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381 Gimarc Accounts of Chemical Research0.023 cm-1 to the previously described Dnss curves.When appended by this spin-orbit contritution, theHarrison-Liedtke37 curve predicts D = 0.745 cm-la t the computed equilibrium angle of 133", whilethat of Langhoff and Davidsonl4 predicts D = 0.807cm-1 a t this same angle. While both estimates are inreasonable agreement with the experimental num-ber,23 0.76 f 0.02 cm-1, it is somewhat disconcert-ing that the more complete theoretical study ofLanghoff and Davidson is outside of the experimen-tal range. Indeed, if we ask what HC H angle the the-oretical curves would predict, given the experimentalD, we see th at the Harrison-Liedtke curve is consis-tent with a range of angles from 145 to 135" whilethe Langhoff-Davidson calculation suggests an angleless than 125". However, given the complexity ofboth the experimental interpretation and theoreticalcalculations, mutual agreement, even at the rt5level, must be regarded as a satisfying accomplish-ment.ConcludingRemarks

While theory and experiment concur on many of

the character istics of CH2, a few loose ends remain.Most notable is the energy of the IB1 - AI 0-0transition. Experiment places this at 20 kcal/molwhile a larger separation, perhaps 33 kcal/mol, issuggested theoretically. Also, while an analysis of thespectrum arising from the 3Az - B1 trans ition fixesthe angle of the 3Bl sta te a t 136", it also demandsthat the 3A2 state be strongly bent with a bond angleof 125". Th e nat ure of this ~ A ztate has not beencharacterized theoretically.45Th at CHZ was predicted3 to be a bent triplet bya b initio calculations at least 10 years before it wasexperimentally substantiated17,19,20 speaks to boththe difficulty of doing the experiments and the reli-ability of current computational techniques andsuggests that similar calculations can play a signifi-cant, if not major, role in elucidating the electronicstructure of other reactive intermediates.46

(45) This problem is being studied in our laboratory b y M r . D av id Wer-(46) J . F. Harr ison and C. W. Eakers, J . A m e r Chem. Soc., 95, 3467ne t t e .

(1973).

Applications of Qualitative Molecular Orbital TheoryBenjamin M . Gimarc

D e p a r t m e n t of Chemis try , U n i v e r s i t y of S o u t h Carolina, C o l u m b i a , S o u t h C a r o l i n a 29208Received A p r i l 26, 1974

Much information of chemical interest can be ob-tained from molecular orbital (MO) theory withoutresort to any calculations at all. Considerations ofsymmetry properties, nodal surfaces, and atomic or-bital (A O ) overlaps in MO's have been used quitefruitfully in recent years by Woodward and Hoff-man nl to elucidate mechanisms of chemical reac-tions. In this Account I will use similar arguments topredict, understand, or at least rationalize theshapes, i . e . the gross geometrical features, of smallpolyatomic molecules in both ground and excitedstate s. In some cases these considerations lead natu-rally to discussions of other properties such as hydro-gen bonding, barriers to inversion and rotation, andhydrogen bridging.In most introductory chemistry textbooks, molecu-lar shapes are explained by an electrostatic model ofrepulsions among lone pairs and bonding pairs of va-lence electrons. This valence-shell electron-pair re-pulsion (VSEPR) model has been summarized atvarious stages in its development by Sidgwick and

Ben j am i n M Gimarc i s Assoc iate Professor and Head of the Depar t -ment of Chemis t ry at the Univers i ty of South Carol ina. He was born inAr izona, but grew up in Texas, pr imar i l y Dal las . After graduat ion f romRice Univers i ty w i th a B .A. degree in 1 9 5 6 , he served w i th the U.S. Navyfor 2 years . He received the P h.D. degree f rom Nor thwes tern Univers ity ,and then spent 2 years at Johns Hopk ins Univers i ty as postdoctoral fe l -low. Dr . Gimar c was on the facul ty at Georgia Ins t i tute of Technology b e-fore mov ing to South Carol ina. H is cur rent rese arch interest i s the devel -opmen t of qual i tati ve concepts of chem ical valence theo ry .

Powell,2 Gillespie and Nyholm,3 and Bartell.4 It hasrecently been extended by Wolfe5 to rationalize rota-tional conformations about single bonds. Schnuelleand Parr6 have discussed a related qualitative modelof molecular shapes based on ideas from crystal fieldtheory.Molecular orbital theory offers an alternative qual-itative model for explaining or rationalizing molecu-lar shapes. It began with the work of Mulliken7 inthe 1930's and '40's. It blossomed in the 1960's whenWalshg published a series of papers containing quali-tative or empirically deduced MO correlation di-agrams showing how orbital energies change withchanges in molecular shape. Recent years have seenan increase in interest in MO-based models for mo-lecular shapes. Gavin9 has discussed some of thequal itative aspects of extended Huckel M O calcula-

(1) R. Hoffmann and R. B . Woodward. Accounts Chem. R e s 1, 17(1968).(2) S V Sidgwick and H . M. Powell, Proc. Roy. Soc., Se r . A , 176, 153(1940).(3) R. J . Gillespie and R . S. Nyholm, Quart. Reo . Chem. Soc. , 11, 339(1957); R . J. Gillespie, J . Chem. E duc . . 17 , 18 (1970).(4) L . S. Bartel l , J . Chem. Ed u c . , 45,754 (1968).5 ) S Wolfe, Accounts Chem. Res . , 5 102 (1972).(6) G. W . Schnuelle and R. G. Parr , J Amer. C he m. .Yoc., 9 4 , 89747 ) R. S. Mulliken, Reti. M o d . P hy s . , 11, 204 (1942); Science, 157, 138 ) A . D . Wals h , J . Chem. Soc., 2260 (1953). and the papers i m m e d i -9) R . M . G av in . J . Chem. E d u c . , 46,41 3 (1969).

