Novel pictorial approach to teaching MO concepts in polyatomic

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D. K. Hoffman, K. Ruedenberg, and J. G. Verkade Ames Laboratory of the ERDA Department of Chemistry Iowa State University Ames. 5001 1

I

I

A Novel Pictorial Approach to Teaching MO Concepts in Polyatomic Molecules

...many students graduate with the impression that the ground rules governing bonding in molecules such as phosphorus trifluoride are somehow different from those which apply to aromatic systems such as benzene. Undergraduate chemistry majors typically find it difficult to formulate MO schemes, especially delocalized ones, for molecules more complicated than diatoms. The major reason for this unfortunate situation is the general impracticability of teaching group theory before students take organic and inorganic chemistry courses where applications of these conrepts art* most beneficial. Consequently many students grnduatr with the impn:ision that the ground rules go\.erning bonding in molecul& such as phosphorus trifluoride are somehow different from those which apply to aromatic systems such as benzene. Furthermore. seniors are usuallv only vaguely, if a t all aware, for example,that sigma bonding (likk extended oi bonding) can be described in a delocalized manner and that 'hybridiz&m need not he invoked to discuss the sigma framework of PF3. In our opinion it would he highly desirable for chemists to realize a t an early stage of their education that both the delocalized and localized honding descriptions are not only highly useful, but when properly employed embody quantitatively identical representations of molecular states. The delocalized MO's convey orbital energy arrangements particularly useful in the discussion of more than one electronic state of a molecule and hence are appropriate for the interpretation of spectroscopic experiments. On the other hand, localized orhitals make evident lone pairs and bond pairs and also help to elucidate molecular shapes and mechanistic pathways in chemical reactions. Both descriptions, however, are eauivalent. In brder to remedy the undergraduate's fragmented appreciation of the general applicability of the fundamental and strongly interrelated concepts of delocalized and localized MO's to polyatomic systems, a one-quarter lecture-problems course has been developed in our department in recent years, which emplovs a novel teaching method designed to equip the beginning undergraduate chemistry and science major with a working skill of both honding approaches. The level of mathema&al maturity expected is that attained in one or two quarters of calculus. However, only the concept of the integral used and no calculus maninulations are carried out exolicitly. Thus the honding course can he offered as early ad the third auarter of the freshman vear. a t which time the student has hah time to be exposed t o i l l the pertinent mathematical concepts. The first part of the course deals with fundamental concepts such as mass, charge, energy, and waves. T h e wave nature of an electron is developed by briefly discussing, qualitatively, standing and traveling electromagnetic waves and extending these concepts to electron waves: Canonical atomic orhitals (i.e., s,p, d, etc.) are then introduced as standing waves. After a presentation of electron spin behavior and the Pauli Principle, one-electron atoms and ions are discussed with particular emnhasis heinp nlaced on the relations between the ~~~~~~ - ~ quantum numbers (nlrn)and the systematic patterns in the nodes of atomic orhitals. The concepts of orthogonality and

is

.~~~~~~ ~

~

~

"

orbital normalization are introduced and the effect of delomlization and orbital nodc,s on orbital energy arc examined. ('onsideration is also mildeof many electron atoms and iuns, and of influences on periodic relationships such as the effect of shielding on atomic sizes and ionization energies. Then the formation of hybrid atomic orhitals by linear combination of canonical AO's is introduced. The equivalence of the delocalired rcanunicd~and the Iocali7~dihybridl descripriun is made plnusil~lehv simply demunsrr,ltiny the invariancr of the total electron density-under any orthogonal transformation among the contributing orhitals. Finally, anumber of ancillary nrincioles oertinent to molecular hondine are discussed, such as the' detkrmination of molecular geometry from VSEPR considerations and the desianation of valence AO's available for bonding. The remainine half of the course is devoted to MO honding concepts appliei to a set of polyatomic molecules offering a wide varietv. of geometrical shapes and sizes. All, however, .. pussess w m r symmetry. The examples in the table are examined from a delocali7ed and iwalizcd point uf view. The repeated use of this dual approach convincw the student that consideration of pyramidal, tetrahedral, and octahedral molecules, for example, is not restricted to localized hybrid AO's on the central atom and that conjugated pi systems can he describnl in a lwalired fashion. T h r prinripal tool which makcti this discussion poisihlc is the idea that MO's. whether o i the localin-d (81 delocali7ed variety. can 116. deduced i r m nn apprnpriate extensiun ui the characteristics d AO's. More speciticallv, rhr 5)mmetr) and dirc~rionnla hrrrrr~lvrirlicsbf A l l ~ bnrv dcril'ed f r m ;ivmmdr, ond dir~crtonnlchnrricturisrics of dO's. T h r appiication"of these principles along with t h e concepts m i -

