Bodie E. Douglas University of Pittsburgh Pennsylvania
A Simple Model of the d Orbitals
Altllough excellent pictures of three-dimensional modrls of the separate 3d orbitals have been presented,' thc models themselves were difficult to co~istruct.~Other very attractive models of the individual 3d orbitals have beer1 described r e ~ e n t l y . ~ The great value of such models i n classmom and research discussions concerning ligand field effects led to the development of inexpensive, easily constructed models. The chief material for the 7-inch model shown in Figure 1 was inexpensive plastic sponges. The 15inch model shown in Figure 2 was made from plastic foam, carved from rigid foam plastic cut from the lower sections of cases used for shipping acid hottles.
Figure
1.
Figure
2.
As a guide to shaping these materials, one can use projections of the houndary surfaces or contour lines of the M orbitals2 or photographs of the models.' For most purposes little value is lost if the "doughnut" lohe of the d,. orbital is omitted; accuracy of the model with respect to size and shape of the orbitals is not critical. The lobes of the orbitals used in the models are somewhat elongated compared to those of Ogryzlo and Porter3 in order to he able to combine the five orbitals in a single model. Scissors are adequat,e to cut teardrop shaped lobes from the sponges. The rough cutting of the rigid foam plastic was done with a hot spatula followed by shaping with a razor blade.
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' PEARSON,R.. (11.) Chem. andEng. News, June 29, 1959, p. 73. ( ' h o ! ~and . Enq. H n w . June 29. 1959. D. 108: Seni.enrl,er 14. ..
3 0 0 ~ ~ z r . 13. o , A., AND PORTER,G. H., J. CHEM.EI>uc.,40, 256 ( I!!W). ' D A Y , M. C., JR., AND SELHIN, J., "The~ret,i~nl Inorganic Chemistry,'' Ileinhold Publishing Corprm.tirm, New Yorlr, 1962,p. 278.
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Journal o f Chemical Education
The lohes were then shaped more uniformly a broad flat file and sanded smooth. Thus shaped, the model represents the boundary surfaces which enclose most of the electronic charge of the d electrons and is related to The signs of $ for the lobes are useful in deciding which combinations of orbitals are expected to contribute to bonding. Hence, it is useful to label or color code the lobes to indicate the sign of $, although the sign has only mathematical, not physical significance. Distinction among orhitals is also important. Three colors were used, one for the positive lobes of the d,. orbital, one for the tl, orbitals, and another for the d,.-,. orbital. The difference between positive and negative lobes of the same orbital was show^^ by shade. I n the sponge model, colors of the d,. and tz, orbitals were chosen from those commercially available in the sponges, and the lobes of the d,.-,. orbital were dyed with methylene bhie. Latex base paints were used for t,he plastic foam model. When cut and painted, the lobes were assembled by mounting with welding rod or braising rod sharpened a t one end and sticking into a cuhe of cork at the origin of the coordinate axes. The coordinat,r axes are easily identified if the rods along thrse clii,rctions project all the way through thc lobes. Also, rods can be inserted along the trigonal axes of thr octahrdrou in order to illustrate t.he sitnation which ohtains i n trtrah~dralcomplexes. I n examining the possihilit,ies of h p pi bonding, it is helpful to show the symmetry of the p orbitals of the lizands in an octahedral comulex. This is done in the &dd of Figure 1 using cardhoard "propellers" labeled p,, p,, etc., with the signs of indicated for each lobe. Ipor considering molecular orhitals, the ligand at,oms can be numbered on the "propellers." Illustrations of the use of symmetry of atomic orbitals in forming molecular orbitals in complexes are given hy O ~ g e l . ~ Students often have difficulty in relating the rrpresentations of the 5 separated orhitals and those of the d orbitals in an unidirectional magnetic field.a.Vl>e model aids in relating these two represent,ations. In an unidirectional magnetic field in the r direction the d,. orbital (m=O) is the same as in the model. The d,, and d, orbitals (m = + 1) are indistingnishahlc and form two cup-shaped lobes with the 7, axis as an axis of rotation. The d,.-,. and d,, orbitals (ml = +2) forma large doughnut in the xy plane.
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s O ~ L. ~E., ~"An~ Introduction , to Tmnsition-Metal Chemistry; Ligxnd Fidd Theory," Methuen and Company, Ltd., 1960, chap. 2. HUME-ROTHERY, W., "Atomic Theory," 2nd ed., The Institute oi Metals, 1952. p. 03
