Excited-state Raman spectroscopy with and without actinic excitation:
S1 Raman spectra of trans-azobenzene
A. L. Dobryakov, M. Quick, I. N. Ioffe,a A. A. Granovsky,b
N. P. Ernsting,* and S. A. Kovalenko*
Department of Chemistry, Humboldt-Universität zu Berlin, Brook-Taylor-Str. 2,
D-12489 Berlin, Germany
Supporting Information
1
A.
THE ELECTRIC FIELD AND BASIC CALCULATION OF THE
TRANSIENT RAMAN SPECTRUM
Nonlinear spectroscopic signals can be calculated with the response function
formalism developed by Mukamel et al.1,2 The basic concepts and key definitions from
Ref. 3-9 are adapted for the present work.
The femtosecond stimulated Raman signal is treated within the fifth-order
perturbation formalism. An useful approximation is the assumption that the interaction
with the actinic pump and the scattering events, induced by the Raman and probe pulses,
are well separated in time with a time delay td. The actinic pump prepares the system with
population in a specific excited state p(a ) . The Raman and probe pulses induce the
optical changes in the third-order of the incoming electric fields. In a previous papers3-9
we developed a unified description of sequential and coherent contributions to optical
transients in the third-order. The sequential contribution generates the background and is
not considered here. We focus on the coherent contribution which arises from the
simultaneous Raman and probe pulses, A(3) (2 ,0) , and generates the background-free
Raman signal. The quantity of interest is the transient spectrum A ( 5) (2 , t d ) induced by
the actinic pump and measured at delay td by the coinciding Raman and probe pulses as
function of probe frequency 2 ; it can be written as
A(5) (2 , t d )  22  p(a , t d )A(3) (2 ,0)
(A1)
a

A(3) (2 ,0)  Im P(3) (2 ) / E 2 (2 )

(A2)
Note that in the absence of actinic excitation p(a,td) corresponds to the timeindependent population in the ground state, and the ground state Raman signal is defined
by Eq.(A2).
The induced polarization P(3) (2 ) which is taken in the probe direction
k 2  k1  k1  k 2 , and which causes the transient signal A(3) (2 ,0) , is captured though
the nonlinear response R ( 3) ( t 3 , t 2 , t1 ) 1-2 and the Fourier transform ( FT(...) ) of the
effective coherent field product E (3) ( t , t 3 , t 2 , t1 ) 4-9



0
0
0


P (3) (2 )   dt 3  dt 2  dt1R (3) ( t 3 , t 2 , t1 )FT E (3) ( t, t 3 , t 2 , t1 ) .
E (3) ( t , t 3 , t 2 , t1 )  E1 ( t , t 3 , t 2 , t1 )  E 2 ( t , t 3 , t 2 , t1 )

E1(t, t 3 , t 2 , t1 )  E1(t  t 3 )E2 (t  t 3  t 2 )E1 (t  t 3  t 2  t1)
(A3)
(A4)
(A5)
2

E2 (t, t 3 , t 2 , t1 )  E1 (t  t 3 )E1 (t  t 3  t 2 )E2 (t  t 3  t 2  t1 )
(A6)
E1 (t ) and E 2 (t ) represent in Eqs.(A5-A6) the Raman/probe pulses,
~
E1, 2 ( t )  E1, 2 ( t ) exp( i1, 2 t )
(A7)
~
where 1, 2 and E1, 2 are the central frequency and temporal envelope of Raman/probe
pulses, respectively. Exact forms for FTE (3) ( t , t 3 , t 2 , t1 )  with transform-limited and
linearly chirped Gaussian envelopes are presented in Ref. 3-8 and the reader is referred to
these papers for details. Here we present a useful form for the calculation of backgroundfree Raman signal with two-sided exponential decay pulse envelope9, i.e,
~
E1, 2 ( t )  exp  1, 2 t  with pulse width 1, 2 . This form is preferred over a Gaussian, since
it allows one to find a simple analytic expression. When the probe is a much shorter than
Raman pulse, 1   2 , the result is obtained as