(1972 .(1967).ately following.


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Vol. 7, 1974 Qualitative M O Theory 385tions as they apply to molecular shapes. Pearsonlohas developed a symmetry rule based on the second-order Jahn-Teller effect and the symmetry of MOs.Other authors have sought to understand the successof Walshs rules from rigorous considerations of MOtheory and the Hellman-Feynman theorem.12Nakatsuji l3 has used the Hellman-Feynman theoremto develop an electrostatic force theory to explainchemical bonding, molecular shapes, and other prop-erties, The relationship between ab in i t io SCF M Ocalculations and Walshs rules has recently been re-viewed by Buenker and Peyerimhoff.14In this Account I will present the rules of qualita-tive MO theory and review some of the highlights oftheir application. I must acknowledge a debt to thepublished work of Walsh.8 The qualitative MOmodel is a practical method which can be easily usedto understand molecular properties. Because of itssimplicity and generality, this approach will soon oc-cupy an important place in chemical education. Itwill also provide a means for the kind of guided spec-ulation abo ut molecular properties tha t could lead tonew experiments or detailed quantum mechanicalcalculations.The basic assumptions , features, and rules of qual-itative M O theory are these.(i) Electrons in molecules are completely delocal-ized and move in molecular orbitals which extendover the entire molecular framework.(ii) For properties that can be explained by quali-tative M O theory, only th e valence electrons need beconsidered.(iii) Satisfactory molecular orbitals can be formedfrom linear combinations of atomic orbitals. This isthe well-known LCAO-MO approximation.(iv) The atoms which form the molecules of a par-ticular series or class contribute the same kinds ofvalence AOs from which molecular orbitals can beconstructed. Therefore the MOs for each series ortype of molecular framework must be qualitativelysimilar and individual molecules differ primarily inthe number of valence electrons occupying the com-mon MO system. A t least for small molecules thisrule holds quite well, and the results of experi-ment15J6 and ab in i t io SCF M O calculations17 sup-port it.(v) The total energy of the molecule is the sum ofthe orbital energies of the individual valence elec-trons, or, more accurately, changes in the total ener-gy parallel those of the orbital energy sum. Allen andcoworkers18 have shown t ha t t his assumption resultsfrom a fortuitous cancellation of energy terms.(vi) No explicit considerations of electron-electron

(10) R G Pearson, J Amer Chem S o c , 91, 1252 , 4947 (1969), J(11) L C Allen, Theor Chim Acta, 24,117 (1973)(12) C A Coulson and B M Deb, Int J Quantum Chem 5 411(13) H N a k a t s u j i , J Amer Chem Soc , 95,3 45, 354, 2084 (1973)(14) R J Buenker and S D Peyerimhoff, Chem Reu , 74,127 (1974)(15) G Herzberg, Molecular Spectra and S tructure 111 ElectronicSpectra a nd Electro nic Struct ure of Polyatomic M olecules, Van Nos-t rand, Pr inceton, NJ ,p 312( 16 ) D W Tur ne r , A D Baker , C Baker , and C R Brundle Molecu-lar Photoelectron Spectroscopy, Wiley-Intersclence, London, 1970(17) L C Snyder and H Basch, Molecular Wave Functions and Prop-

erties, Wiley-Interscience, New York, N Y , 1972(18) S D Peyerimhoff, R J Buenker , and L C Allen, J Chem Phys45,734 (1966 ), L C Allen and J D Russell, cbid , 16,1029 (1967)

Chem Phys , 52,2167 (19701, Chem Phys L e t t , 10.31 (1971)(1971)

\\J n U X

Linear BentFigure 1. Qualitative M O correlation diagram for linear and bentAH2 molecules. Changes in geometry which increase in-phaseoverlaps between AOs lower the M O energy.or nuclear-nuclear repulsions are included in thissimple model.(vii) Molecular orbitals must be either symmetricor antisymmetric with respect to the symmetry oper-ations of the molecule. These symmetry restrictionsseverely limit the number and kinds of AOs thatcombine in a particular MO. This makes the job offorming the MOs even easier, since most small mol-ecules have high symmetry. To use qualitative M Otheory, the only part of group theory necessary is aknowledge of symmetry classifications.(viii) From the properties of AOs available oncomponent atoms it is possible, at least for smallmolecules, to draw pictures of what the MOs mustbe like and to establish the approximate order ofenergies without calculations.(ix) Changes in molecular shape which increasethe in-phase overlap between two or more AOs in aMO tend to lower the energy of that MO. Converse-ly, changes in shape which decrease in-phase overlapor increase out-of-phase overlap among AOs in aMO tend to raise the energy of the MO. Th is can becalled the overlap rule.(x) No a priori assumptions about orbital hybrid-ization are needed.TheAH2 Series

