Chem 3502/4502 Physical Chemistry II (Quantum Mechanics) 3 Credits Spring Semester 2006 Christopher J. Cramer. Lecture 25, March 29, 2006

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1 Chem 3502/4502 Physicl Chemistry II (Quntum Mechnics) 3 Credits Spring Semester 2006 Christopher J. Crmer Lecture 25, Mrch 29, 2006 (Some mteril in this lecture hs been dpted from Crmer, C. J. Essentils of Computtionl Chemistry, Wiley, Chichester: 2002; pp ) Recpitultion of the Vritionl Principle nd the Seculr Eqution Recll tht for ny system where we cnnot determine exct wve functions by nlyticl solution of the Schrödinger eqution (becuse the differentil eqution is simply too difficult to solve), we cn mke guess t the wve function, which we will designte Φ, nd the vritionl principle tells us tht the expecttion vlue of the miltonin for Φ is governed by the eqution where E 0 is the correct ground-stte energy. "!!dr "! # E 0 (25-1)!dr Not only does this lower-limit condition provide us with convenient wy of evluting the qulity of different guesses (lower is better), but it lso permits us to use the tools of vritionl clculus to identify minimizing vlues for ny prmeters tht pper in the definition of Φ. In the LCAO (liner combintion of tomic orbitls) pproch, the prmeters re coefficients tht describe how moleculr orbitls (remember, orbitl is nother word for one-electron wve function contributing to mny-electron wve function) re built up s liner combintions of tomic orbitls. In prticulr, mny-electron wve functions Φ cn be written s ntisymmetrized rtree products i.e., Slter determinnts of such oneelectron orbitls φ, where the one-electron orbitls re defined s N! = # i " i (25-2) i=1 where the set of N tomic-orbitl bsis functions ϕ i is clled the bsis set nd ech hs ssocited with it some coefficient i, where we will use the vritionl principle to find the optiml coefficients. To be specific, for given one-electron orbitl we evlute

2 25-2 E = = = # & # & i!i )%" ( % " j! ( j $ i ' % ( dr $ j ' # &# & ) %" i!i % ( " j! ( j $ i ' % ( dr $ j ' " i j! ) i! j dr i " j )! i! j dr. (25-3) " i j " i j S where the shorthnd nottion nd S is used for the resonnce nd overlp integrls in the numertor nd denomintor, respectively. If we impose the minimiztion condition!e! k = 0 " k (25-4) we get N equtions which must be stisfied in order for eqution 25-4 to hold true, nmely N i i=1 ( ) = 0 " k. (25-5)! ki ES ki these equtions cn be solved for the vribles i if nd only if 11 ES ES 12 L 1N ES 1N 21 ES ES 22 L 2N ES 2 N M M O M = 0 (25-6) N1 ES N1 N2 ES N2 L NN ES NN where eqution 25-6 is clled the seculr eqution. There re N roots (i.e., N different vlues of E) which permit the seculr eqution to be true. For ech such vlue E j there will be different set of coefficients,, which cn be found by solving the set of liner equtions 25-5 using tht specific E j, nd those coefficients will define n optiml ssocited wve function φ j within the given bsis set. The steps in prcticl clcultion re:

3 25-3 (1) Select set of N bsis functions. (2) For tht set of bsis functions, determine ll N 2 vlues of both nd S. (3) Form the seculr determinnt, nd determine the N roots E j of the seculr eqution. (4) For ech of the N vlues of E j, solve the set of liner eqs in order to determine the bsis set coefficients for tht MO. ückel Theory To further illuminte the LCAO vritionl process, we will crry out the steps outlined bove for specific exmple. To keep things simple (nd conceptul), we consider flvor of moleculr orbitl theory developed in the 1930s by Erich ückel to explin some of the unique properties of unsturted nd romtic hydrocrbons. In order to ccomplish steps (1)-(4) of the lst section, ückel theory dopts the following conventions: () The bsis set is formed entirely from prllel crbon 2p orbitls, one per tom. [ückel theory ws originlly designed to tret only plnr hydrocrbon π systems, nd thus the 2p orbitls used re those tht re ssocited with the π system.] (b) The overlp mtrix is defined to be S =! (25-7) Thus, the overlp of ny crbon 2p orbitl with itself is unity (i.e., the p functions re normlized), nd tht between ny two different p orbitls is zero (so we won't wste time computing overlp integrls, we'll just ssume the bsis functions to be orthonorml). (c) Mtrix elements ii re set equl to the negtive of the ioniztion potentil of the methyl rdicl C 3, i.e., the orbitl energy of the singly occupied 2p orbitl in the prototypicl system defining sp 2 crbon hybridiztion. This choice is consistent with our erlier discussion of the reltionship between this mtrix element nd n ioniztion potentil. This energy vlue, which is defined so s to be negtive, is rrely ctully written s numericl vlue, but is insted represented by the symbol α. For those who like working with ctul numbers, α = 9.9 ev. (d) Mtrix elements between nerest neighbors re lso derived from experimentl informtion. A 90 rottion bout the π bond in ethylene removes ll of the

4 25-4 bonding interction between the two crbon 2p orbitls. Tht is, the (positive) cost of the following process, E = E π E = 2E p is ΔE = 2E p E π. The (negtive) stbiliztion energy for the pi bond is distributed eqully to the two p orbitls involved (i.e., divided in hlf) nd this quntity, termed β, is used for between neighbors. (Note, bsed on our definitions so fr, then, tht E p = α nd E π = 2α+2β.) Agin, for those who like numbers, the π bond energy in ethylene is bout 60 kcl mol 1, which is 2.6 ev. Dividing tht up between the two crbon toms results in β = 1.3 ev. (e) Mtrix elements between crbon 2p orbitls more distnt thn nerest neighbors re set equl to zero. 1 C 3 C 2 C! 3 = "%#2$! 2 = " E p C! 1 = "+#2$

5 25-5 The Allyl π System Let us now pply ückel MO theory to the prticulr cse of the llyl system, C 3 3, s illustrted in the figure on the previous pge. Becuse we hve three crbon toms, our bsis set is determined from convention () nd will consist of 3 crbon 2p orbitls, one centered on ech tom. We will rbitrrily number them 1, 2, 3, from left to right for bookkeeping purposes. The bsis set size of 3 implies tht we will need to solve 3 x 3 seculr eqution. ückel conventions (b)-(e) tell us the vlue of ech element in the seculr eqution ( 11 = 22 = 33 = α, 12 = 21 = 23 = 32 = β, 13 = 31 = 0, S 11 = S 22 = S 33 = 1, ll other S vlues re 0) so tht eq is rendered s! E " 0 "! E " 0 "! E = 0 (25-8) The use of the Kronecker delt to define the overlp mtrix ensures tht E ppers only in the digonl elements of the determinnt. Since this is 3 x 3 determinnt, it my be expnded using Crmer s rule s (α E) 3 + (β 2 0) + (0 β 2 ) [0 (α E) 0] β 2 (α E) (α E)β 2 = 0 (25-9) which is firly simple cubic eqution in E tht hs three solutions, nmely E =! + 2",!,! 2" (25-10) Since α nd β re negtive by definition, the lowest energy solution is! + 2". To find the MO ssocited with this energy, we employ it in the set of liner equtions 25-5, together with the vrious necessry nd S vlues lredy noted bove to give [ ] + 2 [" (! + 2" ) 0] + 3 [ 0 (! + 2" ) 0] = 0 [ ( ) 0] + 2 [! (! + 2" ) 1] + 3 [" (! + 2" ) 0] = 0 [ ( ) 0] + 2 [" (! + 2" ) 0] + 3 [! (! + 2" ) 1] = 0 1! (! + 2" ) 1 1 "! + 2" 1 0! + 2" (25-11) (the first eqution comes from k = 1, the second from k = 2, nd the third from k = 3). Some firly trivil, if tedious, lgebr reduces these equtions to 2 = = 1 (25-12)

