Computational Chemistry in the Undergraduate Chemistry Curriculum

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Computational Chemistry in the Undergraduate Chemistry Curriculum: Development of a Comprehensive Course Formula† Zbigniew L. Gasyna and Stuart A. Rice* Department of Chemistry, The University of Chicago, 5735 South Ellis Avenue, Chicago, IL 60637; *[email protected]

The intellectual basis for including computational chemistry in the undergraduate chemistry curriculum can be summarized as follows. Progress in the past decade in the capability for carrying out realistic computations of chemical phenomena and processes has been breathtaking, and that rapid progress is continuing. It is a truism that the availability of modern computer hardware and software has transformed the manner in which many chemical problems are analyzed, making possible the testing of ideas and the design of processes that previously could only be dreamed of. Drawing on these developments, there are already many parts of the undergraduate chemistry curriculum into which computation has been integrated, typically as a tool that eases the burden of didactic calculation, or for molecular modeling, or to illustrate the solutions of, for example, the Schrödinger equation for model systems. Yet, despite being a common component of the undergraduate chemistry curriculum and notwithstanding its various implementations, there is still considerable disagreement as to what is to be achieved by the study of computational chemistry. An excellent survey of the relevant pedagogical issues, and a review of the approaches to teaching computational chemistry in place at several institutions, has been published by DeKock et al. (1). Moreover, particular undergraduate programs in computational chemistry have frequently been described in this Journal (see, for example, articles dealing with electronic structure calculations, refs 2–4). For that reason we will focus attention in this report on the character of the recently introduced junior–senior level Computational Chemistry course at The University of Chicago, using specific examples to illustrate the approach taken. We believe a study of computational chemistry should generate an understanding of the interplay between basic theory and computational methodology, that it should illuminate the circumstances under which computation is the preferred tool for solving problems and illustrate the accuracy with which those problems can be solved. Above all, we believe that a study of computational chemistry should enhance critical thinking about scientific goals which computation can assist in achieving. A course of this type requires both lecture presentation and extensive practice in the computational laboratory. Moreover, the diversity of subject matter to be illustrated and the complexity of the underlying theories imply that the student’s progress needs to be closely monitored and that adequate methods for evaluation of the student’s work must be devised. † Presented at the 30th Great Lakes Regional Meeting of the American Chemical Society, Loyola University of Chicago, Chicago, IL, May 28–30, 1997.

There are, of course, many ways of achieving these goals. Our implementation focuses on problem solving through applications and does not involve any level of code writing by the student. The lecture material is addressed to review of the relevant theoretical basis for various chemical properties, processes, and phenomena and to the teaching of methods of calculation that can be used to carry out numerical evaluation of quantities of physical and chemical interest. The course material has been organized to provide the maximum possible correlation between the laboratory exercises and the lecture material. Since the lecture material in computational chemistry encompasses theoretical topics from several fields of chemistry, completion of a course in physical chemistry is a prerequisite. Course Structure

The Computational Laboratory The computational laboratory hardware consists of a cluster of 12 Silicon Graphics O2 workstations running IRIX 6.3 (Silicon Graphics, Inc., version of UNIX), an HP LaserJet 5M black-and-white printer, and an HP DeskJet 1600CM color printer. The computers are sufficiently powerful (180 MHz R5000SC, 64 Mb memory, 4.3 Gb disk, 17′′ monitors, system CAM) to carry out extensive computations efficiently, and universal enough to have application software available from various vendors. All of the machines are connected via a 100BaseT, 100Mbps Ethernet local area network and share a server that stores a common database of names and passwords as well as some of the software. The design and operation of a laboratory to which students have unlimited access plays an important role in their learning. Instruction relevant to particular applications programs is given in specified laboratory sessions. Other than those scheduled periods, the laboratory is open to students 12 hours each weekday and at other times by special arrangement. The laboratory is supervised by a full-time professional chemist who is qualified to deal with both the computer technology and the tutoring of individual students in Computational Chemistry. In addition, teaching assistants recruited from a pool of experienced and knowledgeable graduate students are directly involved in the instructional laboratory periods.

