Multiphoton spectroscopy

Transcription

1 Multiphoton spectroscopy Kenneth Ruud UiT The Arctic University of Norway I UNIVERSITETET TROMSØ July

2 Outline Why multiphoton absorption? What is multiphoton absorption? Multiphoton absorption cross sections Few-state models for two-photon absorption Multiphoton absorption of centrosymmetric molecules

3 Surface- and interface-specific properties By using multiphoton spectroscopies, we can device nonlinear vibrational spectroscopies that has unique features Focality Surface/interface specificity G. A. Somoraj, Chem. Eng. News 86, 21 (2008 A multitude of other high-order processes can be designed, enhancing specific characteristics of the system being studied

4 Biophotonics: Using complementary information Section of a rabbit aorta. Green is two-photon-induced emission fluorescence (elastin), blue is forwardscattered second-harmonic generation (collagen), red is coherent anti-stokes Raman Scattering (CARS, fatty acids) Huh-7 cells.

5 A quick reminder from response theory Transition moments can be calculated from the residues of response functions The two-photon transition moment is defined as M (2) 0f = 1 n [ 0 ˆµβ n n ˆµ γ f + 0 ˆµγ n n ˆµ ] β f ω n0 ω 1 ω n0 ω 2 Formally, two-photon absorption is related to the cubic response function

6 A quick reminder from response theory Transition moments can be calculated from the residues of response functions The two-photon transition moment is defined as M (2) 0f = 1 n [ 0 ˆµβ n n ˆµ γ f + 0 ˆµγ n n ˆµ ] β f ω n0 ω 1 ω n0 ω 2 Formally, two-photon absorption is related to the cubic response function However, we can identify the two-photon transition matrix element from the residue of the quadratic response function lim (ω f 0 ω 2 )β( ω ω 2 ω σ; ω 1, ω 2 ) f 0 = 1 [ 0 ˆµα n n ˆµ β f f ˆµ γ ˆµ ] β n n ˆµ α f f ˆµ γ 0 (ω n n0 ω 1 ω f 0 ) (ω n0 + ω 1 ) = 1 [ 0 ˆµα n n ˆµ β f ˆµ ] β n n ˆµ α f f ˆµ γ 0 (ω n nf ω 1 ) (ω n0 + ω 1 )

7 Multiphoton absorption cross section General expression for multiphoton absorption probability ω jpa f 0 = 2π ( 2π ) j [ j j 1 ] e2j n i ω i ρ i (E i ) ] j 1[ 1 dω l V 2(j 1) δjpa. i=1 l=1 ρ i (E i ) density of states of photons with energy E i n i the number of photons of energy E i

8 Multiphoton absorption cross section General expression for multiphoton absorption probability ω jpa f 0 = 2π ( 2π ) j [ j j 1 ] e2j n i ω i ρ i (E i ) ] j 1[ 1 dω l V 2(j 1) δjpa. i=1 l=1 ρ i (E i ) density of states of photons with energy E i n i the number of photons of energy E i We can define the photon and energy flux F i = cρ i d ω i n i V [ ] F = photon cm 2 s ; I(ω i ) = 2 ω i cρ i n i V [ ] I = erg cm 2 The general expression for the multiphoton absorption probability is then [ j 1 ] ω jpa = (2π)j+1 j j 1 2j l=1 F l ω l I(ω j ) j e j+1 c j [ j i=1 ω ω i 1 i ] 2(j 1) δjpa i=1

9 Multiphoton absorption transition probabilities ω OPA = (4π2 ) c 2 e2 I(ω 1 ) δ TPA ω TPA = (8π3 ) c 2 3 e4 F 1 ω 1 I(ω 2 ) δ TPA ω 3PA = (16π4 ) c 3 4 e6 F 1 ω 1 F 2 ω 2 I(ω 3 ) δ 3PA ω 4PA = (32π5 ) c 4 5 e8 F 1 ω 1 F 2 ω 2 F 3 ω 3 I(ω 4 ) δ 4PA Describes the transition probability of one photon in the presence of j 1 other photons