rh19

Examples of Compounds Amenable to the Generator Orbital Approach Compound

Geometry

Compound

BeF2 XeF2

linear linear bent bent triangular triangulare trig. planar

SF4 IClrC6H5BrFs PFs C8He

H20

NO2-

H f C3HZt CHQ' BFa

&Fa NH3 PI Tea2+

C& CHI PO$-

trig. planar

shaped pyramidal tetrahedral sq. planar ~ q planard . tetrahedral tetrahedral T

distort,tet. sq. planar pentagonal planara sq. pyramidal

bipyramidal hexagonal planar PFeoctahedral C7H7+ heptagonal planar' I h pentagonal bipyramidal CsH,Z octagonal planar' CsHs cubic CoFe3octahedral Fe(C6H& pseudo octahedral MO~CI,~+ octahedral clustero NbsClr22+ octahedral cluster"

A

590 1 Journal of Chemical Education

Geometry

.Treatment of r molscvlar Orbitals. qreafment of metal cluster molecular orbitals.

trig.

In order to find the proper LCAO-MO's, we introduce a new set of atomic orbitals (in addition to the valence AO's) which we call generator orbitals.

tioned earlier allows a student to generate qualitative delocalized and localized MO expressions and their corresponding mwgy lesel diagrsms. Of course quantiwti\,e treatments ran only be pruvidt!d 11). detailed ralrulatiuns. Valence and Generator Atomic Orbitals Canonical atomic orbitals are conceived to be three-dimensional standine electron waves which have nodal surfaces (spheres, planes, and cones) across which the orbital amplitude changes sign. - The arranaements of these nodal surfaces, which are unambiguously dktermined from the quantum numbers n, 1, m, give rise to the particular symmetries associated with s, p, d, etc. orbitals. For the purpose of MO construction these nodal patterns are the most important features of ~- the AO's. Bv distrihutine the n - 1nossihle total number of nodes for a given principal quantum number n in a systematic manner., anv.orbital shaoe can he deduced. Our objective is to develop a procedure for constructing molecular orbitals as linear combinations of valence AO's. In order to find the proper LCAO MO's, we introduce a new set of AO's (in addition to the valence AO's) which we call generator orbitals (GO's). We imagine these GO'S to be placed at the center of the molecule and we use their symmetry characteristics to deduce LCAO MOS with corresponding properties. After this is accomplished we mentally remove the GO's. By choosing appropriate canonical GO'S we obtain symmetry adapted canonical LCAOMOk which, for the cases we treat here, are also the delocalized MO's. On the other hand, the use of hybrid GO'S yields localized LCAO MO's. Again we emphasize that the GO's serve only as a constructional tool and must be distinguished from the valence AO's. Frequently one requires GO's with quantum numbers higher than those of the valence AO's taking part in the bonding. Therefore an appreciation of orbital shapes beyond d AO's is reauired in order to fullv utilize the Dresent a w proach. However, for the sake of brevity, we will restrict ourselves here to exam~leswherein a knowledge of s, P, and d GO symmetries is sufficient. A fuller and moredetailed discussion of the GO method, including a pictorial technique for deducing the shape of any required GO, is developed elsewhere.' We shall illustrate the mechanics of the approach with ~~~~

r

~

X.

