Table 1. Contour Plots of Hydrogen Orbitals
"Some constants' 1 9.10953x
lo3'
+
1 1.67265x 10.~
so = 8.854188 x 1 0-l2
I
"Reduced mass of H 'Vacuum permittivity"
Surface and Contour Plots of Atomic Orbitals
hsr = 1.05459x lo4 Oe= 1.60219~ lo-''
"Elementary charge"
Yzp(C0) =-
"Normalized wavefunction SI units"
"Mapping Cartesian space into circular coordinates"
"Number of data points: must be even" k = O...N
A variation on this approach is to print out a small matrix (about 15 x 15) and have the students connect points of equal probability density a s advocated by Baughman. I n this case, the values within the matrix should be scaled to fall within the range of 0-9.9. One decimal place should be specified, and trailing zeros should be retained so that all the numbers will occupy the same amount of space.
I=0... N
"Number Bohrs origin to edge" "Origin is (M2, N12)"
A a 1 --
2
The Mathcad (version 5.0 for Windows) document presented in Table 1can he used to prepare surface and contour plots of atomic orbitals. Explanations are set off in quotation marks. The constants are eiven in SI units. The wavefundion gwcn is that for tht: 2p, orhtal. The pararneterNsets thesize of the matrix M rN + 1 k.V T 11. whch wdl contain the ~ n h ability density. Larger values of N give smoother cokour plots but require more calculation. Surface plots with high values ofN have too many lines and are dimcult to interpret. The parameterA is the number of Bohr radii from the origin to the edge of the plot. The subscripted variablesx and Yare the Cartesian coordinates assigned to the matrix locations corresponding to the subscripts. The matrix has 0,O in the upper left comer, so the origin of the Cartesian coordinates must be offset to the center. The atomic orbitals are given in spherical coordinates, so the Cartesian coordinates X and Y are converted t o r and 8.The probability density is then calculated for each point on the matrix. This matrix can be displayed as a contour plot (Fig. 1, N = 80) or as a surface plot (Fig. 4 , N = 16).Matrix G is a scaled version of matrix M with the maximum value of the probability density held to 9.9. This scaled matrix can he printed out as shown in Figure 6 to allow students to prepare their own contour plots. In this case, N should be about 14 to 20. Mathcad and similar mathematics-engineering packages can be valuable tools in allowing students to manipulate atomic orbitals. The same method can be used for hybrid or molecular orbitals. The vivid images and the possibilities for hands-on manipulation can help overcome misconceptions, perpetuated by textbooks, about where the electrons are.
"Special rdes for x axis and origin" 1[ %---I
3.") %n$es are
eXhl+l
counterclockwisefrom y axis." "Normalized Probability Density"
Transformed Probability Density values from 0 to 9.9
and carry out mathematical operations upon them, including integration, solving for variables, finding roots, and preparing various sorts of plots. One example is a program that has been used for a variety of chemistry and chemistry-education applications: Mathcad (Math Soft) (3132).Mathcad can prepare contour plots and surface plots of data presented in a rectangular matrix. The position within the matrix gives the x and y Cartesian coordinates; the value a t that position provides the z coordinate. What is needed is a means of mapping the locations within the matrix to the polar coordin a t e s t h a t a r e n a t u r a l to wavefunctions. Then any wavefundion can be typed into the document, and with some adjustment to the scaling, either a contour plot (Fig. 2) or a surface plot (Fig. 5) of the value of the probability density in the xy plane can be produced.