Manuals The students are provided with a list of textbooks on the theory and practice of numerical analysis and various subjects in theoretical and computational chemistry (see Appendix); only the numerical analysis textbook is required. Notes covering all of the lecture material are distributed to the students following the pertinent lecture.

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To assist a student in carrying out the laboratory exercises we have prepared a manual and written instructions pertinent to each of the course sections. In addition, the laboratory manual is intended to serve three purposes. First, it gives summaries of the most important elements of the theory related to the computational exercises. Second, it reviews some fundamental aspects of the IRIX operating system and introduces the reader to the particular program under the IRIX environment. Third, the manual contains a series of exercises designed to demonstrate how to utilize the program and to illustrate the importance of the proper approach to solving a particular problem.

The Lecture Material The scope and applicability of computational methods in chemistry are too large to be covered in a one-quarter course, so some compromise between depth and breadth of coverage must be reached. As currently constructed, the course in computational chemistry is organized around sample scientific problems (see below). The computational approach to the solution of all such problems requires precise formulation of the mathematical description of the system, the introduction of concepts which permit the selection of appropriate approximations, the transformation of the mathematical representation to a form that is most convenient for numerical solution of the equations, and execution of the computations. The course does not cover the writing of computer code. Instead, emphasis has been placed on the use of available applications programs, some examples of which are Mathematica, the suite of packages associated with MSI/Biosym, and GAMESS. These general applications programs are regarded as tools for solving scientific problems, and attention is focused on the assumptions used in developing the code (e.g., cutoffs in potential functions) and the expected accuracy and systematic limitations of the algorithms used. The lecture and laboratory contents of our Computational Chemistry course include four major topics that are, in our opinion, essential ingredients of the subject. The topics we have chosen to emphasize are Numerical Methods of Analysis, Molecular Mechanics for Conformational Analysis, Molecular Dynamics and Monte Carlo Simulation, and Electronic Structure Calculations. In each of these major areas of computational chemistry, specific application software is used. The application software chosen reflects the expertise available in the Department of Chemistry. In use, each application package is supplemented with utility programs running under the IRIX environment.

In the computational laboratory, instruction in numerical analysis is based on Mathematica (5). This program has found application in a variety of topics in the undergraduate curriculum, many of them reported in this Journal (6–9). Mathematica allows for handling of formulas, numbers, text, and graphics. In its window-based environment, it provides an interactive interface in which the equations can be solved analytically or numerically, and various functions can be introduced from a pull-down menu. Descriptive text may be placed inside the document and two- and three-dimensional plots can be embedded sequentially. The program has an online reference and help system to assist students. The exercises covering numerical methods can be based on other similar programs, of which Mathcad (10) is the most closely related. The students are given homework assignments that focus attention on particular numerical methods. We start with evaluating expressions and simple symbolic computation, as, for example, symbolic differentiation of functions, and then introduce various methods of data manipulation, such as interpolation, and fitting using the linear and polynomial least squares methods. We show how to use the program to plot functions and data. More advanced topics are described below. Iteration Iterations using Mathematica can be performed in many ways. As an example, we employ an iteration method to produce the Fibonacci numbers. Other examples of iteration procedures are then introduced along with more complex applications of the program. Symbolic, Numerical, and Monte Carlo Integration Monte Carlo integration is a method based upon the generation of random numbers and statistical sampling. In order to evaluate an integral b

f x dx

I=

(1)

a

of some particular function f (x) in the interval [a,b] it is approximated by b

b –a N

f x dx ≈

a

N

f Σ i=1

xi

It is pointed out that the error of the integration is of the order of 1/N 1/2, reflecting the fact that the variance in f (x), which is a measure of the extent to which f (x) deviates from its average value over the region of integration, is

