Spectroscopy for the Masses

Transcription

1 Spectroscopy for the Masses Robert J. Le Roy, Scott Hopkins, William P. Power, Tong Leung, Jake Fisher and John Hepburn Department of Chemistry, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada V(x) n = 6 n = 5 n = 4 n = 2 r n = 3 n = 4 e 2 V(r) = 4π ε r l = l =1 l = 2 n = 3 n = 2 n = 1 x L PiB R H n = 1 H atom l = 3 l = 4 l = 5 V(r) = ½ k (r r e ) 2 υ = 5 υ = 4 υ = 3 D e υ = 3 υ = 2 υ = 5 υ = 4 fundamental vibrational band (Stokes) Rayleigh line υ = 2 υ = 1 υ = r e r HO r e υ = 1 υ = Morse r S branch Q branch O branch pure rotation S branch (Stokes) pure rotation S branch (anti-stokes) fundamental vibrational band (anti-stokes) O branch Q branch S branch ν o ν

2 An Introduction to Atomic, Rotational, Vibrational, Raman, Electronic, Photoelectron, and NMR Spectroscopy A general science course for term 1B or 2A Why did we do this? Conventional curricula normally do not teach a Molecular Spectroscopy until year 3 or 4, and it usually presumes a full course in Quantum Mechanics as a pre-requisite.

3 An Introduction to Atomic, Rotational, Vibrational, Raman, Electronic, Photoelectron, and NMR Spectroscopy A general science course for term 1B or 2A Why did we do this? Conventional curricula normally do not teach a Molecular Spectroscopy until year 3 or 4, and it usually presumes a full course in Quantum Mechanics as a pre-requisite. Why was this a problem? This effectively restricted any broad understanding of spectroscopy to the relatively small numbers of students who specialize in Phys. Chem. Why was this a bad thing?

4 An Introduction to Atomic, Rotational, Vibrational, Raman, Electronic, Photoelectron, and NMR Spectroscopy A general science course for term 1B or 2A Why did we do this? Conventional curricula normally do not teach a Molecular Spectroscopy until year 3 or 4, and it usually presumes a full course in Quantum Mechanics as a pre-requisite. Why was this a problem? This effectively restricted any broad understanding of spectroscopy to the relatively small numbers of students who specialize in Phys. Chem. Why was this a bad thing? Students studying, and graduates working in, all areas of science will encounter or use some kind of spectroscopy in their studies or everyday work, and they really should have some understanding of the underlying physical principles. Our students in year-2 complained that they were seeing the same low-level dumbed-down (typically 2-3 lectures) introducing them to spectroscopic concepts in their intro, analytical, organic, inorganic courses, followed by a leap to a particular application, but never get a view of the overall subject. Year-1 courses on most areas of Sciences present much material that students have seen (though often never mastered) in high school, which makes it difficult to motivate them. We felt that presenting new material that combined novelty, physical insight, and broad applicability might excite them!

5 1. Light, Quantization, Atoms and Spectroscopy 1.1 Light and the Electromagnetic Spectrum The wave properties of light c = νλ = m s 1 ν = λ = ν 1 2 c + E E(x,t) oscillation period 1/ν field viewed at fixed point in space E 2 4 time / s E E(x,t) wavelength λ field viewed at fixed point in time E distance 1.5 / µm Figure 1: The electric field of light oscillates in space and in time. ( ) 2πν E(x,t) = E cos x 2πνt + φ c

6 1. Light, Quantization, Atoms and Spectroscopy 1.1 Light and the Electromagnetic Spectrum The Quantum theory of Light Max Planck and black-body radiation 1 13 ν / Hz Intensity Ι(ν,T) / 1-7 J m -2 Rayleigh-Jeans law (15 K) T = 15 K 1 5 T = 3 K ( 1 4 ) T = 6 K ( 1 ) infrared visible ultraviolet Figure 2: Black-body radiation: observed distributions and the Rayleigh-Jeans law prediction. ε = ε(ν) = hν = hc1 2 ν I(ν,T) = 2πhν3 c 2 1 e hν/kt 1

7 1. Light, Quantization, Atoms and Spectroscopy 1.1 Light and the Electromagnetic Spectrum The Quantum theory of Light A Albert Einstein and the photoelectric effect metal cathode with voltage = e - incident light of frequency ν emitted electrons A e - e - negative grid for measuring e - kinetic energy current meter positive anode to collect fast e - - W maximum electron kinetic energy Figure 3: The photoelectric effect: A. The experiment; B. The observations. ν B frequency ν [s -1 ] ε(ν) = hν Arthur H. Compton and bouncing photons p λ = h/λ

