1 HÜCKEL MOLECULAR ORBITAL THEORY In general, the vast maorty polyatomc molecules can be thought of as consstng of a collecton of two electron bonds between pars of atoms. So the qualtatve pcture of σ and π bondng and antbondng orbtals that we developed for a datomc lke CO can be carred over gve a qualtatve startng pont for descrbng the C=O bond n acetone, for example. One place where ths qualtatve pcture s extremely useful s n dealng wth conugated systems that s, molecules that contan a seres of alternatng double/sngle bonds n ther Lews structure lke 1,3,5 hexatrene: Now, you may have been taught n prevous courses that because there are other resonance structures you can draw for ths molecule, such as: that t s better to thnk of the molecule as havng a seres of bonds of order 1 ½ rather than 2/1/2/1/ MO theory actually predcts ths behavor, and ths predcton s one of the great successes of MO theory as a descrptor of chemstry. In ths lecture, we show how even a very smple MO approxmaton descrbes conugated systems. Conugated molecules of tend to be planar, so that we can place all the atoms n the x y plane. Thus, the molecule wll have reflecton symmetry about the axs: Now, for datomcs, we had reflecton symmetry about x and y and ths gave rse to π x and π y orbtals that were odd wth respect to reflecton and σ orbtals that were even. In the same way, for planar conugated systems the orbtals wll separate nto σ orbtals that are even wth respect to reflecton
2 and π orbtals that are odd wth respect to reflecton about. These π orbtals wll be lnear combnatons of the p orbtals on each carbon atom: In tryng to understand the chemstry of these compounds, t makes sense to focus our attenton on these π orbtals and gnore the σ orbtals. The π orbtals turn out to be the hghest occuped orbtals, wth the σ orbtals beng more strongly bound. Thus, the formng and breakng of bonds as mpled by our resonance structures wll be easer f we talk about makng and breakng π bonds rather than σ. Thus, at a basc level, we can gnore the exstence of the σ orbtals and deal only wth the π orbtals n a qualtatve MO theory of conugated systems. Ths s the basc approxmaton of Hückel theory, whch can be outlned n the standard 5 steps of MO theory: 1) Defne a bass of atomc orbtals. Here, snce we are only nterested n the π orbtals, we wll be able to wrte out MOs as lnear combnatons of the p orbtals. If we assume there are N carbon atoms, each contrbutes a p orbtal and we can wrte the th MOs as: N π = c p =1 2) Compute the relevant matrx representatons. Hückel makes some radcal approxmatons at ths step that make the algebra much smpler wthout changng the qualtatve answer. We have to compute two matrces, H and S whch wll nvolve ntegrals between p orbtals on dfferent carbon atoms: H = p Hˆ p d τ S = p p d τ The frst approxmaton we make s that the p orbtals are orthonormal. Ths means that: 1 = S = 0
3 Equvalently, ths means S s the dentty matrx, whch reduces our generaled egenvalue problem to a normal egenvalue problem H c α = E α S c H c = E c The second approxmaton we make s to assume that any Hamltonan ntegrals vansh f they nvolve atoms, that are not nearest neghbors. Ths makes some sense, because when the p orbtals are far apart they wll have very lttle spatal overlap, leadng to an ntegrand that s nearly ero everywhere. We note also that the dagonal (=) terms must all be the same because they nvolve the average energy of an electron n a carbon p orbtal: H = p Hˆ p dτ α Because t descrbes the energy of an electron on a sngle carbon, α s often called the on ste energy. Meanwhle, for any two nearest neghbors, the matrx element wll also be assumed to be constant: H = p Ĥ p dτ β, negbors Ths last approxmaton s good as long as the C C bond lengths n the molecule are all nearly equal. If there s sgnfcant bond length alternaton (e.g. sngle/double/sngle ) then ths approxmaton can be relaxed to allow β to depend on the C C bond dstance. As we wll see, β allows us to descrbe the electron delocalaton that comes from multple resonance structures and hence t s often called a resonance ntegral. There s some debate about what the rght values for the α, β parameters are, but one good choce s α= 11.2 ev and β=.7 ev. 3) Solve the generaled egenvalue problem. Here, we almost always need to use a computer. But because the matrces are so smple, we can usually fnd the egenvalues and egenvectors very quckly. 4) Occupy the orbtals accordng to a stck dagram. At ths stage, we note that from our N p orbtals we wll obtan N π orbtals. Further, each carbon atom has one free valence electron to contrbute, for a total of N electrons that wll need to be accounted for (assumng the molecule s neutral). Accountng for spn, then, there wll be N/2 occuped molecular orbtals and N/2 unoccuped ones. For the ground state, we of course occupy the lowest energy orbtals. 5) Compute the energy. Beng a very approxmate form of MO theory, Hückel uses the non nteractng electron energy expresson:
