The disparate natures of the fields of mathematics and chemistry have led to quite different training frameworks, which in turn continuously influence and are influenced by the research styles and practices that are most common in the two communities. Both fields are highly heterogeneous, containing many distinct research subfields, each with its own style, technical language, and point of view. While sweeping generalities are consequently inaccurate, broad stylistic differences can be identified in the ways in which mathematicians and chemists think, approach problems, and interact with their own and other communities. This chapter attempts to identify specific impediments to interactions between mathematics and chemistry, beyond the general factors affecting interdisciplinary research (see National Research Council and Institute of Medicine, 1990, for example).

Many of the problems that interest and drive computational/theoretical chemists are practical (chemical syntheses, understanding of structure-function relationships in macromolecules, drug design, and so forth). To investigate these problems, chemists must build theories and perform numerical simulations to aid in understanding phenomena that can potentially be verified by experiment or yield new (predictive) data. The importance of mathematics in chemistry is revealed by the success of chemists in developing effective algorithms not only for specialized problems in chemistry, but also for generic mathematical problems—for example, evaluating integrals, calculating matrix eigenvalues, storing and compressing data, finding multidimensional optima and stationary points, and generating numerical solutions to the one-dimensional Schrödinger equation.

The key rule of mathematics in computational chemistry highlights an anomaly: although theoretical chemists understand sophisticated mathematics and make heavy use of the mathematical literature, they have typically not involved mathematicians directly in either the development of models or algorithms or the derivation of formal properties of equations and solutions. In fact, theoretical chemists have become accustomed to self-reliance in mathematics. To date, this system has worked because most leading computational chemists have significantly more training in mathematics than the minimum recommended today for an American chemist (see Box 5.1).

By contrast to the situation in chemistry, the motivation for studying problems in much of pure mathematics does not depend on any connection to an application. Mathematical problems are often studied because of their inherent beauty, richness, and depth, without considering utility or relevance in any time frame. Even when addressing real-world applications, mathematicians tend to view problems as generic rather than specific.

In particular, ''real'' problems, whether from the biological or physical sciences, criminology, or cryptography, are almost always posed incompletely in a strictly mathematical sense. The transformation of a real-world problem into a tractable mathematical form involves increasing levels of abstraction and assumption; the physical terms are defined in a mathematical framework, and various components of the problem may be removed or idealized to build a model that emphasizes what are believed to be the most crucial features.

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5
CULTURAL ISSUES AND BARRIERS TO INTERDISCIPLINARY WORK
The disparate natures of the fields of mathematics and chemistry have led to quite different training frameworks, which in turn continuously influence and are influenced by the research styles and practices that are most common in the two communities. Both fields are highly heterogeneous, containing many distinct research subfields, each with its own style, technical language, and point of view. While sweeping generalities are consequently inaccurate, broad stylistic differences can be identified in the ways in which mathematicians and chemists think, approach problems, and interact with their own and other communities. This chapter attempts to identify specific impediments to interactions between mathematics and chemistry, beyond the general factors affecting interdisciplinary research (see National Research Council and Institute of Medicine, 1990, for example).
Motivation and Connections
Many of the problems that interest and drive computational/theoretical chemists are practical (chemical syntheses, understanding of structure-function relationships in macromolecules, drug design, and so forth). To investigate these problems, chemists must build theories and perform numerical simulations to aid in understanding phenomena that can potentially be verified by experiment or yield new (predictive) data. The importance of mathematics in chemistry is revealed by the success of chemists in developing effective algorithms not only for specialized problems in chemistry, but also for generic mathematical problems—for example, evaluating integrals, calculating matrix eigenvalues, storing and compressing data, finding multidimensional optima and stationary points, and generating numerical solutions to the one-dimensional Schrödinger equation.
The key rule of mathematics in computational chemistry highlights an anomaly: although theoretical chemists understand sophisticated mathematics and make heavy use of the mathematical literature, they have typically not involved mathematicians directly in either the development of models or algorithms or the derivation of formal properties of equations and solutions. In fact, theoretical chemists have become accustomed to self-reliance in mathematics. To date, this system has worked because most leading computational chemists have significantly more training in mathematics than the minimum recommended today for an American chemist (see Box 5.1).
By contrast to the situation in chemistry, the motivation for studying problems in much of pure mathematics does not depend on any connection to an application. Mathematical problems are often studied because of their inherent beauty, richness, and depth, without considering utility or relevance in any time frame. Even when addressing real-world applications, mathematicians tend to view problems as generic rather than specific.
In particular, ''real'' problems, whether from the biological or physical sciences, criminology, or cryptography, are almost always posed incompletely in a strictly mathematical sense. The transformation of a real-world problem into a tractable mathematical form involves increasing levels of abstraction and assumption; the physical terms are defined in a mathematical framework, and various components of the problem may be removed or idealized to build a model that emphasizes what are believed to be the most crucial features.