With only three atoms, the molecules in the seriesAH2 can be only either linear or bent in shape.19Molecules with three or four valence electrons arelinear; those with one or two or five through eightelectrons are bent. Herzberg has recently reviewedthe experimental structural data for several of themembers of thi s series with five electrons or more.20Consider the four valence MOs of lowest energyfor a linear AH2 molecule.21 These c an be made fromthe s and th ree p valence AOs on the central ato m Aby combining them in phase with the Is orbitals onthe hydrogens. Figure 1 shows how thi s is done. Thelowest energy valence MO for linear geometry is 2cgi(19) Most of the s tructural data used in this review come from thesesources: L E . S u t t o n, E d , Chem SOC Spec P ub l, No. 11 (1958), No. 18

(1965); ref 15.(20) G. Herzberg, Science, 177,12 3 (1972)( 2 1) B . M . G i m a r c , J Amer Chem Soc , 93,593 (1971) .


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386 Gimarc Accounts of Chemical ResearchIt is mainly the s A 0 on A plus in-phase overlappingIs AO's from the hydrogens. The pz A 0 points di-rectly at the Is orbitals on the hydrogens and thesecan overlap in phase with pz to form the 10 orbitalwhich is of lower energy than pz alone. In the twol ruorbitals, the hydrogens are located on the nodalsurfaces of the pX and p3 orbitals and hence the hy-drogen Is AO's cannot enter either l ru or l r u v ,These two MO's are pure pl or p,, and they remaindegenerate in energy. Starting with six AO's on thethree separated atoms one can make six MO's forAH2. Only the four of lowest energy are included inFigure 1.Figure 1 correlates orbitals and energies for linearand bent AH2. On bending, the two hydrogen s or -bitals in 2 0 ~ ove closer together in 2al and there-fore into better overlap with each other. Thus the2al orb ita l of bent geometry will have lower energythan the related 2ag orbital of the linear shape. If themolecule contains only one or two valence electrons,these will find lower energy in bent geometry in 2al.The ions H32+, H31, and LiH2+ are indeedbent.22,23 The energy difference between linear 20,and bent 2al is not large, however. To a first approx-imation, these nodeless MO's of lowest energy are al-most spherical 24 When higher energy orbitals areoccupied, the energy changes for the nodeless orbit-als will be negligible by comparison.The energy of the l u u orbital increases on bendingbecause the hydrogen Is AO's are pulled out of over-lap with the lobes of the pz A 0 on the central atomand pushed toward each other across the nodal sur-face of the orbital lbz of ben t geometry. The lau-lb2M O controls the shapes of three- and four-elec tronAH2 molecules such as BeH2+, BeH2, and BHz+which are linear25 because the electrons are atlower energy in the linear l au orbital. The pz A 0 onthe central atom is not required to explain why H3 islinear; the energy of l au -lb2 still increases on bend-ing because the terminal hydrogen Is AO's approacheach other out of phase on opposite sides of thenodal surface.26When l rul : s bent. the hydrogens move on thenodal surface (the xz plane) of l b l ; no change inoverlap is possible and therefore IT and lbl havethe same A 0 composition and the same energy. ThelTu3-lbl M O system has no geometry preferenceand no direct influence on molecular shapes. As 1rUxis bent, the hydrogens move off the nodal surface ofpn (the yz plane), and in-phase overlapping of thehydrogen Is orbitals wi th the bot tom lobe of px low-ers the energy of t he o rbital 3al th us formed. Theshapes of molecules with five through eight electronsare bent because 3al falls rapidly in energy as thevalence angle HAH decreases. In BH2 and AlH2 the3al orbital contains only one electron. Since BH2and AlH2, are both bent it means that the energy ofthe singly occupied 3al orbital falls at least twice as

(22) H . Conr oy and B . L . Br uner , J . Chem.Phys.,47,921 (1967) .(23) R . D . Poshus ta, J . A. Haugen, and D. . Zetik, J . C h e m . Phys . , 31,(24) For accurate M O contour diagrams, see: T. H. Dunning, Jr., R.M.28 ) S . D. Peyer imhoff , R. J . Buenker . and L. C. Allen, J . Chem. Phys.,

( 26 ) B . M . Gimarc, J . Chem. Ph ys . , 53,1623 (1970) .

3343 (1969).P i tzer , and S Aung, J . Chem.P hy s . , 57,5044 (1972).45,734 (1966).

rapidly as tha t of the doubly occupied lb2 risesbelow it. That this might be so can be seen from thefollowing overlap argument. Th e overlap between a porbital and a hydrogen Is orbital is proportional tothe cosine of the angle 4 between the axis of the porbital and the A-H bond axis. The position of max-imum overlap has the hydrogen s A 0 restingsquarely on the p orbital axis 4 = 0 ) . With the hy-drogen tilted 30" away from the p orbital axis, thels,p overlap is still 87% of maximum and at 4 = 60"the overlap is 50% of maximum. At = 90 the 1sA 0is on the p orbital nodal surface and the overlap iszero. If th e size of the energy change is related to theamount of overlap change, then tilting a hydrogen afew degrees away from the p-orbital axis should pro-duce a relatively small energy increase. Moving thehydrogen a few degrees off the p-orbital nodal sur-face should lead to a significant energy lowering. Forexample, imagine decreasing the angle HAH from180 (linear) to 120" (bent). Each of two ls,p over-laps in lb2 decreases from 100% to 87% of maximum,producing a modest rise in energy, while each of twoIs,p, overlaps in 3al increase from 0 to 50% of maxi-mum for a substantial energy lowering or stabiliza-tion. Therefore, even a single electron in 3al canforce molecules such as BH2 and AlHz to be bent .In its ground state BH2 has a valence angle of131". The linear excited state of BH2 is known forwhich the electron configuration is (2al)2-(lb2)2(1b1)1. In this case an electron has been re-moved from the M O 3al that holds ground-stateBH2 bent; the excited electron occupies lbl whichhas no geometry preference and the excited moleculeis therefore linear because of the electron pair in 1 0 ~ -

