Electronic SpectroscopyUV/VIS Spectroscopy

Kenneth RuudUiT The Arctic University of Norway

• U

NIV

ERSITETET

•

I TROMSØ

July 2 2015

Outline

What is spectroscopy?Basic principles of UV/Vis absorptionUV/Vis emission spectroscopyPhosphorescenceVibronic effects

What is spectroscopy?The basic goal of spectroscopy is to unravel the properties of a molecule (nuclearstructure, electronic structure, reactivity) by interpreting the absorption (andemission) of electromagnetic radiation in terms of molecular properties

Different parts of the electromagnetic spectrum probe different parts of themolecular wave function

http://hrsbstaff.ednet.ns.ca/benoitn/chem11/units/1. AtomicTheory/e config/spectra/electromagnetic-spectrum.jpg

What is spectroscopy?The basic goal of spectroscopy is to unravel the properties of a molecule (nuclearstructure, electronic structure, reactivity) by interpreting the absorption (andemission) of electromagnetic radiation in terms of molecular propertiesDifferent parts of the electromagnetic spectrum probe different parts of themolecular wave function

http://hrsbstaff.ednet.ns.ca/benoitn/chem11/units/1. AtomicTheory/e config/spectra/electromagnetic-spectrum.jpg

Jablonski diagram for electronic processes

ESA

ESA

Phosphoresc.Fluoresc.

TPAOPA

ns

ps

ps

μs-ms

ISC

ns-μsIC

IC

S0

S1

Sn

T1

Tn

Lower-energy excitations are always involved, whether they are detectabledepend on the experimental resolutionContribute to broadening of experimental peaksSpecific processes can be explored through time-resolved spectroscopy

Basic principle of absorption spectroscopy

Molecular geometry

Pot

entia

l ene

rgy

In electronic spectroscopy, we will beconcerned with the absorption of light tobring the molecule to an excited electronicstateLeading-order contribution: Electric dipoleoperator

Higher-order contributions can be importantfor high-energy light (X-rays) or fordichroisms

The absorption cross section from the groundto the excited state is

Ai→f =πωfi NA

3ε0~c|µfi |2

We recall that the transition dipole can bedetermined from the single residue of thelinear response function

limω→ωf0

(ωf0−ω) ααβ(−ω;ω) = 〈0|µα|f 〉〈f |µβ |0〉

Important to know what excitation energy isreported.

Basic principle of absorption spectroscopy

Molecular geometry

Pot

entia

l ene

rgy

In electronic spectroscopy, we will beconcerned with the absorption of light tobring the molecule to an excited electronicstateLeading-order contribution: Electric dipoleoperator

Higher-order contributions can be importantfor high-energy light (X-rays) or fordichroismsThe absorption cross section from the groundto the excited state is

Ai→f =πωfi NA

3ε0~c|µfi |2

We recall that the transition dipole can bedetermined from the single residue of thelinear response function

limω→ωf0

(ωf0−ω) ααβ(−ω;ω) = 〈0|µα|f 〉〈f |µβ |0〉

Important to know what excitation energy isreported.

Basic principle of absorption spectroscopy

Molecular geometry

Pot

entia

l ene

rgy

In electronic spectroscopy, we will beconcerned with the absorption of light tobring the molecule to an excited electronicstateLeading-order contribution: Electric dipoleoperator

Higher-order contributions can be importantfor high-energy light (X-rays) or fordichroismsThe absorption cross section from the groundto the excited state is

Ai→f =πωfi NA

3ε0~c|µfi |2

We recall that the transition dipole can bedetermined from the single residue of thelinear response function

limω→ωf0

(ωf0−ω) ααβ(−ω;ω) = 〈0|µα|f 〉〈f |µβ |0〉

Important to know what excitation energy isreported.

One-photon absorption: Symmetry considerations

For molecule with inversion symmetry, parity has to change

Spin has to be preserved

If we only consider electronic, one-photon absorption in the dipoleapproximation, the states we can reach have to be dipole allowed→⟨0∣∣µx/y/z

∣∣ f⟩ have to have a component that transforms as the totallysymmetric irrep

Table : Group multiplication table for the D2h point group.