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At its best—which is usually when the model building is done collaboratively, drawing on the particular insights and strengths of both mathematical and chemical scientists—this diagnostic approach is extremely valuable, leading to both theoretical and numerical understanding of the model, which can then return both quantitative information and conceptual understanding about the original physical problem. Indeed, carefully designed models can suggest and explain properties that are counterintuitive or unexpected to the problem posers themselves. An example is found in macroscopic models of supercoiled DNA, in which higher buckling catastrophes, as extensions to elasticity theory, were recently found. Furthermore, analysis of mathematical models can indicate directions for additional physical and numerical experimentation, as well as for extensions to the model. A potentially negative effect of mathematical abstraction, however, is that the theoretical formulation may lose its relevance to the original application that motivated it.
Effects of Disciplinary Boundaries
Beyond the "cultural" differences just described, attempts to build collaborations between mathematicians and chemists encounter boundaries imposed within most universities as well as within other structures in which disciplinary divisions are strong. These borders are especially difficult to cross early in a scientist's career, ironically when the potential for interdisciplinary work may be greatest because of the appeal of "new frontiers." Institutional practices influence the style of work that is valued (and hence often pursued) in each discipline, as well as the level of regular interaction and communication. Since many computational chemists and mathematicians are faculty members at research universities, both fields are affected by the value system of academia, in which recognition (promotion and tenure) requires a record of individual accomplishment judged as outstanding by one's peers. During the process of creating a portfolio of publications, any perceived dilution of a faculty member's personal contribution through collaborations may be seen as undesirable. This discouragement of collaborative work early in one's career applies to both mathematics and chemistry, although chemists have a strong countervailing tradition of working in groups.
For mathematicians, the potential career damage of collaboration rises when it involves work in a field seen as peripheral to mathematics. In some instances, interdisciplinary work may be regarded by one's mathematical colleagues as "not real mathematics" or as less valuable than traditional mathematics. Most academic mathematicians would agree that it is difficult to obtain accurate and convincing evaluations of "interdisciplinary" work (meaning work that involves significant contributions from other sciences) and research in nontraditional areas of mathematics. In this connection, there is a recent report on the recognition and reward system in the mathematical sciences (Joint Policy Board for Mathematics, 1994). Such issues are particularly worrying for junior mathematicians, since it would be unusual for nonmathematicians to be asked for help in a tenure or promotion evaluation; mathematics departments might well be reluctant to rely on outsiders for judgments and decisions viewed as a departmental prerogative. Another related issue is the value attached to work in which an existing body of mathematics is applied to another scientific problem area; even if the impact is great and the work represents a significant scientific advance, it is not ''new mathematics,'' and hence may be accorded little weight in an evaluation of research contributions.
Because of the tendency to preserve and protect departmental boundaries, mathematics departments are ill-equipped to cope with questions that inevitably arise if mathematicians become seriously interested in interdisciplinary problems. For example, if a young mathematician is hired as a

