the computer bulletin-board Investigating the Harmonic Oscillator Using Mathematica James J. Bruce and Bruce D. ~nderson'
Division of Biological and Physical Sciences Lander University Greenwood,SC 29649 Recently, computer algebra systems, such as MathCad, Mathematica, and Maple, have been adapted to facilitate learning in the quantum mechanics portion of physical chemistry. For example, the graphics capabilities have been used to examine wavefunctions and their probabilities (1,2)as well as atomic and molecular orbitals (3). The computational abilities of computer algebra systems have been used to explore overlap integrals (4) and the solution of the Sehriidinger equation involving the Morse potential (5, 6). Furthermore, Rioux (7)has shown how MathCad can be used throughout quantum mechanics to find numerical solutions to the Schrodinger equation and solve other types of problems as well. As a further example of integrating computer algebra systems into physical chemistry, the use of Mathematica2 to study the one-dimensional harmonic oscillator is discussed. Calculating the Zero-Point Energy for CO
To evaluate the harmonic oscillator, the students should be familiar with the SchrLidinger equation and preferably have experience with Mathematica. As a specific example, consider the calculation of the zero-point energy for CO. Entries for the Diatomic Molecule First, the students should enter the wnstants for the diatomic molecule, including quantum number n force constant k the mass of each atom in kilograms Planck's constant divided by 2n hb Next they enter the expression for the reduced mass of the molecule, and then the time-independent wave function w as shown below. w=U(ZAnn !IA.5(alPiIA.25HemiteH[n,zl Exp[-(axA2Y21 where
Kenyon College Gambier, OH 43022
where F is the energy of the system. The Output The output should be an expression in terms of x and the eigenvalue F. To calculate the eigenvalue, enter a value for x and solve. Because the eigenvalues are known to be independent of x , any value of x will produce the correct answer. For best results, however, exact values of the atomic masses and other constants must be used. For example, if the constants are used with only three significant figures, then values of x that deviate widely from the equilibrium bond length of the molecule will not produce accurate results. This problem can be alleviated if the students enter a reasonable bond length for x. At this point students can repeat the calculation for another value of n to verify that the energy levels of the harmonic oscillator are evenly spaced and nondegenerate. Additlonal Challenges
The problem described could be done as a homework assignment or a computational lab experiment in which each student does the calculations for a different diatomic molecule. In either case, it may be beneficial to combine the assignment with a graphical exploration of the wavefunctions as suggested elsewhere (1,2). Finally, similar computational procedures have been canied out successfully using Mathematica for both the rigid rotor and hydrogen atom problems. It is hoped that using this approach will enhance the students' understanding ofthe importance and validity of the Schrodinger equation. Acknowledgment
The authors would like to thank the Lander Foundation for the funds to purchase the software, and B. Anderson would like to thank the Chem 402 class for their work and their interest in this project.
An I-Strain Prediction via Computer Modeling Keith A. Bellomo, Russell C. Bush and Richard D. sands3
Alfred University
a = (k uP.5 /hb
Alfred, NY 14802
-
and where x is the internuclear separation, and u is the reduced mass. Finally, the Sehrtidinger equation is input as follows. -hbA2/(2u)D[w,(x,2ll+ k xA2w - F w == 0 'Author to whom correspondence should be addressed. 2WolframResearch, PO Box 6059, Champaign. lL 61826. 3Author to whom correspondence should be addressed. 4TriposAssociates, Inc., St. Louis, MO, 1988. A122
edited by
RUSSELLH. BAT^
Journal of Chemical Education
ALCHEMY 11.4 one of several molecular-modeline>urngrams now available, was used by an undergraduate student to studv the conformations and strains of the uossible intermediate ions involved in reactions in which Bve- and six-membered rings (Fig. 1) or six- and seven-membered rings (Fig. 2) must compete to determine the outcomes. Both reactions involve the conversion of a dihalide to a ketone. The mechanism has been conclusively established by Kakis (8).In each case the product depends on which of two possible product-controlling intermediates--2 or 6 and 11or 15--would have less steric energy and thus would be more likely to form. (Continuedon page A124)
the computer bulletin board
Figure 1. The Kakis mechanism applied to competition between fiveand six-membered rings. H. C. Brown's I-strain theory (3,101predicts relief of the inte~nalstrain when a carbon atom in a five or
a seven-membered ring becomes a cation 'increase in the I-strain when a carbon in a six-membered ring becomes a cation It was expected that energies calculated by ALCHEMY I1 would support the I-strain theory. Calculating Steric Energies
ALCHEMY 11uses atomic geometries, bond lengths, and force constants to calculate the steric energies of structures. Each pair of possible intermediate ions in Figures 1 and 2 have the same number and types of atoms and bonds. Their computed steric energies should, therefore, indicate which intermediate in a given pair is less strained. Because there are no parameters for a carbocation as such in the ALCHEMY I1 databank, the aryl carbon, with its three equivalent bonds, was used in place of the carbocation in this exercise. While this may intmduce error, A124
Journal of Chemical Education
