Electronic Spectroscopy UV/VIS Spectroscopy

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Kenneth Ruud UiT The Arctic University of Norway

O MS

July 2 2015

Outline

What is spectroscopy? Basic principles of UV/Vis absorption UV/Vis emission spectroscopy Phosphorescence Vibronic effects

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

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

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

Jablonski diagram for electronic processes Sn IC

ESA

ps

S1

ISC ns-μs

ESA OPA

TPA

IC

Tn

ps

T1 Fluoresc. ns

Phosphoresc. μs-ms

S0 Lower-energy excitations are always involved, whether they are detectable depend on the experimental resolution Contribute to broadening of experimental peaks Specific processes can be explored through time-resolved spectroscopy

Basic principle of absorption spectroscopy In electronic spectroscopy, we will be concerned with the absorption of light to bring the molecule to an excited electronic state Leading-order contribution: Electric dipole operator

Potential energy

Higher-order contributions can be important for high-energy light (X-rays) or for dichroisms

Molecular geometry

Basic principle of absorption spectroscopy In electronic spectroscopy, we will be concerned with the absorption of light to bring the molecule to an excited electronic state Leading-order contribution: Electric dipole operator

Potential energy

Higher-order contributions can be important for high-energy light (X-rays) or for dichroisms The absorption cross section from the ground to the excited state is Ai→f =

πωfi NA |µfi |2 3ε0 ~c

We recall that the transition dipole can be determined from the single residue of the linear response function lim (ωf 0 −ω) ααβ (−ω; ω) = h0|ˆ µα |f ihf |ˆ µβ |0i

ω→ωf 0

Molecular geometry

Basic principle of absorption spectroscopy In electronic spectroscopy, we will be concerned with the absorption of light to bring the molecule to an excited electronic state Leading-order contribution: Electric dipole operator

Potential energy

Higher-order contributions can be important for high-energy light (X-rays) or for dichroisms The absorption cross section from the ground to the excited state is Ai→f =

πωfi NA |µfi |2 3ε0 ~c

We recall that the transition dipole can be determined from the single residue of the linear response function lim (ωf 0 −ω) ααβ (−ω; ω) = h0|ˆ µα |f ihf |ˆ µβ |0i

ω→ωf 0

Molecular geometry

Important to know what excitation energy is reported.

One-photon absorption: Symmetry considerations For molecule with inversion symmetry, parity has to change Spin has to be preserved

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 dipole approximation, the states we can reach have to be dipole allowed

→ 0 µx/y/z f have to have a component that transforms as the totally symmetric irrep

Table : Group multiplication table for the D2h point group. Ag B1g B2g B3g Au B1u B2u B3u

E 1 1 1 1 1 1 1 1

C2 (z) 1 1 -1 -1 1 1 -1 -1

C2 (y) 1 -1 1 -1 1 -1 1 -1

C2 (x) 1 -1 -1 1 1 -1 -1 1

i 1 1 1 1 -1 -1 -1 -1

σ(xy) 1 1 -1 -1 -1 -1 1 1

σ(xz) 1 -1 1 -1 -1 1 -1 1

σ(yz) 1 -1 -1 1 -1 1 1 -1

x 2, y 2, z2 Rz , xy Ry , xz Rx , yz z y x

Emission (fluorescence) spectroscopy Instead of measuring the light absorbed, we can measure the light emitted from an excited state The excited state can be reached in different ways (multiphoton, intersystem crossings....) Requires a difference in the frequency of emitted light compared to that absorbed (which implies a lifetime of the excited state) Through stimulated emission, it can be used to follow excited-state dynamics

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

Emission (fluorescence) spectroscopy Instead of measuring the light absorbed, we can measure the light emitted from an excited state The excited state can be reached in different ways (multiphoton, intersystem crossings....) Requires a difference in the frequency of emitted light compared to that absorbed (which implies a lifetime of the excited state) Through stimulated emission, it can be used to follow excited-state dynamics Computationally, it is determined by the same residue as for absorption, and it can be determined from the ground or the excited state https://www.lifetechnologies.com/content/ dam/LifeTech/Images/integration/Fluorescencediagram-400px.jpg

lim (ωf 0 −ω) ααβ (−ω; ω) = h0|ˆ µα |f ihf |ˆ µβ |0i

ω→ωf 0

Computationally the same residue, but different (excited-state) geometry

Phosphorescence Through intersystem crossings and internal conversions, a molecule in a singlet excited electronic state may be interconverted and end up in a triplet excited state The electromagnetic light has no spin-component → no possibility to emit light and return to the singlet ground state However, there exists (from relativistic theory) a coupling of the spin and orbital motion of the electron, the spin–orbit interaction ˆ SO H


 X X mi +2mj ·lij e m ·l i iK  α2  ZK =− − 3 8π0 me fs riK rij3 iK i6=j

Including these corrections, the transition moment from a specific triplet spin sub-level is k Mα =

∞ ˆ SO |T k i X hS0 |ˆ µα |Sn ihSn |H 1 n=0

E(Sn ) − E(T1 )

