Chapter 8 Approximation Methods, Hueckel Theory

Transcription

1 Witr 3 Chm 356: Itroductory Qutum Mchics Chptr 8 Approimtio Mthods, ucl Thory... 8 Approimtio Mthods... 8 Th Lir Vritiol Pricipl... mpl Lir Vritios... 3 Chptr 8 Approimtio Mthods, ucl Thory Approimtio Mthods A) Th vritiol pricipl For y ormlizd wv fuctio, th pcttio vlu of ˆ, ˆ, th ct groudstt rgy. Proof: c with ˆ If w would msur th rgy w would fid with probbility P This rgumt is bit shy wh Ĥ hs dgrt igvlus. You will do corrct proof i th ssigmts. If would ot b ormlizd w c clcult ˆ *( ) ˆ d P P P ( ) N *( ) ( ) d c d th Ĥ N or i th fil form: Th vritiol pricipl () whr *( ) ˆ ( ) d *( ) ( ) d Domi Domi is th ct groud stt rgy 8

2 Witr 3 Chm 356: Itroductory Qutum Mchics Th vritiol rgy, is ct wh ( ) ( ), th ct groud stt wvfuctio. mpls: us tril wvfuctio tht dpds o o or mor prmtrs, Th miimiz th tril rgy, Som simpl (trivil) mpls: ˆ T tril wvfuctio of th typ Or Q: wht is? Wht is? d hu. T th tril wvfuctio d ( ) N / A: Th ct wvfuctio hs th form / N, by miimizig th rgy w should gt, Q: T th miltoi for th ydrog tom, d l stt / ˆ d d r dr dr r r 4 r - Wht r th itgrls to vlut? - Wht is th optiml vlu for? - Wht is th vlu for? A: 4 ( ) 4 opt r r r dr r r ˆ r dr Miimizig : ( ) 4 gi: ct 4 m You c do thos problms yourslf d s if you gt th corrct swr Notrivil mpl: T th miltoi for -tom, s-orbitl, d us th tril wvfuctio r Chptr 8 Approimtio Mthods, ucl Thory 9

3 Witr 3 Chm 356: Itroductory Qutum Mchics 4 ( ) 4 r r r dr r r ow, sic th tril wvfuctio cot b ct for y ˆ r dr 3 m ( ) 3/ 3 m ( ) 3/ / m 3/ 3( ) opt m m 4 ( opt ) 3 6 ( opt ) Not: Gussi tril orbitls (bsis sts) r widly usd i lctroic structur progrms. This is bcus itgrls r sily vlutd ovr Gussis. This is th origi of th m for th Gussi Progrm: It uss Gussi bsic fuctios! Chptr 8 Approimtio Mthods, ucl Thory

4 Witr 3 Chm 356: Itroductory Qutum Mchics Th Lir Vritiol Pricipl Cosidr tril wv fuctio Lt us ssum for simplicity - Rl cofficits, fuctios, - Orthoorml psio fuctios: Th w c try to optimiz th cofficits Th c ( ) c f( ) c ( ) D c f ( ) c f ( ) d Or m, f f *( ) f ( ) d Domi m m *( ) ( ) d N *( ) ( ) d D N c f ( ) cm fm ( ) d c ( ) ˆ cm f fm ( ) d m, m, c c m, m m m m ccm f ( ) fm ( ) d, m c c c m, m N D D N N c c c D D N c N N D c D c N c m D c c c m m D c c c c Chptr 8 Approimtio Mthods, ucl Thory

5 Witr 3 Chm 356: Itroductory Qutum Mchics mcm c c c m Sic, this is twic th sm qutio. c c m m This hs th form of mtri igvlu qutio! m m d c c m m W c lso writ c m c c This hs th form of lir qutio. m Ac, with A This typ of qutio oly hs solutio which dt A. c dt qutio for sculr dtrmit Lt us discuss mpls ltr. For ow I wt to drw th logy: Schrodigr qutio Ĥ If w m bsis psio c f f f Th w gt mtri typ Schrodigr qutio c c m m With *( ) ˆ ( ) d m m Such igvlu qutio hs M solutios for M M mtri. Thy rprst pproimtios to th groud d citd stts. If th bsis is ot orthoorml th dfi Sm *( ) m ( ) d d c S c (s MQ) Or dt S igvlus. Chptr 8 Approimtio Mthods, ucl Thory

6 Witr 3 Chm 356: Itroductory Qutum Mchics mpl Lir Vritios Cosidr prticl i th bo d m d ( ) si Now dd to lir pottil V Us s tril wv fuctio c si si c i, j, ij V si m d V 6V 9 6V V 9 i d j si m 6 9 m m V Th igvlus of this miltoi r m V / V 4 This hpps to b prtty good solutio, spcilly s is smll V Chptr 8 Approimtio Mthods, ucl Thory 3