10 Multiphoton absorption cross section In order to convert the transition probabilites into cross sections, we need to Divide by the photon flux for each photon Make the substitution F i = I (ω i ) dω i ω i Introduce a lineshape function to account for absorption in an energy band, defined to satisfy j g ω i dω = 1 i

11 Multiphoton absorption cross section The general expression for the multiphoton absorption cross section is σ jpa = ωjpa p 0 ω j dω j j j [ j i=1 F g ω i = (2π)j+1 i ]dω j c j j e 2j ω i g ω i δ jpa i=1 i=1 i=1 More specifically σ TPA = 8π3 c 2 2 e4 ω 1 ω 2 g(ω 1 + ω 2 ) δ TPA σ 3PA = 16π4 c 3 3 e6 ω 1 ω 2 ω 3 g(ω 1 + ω 2 + ω 3 ) δ 3PA σ 4PA = 32π5 c 4 4 e8 ω 1 ω 2 ω 3 ω 4 g(ω 1 + ω 2 + ω 3 + ω 4 ) δ 4PA Units are cm 2j s j 1 /photon j 1

12 An apparently simple problem: Istropic averaging When comparing with experiment, isotropic averaging in most cases required The general expression for a multiphoton absorption cross section δ = A [n] L i 1...i n i1 λ 1...L inλn P [n] i 1...i n,λ 1...λ n L = λ 1...λ n ; P [n] i 1...i 2m = S [m] i 1...i m S[m] i m+1...i 2m cos(φ)cos(θ)cos(ψ) sin(φ)cos(θ)cos(ψ) sin(θ)cos(ψ) sin(φ)sin(ψ) +cos(φ)sin(ψ) cos(φ)cos(θ)sin(ψ) sin(φ)cos(θ)sin(ψ) sin(θ)sin(ψ) sin(φ)cos(ψ) +cos(φ)cos(ψ) cos(φ)sin(θ) sin(φ)sin(θ) cos(θ) Isotropic averaging performed on the coordinate transformation, I (n) = 1 2π π 2π ( i 1...i n,λ 1...λ n 8π 2 L i1 λ 1...L inλn sinθ dφ dθ dψ = f (n)) T M (n) g (n) φ=0 θ=0 ψ=0 f (n) and g (n) are strings with m i (2i 1) elements of Kronecker delta functions D. L. Andrews and W. A. Ghoul, J. Chem. Phys.75, 530 (1981). G. Wagnière, J. Chem. Phys.76, 473 (1982).

13 The case of linear polarization Assuming linear polarization, the only nonvanishing element of A [n] i 1...i n is the all-z component A (f [n] (n)) T N n z M (n) n g (n) P [n] = (M (n) g (n) P [n] ) i i=1

14 The case of linear polarization Assuming linear polarization, the only nonvanishing element of A [n] i 1...i n is the all-z component A (f [n] (n)) T N n z M (n) n g (n) P [n] = (M (n) g (n) P [n] ) i i=1 It can be shown that the rotational average of an n-photon absorption process (involving a 2n-rank tensor) can be written δ npa n = ζ n Ci n Si n Si n i=1 ζ n = n i=1 C n 0 = n!, C n i 1 (2i + 1) = n! 2i 1 (n j) j=0 (i! 2 i ) 2 1 i n+1 2, i N n Ci n C0 n C1 n C2 n C3 n D. H. Friese, M. T. P. Beerepoot, J.. Chem. Phys. 141, (2014).

15 Rotationally averaged n-photon absorption (linear polarization) δ OPA = 1 S a Sa, 3 a δ TPA = 1 ( ) 2Sab Sab + S aa Sbb, 15 ab δ 3PA = 1 ( ) 2Sabc Sabc + 3S aab Sbcc, 35 abc δ 4PA = 1 ( ) 8Sabcd Sabcd + 24S aabc Sbcdd + 3S aabb Sccdd, 315 abcd δ 5PA = 1 ( ) 8Sabcde Sabcde + 40S aabcd Sbcdee + 15S aabbc Scddee, 693 abcde δ 6PA = 1 ( 16Sabcdef Sabcdef + 120S aabcde Sbcdeff + 90S aabbcd Scdeeff abcdef δ 7PA = ) 5S aabbcc Sddeeff, ( 16S abcdefg Sabcdefg + 168S aabcdef Sbcdefgg + 210S aabbcde Scdeffgg + abcdefg 35S aabbccd Sdeeffgg ). D. H. Friese, M. T. P. Beerepoot, J.. Chem. Phys. 141, (2014).