8 1. Light, Quantization, Atoms and Spectroscopy 1.2 The Quantum Theory of Matter The Spectrum of the Hydrogen Atom wavelength λ / nm wavenumber ν / cm -1 Figure 4: The hydrogen atom emission spectrum. If the emitted light is dispersed by a prism, photons of different frequency cause blackening at different locations on a photographic plate. ( 1 ν = R H (2) 1 ) 2 (n 1 ) 2

9 1. Light, Quantization, Atoms and Spectroscopy 1.2 The Quantum Theory of Matter The Bohr Theory of the Atom energy / cm nm nm nm nm Balmer n = n = 4 n = nm nm 194. nm Ritz-Paschen n = 2-4 Lyman nm nm nm nm nm Coulomb potential e 2 V(r) = (4π ε 1 2 hc) r -8-1 n = r / Å Figure 5: Hydrogen atom energy levels and transitions. [ 1 Ẽ(n 1,n 2 ) = Ẽn 1 Ẽn 2 = R H (n 2 ) 1 2 (n 1 ) 2 ) Ẽ A n = R A 1 n 2 = Z2 ( µa µ H R H 1 n 2 ] = ν

10 1. Light, Quantization, Atoms and Spectroscopy 1.3 Wave Mechanics and the Schrödinger Equation A Particle in a One-Dimensional Box: Schrodinger says... ħ2 d 2 y(x) +V(x)y(x) = Ey(x). 2m dx 2 Inside the box where V(x) =, this becomes d 2 ( ) y 2mE d 2 y = y or = k 2 y where k 2 = + 2mE dx 2 ħ 2 dx 2 ħ 2 with solutions y(x) = sin(kx) and y(x) = cos(kx) energy n = 4 V(x) m =1. m = 1. n 7 6 m = 2.25 n = 3 n = n =1 1 x L =1. x L =1.5 x L =1.

11 energy n = 4 n = 3 n = 2 1. Light, Quantization, Atoms and Spectroscopy 1.3 Wave Mechanics and the Schrödinger Equation V(x) A Particle in a One-Dimensional Box: Schrodinger says... ħ2 d 2 y(x) +V(x)y(x) = Ey(x). 2m dx 2 Inside the box where V(x) =, this becomes d 2 ( ) y 2mE d 2 y = y or = k 2 y where k 2 = + 2mE dx 2 ħ 2 dx 2 ħ 2 with solutions y(x) = sin(kx) and y(x) = cos(kx) m =1. m = 1. m = n =1 1 x L =1. x L =1.5 x L =1. n V(x) n = 6 n = 4 n = 3 n = 5 n = 2 n = 1 x L V(r) = ½ k (r r e ) 2 υ = 1 υ = 2 υ = r e υ = 5 υ = 4 υ = 3 r A C R H D e n = 2 n = 1 r e r n = 3 n = 4 e 2 V(r) = 4π ε r υ = υ = 1 υ = 3 υ = 2 r υ = 4 B D υ = 5

12 1. Light, Quantization, Atoms and Spectroscopy 1.3 Wave Mechanics and the Schrödinger Equation Orbital or Rotational Motion: A Particle on a Ring or ħ2 d 2 ψ(φ) 2mr 2 dφ 2 d 2 ψ(φ) dφ 2 = Eψ(φ) ( 2mr 2 ) E = ψ(φ) = b 2 ψ(φ) ħ 2 l = l =1 l = 2 l = 3 l = 4 l = Electronic Structure of Atoms and Molecules Hydrogenic Atomic Orbitals Multi-Electron Atoms and Atomic Spectroscopy Molecular Energies and the Born-Oppenheimer Approximation

13 2. Rotational Spectroscopy 2.1 Classical Description of Molecular Rotation Why Does Light Cause Rotational Transitions? orientation of molecule direction of dipole vertical component of molecular dipole time Behaviour of the vertical component of the dipole field of a polar diatomic molecule rotating clockwise in the plane of the paper Relative Motion and the Reduced Mass KE tot = 1 2 m 1 Ṙ cm m 2 2 ṙ m 1 +m m 2 Ṙ m 1 2 cm + ṙ m 1 +m 2 = 1 2 (m 1+m 2 ) Ṙ ( ) 2 1 m1 m 2 cm + ṙ 2 = KE cm + KE rel 2 m 1 +m Motion of a Rotating Body N KE rot = L2 where I = m i (d i 2I )2 µ(r e ) 2 i=1

