The Droner orientation of wave functions for atomic orbital above orbitals is important when they are added. The Slater-Pauling equations arbitrarily designate the positive r axis to be the positive lobe of the orbital for p, orbitals. The package 0rbitals.m generates contour diagrams of wave funct&nsof both hondingand antibonding o;hitals,depending on the relative signs of the constituent atomic orbitals. Mathematica requires a minimum of four megabytes of main memory. In this worka .Macintosh llcx with four megabytes of memory was used. The graphics each take approximately 2 min to generate. The built-in atomic orbitals of 0rbitals.m are called: s, px, py, dxy, dyz, dxz, dz2, d(x2-y2), and common hybrids, sp, sp2, and sp3. The relationship between the code notation and common orbital descri~tors is clear. Hybrid atomic orbital wave functions other t h k the most common three may. be . generated directly as described earlier.
s
Conclusion The package 0rbitals.m is intended t o encourage the use of and experimentation with a powerful new tool-Mathematica-for undergraduate instruction in chemistry. Many hybrid atomic and molecular orbitals not discussed here may be displayed using tbis package, including single C-C bonds, C-H bonds. The package 0rbitals.m can enhance the understanding of molecular orbital theory by accurate pictorial representations of the wave functions, and should encourage students to experiment with variations on simple hybrid orbitals. Further information regarding this Mathematica package, including a disk copy of the package, documentation, and a display of code is available from the authors. For an electronic copy of the package please send a formatted 3%-in. disk.
Molecular Modeling as an Integral Part of an Advanced Lab Course Stuart Rosenfeld
Smith College Norlhampton. MA 01063 A number of authors have argued recently that the time is right for computational chemistry to assume a significant role in the undergraduate curriculum (7-11). Semiempirical methods and the empiripal molecular mechanics approach provide fast and accurate assessments of geometries and energies. Personal computers with sufficient memory and sneed are increasinelv available. as is the reauisite software fir these calculation;and the associated graphics. The molecular mechanics method (12-15) is both faster and concentually simpler than quantum mechanical methods, makingit a suitable general tool t o apply at various levels throughout the curriculum. Others have provided excellent examples of isolated molecular mechanics modeling exercises tied closely to experimental work (7, 8, 10, 11). We now describe an experience of using structure and energy calculation repeatedly throughout a semester of an advanced undergraduate
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Molecular mechanics structure and energy calculations have been heavily integrated into our junior-level Advanced Laboratory course during the last two years. Our approach has been to do this in three stages; learning the method, using it as an aid in working problems, and using i t as a research tool. Initially we introduce the general background (12-14) on structure modeling with lecture material in class and readings on molecular mechanics followed by an in-lah computer e x e ~ c i s eThis . ~ first lab consists of a tutorialon the use of thesoftware followed by an exercise in which students do calculations on six hydrocarhons for which there exists both experimental and calculated (by molecular mechanics) 488
Journal of Chemical Education
determinations of heats of formation and structure. Students are given a worksheet (several days before tbis lab) with some values listed and a leading reference for them to locate the remaining ones. We use Schleyer's 1973 paper entitled "Critical Evaluation of Molecular Mechanics" for the extensive range of compounds treated. That paper also includes a clear history and background on the method and its limitations and concludes, in agreement with Allinger, that the method had achieved a level of success that made it competitive with the accuracy of experimental determinations of structure and enthalpy (15). The second stage of our students' experience with molecular mechanics occurs in the very next lab in which they receive a problem set with five or six questions. At this point they are reasonably comfortable with the software and are readv t o trv to use i t to gain insiebt into nroblems. In other words, they are ready to see whet6er this method can be used as a tool for understanding chemistrv. The intention is to facilitate recognition that computational models can be used as an aid in problem solving much the same wav that physical models are used. Our students, at this stage, have been quite enthusiastic and anxious to help each other over the practical problems associated with the calculations. Because the software that we use is extremely flexible there are also many options that can be introduced by the instructor when called for by individuals. The following three examples illustrate the sort of question used on thisproblem setf 1. Compare strain energy and hond angles for cyclobexene, cyelopentene and cyclobutene. Wliat do you expect? What is the
principal source of strain?