16 Mind the experiment! We have in principle an expression for TPA cross sections Nevertheless, there is often confusion about TPA cross sections in the literature σ TPA = Nπ3 αa0 5ω2 δ TPA g(2ω, ω 0, Γ) c The formula assumes the frequency of the incoming photon Does the rotational average include the factor 1/15? Is data reported for FWHM or HWHM? Experiment is assumed to be double beam, but most current experiments are single beam N = 8 double beam Which experiment? single beam eq 3 Definition of eq 14 S? N = 4 N = 16

17 Accessing OPA-dark states We noted that the gross selection rule for OPA is that the transition is dipole allowed in molecules with symmetry MPA can allow the OPA-dark states to be detected Table : Group multiplication table for the D 2h point group. E C 2 (z) C 2 (y) C 2 (x) i σ(xy) σ(xz) σ(yz) A g x 2, y 2, z 2 B 1g R z, xy B 2g R y, xz B 3g R x, yz A u B 1u z B 2u y B 3u x

18 Understanding TPA: The two-state model Let us assume a system with only two states, 0 and f, the transition probability reduces to [ S αβ = 1 0 ˆµα f f ˆµ β f + 0 ˆµ ] β f f ˆµ α f ω f 0 ω ω f 0 ω = 2 [ ] 0 ˆµ α f f ˆµ ω β f + 0 ˆµ β f f ˆµ α f f 0 The two-photon absorption cross section can then be written σ0f 2SM = 16π3 αa0 5 1 ( ) c πγ µ 0f 2 µ 0f 2 2 cos 2 ϑ + 1

19 Understanding TPA: The two-state model Let us assume a system with only two states, 0 and f, the transition probability reduces to [ S αβ = 1 0 ˆµα f f ˆµ β f + 0 ˆµ ] β f f ˆµ α f ω f 0 ω ω f 0 ω = 2 [ ] 0 ˆµ α f f ˆµ ω β f + 0 ˆµ β f f ˆµ α f f 0 The two-photon absorption cross section can then be written σ0f 2SM = 16π3 αa0 5 1 ( ) c πγ µ 0f 2 µ 0f 2 2 cos 2 ϑ + 1 Only applicable to TPA excitations to dipole-allowed states TPA enhanced through Maximizing dipole-transition moment Maximizing the change in the dipole moment between the ground- and the excited state Maximizing the overlap between the transition dipole and dipole-moment change vectors

20 Understanding TPA: The two-state model Let us assume a system with only two states, 0 and f, the transition probability reduces to [ S αβ = 1 0 ˆµα f f ˆµ β f + 0 ˆµ ] β f f ˆµ α f ω f 0 ω ω f 0 ω = 2 [ ] 0 ˆµ α f f ˆµ ω β f + 0 ˆµ β f f ˆµ α f f 0 The two-photon absorption cross section can then be written σ0f 2SM = 16π3 αa0 5 1 ( ) c πγ µ 0f 2 µ 0f 2 2 cos 2 ϑ + 1 Only applicable to TPA excitations to dipole-allowed states TPA enhanced through Maximizing dipole-transition moment Maximizing the change in the dipole moment between the ground- and the excited state Maximizing the overlap between the transition dipole and dipole-moment change vectors Charge-transfer transitions in conjugated donor-acceptor systems with large spatial overlap of ground- and excited-state electron densities

