Edited by
textbook forum
RALPH K. BIRDWHISTELL University 01 West Flwida Pensacola. FL 32504 32504
Teaching the Shapes of the Hydrogenlike and Hybrid
Robert D. Allendoerfer State University of New York at Buffalo, Buffalo,NY 14214 For the past several years the instructors in our main General Chemistry course have felt it was necessary to replace the textbook material about the shapes of the hydrogenlike atomic orbitals and spXhybrid orbitals with locally written material. Our survey of the current editions of a dozen mainstream general chemistry texts indicates that the situation would not be improved by changing books since they all give approximately the same treatment. Informal discussions with several of the authors of these texts led us to several conclusions. First, these authors do have difficulty findine eranhic artists c a ~ a h l of e drawine accurate revresenSecond, ;he matationyof tke hydnrgenlikeatomicorbit~ls. -ioritv- of these authors believe that our complainu about inaccurately placed and missing nodes, for example, have no important chemical consequences and therefore need not concern beginning students. The purpose of this article to show that, in this age of computer graphics, the "difficult to obtain" argument no longer has merit and t o give an example of where the standard treatment gives insufficient attention to detail in describing the nodal surfaces of hybrid orbitals, which may lead to conceptual difficulties in more advanced courses. Our Method of Introducing the Hydrogenllke Atomic Orblials After a standard introductory-level discussion of wave mechanics, nodes, quantum numbers, and the Heisenberg uncertainty principle, which concludes with the idea that it is impossible t o track the flight of an electron around the nucleus by experiment, electron density plots similar to the 2p, plot shown in Figure 1are displayed for the class using an overhead projector. These computer generated plots are described as being analogous t o time-lapse photography of the travels of a sinale electron around the nucleus. The mathematics for generating this type of plot has been available to chemistry instructors at least since the publication of Pauling and-wilson's classic text ( I ) , mainframe computer calculations of these diagrams have been published in this Journal (2,3),overhead transparencies were once commercially available (4),minicomputer calculations of this type of diagram are available (5),and recently Liebl (6)published a computer program for a microcomputer that generates the equivalent contour diagrams. Our overhead transparencies are generated from plots made by an IBMPC computer using program DOTORBS (7). When coupled with three-dimensional models of the orbitals, our overhead transparencies enable the instructors to give an informative and stimulating lecture ahour the shapes of theatomic orbitals, but, until rec~nrly,we had not considered how the student was s u ~ ~ o s to e dcarry this information home. The diagrams shown differ significantly from those in the hook and are essentially impossible for students to
sketch accurately during the lecture. Our current solution to this problem is t o give each student a printout of the orbital contour diagrams generated by program CHRORBS (8). These plots (see the sample 2p, orbital in Figure 2) while not as realistic as those of the type in Figure 1because of the use of character graphics, have several advantages. First, the shapes of the contour lines are easily seen which makes contour plots such as those below in Figure 3 easier to explain. Second, since they are all plotted to the same scale with the same characters representing the various levels of electron density in each plot, meaningful comparisons between the orbitals can be made. This is not possible with the Figure l-type plots since they are all necessarily scaled differently. Finally, the source code for program CHRORBS, which is written in Microsoft QuickBASIC 4.00b, is so short and clear that many of our students can read i t without difficulty even though they may have studied or are currently studying a different computer language. Thus, our handout includes a listing of the program, which gives curious students access to the wave functions for the atomic- and - ~ - hybrid orbitals that are not normally in mainstream texts. This enables them to see where the orbital names come from ~~~
~
Volume 67
~
~
Number 1 January 1990
37
~ ~
spherical harmonics or their square. We feel these polar coordinate plots are too complex and of too little value to be introduced into the program at this level, particularly since they cannot he used for the hybrid and molecular orbital discussions which follow in the next chapter. Many instructors may feel that the orbital shielding and effective charge ar~umentsnecessary for discussion of the electron distrihution in many electron atoms and the aufiau principle cannot be taunht without reference to radial distribution functions. We heiieve that this material is better taught by reference to the Figure 2-type diagrams where the electron density near the nucleus for a 3s electron is clearly greater than that for a 3p electron, which is greater than that for a 3d electron, so the variation in effective charge is easily explained. Finally, to facilitate discussion of overlap, drawing diagrams on the blackboard, note taking, etc., the familiar sphere for the s orbital and figure-eight shape for the p
Figve 2. CtmracCer graphics contour plot ofa 2p,wbiml,
Figure 4. Sholmand rotation for the 2p, wbltal.