-

~

I

Figure 1. (left) Canonical GO'S. SO'S, and delocalized MO's for XeF2 Figure 2. (right) Delocalized MO energy level diagram for XeF2,

several examoles selected for the diversitv of hondine Datterns which they cbntain. The following abhrekations wii be used: BMO = hondine - MO.. ABMO = antibondine" MO.. NBMO = non-bonding MO, SO = symmetry orbital, GO = generator orhital. ~elocallzedMO's of XeF2 The linearity of this molecule is apparent from the VSEPR considerations implicit in its electron dash formula.

The essence of the hondina in this molecule can be hroueht out by assuming that all tge valence AO's house lone p&s except the threepz AO's which are concentric about the molecular axis which we take to be the z axis. (This is in agreement with the fact that lone pairs have more s character than bond pairs.) In a complete tieatment by the present method these restrictions would not be made. Let us disregard the Xe atom for a moment and imagine various GO's placed one by one between the fluorine atoms. For each GO we try to find that combination of fluorine orbitals which has the same symmetry until we have as many fluorine svmmetrv orbitals (SO'S) as there are fluorine valence orbitals (two in this case). For instance the symmetry of an s GO is such that it "dictates" a ~ l u sien s in the nearest lobe of each fluorine 2pz A 0 as shownAin~ i g k 1e (a). The outer 2pz lobes are necessarilv in sien since a nodal olane occurs .negative between the lobes of each orhitar If we mentallyremove the s GO, we are left with the SO com~osedof fluorine 2oz AO's with their positive lobes pointed toward one another. If we now try other GO's, we find that only one other SO is produced and it is generated by thepz GO (Fig. l(b)).Because two valence AO's are expected to produce only two SO's we need not search for additional SO's. If we were to try the py and px GO's, we would find that there is no fluorine 2pz orhital combination having their symmetry as illustrated in Figure Uc). We now look for suitable valence AO's on Xe to linearlv combine with these SO's. Although it may seem to be a trivia"^ operation, it is useful to generate the canonical valence AO's of xenon with canonical atomic GO's. Such a procedure gives the complete set of xenon valence AO's of which according to our initial assumption, we use only the 5pz A 0 which is generated by the pz GO. This 5pz A 0 can interact in a bonding and antibonding manner with the fluorine SO generated by the same pz GO, which leads to the BMO and ABMO in Figures l(d) and l(e). Because we reserved the 5s for a lone pair, the fluorine SO generated by the s GO does not interact with any xenon valence A0 and it becomes a non-bonding MO (Fig. I(,?). Figure 2 shows the resulting energy level diagram. Notice that only one electron pair binds the molecule in a delocalized 3 center hond, the other spin-paired electron pair residing in a 3-center NBMO whose only contributions come from the fluorine 2pz AO's. Thus we have an effective hond order of 0.5 for each link.

'Hoffman, D. K., Ruedenberg, K., and Verkade, J. G., Structure and Bonding, (1977). Reprints can he requested from the authors. 2England, W., Salmon, L. S., and Ruedenberg, K., Topics in Current Chemistry, 23,31 (1971).

