Introduction to Vibrational Spectroscopy

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1 Introduction to Vibrational Spectroscopy Harmonic oscillators The classical harmonic oscillator The uantum mechanical harmonic oscillator Harmonic approximations in molecular vibrations Vibrational spectroscopy Normal modes Elementary vibration types Transition moments Selection rules Tools to calculate how energy levels are populated The euipartition theorem The Boltzmann distribution The harmonic oscillator m F m F x x F F = x = mx && F m Hooe s law Newton s nd law x = x cos( ωt + ϕ) 0 && x = ω x cos( ωt + ϕ) = ω x 0 identify = + m m m mx && = x General solution ω = m ω = For diatomic systems, use the reduced mass: For polyatomic systems, use effective mass and generalized vibration coordinates (later). m

2 Quantum mechanical oscillators x V ( x) = = mω x F dv = = x dx The Schrödinger euation: Ĥψ = Eψ The Hamiltonian (total energy operator) for a harmonic oscillator: ˆ ˆ h d x H = E + V ( x) = + m dx h d ψ ( x) x + ψ ( x) = Eψ ( x) m dx ψ 0 when x ± Sip the maths, (see handout) jump straight to to the solutions! Solutions to the Schrödinger e. Normalization constant y / ψ ν = ν ν Hermite polynomials ( x) N e H ( y) Gaussian, ensuring that ψ 0 as x N ν = y ν = 0,,,3... ν α π ν! = x / α 4 α h = m The Hermite polynomials are defined by a recursion formula: yhν = ν Hν + Hν + H 0 = H = y H = 4y 3 H3 = 8y y etc. E ν Energy eigenvalues: ω = = hω ν + ν = 0,,,3... m

3 The wavefunctions Energy eigenvalues: E ν = hω ν + ν = 0,,,3... Quantized and euidistant energy levels, separated by Zero-point energy hω Tunnling into forbidden region Bohr s correspondence principle; classical behaviour recovered at high uantum numbers ν = 0 hω hω hω Probability distributions (Cf. the classical oscillator, where the probability is greatest at the turning points.) The harmonic approximation in vibrating molecules Can we justify the harmonic approximation for interatomic potentials? A Taylor expansion of V(x) around x = r e gives dv ( x) d V ( x) x V ( x) = V ( re ) + x dx dx x= re x= r e d V ( x) V ( x) dx x= r e x but V ( x ) = x = d V dx x= r e d V ( x) Mathematically, is proportional to the curvature of V(x), dx thus large steep potential and stiff spring (bond).

4 Anharmonic approximations Real interatomic (generalized vibrational) potentials are not harmonic! Inclusion of the 3rd order term in the expansion of V(x) gives a better approximation to the potential energy: 3 3 d V x d V x V ( x) + dx dx 6 3 x= r x= r e e A more commonly used potential function is the Morse potential: Converging energy levels! ( ( x x ) ) e V ( x) = D e α E ν ω = h ω = hω ν + ν + 4D m ν = 0,,,3... Harmonic and anharmonic levels in HCl (g) Harmonic potential vs. Morse potential

5 Vibrational spectroscopy Energies for transitions between vibrational states are found in the infrared region. In vibrational spectroscopy, energies are measured in wavenumbers, with the unit reciprocal centimeter, cm -. ν% v % ν = = c λ Now freuency, not uantum number! E = hω = hν = hc% ν - ν [Hz] = ν%c [m ] ν [cm ] = 4-0 λ [ µ m] Mear-IR - ν [cm ] E [mev] Mid-IR Far-IR λ [ µ m] Elementary vibration types The freuencies of these vary according to ν > δ > γ > τ In large molecules (polymers) combined vibrations are commonly labeled separately, e.g. for bending vibrations in CH groups: Rocing, ρ Twist, t Wagging, ω

6 Vibrational modes in CH and CH 3 (The arrows length s indicate relative motion amplitudes) Vibrations in polyatomic molecules Normal modes The spectroscopically accessible vibrations in a molecule are normal modes: Independent, synchronous movements in a group of atoms that can be excited without exciting any other normal mode (NM).. Each normal mode is an independent harmonic oscillation, so that E = ν + hc% ν % ν = π c m. Each NM has a characteristic freuency. 3. The freuency of a NM is mainly determined by the involved atoms. 4. An arbitrary motion in a molecule can always be expressed as a linear combination of normal modes. 5. Momentum and angular momentum are conserved, and no net translation or rotation of the molecule is involved in a NM. 6. There are methods to systematically determine the NMs of a molecule! This is what maes it possible to identify atomic species and functional groups in molecules by vibrational spectroscopy! Effective mass and stiffness, is a generalized coordinate.

