Section 5 Molecular Electronic Spectroscopy (lecture 9 ish)

Section 5 

Molecular Electronic Spectroscopy Section 5 

Molecular Electronic Spectroscopy Molecular Electronic Spectroscopy

(lecture 9  ^ish )

Molecular Electronic Spectroscopy (lecture 9  ^ish )

Quantum theory

Previously: ^{Q y Quantum}  _{V l}

of atoms / molecules

Previously:

Molecular Electronic Spectroscopy

Q

Mechanics Valence

Molecular Electronic Spectroscopy

ƒ Classification of electronic states

ƒ Molecular terms

ƒ Electronic transitions: The Franck‐Condon Principle Electronic transitions: The Franck Condon Principle

ƒ Franck‐Condon factors

ƒ Vibrational structure: Birge‐Sponer extrapolation

ƒ Rotational structure: Bandheads

ƒ Introduction to photoelectron spectroscopy

Molecular  l Molecular 

l Energy Levels Energy Levels

i.e., typically ΔE

>> ΔE

>> ΔE

Different electronic states  (electronic arrangements) (electronic arrangements)

λ Δ ≈

≈

E 2 x 10

– 10

cm

500 – 100 nm

10

– 5 x 10

cm

100  μm – 2 μm

3 – 300 GHz  (0.1 – 10 cm

) Transitions at λ

Vis – UV 

00 μ μ

infrared 10 cm – 1 mm

microwave

5.1 A quick resume 5.1 A quick resume

( ) ( ) ( ) ( )

tot

r ,Q , θ ψ

r ψ

Q ψ θ

Ψ =

The Born Oppenheimer Approximation : _{pp pp}

( ,Q , ) ψ

( ) ( ) ( ) ψ

Q ψ

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:

( ) ( ) ( )

el el el el

ˆH ψ r ,R = E R ψ r ,R

Make orbital approximation for >1 electron:  ψ

= φ

( ) ( ) ( ) ¹ φ

² φ

³ …

Na HF Ar‐HS

Na‐HF

V(R)

Potential energy R

In 1‐dimension

Potential energy curves

Potential energy  surfaces

R

5.2 Classifying molecular electronic states 5.2 Classifying molecular electronic states

Diatomic Term Symbols Diatomic Term Symbols:

Classify according to angular momentum around the internuclear axis, λ.

λ is analogous to m

in atoms: 

e.g., a p orbital has l = 1, m

= 0, ±1  

Two p‐orbital systems yield σ and π molecular orbitals; 

p

(m

=0) combine to yield σ, σ∗ (λ=0) +

p

(m

= ±1) combine to yield π, π∗ (λ=±1)

+

5.2 Molecular term symbols

5.2 Molecular term symbols See “Valence” 

notes HT year 2 Electronic terms are classified according to their overall 

angular momentum on the internuclear axis,  Λ :

Λ ∑ λ λ λ λ

+ + +

Λ

i

λ λ λ λ

= ∑ = + + + ^…

By analogy with atoms we use term symbols:

“Spin multiplicity”= 2S+1

(S is the total spin quantum number

By analogy with atoms we use term symbols:

2 Π

(S is the total spin quantum number  for the molecule)

Π _3/2

Gives  Λ ,  , according to ^g :

Σ for Λ = 0

Π for  Λ = ±1 Spin‐orbit  levels

Ω = | Λ + Σ | Δ for  Λ = ±2

Projection of S on 

internuclear axis

5.3 Additional Symmetry Labels 5.3 Additional Symmetry Labels

For homonuclear diatomics (or symmetric linear molecules e g CO ) it is For homonuclear diatomics (or symmetric linear molecules, 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 n.b.:

g ⊗ g = g 

⊗ Gerade:       g      symmetric

u ⊗ u = g g ⊗ u = u ⊗ g = u

For Sigma terms we denote the symmetry (+/‐) with respect to reflection in a plane For Sigma terms we denote the symmetry ( / ) with respect to reflection in a plane  containing the internuclear axis. 

