# V-1 V. Electronic Spectroscopy. Eigen value: lowest U el is the ground state by electronic dipole selection rule;

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1 V-1 V. Electronc Spectroscopy What we have done so far s use B O approxmaton to elmnate φ el (q,q) For example dd datomc, put asde φ el and focused on vbratonal & rotatonal soluton Now must consder electronc part of molecular state also Descrbe transton φ el χ υ φ el χ vb φ el has a symmetry -- wave/fct are soluton to H el Change electronc state φ el Egen value: lowest U el s the ground state by electronc dpole selecton rule; consder datomc result: ψ µ ψ 0 ψ = φ el χ vb, ψ = φ χ, µ = µ(r,r) = µ(r) + µ(r) Note: these look at nternal coordnates must ntegrate out lab orentaton: dθ dφ dχ yelds J = 0, ±1 S = 0 no effect on spn expand dpole, nclude vbratons: φ el χ vb µ(r) + µ(r) φ el χ vb separate ntegrals = χ φ µ(r) φ χ + φ φ χ µ(r) χ orthogonal-ntegrate = P(R) = 0 electron coordnates due to φ parametrc unless φ = φ' then get pure dependence on R vbraton / rotaton transton = χ P(R) χ Result: P(R) = φ µ(r) φ 0 φ, φ must be dfferent symmetry for datomcs: µ ~ + u, Π u to be seen Datomc: f φ grd = + g φ ex = + u, Π u to be seen (hetero drop g, u) Secondly: have rotaton / vbraton change J = 0, ±1 from vector nature of P(R) υ = 0, ±1, ±2, because χ, χ are solutons to dfferent potental surface Polyatomc: same thng just more coordnates φ el (q, Q,...) χ (Q ) µ (q) φ el χ = χ P(Q) When ntegrate out all the electron coordnates, q, result depends on nuclear coord, Q: P(Q) To observe a transton: P(Q) = φ ex (q, Q) µ(q) φ el(grd) (q,q) 0 so: Most molecules (especally all closed shell) have A 1g ground state (or equvalent) Γ φ el Γ µ Γ φ el ~ Γ xyz (for φ el = A 1g ) Allowed electronc transtons to excted states whose symmetry s that of dpole operator analogous to vbraton dpole moment must change on electronc exctaton φ µ φ 0 χ

2 V-2 1. Pure Rotatonal Spectra (same φ el + χ vb E el + E vb ) E = 2B(J + 1) ; J = ±1 n GHz 10's cm -1 regon 2. Rotaton Vbraton Spectra (same φ el E el ) E R = hν 0 + 2B(J + 1), E p = hν 0 2BJ J = ±1, J-s value n lower vbratonal state ν = ±1, most ntense n cm -1 regon 3. Electronc Spectra (all states dffer) E R = E el + hν 0 ex (υ + ½) hν 0 grd (υ + ½) + 2B(J + 1) E p = analogous to above J = ±1; +1 R, 1 P υ unrestrcted, most ntense are vertcal transtons (Frank-Condon approxmaton) ν 0 dfferent ground and excted states ~ K µ typcal: E > 10,000 cm -1 (no lmt, depends on φ el ) What about Rotaton-Vbraton? Each state of molecule s a new potental surface U el (R ) for vbratonal degrees of freedom. When change states the the χ s and Q s must change also expand P(R) the operator n χ P χ n Q (ground elect. state normal coord) µ P(R) = φ el µ φ el = φ el µ φ el e + φ el φel Q Q0 Q + hgher order terms Evaluate at equlb. Vbratonal dpole excte for φ el =φ el ' Equvalent to φ el (Q 0 ) f dfferent, gves electronc trans. -- vb. dependence The frst term s the largest and carres the requrements of symmetry selecton rule

