2 Learning outcomes from lecture 5 Be able to draw the wavefunctions for the first few solutions to the Schrödinger equation for the harmonic oscillatorBe able to calculate the energy separation between the vibrational levels for the harmonic oscillatorBe able to explain the dependence of this separation on the masses of the atoms and the strength of the bondBe aware of and be able to calculate the zero point energy of a vibrating moleculeBe able to draw the potential energy curve for a real molecule and explain why it is different to the harmonic potentialAssumed knowledgeLight behaves like an oscillating electromagnetic field. The electric field interacts with charges. Two charges separated in space represent a dipole moment which can interact with an electric field. Energy can only be taken or added to the electric field in units of hn (photons).
4 Revision: The Electromagnetic Spectrum Spectral rangel (nm)n (Hz)(cm-1)Energy (kJ/mol)SpectroscopyRadio~1 x 109~108~0.03~10-8NMR/ESRMicrowave~100,000~1012~30~10-2RotationalInfrared~1000~1014~3,000~103VibrationalVisible4-6 x 101414,000-25,0001 – 3x105ElectronicUltraviolet~1015~40,000~5x105X ray<100>1016>100,000>106Core electronic
5 Revision: Fundamental equations QuantitySymbolSI UnitCommon UnitEnergyEJkJ/molFrequencyns-1 or HzWavelengthlmnm or mmWavenumberm-1cm-1ConstantSymbolValueSpeed of lightc3.00 x 108 m/sPlanck constanth6.626 x Js
6 Classical absorption of light Because light is an oscillating electromagnetic field, it can cause charges to oscillate. If the charge can oscillate in resonance with the field then energy can be absorbed.Alternatively, an oscillating charge can emit radiation with frequency in resonance with the original oscillation.For example, in a TV antenna, the oscillating EM field broadcast by the transmitter causes the electrons in your antenna to oscillate at the same frequency.
7 Classical absorption of light What about a molecule as an antenna?How can we get oscillating charges in a molecule?+-+-
8 Classical absorption of light +-Rotating a permanent dipole causes an oscillation of charge
9 Classical absorption of light +-Vibrating a permanent dipole causes an oscillation of charge+-
10 The Quantum Harmonic Oscillator Solution of Schrödinger equation:But:n = 0,1,2,…Oscillation (vibrational) frequencyS.I. unitsNeed to know!G(v) & w in cm-1G(v) is the “energy” from the bottom of the well and w is the “harmonic frequency”
12 The Morse Anharmonic Oscillator all in wavenumber (cm-1) units:This one you DO have to know how to useIn spectroscopy, we tend to use the letter “vee” to indicate the quantum number for vibration. The vibration frequency is indicated by we in cm-1. When solving the general quantum mechanical problem we used the letter n, to minimize confusion with “nu”, the vibrational frequency in s-1.
13 The Morse energy levels Harmonic frequencyAnharmonicity constantV (r)wexe is usually positive, so vibrational levels get closer togetherHarmonic termAnharmonic termThe force constant for the AHO is the same as for the harmonic case:orr
14 Different levels of approximation… General AHO:Morse oscillator:Harmonic oscillator:The level of approximation to use depends on:i) the information you haveii) the information you need
15 The important equations! In wavenumber (cm-1) units:The ones you DO have to know how to use
16 Dissociation energy D0 De D0 = De – G(0) V (r) r This is the energy from the bottom of the well to the dissociation limit.We can estimate the bond dissociation energy from spectroscopic measurements!D0 = De – G(0)But we know about zero-point energy, therefore slightly less energy is required to break the bond.G(0)r(Zero-point energy)
17 vibrational coordinate Selection rulesAll forms of spectroscopy have a set of selection rules that limit the number of allowed transitions.Selection rules arise from the resonance condition, which may be expressed as a transition dipole moment:Upper statelower statemolecular dipolevibrational coordinatepsi(2) is the upper state wavefunction, psi(1) the lower state. mu(x) is the dynamic molecular dipole. mu(21) is the transition dipole.Selection rules tell us when this integral is zero
18 Harmonic Oscillator Selection Rules Dv=±1Dv=+1: absorptionDv=-1: emissionIf an oscillator has only one frequency associated with it, then it can only interact with radiation of that frequency.Selection rules limit the number of allowed transitions
19 Thermal populationAt normal temperatures, only the lowest vibrational state (v =0 ) is usually populated, therefore, only the first transition is typically seen.Transitions arising from v0 are called “hot bands”(Their intensity is strongly temp. dependent)Much of IR spectroscopy can understood from just these two results of the quantum harmonic oscillator:E = (v+½)hn and Dv = ±1
21 More problems with harmonic model…. What are these?COThey have already seen this many times, but ONLY the fundamental has been shown.