14 2. Rotational Spectroscopy 2.1 Quantum Mechanics of Molecular Rotation The Basics E rot (J) = ħ2 [J(J +1)] = B[J(J +1)] 2I Energy Levels, Selection Rules, and Transition Energies ν rot J = F(J) = F(J +1) F(J) = 2B(J +1) ν rot J ν rot J ν rot J 1 =... = 2B rotational energy 56B 42B 3B 2B 12B 6B 2B ν = 14B ν = 12B ν = 1B ν = 8B ν = 6B ν = 4B J = 7 J = 6 J = 5 J = 4 J = 3 J = 2 J = 1 J =

15 2. Rotational Spectroscopy 2.3 Complications! Complication # 1: Isotopologues Complication # 2: Vibrational Stretching and Vibrational Satellites Complication # 3: Rotational Stretching or Centrifugal Distortion: Non-Rigid Rotors 2.4 Degeneracies and Intensities z L z = 2 h 1 h L 1 J pop max (T) g J = (2J+1) h g J e B J(J+1)/k T υ B -1 h -2 h / f J (T) = (2J +1)e F v(j)/k B T Q(T) 5 1 e B J(J+1)/k T υ 1 2 B J

16 2. Rotational Spectroscopy 2.5 Rotational Spectra of Polyatomic Molecules Linear Molecules are (Relatively) Easy to Treat! Case A: A Diatomic Molecule. I = m 1m 2 m 1 +m 2 d 2 = µd 2 = I d Case B: A Symmetric Triatomic Molecule. I = m 2 ( d 12 ) 2 +m 1 () 2 +m 2 (d 12 ) 2 = 2m 2 (d 12 ) 2 Case C: A Symmetric Tetra-atomic Molecule. I = 2m 1 (d d 22) 2 +2m 2 ( 1 2 d 22) 2 Case D: An Asymmetric Triatomic Molecule. I = m 1 (d 12 ) 2 +m 3 (d 23 ) 2 (m 1d 12 m 3 d 23 ) 2 (m 1 +m 2 +m 3 ) Non-Linear Polyatomic Molecules are More Difficult... A B C D d m 1 m 2 d 12 d 12 m 2 m 1 m 2 d 12 d 22 d 12 m 1 m 2 m 2 m 1 d 12 d 23 m 1 m 2 m 3 y I x =... =.6158 [uå 2 ] I A.958 Å x I y =... = [uå 2 ] I B I z = I x +I y = [uå 2 ] I C θ = 14.5

17 3. Vibrational Spectroscopy 3.1 Classical Description of Molecular Vibrations Why Does Light Cause Vibrational Transitions? length of molecule strength of dipole vertical component of molecular dipole time equilibrium dipole moment Dipole moment of a vibrating polar diatomic molecule which is fixed and aligned in space The Centre of Mass and Relative Motion The Classical Harmonic Oscillator V HO (r) = 1 2 k(r r e) 2

18 3. Vibrational Spectroscopy 3.2 Quantum Mechanics of Molecular Vibration and the Harmonic Oscillator Ĥ vib ψ(r) = ħ2 2µ d 2 ψ(r) dr 2 +V(r)ψ(r) = E vib ψ(r) G HO (v) E HO vib (v)/( 1 2 hc ) = ω e (v + 1 /2) [cm 1 ] G HO v+1/2 G HO (v +1) G HO (v) = ω e 1 V(r) = k (r-r e ) 2 ; µ = µ 2 υ = 5 υ = 4 1 V(r) = k (r-r e ) 2 2 ; µ = µ {k = 2 k } υ = 3 V(r) = k (r-r e ) 2 ; µ =.9 µ {k = 2 k } υ = 3 υ = 3 υ = 2 ω e υ = 2 υ = 2 υ = 1 ω e υ = 1 υ = 1 υ = A ω e / 2 υ = ω e B υ = C (r - r e ) (r - r e ) (r - r e )

19 3. Vibrational Spectroscopy 3.3 Anharmonic Vibrations and the Morse Oscillator Eigenvalues and Properties of the Morse Potential [ ] 2 [ ] V Morse (r) = D e 1 e β(r r e) = D e e 2β(r re) 2e β(r re) +1 G Morse (v) = ω e (v + 1 /2) ω e x e (v + 1 /2) 2 2D e β ω e = ħ 2 1 and µ 1 2 hc D Morse e = (ω e ) 2 /4ω e x e V Morse (r) ω e x e = β 2 ħ2 2µ hc υ = 4 υ = 5 υ = 6 } υ = 7 hot bands D e D υ = 2 υ = 3 third overtone second overtone υ = 1 first overtone r e υ = fundamental zero point energy r