Volume 54, Number 10, October 1977

1 591

Locallzed MO's of XeF2

Localized MO's extend over smaller portions of the molecule than do delocalized MO's. The most common examples of such orbitals are 2-center bonds and lone pairs. The description provided by the set of all occupied delocalized orbitals in a molecule is equivalent to that arising from the set of all occupied localized orbitals, even though the individual localized and delocalized orbitals can differ greatly in size and shape.2 The object of the GO method is to generate the electron distributions of MO's about the molecular center. In the preceding section it was shown that canonical Go's serve to establish the svmmetrv of delocalized MO's. For localized MO's the property complementary to symmetry is the directionalitv of the distribution about the molecular center and we therefoie require Go's which indicate directional regions for the localized MO's. The appropriate GO'S for thispurpose are hybridorbitals formed from thosecanonical GO'S which give rise to the occupied delocalized MOS. Such hybrid GO'S establish the radial directions of the localized MO's and identify the major A 0 contributions to the localized orbitals. Localized MO's are formed by taking linear combinations of the occupied delocalized MO's; the quantitative relationship between the two types of orbitals is provided by the hybrid Go's. The coefficient for the contribution of a eiuen delocalized MO to thk linearcombination defining a'lockized MO is identical with the coefficient of the corresponding canonical GO in the appropriate hybrid GO. That is each delocalized MO contrihutes to a localized MO in the same proportion as the corresponding canonical GO contrihutes to the hybrid GO. The present case offers a simple example. Since the occupied delocalized MO's are generated by an .9 and a p canonical GO, the localized MO's are generated by the twosp hybrids, . to the disnamely, (s p ) / 4 and (s - ~ ) / f i Accordina cussion in thti preceding paragraph, the two loc~lirrdMO's w e then the normalized sum and diffrrence of the hondina and non-bonding canonical delocalized MO's of Figure l ( d j and (f). The result of this superposition can he obtained in a pictorial way by allowing the directionality of each hyhrid GO "dictate" the A 0 contributions to the corresponding localized MO. These directional properties indicate that two equivalent

+

localized MO's exist which are oppositely directed from Xe. Each must consist primarily of one fluorine 2pz A 0 and the xenon 5pz AO, with a smaller contribution from the other fluorine 2pz A 0 (Fig. 3(a) and ( b ) ) .In order that the localized MO's he bonding, there must he a honding interaction bez tween the xenon 5 ~ Az 0 and the ~rincinalfluorine 2 ~ AO. Orthogonality l~etkeenthe two lo;alizei MO's requiies that the coefficient of the minor fluorine contribution he o o ~ o s i t e in sign to that of the major one.Vt is thus apparent that each localized MO has a bonding interaction in one link and a somewhat waakernntibondi& interaction in theother. liwe a t t r h t e a bond order of to each localized h10, we are then consistent with our previous conclusion that the bond order is %per link. Within the localized description we can straightforwardly include those occupied valence orbitals in Figure 2 which we assumed to house lone oairs. In the localized MO diaeram in Figure 4 they are given as sp2 hybrids because hybrid P;O's&e more localized than canonical AO's. Delocalized ?r MO's of C3Hli From the electron dash structure of the cyclopropenium ion it is deduced that two electrons are delocalized around the ring in a n system composed of three carbon 2p AO's.

In Figure 5(a) it is seen that an s GO in the center of the ring is incapable of generating SO'S from these carbon valence AO's. Since the oz GO is the onlv one of the D set which eenerates an SO (Fi'p. 5(b)),we mustsearch among the d GO': for two additional GO'S since three AO's are exnected to form three SO's. By trial and error it is easily determined that only the dxz and dyz Go's have the proper symmetry to dictate SO'S (Figs. 5(c) and (d)). Notice that in Figure 5(c) one of the 2pz AO's on carbon is drawn laraer than the others, indicatina a.coefficient twice as large in magnitude. This resuks from t h i requirement that the carbon A 0 contributions on each side of the vertical nodal plane in the GO must halance, which in turn follows from the required orthogonality between the SO's. We now mentally remove the GO'S from Figures 5(b)-(d) and we see that of the three SO's we have constructed, three 2pz carbon AO's contribute in two of them while in the third only two 2pz AO's take part. Since there is no central atom in this molecule ion, the SO'S themselves become the MO's of the system. The MO represented in Figure 5(b) is clearly a bonding one since the adja-

lbl

(a)

Figure 3. Hybrid GO'Sand localized MO's far XeF2. F Hybrid An'l

I*,

lLl%l

F A0'1

\

\

I

Figure 4. Localized MO energy ievel diagram for XeF,. 592 / Journal of Chemical Education

overlap, it is necessary that the contributions of each fluorine to the two MO's be of the same sign. Far this qualitative araument. the overlap between neighboring~o'scan be &nored,