4 E tot = E =1 where E are the MO egenvalues determned n the thrd step. N To llustrate how we apply Hückel n practce, let s work out the energy of benene as an example. 6 1 2 1) Each of the MOs s a lnear combnaton of 6 p orbtals c 5 4 3 1 c 2 6 c ψ = c p c = 3 =1 c 4 c 5 c 6 2) It s relatvely easy to work out the Hamltonan. It s a 6 by 6 matrx. The frst rule mples that every dagonal element s α: α α α H = α α α The only other non ero terms wll be between neghbors: 1 2, 2 3, 3 4, 4 5, 5 6 and 6 1. All these elements are equal to β: α β β β α β β α β H = β α β β α β β β α All the rest of the elements nvolve non nearest neghbors and so are ero:
5 α β 0 0 0 β β α β 0 0 0 0 β α β 0 0 H = 0 0 β α β 0 0 0 0 β α β β 0 0 0 β α 3) Fndng the egenvalues of H s easy wth a computer. We fnd 4 dstnct energes: E 6 =α 2β E 4 =E 5 =α β E 2 =E 3 =α+β E 1 =α+2β The lowest and hghest energes are non degenerate. The second/thrd and fourth/ffth energes are degenerate wth one another. Wth a lttle more work we can get the egenvectors. They are: +1 +1 +1 +1 +1 +1 1 2 0 0 +2 +1 1 +1 1 +1 1 1 1 1 1 +1 1 +1 c 6 = c 5 = c 4 = c 3 = c 2 = c 1 = 6 1 12 +1 4 +1 4 1 12 1 6 +1 +1 2 0 0 2 +1 1 +1 1 +1 1 +1 The pctures at the bottom llustrate the MOs by dentng postve (negatve) lobes by crcles whose se corresponds to the weght of that partcular p orbtal n the MO. The resultng phase pattern s very remnscent of a partcle on a rng, where we saw that the ground state had no nodes, the frst and second excted states were degenerate (sne and cosne) and had one node, the thrd and fourth were degenerate wth two nodes. The one
6 dfference s that, n benene the ffth excted state s the only one wth three nodes, and t s non degenerate. 4) There are 6 π electrons n benene, so we doubly occupy the frst 3 MOs: E 6 =α 2β E 4 =E 5 =α β E 2 =E 3 =α+β E 1 =α+2β 5) The Hückel energy of benene s then: E = 2E + 2E + 2E = 6α + 8β 1 2 3 Now, we get to the nterestng part. What does ths tell us about the bondng n benene? Well, frst we note that benene s somewhat more stable than a typcal system wth three double bonds would be. If we do Hückel theory for ethylene, we fnd that a sngle ethylene double bond has an energy E C=C = 2α + 2β Thus, f benene smply had three double bonds, we would expect t to have a total energy of E = 3E C=C = 6α + 6β whch s off by 2β. We recall that β s negatve, so that the π electrons n benene are more stable than a collecton of three double bonds. We call ths aromatc stablaton, and Hückel theory predcts a smlar stablaton of other cyclc conugated systems wth 4N+2 electrons. Ths energetc stablaton explans n part why benene s so unreactve as compared to other unsaturated hydrocarbons. We can go one step further n our analyss and look at the bond order. In Hückel theory the bond order can be defned as: occ O c c =1 Ths defnton ncorporates the dea that, f molecular orbtal has a bond between the th and th carbons, then the coeffcents of the MO on those carbons should both have the same sgn (e.g. we have p + p ). If the orbtal
7 s antbondng between and, the coeffcents should have opposte sgns(e.g. we have p p ). The summand above reflects ths because c c > 0 f c, c have same sgn c c < 0 f c, c have opposte sgn Thus the formula gves a postve contrbuton for bondng orbtals and a negatve contrbuton for antbondng. The summaton over the occuped orbtals ust sums up the bondng or antbondng contrbutons from all the occuped MOs for the partcular par of carbons to get the total bond order. Note that, n ths summaton, a doubly occuped orbtal wll appear twce. Applyng ths formula to the 1 2 bond n benene, we fnd that: O 2c =1 c =1 + 2c =2 c =2 + 2c =3 c =3 12 1 2 1 2 1 2 +1 +1 +1 +2 +1 0 = 2 + 2 + 2 6 6 12 12 4 4 1 2 2 = 2 + 2 = 6 12 3 Thus, the C 1 and C 2 formally appear to share 2/3 of a π bond [Recall that we are omttng the σ orbtals, so the total bond order would be 1 2/3 ncludng the σ bonds]. We can repeat the same procedure for each C C bond n benene and we wll fnd the same result: there are 6 equvalent π bonds, each of order 2/3. Ths gves us great confdence n drawng the Lews structure we all learned n freshman chemstry: You mght have expected ths to gve a bond order of 1/2 for each C C π bond rather than 2/3. The extra 1/6 of a bond per carbon comes drectly from the aromatc stablaton: because the molecule s more stable than three solated π bonds by 2β, ths effectvely adds another π bond to the system, whch gets dstrbuted equally among all sx carbons, resultng n an ncreased bond order. Ths effect can be confrmed expermentally, as benene has slghtly shorter C C bonds than non aromatc conugated systems, ndcatng a hgher bond order between the carbons. Just as we can use smple MO theory to descrbe resonance structures and aromatc stablaton, we can also use t to descrbe crystal feld and lgand feld states n transton metal compounds and the sp, sp 2 and sp 3 hybrd
8 orbtals that arse n drectonal bondng. These results not only mean MO theory s a useful tool n practce these dscoveres have led to MO theory becomng part of the way chemsts thnk about molecules.