20 3. Vibrational Spectroscopy 3.3 Anharmonic Vibrations and the Morse Oscillator Overtones and Hot Bands Higher-Order Anharmonicity and the Dunham Expansion i G(v) = ω e (v + 1 /2) ω e x e (v + 1 /2) 2 +ω e y e (v + 1 /2) 3 +ω e z e (v + 1 /2) = Y 1, (v + 1 /2)+Y 2, (v + 1 /2) 2 +Y 3, (v + 1 /2) 3 +Y 4, (v + 1 /2) Bond Dissociation Energies and Birge-Sponer Plots υ = V(r) D? A 4 E (υ) vib D? B 5 4 E υ + ½ Birge-Sponer plot C υ υ Vibrations in Polyatomic Molecules 3.6 Rotational Structure in Vibrational Spectra of Diatomics ν P (J ) = Ẽ(v,J ) Ẽ(v,J ) = Ẽ(v,J 1) Ẽ(v,J ) = {G(v )+F v (J 1)} {G(v )+F v (J )} = {ignoring centr. distortion} = ν (v,v ) [B v +B v ](J ) [B v B v ](J ) 2 ν (v,v )...

21 external electric field E 4. Raman Spectroscopy orientation of molecule dipole induced by field dipole induced by field 16 O 18 O 18 O 16 O 18 O 16 O 16 O 18 O 16 O 18 O time 18 O 16 O "virtual level" Stokes scattering ν ν s = ν ν s =ν 1 / ν rot ν δe/h δe Rayleigh scattering anti-stokes scattering ν ν s = ν + δe/h E (υ,j ) E (υ,j ) ν O Stokes (J) = ν {Ẽ(v,J 2) Ẽ(v,J)} = ν {[G(v ) G(v )] + [F v (J 2) F v (J)]} =... = ν { [G(v ) G(v )] (B v +B v )(2J 1) (B v B v )(J 2 J +1) } ν Q Stokes (J) = ν {Ẽ(v,J)},J) Ẽ(v = ν {[G(v ) G(v )] + [F v (J) F v (J)]} = ν {[G(v ) G(v )] [B v B v ]J(J +1)} ν S Stokes (J) = ν {Ẽ(v,J,J)} +2) Ẽ(v = ν {[G(v ) G(v )] + [F v (J +2) F v (J)]} =... = ν { [G(v ) G(v )] + (B v +B v )(2J +3) (B v B v )(J 2 +3J +3) }

22 5. Electronic Spectroscopy (,) (1,) (2,) (3,) (5,) A (,) I /2 (2,) (1,) (3,) (4,) B (,) (1,) (2,) (3,) (4,) (5,) (6,) (8,) C (1,) (12,) (3,) (4,) (5,) (6,) (8,) (2,) (1,) (,) 4 45 ν / cm (1,) D (12,) A B C D ν P (J ) = Ẽ (v,j ) Ẽ (v,j ) = Ẽ(v,J 1) Ẽ (v,j ) = ν (v,v ) [B v +B v ](J ) [B v B v ](J ) 2 ν R (J ) = Ẽ (v,j ) Ẽ (v,j ) = Ẽ(v,J +1) Ẽ (v,j ) = ν (v,v ) + [B v +B v ](J +1) [B v B v ](J +1) 2 ν Q (J ) = Ẽ (v,j ) Ẽ (v,j ) = Ẽ(v,J ) Ẽ (v,j ) = ν (v,v ) [B v B v ]J (J +1)

23 6. Photoelectron Spectroscopy ionization threshold 1 2 m e v 2 e IE = hν KE e = hν 1 2 m e(v e ) 2 2s A hν 2s B H atom Coulomb potential σ u * 1s A 1s B σ g H atom A H 2 H atom B hν = KE e + IE adi + [G + (v + ) G + ()] KE e H(1s) + H + υ + = 6 υ + = 3 υ + = IE(υ + = 2) hν IE adi H(1s) + H(1s) υ =

24 7. NMR Spectroscopy 7.1 Basics of NMR Spectroscopy P = ħ R(R+1) m R = R, R+1, R+2, R+3,..., R 1, R m R P/ h µ = γ I = γħ I(I +1) 1 2 E = E(m I ) = µ z B = (γi z )B = γm I ħb E 5 β(m I = ½) E ( B α ) / α(m I = +½) Chemical Shifts δ A (in molecule) = / B / B α ( ν A (in molecule) ν A ) (in reference species) 1 6 [ppm] (in reference species) ν A

