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Chemistry 2 Lecture 10 Vibronic Spectroscopy Learning outcomes from lecture 9 • Be able to qualitatively explain the origin of the Stokes and antiStokes line in the Raman experiment • Be able to predict the Raman activity of normal modes by working out whether the polarizability changes along the vibration • Be able to use the rule of mutual exclusion to identify molecules with a centre of inversion (centre of symmetry) Assumed knowledge Excitations in the visible and ultraviolet correspond to excitations of electrons between orbitals. There are an infinite number of different electronic states of atoms and molecules. Which electronic transitions are allowed? The allowed transitions are associated with electronic vibration giving rise to an oscillating dipole Electronic spectroscopy of diatomics • For the same reason that we started our examination of IR spectroscopy with diatomic molecules (for simplicity), so too will we start electronic spectroscopy with diatomics. • Some revision: – there are an infinite number of different electronic states of atoms and molecules – changing the electron distribution will change the forces on the atoms, and therefore change the potential energy, including k, we, wexe, De, D0, etc Depicting other electronic states Excited Electronic States 1. Unbound 2. Bound Ground Electronic State There is an infinite number of excited states, so we only draw the ones relevant to the problem at hand. Notice the different shape potential energy curves including different bond lengths… Ladders upon ladders… Each electronic state has its own set of vibrational states. De’ we’ Note that each electronic state has its own set of vibrational parameters, including: - bond length, re - dissociation energy, De - vibrational frequency, we De” we” Notice: single prime (’) = upper state double prime (”) = lower state re” re’ The Born-Oppenheimer Approximation The total wavefunction for a molecule is a function of both nuclear and electronic coordinates: (r1…rn, R1…Rn) where the electron coordinates are denoted, ri , and the nuclear coordinates, Ri. The Born-Oppenheimer approximation uses the fact the nuclei, being much heavier than the electrons, move ~1000x more slowly than the electrons. This suggests that we can separate the wavefunction into two components: (r1…rn, R1…Rn) = elec (r1…rn; Ri) x vib(R1…Rn) Total wavefunction = Electronic wavefunction at × each geometry Nuclear wavefunction The Born-Oppenheimer Approximation (r1…rn, R1…Rn) = elec (r1…rn; Ri) x vib(R1…Rn) Total wavefunction = Electronic wavefunction at × Nuclear wavefunction each geometry The B-O Approximation allows us to think about (and calculate) the motion of the electrons and nuclei separately. The total wavefunction is constructed by holding the nuclei at a fixed distance, then calculating the electronic wavefunction at that distance. Then we choose a new distance, recalculate the electronic part, and so on, until the whole potential energy surface is calculated. While the B-O approximation does break down, particularly for some excited electronic states, the implications for the way that we interpret electronic spectroscopy are enormous! Spectroscopic implications of the B-O approx. 1. The total energy of the molecule is the sum of electronic and vibrational energies: Etot = Eelec + Evib Evib Eelec Etot Spectroscopic implications of the B-O approx. • In the IR spectroscopy lectures we introduced the concept of a transition dipole moment: μ 21 (ri , Ri )μˆ 1 (rj , R j )drdR 0 * 2 |2 transition dipole moment upper state wavefunction lower state integrate dipole moment wavefunction over all operator coords. |1 using the B-O approximation: μ 21 2*vib ( Ri ) 2*elec (ri )μˆ 1elec (rj )1vib ( R j )drdR 2*vib ( Ri )μ( R) 1vib ( R j )dR 2. The transition moment is a smooth function of the nuclear coordinates. Spectroscopic implications of the B-O approx. μ 21 *vib 2 ( Ri ) *vib 2 ( Ri )μ( R) ( R j )dR |2 ( Ri ) ( R j )dR |1 μ0 *vib 2 *elec 2 elec vib ˆ (ri )μ 1 (rj )1 ( R j )drdR vib 1 vib 1 2. The transition moment is a smooth function of the nuclear coordinates. If it is constant then we may take it outside the integral and we are left with a vibrational overlap integral. This is known as the Franck-Condon approximation. 3. The transition moment is derived only from the electronic term. A consequence of this is that the vibrational quantum numbers, v, do not constrain the transition (no Dv selection rule). Electronic Absorption There are no vibrational selection rules, so any Dv is possible. But, there is a distinct favouritism for certain Dv. Why is this? Franck-Condon Principle (classical idea) “Most probable bond length for a molecule in the ground electronic state is at the equilibrium bond length, re.” Energy Classical interpretation: 0 1 2 3 R 4 5 The Franck-Condon Principle states that as electrons move very much faster than nuclei, the nuclei as effectively stationary during an electronic transition. Energy Franck-Condon Principle (classical idea) In the ground state, the molecule is most likely in v=0. 