See “Valence” 

notes HT year 2

5.4 Example molecular term symbols 5.4 Example molecular term symbols

I N ground state (2sσ )

(2sσ )

(2pπ )

(2pσ )

I. N

ground state (2sσ

)

(2sσ

)

(2pπ

)

(2pσ

)

Λ= 0 therefore a  Σ term

S 0 (all electrons paired) hence a singlet term S = 0 (all electrons paired), hence a singlet term 1 Σ +

1

g

Σ +

II. NO ground state (2s σ )

(2s σ∗ )

(2p σ )

(2p π )

(2p π∗ )

^2pπ∗

2pσ∗

n.b. No g, u symmetry because non‐symmetrical

2pσ 2pπ

Λ = ±1 therefore a Π term

2sσ∗

Λ = ±1 therefore a  Π term 2pσ

S = 1/2 (one unpaired electron), hence a doublet term

2 Π

^Π

2sσ

2 Π Giving rise to 

Π

_/

3 2

/

Π

and 

5.5 Example molecular term symbols 5.5 Example molecular term symbols

III. O

ground state (2s σ

)

(2s σ

)

(2p σ

)

(2p π

)

(2p π

)

Λ = 0, or ±2 therefore  Σ, Δ terms arise

S = 0, or 1 singlets and triplets  g  ⊗ g = g all terms gerade

1 1 3 3 1 3

g g g g g g

+ − + −

Σ Σ Σ Σ Δ Δ

∴ expect

But this neglects the  Pauli Principle ^. 

In singlet states ψ

is antisymmetric Hence these can only be paired with symmetric In singlet states, ψ

is antisymmetric. Hence these can only be paired with symmetric  ψ

, i.e.,  g, + states. Likewise triplet states must be paired with g, – states. 

1

Σ

Σ

Δ all violate Pauli and thus do not exist

g g g

Σ Σ Δ all violate Pauli and thus do not exist

1

Σ

Σ

Δ Do exist, of which the triplet state is the lowest in energy 

g g g

Σ Σ Δ ^{p gy}

(spin correlation)

Again, this is only a consideration for multiply occupied (but not full) orbitals

5.6 Molecular electronic states 5.6 Molecular electronic states

3

Σ Σ

, 

Δ Δ

and and 

Σ Σ

states all arise states all arise  from the lowest electronic configuration  ... (2p π

)

(2p π

)

Each is deeply bound and supports vibrational levels. Each can be modelled by a Morse 

potential energy function

Other electronic configurations give rise to  g g additional electronic states correlating with  the same or different dissociation products. 

Transitions between different states are

Transitions between different states are 

accompanied by vibrational band structure.

5.7 Electronic Spectroscopy 5.7 Electronic Spectroscopy

C id ld f i d th T iti Di l M t R ∫

ˆ d 〈 ˆ 〉

Consider our old friend the Transition Dipole Moment R

= ∫ ψ μψ τ

d = 〈 ψ μ ψ

〉

Within the B O approximation Ψ = ψ (r) ψ (R) ^{μ ˆ =} ∑ ^{q r}

^ˆ

Within the B‐O approximation,  Ψ _tot = ψ _el (r) ψ _vib (R) ∑

( ) ( ) ( ) ( )

21 _{el vib}

ˆ

d  d

R = 〈 ψ ψ μ ψ ψ ^{′ ′ ′′ ′′} 〉 = − e ∫∫ ∫∫ ψ ^′

( ) ( ) ( ) ( ) r ψ ^′

R r ψ ^′′ r ψ ^′′ R r R

( ) ( ) ( ) ( )

21 _{el el}

d

d

R = − e ∫ ψ ^′

r r ψ ^′′ r r ∫ ψ ^′

R ψ ^′′ R R

Electronic  transition moment

Vibrational overlap

Transition intensity  ∝ ^R

^∝ ( ( _∫ ^ψ

^′

( ) ( ) ^{r r ψ}

^{′′ r r d} ) ) (

( _∫ ^ψ

^′

^{( ) ( ) R ψ}

^{′′ R R d} ) )

Franck‐Condon factor

(square of the vibrational

Δ Λ = 0, ±1

Δ S = 0 and Δ Σ = 0

overlap integral)

g  ↔ u 

+  ↔ + ;  – ↔ –

5.8 The Franck‐Condon Principle 5.8 The Franck‐Condon Principle

Assumption: electronic transitions take place on such a short timescale that the 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  g y (

overlap of the two vibrational wavefunctions).