FT E (3) ( t , t 3 , t 2 , t1 )  I1 ( t 3 , t 2 , t1 )  I 2 ( t 3 , t 2 , t1 )
(A8)
I1 (t 3 , t 2 , t1 )  exp i2 t 3  i(2  1  i1 )t 2  i(1  i1 )t1 
(A9)
I2 (t 3 , t 2 , t1 )  exp i2 t 3  i(2  1  i1 )t 2  i(2  2i1 )t1 
(A10)
The calculation of P(3) (2 ) in Eq. (A3) involves a triple integration. To calculate
the coherent contribution a reduced response function is used in the general form3-8
~ t  i
~ t  i
~ t ).
R (k3) (t 3 , t 2 , t1 )  iˆ k  exp( i
mn 3
kl 2
pq 1
(A11)
~    i . Two subscripts indicate the frequency
with complex frequency 
ab
ab
ab
difference, ab  a  b , and dephasing rate ab between states a and b . A pathway
k is specified by subscripts mn , kl, pq and ̂k 
indicates the product of dipole matrix
elements for transitions involved in the given pathway. Then the third-order polarization
spectrum P (3) (2 ) is written as the sum over all possible pathways k :
P(3) (2 )  
k 



0
0
0
( 3)
 dt 3  dt 2  dt1R k  (t 3 , t 2 , t1 )  I1 (t 3 , t 2 , t1 )  I2 (t 3 , t 2 , t1 )
(A12)
After substitution of I1, 2 ( t 3 , t 2 , t1 ) and R (k3) (t 3 , t 2 , t1 ) into Eq. (A12), the
integration is carried out analytically and the induced signal is expressed by auxiliary
functions F1, 2 (mn , kl, pq)
3
A (3) 2 ,0   ˆ 1F1 (mn , kl, pq)  ˆ 1F2 (mn , kl, pq) 
k 

F1, 2 mn , kl, pq   Im fmn  gkl  h(pq1),( 2)

(A13)
(A14)
~ ),
fmn  1 /(2  
mn
~  i )
gkl  i /(2  1  
kl
1
(A15)
~  i ) ,
h(pq1)  i /(1  
pq
1
~  2i )
h(pq2)  i /(2  
pq
1
(A16)
Taking the imaginary part in Eq. A14, the Raman signal in the 1   limit can be
presented as
F1 (mn , kl, pq) 
F2 (mn , kl, pq) 
a1x 2  b1x  c1
 (x  mn )2  el2  (x  )2  vib2  (pq1) 2  el2 

a 2 x 2  b2 x  c2
2
( x   mn ) 2  el2  ( x  ) 2  vib
 ( x  (pq2) )2  el2 
(A17)
(A18)
a1  el , b1  el (mn    (pq1) )  vib(pq1) ,
(A19)
c1  el2vib  el(mn  (pq1) )  vibmn (pq1)
(A20)
a 2  2el  vib , b2  el (mn  2  (pq2) )  vib (mn  (pq2) ) ,
(A21)
c2  el2vib  el(mn  (pq2) )  vibmn (pq2)
(A22)
where x  2  1 is the probe-Raman detuning,  is vibrational frequency, and we used
(pq1),( 2)  1  pq , mn  1  mn and el  mn  pq , vib  kl .
The overall transient signal is the sum of all pathways [k] where all vibrational
manifolds in ground gv1, v2 ,... and excited states ev1, v2 ,... , f v1, v2 ,... are included.
As regards the influence of environment, remember that this was already taken into
account by the empirical population and pure dephasing times. The corresponding decay
rates discussed so far are the result of fast perturbations by solvents. The remaining
influence on the chromophore is considered here in the limit of an infinitely long bath
correlation time, i.e. for a frozen solvent. This static inhomogeneous broadening may be
incorporated1 by convolution the homogeneous response functions, Eq. (A13), with a
static (Gaussian) distribution of transition frequencies.
4
B.
CALCULATIONS OF RAMAN DIAGRAMS
Here we present the rules to calculate the Raman diagrams presented in Fig. 2.
The induced polarization P(3) (2 ) which causes the transient signal A (3) 2 ,0 , is the
sum over possible vibronic pathways [k] (see Eq. A13). We focus on the Raman
pathways which are presented in Fig. 2 as energy ladder diagrams. For simplicity
consider response function R 3 (diagram 1 in Fig. 2) and R1 (diagram 3a in Fig. 2) in Fig.
S1. The system is prepared in the S0 state labeled as 1.
R3(t3,t2,t1)
E2
E1
c
c
d
c
a
c
a
b
a
a
E2
E2
E1
h  g12  f 32
3
g
1
E2
c
c
d
c
d
b
d
a
a
a
( 2)
h 31
 g 34  f 31
(1)
13
2
R1(t3,t2,t1)
1
3
E1
E1
e
4
1
3a
Fig S1. Double-side Feyman1 and ladder diagrams for femtosecond stimulated Raman
scattering of a model system with three electronic states. The system is prepared in the S0
state labeled as 1. Double-sided Feynman diagrams are named according to Ref. 1,2,7,
the ladder diagrams are labelled according to Fig.2. The subscripts (1) and (2) denote the
interaction order (see Eqs. A5-A6). Dashed arrows represent bra-side and full arrows ketside interaction. The electronic states are dressed with modes g and e
(1)
The first interaction alters the “bra” q-index ( h13
, F1mn , kl, pq   F1mn , kl,13 ),
( 2)
or the “ket” p-index ( h 31
, F2 mn , kl, pq   F2 mn , kl,31 ) for diagrams R 3 and R1 ,
respectively. We introduce the detuning of the Raman frequency from the electronic
transition induced by this interaction
(pq1)  1  pq , (pq2)  1  pq
(B1)
Analogously the next interaction changes index k or l in g kl , i.e.
F1mn , kl, pq  F1mn ,12,13 and F2 mn , kl, pq  F2 mn ,34,31. The vibrational coherence
5
generated after these interactions is indicated by factors g12 and g 34 and corresponds to
the complex frequency
~    i    i

