Although much of its discovery process is descriptive and qualitative, chemistry is fundamentally a quantitative science. It serves a wide range of human needs, activities, and concerns, a theme forcefully documented in the comprehensive Pimentel report, *Opportunities in Chemistry* (National Research Council, 1985), which presented the status of chemistry as of 1985. The mathematical sciences provide the language for quantitative science, and this language is growing in many directions as computational science in general continues its rapid expansion. A timely opportunity now exists to strengthen and increase the beneficial impacts of chemistry by enhancing the interaction between chemistry and the mathematical sciences.

Computational chemistry is a natural outgrowth of theoretical chemistry, the traditional rule of which involves the creation and dissemination of a penetrating conceptual infrastructure for the chemical sciences, particularly at the atomic and molecular levels. The mathematical sciences have been indispensable allies and have provided vital tools for that rule. Theoretical chemistry has also sought to devise and to implement quantitative algorithms for organizing massive amounts of data from the laboratory, and for predicting the course and extent of chemical phenomena in situations that are difficult or even impossible to observe directly; thus, today it is difficult to classify many lines of research as either "theoretical" or "computational." This report tends toward the term theoretical/ computational—any distinction between the two areas is rather misleading because the subject demands both quantitative characterization and conceptual understanding.

Even before the advent of computers as a major component in physical science research, the theoretical tradition in chemistry had accumulated a substantial membership: in its 1966 report entitled *Theoretical Chemistry, A Current Review*, the Westheimer committee estimated that in 1964, approximately 200 theoretical chemists with faculty appointments in graduate-degree-granting institutions could be identified in the United States (National Research Council, 1966, p. 3).

The subsequent three decades have witnessed a revolutionary expansion in the breadth and capability of theoretical and computational chemistry, as well as in its population. These changes, of course, have been driven by the rapid evolution of computers and by their widespread availability in the scientific community. The resulting impact has been enormous and has expanded the range of research activity in theoretical/computational chemistry to encompass the entire spectrum from purely analytical theory, through simulational study of mathematically well-defined models, to the adroit development of powerful and general computational algorithms. Indeed, for the purposes of this document, the committee takes the viewpoint that theoretical/computational chemistry constitutes a seamless continuum of research activities that deserves to be assessed as a whole.

If the mailing lists of theoretical chemistry conferences can be taken as evidence, the current number of theoretical/computational chemists working in the United States has grown to approximately 1000 (John C. Tully, Chairman of 1993 International Conference on Theoretical Chemistry, personal communication). To some extent, this expansion in population has occurred in the academic community. But more significantly, it represents a major growth in the industrial and government sectors, and reflects an increasing realization that theoretical and computational chemistry contributes to the national economic and security welfare. The last three decades have exhibited a general rise in expectations and optimism surrounding the ability of theoretical/computational chemistry to resolve pressing problems both of fundamental scientific character and of clear practical application. The historical record of these expectations can be seen in reports, for example, of workshops and studies held during the early days of the "supercomputer era" (National Research