+

∞ ˆ SO |Tn ihTn |ˆ X hS0 |H µα |T1k i n=1

E(Tn ) − E(S0 )

ˆ k , Ωii ˆ 0,ω /hf |Ω|0i ˆ = lim ~(ωf 0 − ω)hhˆ µα ; H SO ω→ωf 0

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

Phosphorescence Through intersystem crossings and internal conversions, a molecule in a singlet excited electronic state may be interconverted and end up in a triplet excited state The electromagnetic light has no spin-component → no possibility to emit light and return to the singlet ground state However, there exists (from relativistic theory) a coupling of the spin and orbital motion of the electron, the spin–orbit interaction ˆ SO H


 X X mi +2mj ·lij e m ·l i iK  α2  ZK =− − 3 8π0 me fs riK rij3 iK i6=j

Including these corrections, the transition moment from a specific triplet spin sub-level is k Mα =

∞ ˆ SO |T k i X hS0 |ˆ µα |Sn ihSn |H 1 n=0

E(Sn ) − E(T1 )

+

∞ ˆ SO |Tn ihTn |ˆ X hS0 |H µα |T1k i n=1

E(Tn ) − E(S0 )

ˆ k , Ωii ˆ 0,ω /hf |Ω|0i ˆ = lim ~(ωf 0 − ω)hhˆ µα ; H SO ω→ωf 0

In the relativistic framework, spin is no longer a good quantum number, and the spin–orbit operator mixes states of different spin symmetry Phosphorenscence 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–Fock calculations HF/taug-cc-pVTZ 1.30 Å C

O

H H

1.96 eV 3.46 eV

1.20 Å

O

H

C H

3.46 eV

5.1.10 -7

1.96 eV

7.3.10 -7

24.7 ms

Franck–Condon principle We recall that electronic transitions are largely governed by dipole transition moments Including the full vibronic wave function and expanding the geometry dependence of the dipole moment K

0

hk , K |ˆ µα |0, 0 i = hk

K

0 0 |µK α (Q)|0 i

=

X ∂µK 0 α ∂Qa a

0 K 0 µK α (0)hk |0 i+

Q=0

When exciting from the vibronic ground state to an electronic excited state, the population of the vibrational states of the electronic excited state is determined by hk K |00 i

Potential energy

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

Electronic spectra with vibronic features will be characterized by a progression of lines with fixed separation matching the vibrational frequency Molecular geometry

hk K |Qa |00 i+. . .

Vibronic UV/Vis spectrum of a diatomic molecule

Fluorescence

0 →4 0 →3

Intensity

0 →5

Absorption

3 ←0 4 ←0

0 →2

2 ←0

0 →6

5 ←0 6 ←0

0 →1 1 ←0 0 →7 0 →8 0 →9

7 ←0 0−0

8 ←0 9 ←0

Energy Assumed similar vibrational frequencies in ground and excited states

A hierarchy of vibronic models In general, the equilibrium geometry and vibrational frequencies/normal coordinates of the ground and excited electronic states differ Normal coordinate are linear, and thus we can relate the ground- and excited-state normal coordinates as Q0 = JQK + K

K = L0 ∆q

∆qi =

√

  0 K − xeq,i mi xeq,i

The rotation of the normal coordinates from one set of normal coordinates to another is called Duschinsky rotations  −1 J = L0 LK

A hierarchy of vibronic models In general, the equilibrium geometry and vibrational frequencies/normal coordinates of the ground and excited electronic states differ Normal coordinate are linear, and thus we can relate the ground- and excited-state normal coordinates as Q0 = JQK + K

K = L0 ∆q

∆qi =

√

  0 K − xeq,i mi xeq,i

The rotation of the normal coordinates from one set of normal coordinates to another is called Duschinsky rotations  −1 J = L0 LK

A hierarchy of vibronic models

A hierarchy of vibronic models In general, the equilibrium geometry and vibrational frequencies/normal coordinates of the ground and excited electronic states differ Normal coordinate are linear, and thus we can relate the ground- and excited-state normal coordinates as Q0 = JQK + K

K = L0 ∆q

∆qi =

√

  0 K − xeq,i mi xeq,i

The rotation of the normal coordinates from one set of normal coordinates to another is called Duschinsky rotations  −1 J = L0 LK

A hierarchy of vibronic models Full adiabatic Franck–Condon approximation

A hierarchy of vibronic models In general, the equilibrium geometry and vibrational frequencies/normal coordinates of the ground and excited electronic states differ Normal coordinate are linear, and thus we can relate the ground- and excited-state normal coordinates as Q0 = JQK + K

K = L0 ∆q

∆qi =

√

  0 K − xeq,i mi xeq,i

The rotation of the normal coordinates from one set of normal coordinates to another is called Duschinsky rotations  −1 J = L0 LK

A hierarchy of vibronic models Full adiabatic Franck–Condon approximation Ignore the Duschinsky rotations → Excited-state vibrational frequencies scaled from the ground state

