John W. Moore and William G. Davies Eastern Michigan University Ypsiianti. 48197
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I Illustration of Some Conseauences of the Indistinguishability of ~iectrons
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Use of computer-generated dot-density diagrams
Computer-generated plots in which the density of dots is nronortional to the electron probability function for various . hydrogenic orbitals have bein published previously in this Journal ( I ) , and we have reported the extension of the dotdensity concept to self-consistent field (SCF) wavefunctions of atoms and molecules (2). Such diagrams provide a semiquantitative indication of electron density which is readily comprehended bv students whose mathematical background is limited. ~ e c a & eof their visual impact, dot-density diagrams are an excellent pedagogical tool for elementary chemistry, but thes arcalso capableof illuminating a number of rather suhtle w i n t i , someuiu,hich we shall consider in the remainder of this paper. Readers whose principal interest is pedagogy rather than its theoretical justification, may wish to proceed immediately to the sections headed "Hybrid Orbitals Via Dot-Density Diagrams" and "Ethene Via DotDensity Diagrams" where classroom use of overhead projection materials is discussed.
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Localized and Delocalized Descrlptlons of Electronic Structures I t is a well-known principle of quantum mechanics that electrons are indistinguishable. If the coordinates of two electrons are interchanged the probability of the new configuration must be identical with that of the previous one. Thus the square of the total wavefunction must remain the same, and
en,w= *'?old (1) According to the Pauli Principle the negative sign in eqn. (1) must be chosen. A straightforward means of constructing such an antisymmetric wavefunction for a system of N-electrons, each of whose coordinates x,, y,, z, are represented by integers 1,2,3,.... N, is to use a Slater Determinant as in eqn. 12)
e
=1
A(l)n(l) $,(1),9(1) * l
h ( 2 ) a ( 2 ) ........ . h ( N ) a ( N ) *,(2)@(2)................... *2(2)a(2) ...................
.......................................
(2)
$ ~ 1 2 ( 1 ) @ ( 1$)~ 1 2 ( 2 ) @ (...... 2) $N/N/~(N)B(N)
The functions +I, $2,. .. $ ~ / 2are orbitals, each of which can contain one electron of snin a and one of spin B. Interchanging ... c~mnlinatesof tu80electrons correspa,nds to interchanging columns. chaneinr the wen of the determinant. Ohvio~~sl\. anv set of functions which can be used to build up the sametotal determinantal rIr (or its negative) provides a solution to the Schroedinger equations. Depending upon one's orientation, one or another of such possible solutions may be more useful. From the chemist's ~ o i n of t view there is considerahle evidence that the bonds holding atoms togethcr in molecules nre relatively independent of one another. This implies a pair (or pairs) of electrons moving in orbitals localized~betw&nthe bonded atoms. On the other hand a spectroscopist, whose concern is excitation or ionization, is not inclined to accept removal of an electron from a localized orbital when several other symmetry-equivalent bonds remain unaffected. In this case it is more logical to allow each orbital to extend over the entire molecule. Both of these seemingly antithetical approaches are successful in specific applications and they constitute a duality which is essential to the modern understanding of molecules. A review 426 / Journal of Chemical Education
by Pople (31, based on earlier work of Lennard-Jones (41, gives a number of examples of localized and delocahed descriptions of the same electronic structure. We have chosen two of them for illustration here. Hybrid Orbltals Consider an atom containing two electrons, eachof spin a. T o satisfy the Pauli Principle they must occupy different spatial orbitals, and for this example we choose 2s and 2p. Electron dot-density diagrams derived from Clementi's SCF wavefuctions (5)for 2s and 2p orbitals in a carbon atom are shown in Figures 1 and 2. The antisymmetric total wavefunction for such a system is
are, respectively, Since the one-electron orbitals, J/zs and hp, symmetric and antisymmetric with respect to a horizontal plane (the nodal plane of the 2p function), they can be combined as shown in eqns. (4) to give localized (hybrid) orbitals pointing down and up. Dot-density diagrams corresponding and iLsPare shown in Figures 3 and 4. to il.,p+
kP+ = 1 / 4 5 ($2+ * z p ) Vh- = 1 / 4 5 (*z' - *2J
(4)
The total wavefunction, q ~for , the hybrid orbital case can be written as