Figure 2. The Kakis mechanism applied to the competition between six- and seven-membered rings. the relative energies calculated for the carbocation pairs 2, 6 and 11, 15 should still be mmparable. Minimumization In minimumization the program calculates the steric energy of the structure the operator builds on the screen and then repeats the calculation many times, each with some variation of bond length, bond angle, or rotation of parts of the structure until the conformation with the least energy is ,fieved. ~h~ result is the prefemed and its On the steric-energy surface for a complex molecule, there might be several regions of low steric energy. To help insure that ALCHEMY I1 produces a structure representing the true global minimum on this surface, the operator Can pehurb a structure and then minimumize again. The techniques used to perturb the structures include removing hydrogen atoms, changing from a chair to a boat conformation, or even perhalogenating the molecule. (Continuedon page A126)
Predictions
After minimumization, ALCHEMY 11gave the following total steric energies. For the reaction of 1: -7.7 krallmol far 2 with the five-memberedearbocatlan -4.8 kcabmol for 6 with the srx-memhered carbwation For the reaction of 10: -22.4 k d m o l for 11 with the seven-membered carbocation -17.7 k d m o l for 15 with the six-membered carhocation Therefore, ALCHEMY I1 allows us to predict that 1 would react mth silver nltrak w e v e Ion 2 and eventually ketone 6 instead of forming ion 6 and ketune 9 10 would lead to ion 11 and ketune 14 ~nsteadof ion 15 and ketone 18
Conclusions
The reaction of 1,l'-dibromocyclopentylcyclohexane with silver nitrate usine Kakis' orocedure was recentlv carried out in our laboratory usinisynthetic techniques-from our junior organic preparations course and gas chromatography and mass spectroscopytechniques from our senior analytical course. The ketone formed was, in fact, the predicted ketone 5. Ketone 9 was not detected by a chromatography system that could easily detect 9 in a 2%/98% mixture of 9 in 5. The reaction of 1.1'-dibromocyclohexylcycloheptane with silver nitrate awaits another undergraduate investigator.
Analyze and Identify Surface Colors with a $35 Surface Spectroscopy Interface and Your Personal Computer Joseph ~chnable?Matthew Cheeseman. Chariene ~ e k g e and < Robert Orr
Delaware Valley College Doylestown, PA 18901 Visible or color-reflectance spectroscopy is used in many fields including quality control, matching paints, color formulation, and criminology (11).In most commercial colorreflectance instruments, samples are illuminated with light having wavelenghts in the range 400-700 nm. The amount of light reflected off the sample at each wavelength is recorded. Samples reflect the color that they appear to have, and they absorb complimentarycolors. For example, a red sample might reflect 700-nmred light well, but strongly absorb 400-nmblue to 550-nmgreen light. Uses of the PC and Interface
An IBM-compatible computer interface was constructed to control blue, green, yellow, orange, crimson, and red light-emitting diodes (LED'S) pointed at each sample. The interface also monitored the reflection with a phototransistor and 8-bit analog-to-digital converter. The reflectance probe and interface allowed the computer to 5~uthor to whom correspondence should be addressed. A126
Journal of Chemical Education
Table 1. Reflectance Spectrometer LED'S and Complementary Colors
Wavelength and color emitted by LED
Complementaly color (color of the surface that absorbs the most light)
470-nm blue
yellow-orange
560-nm green
red-purple
590-nm yellow
blue
630-nm orange
greewblue
660-nm crimson
bluwreen
700-nm red
blue-green
match unknown surface colors to the closest of hundreds of standard colors demonstrate complementary color spectroscopy determine the approximate absorption maximum of compounds that impart color to vegetable surfaces monitor temporal hydration of salts Color Matching
The interface that was constructed and the software used allowed scanning and storing reflectance profdes of 370 paint cards. Paint cards presented in a different order were correctly identified by an identification program, although two extremely pale light-pink cards were occasionally identified incorrectly. When abstract samples were scanned, the program seemed to fmd the best color match, even when the sample was more glossy or textured than the paint cards. DemonstratingComplementaryColor Spectroscopy
Each LED used was shone on surfaces of various colors, and the amount of light reflected from each was recorded. For each LED, Table 1lists both the color emitted and the color of the surface that absorbed the most light, that is, the complementary color. For example, for the LED that emits blue light, the yellow-orange surface is the one that reflecta the least light. Table 2 lists & of solutions determined on a W-Vls spectrometer (Perkin-Elmer Lambda 4B),and the percent reflectance of each LED from paper colored by the solntions (or from other solids). Crystal violet stain solution had a & of 590 nm, and paper colored with crystal violet stain reflected 590-nm yellow light least. Green NiClz solution absorbed 723-nm red light best, and paper colored with this solution reflected 560-nmgreen light best. Some analyses were less straightforward. Green food dye is a mixture of blue and yellow food dye. Blue food dye contains a small amount of red dye. This can be confirmed with thin-layer chromatography. Conjugation The J'ungle green" paint chip, the carrot peel, and the tomato skin were only analyzed on the reflectance interface. The carrot peel absorbed 470 nm best. Carrots are orange due to beta-carotene, a compound with 11 conjugated double bonds and a of 497 nm.The tomato skin : (Continued on page A128)