'_,

1281

Figure 5. (len) Canonical GO'S. SO'S, and delocallred MO's far C3Hzt, Figure 6. (right) Delocalizea MO energy level diagram for C3H8+.

cent overlans are all additive. By the same reasoning the MO's in Figures ;(I.) and ( d , are net gntihonding. Furthermore, the two AHMO's are dercenrratr as can easily he shown by a p propriate rotation 07 GO'S (similar to the case of methane below). The delocalized MO diagramof the pi system in Figure 6 reflects these considerations. The two pi electrons are seen to he paired in the lowest MO which is delocalized over three centers giving an average s hond order of 0.33. T h e s* MO's are also delocalized over three centers. Localized r MO's of CsH3+ .. Orbital localization is only possible if there exists more than one occupied delocalized MO. Therefore the 3-center bond just discussed is the most localized description possible for the pi electrons in this molecule

pjjm +

EM0

S H l l l C R LOBES W O I SXOWN

lal

ABMO

IcI

ibl

Figure 9. Hybrid GO'S and localized MO's for CHI.

C AO's

CH.

Locaiimd Mu's

H AO'n

Delocallzed MO's of CH. The Lewis structure and VSEPR considerations lead to the familiar conclusion that methane is a tetrahedral molecule.

Four SO'S can be constructed from the four 1s hydrogen AO's. This is accomplished using an s and three p GO'Sas seen in Figures 7(a)-(dl. The corresponding SO's on the central atom are clearly the 2s and 2p carbon valence AO's. Forming honding and antihonding combinations of the hydrogen and carbon SO's of like symmetry, we obtain four honding and four antihonding MO's. Figures 7(e) and ( f )show the s-generated MO's, for example. From the obvious geometrical equivalence of the three hydrogen SO's in Figure 7(b)-(d), it follows that

Figure 7. Canonical GO'S, SO's, and delocalized MO's for CHa.

Figure 8. Delaolized MO energy level diagram for CH4

l*,i~l*,l.l*,l.l*,l Figure 10. Localized MO energy level diagram for CHI.

the BMO's (and also the ABMO's). generated hv the three D GO'S are degenerate. The delocalized MO enerav level diagram in Figure 8 implies a hond order of 1.0 perlink since &ere are four electron pairs for four links. However, it should be remembered that each of the MO's is 5-centered so that the BMO's represent 2-electron 5-center bonds. Localized MO's of CH4 Using the principles discussed for XeF2we start by constructine GO set from those canonical GO'S which ~, a hvhrid . we just 11st.d to generate the orcupi~11 ilelwd~redhlO'sor CH ,. Hvbridi~ationof ihr s nnd three,, G O ' s \ ~ i r l d s a n s(;Osrt ~' which can he directed toward thk hydrogen atomsas shown in Figure 9(a). Superimposing thissp" GO set on the full set of 1s hydrogen AO's and the carhon valence orbitals shows us where the localization of the BMO's occurs. First of all the directionality of the sp-0 set will generate a set of four equivalent, tetrahedrally oriented, sp" valence AO's on carhon where r is approximatkly 3.4 Secondly in order for honding to occur, the sign in the main lobe of each carbon valence spx-3 hybrid and in the 1s hydrogen A 0 toward which it is directed must be the same. These considerations are depicted in Figure 9(b) for one of the four s p "0's. Mentally removing the GO'S leaves a honding MO largely localized between carbon and one of the hvdroeen atoms. Aw~lvine .. . " the orthoeonality principle with negleit of neighbor overlap (as discussed for X e R ) it can be shown that the minor hvdroaen contrihu. tions i n t h e localized MO's of CH4 are zero. Such "complete localization. in the zero neighbor overlap is . approximation," .. characteristic for a tidl elwtrrm pair bmd in a link. In contrast, the localized M0';fin X e L were still sumewhnr drlw.~lizt,d o w r threr centersand it \\a* this ~leiwnlizntionwhich intro. duced the ~ x t i a l l ~ s n t i h o n. d1har;icter in~ [hilt made them half bonds. 4Thenatation spx for a particular orbital implies that the ratio of its s t o p character is 1:x. It says nothing about its direction. The carhon valence hybrids, although tetrahedrally disposed are not exactly sp%and, hence, somewhat non-orthogonal. This is so because the ratio of the overall carhon 2s to the overall 2p contribution in the delocalized MO set is not determined from symmetry considerations. Volume 54. Number 10, October 1977 1 593

