I. The Connection between Spectroscopy and Quantum Mechanics

Transcription

1 I. Th Connction twn Spctroscopy nd Quntum Mchnics On of th postults of quntum mchnics: Th stt of systm is fully dscrid y its wvfunction, Ψ( r1, r,..., t) whr r 1, r, tc. r th coordints of th constitunt prticls of th systm t tim t. For our purposs, w will focus on th tim-indpndnt wvfunction, r, r 1,...) (,..., t) = ( r, r,...) θ ( ) Ψ( r, r 1 1 t Th nrgy of systm is dscrid y th timindpndnt Schrödingr qution: H = E whr th Hmiltonin oprtor, H, is md up of oprtors dscriing th kintic, T, nd potntil, V, nrgy of th systm, H = T + V.

2 For instnc, considr th quntum mchnicl dscription of th simpl hrmonic oscilltor: T = h m x 1 V = k( x V(x) x ) 0 x x H = T + V H = E Th solution to th Schrödingr qution givs ris to diffrnt wvfunctions tht dscri th nrgy of th systm. Th rquirmnts of th wvfunction rsult in quntiztion of th nrgy lvls of th systm: E v = hν ( v + 1 ) ν = 1 π k m E v v v = 4 v = 3 v = v = 1 0 x v = 0 x

3 Th wvfunction,, is spcifid y st of lls clld quntum numrs (in this cs, v). Th systm is in sttionry stt. Quntum Mchnics: - thorticl dscription of th systm - givs quntizd nrgy lvls nd slction ruls S pctroscopy: - n xprimntl msurmnt of th trnsitions twn nrgy lvls ( spctrum) E v v = 3 v = v = 1 v = 0 0 x

4 Osrvls nd trnsition proilitis An osrvl is msurl proprty of th systm. Osrvls r dscrid y ignvlu qutions: Ωˆ = ω Th mn vlu of n osrvl is clcultd from: Ω = * Ωˆ. dτ whr Ω is trmd th xpcttion vlu of th msurmnt th vrg vlu of lrg numr of msurmnts. In spctroscopy, w r intrstd in trnsitions twn lvls nd. Th quntum mchnicl clcultion of th intnsity of trnsition twn nd dpnds on th trnsition dipol momnt: μ = ˆ μ dτ whr th trnsition intnsity is * μ

5 Th influnc of symmtry on spctrosco py Th trnsition dipol momnt, μ, my ithr zro or non-zro: μ =0 th trnsition is foriddn μ 0 th trnsition is llowd S ymmtry considrtions llow us to dtrmin th slction ruls for trnsitions. Symmtry considrtions lso indict th dgnrcy of nrgy lvls: E.g.: How dos th potntil nrgy chng for vrious orinttions of prlllpipds of diffring symmtry? < < c = < c = = c E c c c c c c c 0 c c

6 Th Born-Oppnhimr Approximtion Considr molcul with nucli ( n) nd lctrons (): Th Hmiltonin for th systm ccounts for ll contriutions to th nrgy of th systm: H = T T + V + V + V + n n nn kintic nrgy of lctrons nd nucli potntil nrgy from lctron-lctron, lctron-nucli, nd nucli-nucli intrctions Compr th mss of n lctron (m ) to tht of nuclus (m n ): m n >> m W cn simplify th nlysis of molcul y trting th nucli s sttionry whil llowing th lctrons to mov in potntil dtrmind y th positions of th nucli. This is trmd th Born-Oppnhimr pproximtion.

7 Now w cn solv th Schrödingr qution for th lctrons lon. H = E Using th Born-Oppnhimr pproximtion, w cn sprt th wvfunction of th molcul into n lctronic wvfunction nd nuclr wvfunction: ( q, Q) = ( q, Q) ( Q) n whr q r th coordi nts of th lctrons nd Q r th nucli coordints. Th totl nrgy of th molcul, E, cn lso sprtd into lctronic, E, nd nuclr, E n, contriutions to th nrgy: E = E + E n

8 W cn lso fctoris th nuclr wvfunction into virtionl wvfunction nd rottionl wvfunction: with = n n vi vi E = E + rot E rot In summry, w trt th systm wvfunction s sprtd into diffrnt, indpndnt, wvfunctions with ssocitd nrgis = vi rot whr th nrgy is th sum of th nrgis of th individul motions E = E + E + vi E rot W will thrfor trt lctronic, virtionl, nd rottionl motions nd thir ssocitd spctr sprtly.

9 first xcitd lctronic stt V(r) virtionl nrgy rottionl nrgy r ground lctronic stt Comprison of nrgis for diffrnt typs of trnsitions: Excitd stt Spctrl Enrgy rgion [cm -1 ] Rottionl microwv Virtionl IR Elctronic visil/uv > 10000