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BOX 5.1 American Chemical Society Curriculum Standards for Mathematical Course Work
"Students should emerge from an ACS-approved program in chemistry with:
A firm foundation in the fundamentals and applications of calculus, including knowledge of differential equations and proficiency with partial derivatives.
An understanding of the basic principles of linear algebra and practical knowledge of statistics with applications to such problems as experimental design, validation of data, and optimization procedures.
Experience with computers, including programming, numerical and non-numerical algorithms, simulations, data acquisition, and use of databases for information handling and retrieval."
SOURCE: Undergraduate Professional Education in Chemistry: Guidelines and Evaluation Procedures, American Chemical Society, Washington, D.C., 1992, p. 11.
numerical analyst and subsequently becomes interested in chemical statistical mechanics, should this be viewed as a loss or a gain?
For academic chemistry departments, analogous principles of departmental autonomy can affect chemists seeking to work with mathematicians. Because theoretical/computational chemists must often demonstrate the applications of their work to experimental areas of chemistry, fundamental work of a mathematical nature—for example, algorithm development or identification of problem features amenable to mathematical attack—may be undervalued. On balance, chemistry departments have more experience in evaluating multidisciplinary research, soliciting judgments as needed from a variety of scientists both inside and outside the department. A further positive effect on interdisciplinary work is that chemistry departments tend to value research that has a significant impact on thinking, research, and practice in chemistry and other areas.
For both fields, the difficulty of interdisciplinary collaboration is exacerbated by the lack of a well-established network of contacts between mathematicians and chemists. On most university campuses, chemistry and mathematics departments are physically separate, so casual daily contact does not occur. An effort is typically required for faculty to attend all the seminars in their own department, let alone in other departments. This reality aggravates the difficulty not only of initiating a collaboration, but also of developing an appreciation of the other discipline's challenges. Faculty members are not immune to misperceptions and stereotypes: chemists may regard mathematicians as unapproachable or uninterested in chemistry problems; mathematicians may not realize that chemistry problems contain interesting and novel mathematics.
There are, however, exceptions. For instance, in the United Kingdom there is a long fruitful history of productive mathematical research being initiated by theoretical scientists ("natural philosophers") employed as faculty members of mathematics departments. This goes back to Newton, but the tradition continues to modern times. D.R. Hartree and P.A.M. Dirac are recent examples at Cambridge, and C.A. Coulson spent most of his career as a professor of mathematics at Oxford. Some departments in the United States also have established atmospheres that are conducive to collaborative work.
Outside of academia (e.g., pharmaceutical companies), cross-disciplinary work between chemists and mathematicians has succeeded because the problems of disciplinary boundaries are less pervasive in many instances, and team efforts are often the norm. Issues of tenure, grants, and promotion are nonexistent or less important. These settings should provide models for collaborative research.

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Effects of the Curriculum
The disparate natures of mathematics and chemistry have led to different training frameworks, which in turn continuously influence and are influenced by the research style and practices in the two fields. The typical curricula encountered by students of chemistry and mathematics, both undergraduate and graduate, do not help to decrease the gaps described above.
In mathematics, basic courses rarely involve exposure to the physical "roots" of problems; mathematicians study idealized problems as exemplars, not for details of the real-world problem. Part of the gap specifically between mathematics and chemistry can be explained by long-standing pedagogical practices in mathematics. Much of classical applied mathematics is based on constructions associated with mechanics and physics: every student of mathematics studies the heat equation, elastic rods, electrical networks, and fluid flow. However, no problems explicitly associated with chemistry are widely taught to or recognized by mathematicians.
There is little time or incentive for mathematics students to learn chemistry at a substantive level, let alone to study interesting chemistry problems. At the undergraduate level, some mathematics curricula require courses in a physical science, but these are more often in physics than chemistry. Although undergraduate mathematicians sometimes take freshman chemistry (frequently a descriptive course), they are unlikely to study physical or organic chemistry. Graduate students in mathematics do not typically take many courses outside their own department and hence have no convenient mechanism for learning about mathematical problems in chemistry.
Education of chemists currently involves little exposure to advanced concepts in modern mathematics (see Box 5.1). Undergraduate chemists take calculus and (perhaps) ordinary differential equations, linear algebra, or numerical methods, but seldom study abstract algebra, differential geometry, numerical analysis, partial differential equations, probability, or topology. Graduate students in chemistry rarely take courses in mathematics departments. Some chemists believe that undergraduate chemistry courses do not require high-level mathematics and prefer instead to build chemical intuition by descriptive methods. For chemistry students and faculty interested in learning modern mathematics, the mathematics curriculum is structured like a tree, with courses of potential interest to chemists at the end of a very long branch of prerequisites; the effect is to discourage chemists from obtaining any knowledge of advanced topics.
Language Differences
Language barriers ranging from conspicuous to subtle must be overcome by anyone who wishes to pursue interdisciplinary work between mathematics and chemistry. At the most obvious level, specialization and the internal communication requirements of mathematics and chemistry have created two technical languages. Thus, fundamental concepts that occur in only one field need to be defined either in the technical language of the other or in a common natural language. Mathematical examples include Pisot numbers, ambient isotropy, and wavelets; chemical examples include ligand, pharmacophore, and racemate.
Within mathematics, each research subdiscipline continually refines concepts and introduces new technical jargon, making it very difficult even for mathematicians in slightly different research areas to communicate with each other. Similar language problems exist in chemistry, though perhaps to a lesser degree. On the mathematical side, the problem is compounded by preferences for an abbreviated writing style. The "abstract minimalism" approach to writing taken in much of the mathematics literature can make graduate texts and research papers in mathematics almost impenetrable except to the most determined readers.