25 7. NMR Spectroscopy 7.2 Spin-Spin Coupling β(m A = ½) ½ J AB α(a), β(a) (m A = ± ½) ν A ν A ½ J AB ν A + ½ J AB α(m A = + ½) ½ J AB no fields external field B / nearby proton α(b) (m B = + ½) nearby proton β(b) (m B = ½) ν A ν B = (δ A δ B ) ν spectrometer J AB J AB ν A ν ν B β(m A = ½) ½ J AB ½ J AB α(a), β(a) (m A = ± ½) ν A ν A J AB ν A ν A + J AB no fields ½ J AB α(m A = + ½) ½ J AB ν A ν B spectrometer = (δ A δ B ) ν external nearby nearby field B protons B protons B m tot m tot B = +1 B = / nearby protons B m tot B = 1 J AB J AB J AB ν A ν ν B

26 What do we expect them to know? all formulae given on a data sheet be able to recognize and understand the symbols in those equations be able to recognize, understand, and use those equations Chem 29 Data Sheet: Exam Dec. 18, 21 Formulae c = λν ν = λ = ν 1 2 c ǫ(ν) = hν = hc/λ KE max (e ) = 1 2 m e(v max e ) 2 = hν W p λ = h/λ ( ) Ẽ n = Z 2 µa RH µ H n 2 ν = R H ( 1 (n 2 ) 2 1 (n 1 ) 2 ) λ p = h/mv = h/p ( ) π 2 C u Ẽ n = m[u] ( L[Å] ) 2 n 2 ( ) C u Ẽ l = m[u] ( r[å] ) 2 l 2 ( )/( Rcm = m i ri i i ) 2 m i ) I = m i (d i i ( ) I d = µ( r) 2 m1 m 2 = ( r) 2 m 1 +m 2 E rot = 1 2 Iω2 = L 2 /2I L = J(J +1) F(J) = B[J(J +1)] or = B[J(J +1)] D[J(J +1)] 2 ν J = F(J) = F(J +1) F(J) = 2B(J +1) or = 2B(J +1) 4D(J +1) 3 B[cm 1 ] = I 1 2 hc = C u I [u Å 2 ] B v = B e α e (v+ 1 2 )+γ e(v+ 1 2 )2 / [ r v (J) = r v (J=) 1 2[J(J +1)] D v B v f J (T) (2J+1)e Fv(J)/kB T J eq max (T) = kb T 2B 1 2 V HO (r) = 1 2 k(r r e) 2 G HO (v) = ω e (v+ 1 2 ) ] 1/2 / ωe HO = 2C u k[cm 1 Å 2 ] µ[u] [ V Morse (r) = D e 1 e β(r re)] 2 ] = D e [e 2β(r re) 2e β(r re) +1 G Morse (v) = ω e (v+ 1 2 ) ω ex e (v+ 1 2 )2 G Morse v+1/2 = GMorse (v+1) G Morse (v) = ω e 2ω e x e (v+1) ( ) / ωe Morse = 4C u D e [cm 1 ] β[å 1 2 ] µ[u] ) / ω e x Morse e = C u (β[å 1 2 ] µ[u] D Morse e = (ω e ) 2 /(4ω e x e ) ν R (J) = ν (v,v )+[B v +B v ](J+1) [B v B v ](J+1) 2 ν P (J) = ν (v,v ) [B v +B v ]J [B v B v ]J 2 ν Q (J) = ν (v,v ) [B v B v ]J(J+1) ν (v,v ) = [T e+g (v )] [T e +G (v )] ν Stokes = ν Ẽint = ν [Ẽ(v,J ) Ẽ(v,J )] ν anti S = ν + Ẽint = ν +[Ẽ(v,J ) Ẽ(v,J )] ẼO int(j) = [G(v ) G(v )] (B v +B v )(2J 1) (B v B v )(J 2 J +1) ẼQ int (J) = [G(v ) G(v )] (B v B v )J(J +1) ẼS int (J) = [G(v ) G(v )] + (B v +B v )(2J +3) (B v B v )(J 2 +3J +3) hν = KE e +IE adiabatic +[G + (v + ) G + ()] = KE e + { [T e + +G+ (v + )] [T e +G (v )] } E(m I ) = µ z B = γm I B[J] E = γ A B = hν [J] ν = γ A B/2π [s 1 ] ν A (molecule) = γ AB eff 2π = γ AB 2π (1 σ A) δ A [ppm] = νa (sample) νa (reference) ν A (reference) 1 6 For Physical Constants, conversion factors, and properties of nuclei... see over