That Ws = Wl is easily verified by evaluating the determinant y that the rou,s of ( 5 )can be obtained by applying or l ~ noting elementar!, row qx,rations ( 6 )to the rows of (31. The identitv of the total wavefunction derived from localized hybrid orbitals with the more familiar one based on spectroscopic (Is, 2s, 2p, etc.) functions is quite evident from the Slater Determinant. The problem which we wish to address here, however, is the presentation of this concept to the freshman 'level, where it js almost invariably intioduced. Knowledge of the properties of determinants or even an understanding of the linear combinations (eqns. (4)) cannot penerallv be assumed a t this level. Even if they could be iniroduced, conaiderahle time which might hetter be devoted to non-mathematical concelm would hv consumed. We hclicve that a graphic approach using dot-density diagrams leads to an accurate description of hybrid formation which will be readily understoodby the majority of freshman students. Hybrid Orbitals Via Dot-Density Diagrams We have found that classroom discussion of hybrid orbitals is best accomplished using color-coded overlay overhead transparencies. Unfortunately these cannot be reproduced on the printed page and will have to he described in terms of Figures 5, 6, and 7. Figure 5 represents the total electron density summed over the 2s and 2p probability functions. On an overhead transparencv it would be produced bv overlavine liepresents the total electron dedsit; Figures 1and 2. summed over and +2,,-.~ and would be produced by overlaying ~ i g & G3 and 4. Our transparencies are constructed with Figures 1and 3 a t
re
~
Figure 1. Elecban dotdensity diagram far carbon atom 2s orbital (deta from ref. (5)).@ Copyright 1975 William G. Davies and John W. Mwre.
Figure 3. Eisctron d o t d h y diagram tor carbon atom hybrid orbital+ $ , (computed as described in ten). @ Copyright 1975 William G. Davis and John W. Mwre.
far oarbon Fioure 4. Electron dotdensihr. diaoram " atom hybrid orollal J S p - (compbtw as osscr Oso in ten) O C w ght 1975 WI Iam G Davier and Jdrn
Figure 2. Electron dotdensity diagram for carbon atom 2p orbital (data from ref. (5)). @ Copyright 1975 William G. Davies and John W. Moore.
Figure 5. Combinedelecbon densny of &,and ,b2* @ Copyright 1975 William G. Davies and John W. Mwre.
h.
Figwe 6. Combined electron density of and $ ,. or+,$ and . ,$ @ Copyright 1975 William G. Davies and John W. Moore.
the top and bottom of one sheet and Figures 2 and 4 at the top and bottom of another. The two sheets are coded hy different colors. When they are overlaid, Figures 5 and 7 are projected a t top and bottom of the screen. At this point the instructor can remind the students that the two electrons are identical and, since the Uncertainty Principle guarantees the impossihility of following each of their trajectories, cannot he distinguished from one another. The color coding is therefore not in accord with nature's law and must be removed. This is
Figure 7. Combined electron density of+,$
and
$w. @ Cap~ight1975William G.Davisand John W. Mmre.
readily done in the transparency by overlaying a third sheet with Figure 6 printed in black, top and bottom. When all of the dots are identical it is seen that Figures 5 and 7are dot-for-dot the same, except for the color coding. I t is immediately apparent to the student that the two figures are different representations of exactly the same atom. There is no magical process by which the 2s and 2p electrons are "changed" into a pair of s p hyhrids. All that has been done is to subdivide the total electron density in two different ways. Volume 53, Number 7, July 1978 / 427
The localized, hybrid representation may he more useful in some cases and the delocalized, spectroscopic orbitals in others. The duality between them is much like the waveparticle duality previously encountered by the student in discussions of the quantum theory. Partlal Hybrldlzation
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The formation discussed above avolies strictly - ~~-s n hvhrid .. , to a two-electron atom only, since the Slater Determinant is 2 x 2. - However. it isclear that the hvhrid orbitalscould each hold a pair of eiectrons (3)without changing any of the conclusions we have drawn. Moreover, not all of the spectroscopic orbitals need he converted to hybrids. One would not consider incorporating a carbon 1s orbital in the s p hybrid, for example, and often one or morep orbitals are left in spectroscopic form as well. The simplest demonstration that this is permissible requires three electrons. Let us choose 2s, 2p, and 2p, orhitals, each containing an electron of u spin. The Slater Determinant is then ~
~
