Quantitative description of two-photon absorption in dipolar molecules with two-level model

Nikolay Makarov, Mikhail Drobizhev, Zhiyong Suo, Aleks Rebane E. Scott Tarter, Benjamin D. Reeves, Brenda Spangler Fanqing Meng, Charles W. Spangler Craig J. Wilson, Harry L. Anderson

Department of Physics, Montana State University, Bozeman, MT 59717 Sensopath Technologies, Inc., Bozeman, MT 59715 MPA Technologies, Inc., Bozeman, MT 59715 Department of Chemistry, University of Oxford, Mansfield, Oxford, UK

ABSTRACT

High demand for efficient two-photon absorbing (2PA) chromophores requires better understanding of what molecular parameters are responsible for the enhancement of 2PA cross section.

Here we present a systematic approach for quantitative description of 2PA cross section by using two-level approximation in low-lying transitions of dipolar molecules. In these molecules, the lowest energy transition is simultaneously allowed for 1PA and 2PA. The 2PA cross section is proportional to the square of the transition dipole moment (| 

01

|), square of the difference in permanent dipole moments (| 

01

|), and is inverse proportional to the absorption linewidth FWHM. The 2PA cross section also depends on the angle between 

01

and 

01

and is maximum if they are parallel.

Three different types of molecules were studied: substituted linear diphenylaminostilbenes, linear carbazolyl-stylbenes, and push-pull porphyrins. In these types of molecules we measured 2PA cross sections, 

01

, 

01

, and linewidth. The measured 2PA cross sections do not exceed 150 GM and quantitatively agree with the quantum-mechanical expression for two-level system within experimental errors.

This work shows for the first time the quantitative structure-to-property relationship for 2PA in dipolar molecules.

Ideally, if the molecule has particularly large dipole moments 

01

= 

01

=15 D and linewidth FWHM=1000 1/cm, the value of 2PA cross section could reach 900 GM. Higher cross sections are also possible if the higher energy levels of the molecule contribute to 2PA.

Ground state dipole moment vs. permanent dipole moment difference Solvatochromic shifts of some compounds Fluorescence decay of some compounds 1 0.1

9

,  =3.11 ns

1

,  =1.71 ns 5000 4500 4000 3500 1 4 5 10 16 14 12 2 1 5 4 3 6 0.01

10 -3 0 1 2

11

,  =0.86 ns 3

Time, ns

4 5 6

6

,  =1.46 ns

4

,  =1.28 ns

12

,  =1.14 ns 7 3000 2500 2000 1500 1000 500 0 0.0

0.1

7 8 9 10 11 12 2 3 4 5 6

Compound

1

 , M -1 cm -1 24400 30900 43100 42000 33300 32600 22000 8250 24900 16683 56633 65696

r

0.022

0.031

0.030

0.051

0.053

0.031

0.055

0.071

0.029

0.031

0.150

0.128

 , ns 1.71

1.67

1.57

1.28

1.46

1.46

2.73

1.72

3.11

2.99

0.86

1.14

 F 0.72

0.34

0.67

0.72

0.64

0.51

0.11

0.25

0.19

0.18

0.27

0.34

38 116 44 109 12 7.2

 FM , ns 4.6

3.8

18 2.7

3.5

4.9

34 510 149 134 489 538  S1 , cm -1 3243 3767 3430 3230 2967 3040 114 719 150 176 617 594  S2 , cm -1 4520 6030 4210 3976 4194 4238 23 20 12 5.5

145 100  2 , GM* 47 136 69 49 70 40 2.9

20 5.8

13 2.1

2.5

 , ns ** 3.3

1.3

12 1.9

2.2

2.5

a, Å *** 5.3 (5.2) 6.0 (5.3) 5.8 (8.1) 6.6 (5.7) 7.0 (6.0) 5.7 (5.2) 8.3 (7.3) 8.2 (7.3) 7.1 (6.9) 7.2 (7.0) 9.2 (8.2) 9.3 (10.0) 4.0