Course Material

Numerical Methods of Analysis The lecture material dealing with numerical analysis is intended to sample several topics that are ubiquitous components of applications software. The goal of the presentation is to make the student aware of the mathematical basis for an algorithm and of the trade-offs between accuracy and convenience of use of an algorithm, and to provide a basis for understanding how algorithms for solving particular problems can be selected. The topics covered are computer arithmetic, interpolation and extrapolation, numerical differentiation, numerical integration, and the solution of ordinary differential equations. 1024

(2)

σ I2 ≈ 1 σ f2 = 1 1 N N N

N

2 fi – Σ i=1

1 N

N

fi Σ i=1

2

(3)

As an example, we show how to implement the Monte Carlo procedure in Mathematica to evaluate the integral and standard deviation for various large values of N. The example is integration of 1

I =4 0

dx 1 + x2

(4)

This integral is equal to π and is also calculated, for com-

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parison, using the internal Mathematica integrating function. Students are asked to tabulate the computed values of π and their standard deviations for different values of N. They should then be able to notice the improvement in the accuracy of the calculated value with increasing N and to examine the tradeoff of accuracy for computational time. Other mathematical functions are then integrated using the Monte Carlo method, and techniques for reducing the variance and improving the efficiency of the calculations are described and discussed. Complex Numbers and Fourier Transforms Mathematica allows the execution of operations on complex numbers. Fourier transformation of data, which inherently involves operations on complex numbers, is one of the meaningful applications, because Fourier transform methods have become important in chemical applications since the advent of FTIR and FT-NMR. Mathematica supports calculation of both the forward and the inverse Fourier transform of discrete data. The purpose of this exercise is to calculate the Fourier transform of a spectrum and to carry out the inverse transform to reproduce the spectrum. The original data can be modified in many ways, including the addition of random noise. The noisy spectral data are then transformed by applying a filter function that passes only the “low-frequency” information, allowing the student to remove the noise. The result of the inversion calculation is then compared with the starting spectrum. Matrices: Hückel Molecular Orbital Calculations Mathematica provides an excellent vehicle with which to illustrate the application of Hückel theory in the calculation of molecular orbital energies and the coefficients in the wave function representation, because the program handles matrices and operations on matrices with great ease. A good model compound for such calculations is benzene. Hückel theory calculations of the π-electronic structure of the benzene molecule deal with a general case of degenerate energy levels, which requires special attention. When there is degeneracy, the standard Mathematica procedure for solving the secular matrix only guarantees that the eigenvectors are linearly independent. To use the Hückel eigenvectors as the wave function coefficients, they all must be mutually orthonormal. The orthonormalization of the eigenvector matrix for benzene is accomplished by application of a suitable procedure from the Mathematica online linear algebra package. Solving Equations Solving equations with Mathematica can be carried out in various ways. There are simple internal procedures which solve polynomial equations. The limitations in the application of these procedures to certain equations is demonstrated with suitable examples. For solving simultaneous linear equations, the matrix method is implemented. Differential Equations in Chemical Kinetics There are many procedures that can be used to solve differential equations numerically. We have selected the fourth-order Runge–Kutta method for demonstration purposes. The examples demonstrate the usefulness of this technique in studies of chemical kinetics. A rate equation for a simple first-order reaction A → B, for which dx/dt = {kx, is solved and compared with the exact solution. The method is then applied to coupled differential equations that describe various

chemical processes; for example, calculation of the rate of product formation for the consecutive chemical reactions A → B, B → C is investigated. The time-dependent consumption of the substrate A and the evolution of the intermediate B and the product C are computed. Several more complex chemical processes are also studied using this technique, among them oscillatory chemical reactions. Three-Dimensional Plots—Visualization of Atomic Orbitals The graphical display of atomic and molecular orbitals is a focus of attention throughout the chemistry curriculum. With Mathematica the wave functions of simple atomic orbitals (s, p, d) can be computed and displayed. An example of such a wave function representation is a mathematical description of the 2p orbital ψ 2pz ∝ ze { kr

(x 2

y2

(5)

z 2)1/2,

with r = + + and k a constant. The wave function isosurface (i.e., the surface on which |ψ 2pz| has a constant value) can be plotted employing graphics functions from the Mathematica add-on package. Introduction of different shading or color schemes for the positive and negative regions of the wave function provides an additional challenge in generating this graphical representation.

