LCAO APPROXIMATIONS OF ORGANIC Pi MO SYSTEMS The allyl system (cation, anion or radical).

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1 Principles f Organic Chemistry lecture 5, page LCAO APPROIMATIONS OF ORGANIC Pi MO SYSTEMS The allyl system (catin, anin r radical).. Draw mlecule and set up determinant C C 2 = 0 C = ( 2 -) = 0 = 3-2 = ( 2-2); = 0, = SQRT(2), = SQRT(2) These are the energies f the rbitals in terms f beta: bnding, SQRT(2)β; nnbnding, 0; antibnding, SQRT(2)β = discuss the catin anin and radical with regard t the slutins C C C pi MOs f Allyl anin C C C pi MOs f Allyl radical C C C pi MOs f Allyl catin The bnding rbital cntributes 2 SQRT(2)β t the energy f the mlecule. The nn-bnding rbital cntributes nthing t the energy f the mlecule. It shuld be relatively easy t put electrns in and take electrns ut f the allyl system. In general we bserve aninic, radical and catin chemistries fr allyl. This is similar t the simplest nn-bnding rbital f atmic hydrgen. This why the chemistry f is (+), neutral and ( ). Prtn, hydrgen atm (radical) dt, and hydride. Des this mean that the allyl anin and catin are really isenergetic? N, there are e/e repulsin terms that are nt cnsidered in the ückel frmulatin. There are ther effects that ückel des nt include. ückel is apprximating what the rbital spacing is. When we put electrns int the system the ückel levels dn t change, but the verall energy f the mlecule certainly des.

2 Principles f Organic Chemistry lecture 5, page 2 ückel LCAO is nly a first apprximatin. Yu will ntice that all the atmic rbitals that make up the MOs are nt the same size. This is the stuff f C nm, the weighting cnstants in the MOs wave functin, r the AO cefficients. These AO cefficients can be btained by expansin f minrs f the secular determinant int cfactrs. Fr example: 2 3 A = 2 - likewise A 2 = A 3 = A = 0 We can determine the relative ratis f the cefficients. c /c 3 = A / A 3 = 2 - c 2 /c 3 = A 2 / A 3 = as befre, Lets arbitrary set c 3 = FROM A Fr ψ, = SQRT(2), s c = Fr ψ 2, = 0, s c 2 = - Fr ψ 3, = SQRT(2), s c 3 = FROM A 2 Fr ψ, = SQRT(2), s c 2 = SQRT(2) Fr ψ 2, = 0, s c 22 = 0 Fr ψ 3, = SQRT(2), s c 23 = -SQRT(2) FROM A 3 Fr ψ, = SQRT(2), s c 3 = Fr ψ 2, = 0, s c 23 = Fr ψ 3, = SQRT(2), s c 33 = A ( 2 -) A 2 (-) A 3 () SQRT(2) SQRT(2) 0-0

3 Principles f Organic Chemistry lecture 5, page 3 SQRT(2) SQRT(2) Nw we can write ut the MOs ψ = χ + SQRT(2)χ 2 + χ 3 ψ 2 = -χ 2 + χ 23 ψ 3 = χ 3 SQRT(2)χ 32 + χ 33 These are the cefficients; Nrmalize and yu are dne. T d this we need t find the nrmalizatin factr t apply t the MO abve. fr ψ and ψ 3 fr ψ 2 SQRT( 2 + SQRT(2) ) = 2 SQRT( ) = SQRT(2) Thus ½ and / SQRT(2) are the nrmalizatin factrs fr ψ, ψ 3 ; and ψ 2 respectively. ψ = ½χ + /(SQRT(2)χ 2 + ½χ 3 ψ 2 = -/(SQRT(2)χ 2 + /(SQRT(2)χ 2 ψ 3 = ½χ 3 /(SQRT(2)χ 32 + ½χ 33 Bnd rder and Atmistic Electrn Density can be apprximated by the LCAO methd. The cncept f bnd rder. R 3 C CR 3 has C/C bnd rder f ne. R 3 C=CR 3 has C/C bnd rder f tw. R 3 C=O has C/C bnd rder f tw. R 3 C=O (+) has C/C bnd rder < tw. O O O O O w might we try t gather experimental evidence fr the abve shift in bnd rder? The resnance (secnd frm the right) is mre imprtant when the O atm is prtnated. Likewise ne might wnder abut the CC bnd rder in allyl catin. Bnd rder: P Nij = Sum(Nc i c j ) N is the number f electrns in the MO c i c j is the prduct f the cefficients fr the bund atms.