Chemical Education via MOLGEN C. Benecke, R. Grund, A. Kerber, R. Laue, and T. Wieland Depaltment of Mathematics University of Bayreuth D-95440Bayreuth. Germany A main topic a t certain school classes, university lectures, or seminars is the very large variety of hydrocarbon molecules, such a s t h e well-known benzene. Therefore, i t i s important to show students t h a t the molecular formula CsH6 corresponds to more t h a n just t h e famous benzene ring; there are many isomers of t h a t particular formula. Few students know of the existence of the 217 structural formulas, and they will be astonished to hear t h a t about 50 of them have been observed i n n a t u r e o r laboratory. Ideally t h i s fact could be demonstrated with a n example on the screen of a computer monitor. MOLGEN provides a solution to t h a t problem. I t was awarded the German-Austrian University Software Prize 1993 for excellent educational software in chemistry. Several schools, universities, and chemist r y companies in Germany use this program, which i s the result of a research project supported by the Deutsche Forschungsgemeinschaft for several years. We will describe MOLGEN by presenting short examples taken from chemical education and research. Volume 72 Number 5
May 1995
403
Applications of MOLGEN Learning with MOLGEN
Consider, for example, benzene. The 217 stmctural formulas corresponding to its chemical formula CsHs are obtained by MOLGEN within a fraction of a second. After generation a visualization of the result is needed. I n set-
ting up the display a two-dimensional placement function is called for drawing the molecules. Having done this for the 217 isomers of benzene, you will see the well-known benzene ring showing up as number 214 (see Fig. 7). Molecule no. 215 is benzvalene, which was discovered just 25 years ago, though i t s existence was by E. Hiickel in 1937. Teaching with the Help of MOLGEN
21Wil
217R17
HW-CI~
HR-CR
1 Figure 7. Some of the connectivity isomers of benzene. 1122 SOLUTION 1 H[6]-0[6]-C[4]
4122 SOLUTION 2 H[6]-0[6]-C[4]
Figure 8. The solutions of the high school exercise 404
Journal of Chemical Education
Often we want to show or even calculate only t h e isomers belonging to a given class of chemical substances. For example, the chemical formula for alcohols is C,H2,+10H, and one structural element, the hydroxyl group, is emphasized. Suppose we want to compute or see only structural formulas with n carbon atoms with 2n+2 hydrogen atoms and a single oxygen atom that belongs to a hydroxyl group. Then we can easily enter the chemical formula C3H80 with the substmcture OH and view two connectivity isomers, whereas three are shown if the hydroxyl group is not prescribed.
Exercise A corresponding exercise taken from a n A-level examination in chemistry a t B a v a r i a n high schools in 1992 further demon-
strates the use of MOLGEN. I n a first question t h e molecular formula for tartaric acid was derived as C4HsOs. Then the following challenge i s presented. .During an acid-base titration it was discavered that 1mol of tartaric acid is equivalent to 2 mol of sodium hydroxide. Derive all possible constitutional structures compatible with this result and the chemical formula derived above. The hint should be interpreted a s the exist? ence of two substructures bf the form COOH, which have to be prescribed. After entering all these together with t h e molecular formula, MOLGEN produces 22 solutions. Several of them contain substructures that a r e quite uncommon. F o r example, t h e solutions numbered 17 and 18 (among others) contain two bonded oxygens (peroxo groups). If vou want to abandon such cases. vou mav proceed as follows: I n addition t h e t w o subs t r u c t u r e s COOH res scribe two hvdroxvl groups. Thus, after another r u n of the gener- Figure 9. Stereois ator. onlv four structural formulas remain. (All solutions containing two oxygen atoms between two carbon atoms were skipped.) Two of these remaining four solutions shbw the hydroxyl groups attached to t h e same carbon atom, so we may cancel them by a n application of t h e Erlenmeyer Rule, which is still used i n undergraduate chemistry. Finally we arrive a t the two structural formulas suggested by t h e people who designed t h a t exercise. Recall t h a t we started using MOLGEN with 22 isomers, and the final solution of the problem is isomers 1a n d 4 (see Fig. 8). I n practice, the chemist will see by inspection t h a t several further restrictions must be imposed. Therefore the built-in graphical editor allows us to isolate and define substructures t h a t a r e either prescribed or forbidden. With another r u n under these conditions, t h a t is, i n a n interactive way, the total amount of isomers can be reduced to a significantly smaller set of candidates to be examined.
-
'a
Research using MOLGEN MOLGEN allows u s to handle cases t h a t were impossible before efficient eenerators became available. Here are a few remarks concerning applications i n research ~~~~~
-
.
Q
MOLGEN provides all mathematically possible solutions, several ofwhich may be plausible chemically but are yet unknown and thus may correspond to new substances. The identification of chemical substances using spectroscopic data is a central problem in chemical laboratories. The wish for automatization may arise because many such problems occur daily, for example, in environmental research. This identification clearly needs a generator that provides all the possible cases from which the correct ones are taken! A major task is to reduce the number of candidates using sophisticated theoretical and practical methods.
In the first approximation, molecules are described by their molecular formula. The second approximation is its structural formula. MOLGEN allows us to do this second step in an efficient way. Moreover, another module of the MOLGEN program system allows us to generate the corresponding stereoisomers, as well as three-dimensionalrepresentations of them using Allinger's MM2 energy model (a simplified version). See tetramethylcyclohutane in Figure 9 as an example. The central part of an expert system for molecular-strueture elucidation therefore is a generator like MOLGEN. It generates all mathematically possible structural formulas and the possible stereoisomers compatible with a given set of chemical conditions. Molecular-structure elucidation thus may proceed in the following way: Spectroscopic data, ahtained from MS, IR, or NMR spectra, are translated to a chemical formula together with a set of conditions used as input for MOLGEN. After a run of the generator the resulting set of candidates is examined by a chemist. Usually additional restrictions are found for a next run of the generator. This procedure is iterated until the set of candidates is small enough to find the solution by visual inspection. For example, if the molecular formula C1202C14H4is entered together with t h e single substructure of the dioxin skeleton (substitutes removed). within a second MOLGEN provides the 22 constitutional'isomers of the dioxin, three of which a r e shown in Figure 10. Industrial Problem I n a complex problem from molecular-structure elucidation i n industry that we solved using MOLGEN, the molecular formula given was C22HZ5N3O3.We were told that t h e molecule i n q u e s t i o n c o n t a i n s t h e following six
Figure 10. Three isomers of dioxin.