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Mathematical Challenges from Theoretical/Computational Chemistry
1
INTRODUCTION
Although much of its discovery process is descriptive and qualitative, chemistry is fundamentally a quantitative science. It serves a wide range of human needs, activities, and concerns, a theme forcefully documented in the comprehensive Pimentel report, Opportunities in Chemistry (National Research Council, 1985), which presented the status of chemistry as of 1985. The mathematical sciences provide the language for quantitative science, and this language is growing in many directions as computational science in general continues its rapid expansion. A timely opportunity now exists to strengthen and increase the beneficial impacts of chemistry by enhancing the interaction between chemistry and the mathematical sciences.
Computational chemistry is a natural outgrowth of theoretical chemistry, the traditional rule of which involves the creation and dissemination of a penetrating conceptual infrastructure for the chemical sciences, particularly at the atomic and molecular levels. The mathematical sciences have been indispensable allies and have provided vital tools for that rule. Theoretical chemistry has also sought to devise and to implement quantitative algorithms for organizing massive amounts of data from the laboratory, and for predicting the course and extent of chemical phenomena in situations that are difficult or even impossible to observe directly; thus, today it is difficult to classify many lines of research as either "theoretical" or "computational." This report tends toward the term theoretical/ computational—any distinction between the two areas is rather misleading because the subject demands both quantitative characterization and conceptual understanding.
Even before the advent of computers as a major component in physical science research, the theoretical tradition in chemistry had accumulated a substantial membership: in its 1966 report entitled Theoretical Chemistry, A Current Review, the Westheimer committee estimated that in 1964, approximately 200 theoretical chemists with faculty appointments in graduate-degree-granting institutions could be identified in the United States (National Research Council, 1966, p. 3).
The subsequent three decades have witnessed a revolutionary expansion in the breadth and capability of theoretical and computational chemistry, as well as in its population. These changes, of course, have been driven by the rapid evolution of computers and by their widespread availability in the scientific community. The resulting impact has been enormous and has expanded the range of research activity in theoretical/computational chemistry to encompass the entire spectrum from purely analytical theory, through simulational study of mathematically well-defined models, to the adroit development of powerful and general computational algorithms. Indeed, for the purposes of this document, the committee takes the viewpoint that theoretical/computational chemistry constitutes a seamless continuum of research activities that deserves to be assessed as a whole.
If the mailing lists of theoretical chemistry conferences can be taken as evidence, the current number of theoretical/computational chemists working in the United States has grown to approximately 1000 (John C. Tully, Chairman of 1993 International Conference on Theoretical Chemistry, personal communication). To some extent, this expansion in population has occurred in the academic community. But more significantly, it represents a major growth in the industrial and government sectors, and reflects an increasing realization that theoretical and computational chemistry contributes to the national economic and security welfare. The last three decades have exhibited a general rise in expectations and optimism surrounding the ability of theoretical/computational chemistry to resolve pressing problems both of fundamental scientific character and of clear practical application. The historical record of these expectations can be seen in reports, for example, of workshops and studies held during the early days of the "supercomputer era" (National Research

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Council, 1974, 1975, 1976; Schatz, 1984; Berne, 1985). Not surprisingly, physics and engineering manifested similar experiences at the same time (National Science Foundation (NSF) Advisory Committee for Physics, 1981; Lax, 1982; NSF Working Group on Computers for Research, 1983; National Research Council, 1984).
The pervasive significance and widespread applicability of theoretical and computational chemistry may not always be immediately obvious to those not frequently concerned with this activity. Nevertheless, it is central to rational drug design, it contributes to the selection and synthesis of new materials, and it guides the design of catalysts. New quantum mechanical techniques underlie the understanding of electronic properties of materials and have advanced the level of precision at which molecules of at least moderate size can be modeled. Furthermore, computational chemistry software is a set of tools used increasingly by chemists of many persuasions. These various abilities and facilities have proved to be very important to American industry, and their advancement would generate even further industrial benefits. Engaging problems and deep challenges for mathematical scientists are posed by the needs of theoretical and computational chemists, and the products of mathematical research in these areas can have far-reaching ramifications.
The marked growth of theoretical/computational chemistry inevitably has involved a substantial national investment of skilled human resources and of expensive computing resources (both hardware and software). Both of these types of commodities are relatively scarce and are subject to competition between alternative scientific and technological disciplines. Table 1.1 shows, for instance, that software for theoretical and computational chemistry claims much of the cpu usage on the Cray Y-MP at the San Diego Supercomputer Center. Data from other NSF supercomputer centers reveals similar patterns. What Table 1.1 does not show is the heavy dependence of these chemistry codes on mathematical software such as LINPACK and EISPACK. The productivity of these computational resources, broadly construed, must be an issue for continual analysis and informed action by policymakers. In particular, the strong mathematical flavor of theoretical/computational chemistry leads to a natural examination of the efficacy of links between the mathematical and the chemical sciences, and to the past, present, and future rules of interdisciplinary research at the interface between these subjects. These issues constitute basic concerns for the present study.
The 14 chemists, biochemists, and mathematical scientists from industry, government, and academia who attended a 1991 workshop at the National Research Council (NRC) decided that the interface of the mathematical sciences and theoretical/computational chemistry was an area that deserved encouragement, and that a fuller study of the issues was warranted. Subsequently, the Board on Mathematical Sciences and the Board on Chemical Sciences and Technology of the NRC jointly proposed a study to identify research opportunities for the mathematical sciences relevant to computational chemistry, with the goal of engaging the talent of more mathematical scientists in the problems of computational chemistry, which should produce advances of benefit to both the mathematical and the chemical sciences. The phrase "computational chemistry" was to be interpreted to include those areas related to molecular structure and its determination, broadly defined; it was felt that there was less need to promote greater participation by mathematical scientists in the areas of computational chemistry on the macroscopic scale—including such topics as reaction/diffusion modeling and most of chemical engineering. On securing approval and funding for this study, a Committee on Mathematical Challenges from Computational Chemistry was selected, with its first meeting held in Washington, D.C. on March 29–30, 1994. Two subsequent meetings took place: June 910, 1994, in Washington, D.C., and September 911, 1994, in Woods Hole, Massachusetts.
The statement of task given to the Committee on Mathematical Challenges from Computational Chemistry reads as follows: "The committee will investigate and report on opportunities for collaborative and synergistic research in the mathematical sciences that can accelerate progress in theoretical and computational chemistry and their applications, and make recommendations for promoting this research." It was clear from the outset that the study could not presume to be