orbital (Fig. 4) are introduced as convenient shorthand notation for the Figure 3-type diagrams. Teaching the Shapes ot the Hybrid Atomic Orbnals
Our method of teachine the shaves of the so. so2. and S D ~ hybrid atomic orbitals is &rallel t o that described above fbr the hvdroeenlike orhitals. First. the accurate electron densi" " ty plots are shown as overhead transparencies; Figure 5 shows a typical sp orbital plot. Then the contour plot from program CHRORBS is introduced (see Fig. 6),and, thirdly, single contour surface diagrams containing 90%of the electron density are drawn, as shown in two dimensions in Figure 7. The final step is to develop a shorthand notation for the hyhrid orbitals, which keeps the important features of the Figure 6-type diagrams intact. Our shorthand notation
Figure 3. Slngb canow line pi@ of a 2p, wbhl.
and to investigate the effects of varying the effective charge if they.have access to a microcomputer. Once the students are comfortable with the program CHRORBS plots, the idea of finding the orbital contour that encloses some arbitrary fraction, say 90%, of the total electron density can be introduced. Figure 3 shows such a single contour plot, and many texts give excellent 3-dimensional representations of such contour surfaces. Note that to this staee we have not mentioned either radial distribution functions or polar coordinate plots of the spherical harmonics. Texts that show these 90% contour diamams of p orbitals as two tangent spheres or as a figure-eight-shaped surface are either unnecessarily distorting the actual shape or are mislabeling plots that are in reality polar coordinate plots of the 38
Journal of Chemical Education
Figwe 5. Eiectmn density p b l of a sp m i d Orbhi.
F l w e 6. Charsnw gaphi-
Figure 7. Single urnlour lice plDt of a sp h y k M orbital.
cornow p l d ot a sp h y k M orbital.
for the sp hybrid orbital, which differs only slightly from that for the sp2 and sp3 orbitals, is given in Figure 8. This
The strikine correlation seen in the table between the Dercent s character and the spin-spin coupling constantis a direct consequence of the electron density at the carbon nucleus as shown in our Figure 6-type di&ams. If the sp hybrid orbitals are drawn as in Figure 8, then the sigma bonding orbitals in acetylene should be drawn as in Figure 9,
F l w e 8. Shorthand m t i c n ta a sp hybrid orbital.
notation differs from that in all modern textbooks that we have been ahle to review in that the nucleus is correctly depicted within the smaller high-electron-density lobe of the orbital and not a t the node between the lobes. It has been argued by several authors that their small and convenient artistic license taken with the placement of the nucleus has no conceptual consequences of importance and thus the ~racticeshould be continued. We disagree. The physical properties of molecules that depend o n the interaction of electron and nuclear spin, depend directly on the s-electron density at the nucleus, and that density is incorrectly depicted in the standard textbook diagram as being zero. The most common places where the variation of this electron-nuclear hyperfine interaction with hybridization are observed are in electron s ~ i resonance n hyperfine coupling constants in hydrocarbon>adicals and in nuclear magnetic resonance s ~ i n - s ~ icou~linp n constants where the coupling 3 and a proton. The tahle measured is.between a ~ 1 1 nucleus gives the C-13 to H - l NMR spin-spincoupling constants for simple hydrocarhons in the three hybridization states (9). ~
C-IS M H Spin-Spin Coupllng Constants and Hybridization
Figure 9. Repre~8ntaticnof ttm atomic orbitals owlapping to form sigma bonds In acetylene.
which makes the origin of the spin-spin coupling clear. The standard textbook diaeram with the carbon nucleus at a hybrid orbital node p k c t s that the spin-spin coupling should he zero and inde~endentof hvbridization. As increasingly detaiied discussi& of NMR are introduced into elementary organic chemistry courses, it becomes increasingly important that theshapen the hyhridorbitals be taught corrertly in General Chemistry.
of
Literature cnd
1. Fading, L.;Wilson. E. 6.Infmdution to Qvonlum Mechanics; McCraa-Hill: York. 1935. 2. Cromer, D. T. J Cham. Edur. 1968.45.626632. 3. Cohcn. I.;Bust8rd.T. J. Chrm. Educ. 1966,43,167-193.
New
4. Moore,J.W.:Davi~s.W.G.InlroduefionfoAtami~Strulure:ASerieaafTenCom~uler.Genemted T m ~ p r e n c i e s Utilizing the D o l - D m d y Technique; Science Rekt
ed Materials, 1975. 5. Wise. J. H. Warhington and L e U n i ~ n i t y Pint : prhs. Compubr Graphic. Conteat: Eighth Biennial Conferencs on Chemical Education, 1985. 6. Liebl. M . J. Chem.Edu. 1988.65.23-24. 7. Thoorbitalplotfingmethduaed in programDOTORBSaassuggestedby J. W. Mmre, University of Wisconsin-Madieon, p e m n s l eommuni&ion. December 1981. The program is ~rrentlyavailablefrom PmjectSERAPHlM at tho University of Wiamn-
9. Jackman, .I
M.; Stemhell, S. A p p l i c o f i o ~of Nuclear Mognrtic Resouace SpectmsPergamon: Oxford, 1969; p 346. copy in Organic Chemistry. 2nd
ed.:
Volume 67
Number 1 January 1990
39