8.0

5.1

5.1

12.2

13.7

|  01 |, D 6.4

5.4

6.5

7.3

6.6

6.0

80 267 1 42 128 56  S , cm -1 1277 2263 780 746 1227 1198 |  01 |, D*** 11.2 (10.8) 14.9 (12.6) 8.8 (14.5) 12.9 (10.6) 14.7 (11.7) 10.7 (9.3) 4.5 (4.8) 7.1 (6.2) 0.4 (0.4) 2.6 (2.5) 6.8 (5.8) 4.6 (5.1)  S1 is measured in toluene (

D

=2.4) for compounds 1, 3-6 and in

n

-octane (

D

=2.0) for compounds 2, 7-11; is measured in tetrahydrofuran (

D

=7.58) for compounds 1, 3-6 and in 2-chlorobutane (

D

=8.06) for compounds 2, 7-11.

*Cross section at the doubled wavelength of the lowest dipole-allowed 1PA transition.

**Calculated from  FM and  F .

***Calculated from the density of the molecules.

10 8 8 11 6 12 7 4 10 0.2



f(D)

0.3

0.4

0.5

2 9 

F

01 2  4 2  3

h

  4 

FM

  01 2 

hc

  

f S

 

a

3

f

  

2

D D

 1



 1  

FR n n

2

R

2  

F F R

   

d



d



a

   3

M

4 

N A

   1 3

n

 01  1

fl

2

f

2 0 1

|



00 |

10 

FM

 00

a

3     

F

4 

hc

    01 

A a

3   1  10 

OD R

1  10 

OD

3 

kT

0 .

4

r

 1 

r

 Spearman correlation coefficient for the data is  = –0.69, which suggests existence of anti-correlation with ~98% confidence.

This can be explained as follows: the absolute value of permanent dipole moment is limited. If the ground state dipole is high, the excited state dipole can be only increased up to the limiting value, so the dipole difference is low. If the ground state dipole is low, the difference can be much higher.

I

||

I

||  

I

 2

I

  2 300  2   15 2  4  

f

2 4 01 2    01 2  2 cos 2   1 

g

200 100 80 60 40 20 0 600 300 32000 320 30000 340 28000 360 26000 380 24000 400 420 22000 0 440 20000 500 400 300 200 100 0 300 150 18000 100 50 32000 30000 340

Frequency, cm -1

28000 26000 17000

Frequency, cm -1

16000 15000 0 560 250 18000 17000 600 16000 200 150 100 50 380 420

Wavelength, nm

11 640 15000 Q x 680 14000 Q x 24000 460 1 14000 2 720 13000 12 5 3.75

2.5

1.25

12 9 6 3 500 0 5 3.75

2.5

1.25

0 8 6 4 2 0 650 700 750

Wavelength, nm

800 850 0

Push

-

pull porphyrin monomer and push

-

pull porphyrin dimer

2 11 100 80 60 40 20 120 100 80 60 40 32000 30000

Frequency, cm -1

28000 26000 0 140 300 32000 320 30000 340 28000 360 380 26000 20 0 300 320 340 360 380 400

Wavelength, nm

420

Substituted diphenylaminostilbenes

440 0 2 400 300 200 100 0 300 500 34000 400 300 200 100 0 280 32000 30000

Frequency, cm -1

28000 26000 24000 300 350 32000 30000 400 28000 24000 400 24000 420 4 3 22000 450 26000 4 3 2 1 0 440 8 6 4 14 20000 100 13 4 3 2 1 80 60 40 20 500 0 4 3 2 32000 30000

Frequency, cm -1

28000 26000 0 80 300 32000 320 340 30000 360 28000 70 60 50 40 380 26000 400 420 24000 6 440 0 4 3 30 20 10 0 300 320 340 360 380 400

Wavelength, nm

420 440 2 1 0 1200 32000 1000 800 600 400 200 300 250 200 150 24000 30000

Frequency, cm -1

28000 26000 24000 0 300 34000 32000 350 30000 350 15 28000 400 26000 5 100 1 50 320 340