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TABLE 1.1 Top ten applications in terms of percentage of CRAY C90 usage at the San Diego Supercomputer Center for the period December 1, 1993, to August 17, 1994
Time Used (%)
Application
Description
7.1
ESP
Molecular dynamics
6.7
Gaussian
Quantum chemistry
5.4
AMBER
Molecular dynamics
2.6
TREESPH
Galactic dynamics
2.1
GAMESS
Quantum chemistry
2.0
ARGOS
Molecular dynamics
1.5
CGCM
Coupled ocean-atmosphere global climate model
1.5
DMOL
Quantum chemistry
1.3
COULMETL
Materials science
1.2
DIEL
Materials science
SOURCE: Wayne Pfeiffer, San Diego Supercomputer Center, personal communication.
exhaustive. However, it seemed realistic to strive for representative sampling of the two communities involved and to identify instructive examples of past collaborative successes, likely prospects for interdisciplinary synergy, and barriers to joint research that could be removed or at least lowered.
In order to supplement its own breadth of expertise, as well as to reach out to the mathematical sciences community, the committee invited guests to its first two meetings to learn from their perspectives. At its first meeting, the committee engaged in a lengthy discussion with Richard Herman, chair of the Joint Policy Board for Mathematics, learning about the range of attitudes in that community toward interdisciplinary research and about efforts to adjust the community's priorities on many fronts (Joint Policy Board for Mathematics, 1994). At its second meeting, the committee invited an optimization researcher (Margaret Wright, of AT&T Bell Laboratories, incoming president of the Society for Industrial and Applied Mathematics), a statistician (Douglas Simpson of the University of Illinois at Urbana-Champaign), and a researcher in computational fluid dynamics (David Keyes from the National Aeronautics and Space Administration's Langley Research Center). These guests were invaluable, both for their insights about interdisciplinary research opportunities and for their perspectives on how the committee might influence the mathematical sciences community.
In scanning the research needs of theoretical/computational chemistry, the committee found opportunities for synergistic research with almost the entire mathematical sciences community, where that term is used in its broadest sense to include core and applied mathematicians, statisticians, operations researchers, and theoretical computer scientists in academe, government laboratories, and industry. The common denominator shared by mathematical scientists who have contributed or could contribute to progress in chemistry is not a particular background; rather, it is a willingness to truly collaborate.
Readers may wish to note that two other recently issued reports have a strong bearing on matters