22 What are the new peaks? Three peaks… 2143 cm-1 4260 cm-1 6352 cm-1 v=4 These are almost 1 : 2 : 3v=2which suggests transitionsmight be0 10 20 3v=1v=0
23 Anharmonic oscillator (A.H.O.) selection rules: There are none!But!...Harmonic and anharmonic models are very similar at low energy, so selection rules of AHO converge on HO as the anharmonicity becomes less:A.H.O.A.H.O. selection rule:Dv=±1,±2, ±3Intensity gets weaker and weaker (typically 10× weaker for each)
24 Anharmonic oscillator (A.H.O.) A.H.O. selection rule:Dv= ±1,±2, ±3Dv = 1 : fundamentalDv = 2 : first overtoneDv = 3 : second overtone, etcNo, it is not a breakdown of the resonance condition. Continue the music analogy. If you pluck a string, or blow a wind instrument hard enough you can generate overtones (higher harmonics). Same thing with molecules. If the light is twice the frequency then harmonics can be generated.
25 Typical Exam QuestionConsider the infrared absorption spectrum of CO below.From the wavenumber measurements on the spectrum, assign the spectrum, hence determine the harmonic frequency, we (in cm-1) and the anharmonicity constant wexe (in cm-1).Estimate the bond dissociation, D0, for this molecule. There is no absorption below 2000 cm-1.Fundamental:2143 = G(1) – G(0),Overtone:4260 = G(2) – G(0)Draw AHO on blackboard and indicate transitions, G(v) and DG(v)
26 Using the spectra to get information… G(1)-G(0) = [(1.5)we – (1.5)2 wexe] - [(0.5)we – (0.5)2 wexe]2143 = we – 2wexe …(1)G(2)-G(0) = [(2.5)we – (2.5)2 wexe] - [(0.5)we – (0.5)2 wexe]4260 = 2we – 6wexe …(2)Two simultaneous equations (simple to solve)→ we = 2169 cm-1, and wexe = 13 cm-1
27 Using the spectra to get information… V (r)DeD0G(0)(Zero-point energy)89,400 cm-1 = 1069 kJ/molc.f. exp. value: kJ/molWhy the difference?Remember Morse is still an approx. to the true intermolecular potential. Still 2% error is pretty good for just 2 measurements!
28 Equations to know how to use… m m1 × m2m1 + m2
29 Learning outcomesBe able to manipulate and use the key equations given in the green box at the end of the lecture.Utilize the harmonic oscillator and anharmonic oscillator as a model for the energy level structure of a vibrating diatomic molecule.
30 Next lecture Week 11 homework The vibrational spectroscopy of polyatomic molecules.Week 11 homeworkWork through the tutorial worksheet and the practice problems at the end of the lectures and check your answers with those available onlinePlay with the “IR Tutor” in the 3rd floor computer lab and with the online simulations:
31 Practice Questions HBr b) H2 c) CO d) I2 1. Which of the following diatomic molecules will exhibit an infrared spectrum?HBr b) H2 c) CO d) I22. An unknown diatomic oxide has a harmonic vibrational frequency of ω = cm−1 and a force constant of 1607 N m −1. Identify the molecule.CO b) BrO c) NO d) 13CO3. As the energy increases, the vibrational level spacing for a harmonic oscillator is:increases b) decreases c) stays constant4. As the energy increases, the vibrational level spacings for a Morse oscillator usually:increase b) decrease c) stay constant5. As the energy increases, the vibrational level spacings for an anharmonic oscillator usually:
32 Practice Questions6. For a Morse oscillator the observed dissociation energy, D0, is related to the equilibrium vibrational frequency and the anharmonicity by the following expression:a) ωe2/4ωexe b) [ωe2/4ωexe]-G(0) c) (v+½)ωe+(v+½)2ωexe d) (v+½)ωe7. Which of the following statements about the classical and quantum harmonic oscillator (HO) are true (more than one possible answer here)?a) The classical HO frequency is continuous, whereas the quantum frequency is discrete.b) The classical HO has continuous energy levels, whereas the quantum HO levels are discrete.c) The classical HO depends on the force constant, but the quantum HO does not.d) The classical HO may have zero energy, but the quantum HO may not.e) The classical HO does allow the bond to break, whereas the quantum HO does.
33 Practice Questions8. Which of the following statements correctly describe features of the quantum Morse oscillator and energy levels associated with it?a) The vibrational energy levels get more closely spaced with increasing v.b) The vibrational energy levels approach a continuum as the dissociation energy is approached.c) The Morse oscillator and HO are nearly the same at very low vd) The Morse potential is steeper than the HO for r < ree) The Morse potential exactly describes the interatomic potential.