cannot commute.) this idea, we can claim that the average value of the energy is the sum of such terms over all points in space:

Transcription

1 Che 441 Quatu Cheistry Ntes May, 3 rev VI. Apprxiate Slutis A. Variati Methd ad Huckel Mlecular Orbital (HMO) Calculatis Refereces: Liberles, Ch. 4, Atkis, Ch. 8, Paulig ad Wils Streitweiser, "MO Thery fr Orgaic Cheists" Bauer, H; Rth, K. "The Nueric Sluti f igevalue Prbles", J. Che. duc., 198, 57, Csider the Schrdiger quati: H e Where is a apprxiate sluti t a particular prble adh is a accurate Hailtia peratr fr that prble. A key differece i this apprxiate case is that the eergy e is t cstat (uless, f curse, the apprxiate sluti happes t be exact!). Nw, csider the fllwig peratis : Multiply bth sides by : H e e (te that e ad cute, but H ad cat cute.) dτ Nw te that (3) d τ is the prbability f fidig a particle i the regi dτ. (The itegral i the deiatr is ver the etire regi f space where the particle described by culd be fud. It has the value 1 if is ralized.) dτ Thus it is reasable t clai that e represets the prbability f fidig the particle with eergy e. Usig d τ this idea, we ca clai that the average value f the eergy is the su f such ters ver all pits i space: e dτ H < H > dτ < > (The <...> sybls i < > are siply a shrthad tati fr itegrati ver the etire space where the particle described by ight be fud.) A iprtat there abut <> is the Variati There: < >, exact (5) Prf: let c where is a exact sluti with eigevalue. Pluggig this it (4) we have: ch c cch c c c () (4) c c c c c c c (6) Page 1

2 which ca be rearraged t: c c where we have assued that the apprxiate eergy will be a apprxiati t the lwest level. Csider w: exact c exact c Hece, <> >_,exact QD. Prble 1. Prve that all the steps i (6) are crrect.,, Prble : The substituti: c is reasable if it is a geeral way f describig ay pssible apprxiate wave fucti. We ca prve that this is the case by shwig hw t btai a geeral frula fr ck. Shw that: c k < > k (7) (Hit: ultiply the startig equati fr the left by ad use the rthgality f the basis set slutis ). Nte: I a re careful treatet f this aterial, c shuld be replaced by c c. There is differece betwee the tw cases whe c is real, as will be the case i the exaples we exaie. Applicatis t Mlecular Orbitals Let us suppse that we are tryig t fid apprxiate eergies ad wave fuctis fr the butadiee p electrs. The usual way f picturig this is shw i Figure 1 the fllwig page, where the (+) ad (-) sybls idicate relative phases. The fur pictures are relatively easy t cstruct withut ay algebra, ad the assigets t π ad π categries are als easy t ratialize. All fur MO's als ca be writte i a algebraic way as: MO c where we take t be p carb ats 1-4 ( 1,,3,4). Whe we cstruct lecular rbitals f this type, we are usig a liear cbiati f atic rbitals (LCAO) apprach. The gal f this apprxiate prcedure is t deterie, first, the best set f eergy levels fr the LCAO 's, ad secd, the shapes f the assciated wavefuctis. Figure 1. π Mlecular Orbitals fr Butadiee Page