Molecular Mechanics—Multidimensional Minimization The lecture material dealing with molecular mechanics examines the various contributions to the potential energy of an isolated molecule and the definition of suitable potential energy functions. The use of interpolation procedures and tables in the calculation of components of the potential energy function, the treatment of solvent effects on molecular conformation, and multidimensional minimization are then discussed. Primitive versions of Monte Carlo and molecular dynamics augmentations of conformation searches are addressed. Attention is called to the differences between the energies and conformations of molecules predicted by the several variants of molecular mechanics applications software. In the computational laboratory, molecular mechanics and molecular dynamics exercises are carried out using the Insight II (Molecular Simulations, Inc.) suite of programs. Some examples of analyses carried out by the students are cited below. Cyclohexane and 1,4-Cyclohexadiene Conformational Analyses Using the Builder or Biopolymer module of Insight II, students assemble cyclohexane and 1,4-cyclohexadiene molecules and analyze their conformations. Using the Discover module, the students optimize the structure of the molecule under different force fields (Amber, cvff, cff91) and explore various minimization algorithms, such as steepest descent and conjugate gradient. The effects of including cross terms and of use of the Morse potential are investigated. Cyclooctane Conformational Analysis See the discussion under this subheading in the following section.

Monte Carlo and Molecular Dynamics Simulations Monte Carlo methods are used for many purposes. We have focused attention on the calculation of the equilibrium


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properties of a many-body system of interacting particles. One example of a Monte Carlo procedure, namely Monte Carlo integration, is discussed in the numerical analysis part of the lectures; it is developed with Mathematica. In this part of the course we show that similar mathematical problems can be solved using simple Fortran programs compiled under IRIX. The general lecture material dealing with Monte Carlo simulations discusses all the major procedures associated with calculating the thermodynamic properties of many-molecule assemblies and includes the discussion of various sampling methods and of simulations in several different ensembles. The analysis of molecular dynamics methods includes the discussion of algorithms for the integration of the equations of motion, simulations in various ensembles, and formulations of the equations of motion that permit the definition of thermodynamic boundary conditions. The computational laboratory exercises carried out by the students are listed below. Monte Carlo Simulations Typical exercises deal with the simulation of a simple molecular liquid in the constant-NVT ensemble; the example calculations are based on Fortran programs. The differences between the structures (i.e., the pair correlation functions) of fluids that interact with hard-sphere and Lennard–Jones potentials are examined. Molecular Dynamics Simulations A molecular dynamics simulation of the structure of a simple liquid (e.g., H2O) is carried out using the Discover program of Insight II. Cyclooctane Conformational Analysis Starting from one of the conformations of the cyclooctane molecule the student runs a conformational search using the unrestrained Molecular Dynamics procedure of Insight II. This procedure generates many cyclooctane conformers. One of the parameters that affect the molecular dynamics simulations is temperature. We suggest using temperatures of 300, 600 and 900 K in these simulations. The analysis of the Molecular Dynamics runs requires construction of graphs of the trajectories—that is, system energy vs frame (or time) and cluster graphs. The cluster graphs are indispensable for the identification of new conformers formed in the simulation process. The completed assignment requires screen printouts containing graphs and molecular structures, identification of different conformers obtained in the simulations, and explanation of the effect of temperature based on the analysis of the runs output. Folding a DNA Double Helix Because some students enrolled in the course are concentrating in one or another Biological Chemistry/Molecular Biology program, we have introduced exercises that pertain to biological systems. A typical example is the exercise in which a student attempts to restore the double helix structure of a partially denatured oligonucleotide double strand using the Molecular Dynamics procedure of Insight II. Denaturing of DNA structures is associated with the breakage of hydrogen bonds connecting the complementary nucleotide bases and partial to complete unfolding of the strands. The hydrogen bonds of interest and their typical lengths are provided for the complementary bases C–G and T–A. This Molecular 1026

Dynamics analysis requires that certain constraints be imposed on the structure during the computations in order to subject the system to folding. The superposition of the calculated structure on the double helix of the reference native structure gives the means to evaluate the outcome of calculations.