7 Normal modes in amides In practice, normal modes (i.e. experimentally accessible vibrations) are combinations of different vibration types cm - Primarily C=O stretching, with contributions from N H bending and C N stretching cm - Primarily N H bending, with some contribution from C N stretching. Amide III: About 30 cm - from N H bending and C N stretching. DeFlores et al., J. Phys. Chem. B 0, 8973 (006) Vibrational modes in polyatomic molecules The number of modes in a molecule with N atoms is: 3N 6 in general 3N 5 for linear molecules Thus more atoms gives more complex spectra!

8 Water and CO in IR spectra Absorbance Water Peptide Rotational transitions combined with the stretching vibrational transitions. CO The details of the water vib-rot spectrum will also vary with the temperature! Wavenumber (/cm) Symmetric stretching, ν Asymmetric stretching, ν Bending, ν 3 HO (g) 3657 cm cm cm - HO (l) 3490 cm cm cm - HO (s) 377 cm - Water must be avoided! Vibrational modes in water Symmetric stretching, ν Asymmetric stretching, ν Bending, δ 3 HO (g) 3657 cm cm cm - HO (l) 3490 cm cm cm - HO (s) 377 cm - DO (g) 77 cm - Isotope exchange can be used to move or identify specific vibrations! DO (l) 67 cm cm - 78 cm - In condensd phases intermolecular coupling (e.g. via hydrogen bonding) introduce additional modes which complicate spectra; overtones, combination bands, torsional and cluster vibrations etc. Rocing modes In ice or H-bonded environments x y z

9 Identification of unnown substances Carbonyls on next slide! Carbonyls O R C R R, R aliphatic substituents

10 Total integrated absorption: IR intensities Integration Lambert-Beer s law: ( ) di = α( ν ) ICdl I = I e α ν α ( ν I ) ν ln Cl I ν 0 A = d = d band band 0 C Total absorption of a vibration can be obtained via: Einstein s transition probability: B = const µ if if A = 0N A hν if Bif c µ = ψ µψ is the transition moment * if i f Calculation of the transition moment yields selection rules (permitted and forbidden transitions), and probabilities for specific transitions. 0 ν transitions in the harmonic oscillator ψ zψ * 5 0 ψ zψ * 3 0 For these wavefunctions (and all other with ν > 5) the areas above and under the horizontal axis cancel each other ψ zψ * 0 z

11 The euipartition principle Each degree of freedom has, on average, the thermal energy B T/. Degree of freedom mode of motion, e.g. vibration, rotation, translation... E T T E = hν h = B B ν = = { T 300K} = = 0 Hz T = 4, x 0 - J = 6 mev = 0 cm - Mid-IR range: cm - most vibrational modes are in their ground states at room temperature (RT). Normally in ground state at RT. BT Significant fraction in excited states at RT. When the differences between energy levels << B T uantum mechanical results approach those of classical physics (continuous spectra). The Boltzmann distribution What is the probability of finding a particular vibrational mode in an excited state? Consider N identical systems (e.g. vibrating molecules ) with energy levels ε i, and where n i is the number of systems found with energy ε i. (Note that here ε 0 0, thus zero-point energy is ignored!) At euilibrium, the total energy is distributed over the systems according to the Boltzmann distribution, i.e. the fraction of systems in state i is: 0 E ε 4 ε ε 3 ε ε 0 p i n e e = = = εi / T εi / T i εi / T N e i i / T = e ε is the molecular partition function (a measure of the number of i thermally accessible states)

12 The Boltzmann distribution with euidistant energy levels For euidistant energies, separated by ε, the relations can be simplified: ε ε Using that We can write + x + x + x +... = for x < x 3 i / T = e ε = i ε / T ε / T 3 ε / T = + e + e + e +... = e ε / T giving the populations directly as: n N e εi / T i εi / T pi = = = e e ε / T ( ) Distribution between states

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