E C2(z) C2(y) C2(x) i σ(xy) σ(xz) σ(yz)Ag 1 1 1 1 1 1 1 1 x2, y2, z2

B1g 1 1 -1 -1 1 1 -1 -1 Rz , xyB2g 1 -1 1 -1 1 -1 1 -1 Ry , xzB3g 1 -1 -1 1 1 -1 -1 1 Rx , yzAu 1 1 1 1 -1 -1 -1 -1B1u 1 1 -1 -1 -1 -1 1 1 zB2u 1 -1 1 -1 -1 1 -1 1 yB3u 1 -1 -1 1 -1 1 1 -1 x

One-photon absorption: Symmetry considerations

For molecule with inversion symmetry, parity has to change

Spin has to be preserved

If we only consider electronic, one-photon absorption in the dipoleapproximation, the states we can reach have to be dipole allowed→⟨0∣∣µx/y/z

∣∣ f⟩ have to have a component that transforms as the totallysymmetric irrep

Table : Group multiplication table for the D2h point group.

E C2(z) C2(y) C2(x) i σ(xy) σ(xz) σ(yz)Ag 1 1 1 1 1 1 1 1 x2, y2, z2

B1g 1 1 -1 -1 1 1 -1 -1 Rz , xyB2g 1 -1 1 -1 1 -1 1 -1 Ry , xzB3g 1 -1 -1 1 1 -1 -1 1 Rx , yzAu 1 1 1 1 -1 -1 -1 -1B1u 1 1 -1 -1 -1 -1 1 1 zB2u 1 -1 1 -1 -1 1 -1 1 yB3u 1 -1 -1 1 -1 1 1 -1 x

Emission (fluorescence) spectroscopy

https://www.lifetechnologies.com/content/dam/LifeTech/Images/integration/Fluorescence-diagram-400px.jpg

Instead of measuring the light absorbed, wecan measure the light emitted from an excitedstateThe excited state can be reached in differentways (multiphoton, intersystem crossings....)

Requires a difference in the frequency ofemitted light compared to that absorbed(which implies a lifetime of the excited state)

Through stimulated emission, it can be usedto follow excited-state dynamics

Computationally, it is determined by thesame residue as for absorption, and it can bedetermined from the ground or the excitedstate

limω→ωf0

(ωf0−ω) ααβ(−ω;ω) = 〈0|µα|f 〉〈f |µβ |0〉

Computationally the same residue, butdifferent (excited-state) geometry

Emission (fluorescence) spectroscopy

https://www.lifetechnologies.com/content/dam/LifeTech/Images/integration/Fluorescence-diagram-400px.jpg

Instead of measuring the light absorbed, wecan measure the light emitted from an excitedstateThe excited state can be reached in differentways (multiphoton, intersystem crossings....)

Requires a difference in the frequency ofemitted light compared to that absorbed(which implies a lifetime of the excited state)

Through stimulated emission, it can be usedto follow excited-state dynamics

Computationally, it is determined by thesame residue as for absorption, and it can bedetermined from the ground or the excitedstate

limω→ωf0

(ωf0−ω) ααβ(−ω;ω) = 〈0|µα|f 〉〈f |µβ |0〉

Computationally the same residue, butdifferent (excited-state) geometry

PhosphorescenceThrough intersystem crossings and internal conversions, a molecule in a singletexcited electronic state may be interconverted and end up in a triplet excitedstateThe electromagnetic light has no spin-component→ no possibility to emit lightand return to the singlet ground stateHowever, there exists (from relativistic theory) a coupling of the spin and orbitalmotion of the electron, the spin–orbit interaction

HSO = −e

8πε0meα2

fs

∑iK

ZKmi ·liK

r3iK

−∑i 6=j

(mi +2mj

)·lij

r3ij

Including these corrections, the transition moment from a specific triplet spinsub-level is

Mkα =

∞∑n=0

〈S0|µα|Sn〉〈Sn|HSO|T k1 〉

E(Sn)− E(T1)+∞∑

n=1

〈S0|HSO|Tn〉〈Tn|µα|T k1 〉

E(Tn)− E(S0)

= limω→ωf0

~(ωf0 − ω)〈〈µα; HkSO, Ω〉〉0,ω/〈f |Ω|0〉

In the relativistic framework, spin is no longer a good quantum number, and thespin–orbit operator mixes states of different spin symmetryPhosphorenscence transition rates and lifetimes determined as single residues,but small probabilites

PhosphorescenceThrough intersystem crossings and internal conversions, a molecule in a singletexcited electronic state may be interconverted and end up in a triplet excitedstateThe electromagnetic light has no spin-component→ no possibility to emit lightand return to the singlet ground stateHowever, there exists (from relativistic theory) a coupling of the spin and orbitalmotion of the electron, the spin–orbit interaction

