Section 5 Molecular Electronic Spectroscopy (lecture 9 ish)

Transcription

1 Section 5 Molecular Electronic Spectroscopy (lecture 9 ish) Previously: Quantum theory of atoms / molecules Quantum Mechanics Vl Valence Molecular Electronic Spectroscopy Classification of electronic states Molecular terms Electronic transitions: The Franck Condon Principle Franck Condon factors Vibrational structure: Birge Sponer extrapolation Rotational structure: Bandheads Introduction to photoelectron spectroscopy

2 Molecular Energy Levels i.e., typically ΔE el >> ΔE vib >> ΔE rot Different electronic states (electronic arrangements) ΔE x cm 1 Transitions at λ nm Vis UV 10 5 x 10 3 cm μm μm infrared GHz ( cm 1 ) 10 cm 1 mm microwave

3 5.1 A quick resume The Born Oppenheimer Approximation : Ψ ( r,q, θ ) = ψ ( r ) ψ ( Q ) ψ ( θ tot el vib ) But this assumes there is a single ψ el for a given electronic configuration. In fact, we should solve the electronic Hamiltonian at each nuclear configuration: Ĥ ψ ( r,r el el ) = Eel ( R) ψel ( r,r) Make orbital approximation for >1 electron: ψ φ ( 1) φ ( ) φ ( 3 el a b c ) = Ar HS Na HF V(R) In 1 dimension R AB Potential energy curves Potential energy surfaces

4 5. Classifying molecular electronic states Diatomic Term Symbols: Classify according to angular momentum around the internuclear axis, λ. λ is analogous to m l in atoms: e.g., a p orbital has l = 1, m l = 0, ±1 Two p orbital systems yield σ and π molecular orbitals; p z (m l =0) combine to yield σ, σ (λ=0) + p x,y (m l = ±1) combine to yield π, π (λ=±1) +

5 5. Molecular term symbols See Valence notes HT year Electronic terms are classified according to their overall angular momentum on the internuclear axis, Λ: i 1 3 i Λ = λ = λ + λ + λ + By analogy with atoms we use term symbols: Spin multiplicity = S+1 (S is the total spin quantum number for the molecule) Gives Λ,, according to: Σ for Λ = 0 Π for Λ = ±1 Δ for Λ = ± Π 3/ Π Spin orbit levels Ω = Λ + Σ Projection of S on internuclear axis

6 5.3 Additional Symmetry Labels For homonuclear diatomics (or symmetric linear molecules, eg e.g., CO ) it is convenient to label molecular orbitals and terms according to symmetry (g,u) with respect to inversion through the centre of symmetry. Ungerade: u Anti symmetric Gerade: g symmetric n.b.: g g = g u u = g g u = u g = u For Sigma terms we denote the symmetry (+/ ) with respect to reflection in a plane containing the internuclear axis. See Valence notes HT year

7 5.4 Example molecular term symbols I. N ground state (sσ g ) (sσ u ) (pπ u ) 4 (pσ g ) Λ= 0 therefore a Σ term S = 0 (all electrons paired), hence a singlet term 1 Σ + g II. NO ground state (sσ) (sσ ) (pσ) (pπ) 4 (pπ ) 1 n.b. No g, u symmetry because non symmetrical Λ= ±1 therefore a Π term S = 1/ (one unpaired electron), hence a doublet term Π Giving rise to Π and 1/ Π 3 / pσ pπ pπ pσ sσ sσ

8 5.5 Example molecular term symbols III. O ground state (sσ g ) (sσ u ) (pσ g ) (pπ u ) 4 (pπ g ) Λ= 0, or ± therefore Σ, Δ terms arise S = 0, or 1 singlets and triplets g g = g all terms gerade expect Σ Σ Σ Σ Δ Δ g g g g g g But this neglects the Pauli Principle. In singlet states, ψ spin is antisymmetric. Hence these can only be paired with symmetric ψ space, i.e., g, + states. Likewise triplet states must be paired with g, states. Σ Σ Δ allviolate Pauli andthusdonotexist do not g g g Σ Σ Δ Do exist, of which the triplet state is the lowest in energy g g g (spin correlation) Again, this is only a consideration for multiply occupied (but not full) orbitals

9 5.6 Molecular electronic states Σ, Δ and Σ states allarise arise g g g from the lowest electronic configuration... (pπ u ) 4 (pπ g ) Each is deeply bound and supports vibrational levels. Each can be modelled by a Morse potential energy function Other electronic configurations give rise to additional electronic states correlating with the same or different dissociation products. Transitions between different states are accompanied by vibrational band structure.

10 5.7 Electronic Spectroscopy R = ψ * μψ ˆ dτ = ψ μˆ ψ Consider our old friend fi dthe Transition Dipole Moment μ ˆ = q rˆ i i Within the B O approximation, Ψ tot = ψ el (r)ψ vib (R) = = ( ) ( ) ( ) ( ) R ψψ μψψ ˆ e ψ r ψ R rψ r ψ R d r dr 1 el vib el vib el vib el vib ( ) ( ) ( ) ( ) R e ψ r rψ r dr ψ R ψ R dr 1 el el vib vib = i Electronic transition moment Vibrational overlap Transition intensity ( ) ( ) ( ψ ψ ) ( ψ ( ) ψ ( ) ) R d d 1 r r r r R R R el el vib vib ΔΛ = 0, ±1 ΔS = 0andΔΣ = 0 g u (where g, u exist) + + ; (for Σ Σ transitions) Franck Condon factor (square of the vibrational overlap integral)

11 5.8 The Franck Condon Principle Assumption: electronic transitions take place on such a short timescale that the nuclei remain frozen (R unchanged) during the transition. We talk of vertical transitions between potential energy curves. There is no selection rule governing the allowed vibrational changes accompanying an electronic transition. Instead, the probability of undertaking a v v transition is governed by Franck Condon factors (the overlap of the two vibrational wavefunctions).