5.9 Vibrational Structure: Franck‐Condon factors 5.9 Vibrational Structure: Franck‐Condon factors

The overlap of the vibrational wavefunctions is governed by the nature of the I. If the bonding character of the 

t t t i i il

II. If the bonding character of the  two states is very different

The overlap of the vibrational wavefunctions is governed by the nature of the  electronic states involved

two states is similar: two states is very different:

0 1 ‐0 V’ ‐0

0 ‐0 1

R

’  ∼ R

’’ R

’ ≠ R

’’

Best overlap (v’ =0) – (v’’=0)

“Short progression” 

Best overlap (v’ > 0) – (v’’=0)

“Long progression” 

5.10 Determining dissociation energies 5.10 Determining dissociation energies

In some cases, Franck‐Condon overlap extends above  the dissociation limit and excited‐state dissociation  energies are measured directly

energies are measured directly.

When this is not the case but several vibrational levels  are excited it is possible to extrapolate to find the

are  excited it is possible to extrapolate to find the  dissociation limit.

Recall, Morse oscillator:

0 1 2 3

v e

x

e e

G

v

v v x

v , , , ,

ω ω

= + − +

= …

(

)

2

v

e e e

dG x v

dv = ω − ω ( + ) Which, at the dissociation limit (v+½)

becomes zero

dv

(

)

0 2

dG

ω ω x v

= = − +

1

v ω

= −

2

G D ω

= =

(

)

0

2

x v

dv ω ω +

2 2

max

e e

v ω x

⁴ ω

x

5.11 Determining dissociation energies: Birge‐Sponer Extrapolation 5.11 Determining dissociation energies: Birge‐Sponer Extrapolation

Alternatively, the experimental dissociation energy is the sum of all the vibrational energy level spacings,  Δ G

:

D

Δ G

Δ G

Δ G

D = Δ G + Δ G + Δ G + ....

Area under the plot yields D

Plot  Δ G

as a function of (v+½) Area under the plot yields D

Most such plots deviate from linearity at high v as the Morse Potential function becomes an as the Morse Potential function becomes an  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.

vibrational bands in electronic spectra.

5.12 Rotational Structure in Electronic Spectra 5.12 Rotational Structure in Electronic Spectra

e v

E T G F hc ^{= + +}

Total term values:

Transition wavenumbers:  ^{ν =} ( ^{T T}

^{′ −}

^′′ ) ( ^{+ G G}

^{′ −}

^′′ ) ( ^{+ F}

^{′ − F}

^′′ )

Electronic Vibrational Rotational Electronic

Transition 

Vibrational transition

FCF

Rotational transition (Δ J rules)

Example:   

Σ ‐

Σ Δ J = ± 1, leads to P(J) and R(J) branches as in vib‐rot spectra 

[See section 4.7]

[ ]

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.

5.13 Band Heads in Electronic Spectra 5.13 Band Heads in Electronic Spectra

( ) ( )( ) ( )( )

( )

( )( ¹ ) ( )( ¹ )

R J ⁼ ν

⁺ B B ^{′ + ′′} J ^{+ +} B B ^{′ − ′′} J ⁺

( )

( ) ( )

P J = ν

− B B ^′ + ^′′ J + B B ^′ − ^′′ J

(1)

( )

( ) ( ) (2)

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.