Simply assigning electrons to the lowest availableM O energy levels for AH2 leads one to the conclusionth at for six-valence-electron molecules the (3a# sin-glet configuration should have lower energy than the(3a1)l(1b l) l triplet. Both experiment27 and ab initioSCF MO calculations28 show that this conclusion iswrong for CH2 and apparently for NH2+ as well.29The ground states of these species are, in fact, trip-lets, ( 3a l) l( lb l) l. The valence angles for theseground-state triplets are 136" (CH2) and 140-150"( S H z t ) , rather wide angles, comparable t o that forthe BH2 ground state and consistent with the notionth at 3al is singly occupied. Each of these species hasa singlet state at higher energy with 3al doubly oc-cupied, and hence a sharper valence angle. For the(3al)2 singlets, the angles are 105 (CH2) and 115-120 (NH2-). The singlets of still higher energy, the(3al)l( bl)l configuration, have wide valence anglessimilar to those of the g round-state triplets : 140-150"for CH2 and around 150 for NH2f. The groundst at e of SiH2 is a singlet a12 and has a valence angleof 97O.30 Higher in energy is the triplet configurational lbl l with th e wider angle of 124". Stil l higher is

lb2.

(27) E Wasserman, W A Yager and V J Kuck, Chem Ph3s L e t t 7,409 (1970), G Herzberg and J W C Johns , J Chem P h y s 54, 2276(1971)(28) J F Harr ison and L C Allen, J Amer Chem Soc 91, 807 (19691,S V O'Neil, H F Schaefer 111, a n d C F Bender J Chem P hj s 5 5 162(1971)(29) S Y Chu 4 K Q Siu , and E F H ayes , J Amer Chem Coc 94,2969 (1972)(30) B Wirsam, Chem Phqs Let t 14,21 4 (1972)


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Vol. 7, 1974 Qualitative M O Theory 387the allbll singlet with a comparable angle, 126.While the simple notion of sums of orbita l energiesfails to give the proper energy order of states of CH2and NHz+, the qualitative model does represent thecorrect trends in bond angles for those states.The ground-state electron configuration of theseven-valence electron molecule NH: is (3a1)2(1bl)land the valence angle of 103 is consistent with dou-ble occupancy of 3a l. Excitation to th e configuration(3 al )l (l b1 )~ oves one electron from the orbital tha tholds NH2 bent to an orbital with no geometry pref-erence. A single electron in 3al is enough to keep ex-cited NH2 bent, but at the wider valence angle of144. PH2 shows a similar increase of valence anglefrom the ground s tate (92) o excited stat e (123).AH2 molecules with eight-valence electrons, ofwhich water is an example, are bent due to doubleoccupancy of t he 3a1 orbital.The AH3 Serie s

Molecules of the general formula AH3 with eightvalence electrons (NH3, PH3, H30+, CH3-) are py-ramidal or nonplanar. Those with six electrons(BeH3-, BH3, CH3f) are planar and equilateral tri -angular. The CH3 radical and the NH3+ radicalion31 are apparently also planar. Figure 2 is a quali-tative M O correlation diagram for planar and pyra-midal shapes. Only the four valence MOs of lowestenergy appear in Figure 2. The lowest of these is thenodeless 2al-2a1 system. Because the Is,1s overlapsare greater in 2al (pyramidal) , this orbital is slightlymore stab le tha n 2al (planar). Therefore the two-electron ion H42+ should have lowest energy in thetetrahedral (pyramidal) shape32 since this arrange-ment would give the maximum in-phase overlapamong four hydrogen 1s orbitals.26 For moleculesand ions with more than two electrons, the 2al-2alM O system does not control molecular geometry.Doubly degenerate MOs occur for molecules withgeometries which have a symmetry axis of threefoldor greater. For planar and pyramidal AH3 moleculesthese degenerate orbitals are le-le. As shown inFigure 2, one of the two e MOs for planar geometrycontains a p orbi tal in 100% overlap with one hydro-gen 1s A 0 and 50% overlap with each of the twoother 1s orbitals. The other e MO has its p orbitalin about 87% overlap with each of two hydrogen 1sorbitals while the third hydrogen lies on the nodalplane of the MO. Thus even without symmetry argu-ments one can see that the two MOs designated eshould have nearly equal energies. Thi s is also a goodplace t o emphasize tha t t he simple overlap consider-ations are only qualitative for comparisons betweenMOs with different A 0 compositions. Suppose thevalence angle decreases from 120 (planar) to 110(pyramidal ). The ls ,p overlap decreases are small,but they are large enough to hold molecules withthree through six valence electrons planar. In theplanar-pyramidal transformation, the three hydro-gens in laz move off the nodal surface. As they doso their Is AOs can enter the resulting 3al M O andoverlap one of the lobes of the central atom px AO.Since this overlap starts from zero, it increases rap-(31) W . R. H ar s hbar ge r , J . Chem. Phys 56,177 (1972) .(32) H . Conroy and G . Malli , J . Chem. Phys., 50,5049 (1969)

