6. Molecular structure and spectroscopy I

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1 6. Molecular structure and spectroscopy I 1 6. Molecular structure and spectroscopy I 1 molecular spectroscopy introduction 2 light-matter interaction

2 6.1 molecular spectroscopy introduction 2 Molecular energy domains Born-Oppenheimer approximation Ĥ Ĥe + Ĥn { Ψ ψφ ψe Φ E V + E n E e + E n Ignoring nuclear spin and transforming to centre-of-mass system Ĥ n Ĥv + Ĥr + Ĥt CM transf Ĥv + Ĥr Ĥ Ĥe + Ĥv + Ĥr { Ψ ψe ψ v ψ r E E e + E v + E r For EM transition between molecular states hν = E = E e + E v + E r E e E v E r

3 6.1 molecular spectroscopy introduction 3 Molecular spectral domains overview

4 6.1 molecular spectroscopy introduction 4 Molecular spectra physical origin Spectra are associated with primarily varying electric or magnetic dipole moments Radiowave region Flipping of nuclear or electron spin magnetic moments interacting with the magnetic component of the EM field Microwave region Rotational motion requires permanent molecular electric dipole moment

5 6.1 molecular spectroscopy introduction 5 Molecular spectra physical origin Infrared region Vibrational motion requires varying molecular electric dipole moment

6 6.1 molecular spectroscopy introduction 6 Molecular spectra physical origin Visible and ultraviolet Rearrangement of valence electronic charge density in the interaction with the electric component of the EM field

7 6.1 molecular spectroscopy introduction 7 Molecular spectra physical origin Raman spectra are different caused by varying molecular polarisability. The total radiated energy from an oscillating dipole, ignoring permanent dipoles, I = 2 ( ) 2 d2 P ; P = αe 3c 2 dt 2 0 sin(ωt) µ ind There are two main contributions vibration α α 0 + α 1v sin(ω v t) rotation α α 0 + α 1r sin(2ω r t) (anisotropic α) (Product to sum :sin(a) sin(b) = 1 {cos(a b) cos(a + b)}) 2 P = α 0 E 0 sin(ωt) + α 1vE 0 {cos[(ω ω v )t] cos[(ω + ω v )t]} 2 α 0 E 0 sin(ωt) + α 1rE 0 2 {cos[(ω 2ω r)t] cos[(ω + 2ω r )t]} Rayleigh Stokes anti-stokes

8 6.1 molecular spectroscopy introduction 8 Molecular properties from spectroscopy

9 6. Molecular structure and spectroscopy I 9 6. Molecular structure and spectroscopy I 1 molecular spectroscopy introduction 2 light-matter interaction

10 6.2 light-matter interaction 10 Basic processes single -photon multiphoton The lifetime of these virtual states in multi-photon processes is important

11 6.2 light-matter interaction 11 Einstein coefficients For non-degenerate two-level (N=2,1) system at thermal equilibrium ( ν ν/c) noting that absorption ( B 12 ) and stimulated emission ( B 21 ) depend on the radiation density, ρ( ν 21 ), while spontaneous emission ( A 21 ) does not, the rate of change of particles in state N 2 is: equil N 2 = N 1 B 12 ρ( ν) N 2 B 21 ρ( ν) N 2 A 21 = 0 A 21 ρ( ν) = N 1 N 2 B 12 B 21 N 1 = e E1/kBT N 2 e = E 2/k B T e(e 2 E 1 )/k B T = e hν/k BT ρ( ν) Planck = 8πh ν 3 21 e hc ν/k BT 1 = A 21 e hν/k BT B 12 B 21 [ B 12 = B 21 A 21 = 8πh ν 3 B 12

12 6.2 light-matter interaction 12 Beer-Lambert law For a system of N 1 molecules/m 3 in N 1 and N 2 molecules/m 3 in N 2. A flux of photons, F = I 0 /hν (photons/m 2 s) from left enters the system, and can be absorbed or stimulate emission only. What is F after a distance l? ρ( ν) = I 0 /c = h νf N 2 = B 21 h νfn 2 + B 12 h νfn 1 σf N ; N = N 1 N 2 Change in F in passing element dx F df df = σf Ndx F 0 F = σ N ( ) ( ) F I ln = ln = σ Nl F 0 I 0 l 0 dx I = I 0 e σ Nl

13 6.2 light-matter interaction 13 Time-dependent perturbation theory i h t Ψ = ĤΨ Semiclassically quantum two-level system interacting with classical EM field Split Ĥ in time dependent Ĥ and independent Ĥ parts Ĥ = Ĥ + Ĥ i h t Ψ = [Ĥ + Ĥ ]Ψ then, without Ĥ i h t Ψ = ĤΨ Ψ = n ψ n e ie nt/ h = n ψ n e iω nt where Ĥψ n = E n ψ n For the perturbed system, sol n. is a linear combination: Ψ(t) = n a n (t)ψ n e iω nt = n a n ψ n e iω nt Plug this into the TDSE: i h n a n ψ n e iωnt = Ĥ n a n ψ n e iω nt And multiply by n ψ ne iω nt as appropriate and integrate over all space, for a two-state system (ψ 0,1 ) where ψ 0 Ĥ ψ 0 = ψ 0 Ĥ ψ 0 dτ

14 6.2 light-matter interaction 14 TDPT electric dipole approximation In the electric dipole approximation λ l system Ĥ = µ z E z cosωt eze z cosωt Ĥ is an odd function, since µ = ez ψ n 2 = even, ψ n Ĥ ψ n = odd, ψ n Ĥ ψ n = 0 The solutions reduce to: a 0 = ia 1 h M 01E z e iω10t cosωt, a 1 = ia 0 h M 10E z e iω10t cosωt M 01 = M 10 = ψ 1 µ ψ 0 Weak field, so a 1 0, a 0 1. a 1 = ie zm 10 ( ) cos(ωt)e iω 10 t = ie zm 10 h 2 h (e i(ω 10+ω)t + e i(ω 10 ω)t ) Rotating Wave Approximation, terms in Ĥ which oscillate rapidly are neglected while slow oscillations kept. (ω 10 +ω) (ω 10 ω): RWA a 1 ie zm 10 2 h ei(ω 10 ω)t = ie zm 10 2 h e i t ; = ω ω 10 a 1 = t 0 a 1 = E zm 10 2 h a 1 dt = ie zm 10 2 h ( e i t 1 ) t 0 e i t dt transition dipole moment M mn { selection rules line intensities

15 6.2 light-matter interaction 15 TDPT Einstein coeffients Transition probability is given by a m (t), in principle. P mn = a m (t) 2 = E2 z 4 h 2 2 M mn 2 e i t 1 2 P mn = E2 z h 2 M mn 2 sin2 [(ω ω mn )t/2] (ω ω mn ) 2 However, monochromatic short times. Remember E t h ω t 1 integration over ω required. With broadband radiation density ρ = ε 0 Ez 2 /2 P mn = a m (t) 2 dω = 2 ε 0 h 2 M mn 2 = 2 ε 0 h 2 M mn 2 ρ(ω mn ) = 1 ε 0 h 2 M mn 2 ρ(ω mn )πt ρ(ω) sin2 [(ω ω mn )t/2] dω (ω ω mn ) 2 sin 2 [(ω ω mn )t/2] dω (ω ω mn ) 2 The absorption rate per molecule is then dp mn dt = π ε 0 h 2 M mn 2 ρ(ω mn ) = 1 d(n m /N) 3 dt = 2π 3 B mnρ(ω mn ) B mn = 2π2 3ε 0 h 2 M mn 2 A mn = 16π3 ν 3 3ε 0 hc 3 M mn 2