21 Understanding TPA: The three-state model To access dipole-forbidden states, we need to consider a three-state model S αβ = 1 [ 0 ˆµα i i ˆµ β f ω i0 ω f 0 /2 For which we then obtain the TPA cross section + 0 ˆµ ] β i i ˆµ α f ω i0 ω f 0 /2 σ0f 3SM = 16π3 αa0 5 1 µ 0i 2 µ if 2 ( 2 cos 2 ϑ + 1 ) c πγ ω i0 ω f 0 /2

22 Understanding TPA: The three-state model To access dipole-forbidden states, we need to consider a three-state model S αβ = 1 [ 0 ˆµα i i ˆµ β f ω i0 ω f 0 /2 For which we then obtain the TPA cross section + 0 ˆµ ] β i i ˆµ α f ω i0 ω f 0 /2 σ0f 3SM = 16π3 αa0 5 1 µ 0i 2 µ if 2 ( 2 cos 2 ϑ + 1 ) c πγ ω i0 ω f 0 /2 Dipole-forbidden states can be reached through an intermediate, dipole-allowed state Maximizing the two different dipole-allowed transitions and their alignment will maximize the TPA cross section

23 Understanding TPA: The three-state model To access dipole-forbidden states, we need to consider a three-state model S αβ = 1 [ 0 ˆµα i i ˆµ β f ω i0 ω f 0 /2 For which we then obtain the TPA cross section + 0 ˆµ ] β i i ˆµ α f ω i0 ω f 0 /2 σ0f 3SM = 16π3 αa0 5 1 µ 0i 2 µ if 2 ( 2 cos 2 ϑ + 1 ) c πγ ω i0 ω f 0 /2 Dipole-forbidden states can be reached through an intermediate, dipole-allowed state Maximizing the two different dipole-allowed transitions and their alignment will maximize the TPA cross section Opens TPA-active molecules based on conjugated donor-donor and acceptor acceptor systems

24 Understanding TPA: The three-state model To access dipole-forbidden states, we need to consider a three-state model S αβ = 1 [ 0 ˆµα i i ˆµ β f ω i0 ω f 0 /2 For which we then obtain the TPA cross section + 0 ˆµ ] β i i ˆµ α f ω i0 ω f 0 /2 σ0f 3SM = 16π3 αa0 5 1 µ 0i 2 µ if 2 ( 2 cos 2 ϑ + 1 ) c πγ ω i0 ω f 0 /2 Dipole-forbidden states can be reached through an intermediate, dipole-allowed state Maximizing the two different dipole-allowed transitions and their alignment will maximize the TPA cross section Opens TPA-active molecules based on conjugated donor-donor and acceptor acceptor systems TPA enhancement also possible through resonance enhancement, ω i0 ω f 0 /2

25 TPA-active intermolecular charge-transfer states S. Chakrabarti and K. Ruud, Phys. Chem. Chem. Phys. 11, 2592 (2009).

26 TPA and OPA results Geometry optimzied using the 6-311G** basis and Truhlars dispersion corrected MPW1B95 functional OPA and TPA calculated using the CAMB3LYP functional and the cc-pvdz basis set as residues of linear and quadratic response functions All OPA cross sections less than 0.01 in the complex, and Λ = Molecule TPA excitation δ a.u δ GM Wave length(nm) Tweezer x x x x x10 04 TNF x x x x x10 04 Tweezer-TNF x x10 06 complex x x x x10 03 S. Chakrabarti and K. Ruud, Phys. Chem. Chem. Phys. 11, 2592 (2009).

27 Multiphoton absorption: XC functional dependence pna 2-photon absorption strength in a.u HF BLYP B3LYP CAM-B3LYP 1e No. of excited state HF 1e+06 HF 3-photon absorption strength in a.u BLYP B3LYP CAM-B3LYP 4-photon absorption strength in a.u BLYP B3LYP CAM-B3LYP No. of excited state No. of excited state Basis set dependence of the multiphoton absorption behaviour of the pna molecule for TPA (top), 3PA (bottom left) and 4PA (bottom right). All data were calculated using the aug-cc-pvdz basis set and the CAM-B3LYP density functional. Important: Not all functionals generate the same excited states or in the same order, examine the states if comparing different calculations