Wavelength, nm

360 380 400 0 0 280 320 360

Wavelength, nm

400

Substituted carbazolyl-stylbenes and diphenylaminostilbenes

450 24000 2.5

1.25

22000 16 440 300 20000 19000

Frequency, cm -1

18000 17000 16000 Q x (2) 250 200 150 100 50 0 500 200 20000 150 100 50 0 500 20000 5 3.75

2.5

500 22000 0 5 1.25

3.75

2.5

1.25

0 15000 7 4 150 20000 19000

Frequency, cm -1

18000 17000 16000 Q x (2) 15000 9 5 100 9 7 6 3 10 1 8 12 4 5 0 100 3 19000 540 18000 580 17000 Q x (2) 620 16000 Q x (1) 660 15000 2 1 700 0 50 0 500 60 20000 19000 540 580 18000 Q x (2) 17000 620 16000 Q x (1) 660 15000 3.75

2.5

1.25

700 0 0 100 

2 , GM

200 300 Dipole moments were measured, and the line shape function was assumed both Gaussian and Lorentzian.

More uncertainty on horizontal axes.

 2  0 .

96  10 20   

S

 01 

f

  01  

a

3

f n

2 2

PA

2 4

PA n

1

PA f

2 1

PA

300 8 10 200 50 11 2 2 40 4 Q x (1) 1 30 20 Q x (1) 3 2 100 0 9 6 7 10 8 12 1 3 14 5 4 300 10 1 540 580 620

Wavelength, nm

660 700 0 0 500 540 580 620

Wavelength, nm

Meso

-

DPAS and BDPAS

-

substituted porphyrins

400 Experimental 2PA spectrum Experimental 1PA spectrum Theoretical 2PA spectrum Theoretical 1PA spectrum 660 700 0 

R 2 f L f O

For molecule density  2 (a)  2 (b) 0.8

3.3

0.6

1.8

0 100 

2 , GM

200 300 Extinction, central frequency, solvatochromic shifts and the molecular radii were measured, and the line shape function was assumed to be the same as in 1PA. Less uncertainty on horizontal axes.

For anisotropy  2 (a)  2 (b) 1.4

1.4

0.8

0.9



R

2 

N

1  1 

i

 

y i

mod 

el i



y i

  2

f L



n

2  2 3

n

 1 .

5 

f L f O

 1 .

15 ;

f O

 2 3

n

2

n

2  1

f L f O

4  1 .

78 200 100 0 250 300 350 400 450

Wavelength, nm

500 550 Theoretical fit of 2PA (thick solid line) and 1PA (dashed) of

13

using three level density matrix model. Thin solid line and squares are the normalized 1PA spectrum and 2PA spectrum.

The molecular parameters from linear measurements are: 01  10

D

; 02  8

D

;    01

T

01  2 .

6

fs

;

T

02  1 .

8

fs

;

T

12  7

D

;  100    02  2

D

;

fs

;

T

11  1000    12

fs

;

T

22  9

D

;  1000

fs

Transition dipole moment between the excited states | 

12

|=13.5 D is obtained from the best fit.

83% of the experimental and theoretical values coincide within the error margins. To our best knowledge, this is the first demonstration of quantitative correspondence between experimental and theoretical two level model based 2PA cross section for a broad range of different dipolar compounds.

The combination of the expression for  2 (b) with the molecular radius data obtained from fluorescence anisotropy and the Onsager local field factor gives the best correlation between the experimental and theoretical cross sections.

Conclusions

• We show that perturbation theory applied for two-level system quantitatively predicts the 2PA cross sections, provided that the necessary molecular parameters such as transition- and permanent dipole moments are independently measured.

•In most cases, the discrepancy between theory and experiment was less than 20%, and always less than 50%. This is the first time that such direct quantitative correspondence is demonstrated for a wide range of dipolar molecules.

• The overall significance of this work demonstrates a practical way how a set of relatively straightforward linear spectroscopic measurements can be used to study and predict nonlinear 2PA properties.