Configurations versus States; Vibronic Coupling

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Lynne Wheeler
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Confirations verss States; Vibronic Coplin C734b 008 C734b Spectroscopy-I Electronic Spectrm of Benzene The flly allowed A ---> > E transition is assined to the most intense transition which occrs at

1 Confirations verss States; Vibronic Coplin C734b 008 C734b Spectroscopy-I Electronic Spectrm of Benzene The flly allowed A ---> > E transition is assined to the most intense transition which occrs at 80 nm. The vibronically assisted A --> > B and A --> > B transitions are assined to the less intense bands at 00 and 60 nm, respectively. The spin-forbidden A --> 3 B is assined to the lowest enery and lowest intensity transition at 340 nm. C734b Spectroscopy-I

2 Why? From previos work on the MOs of benzene, we dedced that that its rond state electronic confiration was (a (e b α-β anti-bondin e α-β non-bondin e α+β bondin a α+β This is a closed shell confiration which satisfies the Pali Exclsion Principle. As sch the rond state wavefnction forms a basis for the totally symmetric IR ( A * This arment is eneral for closed shell atoms and molecles. C734b Spectroscopy-I 3 a a IR (a can be fond from: (e 4 can be viewed as (e (e. Consider e e D 6h E ' " C6 C3 C 3C 3C i S3 S6 σ 3σ 3σ h d v a a a = A e e e Redction yields: A A E Same arment holds for (t 6 confirations which can be viewed a (t (t (3t. C734b Spectroscopy-I 4

3 The first excited state of benzene is (a (e 3 (e The dobly occpied a orbital and one of the e orbital are closed so they transform as A. Only the sinly occpied orbitals need to be considered: Excited state can be written as (e (e Can show that in D 6h e e B B E This means that the confiration (e (e leads to 3 electronic states with B, B, and E symmetry. The only system where confirations states are those with only one electron. Next: examine the dipole moment operator. In D 6h (x,y transform accordin to E and z transforms accordin to A C734b Spectroscopy-I 5 Recall: ex Oˆ B B = = A Therefore need only find ot if A = E = B to be allowed ˆ ex A O forbidden forbidden B B A = E = B forbidden forbidden E E A = A = E A E allowed forbidden C734b Spectroscopy-I 6 3

4 A E only symmetry allowed transition; fond at λ ~ 80 nm Aside: what is the sperscript on the state labels? This is the spin mltiplicity S+. Althoh each electron has spin s = ½, the total spin of the system is S = 0. Therefore S+ =. These are called sinlet states If the electron flips its spin drin a transition (violation of ΔS = 0 selection rle, then the total spin of the system =. Therefore S+ = 3. These are called triplet states. C734b Spectroscopy-I 7 Not shown is a very weak band near 350 nm which is assined to made partially allowed throh spin-orbit coplin. 3 A E The transitions: A B and A B are symmetry-forbidden bt can be made allowed throh vibronic coplin Vibronic Coplin r r Ψ, R The electronic wave fnction a ( {} r bt also on the nclear coordinates { } depends not only on the electron coordinates R r. Here a are the electronic qantm nmbers. Since the mass of the electron << mass of the nclei, the electronic motion follows the nclear motion adiabatically. C734b Spectroscopy-I 8 4

5 Therefore, we can adopt the Born-Oppenheimer approximation and write the wave fnction as: Ψ a r r r (, R =ψ ( χ ( R a, R where v is a vibrational qantm nmber. a,v r The electronic wave fnction depends parametrically on the positions of the nclei. The electronic enery E a (R is calclated at a series of nclear displacements, R, which provides the potential enery V a (R for the vibrational motion. V a (R depends on the electronic state, a adiabatic potential. It is invariant nder symmetry operations. C734b Spectroscopy-I 9 Matrix element for vibratin molecle becomes: Ψ a' Dˆ Ψ a l k ψ χ j i Dˆ ψ χ l k, = 0 nless ( x y, x j i Since for the fndamental vibration i = and k one of the IRs (Q k to which the normal mode belons, the qestion redces to: j does l ( ( Q? x, y, z k To find the symmetry forbidden transitions that are vibronically allowed yo mst do a normal mode analysis. At this point it is only relevant to know that benzene has modes with B and E symmetry C734b Spectroscopy-I 0 5