where dl).d 2 ) , and 4 3 ) have heen omitted in the interest of briefer notation. If hybrids are formed as in eqns. (4) with J 2 p = ,$2p,, the total wavefunction containing the hybrids is
in spectroscopy or reaction mechanisms. Only three of the four valence orhitals of carhon are hyhridized, a situation which is permissible as we have noted ahove, but sometimes confusing to students who wonder why the p, orhitals are aloof from the rest. The intuition of chemists who first described the bonding in ethene (8)was to localize orhitals to the maximum degree, and many types of molecular models used today incorporate bent or "banana" bonds between tetrahedral carbon atoms. Although the popularity of molecular orhital theory and discoverv of the laree (117°) H-C-H angle in ethene have caused man; chemistsvto abandon descriptions of its electronic structure based on bent bonds between sp3 hyhridized carbon atoms, the hanana bond remains a very useful tool for the introduction of concepts such as rotational harriers, isomerism, reactivity, etc. a t an elementary level (9). Using the concepts already developed i t is easy to show mathematically that the sigma-pi and bent-bond descriptions of ethene are equivalent. Wavefunctions for the sigma and pi orhitals may be written as *m, = 1/45(*** + *us) (9) *ms = 1/45 ( * P A + *p,J where #*, represents an sp2hybrid on carhon atom A. Bent honds may be constructed as *bsnt+ = *bent-
1/45
+ ILee,)
= 1/& (tee. - *ccn)
(10)
It is obvious that bent+ and #bent- hear the same relationship to fiec,and ,#, that ILSp+ and #8p- in eqns. (4) hear to 2s and 2p atomic orbitals. Again i t is readily shown that q7 = q8by expansion or by noting that the first two rows of (7) can he derived from those of (6). Finally it should he emphasized that unless required by the svmmetrv of the molecule, linear combinations such as eqns. (k) need not contain equal proportions of s a n d p character. That is, more than just the usual tetrahedral (sp3),trigonal (sp2), and linear (sp) hybridizations are possible, should molecular symmetry permit non-equivalent orhitals. Techniques have heen developed (7) for determining hyhridizations which can produce any desired comhination of bond aneles. In an unsvmmetric molecule the onlv on . requirement . eqns. (4, is that they he orthogonal and normalized. This can be \,erified bv comoarine a Slater Determinant constructed from
-
#aeP*sp6+
a
= 5b% = 114 *%
- 116PZ'$ + a*zp
(8)
Ethene Vla Dot-Denslty Diagrams
To illustrate graphically this equivalence we have produced the dot-density diagrams shown in Figures 8-10, using the completely localized wavefunctions of Kaldor and Shavitt (10). Figure 8 shows total electron density in the molecular ( x y ) plane. As an overhead transparency it would he separated into four overlays-one containing the two carbon is pairs, one showing the four C-H honds and two for the hanana honds. (If the sigma-pi formulation were used a total of three overlavs would suffice since the pi orbital has a node in the moleckar plane.) Figures 9 and 10 show electron density in the plane containkg the two carhon 2p, orhitals. In this plane we have constructed an overhead transparency from six color-coded overlays. The first two show the two carhon 1s pairs and the four C-H bond pairs. They have been omitted from Figures 9 and 10 for clarity. Figure 9 shows sigma (gray) and pi (black)
with qsor q3.Hybridization does not require equal participation of the s orbital or all of the p orhitals in each linear comhination. Representationof Bonding In Ethene
Manv of the remarks in the oreceding section are orefatory to a discussion of binding in ethene. Its electronic structure is usuallv described in terms which include both localized and delocaliied orhitals. The planar molecular structure permits classification of orhitals as symmetric (a)or antisymmetric (a)with respect to reflection. If the molecular plane is taken perpendicular to the z axis, linear combinations of the hydrogen 1s and carbon 2s, Zp,, and 2p, orhitals may be used to construct a molecular orbitals. The only valence orhitals having a symmetry are the carbon Zp,. The 12 valence electrons would thus occupy five sigma orhitals (delocalized over the six nuclei) and one pi orhital. A transformation of the Slater Determinant which localizes the sigma MO's into four C-H bonds and one C-C bond is usually implicit in any discussion of ethene. These honds are usuallv described as heine formed hv overlao of H 1s and carbon sp2 (tr, trigonal) k r i d orbitals. ~ t t k t i o nis then focused on the a and a * orbitals, especially if one is interested 428 / Journal of Chemical Education
Figure 8. Total electron density In the molecular plane of ethene (data from ref.
(lo)).@ Copyright 1975 Wllllam G. Davier and John W. Moore.
