Chemistry 2. Assumed knowledge. Learning outcomes. The particle on a ring j = 3. Lecture 4. Cyclic π Systems

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1 Chemistry Leture QuatitativeMO Theoryfor Begiers: Cyli Systems Assumed kowledge Be able to predit the umber of eletros ad the presee of ougatio i a rig otaiig arbo ad/or heteroatoms suh as itroge ad oxyge. Be able to overt betwee eergy ad spetrosopi uits (J, ev, Hz ad m) Learig outomes Reall ad apply the rule for aromatiity Reogize ad iterpret the polygo memoi for the eergy levels of a ougated yli ompoud. Be able to apply moleular orbital theory as a model for the eletroi struture of a ougated rig moleule (give equatio for E ). The partile o a rig 3 All sigly degeerate Doubly degeerate above 0 3 box rig 0

2 Iteratig orbitals. The extet of orbital mixig is give by the itegral H pa pb pa pb The p orbital o oe atom a iterat with the p from the other atom. Sie they have the same eergy this mixig is omplete. Iteratig orbitals p A p B pa pa H H pa pb pb pb H H pb pa The Coulomb itegral, is kow as the Coulomb itegral ad takes aout of the self eergy of a orbital ad the attrative potetial of the other ulei. Sie this is attrative, the Coulomb itegral is egative. At large iteratomi distaes, the value of this itegral is the atomi orbital eergy, whih is also egative. pa r pah pa pbh pb

3 The Resoae itegral, is a measure the stregth of the bodig iteratio as a result of the overlap of orbitals pa ad pb. It is egative (attrative) where the orbitals ostrutively overlap, ad is zero at large separatio. pa Ĥ pb H pb r H pa pb pa Iteratig orbitals p A p B Whe two p-orbitals iterat, they are split by. The bodig orbital has a eergy of, where both itegrals are egative. MOs of Bezee Maybe we a model the eletros as orbitals multiplied by waves? 3

4 MOs of Bezee os(3θ) where is si(3θ)?? os(θ) si(θ) os(θ) si(θ) os(0θ) θ /3 θ 0 Ψ MOs o a rig these are the uhybridized p orbitals this is here to keep the total probability to also 3 Ψ si this is the partile o a rig wave θ Costrut MOs as atomi orbitals multiplied by the partile o a rig wavefutios. orbital eergies Coeffiiets of p orbitals give by Partile o a rig wavefutio ΨHΨ (... ) H (... ) for bezee, see appedix for derivatio

5 orbitals of bezee 3 The six p orbitals all have the same eergy iterat ad mix 0 3 a geometrial memoi 3 / 0 a geometrial memoi for membered rigs ylopropeyl radial ylobutadiee ylopetadieyl radial bezee

6 a geometrial memoi for membered rigs etred at, radius 3, 0,,, membered rigs are stable, ad are refered to as aromati. bezee Questio: what is the eergy of the HOMO LUMO trasitio, give the formula for the eergy levels? Aswer: We fid that the trasitio is from to. Substitutig, The HOMO LUMO trasitio is.? 0 bezee Questio: from the alulated ressio of the HOMO LUMO trasitio, alulate a erimetal value for the resoae itegral,, takig ) the itese trasitio i the figure for alibratio, ) the lowest eergy trasitio for alibratio. Aswer: The alulated eergy of the HOMO LUMO trasitio is. This orrespods to photos of wavelegth aroud 80 m, for the itese trasitio, or 0m for the lowest trasitio. Thus, h/( ) J.89 ev ) 3. ev or ).38 ev 80 m 0 m 0 3 Hiraya ad Shobatake, J. Chem. Phys. 9, 7700 (99)

7 Questio: [] aulee ad [] aulee are pitured. Whih is aromati? aulees Aswer: The rule applies to [] aulee, 3, but ot [] aulee, whih is atiaromati. Questio: How may eletros i [8] aulee? What are the values of for the HOMO ad the LUMO? [] aulee [] aulee Aswer: There are 8 eletros, so the trasitio is from to. (draw a piture) [8] aulee aulees Questio: Predit the lowest eergy at whih [8] aulee might absorb strogly, give your smaller alulated value for from bezee. Aswer: ev [8] aulee This orrespods to 70 m, or 3300 m, whih is a value supported by erimet. some aulee MOs 7

8 Summary By multiplyig the uhybridized p orbitals by the partile o a rig wave, oe obtais moleular orbitals for yli systems. This gives a eergy formula whih a be easily remembered by drawig a polygo etred o with radius. ext leture Vibratioal spetrosopy: the simple harmoi osillator Week 0 tutorials Partile i a box approximatio you solve the Shrödiger equatio. Pratie Questios. Bezee absorbs at 0 m, orrespodig to the HOMO LUMO trasitio. (a) What is the spetrosopi value of i ev ad Joules. (b) Calulate the total eergy of the eletros i bezee usig this value () A isolated CC C bod has eergy σ. What is the total eergy of the eletros i three CC bods (d) Usig your aswer to (b) ad (), what is the aromatizatio eergy?. Draw a irle ad isribe a equilateral triagle iside suh that oe vertex lies at the o lok positio. The poits at whih the two figures touh are the eergy levels. 3. Repeat for, ad membered rigs. 8

9 9 Appedix The followig is a derivatio of the eergy formula for moleular orbitals o a rig. 3 i ± ± Ψ omplex wavefutios ( ) ( ) ( ) θ θ θ i i si os ± ± i ± Ψ m Sie the si ad os wavefutios are degeerate, we may make liear ombiatios of these to make ew solutios. It will simplify the mathematis to use these omplex forms of the wavefutios. We a ow distiguish betwee the two degeerate solutios as ±, whih is like goig aroud the rig oe way, or the other. ( ) ( ) Ψ Ψ τ τ τ d H d i H i d H eergies The iteratio betwee a orbital ad itself is, the Coulomb itegral. The iteratio betwee a orbital ad a adaet oe is the Resoae itegral,. The iteratio betwee o adaet orbitals is igored. igored

10 0 ( ) ( ) i i d H d i H i τ τ eergies i i os eighbours to the left to the right os