HSO = −e

8πε0meα2

fs

∑iK

ZKmi ·liK

r3iK

−∑i 6=j

(mi +2mj

)·lij

r3ij

Including these corrections, the transition moment from a specific triplet spinsub-level is

Mkα =

∞∑n=0

〈S0|µα|Sn〉〈Sn|HSO|T k1 〉

E(Sn)− E(T1)+∞∑

n=1

〈S0|HSO|Tn〉〈Tn|µα|T k1 〉

E(Tn)− E(S0)

= limω→ωf0

~(ωf0 − ω)〈〈µα; HkSO, Ω〉〉0,ω/〈f |Ω|0〉

In the relativistic framework, spin is no longer a good quantum number, and thespin–orbit operator mixes states of different spin symmetryPhosphorenscence transition rates and lifetimes determined as single residues,but small probabilites

Phosphorescence: A case study

Geometry optimization at the B3LYP/cc-pVTZ level of theory

Transitions moment calculations are four-component Dirac–Hartree–Fockcalculations HF/taug-cc-pVTZ

H

O C

H

H

OC

H

1.20 Å

1.30 Å

3.46 eV

1.96 eV

3.46 eV 5.1.10-7

7.3.10-71.96 eV 24.7 ms

Franck–Condon principle

We recall that electronic transitions are largely governed by dipole transitionmomentsIncluding the full vibronic wave function and expanding the geometrydependence of the dipole moment

〈kK ,K |µα|0, 00〉 = 〈kK |µK 0α (Q)|00〉 = µK 0

α (0)〈kK |00〉+∑

a

∂µK 0α

∂Qa

∣∣∣∣∣Q=0

〈kK |Qa|00〉+. . .

Franck–Condon: Truncation after first termHerzberg–Teller: Second term in expansion

When exciting from the vibronic ground stateto an electronic excited state, the populationof the vibrational states of the electronicexcited state is determined by 〈kK |00〉Electronic spectra with vibronic features willbe characterized by a progression of lineswith fixed separation matching thevibrational frequency

Molecular geometry

Pot

entia

l ene

rgy

Vibronic UV/Vis spectrum of a diatomic molecule

Energy

Inte

nsi

ty

Fluorescence Absorption

0−0

0→1 1←0

0→2 2←0

0→3 3←00→4 4←0

0→5 5←0

0→6 6←0

0→7 7←0

0→8 8←0

0→9 9←0

Assumed similar vibrational frequencies in ground and excited states

A hierarchy of vibronic modelsIn general, the equilibrium geometry and vibrational frequencies/normalcoordinates of the ground and excited electronic states differ

Normal coordinate are linear, and thus we can relate the ground- andexcited-state normal coordinates as

Q0 = JQK + K K = L0∆q ∆qi =√

mi

(xK

eq,i − x0eq,i

)The rotation of the normal coordinates from one set of normal coordinates toanother is called Duschinsky rotations

J = L0(

LK)−1

A hierarchy of vibronic modelsFull adiabatic Franck–Condon approximationIgnore the Duschinsky rotations→ Excited-state vibrational frequenciesscaled from the ground stateUse the ground-state vibrational force field also for the excited state, butuse a shifted geometryLinear coupling model (vertical gradient approximation): As above, butapproximate the excited-state equilibrium geomtry by the excited-stategradient projected on the ground-state force field

A hierarchy of vibronic modelsIn general, the equilibrium geometry and vibrational frequencies/normalcoordinates of the ground and excited electronic states differ

Normal coordinate are linear, and thus we can relate the ground- andexcited-state normal coordinates as

Q0 = JQK + K K = L0∆q ∆qi =√

mi

(xK

eq,i − x0eq,i

)The rotation of the normal coordinates from one set of normal coordinates toanother is called Duschinsky rotations

J = L0(

LK)−1

A hierarchy of vibronic models

Full adiabatic Franck–Condon approximationIgnore the Duschinsky rotations→ Excited-state vibrational frequenciesscaled from the ground stateUse the ground-state vibrational force field also for the excited state, butuse a shifted geometryLinear coupling model (vertical gradient approximation): As above, butapproximate the excited-state equilibrium geomtry by the excited-stategradient projected on the ground-state force field

A hierarchy of vibronic modelsIn general, the equilibrium geometry and vibrational frequencies/normalcoordinates of the ground and excited electronic states differ

Normal coordinate are linear, and thus we can relate the ground- andexcited-state normal coordinates as

Q0 = JQK + K K = L0∆q ∆qi =√

mi

(xK

eq,i − x0eq,i

)The rotation of the normal coordinates from one set of normal coordinates toanother is called Duschinsky rotations