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BOX 5.2 Information Sources About Theoretical/Computational Chemistry
For a mathematician who wants to get involved with theoretical and computational chemistry, the best source of information should be the chemists at his or her own institution or research center. The American Chemical Society (ACS) publishes a directory (Directory of Graduate Research) that lists the research interests and recent publications of academic chemists, if one wishes to look further afield. Those mathematical scientists who have established productive collaborative or interdisciplinary lines of research often observe that one must be a good listener and be willing to devote time and energy to learning nuances of language and concepts.
There are a number of printed reviews available. For molecular modeling, the book series Reviews in Computational Chemistry (D. Boyd and K. Lipkowitz, eds., VCH Publishers, New York), gives a good overview of the field. Also, a visit to the exposition at an ACS national meeting (held twice a year) will give a feeling for the large number of software vendors in this field, the type of software available, and the types of problems of interest to chemists. The November 1993 issue of Chemical Reviews was also devoted to this subject. A variety of approaches to computer-based drug design are discussed in the series Comprehensive Medicinal Chemistry (Pergamon Press, Oxford).
For electronic structure problems, the literature is very scattered. The book series Advances in Quantum Chemistry reviews the more mathematical aspects of the field along with some very applied results. The July/August 1991 issue of Chemical Reviews contained reviews on a wide range of applications. There is unfortunately no comprehensive review of the algorithms involved in popular programs, although the user's guides to GAUSSIAN, MELD, and HONDO list many of the papers on which these programs are based. Also, the Modern Techniques in Computational Chemistry reports (E. Clementi, ed., ESCOM Science Publishers, The Netherlands) discuss many algorithms. The book series Relativistic and Electron Correlation Effects in Molecules and Solids (G.L. Malli, ed., Plenum Press, New York) and the series Methods in Computational Molecular Physics (G.H.F. Diercksen and S. Wilson, eds., Reidel Publishing, Dordrecht) also contain several volumes devoted to methods for quantum chemistry. The annual "Sanibel" meeting organized by the Quantum Theory Project at the University of Florida (and now held at St. Augustine, Florida) is a good place to meet quantum chemists. The papers from that meeting are published annually in a special symposium series from the International Journal of Quantum Chemistry.
An excellent discussion of the molecular dynamics method, Monte Carlo calculations, and related methods for computer simulation studies of materials is contained in Computer Simulation of Liquids (M.P. Allen and D.J. Tildesley, Oxford University Press, 1987). This monograph is a useful resource for learning about the theory of such simulations as well as the algorithms used in research.
Finally, the Computational Chemistry List is a very active electronic clearinghouse for information on that subject. Interested readers may subscribe by sending their name, affiliation, and electronic mail address to chemistry-request@osc.edu.
In addition to separate sets of terminology, mathematicians and chemists face several varieties of linguistic confusion. Similar concepts in both disciplines are sometimes denoted by different words. For example, a (nontrivial) "link" to a mathematician is a collection of elastic circles that are mutually entangled and cannot be separated spatially into subcollections; to a chemist, a "catenane" (from the Latin "catena" or chain) is a collection of circular molecules held together by topological bonds, not by chemical bonds. Mathematics invokes the concept of graph isomorphism type, whereas chemists speak of connectivity of the molecular graph.
At the other extreme, identical technical names may be used by the two disciplines for different concepts or for concepts that are similar but vary in precision. For example, the term "topology" is