In the present case we can localize the ABMO's by the same procedure and this gives rise to an ABMO for each BMO as shown in Figure 9(c). The localized MO energy level diagram is shown in Figure 10. Delocalized MO's of CO, Our final example involves the simultaneous consideration of a and a bonding which leads to some useful hut rarely appreciated aspects of the localized view. The linearity of this molecule can be deduced from its electron dash structure and VSEPR ideas. The presence of lone pairs on the oxygens prompts us to reserve an s orbital on each oxygen for such a pair. /

For o honding, the 2pz AO's will he directed toward the carbon as was done for XeF2. Since there is no lone pair on carhon, its s A 0 participates in the bonding as well as its 2pz AO. As in XeF2, the SO's resulting from these four valence AO's are generated by the s and pz GO's. From these SO'S shown in Figure l(o) and (b), the two o BMO's and corresponding ABMO's shown in Figure - l ( d ) and ( e )and l l ( o ) and (b) can he formed. In order to develop the delocalized a MO's of COz, we first generate SO's from the 2px oxygen AO's. Two SO'S are exoected and some simple trials with GO's soon reveal that the GO'S which are ahleto generate SO's from the oxygen 2px valence AO's are the px and the dxz GO (Fig. l l ( c ) and (d)). On carhon, the 2px GO also generates the 2px carhon valence AO, hut the dxz GO is ineffective since there are n o d orbitals in the valence shell. Consequently a a BMO and a a ABMO are formed by linearly combining the SO in Figure l l ( c ) with the carhon 2px A 0 (Fig. 1l(e) and (g)).But the SO in Figure I l ( d ) becomes a non-bonding MO (Fig. Il(f)). An identical a system at right angles to the one just discussed can be formulated from the 2py AO's of the oxygens and carbon. The delocalized energy level diagram in Figure 12 shows an average hond order of 2.0 per link as expected from the Lewis structure. Note, however, that all the BMO's are delocalized over 3-centers and the same is true for NBMO's. Because there is no carhon contrihution to the a NBMO's (there being a nodal plane containing zero electron density in the center as seen in Figure Il(f)) half the electron pair density in each is localized on each of the oxygens. I t must be realized however, that each of these r NBMO's is a single delocalized MO occupied by twb paired electrons, and not two localized oxygen AO's each occupied by one electron. This distinction is important in understanding molecular properties such as the diamagnetism of C02.

__m_----_C12al Figure 12. Oelocalized MO energy diagram for 602. A

I bl

Icl

Figure 13. Hybrid GO'Sand localized r MO's for C02.

Ibl

111

Figure 14. Hybrid GO's and localized banana bonds for C02.

Localized MO's of COz

Carbon dioxide is an example of a molecule for which several alternative localization schemes are instructive. We first discuss the localized MO formulation which is closest to the commonly depicted localized binding model. To this end we localize the rr and a system separately. The two delocalized a MO's arose from thes and p GO's. Hence the s p hyhrid GO set introduced in XeF2 also yields the localized a MO's of C02. In contrast to XeF2, however, both the s and p valence AO's on the central atom participate in the bonding. The directionality of the hyhrid s p GO'S therefore gives rise to two diagonal spx carhon hyhrid A 0 contributions to the localized MO's, where x is approximately unity. Furthermore, eachsp hybrid GO generates a 2pz A 0 on the oxygen toward which i t points. Consequently there will be a localized bonding MO in the right link and an equivalent one in the left. In the zero neighbor overlap approximation, the localized MO on the right has no contribution from the left oxygen and vice versa. As mentioned for CHI, this implies a a bond order of 1.0 per link. The occupied T delocalized MO's in the xz plane are generated by the px and dxz GO's and, hence, localization is accomplished with the GO hybrids (px dxz)l\/2 and (px d x z ) l d shown in Figure 13(a). Since we have a 3-center BMO and a 3-center NBMO in the delocalized description, the localization process for this a system is identical to that used for the a system of XeF2. This leads to the two localized a MO's as shown in Figure 13(b) and (c), each of which has a hond order of '1%.As a result we have a hond order of % per link per a bond in the xz plane. Identical considerations apply to