J = L0(

LK)−1

A hierarchy of vibronic modelsFull adiabatic Franck–Condon approximation

Ignore the Duschinsky rotations→ Excited-state vibrational frequenciesscaled from the ground stateUse the ground-state vibrational force field also for the excited state, butuse a shifted geometryLinear coupling model (vertical gradient approximation): As above, butapproximate the excited-state equilibrium geomtry by the excited-stategradient projected on the ground-state force field

A hierarchy of vibronic modelsIn general, the equilibrium geometry and vibrational frequencies/normalcoordinates of the ground and excited electronic states differ

Normal coordinate are linear, and thus we can relate the ground- andexcited-state normal coordinates as

Q0 = JQK + K K = L0∆q ∆qi =√

mi

(xK

eq,i − x0eq,i

)The rotation of the normal coordinates from one set of normal coordinates toanother is called Duschinsky rotations

J = L0(

LK)−1

A hierarchy of vibronic modelsFull adiabatic Franck–Condon approximationIgnore the Duschinsky rotations→ Excited-state vibrational frequenciesscaled from the ground state

Use the ground-state vibrational force field also for the excited state, butuse a shifted geometryLinear coupling model (vertical gradient approximation): As above, butapproximate the excited-state equilibrium geomtry by the excited-stategradient projected on the ground-state force field

A hierarchy of vibronic modelsIn general, the equilibrium geometry and vibrational frequencies/normalcoordinates of the ground and excited electronic states differ

Normal coordinate are linear, and thus we can relate the ground- andexcited-state normal coordinates as

Q0 = JQK + K K = L0∆q ∆qi =√

mi

(xK

eq,i − x0eq,i

)The rotation of the normal coordinates from one set of normal coordinates toanother is called Duschinsky rotations

J = L0(

LK)−1

A hierarchy of vibronic modelsFull adiabatic Franck–Condon approximationIgnore the Duschinsky rotations→ Excited-state vibrational frequenciesscaled from the ground stateUse the ground-state vibrational force field also for the excited state, butuse a shifted geometry

Linear coupling model (vertical gradient approximation): As above, butapproximate the excited-state equilibrium geomtry by the excited-stategradient projected on the ground-state force field

A hierarchy of vibronic modelsIn general, the equilibrium geometry and vibrational frequencies/normalcoordinates of the ground and excited electronic states differ

Normal coordinate are linear, and thus we can relate the ground- andexcited-state normal coordinates as

Q0 = JQK + K K = L0∆q ∆qi =√

mi

(xK

eq,i − x0eq,i

)The rotation of the normal coordinates from one set of normal coordinates toanother is called Duschinsky rotations

J = L0(

LK)−1

A hierarchy of vibronic modelsFull adiabatic Franck–Condon approximationIgnore the Duschinsky rotations→ Excited-state vibrational frequenciesscaled from the ground stateUse the ground-state vibrational force field also for the excited state, butuse a shifted geometryLinear coupling model (vertical gradient approximation): As above, butapproximate the excited-state equilibrium geomtry by the excited-stategradient projected on the ground-state force field

Different models graphically illustrated

Evaluating Franck–Condon factorsIn order to evaluate Franck–Condon factors, we need to evaluate the overlap ofvibrational wave functions of the form

|n0〉 =ω

140

π14√

2nn!Hn(√ω0Q0)e−

12ω0Q2

0 ,

|kK 〉 =ω

14K

π14√

2k k!Hk (√ωK QK )e−

12ωK Q2

K

Solving the general case is complicated, involves the use of so-called generatingfunctionsFor the special case of a diatomic molecules with ωK = ω0, we can write

P(T ,U) =∑k,n

T k Un

√2k 2n

k!n!〈kK |n0〉 = I0eBT +DU+ETU

I0 = e−14ω0∆2

B = −√ω0∆

D =√ω0∆

E = 2

B and D are coefficients for the ground and excited electronic state, respectively→ Absorption and emission vibronic spectra are each others mirror images

Franck–Condon factors for diatomic molecules

We assume only the ground vibrational state of the electronic ground-state ispopulated→ D = 0

We expand the left- and right-hand sides

〈0K | 00〉+ T√

2〈1K | 00〉+ T 2√

2〈2K | 00〉+ T 3 2√

3〈3K | 00〉+ . . .

= I0

[1 + TB +

12!

T 2B2 +13!

T 3B3 + . . .