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BOX 5.3 Information Sources About the Mathematical Sciences
Journals and Newsletters
Acta Numerica contains an annual review article on important developments in numerical analysis.
Bulletin of the Institute of Mathematical Statistics contains information on meetings and articles on items of current interest in statistics.
Mathematical Reviews contains short reviews of articles on a wide spectrum of mathematical topics.
Proceedings of the International Congress of Mathematicians is published every four years and contains invited survey articles by leading mathematicians on recent mathematical progress.
Notices of the American Mathematical Society contains information on meetings and articles on items of current interest in core mathematics.
SIAM News is the news journal of the Society for Industrial and Applied Mathematics, containing information on meetings and items of current interest in applied and applicable mathematics.
SIAM Review contains technical review papers on subjects in applied and numerical mathematics.
Electronic Information
e-MATH is the electronic information connection to the American Mathematical Society. Information is available on meetings, membership, electronic sources of mathematics information, etc. Telnet e-math.ams.org; login and password are both "e-math" (lower case).
NA DIGEST is a numerical analysis electronic newsletter; for information, send e-mail to na.help@na-net.ornl.gov.
NETLIB is a database of public-domain computer programs with e-mail address netlib@ornl.gov.
used in science as a catchall word describing shape phenomena, but it has an exact technical meaning in mathematics. The word "homotopic" can refer to deformation of paths in mathematics and to interconvertible protons in nuclear magnetic resonance (NMR) spectroscopy.
Toward a Fruitful Collaboration
Because research inherently moves into the unknown, there is no way to predict reliably which areas of mathematics and chemistry might work together effectively. Interactions between computational chemists and computational mathematicians are perhaps the most obviously rewarding today for addressing large-scale computational problems that occur in quantum chemistry, molecular mechanics, and molecular dynamics; these areas seem promising for serious collaborations since progress is likely to be made only by combining significant expertise in chemistry, mathematics, and computer science. However, the ranges of opportunity and success stories are very broad (see Chapters 3 and 4 of this report), and fruitful interactions may emerge between chemists and mathematicians in any subfield.
The ideal interdisciplinary collaboration often begins with personal contacts between two or more scientists who share an interest in a particular problem. Just as the experimental chemist might tend to approach the theoretical/computational chemist for assistance in certain areas, it is perhaps more

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common for a chemist to initiate a mathematics-chemistry collaboration. Recently, however, there has been an increasing emphasis on involvement by mathematical scientists in "grand challenge" problems, and some applied mathematicians have actively sought connections with chemistry.
Once a contact has been initiated, the success of a collaboration depends on a strong sense of mutual respect and benefit among the participants. These feelings are essential so that each partner is willing to learn new science as needed (e.g., protein chemistry for the mathematician, optimization theory for the chemist) and to adapt to a somewhat different style than he or she is accustomed to, welcoming a combination of theory, computation, and physical intuition toward solution of a problem. Such a synergistic process allows an "evolution" of solutions that can progressively address more of the complexity of the realistic problem and incorporate new physical data as they become available.
References
Joint Policy Board for Mathematics, 1994, Recognition and Rewards in the Mathematical Sciences, American Mathematical Society, Providence, R.I.
National Research Council and Institute of Medicine, 1990, Interdisciplinary Research: Promoting Collaboration Between the Life Sciences and Medicine and the Physical Sciences and Engineering, National Academy Press, Washington, D.C.

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