Computational chemistry has its roots in the early attempts by theoretical physicists, beginning in 1928, to solve the Schrödinger equation (see Box 2.1) using hand-cranked calculating machines. These calculations verified that solutions to the Schrödinger equation quantitatively reproduced experimentally observed features of simple systems such as the helium atom and the hydrogen molecule. Approximate solutions for larger systems and exact solut ions to simple model problems allowed chemists and physicists to make qualitative explanations of spectra, structure, and reactivity of all types of matter.
During the Second World War, electronic computers were invented, and in the decade after the war these became available for general use by scientists. At the same time, physicists generally became more interested in nuclear structure and lost interest in the details of molecular structure and spectra. Hence, beginning in the mid-1950s, a new discipline was developed, primarily by chemists, in which serious attempts were made to obtain quantitative information about the behavior of molecules via numeri cal approximations to the solution of the Schrödinger equation, obtained by using a digital computer. The present success of this field has come largely from the enormous increase in speed, and decrease in cost, of computers, with significant improv ements also attributable to many developments in algorithms and methodology. During the 1960s, three major developments in algorithms and methodology made quantum chemistry a useful tool: computationally feasible, accurate basis sets were developed; reas onably accurate approximate solutions to the electron correlation problem were demonstrated; and formulas for analytic derivatives of the energy with respect to nuclear position were derived. These developments were incorporated into several software pac kages that were made readily available to most chemists in the early 1970s, leading to an explosion in the literature of applications of computations to chemical problems. These programs are used to predict and explain the structure and reactivity of mol ecules and to complement the information obtained from many types of spectral measurements. Refinement of the program packages has, of course, continued, with emphasis on increased accuracy, increased size of molecules that can be studied, and adaptation to new computer hardware. The present 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 resu lts for molecules with up to 10 to 20 atoms. Much of the current research in new methods is aimed at developing methods that are feasible for even larger molecules.
A classic example of the power of the theoretical/computational approach is the work in the 1960s by W. Kolos and L. Wolniewicz. Explicit r12 calculations had been introduc ed for the hydrogen molecule in 1933 by James and Coolidge, and Kolos and Roothaan, working together in Mulliken's lab, improved these calculations in 1960. Subsequently, Kolos teamed up with Wolniewicz to author a sequence of papers of increasing accura cy. Their results diverged from the accepted (experimentally derived) dissociation energy of H2. When all known corrections were included, Kolos and Wolniewicz's best estimate of the discrepancy (in 1968) was 3.8 cm-1. Thus prodded, experimentalists reexamined the issue and in 1970 a new spectrum of better resolution and a new assignment of the vibrational quantum numbers of the upper electronic state were published. Both of the se results were within experimental uncertainty of the best theoretical result.
While the emphasis of one aspect of computational chemistry has been on solving the many-body electronic structure problem, another group of chemists has focused on using the resulting potential energy surface for studying nuclear motion. This has led to a collection of programs for doing classical, semiclassical, and quantum calculations. Since 1980, use of these programs has become a routine tool for modeling molecules and gas-phase chemical reactions. These computations yield collision cross secti ons, both differential and integral, for elastic, inelastic, and reactive events. These approaches require, as with transition state theory, potential energy surface(s) obtained using quantum chemical methods of solution of the electronic Schrödinger equation. The Schrödinger equation for nuclear motion is solved subject to a scattering boundary condition, which takes the form of coupled differential, integro-differential, algebraic, or integral equation systems. The methods used to solve the se coupled systems of equations are drawn from the applied mathematics literature as well as from algorithm improvements developed by computational chemists.
Meanwhile, simpler approximations have long been used by chemists to estimate the energy of molecules near their equilibrium geometry. In the molecular mechanics approach (see Box 2.2) the total energy of a chemical system is approximated by a sum of simple terms involving distances between atoms, bond angles, and dihedral angles. These terms involve estimated parameters that are assumed to have the same values as similar parameters obtained by data fitting for simpler molecules. (Chemists have long known that many structural and energetic features of molecules are nearly transferable between similar subfragments of molecules.) This representation of the energy has made possible the modeling of biological systems and rational drug design. It is also at the heart of the computational engine of many programs that produce three-dimensional computer graphics images of molecules. Molecular mechanics has become so prevalent that many chemists now equate it wi th computational chemistry. This approach has allowed the modeling of molecules with thousands of atoms. The practical disadvantage is that only structural types previously encountered in smaller molecules can be parameterized for larger molecules, so m any parameters remain unknown. The conceptual disadvantage is that this is no longer a first principles theory and the connection to the Schrödinger equation is unclear. Hence, there can be no rigorous estimate of the potential errors in this appro ach and its success relies on chemical intuition for finding suitable molecules from which to develop the "transferable" parameters.
