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. 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.

Computational chemistry has its roots in the early attempts by theoretical physicists, beginning in 1928, to solve the Schrödinger equation using hand-cranked calculating machines. By the 1950s, with the appearance of digital computers, serious attempts were being made to obtain highly accurate quantitative information about the chemical behavior of molecules via numerical approximations to the solution of the Schrödinger equation. In subsequent years, thanks to leaps in computing power and algorithms, methods have evolved from those that were used to study 1- and 2-atom systems in 1928, through those that were used to study 2- to 5-atom systems in 1970, to the present programs that produce useful quantitative results for molecules with up to 10 to 20 atoms. Other chemists, whose research can be accomplished with cruder models of the atom, have pushed this limit much higher. For instance, simpler approximations have long been used in the molecular mechanics approach that make possible the modeling of biological molecules with thousands of atoms.

Recent decades have witnessed a revolutionary expansion in the breadth and capability of theoretical and computational chemistry—with a commensurate rise in optimism regarding the ability of theoretical/computational chemistry to resolve pressing problems both of a fundamental scientific character and of clearly practical interest. Those outside the field may not realize that theoretical/computational chemistry, broadly defined, underpins rational drug design, contributes to the selection and synthesis of new compounds, and 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 U.S. 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 could have farreaching ramifications.

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

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Mathematical Challenges from Theoretical/Computational Chemistry
EXECUTIVE SUMMARY
Overview
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. 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.
Computational chemistry has its roots in the early attempts by theoretical physicists, beginning in 1928, to solve the Schrödinger equation using hand-cranked calculating machines. By the 1950s, with the appearance of digital computers, serious attempts were being made to obtain highly accurate quantitative information about the chemical behavior of molecules via numerical approximations to the solution of the Schrödinger equation. In subsequent years, thanks to leaps in computing power and algorithms, methods have evolved from those that were used to study 1- and 2-atom systems in 1928, through those that were used to study 2- to 5-atom systems in 1970, to the present programs that produce useful quantitative results for molecules with up to 10 to 20 atoms. Other chemists, whose research can be accomplished with cruder models of the atom, have pushed this limit much higher. For instance, simpler approximations have long been used in the molecular mechanics approach that make possible the modeling of biological molecules with thousands of atoms.
Recent decades have witnessed a revolutionary expansion in the breadth and capability of theoretical and computational chemistry—with a commensurate rise in optimism regarding the ability of theoretical/computational chemistry to resolve pressing problems both of a fundamental scientific character and of clearly practical interest. Those outside the field may not realize that theoretical/computational chemistry, broadly defined, underpins rational drug design, contributes to the selection and synthesis of new compounds, and 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 U.S. 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 could have farreaching ramifications.
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

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Mathematical Challenges from Theoretical/Computational Chemistry
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 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.
A number of fruitful collaborations between mathematical scientists and theoretical/computational chemists have occurred in the past. Noteworthy examples include the Nobel prize-winning work of Hauptman and Karle to advance the science of X-ray crystallography, now a basic tool; quantitative structure-activity relationships have led to the development of at least four commercially successful products (an antibacterial compound, two herbicides, and one fungicide); and insights into molecular structure have been gained from mathematical results in group theory and topology.
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. Many of the mathematical lines of research that, if reoriented, could contribute to chemistry are already being pursued in other contexts. The matrix in Figure ES.1 displays a subjective assessment of the depth of potential cross-fertilization between major challenges from theoretical and computational chemistry and relevant 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.
Conclusions and Recommendations
As a result of its investigations and collective evaluation of the available information, the committee has reached the following conclusions.
Several notable "success stories" can be identified, illustrating the value of interdisciplinary stimulation and synergistic research collaboration involving cooperation between the mathematical sciences and the theoretical/computational chemistry communities.
Many opportunities appear to exist for further collaborations between the mathematical and chemical sciences that could result in high-quality scholarship and research progress that would advance national interests. The productivity of applied computational chemistry would likely be enhanced as a result, which could be potentially significant for industry.
Active encouragement of further collaborations is warranted because it would likely result in an acceleration of such research progress.
Cultural differences between the mathematics and the chemistry communities, involving language, training, aesthetics, and research style, have tended to act as barriers to collaboration, even in circumstances that might otherwise suggest the benefit of cooperation.
Institutional structures and reward systems in the academic community have often placed significant difficulties in the way of collaborative research across traditional disciplinary boundaries, which can be especially inhibiting to those in early career stages.

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Mathematical Challenges from Theoretical/Computational Chemistry
Adaptive and multiscale methods
H
M
M
M
M
M
M
M
Special bases
H
M
M
H
Differential geometry
M
M
M
M
H
H
Functional analysis
H
M
H
H
H
M
M
M
Graph theory
M
M
M
H
M
M
M
Group theory
H
M
M
M
M
M
M
Optimization
H
H
H
H
M
H
H
M
H
H
H
Numerical linear algebra
H
H
M
M
M
Number theory
M
M
M
Pattern recognition
M
M
H
H
M
H
H
H
Probability and statistics
H
H
M
H
M
H
H
H
Several complex variables
H
H
M
M
M
M
Topology
M
M
H
H
M
M
H
H
H
Dynamical systems
M
H
H
M
H
M
Quantum electronic structure
Molecular mechanics
Condensed-phase simulations
Density functionals
N-representability
Design of molecules
Construction of potential energy functions
Gasp-hase dynamics
Polymers
Topography of potential energy surface
Biological macromolcules (including protein folding)
Figure ES.1
Subjective assessment of depth of potential cross-fertilization between major areas of the mathematical sciences and theoretical and computational chemistry. An "H" implies an overlap that is clearly promising and an "M" suggests that some synergy is likely.