Electronic Structure Calculations The lecture material deals with the electronic structure of atoms and molecules from the point of view of approximations to the solution of the Schrödinger equation. The calculation of the electronic structures of atoms is used as the vehicle for introducing the independent electron model, the self-consistent field scheme, the Hartree–Fock method, and the importance of correlation energy. Quantum mechanical perturbation theory and the variational principle are discussed. The Born–Oppenheimer approximation is introduced, along with a qualitative discussion of the circumstances under which it breaks down. The role of symmetry in restricting the class of solutions to the Schrödinger equation is illustrated for a simple molecule. The student is also introduced to the principal ideas involved in several semiempirical theories of molecular electronic structure. The computational laboratory exercises are based on use of the GAMESS (General Atomic and Molecular Electronic Structure System) program (11) (http://www.msg.ameslab.gov/ GAMESS/GAMESS.html ). GAMESS is a very flexible application program that can be used to calculate the equilibrium geometries, vibrational frequencies, electric dipole moments, and properties of the excited electronic states of molecules. It can also be used to calculate reaction coordinates and transition states and classical trajectories on a molecular potential energy surface. GAMESS has built-in a wide range of methods for the calculation of the electronic structure of a molecule. Other similar ab initio programs can be adopted in these calculations, one obvious choice being Gaussian (12). The GAMESS input and output formats are not particularly user-friendly, and the program has only a text interface. A companion program Molden (http://www.caos.kun.nl/ ~schaft/molden/molden.html) is available under UNIX for viewing the results of the calculations. This freeware program reads the GAMESS output files and displays data in twoand three-dimensional formats. Molden is particularly easy to use and it has proved to be an indispensable graphical interface to GAMESS. One of the difficulties in formulating the input data for the ab initio calculation of the electronic structure of a molecule is associated with the necessity of providing the initial coordinates for the molecule, most commonly in the form of a Z-matrix. In some cases, the students can take advantage of their previous knowledge of Insight II to generate the Zmatrix in the MOPAC format, which is one of the input data formats recognized by GAMESS. Several of the exercises that have been implemented in the course are described below. The Structure of Water We begin the hands-on exercises for GAMESS with a study of the structure of the water molecule. The student calculates the geometry, electric dipole moment, and normal mode frequencies of water in the ground electronic state using three different basis sets, and compares the results to see how each quantity converges (or fails to converge) to the experimental



value as the quality of the representation improves. The calculations start with the STO-3G basis set and are repeated using a larger basis set, such as the 3-21, 6-31G and 6-31 G* basis sets. The question to be answered is if the larger basis set or the inclusion of the set of diffuse functions gives a significant improvement when the results are compared to experimental values. Looking at the calculated relative frequencies and the X-Y-Z components of the normal modes, students should be able to identify the symmetric and antisymmetric stretching and the bending normal modes. We ask the students to comment on which calculated quantities are most reliably inferred from electronic structure calculations, and also which are most accurate, and how strongly they depend on the choice of basis set. GAMESS allows students to calculate the properties of excited electronic states of molecules as well as those of the ground electronic state. In this case, we use an MCSCF (CASSCF) approach to get the energy of an excited electronic state of water. Students are required to define the configuration interaction space—that is, to use their own judgment in describing the ground state and the excited electronic states in terms of the number of singly occupied α - and β-spin orbitals, assigning the number of frozen core orbitals and allocating the number of occupied and virtual orbitals. HCN Isomers and the Dissociation Energy of the H–CN Bond In this exercise the students employ GAMESS to estimate the relative energies of HCN and HNC and to estimate the dissociation energy for the H–CN bond. The restricted Hartree–Fock level of description, based on an STO-3G basis set, is used for simplicity. The linearity of HCN allows the use of Cartesian coordinates rather than the Z-matrix in the input file for the geometry optimization. Calculations are repeated for the case of C 1 symmetry and the results are compared. The input file is rearranged so that it represents linear HNC and the energies and bond lengths and angles are compared for the optimized geometries of HCN and HNC. The calculation of the dissociation energy of the H–CN bond provides the opportunity to explore the large disparity between the results obtained for the character of the dissociation limit from the restricted Hartree–Fock and the unrestricted Hartree–Fock levels of calculation. In this case, the former extrapolates to a dissociation limit which is H+ + CN {, whereas the latter extrapolates to the correct dissociation limit H? and CN?.