kl
kl
kl
vib
(B2)
The last interaction changes the m-index for R 3 and n-index for R1 , i.e.,
F1 mn , kl, pq  F1 32,12,13 and F2 mn , kl, pq  F2 31,34,31 . The corresponding
electronic detuning from the Raman frequency is given as
mn  1  mn
(B3)
Taking the imaginary part in Eq. A14, the Raman signal in the 1   limit can be
presented as
F1 (mn , kl, pq) 
F2 (mn , kl, pq) 
a1x 2  b1x  c1
(B4)
 (x  mn )2  el2  (x  )2  vib2  (pq1) 2  el2 

a 2 x 2  b2 x  c2
2
 ( x  (pq2) ) 2  el2
( x   mn ) 2  el2  ( x  ) 2  vib
(B5)

a1  el , a 2  2el  vib
(B6)
b1  el (mn    (pq1) )  vib(pq1) , b2  el (mn  2  (pq2) )  vib (mn  (pq2) )
(B7)
c1  el2vib  el(mn  (pq1) )  vibmn (pq1) , c2  el2vib  el(mn  2pq )  vibmn (pq2) (B8)
where x  2  1 is the probe-Raman detuning and el  mn  pq , vib  kl .
Then, by some rearrangements, the contributions from R3/R1 diagrams is captured
through


(1)
A1  F1 32,12,13  Im f 32  g12  h13
 A 0  ( g )  L(vib ,  g )


( 2)
A 3b  F2 31,34,31  Im f 31  g 34  h 31
 A 0  ( e )  L(vib ,  e )
(B9)
(B10)
Here we have introduced the normalization factor A0  1/ el / vib and the auxiliary
functions can be evaluated as
2

vib

L(vib ,  g )  2 el 2 
 2 el 2
2
2
eg  el ( x   e )  vib eg  el
2
2
2
(B11)
6
el
vib


 2 el 2
2
2
2
2
eg  el ( x   e )  vib eg  el
2
L(vib , , e ) 
 eg  el
2
 ( g ) 
 ( e ) 
el
2
2
(B12)
2
(B13)
2
( eg   e ) 2  el
el
2
2

2( eg   e )(el  vib )
el vib
2
(x   e )
(B14)
Here we introduce detuning of the Raman frequency from the electronic transition
g  e , i.e. eg  1  eg .
Then, by some rearrangements, the contributions from R3 diagram is captured through

vib
A1
  2 el 2
2
4A 0
el  1 (   g ) 2  vib
2
2
(B15)
The contributions from diagrams 3a and 3b in Fig.2 together with the contributions from
excited state e ( diagrams 2a and 2b, see Fig.2) are obtained in a similar manner
 ( eg   e   g / 2)(   g )  el 2
A 3a  A 3b

 1 
2
2
4A 0


el
vib

 el   eg
2
2

vib

(B16)
 (   ) 2   2
e
vib

2
2
2
A 2a  A 2 b   fe (   g )  el
el
vib

 1 
4A 0
el vib  el 2   eg 2 el 2   fe 2 (   e ) 2  vib 2

(B17)
Here we used the detuning of the Raman frequency from the electronic transition
e  f , i.e. fe  1  fe .
Collecting all S0-diagrams in Fig.2 we obtain the S0-Raman (for simplicity we consider
here that ge  fe   ) signal as