]Collecting terms with the same order in T , and we will find

〈kK |00〉 = I0

√1

2k k!

[√ω0∆

]k

Assuming different vibrational frequencies in the ground and excited electronicstates complicates the formulas significantly

P(T ,U) =∑k,n

T k Un

√2k 2n

k!n!〈kK |n0〉 = I0eAT 2+BT +CU2+DU+ETU

The linear coupling model

Two approximationsAssume the excited-state force field is the same as the ground-state forcefield: LK = L0, J = I, and frequency matrix ΛK = Λ0

Assume that the shift in excited-state equilibrium geometry can beapproximated by the excited-state gradient projected onto theground-state normal modes

K = L0 ∂EK

∂∆

Reduced to a generalization to polyatomic molecules of the case of a diatomicmolecule with the same vibrational frequencies and force fields

I0 =4√|Γ0Γ0|4m√|(2Γ0)|

exp[−

14

K†Γ0K],

B =√

Γ0K

D =−√

Γ0K,E =2I

Herzberg–Teller couplings

If we go beyond the Franck–Condon approximation, the leading-ordercorrection is the Herzberg–Teller contribution

〈kK ,K |µα|0, 00〉 =∑

a

∂µK 0α

∂Qa

∣∣∣∣∣Q=0

〈kK |Qa|00〉

Being of higher order, normally a small correction

Larger dipole gradient transition moment will increase Herzberg–TellercontributionMore importantly: Can induce transitions to dipole-forbidden states, as the

irreps spanned by ∂µK 0α

∂Qamore more diverse than the dipole moment

A case study: Octahedral metal d → d transitions

All d orbitals have gerade symmetry

All dipole components have ungeradesymmetry

The 15 vibrational modes in anoctahedral complex transform asA1g , Eg , T2g , 2T1u , and T2u

Focus on modes that have ungeradesymmetry

Direct product of dipole operatorwith relevant vibrational modes

T1u ⊗ T1u = A1g ⊕ Eg ⊕ T1g ⊕ T2g

T2u ⊗ T1u = A2g ⊕ Eg ⊕ T1g ⊕ T2g

A1g now in set of irreps spanned→allows for coupling of d orbitals

A case study: Octahedral metal d → d transitions

All d orbitals have gerade symmetry

All dipole components have ungeradesymmetry

The 15 vibrational modes in anoctahedral complex transform asA1g , Eg , T2g , 2T1u , and T2u

Focus on modes that have ungeradesymmetry

Direct product of dipole operatorwith relevant vibrational modes

T1u ⊗ T1u = A1g ⊕ Eg ⊕ T1g ⊕ T2g

T2u ⊗ T1u = A2g ⊕ Eg ⊕ T1g ⊕ T2g

A1g now in set of irreps spanned→allows for coupling of d orbitals

A case study: Octahedral metal d → d transitions

All d orbitals have gerade symmetry

All dipole components have ungeradesymmetry

The 15 vibrational modes in anoctahedral complex transform asA1g , Eg , T2g , 2T1u , and T2u

Focus on modes that have ungeradesymmetry

Direct product of dipole operatorwith relevant vibrational modes

T1u ⊗ T1u = A1g ⊕ Eg ⊕ T1g ⊕ T2g

T2u ⊗ T1u = A2g ⊕ Eg ⊕ T1g ⊕ T2g

A1g now in set of irreps spanned→allows for coupling of d orbitals

A case study: Octahedral metal d → d transitions

All d orbitals have gerade symmetry

All dipole components have ungeradesymmetry

The 15 vibrational modes in anoctahedral complex transform asA1g , Eg , T2g , 2T1u , and T2u

Focus on modes that have ungeradesymmetry

Direct product of dipole operatorwith relevant vibrational modes

T1u ⊗ T1u = A1g ⊕ Eg ⊕ T1g ⊕ T2g

T2u ⊗ T1u = A2g ⊕ Eg ⊕ T1g ⊕ T2g

A1g now in set of irreps spanned→allows for coupling of d orbitals

Summary

Electronic spectroscopy in the UV/Vis range can provide detailed informationabout the electronic excited states of a moleculeOnly information about dipole-allowed excited states possible

In cases of high resolution, also the vibrational fine structure can be observed

Franck–Condon approximation assumes that electronic excitations happen whilethe nuclei remain fixedObserved absorption maximum higher in energy than vertical and 0-0transitionsFluorescence signal appears at lower energies than the absorption frequencies

Long-lived excited states are normal triplet state generated by intersystemcrossing, detected through phosphorescent emission