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considered herein. The NRC has completed a parallel study entitled Mathematical Research in Materials Science, which examines many of the same kinds of prospects, barriers, and cures discussed below, although some key distinctions become clear (National Research Council, 1994). The present report gives a somewhat heavier emphasis to biological applications of computational chemistry to avoid excessive overlap with that earlier report. The second is Recognition and Rewards in the Mathematical Sciences by a committee of the Joint Policy Board for Mathematics (1994), the recommendations of which are consistent with those contained herein.
The committee believes that this report has relevance and potentially valuable suggestions for a wide range of readers. Several important target audiences and the kinds of benefits they might expect to derive are the following:
Graduate departments in the mathematical and chemical sciences could glean suggestions for promising research directions for graduate students and young scientists, ideas about how to foster interdisciplinary collaborations, and insight into new types of job opportunities that may appear in the future.
Federal and private agencies that fund research in the mathematical and chemical sciences—including federal policymakers involved in the high-performance computing and communications, materials science, and biotechnology initiatives—can find suggested topics that provide links between the fields, high-priority research topics at the interface, and suggestions for fostering collaborations.
Selected industrial and government research and development laboratories can learn of ways in which research from the mathematical sciences could be used to improve the productivity of theoretical and computational chemists.
Developers of software and hardware for computational chemistry can gain more insight into the rule that the mathematical sciences could play.
Selected individual researchers can find inspiration and background for promising research directions (especially for graduate students and young researchers), ways in which their existing lines of research may have parallels or applications in another field, and suggestions for initiating collaborations.
Chapter 2 of this report covers some history of computational chemistry for the nonspecialist, while Chapter 3 illustrates the fruits of some past successful cross-fertilization between mathematical scientists and computational/theoretical chemists. In Chapter 4 the committee has assembled a representative, but not exhaustive, survey of research opportunities. Most of these are descriptions of important open problems in computational/theoretical chemistry that could gain much from the efforts of innovative mathematical scientists, written so as to be accessible introductions to the nonspecialist. Chapter 5 is an assessment, necessarily subjective, of cultural differences that must be overcome if collaborative work is to be encouraged between the mathematical and the chemical communities. Finally, the report ends with a brief list of conclusions and recommendations that, if followed, could promote accelerated progress at this interface. Recognizing that bothersome language issues can inhibit prospects for collaborative research at the interface between distinctive disciplines, the committee has attempted throughout to maintain an accessible style, in part by using illustrative boxes, and has included at the end of the report a glossary of technical terms that may be familiar to only a subset of the target audiences listed above.
References
Berne, Bruce J., 1985, organizer, Supercomputers in the Simulation and Modeling of Chemical Systems, National Science Foundation workshop, Arden House, Harriman, N.Y., April 26–28.

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Joint Policy Board for Mathematics, 1994, Recognition and Rewards in the Mathematical Sciences, American Mathematical Society, Providence, R.I.
Lax, Peter D., 1982, chairman, Large Scale Computing in Science and Engineering, sponsored by the Department of Defense and the National Science Foundation in cooperation with the Department of Energy and the National Aeronautics and Space Administration.
National Research Council 1966, Theoretical Chemistry, A Current Review, National Academy Press, Washington, D.C.
National Research Council 1974, A Study of a National Center for Computation in Chemistry, National Academy Press, Washington, D.C.
National Research Council 1975, A Proposed National Resource for Computation in Chemistry: A User Oriented Facility, National Academy Press, Washington, D.C.
National Research Council 1976, Needs and Opportunities for the National Research for Computation in Chemistry (NRCC), National Academy Press, Washington, D.C.
National Research Council 1984, Computational Modeling and Mathematics Applied to the Physical Sciences, National Academy Press, Washington, D.C.
National Research Council, 1985, Opportunities in Chemistry, National Academy Press, Washington, D.C.
National Research Council, 1994, Mathematical Research in Materials Science, National Academy Press, Washington, D.C.
National Science Foundation Advisory Committee for Physics, 1981, Prospectus for Computational Physics, National Science Foundation, Arlington, Va.
National Science Foundation Working Group on Computers for Research (Kent K. Curtis, Chairman), 1983, A National Computing Environment for Academic Research, National Science Foundation, Arlington, Va.
Schatz, George C. (organizer), 1984, Future Directions fr Supercomputer Use in Chemistry, National Science Foundation workshop, Evanston, Ill., October 15–17.

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