3 V-3 Addng the vbratonal part: Π χ P χ ~ P(Q 0 ) e Π χ Π χ Const. Vbratonal overlap ntegral = P(Q 0 ) e χ χ snce ntegral over 3N-6 orthogonal coordnate need to adjust for 2 potental surfaces, dfferent Q each functon Q - makes product of ndependent ntegrals but χ expand n χ to get actual value Observed ntensty (absorbance or fluorescence) ~ ψ µ ψ 2 ~ [ P(Q) e ] 2 χ χ 2 Consder datomc ψ µ ψ 2 = P 2 χ υ χ υ 2 χ υ χ υ 2 = Frank-Condon factor ~ c j χ Ths tells us the relatve ntensty n each quantum of vbraton.e. Typcally υ = 0, grd state υ 0 2 χ 2 = c j 2 Franck Condon terms υ υ 0 2 = 1 gves fracton of ntensty of electronc trans n each υ f more than one υ populated: Total ntensty: ψ µ ψ 2 α υ = Eυ kt e e Eυ kt υ ~ P 2 υ,ν α υ υ υ 2 ~ P 2 Thermal populaton Polyatomcs same dea, just need υ υ 2 these wll dffer by symmetry (see below) populaton dstrbuted on all the modes Frst some Datomc examples of what you mght see n electronc spectra: Spectra wll depend on potental surfaces Spacng of levels wll gve the frequences of transtons Relatve shape of surfaces wll gve ntensty dstrbutons n vbratons Snce Dfferent surfaces, the B values can vary a lot, can get new knds of shapes n PQR branches

4 V-4

5 V-5

6 For excted states wth substantally changed geometry the rotatonal branches can collapse or even turn around, creatng a band head, whch you see below n the R branches. If the excted state s larger (less bound) the B value wll be smaller and the gan n B from the excted state wll not offset the loss from the ground state V-6

7 V-7 ν 2 1 Hot bands Polyatomcs same dea, just need υ υ 2 these wll dffer by symmetry (see below) populaton dstrbuted on all the modes Consder each coordnate Cases Allowed electronc transtons (1) Parallel potental surfaces U e = φ H el φ and U e = φ H el φ f surface parallel U (Q) = U(Q) + U (Q e ) constant so wave/fct solvng [T N + U (Q)] wll be equal to soluton of [T N + U(Q)] but energy wll hae constant offset U (Q e ) hence: χ υ orthonomal χ υ υ υ = δ υυ ψ µ ψ = P 2 α υ υ and each of these υ υ = υ wll be degenerate - so see only one transton However, f U e and U e have no dsplacement but dfferent curvature most ntensty wll be n 0 0 polyatomc can occur n asymmetrc modes

8 V-8 then f U slghtly dsplaced then progressons appear only n the total symmetrc mode, A 1g Fluorescence should be reverse form parallel surfaces overlap at 0 0 see progresson of weak sde bands υ = 0, 1, 2, 3, -- f strongest one at υ = 0 υ = 0 parallel surfaces -- thnk of as expand χ υ = and c υ cυ χ υ υ would be smaller and smaller = 0 Note: n polyatomc only A 1g modes can shft equlbrum (Q 0 ) [.e. f Q 0 shft n non-a 1g, the molecule loses symmetry] Example T d molecule U(BH 4 ) 4

9 V-9 Butadene bt trcky, π π* General case shft and change curvature vertcal transton Frank-Condon Prncple electronc transton happens very fast, nucle effectvely do not move maxmum ntensty for modes maxmum overlap χ υ χ υ maxmum when lobes overlap ntegral s area of product -- FC: υ υ 2 see progresson n totally symmetrc modes f anharmonc, frequency spacng gets smaller maxmum concdes wth largest υ υ Soluton data: Vapor spectra: approxmate example (actually forbdden, no 0-0) Benzene 1 A 1g 1 B 2u,

10 V-10 MnO 4 {mode must strongly affect electronc state, most ntense F-C terms): Note χ υ and χ υ now soluton to dfferent U, U, so not orthonomal f potental surface very offset, can have spectra wthout O O

11 V-11 Example I 2 Clearly measure of peak and shape nformaton on surface/states Frank-Condon profles McHale, excellent pctures 11.8, 11.9, 11.10, 11.11, 11.12

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