Volume 72 Number 5
May 1995
405
nonoverlapping substructures (macro atoms, in terms of MOLGEN) F1, ..., F6.
Using these substructures, the molecular formula reduces to F1F2F3F4F5F6N2. MOLGEN obtains (after several steps not described in detail here) 2,337 structural formulas for that reduced molecular formula, and after expansion it constructs 8,916 isomers. We used the following restrictions: no triple bond and ring sizes between 5 and 6. Only 201 candidates remained from which a n expert easily obtained the correct solution.
relatively inexpensive personal computers based on the 80486 (or 80386 with a math coprocessor) microprocessor or RISC workstations. In this article, we discuss how three-dimensional maphical representations of hvdrogenic atomic orbital; m i y be generated (and viewedkom any perspective) using the software package Mathematica. Generating lsosurfaces The basis of this exercise is a program we have developed, which serves as input to Mathematica. Our program finds and plots the points in three-dimensional Cartesian space where I yf.~, I has a constant value, that is, isosurfaces. Both the positive and negative regions of the functions are plotted by using the absolute value of the wave function for selecting the points. The sign of the wave function in the different regions is represented by different colors (or different gray scaling). We illustrate our method using the case of the 2p, orbital, which for the hydrogen atom (atomic number Z = I), is 1 vzPp,e,$) = -re'"2'cos e (1) 44%
1 CH3
MOLGEN runs under DOS on a PC. Versions for Windows, OSl2, Macintosh and various workstations are in preparation. There exists a limited version for education and a full version for science and research. For more information contact the authors. Telephone (0921) 55-3388, Fax (0921) 55-3385, E-mail: k e r b e r @ b t m 2 ~ 2 . m a t . u n i bayreuth.de
Three-Dimensional Graphical Visualization of One-Electron Atomic Orbitals 6. Ramachandran
Department of Chemistry and P.C. Kong Department of Mathematics Louisiana Tech University Ruston, LA 71272
where the atomic units of length (a0= 1) are used. The isoy, z ) surface to be plotted is the set of points lr, 8, $1 or [z, for which the absolute value of the function in eq 1 is a certain positive constant, K. I n other words, we seek all points that satisfy the equation K-
I re'~"2'eos e I = 0
(2)
where the normalization constant has been absorbed into K. A graphical representation of the resulting surface is generated a s a parametric d o t of one of the variables expressed as a function of the other two. Due to the nonlinear dependence of the wave function on r, it is easier to express cos 0 in terms of the radial distance r a s K cos 9 = -e"12'
(3)
r
K
< 2e-'; r E (r1,r2)
where rl and rz are the roots of eq 2 when 0 = 0. This is schematically shown in Figure 11. The following relationships for the set lx, y, zl are then obtained in terms of lr, $I x = min
@ =r
w
c
o
s
4
Discussions of the one-electron atomic orbitals yfnl,(r, 8,
4) in physical chemistry courses typically cover the coordinate dependence of the radial functions and the spherical harmonics separately. These ideas are then combined to obtain the shapes and the nodal patterns of atomic orbitals, which are illustrated by schematic sketches of the "shapes" of the orbitals. Contour diagrams are sometimes used (33) in order to clearly show the positions of the radial and angular nodes. On the contour plots, radial nodes appear as circles, and the angular nodes appear a s straight lines. Some familiarity with the concept of contour mapping and imagination are then necessary to visualize the radial nodes a s spheres and the angular nodes as planes in three-dimensional space. All this is necessary before students can visualize the spatial distribution of electron densities around the nucleus. The shapes and the nodal structures of atomic orbitals can be studied and taught very effectively if the students (and the teacher) generated and viewed three-dimensional graphical representations of atomic orbitals. I t is now entirely practical to carry out the computations and the rather sophisticated plotting required for this exercise on 406
Journal of Chemical Education
Figure 11. Schematic representation of the roots r j , lz for the 2pz orbital (9 = 0) for an arbitrary value of K. The normalization constants were dropped from the radial function.