Planar D,,r

Pyramidal C,

Figure 2 . Qualitative M O correlation diagram for an AH3 mole-cule in planar and pyramidal shapes.idly; hence the energy of 3al must fall sharply, muchfaster than the two le orbitals which rise below it.For eight-electron molecules such as NH3, the la2-3al orbital is doubly occupied and the molecules arepyramidal rather than planar. For the seven-electronmethyl radical, with only one electron in la2-3a1but two pairs in le-le, the total overlap increases in3al are comparable with the decreases in l e and thequalitative M O model is inconclusive. However, ex-tended Huckel calculations and the more rigorous a binitio SCF M O calculations agree that CH3 is pla-nar.Still other geometries are accessible to AH3 mole-cules. The transformation Y -shape (CzL)-equilateraltriangular (D3h)-T-shape CzL) s a geometry changeth at intersects the planar-pyramidal change.v Y 71

Ph1CS

Intersecting correlation diagrams are available whichpermit energy comparisons.21 These comparisons donot change the conclusions we have already reachedconcerning one and two or five, six, seven, and eightelectron molecules, but they do predict that three-and four-electron AH3 molecules should be Y-shaped. Valence bond calculations for the three-elec-tron radical ions Hq+ and LiH3f and for the four-electron ion HeH3+ offer evidence that these hypo-thetical ions should be Y-shaped.23Th e AH4 Series

The tetrahedral structure of methane can be ex-plained without assuming tetrahedral hybrid orbit-als. Figure 3 correlates valence MOs for tetrahedralTd ) nd square-planar (D4h) structures for AH4.The lowest energy or nodeless orbital al-alg does not


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388 Gimarc Accounts of Chemical Research

Tetrahedra SqGt-e P l a n a r

Figure 3 Molecular orbital pictures for AH4 in tetrahedral andsquare-pl anar geometry.

/ eFigure 4 . Comparison of higher energy valence levels of AH3 inpyramidal and planar shapes and for AH4 in tetrahedral andsquare-planar shapes. The relatively small gap between 3al andlaz permits inversion of eight-electron AH3 molecules. Inversionof eight-electron AH4 molecules is much harder because of thelarge gap between f and a,.influence the shape of the molecules to be consideredhere. Tetrahedral geometry requires a set a triply de-generate MOs, the f orbitals of Figure 3. While theparticular set of A 0 composition pictures in Figure 3is not the one that shows the degeneracy most clear-ly, that set does demonstrate most conveniently theorbital energy differences between tetrahedral andsquare-planar shapes. These energy changes can bereadily deduced from the overlap rule. In the tetra-hedral orbitals f, and f, the hydrogen 1s AOs arealready in good overlap with the lobes of the adja-cent p orbital. In square-planar shape the overlapsare even bette r though the increases are small, lead-ing to a small lowering of energy. For f,-aZu the over-lap decrease is large, all overlaps going to zero foraZu, producing a correspondingly large energy rise,For BH4-, CH4, and NH4+, each with eight valenceelectrons, the high energy of aZu prohibi ts square-planar geometry and makes these species tetrahe-

Linear BentFigure j Qualitative correlation diagram for linear and bentHAB molecules. Dashed arrows indicate intended correlationswhich are blocked by the noncrossing rule.dral. Molecules with six electrons, BeH4 for instance,should be square planar. Th at the pair of square-pla -nar MOs eux and euv must be doubly degenerate iseasy to see from the diagrams in Figure 3.Inversion

Qualitative M O pictures can show why ammoniainverts but methane does not. The most straightfor-ward comparison for this purpose would be betweenNH3 and N H ~ T . igure 4 contains energy levels inpyramidal C3L)and planar-triangular D 3 h ) shapesfor AH3 compared with those for AH4 in tetrahedralT d ) nd square-planar 0 ~ )hapes. Figures 2 and3 form the basis for Figure 4. The aZu level of planarAH4 is prohibitively high in energy relative to the forbitals of tetrahedral geometry, while the 3al orbit-al of pyramidal AH3 is much closer to laz of planarshape, making the inversion of AH3 much cheaperenergetically.The H A B Series

Molecules of the series HAB are either linear orbent. The occupied valence M O s for linear HARmolecules can be derived from those for the diatomicmolecule AB by combining a hydrogen Is A 0 in-phase with the o MOs of AB.33 The nodal planes ofthe 7r orbitals prevent any 1s contributions to thoseMOs . Figure 5 shows schematic MO pictures corre-lating orbitals for linear and bent HAB. Orbitals forbent HAB must be either symmetric (a) or antisym-metric (a) with respect to reflection in the symme-try plane, Orbital energy changes are easy to inter-pret by the overlap rule with the exceptions of 50-(33) B . M. Cimarc,J.Amer. Chem. Soc. , 93 ,816 (1971) .