In D 6h symmetry the dipole moment forms a basis for the representations: A E =, ( x, y z Since the rond state is A ' ( ( = B ( A E ψ e x, y, z = B = B E ( A E = B E Since B and E are normal modes A

6 In D 6h symmetry the dipole moment forms a basis for the representations: A E =, ( x, y z Since the rond state is A ' ( ( = B ( A E ψ e x, y, z = B = B E ( A E = B E Since B and E are normal modes A B ( nm ; A B ( 60nm 00 become allowed throh vibronic coplin C734b Spectroscopy-I Transitions which are symmetry forbidden bt vibronically allowed can be expected to be weaker than symmetry allowed transitions, and are enerally broadened (de to vibrational fine strctre 3 Note: there is an even weaker spin-forbidden transition near 35 nm: A B ( 35nm Note: C734b Spectroscopy-I 6

7 abel vibronic transitions as: ' " ' v v v v " Here,, nmber which normal mode is involved, v is the rond state vibrational qantm nmber and v is the excited state vibrational qantm nmber Hih resoltion spectrm of A B ( 60nm 6 means ν 6 which has E symmetry. means ν which is the rin breathin mode and has A symmetry 0 indicates the oriin line (electronic transition with no vibrations. Vibronic coplin also called the Herzber-Teller effect and represents a breakdown of the Born-Oppenheimer Approximation. C734b Spectroscopy-I 3 Electronic Spectrm of Ti(H O 6 3+ The rond state of this nd O h complex is t and its first excited state is e. This means that this is a d-d transition which is symmetry forbidden by parity. Still Ti(H O 3+ 6 has an absorption at 0,000 cm - (500 nm and a sholder jst to lower eneries. Why? C734b Spectroscopy-I 4 7

8 ψ ψ = e t = T T First examine ( ex ( in O h Does this direct prodct contain ( = T? x, y, z (Read off character table No! t e is symmetry (parity forbidden Next: examine ( ψ ex ( ψ ( x, y, z = E T T = A A E T T Can also show that normal modes for a M 6 complex form bases for: A E T T T Therefore parity forbidden transition can become vibronically allowed throh coplin with odd parity T and T normal modes C734b Spectroscopy-I 5 Note: as this is a d system, the spin mltiplicity is S+ = (½+ =. These are called doblet transitions T E acconts for the absorption band. Why is there additional strctre? The additional band is associated with a lowerin of the symmetry de to the Jahn-Teller effect. Here the symmetry of the O h complex is redced to D 4h by elonation of the axial liand bond lenths. C734b Spectroscopy-I 6 8

9 M Jahn-Teller Distortion M z O h D 4h Use physical reasonin and Character Tables to dedce how the d-orbitals split when there is a Jahn-Teller distortion (alon the z-axis as shown C734b Spectroscopy-I 7 (d z, d x-y e (d x-y (d z b a Jahn-Teller Distortion ( d xy b (d xz, d yz, d xy t (d xz, d yz e O h D 4h Three possible transitions possible in D 4h C734b Spectroscopy-I 8 9

10 First: e = e b symmetry forbidden Q = A E ( x, y, z in D 4h Similarly: e a = e symmetry forbidden e b = e symmetry forbidden Can show that the 5 normal modes of a D 4h complex transform as: vib = B B E A B A 3E Have to mix in vibrational modes to make the transition vibronically allowed. Which ones? C734b Spectroscopy-I 9 Examine: ( ψ ( ψ ( Q = E ( Q ex k k Does it contain ( x, y, z = component of dipole moment operator? = A E Recall: ( x, y, z A = E vibronically allowed B = E vibronically allowed E = A A B B vibronically allowed C734b Spectroscopy-I 0 0

11 Hihest enery transition: E B The third E B lies in the infrared ower enery sholder: E A C734b Spectroscopy-I

In the last lecture we have seen the electronic transitions and the vibrational structure of these electronic transitions.

In the last lecture we have seen the electronic transitions and the vibrational structure of these electronic transitions. Title: Term vales of the electronic states of the molecle Pae-1 In the beinnin of this modle, we have learnt the formation of the molecle from atoms. We have also learnt the moleclar orbital and the electronic