Band heads occur when lines in a branch coalesce, i.e.,   d 0 d

ν ₌

J ( ^{B B ′ ′′} )

In the R branch:  ^d _d ^{ν =} ( ^{B B ′ + ′′} ) ( ^{+ 2 B B ′ − ′′} )( ^{J + 1} )

J ( ) ( )

( )

1

2

B B B B

′ + ′′

= − ′ − ′′

J

In the P branch:  ^d _d ^{ν = −} ( ^{B B ′ + ′′} ) ^{+ 2 J} ( ^{B B ′ − ′′} )

J

( )

( )

head

2

B B B B

′ + ′′

= ′ ′′

( ) ( ) J

d J ² ( ^{B B −} )

5.13 Band Heads in Electronic Spectra 5.13 Band Heads in Electronic Spectra

Large change in R

in transition  with result that B’<<B” –

with result that B <<B  –

rotational levels more closely  spaced in upper state.

Bandhead in the R‐branch: 

increased spacing in P‐branch

CuH

Σ ‐

Σ transition

5.14 Q‐branches 5.14 Q‐branches

The Δ J = ±1 selection rule arises from conservation of angular momentum and The  Δ J = ±1 selection rule arises from conservation of angular momentum and 

symmetry. However, if additional angular momenta are present we can also observe  Q‐branches  ( Δ J = 0 transitions).  ^Q ( ) ^{J = ν}

⁺ ( ^{B B ′ − ′′} ) ( ^{J J + 1} )

This angular momentum could be electronic: e.g.,  

Π –

Σ transition (or indeed any  transition involving ΔΛ > 0)...

( )

( ) ( ¹ )

Q J ν

+ B B J J +

transition involving  ΔΛ > 0)...

...or vibrational if the transition involves a degenerate vibrational mode (e.g., a  bending mode of a linear molecule)

AlH

Π –

Σ

emission spectrum  showing Q branch

showing Q‐branch 

5.15 Photoelectron Spectroscopy (gas phase) 5.15 Photoelectron Spectroscopy (gas phase)

Photoionization is the limiting case of exciting a single electron to higher electronic 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.

Ionic states

K.E.

1) Excite with fixed  λ high above the I.E.

(e.g., with a He(I) lamp at 21.22 eV) Ionic states (e.g., with a He(I) lamp at 21.22 eV)

2) Measure K.E. of ejected photoelectrons

Adiabatic 

ionization energy

As energy is conserved: 

h ν = I + E

+ K.E.

+ K.E.

h ν

where I is the adiabatic ionization energy  and E

is the internal energy of the ion.

h ν

K.E.

>> K.E.

≈ 0

K.E.

= h ν – I – E

5.16 Photoelectron Spectroscopy of atoms 5.16 Photoelectron Spectroscopy of atoms

2

S

2

S

2s

1s

2

P S

2

S

(Spin‐orbit splitting 

unresolved here) 2s

2p

2p

2

P

1s

Now resolve spin‐

bit li i i i

orbit coupling in ionic 

states

5.17 Photoelectron Spectroscopy of H ₂ 5.17 Photoelectron Spectroscopy of H ₂

With a He(I) lamp only the first ionization threshold  of H

(1s σ

)

is accessible.

Removal of a strongly bonding electron results  in a substantial reduction in bonding character

(H

kl b d th H ) d R

R ”

H

ionic  state

(H

more weakly bound than H

) and R

> R

”

Long vibrational progression

(Franck‐Condon principle)

5.18 Photoelectron Spectroscopy of N ₂ 5.18 Photoelectron Spectroscopy of N ₂

A A B

C

3 bands in He(I) PES:

A:  (2p σ

)

:  weakly bonding electron removed, short progression,  N

² Σ _g ⁺

B:  (2p π

)

:  strongly bonding electron removed, longer progression,  N

² Π _u

C:  (2s ( σ

) )

:  weakly anti‐bonding electron removed, short progression, N y g , p g ,

² Σ _{u u} ⁺

i.e., band sub‐structure represents vibrational levels of each ionic state