correction term. A comparison of the optimized molecular geometries is used to see if there are any significant changes due to the correlation energy. The Barrier for Inversion in NH3 The ammonia molecule has a C3v pyramidal structure, which is known to undergo inversion through an umbrellalike motion. In this exercise the student estimates the barrier for the isomerization using GAMESS. First, the energy is calculated for the pyramidal configuration, then the energy assuming a planar D3h configuration; the difference between these energies is taken as an estimate of the barrier for inversion. Using the known literature value of the barrier energy (2009 cm {1) students are asked to suggest possible reasons for the discrepancy between their result and experiment. In addition to purely computational aspects, this exercise has features that force the student to deal with the molecular symmetry, with encoding the symmetry using the symmetryunique atoms, and with a series of technical problems pertinent to the program. Computational Efficiency (Optimization of the Structure of Benzene) In this exercise, the benzene structure is optimized using different symmetries and different coordinate representations, but the focus is not on the final results of the calculations, which should be identical, but on the relative computational efficiency of the calculations. The system geometry is optimized under its full D 6h symmetry, and then under C 2v and C1 symmetries. In addition to noting any changes in the numerical results, the total CPU time for each run is recorded. Subsequently, the student checks whether it is more efficient to perform optimization in Cartesian coordinates or in internal coordinates for the benzene ring system.

The Electronic Structure of AgCO+ Transition metals and other heavy elements represent a challenge to electronic structure theory because they contain large numbers of electrons. For this exercise we suggest that the student use the effective core potential (ECP) method to approximate the effect of core electrons on valence electrons. The basis set proposed by Stevens, Basch, Krauss, Jasien, and Cundari (SKB) is used. The ECP method treats valence orbitals in a 6-31G* basis and fits core orbitals using an electrostatic potential represented as a sum of four empirically fitted Gaussians. It is also often the case with heavy elements that the correlation energy is a very significant contribution to the total energy. The student is asked to carry out two calculations on of the structure of AgCO+, one with the restricted Hartree–Fock method, and one with the MP2

Molecular Mechanics 1997: Determine, by the method of Molecular Mechanics, the equilibrium conformation of perfluoroeicosane, F(CF2)20F, at T = 0 K. Using the same potential energy surface, estimate the number of defects in a chain at T = 300 K. 1998: Calculate the relative energies and conformational geometries of all of the conformers of cyclodecane and cyclodecyne.

Student Assessment The course grade is based on the results of completed laboratory assignments and a computational project. Possible topics for the computational project are distributed early in the quarter, and a date for completion (near the end of the quarter) is assigned. The student is asked to choose one of the computational projects, carry it out, and prepare a report describing the basic theory used, the results, and the implications for other systems that might be studied. The following topics were proposed to the students in Spring 1997 and Spring 1998.

Molecular Dynamics 1997: Carry out a Molecular Dynamics simulation of the structure of water in the solvation shells surrounding a Ne atom and a Na+ ion, and compare these structures. Alternative choices for the pair of atom and ion solutes whose solvation shell structures are to be compared are Ar and K+ and Kr and Rb +.


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Information • Textbooks • Media • Resources 1998: Calculate the differences between the equilibrium constants for the reaction n-butane (all trans) → n-butane (gauche) when the n-butane molecule is in its own vapor at 165 K and p = 0.01 torr, and when it is dissolved in dense fluid Xe at 165 K and p = 1 atm. Electronic Structure of Molecules 1997: Compute the barrier to rotational motion in the ground electronic state of ethylene from calculations of the electronic energy as a function of the angle between the CH2 groups. 1998: Calculate the geometries of acetylene in its ground and first excited singlet electronic states.

