Born Oppenheimer Approximation and Beyond

Transcription

1 L Born Oppenhemer Approxmaon and Beyond aro Barba A*dex Char Professor amu.fr Ax arselle Unversé, nsu de Chme Radcalare

2 LGHT AD Adabac x dabac x nonadabac

3 LGHT AD From Gree dabaos: o be crossed or passed dabac = wh crossng a-dabac = whou crossng non-a-dabac = wh crossng!? 3

4 LGHT AD whou exchangng (cross) hea or energy wh envronmen 4

5 LGHT AD A physcal sysem remans n s nsananeous egensae f a gven perurbaon s acng on slowly enough and f here s a gap beween he egenvalue and he res of he Hamlonan's specrum. Adabac heorem (Born and Foc, 198). n hs example (adabac process), he sprng consan of a harmonc oscllaor s slowly (adabacally) changed. The sysem remans n he ground sae, whch s adjused also smoohly o he new poenal shape. s sae s always an engensae of he Hamlonan a each me ( no crossng ). x 5

6 LGHT AD A physcal sysem remans n s nsananeous egensae f a gven perurbaon s acng on slowly enough and f here s a gap beween he egenvalue and he res of he Hamlonan's specrum. Adabac heorem (Born and Foc, 198). n hs example (dabac process), he sprng consan of a harmonc oscllaor s suddenly (dabacally) changed. The sysem remans n he orgnal sae, whch s no a engensae of he new Hamlonan. s a superposon ( crossng ) of several engensaes of he new Hamlonan. x 6

7 LGHT AD The nuclear vbraon n a molecule s a slowly acng perurbaon o he elecronc Hamlonan. Therefore, he elecronc sysem remans n s nsananeous egensae f here s a gap beween he egenvalue and he res of he Hamlonan's specrum. Ths s anoher way o say ha: The elecrons see he nucle nsananeously frozen 7

8 LGHT AD S S adabac n* * dabac nergy S 1 n* S 1 * Reacon coordnae 8

9 LGHT AD Beyond Born-Oppenhemer : Tme-ndependen formulaon 9

10 LGHT AD H U H T H e T Knec energy nucle H e poenal energy erms whch solves: H r;r (adabac bass) e depends on he elecronc coordnaes r and paramercally on he nuclear coordnaes R. Snce s a complee bass, any funcon n he Hlber space can be exacly wren as a lnear combnaon of. 1

11 s, ; rr R rr H T 1 H e H U nuclear wave funcon LGHT AD ulply by a lef and negrae n he elecronc coordnaes s U 1 T n T a 1 AB A B AB A B on-adabac couplng erms Prove! s U T T 1 11

12 1 s T T U 1 f non-adabac couplng erms = U T uclear vbraonal problem. f s expanded o he second order around he equlbrum poson: a a l l eq l eq q q q q R eq x x q, 1/ can be reaed by normal mode analyss.

13 LGHT AD Beyond Born-Oppenhemer : Tme-dependen formulaon 13

14 LGHT AD H( r, R, ) H T H e T Knec energy nucle H e poenal energy erms whch solves: H e (adabac bass) r;r depends on he elecronc coordnaes r and paramercally on he nuclear coordnaes R. Snce s a complee bass, any funcon n he Hlber space can be exacly wren as a lnear combnaon of. 14

15 15 s 1 ;, R r R R r nuclear wave funcon Prove! ulply by a lef and negrae n he elecronc coordnaes Tme dependen Schrödnger equaon for he nucle s T T 1 H e H T ),, ( R r H

16 onadabac couplng erms s T 1 T Frs suppose he couplngs are null (adabac approxmaon): T ndependen equaons for each surface. () ( ) me () 16

17 LGHT AD Classcal lm of nuclear moon 17

18 LGHT AD 18 ), ( )exp, ( ), ( S A R R R Wre nuclear wave funcon n polar form Ld S ' ) (R, The phase (acon) s he negral of he Lagrangan T Adabac approxmaon A A S S S S Classcal lm Tully, Faraday Dscuss. 11, 47 (1998)

19 Hamlon-Jacob quaon S S To solve he Hamlon-Jacob equaon for he acon s oally equvalene o solve he ewon`s equaons for he coordnaes! ewon equaon d d R n he classcal lm, he soluons of he me dependen Schrödnger equaon for he nucle n he adabac approxmaon are equvalen o he soluons of he ewon`s equaons. n whch cases does hs classcal lm lose valdy? 19

20 LGHT AD A A S S adabac quanum erms 1 nonadabac couplng erms s T T 1 n whch cases does hs classcal lm lose valdy?

21 on-adabac couplng erms s T 1 T 1 1 () x (a ) () x 1 (a ) 1

22 LGHT AD s rr, R rr ; Born-Huang odel 1, ; rr R rr Adabac approxmaon He U T Born-Oppenhemer Approxmaon s U T T 1 onadabac couplng erms