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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
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
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