vib
A
  2 el 2

2
4A 0
el   (   g ) 2  vib
2
2
( e 
g
)(   g ) 
2
el
2

  2  2
el vib
 el
2
2

vib

 (   ) 2   2
e
vib

(B18)
7
The contributions from diagrams 4-8 in Fig.2 may be obtained is the similar manner.
C. DOUBLE-SIDED FEYNMAN DIAGRAMS AND
DIAGRAMS, AND THE TRANSIENT RAMAN SIGNAL
ENERGY
LADDER
We consider the electronic ground state g , the first excited state e , and a
higher electronic state f . To treat vibrational activity we dress each electronic state with
modes  g ,  e and  f . The (dimensionless) mode displacements between g and e are
2 . For simplicity, the same displacements are assumed between modes in e and f
and transition dipole momets are eg  fe  1 . The detuning of the Raman frequency is
equal for g  e and e  f
electronic transitions, i.e.   eg  fe . Dephasing
times contain the electronic ( el  1/ 5 fs ) and vibrational ( vib  1 / 5 ps ) dephasing
times. Simulations of transient Raman signal for the model system with three electronic
level are shown in Figs.S2-S5.
8
R6(t3,t2,t1)
R3(t3,t2,t1)
E2
c
E1
E2
c
d
c
a
c
a
E2
a
E2
_
c
b
d
b
d
a
a
a
E2
E1
2a
1
-1000
-500
0
+500
+1000
-1.0
-1.5
c
c
d
c
d
c
E1
b
d
b
d
b
a
b
d
a
a
b
a
a
a
a
a
a
E1
E2
E1
_
3a
3b
-1000
-500
0
+500
-1000
E1
E1
0.0
-1
RA2b
E2
_
detuning, (cm )
1.0
0.0
-0.5
R2
RA3a
-1
detuning, (cm )
-1000
-500
0
+500
-1000
0.5
0
-2.0
c
d
-1
-1000
-500
0
+500
+1000
E2
c
b
2b
detuning, (cm )
RA2a
-1
detuning, (cm )
 A1
R2(t3,t2,t1)
c
+
0.0
-0.5
E2
b
E1
+
1
differential absorbance A (a.u.)
b
b
E1
E1
R1(t3,t2,t1)
E 2
b
b
a
R5(t3,t2,t1)
-1.0
-1
detuning, (cm )
-0.5
-1000
-500
0
+500
+1000
-1.0
0.0
-2.5
-1120
-1100
-1080
-1
Stokes wavenumber (cm )
-920
-900
-880
-1
Stokes wavenumber (cm )
-920
-900
-880
-1
Stokes wavenumber (cm )
-920
-900
-880
-1
Stokes wavenumber (cm )
-920
-900
-880
-1
Stokes wavenumber (cm )
Fig. S2. Double-sided Feynman1 and energy ladder diagrams representing the main
pathways for stimulated Raman scattering (starting in S0, without actinic excitation) of a
model system with three electronic level. Double-side Feynman diagrams are named
according to Ref. 1,2,7, the ladder diagrams are labelled according to Fig.2.
9
differential absorbance A (a.u.)
1
0