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Vol. 7, 1974 Qualitative M O Theory 389

F,2 FHF

QS F-H-Fc6Figure 6. Relative energy levels and M O pictures for F& and FHF -. Energy levels shown were obtained by extended Hdckel calculations.5a and lax-6a. Both 5a and la, must yield orbitalsof a symmetry in b ent geometry. Intended orbitalcorrelations are shown by dashed arrows in Figure 5 .The crossing of two a orbitals is forbidden by thenoncrossing rule. Instead, these two orbitals mix anddiverge a s shown by the solid arrows for 5a-5a andla,-6a.With ten valence electrons, HCN is linear, thehighest occupied M O being la,-6a. The first excit-ed state of HCN is bent. Excitation transfers anelectron from 1ax-6a, the orbital that holds themolecule linear, to 2ax-7a, an orbital th at goes tosignificantly lower energy in the bent shape. Thus,ten-electron molecules should have linear groundstates and bent excited states. The 11-electron radi-cal HCO should be bent in its ground state with oneelectron in 7a. Excita tion removes this electron from7a, the orbital that makes HCO bent, and adds it to2ay-2a, an orbital having no geometry preference.Lower occupied orbitals, the same ones that makethe HCN ground state linear, give excited HCO lin-ear geometry. Molecules with 12 through 14 electrons(HNO, HNF, HOC1) should be bent in both groundand excited states because at least one electron willbe in 7a.Hydrogen Bonding

Unlike the o ther HAB species, t he 16 electron ionsFHF-, ClHC1-, etc., are linear and symmetric withthe hydrogen on the axis and midway between thetwo halogens. One can think of an ordinary HABmolecule (14 electrons or less) as an AB- ion with aproton attached to one end. In fact, HCN, HNO,and HOC1 are all weak acids, and the correspondingAB- ions are known. One would not expect the ionF22- (16 electrons, isoelectronic with Ne2) to bebound because equal numbers of bonding and anti-bonding orbitals are occupied. For the FHF- system,the insertion of a hydrogen with its Is A 0 between

the two halogens stabilizes the antibonding oU andrg orbitals (particularly the highest occupied 3auwhich has pz AOs pointing directly at each otherout of phase) by separating the antibonding or out-of-phase fragments, thereby lowering their out-of-phase overlaps and making them nonbonding. At thesame time, the central hydrogen Is orbital providesin-phase overlaps to maintain the bonding characterof the ug MOs. Figure 6 contains schematic di-agrams of these orbitals. Only the bonding r rbit-als are raised in energy (to become nonbonding) bythe increased separation of the two fluorine atoms.In the rUMOs the hydrogen is on the nodal surface,and its Is A 0 cannot participate in any overlap sta -bilization. The result is a change from four bondingand four antibonding MOs for X22- to two bondingand six nonbonding MOs for XHX- .Pimentel and Spratley34 predicted that FHeFshould be stable, arguing from analogy with the iso-electronic ion FHF- an d FXeF. On the basis of a b in-itio valence bond and SCF MO calculations Allenand coworkers35 and Noble and Kortzeborn36 con-clude that, while FHF- is stable, FHeF should beunstable. Those conclusions can be rationalized bymeans of the qualitat ive MO model. Hydrogen bond-ing results from antibond ing or out-of-phase pieces ofMOs being separated far enough to make all anti-bonding orbitals nonbonding while at the same timesome of the bonding orbitals retain their bondingcharacter through new overlaps provided by the in-serted Is AO. Now the larger charge of the He nucle-us makes a He Is A 0 considerably smaller than ahydrogen 1s. In order to maintain effective in-phaseoverlaps in the two cg MOs, a He Is orbital cannot

(34) G. C. P imen te l and R. D. Spratley. J Amer Chem Soc 85. 8261963) ..(35) L. C . Allen, R . M. Er dah l , and J . L. Whi t t en , J Amer. Chem. Soc. ,(36) P. N. Noble and R. N. Kortzehorn, J. Chem. Phys. 52,5375 (1970).87.3769 ( 1965).


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390 Gimarc Accounts of Chemical Research

t rans l i n e a r c i sigure 7 . Qualitatike correlation diagram for HAAH In h e a rand nonlinear CISand trans) shapes

separate the F atoms far enough to stabilize signifi-cantly the antibonding MOs. Therefore net bondingin FHeF is not likely.C Z HZ , zHz,and HzOz

The A2Hz series is a simple extension of the HABseries.37 Figure 7 displays the valence MOs for lin-ear and nonlinear, cis and tra ns, geometries of a typ-ical A2H2 molecule. Orbital energy changes can beinterpreted by the overlap rule. A case of noncrossingof orbitals of the same symmetry for cis geometry ex-actly parallels the HAB model. With ten electrons,acetylene (CzH2) is linear. Diimide (NzHz;1 2 elec-trons) has planar cis and trans isomers because ofthe lower energies of bz (cis) and a, (tr ans ) thatstem from the 7 r gX M O of linear geometry. Hydrogenperoxide (H202; 14 electrons) should also be nonlin-ear. To see why H202 is actually nonplanar, orgauche, look at Figure 8 which contains pictures ofthe two highest occupied MOs for HzOz and showshow their energies change as the molecular frame-work is twisted from planar, trans, through nonpla-nar, gauche, to planar, cis. For the purposes of Fig-ure 8, this twisting takes place in such a way thatthe molecular twofold symmetry axis remains per-pendicular to the page, passing through the mid-point of the A-A bond. Also the p AOs are lockedparallel to the Cartesian axes and are not twisted orrotated. Only the hydrogens move. In the orbitalb,-b-bz, the hydrogens move away from the nodalsurface in be (tr ans ) and into positions where their1s orbitals can overlap well with the p orbital lobesin bz (c is). This produces a lowering of energy fromb, to b to bz. At the same time, the hydrogens ina,-a-az move out of good overlap in a, (t ra ns ) andtowards the nodal surface of a2 (cis) for an energy in -(37) B M G i m a r c , J Amer Chem Soc 9 2 , 2 6 6 (1970)