Have We Achieved the Goals of the Course? Consider first the view from the student’s world. Student evaluations of the content of the Computational Chemistry course, of the division between lecture and laboratory material, and of the amount they have learned have all been strongly favorable, notwithstanding the pace at which the course proceeds. These evaluations also identify two issues that are pertinent to our understanding of undergraduate education. First, a reasonable fraction of the student evaluations made the comment that the instructor expected them to have remembered and to be able to use what they learned in other courses. Aside from the amusing character of this remark, it in part reflects the extent to which the instructor pushed the students to learn rapidly, and in part suggests that our undergraduate curricula do not sufficiently emphasize integration of subject matter and evolution of understanding through increasing depth of study. Second, the breadth of the material covered in the course cannot be found in any one textbook. Because the course focused on developing understanding of the character of the underlying numerical methods of computation, a single text, dealing with numerical analysis, was chosen. It was assumed that the combination of distributed lecture notes, other books the student retained from previous courses, and the reference texts held at the library, would adequately supplement the chosen text. However, it appears that this textbook strategy is not optimal; the student evaluations report that the text was rarely used. In principle, the availability of a textbook with requisite breadth would correct this difficulty. On the other hand, it is plausible to interpret lack of appreciation of the usefulness of the assigned text as an example of the insufficiency of the current curriculum with respect to intellectual integration of the various disciplines that contribute to chemistry. We believe that one of the advantages of a computational chemistry course is that it forces confrontation with the need for that integration. Consider now the view from the instructor’s world. In the first section we posited several goals for a course in computational chemistry. Specifically, we suggested that this course should generate an understanding of the interplay between basic theory and computational methodology, that it should illuminate the circumstances under which computation is the preferred tool for solving problems and illustrate the accuracy with which those problems can be solved. We also argued that a study of computational chemistry should enhance critical thinking about scientific goals which computation can assist in achieving. How well does the implementation of 1028

Computational Chemistry we have taught for two years meet these goals, and what criteria can be used to make that assessment? One feature of contemporary education is the extent to which technology has interfered with the employment of primary tools and principles in didactic exercises. It is the case that modern laboratory instrumentation enables a student to carry out sophisticated experiments without having to think about the principles that define the method of measurement. Similarly, and more relevant for present purposes, calculations using applications programs enable a student to visualize complex molecular structures, for example, a protein, without knowing anything about the underlying theory used and without clearly thinking through the uncertainties generated by the method of calculation. The emphasis we have placed on the methods used for computation, their accuracy and their limitations, are intended to redress the balance between understanding of fundamental principles and obtaining results. It has been our experience that a very large fraction (usually more than half ) of the students who have taken the Computational Chemistry course at The University of Chicago have worked with scientific applications software prior to enrollment in the course. This experience comes by way of undergraduate research opportunities, most often associated with biomedical problems. Before enrollment in Computational Chemistry, by their own testimony, the students believed everything they saw as output on the monitor screen. At the end of the course, again by their own testimony, these students have become much more sophisticated and critical in their interpretation of the results of computations. Judging by the quality of their term papers, they have also grasped the most important elements of the science underlying the problems they addressed, and they have developed an understanding of the nature of approximation in calculations and of the trade-off between computational efficiency and accuracy. At least in these senses we believe our Computational Chemistry course has been successful. We close with one example that leads us to the conclusion just cited, namely, the response of a student who undertook, as a quarter project, the calculation of the equilibrium conformation of perfluoroeicosane. Prior to enrollment in Computational Chemistry this student had three years experience in modeling peptide and protein molecules using commercially available applications software. He was one of those characterized above as believing whatever appeared on the monitor screen. When he discovered that the potential parameters in the Molecular Mechanics application program he was using predicted that perfluoroeicosane could not exist he was stunned. He spent many hours trying to revise the potential parameters and the calculation so as to obtain a result he could believe was realistic. His report contained the following emphatic statement: “What Dr. Rice has been telling us all quarter is true! These programs contain hidden assumptions that can lead to calculated results that are nonsense. It is necessary to know why a calculation is done in a specific way”. Acknowledgments We thank the National Science Foundation’s Division of Undergraduate Education for financial support through the Instrumentation and Laboratory Improvement program (DUE-9751224). We thank Dima Chekmarev, Mark Kobrak, Jeanne Siemion, R. Michael Townsend, and Meishan Zhao