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Government funding agencies have for the most part made constructive efforts to identify and fund worthy interdisciplinary and collaborative research. However, this process is still somewhat haphazard. Agencies tend to be organized along traditional disciplinary lines, and the evaluation of interdisciplinary proposals relies on personal contacts between program managers and on timely and comprehensive responses from what is typically a small pool of qualified reviewers. The time lapse involved in the proposal evaluation process thus has often been anomalously long.
To a large extent, both mathematical scientists and theoretical/computational chemists are relatively unaware of the most exciting recent advances in each others' fields. Consequently both groups tend to be insensitive to the opportunities for interdisciplinary cross-fertilization that could produce intellectual novelty and productivity enhancements on both sides.
The system of prizes and awards administered by the mathematical sciences and chemistry professional societies is currently not geared to recognize and reward interdisciplinary collaborative research advances.
The national environment—including Congress, funding agencies, and the professional societies (see, e.g., Joint Policy Board for Mathematics, 1994)—has become perceptibly more conducive to encouraging and supporting interdisciplinary and collaborative research, particularly as it may concern industrial innovation and productivity. Government agencies in particular are currently in a mood to actively encourage joint industrial-academic research, even though proprietary rights barriers to free collaboration are recognized to exist.
The overwhelming volume of specialized technical literature aggravates the communication problems between fields and occasionally leads to wasted effort, redundancy, and rediscovery. It appears that well-researched and well-written review articles spanning normally disconnected specialties in the mathematical sciences and in theoretical/computational chemistry represent a disproportionately small fraction of the technical literature, in spite of the fact that they can eliminate redundant effort.
In response to these conclusions and to the insights gained from its study, the committee makes the following recommendations:
Undergraduate Education. The best way to attract scientists to interdisciplinary work is to get them interested as undergraduates. It is recommended that universities encourage undergraduate interdisciplinary research courses, seminars, and summer programs.
Graduate Education. Departments in the mathematical and chemical sciences should encourage graduate degrees (both M.S. and Ph.D.) that involve dual (mathematics and chemistry) mentoring. Dual mentoring activity between chemistry and physics and chemistry and biology has been successful in many universities. The committee recommends that mathematics graduate students consider a minor in chemistry instead of a minor in an area of mathematics related to their research specialty. Theoretical and computational chemistry graduate students should consider a minor in mathematics or, alternatively, take a core of mathematical courses appropriate to their interest (perhaps in the framework of a special "interdisciplinary track").
Faculty Interaction. Mathematics and chemistry departments should on occasion invite a person from the other area to speak in a research seminar or a colloquium. Lists of speakers of potential interest to industry should be circulated to local industrial laboratories, and vice versa.

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Interdisciplinary Research. The committee recommends that mathematics and chemistry departments encourage and value individual and collaborative research that is at the interface of the two disciplines. Such work has the potential for significant intellectual impact on computational chemistry, and hence on the future evolution of chemical research and its applications to problems of importance in our society.
Professional Societies. Professional meetings in mathematics and chemistry—for instance, those of the American Mathematical Society (AMS), American Chemical Society, Society for Industrial and Applied Mathematics (SIAM), and the Chemical Physics Division of the American Physical Society—would benefit from talks very much like the seminar and colloquium talks described in the recommendation for faculty interaction above, from shorter presentations in special sessions, and from panel discussions. There are already some promising moves in this direction as reflected, for example, by recent AMS sessions on mathematics and molecular biology or SIAM sessions on molecular chemistry problems and optimization. These sessions at national and regional professional society meetings could ultimately lead to focused interdisciplinary meetings.
Prizes and Awards. The committee recommends that professional societies in the mathematical and chemical sciences examine the feasibility of establishing awards and named lectureships for work at the mathematics-chemistry interface. High-level public recognition by peers would be a major step toward breaking down interdisciplinary barriers.
Expository Articles and Books. Professional journals in mathematics and chemistry could enhance their quality, appeal, and influence by publishing expository articles on work at the mathematics-chemistry interface. There is a shortage of books written for someone who is mathematically (chemically) sophisticated and desires fairly precise but nonrigorous chemical (mathematical) explanations.
Interdisciplinary and Industrial Postdoctorals and Sabbaticals. Mathematics and chemistry departments should encourage postdoctoral and faculty sabbatical study at the mathematics-chemistry interface. The committee recommends that the chemical software, pharmaceutical, and chemical industries expand their use of mathematics postdoctorals and faculty on sabbatical leave, and increase their cooperation with and utilization of existing National Science Foundation (NSF) programs such as the University-Industry Cooperative Research Program in the Mathematical Sciences; Industry-Based Graduate Research Assistantships and Cooperative Fellowships in the Mathematical Sciences; Mathematical Sciences University-Industry Postdoctoral Research Fellowships; and Mathematical Sciences University-Industry Senior Research Fellowships. Another opportunity in this regard exists at the Institute for Mathematics and Its Applications at the University of Minnesota, which has an active industrial postdoctoral research program with the aim of broadening the perspectives of recent doctoral recipients in the mathematical sciences and preparing them for research careers involving industrial interaction.
Reference
Joint Policy Board for Mathematics, 1994, Recognition and Rewards in the Mathematical Sciences, American Mathematical Society, Providence, R.I.

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