+

Figure 11. Canonical GO's. So's and delocalized MO's for COI.

594 1 Journal of Chemical Education

the r MO's in the yr plane, yielding an additional hond order of ' 2 in each link. The total hond order ver link is thus 1 + ' 9 '1; = 2 in agreement with the conclusion reached from the delocalized view. Another localized honding scheme results from equivalently hybridizing the GO'S which generate all the occupied delocalized BMO's, namely s, px, py and pz. Figure 14(a) shows the tetrahedrally disposed GO hyhrid set as well as the valence AO's which contribute to the resulting four localized MO's. The latter are a set of four equivalent hanana honds in two mutually perpendicular planes as indicated in Figure 14(b). Each of these has a hond order of 1.0 thus accounting for the bond order of 2.0 in each link. With this localization of the BMO's, no localization is possible for the a NBMO's, since hoth NBMO's are generated by d GO'S and mixing these GO'S results in rotation but not in hybridization. A localization victure intermediate to the two alreadv introduced is arrivkd a t by using several hybrid GO s e t s . ~ h e first hyhrid GO we choose is the pxdxz hyhrid in the left link and the second is the pydyz hyhrid in the right link, hoth of which were introduced in our first localization scheme. From the remaining two p d hyhrid GO'S and the s and pz GO'S,one can form four equivalent ~ p b dh )y h~~ i d sAs . ~before, the two p d hybrids give rise to a a bond in each link (at right angles to each other). The four equivalent sp(pdI2hybrids produce

+

T h e ~ p ( p dhybrids )~ are given by (s

+ pz + px + d x z ) / 2 , (s + pr

- px - d r z ) / 2 , (s - pz + py + d y z ) / 2 ,and (s - pz - py

- dyz)/2.

two hanana honds of hond order 0.75 in each link, which are ~ e r ~ e n d i c u lto a rthe % a hond in that link. It is instructive to k i p a r e the second lo&lization picture with the present one. In the preceding scheme the NBMO's have their charge entirely on the oxygens, are non-bonding, and are completely delocalized. By contrast in the present scheme, left-right localization has been partially accomplished but a t the expense of changing the NBMO's into half-bonding, a MO's with a corresponding weakening of the hanana honds. I t is therefore unjustified to conceive of COz as having two full a honds and two 2p lone pairs as is commonly inferred from the Lewis dnlrt,,,~

Finally, it is possihle toconstrt~ctsix equivalent sp.'d%yhrid GO'i from the six canonical GO'S which give rise toall the occupied delncnlrzrd MO's.'l'hese generaw three equivnlenr trigunal prismatically oriented banana hondi 01' hond order '.j in each link. This somewhat unusual \,irw uf ('0,is inrrresting in that the bonding in each link issimilnr tu the localized r~ictureof CO which nlso contains three eauivnl(nt hanana-bonds. Concluding Remarks The examples discussed here show some [IT tht, pedagogical potenrinl 111the generator irl~itiili~ppruach.A more cxtcnsi\,e discussion and further examoles o f t h e method are eiven in another article.' For an introductory exposition of the subject of localized orbitals, the reader is referred to Eneland. " ~.Salmon and R e u d e n b e ~ g . ~

-

~

Volume 54. Number 10. October 1977 / 595