Another important thread in theoretical chemistry has been the study of many-particle systems such as liquids, solid materials, and biological macromolecules. The major framework for this study has been statistical mechanics--a subject with its formal roots in the nineteenth century. In the 1930s, the study by physical chemists of structure and thermodynamics accelerated with the advent of simple ideas about intramolecular and intermolecular forces. Equilibrium statistical mechanics has offered many questions of principle--for example, the question of the nature, and even definition, of phase transitions. These questions fostered a long-standin g cross-fertilization between workers in both the mathematical and chemical communities (see Box 2.3). Similarly the study of phenomena away from equilibrium (e.g., the transport phenomena of hydrodynamics and the chemical rates) attracted the fundamental thinkers in statistical mechanics starting in the 1950s. Recently, corresponding deep questions of principle about disordered systems such as glasses have attracted workers from both communities.
Although a large part of statistical mechanics can be studied without computers, machine calculations for many-body simulation made an early impact in the 1950s and have grown to be the dominant mode of investigation. Monte Carlo methods, invented at t he weapons laboratories by workers such as Fermi, Ulam, von Neumann, Metropolis, and Teller, were used immediately to address the many-body problems relevant to the thermodynamics of liquids. Such Monte Carlo approaches were adapted quickly to the study of polymers as well. The numerical solution of Newton's laws for many-particle models, so-called molecular dynamics (see Box 2.3), was also first carried out by theoretical chemists in the late 1950s and early 1960s. The application of molecular dynamic s and Monte Carlo methods to proteins and other biomolecules in the 1970s has led to their widespread use throughout the theoretical and experimental chemical communities. Since significant advances in the efficiency of the algorithms used in molecular d ynamics and Monte Carlo simulation are needed to address the forefront questions such as protein folding, a renewed contact of theoretical chemists with the numerical mathematics community has recently involved collaborative efforts of mathematicians, che mists, and physicists.
The advent of molecular quantum mechanics was followed by a very successful theory of chemical reaction rates that modeled a reactive event as passage over a reaction barrier on a multidimensional potential energy surface representing the energy as a fu nction of the internal coordinates of the reacting system. In its simplest form, the model corresponds to the system moving from reactants to transition states (the critical configuration), from which the system moves to reaction products. This conceptua lly simple model has remained the predominant approach for estimating rates of chemical reactions. Because of the multidimensionality of the reactive system, however, it is computationally difficult to implement rigorously. Over the years, efforts have focused on improving methods to estimate reaction barriers and properties of the reactants, and these have required better solutions of the electronic and nuclear transition states.
The roots of much of the mathematics now finding application to computational chemistry extend back at least to the eighteenth or nineteenth century, although, as illustrated in Chapters 3 and 4 of this report, the most up-to-date developments in the ma thematical sciences can also be very natural tools. Group theory traces its origin to fundamental studies of geometries, but from it has come the theory of groups of motions, continuous groups, Lie groups, and Lie theory. The need to understand function s on the sphere and other surfaces led to the representation theory of groups and to various kinds of function theory. These theories grew up with the creation of quantum mechanics and fed, and were fed by, quantum mechanics. Much of operator theory and integral equations came from physics and engineering, as did the general theory of harmonic analysis. Numerical linear algebra and numerical analysis developed largely as tools for fluid mechanics and military applications, but their usefulness is vastl y more widespread than that.
After World War II the mathematics community entered a period of intense development of its core, accelerating the growth of fields such as topology, number theory, algebraic geometry, and graph theory. Advances were largely motivated by questions gene rated by the internal structure of mathematics and not by contact with the outside world. In recent years, however, attention has once again turned outward, and the products of this intense period are now being applied widely in novel ways. The advent o f modern computing capacity has enabled mathematicians to generate computational algorithms that yield answers--when combined with proper modeling techniques--to important practical problems. Success has been achieved in signal processing, sound and imag e compression, flow problems, and electromagnetic theory. Historically, mathematical scientists have worked more closely with engineers and physicists than with chemists, but recently many fields of mathematics such as numerical linear algebra, geometric topology, distance geometry, and symbolic computation have begun to play roles in chemical studies.
Before proceeding to accounts of past and potential contributions that mathematics can make to progress in chemistry, it should be emphasized that the challenge of interdisciplinary research is not one of scientific content alone, but also one of scient ific process. Neither the chemist nor the mathematician is generally the optimal person to construct a mathematical model, as the model by its very nature lies at the interface between theory and observation. To build the model, an iterative process of refinement is required, in which mathematical considerations motivate approximations that need to be checked against reality, and in which key chemical insights necessarily force levels of mathematical complexity. It is exactly this need for iterative mo del construction that may motivate the collaboration of mathematicians and chemists, against the self-referential and conservative tendencies of each discipline. Focusing on this process of iterative model construction can help clarify the roles of the c ollaborators in interdisciplinary research, and by extension illustrate the goals for their respective disciplines as attempts are made to lower the hurdles to such collaborations. The model is both the interface between the disciplinary boundaries and t he lingua franca between the cultures.
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