6 25-6 While there re infinitely mny vlues of 1, 2, nd 3 which stisfy eq , imposing the requirement tht the wve function be normlized provides finl constrint in the form of 3 2 i i=1 The unique vlues stisfying both eqs nd re! =1 (25-13) 11 = 1 2, 21 = 2 2, 31 = 1 2 (25-14) where we hve now emphsized tht these coefficients re specific to the lowest energy moleculr orbitl by dding the second subscript 1. Since we now know both the coefficients nd the bsis functions, we my construct the lowest energy moleculr orbitl, i.e.,! 1 = 1 2 p p p 3 (25-15) which is illustrted in the figure bove. By choosing the higher energy roots of eq. 25-8, we my solve the sets of liner equtions nlogous to eq in order to rrive t the coefficients required to construct φ 2 (from E = α) nd φ 3 (from E =! 2"). Although the lgebr is left s homework problem, the results re 12 = 2 2, 22 = 0, 32 = = 1 2, 23 = 2 2, 33 = 1 2 (25-16) where these orbitls re lso illustrted in the figure bove. The three orbitls we hve derived re the bonding, non-bonding, nd ntibonding moleculr orbitls of the llyl system with which ll orgnic chemists re fmilir. Importntly, ückel theory ffords us certin insights into the llyl system, one in prticulr being n nlysis of the so-clled resonnce energy rising from electronic delocliztion in the π system. By delocliztion we refer to the prticiption of more thn two toms in given MO. Consider for exmple the llyl ction, which hs totl of two electrons in the π system. If we dopt moleculr ufbu principle of filling lowest energy MOs first nd further mke the ssumption tht ech electron hs the energy of the one-electron MO tht it occupies (φ 1 in this cse) then the totl energy of the llyl ction π system is 2(! + 2"). Consider the lterntive fully loclized structure for the

7 25-7 llyl system, in which there is full (i.e., doubly-occupied) π bond between two of the crbons, nd n empty, non-intercting p orbitl on the remining crbon tom (this could be chieved by rotting the ctionic methylene group 90 so tht the p orbitl becomes orthogonl to the remining π bond, but tht could no longer be described by simple ückel theory since the system would be non-plnr the non-interction we re considering here is purely thought-experiment). The π energy of such system would simply be tht of double bond, which by our definition of terms bove is 2(α + β). Thus, the ückel resonnce energy, which is equl to π loclized, is 0.83β (remember β is negtive by definition, so resonnce is fvorble phenomenon). Reclling the definition of β, the resonnce energy in the llyl ction is predicted to be bout 40% of the rottion brrier in ethylene roughly 25 kcl mol 1. We my perform the sme nlysis for the llyl rdicl nd the llyl nion, respectively, by dding the energy of φ 2 to the ction with ech successive ddition of n electron, i.e., π (llyl rdicl) = 2(! + 2") + α nd π (llyl nion) = 2(! + 2") + 2α. In the hypotheticl fully-π-loclized non-intercting system, ech new electron would go into the non-intercting p orbitl, lso contributing ech time fctor of α to the energy (by definition of α). Thus, the ückel resonnce energies of the llyl rdicl nd the llyl nion re the sme s for the llyl ction, nmely, 0.83β. Unfortuntely, while it is cler tht the llyl ction, rdicl, nd nion ll enjoy some degree of resonnce stbiliztion, neither experiment, in the form of mesured rottionl brriers, nor more complete levels of quntum theory support the notion tht in ll three cses the mgnitude is the sme. So, there is some spect of ückel theory tht renders it incpble of ccurtely distinguishing between these three llyl systems. We will exmine this issue shortly. omework To be solved in clss: Find the shpes nd energies of the 3 moleculr orbitls for the cyclopropenium system. Note tht this system differs from the llyl system in tht it hs n dditionl connectivity between toms 1 nd 3. ow does this qulittively chnge the MO picture for cyclopropenyl compred to llyl? Bsed on your nlysis, will the cyclopropenyl nion be singlet or triplet? (int: to quickly solve the somewht chllenging cubic eqution in E tht comes up in the determinnt, try setting E = α + cβ nd solve for c.) To be turned in for possible grding Apr. 7: Using the third (ntibonding) root of the seculr eqution for the llyl system, verify the orbitl coefficients given in eq