Molecules and electronic, vibrational and rotational structure

Transcription

1 Molculs and ctonic, ational and otational stuctu Max on ob 954 obt Oppnhim Ghad Hzbg ob 97 Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs

2 Hamiltonian fo a molcul h h H i m M i V i fs to ctons, to li; Potntial ngy tms: (, on-oppnhim: th diati of ctonic wa function w..t la coodinats is small: ψ 0 Z ( Z Z V, ucli can b considd stationay. 4πε i i > 0, 0 4πε 4πε i> j 0 ij Thn: ψ χ ψ χ ssum that th wa function of th systm is spaabl and can b wittn as: Spaation of aiabls is possibl. Ψmol ( i, ψ ( i; χ( Inst sults in th Schöding quation: ssumd that th ctonic wa function ψ ( i; can b calculatd fo a paticula H molψ χ molψ χ Thn: iψ ( i; χ( χ( iψ ( i; ψ χ ψ χ ( ψ ( χ ψ Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs

3 Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs mol total 0, 0 0 mol Ψ Ψ > > M Z Z Z m H i i j i ij i i χ πε ψ ψ πε πε χ h h Spaation of aiabls in th molcula Hamiltonian Th wa function fo th ctonic pat can b wittn spaaty and sold ; consid this as a poblm of molcula binding. ( ( Z m i i i i j i ij i i h ; ; 4 4, 0 0 ψ ψ πε πε > Sol th ctonic poblm fo ach and inst sult in wa function. This yids a wa quation fo th la motion: total 0 ( 4 χ χ πε Z Z M > h

4 Schoding quation fo th la motion Th pious analysis yids: h M > Z 4πε 0 Z ( χ total χ This is a Schöding quation with a potntial ngy: V Z Z ( ( > 4πε 0 Typical potntial ngy cus in molculs la pulsion chmical binding ow ty to find solutions to th Hamiltonian fo th la motion h M χ ( V ( χ ( χ ( Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs

5 Quantizd motion in a diatomic molcul Quantummchanical two-paticl poblm Tansf to cnt-of-mass systm M M M M Singl-paticl Schöding quation h Δ χ ( V ( χ ( χ ( Consid th similaity and diffncs btwn this quation and that of th H-atom: - intptation of th wa function - shap of th potntial Laplacian: Δ sinθ sinθ θ θ sinθ φ ngula pat is th wl-know quation with solutions: ngula momntum opatos Sphical hamonic wa functions! Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs

6 ngula momntum in a molcul Solution:, M h ( z, M hm, M with 0,,,3... M,,...,, M nd angula wa function, M YM ( θ,φ Hnc th wa function of th molcul: (, θ, φ Ξ( Y ( θ φ χ, M duction of molcula Schöding quation h V ( χ (,ot χ ( Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs

7 ignngis of a igid oto igid oto, so it is assumd that constant Choos: V ( V ( 0 ll diats yid zo Inst in: h V ( χ (,ot χ ( (,otχ( So quantizd motion of otation: ( ot h ( With th otational constant χ Dduc fom spctoscopy isotop ffct Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs

8 Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs Vibational motion ( ( (,ot V χ χ h on-otation: 0 Inst : ( Q ( Ξ ( ( ( Q Q V d d h Mak a Taylo sis xpansion aound ρ-... ( ( ρ ρ d V d d dv V V 0 ( V by choic 0 d dv at th bottom of th wl Hnc: ( ( k V hamonic potntial

9 Vibational motion d h dρ kρ Q( ρ Q( ρ So th wa function of a ating molcul smbls th -dimnsional hamonic oscillato, solutions: Q ( ρ / α! π / 4 / 4 xp αρ H ( α ρ with: α ω h and ω k ngy ignalus: hω Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs

10 Fin dtails of th oational motion Cntifugal distotion: ot ( D ( nhamonic ational motion ω ω x... Dunham xpansion: k l Ykl k, l ( Vibational ngis in th H -molcul Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs

11 ngy ls in a molcul: gnal stuctu Ψ J 0 oational stuctu supimposd on ctonic stuctu Ψ J 0 Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs

12 adiati tansitions in molculs Th dipol momnt in a molcul: i Z i In a molcul, th may b a: - pmannt o otational dipol momnt - ational dipol momnt 0 d d ρ d d ρ In atoms only ctonic tansitions, in molculs tansitions within ctonic stat Ψ (, ψ ( ψ ( mol i i; Dipol tansition btwn two stats if if Ψ' Ψ " dτ Two diffnt typs of tansitions ' " ( ψ ψ d ' ψ " ' ψ ' ψ ψ d ψ ( ψ ' ' ψ ψ " " " ψ ψ d d " dτ lctonic tansitions oational tansitions Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs

13 Th Fanck-Condon pincipl fo ctonic tansitions in molculs st tm: ψ if ' " ' ( ψ d ψ ψ d " Intnsity of ctonic tansitions I if ψ ψ d ' " Only contibutions if (paity sction ul ψ ' ψ " Fanck-Condon appoximation: Th ctonic dipol momnt indpndnt of intla spaation: Hnc M if ' " ( ψ ψ d ' " ( ψ ψ d M Intnsity popotional to th squa of th wa function oap Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs

14 otational tansitions in molculs nd tm in th tansition dipol if Ψ Ψ ' M ' M Pojction of th dipol momnt cto on th quantization axis. Fo otation tak th pmannt dipol. if ' 0 sinθ cosφ 0 sinθ sinφ cosθ 0 0 Y ' M ' Y ' 0 M ' m Y m M 0 Y m dω M Sction uls Δ ± ΔM 0, ± Puy otational spcta L ngis: F ( D ( Tansition fquncis: F ( ' F ( " [ '( ' "( " ] D ' ( ' " ( " [ ] Gound stat with and xcitd bsoption in otational ladd: 3 abs ( " 4D ( " spacing btwn lins ~ Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs

15 otational spctum in a diatomic molcul Gound stat with and xcitd bsoption in otational ladd: 3 abs ( " 4D ( " spacing btwn lins ~ In an absoption spctum: -lins In an mission spctum: P-lins Homola molcul Z Z ( 0 Fo pmannt dipol o otational spctum λ Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs

16 Vibational tansitions in molculs nd tm in th tansition dipol if ' " ' d d ψ ψ ψ ψ " Within a ctain ctonic stat: ' " ψ ψ ψ ψ d ' " Lin intnsity: if Pmannt and inducd dipol momnts; 0 d d ρ d d ρ Pmannt dipol dos not poduc a ational spctum if Wa functions fo on ctonic stat a othogonal. if d d ρ ρ Dipol momnt should ay with Displacmnt ational spctum Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs

17 Vibational tansitions in molculs Pmannt and inducd dipol momnts; aρ bρ if Fist od: th ating dipol momnt if ρ In th hamonic oscillato appoximation; n ρ k Q n ( ρ ρq ( ρ dρ h n n δk, n δk ω k, n Sction uls: Δ ± Homola molcul Z Z ( 0 Fo all diatis o ational dipol spctum High od tansitions (otons fom: - anhamonicity in potntial - inducd dipol momnts Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs

18 oational spcta in a diatomic molcul L ngis: F T G( F ( with: ( D ( ( G ω ω x Tansitions fom (gound to (xcitd σ with ( σ F ( ' F ( " ' 0 ( " σ0 G( G th band oigin; th otationlss tansition (not always isibl Tansitions -banch ( nglct D σ σ 0 3 Fo ( ( σ σ ( 0 ' P-banch ( - nglct D σ P σ ( ( 0 Fo σ σ ( 0 ' Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs

19 oational spcta σ σ P σ σ 0 3 ( ( ( ( 0 Mo pcisy spacing btwn lins: σ σ ( σ ( 3 < P( σ P( > ν' -banch ν" 0 σ0 P P P P-banch " If, as usual: < otational constant in xcitd stat is small. Spacing in P banch is lag and had fomation in -banch Spacing btwn (0 and P( is 4 band gap Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs

20 xampl: oational spctum of HCl; fundamntal ation Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs

21 xampl: oational spctum of H; fundamntal ation Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs

22 oonic spcta Ψ Vibations gond by th Fanck-Condon pincipl otations gond by angula momntum sction uls T J 0 Tansition fquncis T ' T" T ' T G'( F '( ' T" T G"( F "( " and P banchs can b dfind in th sam way σ σ 0 3 ( ( Ψ σ P σ ( ( 0 T J 0 Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs

23 and had stuctus and Fotat diagam σ σ P σ σ 0 3 ( ( ( ( 0 Dfin: m fo th -banch m fo th P-banch thn fo both banchs: σ σ ( m ( 0 m if: < σ σ 0 αm βm paabola psnts both banchs -no lin fo m0 ; band gap - th is always a band had, in on banch Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs

24 Population distibutions; ations Pobability of finding a molcul in a ational quantum stat: P(/P(0 P( Z ( / kt ( / kt ω ( / kt oltzmann distibution ot: not always thmodynamic quilibium Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs

25 Population distibutions; otational stats in a diatomic molcul Pobability of finding a molcul in a otational quantum stat: ( J P( J (J ' Z ot J ' (J ot / kt ot / kt J ( J DJ ( J HCl T300 K Find optimum ia dp( J 0 dj Lctu ots Stuctu of Matt: toms and Molculs; W. Ubachs