t r a n s gauche i sp l a n a r n o n p l an a r p l a n a r

igure 8. Correlations for the highest occupied pair of MOs inHsOz on rotation from trans to cis through skewed or nonplanargeometry.crease from a, to a t o a2. Now the energy of b,-bshould drop rapidly because the overlap increasesfrom zero while the energy of a,-a should increasemore slowly because the overlap decreases from nearmaximum. Therefore the energy at which orbitals aand b cross in gauche geometry is lower tha n the av-erage of ei ther b, an d aRr or tran s or a2 and b2 forcis. (It turns ou t th at a similar crossing of a and bMOs for hydrazine (NzH4) makes that moleculegauche als0.)38 While this naive model gives an ap-pealing rationale for the gauche shape of H202. theactual situation is more complicated. Extensive mix-ing among orbitals of a symmetry an d among orbit-als of b symmetry for the gauche shape makes thecrossing of these a and b orbitals occur at higher en-ergy. Gauche geometry of HzOz arises from a deli-cate balance among all the occupied a and b valenceorbitals.Orbitals for Ethylene

Qualitative MOs for the planar AzH4 moleculecan be constructed by taking in-phase and out-of-phase combinations of the MOs for separated, bentAH2 fragments.38 Figure 9 correlates relative energylevels and MOs for planar A2H4 and two isolatedAH2 molecules. Compared with 3al (AHz) the ener-gy of 3a, (AzH4) should be quite low and that of3b3, (not shown, off the top of Figure 9) should bequite high because of the bonding and antibondingp-cr type overlaps th at result as the two AH2 frag-ments are joined. The splittings that occur as a re-sult of the two lbz and the two lbl combinations aremuch smaller because of their 7r-type overlap. Onemight be concerned that the A-A antibonding orbit-al lb le lies below the bonding orbital l b lu . The fourls,p overlaps available to Ibl, but not to lbl, givelbl, a lower energy. For ethylene (12 electrons), thebonding 7r-orbital lbl, is the highest occupied M O .The lowest unoccupied M O is the antibonding 7r or-bital lbzR.Now consider twisting one AH2 group relative tothe other about the A-A bond. Figure 10 is the cor-relation diagram for this process. For convenience,the two valence MOs of lowest energy, 2ae-2al and

(38) J R Durig, B M Gimarc, and J D Odom in Vibrat ional Spectraand Structures , Vol 11, J R Durig, Ed Marcel Dekker , Yew York, NY ,1973


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Vol. 7, 1974 Qualita tive M O Theory 39 12 AH,

Bent c,, A2H4Planar D,,Plan ar Twirtod Staggored

DZ h D2 D2d

1b z g

Figure 9 . Formation of MO's for planar AzH4 by taking in-phaseand out-of-phase combinations of the MO's of bent AH2 frag-ments.2b3u-2bl, are not shown in Figure 10. These orbitals,as well as 3ae-3al, are axially symmetric and thei renergies change very little on rotation. Symmetry re-quires tha t each bl,b2 pair of Dah or D2 geometry be-comes degenerate in D2d . Both lbl, and lbl, of D2hbecome bl in intermediate D2 geometry. Intendedcorrelations, indicated by dashed arrows in Figure10, are lbl,-2e and 1blu-le . Because lb l, lies abovelbl, in energy, the intended correlations are blockedby the noncrossing rule. It is the rising energy of theM O lblu-bl-2e th at makes ethylene planar . OtherM O correlation diagrams are available which sum-marize the various shapes of the remaining membersof the AzH4 series.38Ethane and Diborane

Figure 11 correlates the orbital energies of an A2Hsmolecule of the staggered ethane conformation withthose for a bridged or diborane shape.39 For simplici-ty, the pair of valence MO's of lowest energy has notbeen included in the diagram. Those two orbitalshave no significance for the ethane-diborane struc-tural transformation. In fac t, only two MO's undergolarge energy changes. Look at 3alr-3a,. On rockingfrom the staggered form to the bridged shape, fourhydrogens in 3ale move to terminal positions in 3a,and two hydrogens move to bridging positions. Allsix move to positions of increased overlap. Each ofthe bridging hydrogens overlaps two p orbitals. This(39) B. M. Gimarc, J . Amer . C h e m . Soc., 96, 1417 (1973); R. Hoffmannand J. E. Williams, Jr.,Helu. Chim .A c t a , 55 67 (1972).

Figure 10. Qualitative correlation diagram for planar andstaggered ethane. The degenerate orbital pair l e has about thesame energy as 3al (staggered).

Staggorod Dad c2 Bridged D,,

/'

Figure 11. Qualitative correlation diagram for an AzHs moleculein staggered Dad and bridged D z ~hapes.