3 First let us lk at eergies. We begi with the first versi f (6): c H c cc cch cc ccs H cc (8) where H H ad S (9) Our gal is, first f all, t fid the iiu eergy, which will be the best apprxiati t the crrect. We ca esure that <> is iiized by takig its derivative with respect t a particular cstat ci ad settig it equal t : ch cch cs ch cs c c c S c c S i i i i i ccs r: ch cs i i (11) (which is usually writte: c Hi Si (1) This equati ust be true fr all values f i. The set f all such equatis is fte expressed i atrix fr: (1) H11 < > S11 H1 < > S1 H13 < > S13... H1 j < > S1 j c1 H1 < > S1 H < > S H3 < > S3... H j < > S j c H31 < > S31 H3 < > S3 H33 < > S33... H3 j < > S 3 j c H < > S H < > S H S... H S c i < > < > i1 i1 i i i3 i3 ij ij (13) These equatis are kw cllectively as the secular equatis. Se further siplificatis: 1) Chse Sij 1 if i j, if i / j ) Call Hij α if i j (α is the eergy f a -bdig p rbital). 3) Call Hij β if i j +_ 1 (β is kw as the resace itegral). 4) Take Hij fr all ther i,j cbiatis. Prble: Shw that α des ideed crrespd apprxiately t the eergy f a p rbital i a carb at. Page 3

4 Fr butadiee, usig these assuptis, the secular atrix equati beces: α β c1 β α β c β α β c 3 β α c 4 (14) where <> has bee used fr tatial siplicity. This equati ca be siplified by dividig bth sides by β ad α the defiig x (Nte that: α - βx) β x 1 c1 1 x 1 c Whe this is de, (14) reduces t: (16) 1 x 1c 3 1 x c 4 This equati is valid either if all the ci are (a sluti f physical iterest) r if the deteriat f the square atrix is. Slvig this deteriat leads t fur values f x, which i tur leads t fur values f. Slvig the Deteriat: x 1 1 x 1 1 x 1 1 x x ( ) ( ) 3 4 x 1 x 1 1 x 1 x x x x x 1 x 3x + 1 y 3y+ 1 (17) 1 x 1 x where y x 3± 5 3± 5. This has the rts: y s x , -.618, +.618, (18) Prble: Write ut the 4 eergy values fr i ters f α ad β usig (15) Nte: MacDald s There csiders the relatiship betwee the variatial eergy apprxiati fr the th level ad the exact eergy value fr this level. The result is a extesi f the variatial there stated i (5) abve: (cf. Paulig ad Wils, p. 188) (19) exact The MO's assciated with the eergy values are derived by the fllwig recipe: 1) put a particular x value it the secular deteriat, e.g. x : ) write dw the cfactrs f each clu i the deteriat: A 11 (1+ 1) ( 1) Page 4

5 A 1 (1+ ) 1 1 ( 1) A A 13 (1+ 3) ( 1) (1+ 4) ( 1) ) set 1i 1i A A ci j ( A ) j 1 givig c 1.37 c.61 c 3.61 c 4.37 () Prble: Sketch a picture f the MO assciated with these cefficiets. Prble: Derive the cefficiets ad pictures fr the ther three MO's Prble: Shw that the first MO is ralized. Prble: Shw that c11 + c1 + c13 + c14 1 where "11" refers t the cefficiet fr the first MO fr carb 1, "1" is the cefficiet fr the secd MO fr carb 1, etc. Mre cplex prbles ad cputer cdes It is ipractical t wrk ut slutis "by had" fr atrix prbles with a diesi greater tha 4, ad s we tur t cputer prgras t slve atrix equatis aalgus t (16). There is a gd discussi f the geeral ethd ("Jacbi rtati") i the J. Che. Surce cited at the begiig f this uit, ad als i a prgraig surce such as "Nuerical Recipies" by Press, et al. The prcedures geerally trasfr the square atrix f (16), with x set equal t zer, by a series f Jacbi rtatis it a diagal fr (that is, a set f trasfratis that akes all ff-diagal eleets arbitrarily clse t zer). Whe the square atrix f (16) is "diagalized" i this way, the diagal eleets cstitute the set f "x" eigevalues, ad the prduct f all the successive Jacbia rtati atrices is a square atrix with the prperty that each rw (r clu) ctais the set f LCAO cefficiets. We have a FORTRAN ipleetati f this prcedure. Applicatis f the siple HMO ethd 1) the 4 + rule (cf. Streitweiser, MO Thery fr Orgaic Cheists) ) epr spectra fr cjugated radical ais ad catis f cjugated systes. We will d se spreadsheet exaples. Page 5