Flgure 9. ElecWon dsnslty of the carboncarbon double bond-sigma (gray) and pi (black) formuUian. @ Copyright 1975 William G. Davies and John W. Mwre.
Figure 10. Elecwon denslty of the carbon-carbon doublb bond-banana bond forrnulatlon. @ Copyrlght 1975 Willlam G. Davles and John W. Mwre.
electron density and Figure 10 shows two banana bonds. Each of these is color-coded on a separate overlay so that any one or all of them can be shown to the class. The careful observer will find that Figures 9 and 10 are dot-for-dot the same when shadine is ienored. sbowine that thev are two different wavs of divi&ngthe s a k e total ilectron density. As in the case bf hybrid orbitals this can be demonstrated clearly and accurately without the requirement of any mathematical preparation. If a t a later date the mathematical techniques are to be taught, the student who has heen exposed to dot-density diagrams ought to be receptive to them and will not have been misled regarding their end result. Two further points need to be made. First the Kaldor and Shavitt wavefunctions are completely localized and contain bent rather than sigma-pi bonds. Thus the electron density displayed in Figure 9 was calculated from linear combinations of the bent-bond orbitals whose probability functions are shown in Figure 10. (This process is essentially the reverse of that used to obtain eqns. (10) from eqns. (9)). The second point is that bent bonds do not requiresp3 hybridization on the carbon atoms in ethene, because tetrahedral symmetry is not necessarily maintained. (The bent bonds in eqns. (10) contain wavefunctions originally derived from sp2 hybrids.) Neither do sigma and pi bonds requiresp2 hybridization. In fact, since the H-C-H angle in ethene is somewhat less than 120". the hvbrids used to form C-H bonds must hat,e somewhat more p character than puresp'. Non-equivalent hybrid orbirnls of the type mentioned earlier (71 must he involved. This is, of course, permissible, and in no way in\,alidatea the euuivnlence of bent and siema-oi - . bonds. ~ c t u a l ethene l~ is somewhat unusual in having a larger H-C-H angle than the R-C-R' angle adjacent to many C=C bonds (11). The C-R bonds in such substituted molecules can be described in terms of hybrid orbitals much nearer to sp3 than sp2. We would argue that the bent bond is not only theoretically equivalent to the sigma-pi description hut also is more readily understood a t an elementary level and quite capable of accounting for phenomena such as rotational barriers, isomerism, and reactivity. Theoretical concepts such as orbital overlap, sigma and pi symmetry, positive and negative lobes of orbitals, etc. are necessary precursors to discussions of
spectroscopy and reaction mechanisms, but they need not be introduced into the curriculum before these topics are encountered. At such a time electron dot-density diagrams, which require little or no mathematical background of the student.. can . orovide a clear and accurate transition to delocalized orbitals. Extensive discourse on molecular orbital theorv is not a necessarv orereauisite for understandine localizdd electron pair bonds which can he used to interp;et a ereat manv chemical ~henomena.Introduction of MO's mieht well be pokponed to-make room in the curriculum for more descriptive topics. We wish to thank Professor Henry A. Bent for suggesting some of the references and approaches used in this paper. Computations for the dot-density diagrams were made on the DEC SYSTEM-10 of Instructional Computer Services, Eastern Michigan University, Edmond Goings, Director. The 'plotter subroutines were written by Mr. Gary Frownfelter and Professor Jerald Griess. We also wish to thank Mr. Richard Oltmanns of Eastern Michigan's Audio-visual Production Services for his aid in constructing the overlay overhead transparencies. Since computer plotting programs for the oroduction of dot-densitv diaerams are not readilv transportable from one machine to another, we are unable to provide listings or card decks. However, overhead transparencies of the type described here are available commercially from Science Related Materials, Inc., Janesville, Wisconsin, 53545.
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Literature Clted 11) Cmmer.DonT..J.CHEM. EDUC.,45(101.626 11966). (2) Davias, William G.,and Maore, John W., in T"eeeding. ofthe Fifth Conference on Computers in the Undergrnduata Curricula: Department of computer science. Washington State Uniuer~ity.1974. pp. 67-76. 13) Popie. J. A,. Quart, Rru.. 11.273 119571;rceslroEngland. W..J. CHEM.EDUC..62 17). 427 119731 for an up-to-date review. 141 1.ennsrd-Jones, J.,Pror. Roy. S o l . A,. 198.1 11949). (51 Clementi, E.. IRA4 J