28 Multiphoton absorption: Basis set dependence pna 2-photon absorption strength in a.u aug-cc-pvdz aug-cc-pvtz 1e No. of excited state aug-cc-pvdz 1e+06 aug-cc-pvdz 2-photon absorption strength in a.u aug-cc-pvtz 2-photon absorption strength in a.u aug-cc-pvtz No. of excited state No. of excited state Multiphoton absorption behaviour of the PNA molecule using the CAMB3LYP functional. Diffuse functions important for correct results

29 Multiphoton absorption in centrosymmetric systems MPA cross section in specific units 1.2e-19 1e-19 8e-20 6e-20 4e-20 2e-20 OPA TPA 3PA 4PA 5PA Excited state Multiphoton absorption behaviour of para-dinitro-benzene calculated using the aug-cc-pvdz basis set and the CAM-B3LYP density functional.

30 Rationalizing MPA in centrosymmetric molecules Inversion symmetry implies states are gerade or ungerade Dipole transitions only allowed between states of different inversion symmetry Gross selection rule: Absorption processes involving an odd number of photons only allowed to states of ungerade symmetry, processes involving an even number of photons only to states of gerade symmetry

31 Rationalizing MPA in centrosymmetric molecules Inversion symmetry implies states are gerade or ungerade Dipole transitions only allowed between states of different inversion symmetry Gross selection rule: Absorption processes involving an odd number of photons only allowed to states of ungerade symmetry, processes involving an even number of photons only to states of gerade symmetry We divide all excited states into their inversion symmetry properties In the symmetry of the final state, we introduce a two-state model, keep all states in the other symmetries

32 Rationalizing MPA in centrosymmetric molecules Inversion symmetry implies states are gerade or ungerade Dipole transitions only allowed between states of different inversion symmetry Gross selection rule: Absorption processes involving an odd number of photons only allowed to states of ungerade symmetry, processes involving an even number of photons only to states of gerade symmetry We divide all excited states into their inversion symmetry properties In the symmetry of the final state, we introduce a two-state model, keep all states in the other symmetries M0f 3PA,αβγ = 1 n U n G 0 ˆµ α U k U k ˆµ β G l G l ˆµ γ U f Pαβγ 2. (ω α ω k=1 l=1 Uk )(ω α + ω β ω Gl )

33 Rationalizing MPA in centrosymmetric molecules: 4PA M0f 4PA,αβγδ = 1 n U n G n U 0 ˆµ α U k U k ˆµ β G l G l ˆµ γ U m U m ˆµ δ G f Pαβγδ 3 (ω k=1 l=1 m=1 α ω Uk )(ω α + ω β ω Gl )(ω α + ω β + ω γ ω Um ) 1 n U n U 0 ˆµ α U k U k ˆµ β G f G f ˆµ γ U m U m ˆµ γ G f Pαβγδ 3 (ω k=1 m=1 α ω Uk )(ω α + ω β ω Gf )(ω α + ω β + ω γ ω Um ) 1 Pαβγδ MTPA n U 0f,αβ G ) (ω 2 f ˆµ γ U m U m ˆµ δ G f ωβ (ω ω Gf m=1 α + ω β + ω γ ω Um ) Unless lower-order forbidden, even-order MPA cross sections are proportional to TPA cross sections, and odd-order MPA proportional to OPA. Proportionality constant is the excited-state polarizability Resonance-enhancements can be achieved when there are intermediate states close to multiples of the incoming photon frequency. 3PA: 2/3ω f 0 4PA: 3/4ω f 0 5PA: 2/5ω f 0 and 4/5ω f 0

34 Summary Multiphoton absorption allows you to use longer wavelengths (less energy) to excite molecules into excited electronic states Longer wavelengths penetrate deeper in skin and are less harmful Multiple photon sources allows for focality Conjugated systems with D-D, D-A or A-A substitution will in general support multiphoton absorption Multiple photons also allows OPA-dark states to be explored There is strong similarities in intensities for odd- and even-order processes.