6 B. Apprxiate Slutis - Perturbati Thery Perturbati thery is a alterative rather useful way f gettig apprxiate results fr cplex prbles i quatu echaics. It is develped as fllws: Assue that the Hailtia peratr ca be brke up it tw parts: H H +λv (1) where H is the Hailtia fr a exactly slvable prble ad v is a additial ptetial eergy ter (a sall e we hpe!) which akes the prble difficult r ipssible t slve exactly. We the assue that the eigefuctis f H ca be writte i the fllwig fr: λ λ λ () 3 (3) () () 3 (3) ad that + λ + λ + λ +... (3) The ad are slutis t the exact prble (v ). ad are kw as first-rder perturbatis, () are secd rder perturbatis, etc.. The thery is useful i cases where the series ters quickly get vaishigly sall. We als usually assue that λ 1 fr practical calculatis, but we d t wat t d this just yet. Nw csider the crrect S: H which beces, with substituti: ( H ) ( ) + λ + λ + λ + λ + () 3 (3) v... () 3 (3) () 3 (3) + λ + λ + λ λ + λ + λ +... The gae we play w ivlves cllectig like ters i λ i the prducts f (4): ( 1) ) ( H ) + v + H λ () () () 1 () 1 ( ) λ + v + H λ [ ] The, sice λ is a variable, we require the cefficiet f each λ ter t (we ecutered this prperty earlier i the () () ter with the haric scillatr prble). This allws us t deterie, i successi,,,, etc. () ad (4) (5) : Settig the cefficiet f λ i (5) equal t gives: H v + (6) Rearrage this t: v + H (7) ad the ultiplyig bth sides by fr the left ad itegratig ver all space gives: v + H (8) There is a prperty f wavefuctis which we have t itrduced r explred kw as the Heritia prperty which is stated as fllws: a b b a We ca use this prperty t eliiate the last tw ters: H H (9) v + H v v (3) s: v assuig ralizati f. (31) v Page 6

7 : fr a particular eergy level is st fte treated as a liear cbiati f the exact slutis: c (3) If we lk at equati (6) ad fcus the iddle ters, bth f which ivlve it is easy t shw that if (1') is a sluti t (6), the + will be a equally gd sluti. (1') Prble: Prve that this clai is true. I ther wrds, a is t deteried exactly by (6). We ca take advatage f this by tig that: + a (33) (1') if we chse the arbitrary cstat as: (1') a (34) This eas that i geeral we ca assue that will be chse s that it is rthgal t. Let us w g back t (3) ad slve it fr c: First we te that the rthgality f ad Prble: Shw that if c the ad w requires c. are i fact rthgal. Let us w plug (3) it (6). This leads t a frula fr the c cefficiets as fllws: ( ) v + c H c v + c c (35) v + c c v (36) which rearrages t: ( ) () T get a particular cefficiet ck, ultiply bth sides f (36) by k ad itegrate ver all space. Takig advatage f the rthgality f the fuctis, we get: ( ) ( ) c k ck k v k k v (37) ad slvig fr ck thus gives: Fially, this gives a explicit frula fr c k v ( k ) k (38) v ( ) (39) : Settig the cefficiet f λ, we ultiply by ce agai fr the left, fllwed by itegrati. The Heritia prperty is w used t eliiate the secd ad third ters i the λ cefficiet i (5), givig: Page 7

8 () v v (4) Prble: Shw the steps i the derivati f (4). (Orthgality accuts fr the ueratr siplificati.) If is writte as a expasi ce re (cf. (3)), equati (4) ca be writte as: () Agai, ralizati f the v v c v (41) ( ) values has bee assued. Applicatis: 1. Perturbed Haric Oscillatr. lectric trasitis: e.g. IR spectrscpy f HCl Page 8