7 Witr 3 Chm 356: Itroductory Qutum Mchics Othr istructiv mpl: cosidr prticl o th rig mr, with th dgrt m solutios cos si Now pply mgtic fild, which dds to th miltoi L ˆz Udr th ifluc of th prturbtio th lvls split. Clcult th rgy splittig. mr B L ˆ L ˆ m z z i I usd wrog formul; sig is wrog L m z c cos c si ˆ zlz cos i si ˆ Lz si i cos mr = i i mr dt Z mr mr mr Chptr 8 Approimtio Mthods, ucl Thory 4

8 Witr 3 Chm 356: Itroductory Qutum Mchics mpls: C I fid igfuctios? Wht r th igfuctios th? Bottom li: Digoliz -tom: W c idict prturbtio ovr dgrt stts. Digoliz Ĥ ovr p orbitls p3, p igfuctios from digolizig 6 6 miltoi. vrythig coms out by brut forc. mpl : i mr i mr i mr dd i dditiol mgtic itrctio ( ) ˆ g() v L S B L B S m m digoliz ovr p d s orbitls ll th splittig from digoliztio i i mr i mr i i mr cos si ~ i i i cos isi ~ i i (w ormliz, fctor) i i mr mr i i mr mr g v L S ( ) () Th lir vritiol pricipl is vry powrful tool to clcult pproimt igfuctios i i V z z z z Chptr 8 Approimtio Mthods, ucl Thory 5

9 Witr 3 Chm 356: Itroductory Qutum Mchics It is widly usd to clcult th splittig of rgis i dgrt mifold, wh ddig prturbtio. Wh th rgis of miltoi r ot dgrt, o c gt good stimt of th rgy corrctio du to prturbtio V, by clcultig V. () c if, th igvlus of ˆ ˆ Vˆ r giv to first pproimtio by ˆ *( ) V ( ) d Ths r just th digol lmts of th miltoi mtri = First ordr Prturbtio Thory: If w go bc to bo + lir fild If zro-ordr stts r dgrt, first-ordr prturbtio thory is uslss. Istd us lir vritiol pricipl *( ) V ( ) d Vˆ () d m d Ĥ si () V ˆv m V si d V V V ll rgis r shiftd by () V mpl L i si( m) bsis ˆz im choos othr bsis: rsults im lwys digoliz ovr zroth-ordr stts: dgrt first-ordr prturbtio thory Aothr mpl of c c : ücl -lctro thory I orgic chmistry, my molculs r sstilly plr. Th pl cotis crbo, oyg, itrog toms. Th out of pl P -orbitls costitut th -orbitls. Th molcul s - orbitls r z sp Chptr 8 Approimtio Mthods, ucl Thory 6

10 Witr 3 Chm 356: Itroductory Qutum Mchics lir combitios of th tomic miltoi mtri s follows. Lt us rstrict ourslvs to P z -orbitls. O c prmtriz o-lctro ffctiv sp crbos. Rul: o digol for y two djct toms coctd by -bod Followig th vritiol pricipl w ) Digoliz th miltoi orbitl rgis, igvctors b) Fill up orbitl lvls from th bottom up puttig d lctro i ch lvl. Occupy s my lvls s you hv -lctros c) If lvls r dgrt, fill thm up with -lctros first, th dd dditiol - lctros d) Totl rgy: ) Dsity Mtri ~ D occupid orbitls () N () z z l l, occupid D P V P d c c l Dl (s McQurri) l c Chptr 8 Approimtio Mthods, ucl Thory 7

11 Witr 3 Chm 356: Itroductory Qutum Mchics This procdur would wor fi i MthCd ow do w do it o ppr? T dt thyl - lctros Usig d is bit tdious for lrgr problms divid ch colum by d dfi Or Alwys: i i TrN is doubl solutio. 3 3 o Chptr 8 Approimtio Mthods, ucl Thory 8

12 Witr 3 Chm 356: Itroductory Qutum Mchics 3 -lctros (4-fold dgrt) Triplt (3-fold dgrt) Siglt Wht if w would loo t th siglt stt of th io? This is ot stbl structur, th molcul would distort Jh-Tllr Distortio! I might s qustios of 4*4 dtrmit too hrd to solv Chptr 8 Approimtio Mthods, ucl Thory 9

13 Witr 3 Chm 356: Itroductory Qutum Mchics I would giv you th solutio,, 3, 4 You show tht ( )( )( )( ) is your sculr dtrmit 3 4 You c guss th orbitls (phss) from symmtry rgumts: Th orbitls r lwys symmtric or tisymmtric with rspct to pl or is of symmtry If you ow vlu of you c solv b c : b c ( - ) orthogol combitios - : Chptr 8 Approimtio Mthods, ucl Thory