscale methods	H	M	M
Special bases	H		M	M
Differential geometry		M
Functional analysis	H		M	H	H
Graph theory	M	M	M
Group Theory	H		M		M
Optimization	H	H	H	H	M
Numerical linear algbra	H	H		M
Number Theory			M
Pattern Recognition	M	M	H
Probability and Statistics	H		H		M
Several complex variables	H		H	M
Topology	M	M	H
Dynamical systems		M	H

	Design of molecules	Construc- tion of potential energy functions	Gas- phase dynamics	Poly- mers	Topo- graphy of potential energy surfaces	Biological macro- molecules (including protein folding)
Adaptive and Multiscale methods		M	M	M	M	M
Special bases			H
Differential geometry	M	M		M	H	H
Functional analysis		H	M		M	M
Graph theory	H			M	M	M
Group Theory	M		M	M		M
Optimization	H	H	M	H	H	H
Numerical linear algbra		M			M
Number Theory			M		M
Pattern Recognition	H		M	H	H	H
Probability and Statistics	H		M	H	H	H
Several complex variables			M		M	M
Topology	H	M	M	H	H	H
Dynamical systems			H	M	H	M

Two topics, polymers and protein folding, are consciously underrepresented in this report because the mathematical research opportunities related to these topics have been surveyed very recently by other reports from the National Research Council. For a discussion of mathematical opportunities related to the frontiers of polymer science, see pages 153-168 of Polymer Science and Engineering: The Shifting Research Frontiers (National Research Council, 1994). A chapter devoted to mathematical res earch into protein folding may be found in Calculating the Secrets of Life (National Research Council, 1995).

References

National Research Council, 1994, Polymer Science and Engineering: The Shifting Research Frontiers, National Academy Press, Washington, D.C.

National Research Council, 1995, Calculating the Secrets of Life, National Academy Press, Washington, D.C., in press.

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