12 5.9 Vibrational Structure: Franck Condon factors The overlap ofthe vibrational wavefunctions is governed by the nature of the electronic states involved I. If the bonding character of the two states tt is similar: il II. If the bonding character of the two states is very different: V 0 R e R e R e R e Best overlap (v =0) (v =0) Short progression Best overlap (v > 0) (v =0) Long progression

13 5.10 Determining dissociation energies In some cases, Franck Condon overlap extends above the dissociation limit and excited state dissociation energies are measured directly. When this is not the case but several vibrational levels are excited it is possible to extrapolate to find the dissociation limit. Recall, Morse oscillator: G v = v ω v ω x e e e v = 013,,,, v ma x dg v dv 1 ( v ) = ω ω x + e e e Which, at the dissociation limit (v+½) max becomes zero dg v ω 1 e 1 = 0= ω ω x = G ( v + ) v = D = e e e max max dv ω x v max e 4 e e ω e e ω x e

14 5.11 Determining dissociation energies: Birge Sponer Extrapolation Alternatively, the experimental dissociation energy is the sum of all the vibrational energy level spacings, ΔG v+1/ : D =Δ G + Δ G + Δ G / 3/ 5/ Plot ΔG v+1/ as a function of (v+½) Areaundertheplotyields D 0 Most such plots deviate from linearity at high v asthemorsepotential functionbecomesan increasingly poor representation of the real potential. Several vibrational energy levels are required for such a fit and so they are generally only used for vibrational bands in electronic spectra.

15 5.1 Rotational Structure in Electronic Spectra Total term values: E T G F e v hc = + + J Transition wavenumbers: ν = ( T T ) + ( G G ) + ( F F ) e e v v Electronic Vibrational Rotational Transition transition transition FCF (ΔJ rules) Example: 1 Σ 1 Σ ΔJ = ±1, leads to P(J) and R(J) branches as in vib rot spectra [See section 4.7] J J However, much larger changes in rotational constants, of both sign, are now possible band heads are commonly observed in electronic spectra, and may occur in either branch.

16 5.13 Band Heads in Electronic Spectra ( J ) = ν ( )( 1 ) ( )( 1 ) el + + J + + J + vib ( J ) = ν ( + ) J + ( ) J el vib R B B B B P B B B B (1) () In vibration rotation spectra, B generally decreases slightly with v leading to bunching of lines in the R branch. In electronic transitions, the change in <R > can be large depending on the bonding character of the two orbitals between which the electron moves. dν Band heads occur when lines in a branch coalesce, i.e., = 0 d J dν d In the R branch: = ( B + B ) + ( B B )( J + 1) dν d J ( J 1) J In the P branch: = ( B + B ) + J ( B B ) J head head = = ( B + B ) ( B B ) ( B + B ) ( B B )

17 5.13 Band Heads in Electronic Spectra Large change in R e in transition with result that B <<B rotational levels more closely spaced in upper state. Bandhead in the R branch: increased spacing in P branch CuH 1 Σ 1 Σ transition

18 5.14 Q branches The ΔJ = ±1 selection rule arises from conservation of angular momentum and symmetry. However, if additional angular momenta are present we can also observe Q branches (ΔJ = 0 transitions). Q J = ν + B B J J + 1 ( ) ( ) ( ) This angular momentum could be electronic: e.g., 1 Π 1 Σ transition (or indeed any transition involving ΔΛ > 0)......or vibrational if the transition involves a degenerate vibrational mode (e.g., a bending mode of a linear molecule) el vib AlH 1 Π 1 Σ + emission spectrum showing Q branch

19 5.15 Photoelectron Spectroscopy (gas phase) Photoionization is the limiting case of exciting a single electron to higher electronic states. Photoelectron spectroscopy, which records the ionization energies for removal of electrons, provides a measure of the energy of molecular orbitals. K.E. e Ionic states 1) Excite with fixed λ high above the I.E. (e.g., with a He(I) lamp at 1. ev) ) Measure K.E. of ejected photoelectrons hν Adiabatic ionization energy As energy is conserved: hν = I + E M+ + K.E. e + K.E. M+ where I is the adiabatic ionization energy and E M+ is the internal energy of the ion. K.E. e >> K.E. M+ 0 K.E. e = hν I E M+

20 5.16 Photoelectron Spectroscopy of atoms S S 1/ S 1/ (Spin orbit splitting unresolved here) 1s 1 s 1 P 1/,3/ p s p 1 1s Now resolve spin orbit coupling in ionic i states

21 5.17 Photoelectron Spectroscopy of H With a He(I) lamp only the first ionization threshold of H (1sσ g ) 1 is accessible. H + ionic state Removal of a strongly bonding electron results in a substantial reduction in bonding character (H + dth + more weakly bound than H ) and R + e > R e Long vibrational progression (Franck Condon principle)

22 5.18 Photoelectron Spectroscopy of N B A C 3 bands in He(I) PES: A: (pσ g ) 1 : weakly bonding electron removed, short progression, N + Σ g + B: (pπ u ) 1 : strongly bonding electron removed, longer progression, N + Π u C: (sσ u 1 : weakly anti bonding electron removed, short progression, N + Σ + u) u i.e., band sub structure represents vibrational levels of each ionic state