for their invaluable contribution to the Computational Chemistry project at the University of Chicago.

D. J.; Baker, J.; Stewart, J. P.; Head-Gordon, M.; Gonzalez, C.; Pople, J. A. Gaussian 94; Gaussian, Inc.: Pittsburgh PA, 1995.

Appendix

Literature Cited 1. DeKock, R. L.; Madura, J. D.; Rioux, F.; Casanova, J. Reviews in Computational Chemistry, Vol 4; VCH: Weinheim, 1993; pp 149–228. 2. Duke, B. J.; O’Leary, B. J. Chem. Educ. 1995, 72, 501–504. 3. Williams, D. L.; Minarik, P. R.; Nibler, J. W. J. Chem. Educ. 1996, 73, 608–613. 4. Lehman, J. J; Goldstein, E. J. Chem. Educ. 1996, 73, 1096–1098. 5. MATHEMATICA; Wolfram Research, PO Box 6059, Champaign, IL 61826. 6. Healy, E. F. J. Chem. Educ. 1995, 72, A120–A121. 7. Ramachandran, B.; Kong, P. C. J. Chem. Educ. 1995, 72, 406–408. 8. David, C. W. J. Chem. Educ. 1995, 72, 995–997. 9. Lang, P. L. Towns, M. H. J. Chem. Educ. 1998, 75, 506–509. 10. Mathcad; MathSoft, One Kendall Square, Cambridge, MA 02139. 11. Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S. J.; Windus, T. L.; Dupuis, M.; Montgomery, J. A. J. Comput. Chem. 1993, 14, 1347–1363. 12. Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.; Johnson, B. G.; Robb, M. A.; Cheeseman, J. R.; Keith, T.; Petersson, G. A.; Montgomery, J. A.; Raghavachari, K.; Al-Laham, M. A.; Zakrzewski, V. G.; Ortiz, J. V.; Foresman, J. B.; Cioslowski, J.; Stefanov, B. B.; Nanayakkara, A.; Challacombe, M.; Peng, C. Y.; Ayala, P. Y.; Chen, W.; Wong, M. W.; Andres, J. L.; Replogle, E. S.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Binkley, J. S.; Defrees,

Required Textbook

Burden, R. L.; Faires, J. D. Numerical Analysis, 6th ed.; Brooks/Cole: Pacific Grove, CA, 1997; ISBN 0-534-95532-0. Recommended Books

Wolfram, S. The Mathematica Book, 3rd ed.; Wolfram Media/ Cambridge University Press, 1996; ISBN 0-9650532-0-2. Koonin, S. E.; Meredith, D. C. Computational Physics (FORTRAN ed.); Addison-Wesley: Redwood City, CA, 1989; ISBN 0-201-12779-2. Haile, J. M. Molecular Dynamics Simulations: Elementary Methods; Wiley: New York, 1992; ISBN 0-471-81966-2. Kalos, M. H.; Whitlock, P. A. Monte Carlo Methods; Wiley: New York, 1986; ISBN 0-471-8~839-2. Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applications; Academic: San Diego, 1996; ISBN 0-12-267370-0. Hasanein, A. A.; Evans, M. W. Computational Methods in Quantum Chemistry; World Scientific: River Edge, NJ, 1996; ISBN 981-02-2611-X. Szabo, A.; Osthend, N. S. Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory; Free Press: New York; Collier Macmillan: London, 1982; ISBN 0-486-69186-1.


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Computational Chemistry in the Undergraduate Chemistry Curriculum

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