0 1 2 3 R 4 5 •The Franck-Condon Principle states that as electrons move very much faster than nuclei, the nuclei as effectively stationary during an electronic transition. The electron excitation is effectively instantaneous; the nuclei do not have a chance to move. The transition is represented by a VERTICAL ARROW on the diagram (R does not change). Energy Franck-Condon Principle (classical idea) 0 1 2 3 R 4 5 •The Franck-Condon Principle states that as electrons move very much faster than nuclei, the nuclei as effectively stationary during an electronic transition. The most likely place to find an oscillating object is at its turning point (where it slows down and reverses). So the most likely transition is to a turning point on the excited state. Energy Franck-Condon Principle (classical idea) 0 1 2 3 R 4 5 Quantum (mathematical) description of FC principle *vib vib μ 21 μ 0 2 ( Ri ) 1 ( R j )dR approximately constant with geometry Franck-Condon (FC) factor μ21 = constant × FC factor FC factors are not as restrictive as IR selection rules (Dv=1). As a result there are many more vibrational transitions in electronic spectroscopy. FC factors, however, do determine the intensity. Franck-Condon Principle (quantum idea) In the ground state, what is the most likely position to find the nuclei? 3 v Prob 2 2 1 Max. probability at Re (0) 0 v=0 0 1 2 R 3 4 Franck-Condon Factors If electronic excitation is much faster than nuclei move, then wavefunction cannot change. The most likely transition is the one that has most overlap with the excited state wavefunction. 2 1 v’ = 0 0 1 2 Wave number 3 4 v” = 0 Look at this more closely… Negative overlap in middle Positive overlap at edges overall very small overlap Negative overlap to left, postive overlap to right overall zero overlap • Excellent overlap everywhere Franck-Condon Factors 1 2 0 3 Wave number 4 Franck-Condon Factors v=10 Note: analogy with classical picture of FC principle! v. poor v=0 overlap Electronic Absorption There are no vibrational selection rules, so any Dv is possible. Relative vibrational intensities come from the FC factor μ21 = constant × FC factor Absorption spectrum of binaphthyl •Example of real spectra showing FC profile 16 17 15 18 19 20 21 22 14 23 13 24 12 25 11 26 27 28 10 9 (3) (4) (5) 30100 6 7 30200 8 30300 30400 30500 30600 -1 Wave number (cm ) 30700 30800 30900 Absorption spectrum of CFCl = CCl2 peaks (0,n,0) (0,n,1) (0,n,2) (1,n,0) 3 3 2 (0,n,0) (0,n,1) (0,n,2) 5 5 4 17000 18000 } 10 (0,0,1) hot bands 19000 20000 -1 Wave number (cm ) 21000 Unbound states (1) If the excited state is dissociative, e.g. a p* state, then there are no vibrational states and the absorption spectrum is broad and diffuse. Unbound states (2) Even if the excited state is bound, it is possible to access a range of vibrations, right into the dissociative continuum. Then the spectrum is structured for low energy and diffuse at higher energy. Some real examples… HI A purely dissociative state leads to a diffuse spectrum. Some real examples… I2 The dissociation limit observed in the spectrum! 0.25 I2 Absorbance 0.20 0.15 0.10 0.05 0.00 16000 18000 20000 -1 Wave number (cm ) 22000 Analyzing the spectrum… All transitions are (in principle) possible. There is no Dv selection rule Vibrational structure 0.25 I2 Absorbance 0.20 0.15 0.10 0.05 0.00 16000 18000 20000 -1 Wave number (cm ) 22000 Analysing the spectrum… v” 0 v’ 25 cm-1 18327.8 0 26 18405.4 0 27 18480.9 0 28 18555.6 0 29 18626.8 0 30 18706.3 0 31 18780.0 0 32 18846.6 0 33 18911.5 0 34 18973.9 0 35 19037.5 ( ( 2 1 1 G( v) v 2 we v 2 we xe How would you solve this? (you have too much data!) 1. Take various combinations of v’ and solve for we and wexe simultaneously. Average the answers. 2. Fit the equation to your data (using XL or some other program). Analyzing the spectrum… Etot Eelec Evib 2 Etot Eelec G( v) Eelec (v 1 2 we (v 1 2 we xe v” 0 v’ 25 cm-1 18327.8 0 26 18405.4 0 27 18480.9 0 28 18555.6 0 29 18626.8 0 30 18706.3 0 31 18780.0 0 32 18846.6 0 33 18911.5 0 34 18973.9 10 0 35 19037.5 Dissociation energy = 19950 cm-1 20000 -1 Wave number (cm ) 19500 19000 18500 Eelec = 15,667 cm-1 18000 we = 129.30 cm-1 wexe = 0.976 cm-1 17500 17000 20 30 40 50 v' 60 70 Learning outcomes • Be able to draw the potential energy curves for excited electronic states in diatomics that are bound and unbound • Be able to explain the vibrational fine structure on the bands in electronic spectroscopy for bound excited states in terms of the classical Franck-Condon model • Be able to explain the appearance of the band in electronic spectroscopy for unbound excited states The take home message from this lecture is to understand the (classical) Franck-Condon Principle Next lecture • The vibrational spectroscopy of polyatomic molecules. Week 12 homework • Vibrational spectroscopy worksheet in tutorials • Practice problems at the end of lecture notes • Play with the “IR Tutor” in the 3rd floor computer lab and with the online simulations: http://assign3.chem.usyd.edu.au/spectroscopy/index.php Practice Questions 1. Which of the following molecular parameters are likely to change when a molecule is electronically excited? (a) ωe (b) ωexe (c) μ (d) De (e) k 2. Consider the four sketches below, each depicting an electronic transition in a diatomic molecule. Note that more than one answer may be possible (a) Which depicts a transition to a dissociative state? (b) Which depicts a transition in a molecule that has a larger bond length in the excited state? (c) Which would show the largest intensity in the 0-0 transition? (d) Which represents molecules that can dissociate after electronic excitation? (e) Which represents the states of a molecule for which the v”=0 → v’=3 transition is strongest?