2. Compare AHfandthe overall shape of the cis and trans isomers of
cyclotetradecene ( C M H ~ ~Does ) . the double bond stereoehemistry affect the overall shape of the molecule? 3. How does an additional double hond alter the C-13 bond length in an aidrhvdeTrs this for batheoniueatedand isoiacrd douhle bonds. F& the eoijugated system, haw do the carbon-carbon bond lengths (both single and double) compare to typicdvalues? Finally, students use molecular mechanics modeling in a t least one more lab during the semester. For example, we have used modeling as a complement to a lab in which a tautomeric compound is synthesized and the tautomeric equilibrium is evaluated by NMR (16). Independent projects are incorporated a t the end of that lab and students have often included structure and energy calculations in those nroiects as well. w e - u s e Serena Software's program PCMODEL', and some of the features of tbis software warrant comment since they have direct impact on the quality of the experience. PCMODEL features encompass structure input, energy
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BThiswork is done in a computer lab equlpped with 20 IBM computersthat are currently mainly PSI2 models 30-286 and 55. The minimum system requirements tor this software are an MS-DOS machine with 640K, math coprocessor, and a mouse. it should be noted that calculations on molecules with conjugated pi systems c a n be too costly in time, and therefore problems should be selected with anention to the speed of the available computers. The first lab meeting in this sequence is for 4 h and the second Is for 3 h. Well within the time constraints of our first meeting students begin to work comfortablywith the software, asking questions, and sharing informatlon with each other as the need arises. By the end of the second meeting all students have been able to use the software independently. Ail of their work on modeling after that point is self-scheduled. Our computer lab is open seven days a week and most evenings. PCMODELand the companion program PCDiSPLAY are available from Serena Software. P.O. Box 3076. Bloomington, IN 47402. PCMODEL incorporates the MMX force field, which is a derivative of Aliinger's MM2 force field with pi-VESCF routines added. Serena Software has an extremely liberal policy with respect to the use of this software in a classroom sening thereby making this approach affordable.
minimization, and screen and hard copy display of calculated structures. One moves easily among the three screens that represent these three functions. Structure input (like most operations) is via a mouse and drawing; correcting errors and aitering previously drawn structures is fast and simple. Students alert to the difficulties associated with locating global minima will wish to use two or more starting geometries for some calculations. The ability, for example, to alter an existing structure in a predictable way by rotation about a particular bond is very useful. During the energy minimization, the screen shows the individual components of the calculated steric energy for the starting geometry, the structure itself (mono or stereo) updated periodically to show changes and the new steric energy after each set of five iterations. Therefore, one sees the quantitative and qualitative slide of the system into a potential energy minimum. When the structure is displayed on the screen following minimization, one can easily tumble i t into the desired position, extract structural parameters (distances between selected atoms, bond angles, dihedral angles), and even throw a second previously minimized structure onto the screen for a visual comparison of shape. Although onemay plot structures from PCMODEL, we use the companion program PCDISPLAY for its ability t o generate PLUTO and ORTEP drawings on the screen and paper via a plotter or laser printer. Students often incorporate reduced copies of these drawings in papers. The power and flexibility of this software clearly exceeds the needs of students a t this level, and one might reasonably focus on the disadvantages of the associated complexity. However, students seem to enjoy wielding a true research tool, accepting, as they do in using our FTNMR spectrometer, that they need not use every feature to use it effectively. Moreover, i t is a pleasure for the instructor to show a new feature to a student that has a prohlem that she recognizes is amenable to a more sophisticated approach. And, students pass that new expertise on with obvious enthusiasm. Finally, the increasingly routine use of structure and energy calculatinns hv chemists is sufficient reason to incorporate them in the undergraduate curriculum. However, from a personal perspective, the central educational benefit here is in giving a chemistry student a sophisticated tool, showing her how to useit togain insight, and then watching her skill andmotivation grow.
where i = 1,2,. . .,Nspecies, C; is the concentration a t time t At, and Ci is the concentration a t time t. For a first-order monoelectronic transfer 0 e a R ( 0 and R being soluble species), current density a t each instant for a semi-infinite linear diffusion process is given by eqs 3 and 4:
+
+
The transformation of eq 4 into two finite-differences equations yields eqs 5 and 6:
The resolution of the system formed by eqs 2 , 3 , 5 , and 6 allows one to calculate the current density a t each instant and the new concentrations a t the electrode surface ( x = 0). For mechanisms where first-order chemical reactions are coupled to the electron transfer, eq 1is transformed into eq 7, and their effect on eq 2 must be considered.
d. l* RWERSE PRINT HOLD SlbRI PROFILESO I - W S PbUSE
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A Voltammetry Experiment by Digital Simulation Gaspar SBnchez, Gulllermo Codlna, and Antonlo Aldaz Universidad de Allcante Aptdo. 99 03080 Allcante. Spain Cyclic voltammetry (17-20)is perhaps the most widely used electrochemical technique for the study of the kinetics of electrodic systems. The technique is based on the variation of the potential of the working electrode in a triangular cyclic form, the resulting current being measured. However, the analytical solution of the differential second-order equations necessary to obtain the voltammetric curves is very difficult if not impossible. Thus, it is not easy for the student to understand the influence on the voltammetric curves either of the different kinetic parameters of the reactions (electrocbemical rate constant, diffusion coefficient, etc.) or of the experimental conditions (scan rate, initial potential, etc.). For these reasons, the aim of this work is to provide a suitable comnuter oromam to be used both for educational . .. purposes and for approximate evaluarion of kinetic parameters. The Droeram developed has been called SIMCLA. he basic equation to solve is the Fick's diffusion equation (eq 1).The model is based upon the transformation of this equation from its differential form into a finite-differences equation using the Cranck-Nicholson's method.
Figure 4. Screen obtained fore simple electron hansfer.
Figure 5. Screen obtained for the ECE mechanism. Volume 68 Number 6 June 1991
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