A hierarchy of vibronic models In general, the equilibrium geometry and vibrational frequencies/normal coordinates of the ground and excited electronic states differ Normal coordinate are linear, and thus we can relate the ground- and excited-state normal coordinates as Q0 = JQK + K

K = L0 ∆q

∆qi =

√

  0 K − xeq,i mi xeq,i

The rotation of the normal coordinates from one set of normal coordinates to another is called Duschinsky rotations  −1 J = L0 LK

A hierarchy of vibronic models Full adiabatic Franck–Condon approximation Ignore the Duschinsky rotations → Excited-state vibrational frequencies scaled from the ground state Use the ground-state vibrational force field also for the excited state, but use a shifted geometry

A hierarchy of vibronic models In general, the equilibrium geometry and vibrational frequencies/normal coordinates of the ground and excited electronic states differ Normal coordinate are linear, and thus we can relate the ground- and excited-state normal coordinates as Q0 = JQK + K

K = L0 ∆q

∆qi =

√

  0 K − xeq,i mi xeq,i

The rotation of the normal coordinates from one set of normal coordinates to another is called Duschinsky rotations  −1 J = L0 LK

A hierarchy of vibronic models Full adiabatic Franck–Condon approximation Ignore the Duschinsky rotations → Excited-state vibrational frequencies scaled from the ground state Use the ground-state vibrational force field also for the excited state, but use a shifted geometry Linear coupling model (vertical gradient approximation): As above, but approximate the excited-state equilibrium geomtry by the excited-state gradient projected on the ground-state force field

Different models graphically illustrated

Evaluating Franck–Condon factors In order to evaluate Franck–Condon factors, we need to evaluate the overlap of vibrational wave functions of the form 1

ω4 2 1 √ |n i = 1 √0 Hn ( ω0 Q0 )e− 2 ω0 Q0 , n π 4 2 n! 0

1

ω4 2 1 √ |k i = 1 √K Hk ( ωK QK )e− 2 ωK QK k 4 π 2 k! K

Solving the general case is complicated, involves the use of so-called generating functions For the special case of a diatomic molecules with ωK = ω0 , we can write s P(T , U) =

X

k

T U

n

k,n 1

2k 2n K 0 hk |n i = I0 eBT +DU+ETU k !n!

2

I 0 = e − 4 ω0 ∆ √ 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 is populated → D = 0 We expand the left- and right-hand sides h0K | 00 i + T

√

√ 2 2h1K | 00 i + T 2 2h2K | 00 i + T 3 √ h3K | 00 i + . . . 3   1 1 2 2 = I0 1 + TB + T B + T 3 B 3 + . . . 2! 3!

Collecting terms with the same order in T , and we will find r hk K |00 i = I0

 k 1 √ ω ∆ 0 2k k!

Assuming different vibrational frequencies in the ground and excited electronic states complicates the formulas significantly s P(T , U) =

X k,n

T k Un

2 2 2k 2n K 0 hk |n i = I0 eAT +BT +CU +DU+ETU k!n!

The linear coupling model Two approximations Assume the excited-state force field is the same as the ground-state force field: LK = L0 , J = I, and frequency matrix ΛK = Λ0 Assume that the shift in excited-state equilibrium geometry can be approximated by the excited-state gradient projected onto the ground-state normal modes K = L0

∂E K ∂∆

Reduced to a generalization to polyatomic molecules of the case of a diatomic molecule with the same vibrational frequencies and force fields p   4 |Γ0 Γ0 |4m 1 I0 = p exp − K† Γ0 K , 4 |(2Γ0 )| p 0 B= Γ K p D = − Γ0 K, E =2I

Herzberg–Teller couplings

If we go beyond the Franck–Condon approximation, the leading-order correction is the Herzberg–Teller contribution hk K , K |ˆ µα |0, 00 i =

X ∂µK 0 α ∂Qa a

hk K |Qa |00 i

Q=0

Being of higher order, normally a small correction Larger dipole gradient transition moment will increase Herzberg–Teller contribution More importantly: Can induce transitions to dipole-forbidden states, as the irreps spanned by

0 ∂µK α ∂Qa

more more diverse than the dipole moment

A case study: Octahedral metal d → d transitions

All d orbitals have gerade symmetry

A case study: Octahedral metal d → d transitions

All d orbitals have gerade symmetry All dipole components have ungerade symmetry

A case study: Octahedral metal d → d transitions

All d orbitals have gerade symmetry All dipole components have ungerade symmetry The 15 vibrational modes in an octahedral complex transform as A1g , Eg , T2g , 2T1u , and T2u

A case study: Octahedral metal d → d transitions

All d orbitals have gerade symmetry All dipole components have ungerade symmetry The 15 vibrational modes in an octahedral complex transform as A1g , Eg , T2g , 2T1u , and T2u Focus on modes that have ungerade symmetry Direct product of dipole operator with 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 information about the electronic excited states of a molecule Only 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 while the nuclei remain fixed Observed absorption maximum higher in energy than vertical and 0-0 transitions Fluorescence signal appears at lower energies than the absorption frequencies Long-lived excited states are normal triplet state generated by intersystem crossing, detected through phosphorescent emission