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392 Gimarc Accounts of Chemical Researchconsiderable increase in overlap makes the bridgedform 3a, much lower in energy than the staggeredform 3a1,. Now consider the le,-lb2, orbital . In thestaggered form (le,) all six hydrogens are in goodoverlap with the p orbitals of th e principal atoms . Inthe bridged form ( lb ae ) all six hydrogens lie on nodalsurfaces of the MO and the overlap is zero. There-fore the energy of lbzn is considerably higher thanth at of le,. Thi s increase is even larger than the de-crease of 3a, rela tive to 3al,. While individual over-laps in le,-lb3, do change, they change in such away as to cancel each other . Overlap changes in le,-lb zu and lee- lbsu are negligible compared to thosedescribed already.Diborane, with 12 valence electrons, has lb3, as itshighest occupied M O . The favorable overlap ar-rangement of the bridging hydrogens in 3a, is re-sponsible for the bridged s truc ture of diborane. Forethane (14 electrons) the steeply rising energy of theoccupied le,-lbz, orbi tal holds eth ane in staggeredgeometry. In M O theory, bridge bonds arise in a nat-ural way. For the valence bond model the basic rulesmust be amended and additional concepts such aselectron deficiency must be introduced to account forhydrogen bridging.Conclusion

Are these qualitative speculations and interpreta-tions verifiable? The correlation diagrams presentedhere compare very favorably with those obtainedfrom extended Huckel M O calculations21 ,33,37-39and, where available, ab initio SCF M O calcula-tions.l1340 The relation between the qualitati ve M Omodel a nd semiempirical M O methods is qui te close.The overlap rule en ters the extended Huckel methodthrough the overlap proportionality chosen to repre-sent the off-diagonal elements of the Hamiltonianmatrix in the secular equation.41 Thus, the qualita-tive arguments can be checked by comparable quan-titative methods. Unfortunately this means that ifthe calculations give the wrong answer the qualita-tive model will also lead to the wrong conclusions.An example is the failure of both qualitative M Otheory and the extended Huckel calculations to pre-dict the correct geometry for hydroxylamine(H2NOH), presumably because of t he failure of ei -ther method to include changes in internuclear re-pulsions (assumption vi) 33 However, it should stillbe considered an advance in valence theory t o have a

(40) D. C . P an and L. C . Allen, J . Chem. Phys., 46, 1797 (1967); R. J.Buenker , ibid. 48, 1368 (1969); W . H . Fink, D . C . P a n , a n d L. C. Allen,i b id . , 47, 896 (1967): R . J . Buenker , S. D. Peyerimhoff, L. C. Allen. an d J .L. Whi t t en , ibid. 5,2835 (1966).(41) R . Hoffmann, J . Chem.Phys. , 39,1397 (1963).

qualitative model that corresponds so closely to easi-ly performed calculations.It. is hard to suppress the urge to refine the quali-tative pictures. Streitwieser and Owens42 have com-piled a book of computer-drawn M O contour di-agrams for small molecules. Surely, those diagramspresent a more accurate picture of the MOs than dothe qualitative A 0 composition diagrams tha t ac-company this Account but energy changes arisingfrom changes in molecular shape are easier to inter-pret with th e qualitative A 0 composition diagrams.What about qualitative MOs for larger molecules?The same overlap and symmetry arguments shouldapply, but unfortunately problems rapidly becomemore complicated because of the larger number ofMOs involved and the fact that these orbitals liemuch closer together in energy, and the extent ofmixing among orbitals of the same symmetry istherefore likely to be greater. This M O mixing to-gether with energy differences due to different atom-ic electronegativities may cause a reordering of close-lying M O energies (failure of rule iv). But even thispresents no difficulty as long as highest occupied andlowest unoccupied orbitals are not reversed. Severalapproaches for extension to larger systems may beuseful. First is the formation of MOs of large mole-cules from those of small molecules as we have donefor ASH^ and A2Hs. Another possibility is that onemight hope to resolve only those MOs of chemicalinterest from the traditional valence bond structuresmuch as Woodward and Hoffmann have done intheir studies of mechanisms of concerted organicreactions.1 Finally, it is always possible to use thequalitative M O model t o in terpret t he results of M Ocalculations.The familiar, remarkably simple, and often power-ful qualitative concepts of M O theory lead to somevery appealing interpretations of chemical proper-ties. The model is applicable to molecular excitedstates as well as ground states and it can account ina very natural way for such phenomena as hydrogenbridging in diborane and the skewed geometry of hy-drogen peroxide, problems not so easily explained byvalence bond theory or the VSEPR model. Finally,the qualitative M O explanations can easily bechecked by performing M O calculations.

I a m p l ea s e d t o a c k n o w l e d g e t h e g r a d u at e s t u d e n t s a n d f a c u l t yof t h e C h e m i s t r y D e p a r t m e n t a t t h e C n z ue r si t y of S o u t h C a r ol i n aw h o s e u n r e a s o n a b l e d e m a n d s f o r e x p l a n a t i o n s of c h e m i c a l f a c t sf or ce d m e t o t h i n k a b o u t M O theorx in a q u a l i t a t i v e u a y Thisuork has bee n suppor ted bg a gran t f ro m th e N a t iona l Sc ienLeF o u n d a t i o n

(42 ) A Streitwieser J r and P H Owens , Orbital and Electron Dens it y Diagrams, Macm il lan, Riverside, U J , 1973

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