A1
-1
A2a+A2b
A3a+A3b
-2
-1120
detuning
-1100
-920
-900
-880
-1
Stokes wavenumber (cm )
Fig. S3. Resonant femtosecond stimulated Raman signal A   Ai (starting in S0,
without actinic excitation) of a model system with three electronic level
( el  1/ 5 fs , vib  1 / 5 ps ) with vibrational mode g  1100 cm1 and  e  900 cm 1 in
the ground and excited electronic states, respectively.
10
R8(t3,t2,t1)
R3(t3,t2,t1)
R3(t3,t2,t1)
R8(t3,t2,t1)
R5(t3,t2,t1)
R6(t3,t2,t1)
E2
E2
d
d
a
d
a
c
a
b
a
a
E2
E1
c
E1
E1
E2
E2
c
d
c
a
c
E1
a
b
E2
a
a
c
E1
d
d
a
d
d
c
a
c
a
b
a
a
a
E2
E 2
E1
E1
c
a
b
a
a
E2
E1
+
_
4
b
b
c
b
d
b
a
b
a
a
detuning, cm
1
6
7
0
detuning, cm
detuning, cm
 A5
-1
-1000
-500
0
+500
+1000
-2
-900
-880
-1
Stokes wavenumber (cm )
-920
b
E1
d
a
a
a
8b
-1
 A6
-1
-1000
-500
0
+500
+1000
-2
2
detuning, cm
7
E2
1
-1000
-500
0
+500
+1000
8a
-1000
-500
0
+500
+1000
1
-1
-1
-900
-880
-1
Stokes wavenumber (cm )
-920
-900
Stokes wavenumber [1/cm]
-880
1
detuning, cm
 A8b
-1
-1000
-500
0
+500
+1000
0
0
0
-920
d
8a
detuning, cm
0
-1
b
E1
+
-1
-1000
-500
0
+500
+1000
b
c
_
5
 A4
b
+
0
2
E2
E2
E1
+
differential absorbance A (a.u.)
E1
c
-920
-900
-880
-1
Stokes wavenumber (cm )
-1120
-1100
-1080
-1120
-1
-1100
-1080
-1
Stokes wavenumber (cm )
Stokes wavenumber (cm )
Fig. S4. Double-sided Feynman1 and energy ladder diagrams representing the main
pathways for stimulated Raman scattering (starting in S1, with actinic excitation) of a
model system with three electronic level. Double-side Feyman diagrams are named
according to Ref. 1,2,7, the ladder diagrams are labelled according to Fig.2.
11
differential absorbance A (a.u.)
4
2
0
A
A8a+A8b
-2
A5+A6
A4+A7
-4
-1120
-1100
-1080
detuning =0
-920
-900
-880
-1
Stokes wavenumber (cm )
Fig. S5. Resonant femtosecond stimulated Raman signal A   Ai (starting in S1,
with actinic excitation) of a model system with three electronic level
( el  1/ 5 fs , vib  1 / 5 ps ) with vibrational mode g  1100 cm1 and  e  900 cm 1 in
the ground and excited electronic states, respectively.
12
References
1
S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University: New
York, 1995).
2
Y.J. Yan, S. Mukamel, Phys. Rev. A 41, 6485 (1990); S. Mukamel, Annu. Rev. Phys.
Chem. 41, 647 (1990); R.F. Loring, Y.J. Yan, S. Mukamel, J. Chem. Phys. 87, 5840
(1987); Y.J. Yan, L.E. Fread, S. Mukamel, J. Phys. Chem. 93, 8149 (1989).
3
S.A. Kovalenko, A.L. Dobryakov, J. Ruthmann, N.P. Ernsting, Phys. Rev. A 59, 2369
(1999).
4
A.L. Dobryakov, S.A. Kovalenko, N. P. Ernsting, J. Chem. Phys. 119, 988 (2003).
5
A.L. Dobryakov, S.A. Kovalenko, N.P. Ernsting, J. Chem. Phys. 123, 044502 (2005)
6
A.L. Dobryakov, N.P. Ernsting, W. Gawelda, C. Bressler, M. Chergui, in: O. Kühn, L.
Wöste (Eds) Analysis and Control of Ultrafast Photoinduced Reactions, Springer Series
in Chemical Physics, Vol. 87, Springer, Heidelberg, 2007, pp.689.
7
A.L. Dobryakov, J.L. Perez Lustres, S.A. Kovalenko, N. P. Ernsting, Chem. Phys. 347,
127 (2008).
8
A.L. Dobryakov, N. P. Ernsting, J. Chem. Phys. 129, 184504 (2008).
9
A. Weigel, A. Dobryakov, B. Klaumuenzer, M. Sajadi, P. Saalfrank, N.P. Ernsting, J.
Phys.Chem. B 115, 3656 (2011).
13
FSR spectra of trans-stilbene in n-hexane
differential absorbance A (mOD)
0
S0
measured
calculated
-1
0
-1
S1
measured
calculated
-2
200
400
600
800
1000 1200 1400 1600
-1
Stokes wavenumber (cm )
Fig. S6. FSR spectra of trans-stilbene in n-hexane compared to calculations. Note that the
spectral pattern in S0 and S1 state are different. At low frequencies, Raman activity is
stronger in S1 than in S0 because the molecule is less rigid in the excited state.
14
experimental and calculated Raman spectra
0.0
-0.2
differential absorbance (mOD)
-0.4
x20
-0.6
S0
-0.8
measured
calculated
-1.0
-1.2
200
400
644
0.04
0.02
600
202
294
800
1000
1200
1400
1600
847
0.00
-0.02
190 305
881
-0.04
-0.06
without actinic exc.
with actinic exc.
1526
calculated
1566
657
-0.08
-0.10
200
400
600
S1
800
1000
1200
1400
1600
-1
Stokes wavenumber (R )(cm )
Fig. S7. Enlarged copy of Fig. 5 shows that calculated Raman frequencies (green) are in reasonable
agreement with experiment. (These details are difficult to see in Fig. 5 which is of one column width).
15
S0(14N)
S1(14N)
S0(15N)
S1(15N)
Raman shift (cm-1)
Fig. S8. S0 and S1 spectra of trans-azobenzene in solution for two isotopomers
15
14
N and
N from Ref. 23. Strong similarity between the S1 and S0 spectral is possibly due to
imperfect subtraction of the S0 contributions.
16