4 Principles f Organic Chemistry lecture 5, page 4 We can get these parameters frm calculatins. The better the calculatins, the mre agreement there is with experimental results. Why shuld this calculatin give the bnd rder? Think abut interactin between AOs in the mlecule. It is a zer sum game due t nrmalizatin. If an rbital grws anther has t shrink elsewhere. Pi Bnd Order in ALLYL The extent f interactin (bnding) between the p rbital abve is a functin f hw much vlume is gained fr the electrn ccupying bth versus nly ne rbital. The interactin (bnding) scales linearly with the cefficient f the AO in the MO. P cc (allyl catin) = 2 /2 /(SQRT(2) = /(SQRT(2) = 0.7 There are 2 electrns. Cefficient = /2. Cefficient 2= /(SQRT(2). This is the pi bnd rder. There is als a sigma bnd s the actual bnd rder is.7. The number is the same fr the radical and anin by this methd because these added electrns are in the NBO. This is nt always the case. When the next electrn up is but in a bnding rbital we get mre bnd rder frm mre ccupatin. Mlecular Orbitals f pi systems and their applicatin t rganic chemistry Instructr cmpares the C 2 bnd rder in allyl catin with tw hypthetical electrnically excited states f allyl catin. Instructr talks abut the differences in dynamics that ne wuld expect between grund and excited states. [If asked culd yu predict which electrnic state f allyl catin wuld be mre cnfrmatinally stable] [Culd yu d the same thing fr ally radical?] WAT IF TE COEFFICIENTS JUST DISTRIBUTED e(-) density unifrmly? ψ = /(SQRT(3)χ + /(SQRT(3)χ 2 + /(SQRT(3)χ 3

5 Principles f Organic Chemistry lecture 5, page 5 P cc (allyl catin) = 2 /(SQRT(3) /(SQRT(3) = 2/3 = 0.66 The real distributin leads t mre bnding in the lwest energy rbital than Ntice that the lw-energy rbital is als mre spherical. What is the pi bnd rder frm resnance? Instructr draws resnance structures and talks abut weighting. C C 3 Thus, the ückel LCAO slutins give us a clear picture why cnjugatin stabilizes mlecular systems! Instructr explains... this effect is nt bvius frm resnance structures withut a lt f hand waving. The duble bnd in ethene has a 66 kcal/ml barrier t rtatin. The bnd is rder is 2 in ethene. Therefre the rtatin abut allyl catin accrding t ückel shuld be kcal/ml = 47 kcal/ml f the naked gas phase species. This is really slw n the mlecular time scale! Remember that yu need a barrier f abut 24 kcal/ml between mlecules and 2 t islate mlecule r mlecule 2 at rm temperature. Checking the number against literature values give sme discrepancy; the literature values are interesting. Experimental value fr the allyl radical is ~5 kcal/ml Krth, G.-G.; Trill,.; Sustmann, R., "[ l- 2 ]Allyl Radical: Barrier t Rtatin and Allyl Delcalizatin Energy" J. Am. Chem. Sc 98, 03, igh level calculatin fr the allyl radical gives ~3 kcal/ml Gbbi, A.; Frenking, G., "Resnance Stabilizatin in Allyl Catin, Radical, and Anin" J. Am. Chem. Sc 994, 6, nte: the year, this stuff is just being figured ut currently. It makes yu wnder hw mature Organic Chemistry is if the fundamental behavir f such simple mlecules is unknwn. igh level calculatin fr the allyl catin gives ~39 kcal/ml igh level calculatin fr the allyl anin gives ~23 kcal/ml One might hypthesize that the massive difference in electrnic distributin between grund and transitin states in the anin and the catin versus the radical culd accunt fr the difference in barrier t rtatin.

6 Principles f Organic Chemistry lecture 5, page 6 Demnstrate this n the bard.