I. Introduction
Recently experimental and theoretical investigations of the vibrational relaxation dynamics of water and
other molecules in condensed phases have attracted a considerable attention. In comparison the vibrational
relaxation dynamics of isolated molecules has received much less attention. A theoretical approach based on the
adiabatic approximation model of vibrational relaxation has been developed. In this report we shall review the
molecular theories and applications of vibrational and electronic SFG.
In addition, we shall present recent
progress in our theoretical works on both vibrational and electronic SFG spectroscopy.
II. Theory
1. Susceptibility Method
To treat nonlinear optical processes, the susceptibility method is commonly used. The central problem in
the susceptibility method is to calculate the polarization P which is an ensemble average of dipole moment
 , i.e.



P  Tr ˆ   Tr ˆ 
(1-1)
where ̂ denotes the density matrix of the system. The diagonal and off-diagonal elements of ̂ describe
the population and phase (or coherence) of the system, respectively. Notice that the equation of motion for the
density matrix ˆ (t ) of the system is given by the stochastic Liouville equation



ˆ
i
i
 iLˆ 0 ˆ  ˆ ˆ  iLˆ ' t ˆ   Hˆ 0 , ˆ  ˆ ˆ  Vˆ t , ˆ
t


i
 iLˆ'0 ˆ  Vˆ , ˆ

 

(1-2)
where iLˆ0  iLˆ0  ˆ , and L̂0 and Lˆ (t ) denote the Liouville operators corresponding to the zeroth-order
Hamiltonian and Hamiltonian for the interaction Vˆ between the system and radiation field, respectively. ̂
denotes the damping operator describing the relaxation and dephasing of the system. To evaluate P , it is
necessary to solve Eq. (1-2). For this purpose, for the weak field case the perturbation method is often used.
This can be carried out by regarding L̂  in Eq. (1-2) as a perturbation and solving Eq. (1-2) directly.
For SFG, we have to evaluate the second-order polarization P (2) (t ) given by


P 2  t   Tr ˆ 2  .


(1-3)
In the dipole approximation, one can write Vˆ (t ) as
2
 
Vˆ t      E  ii e  it i i li t 
i
(1-4)
i
and
li t   e


 t p  t Ti
(1-5)
where  represents the dipole operator,  i takes the value 1 or –1 and li (t ) denotes the laser pulse-shape
function with pulse duration Ti . The summation over i in Eq. (1-4) describes the situation that in SFG two
lasers are involved.
Eq. (1-3) is general and can be applied to study difference-frequency and sum-frequency generations. For
SFG, for convenience, Eq. (1-3) can be written as



P2  t p   P2  1  2 exp  it p 1  2   P2   1  2 exp it p 1  2 




(1-6)
where








i j

 k 'l  Ei i   kk '  E j  j 
 lk  lk '


 i  '    1 T
'
k
k'
l 
k 'l
i
i
 i  kl  i   j  1 T1  1 T2 i j





i j
 k 'k  Ei i   k 'l  E j  j 
 lk  kk'


'
i  kl  i   j  1 T1  1 T2 i j
i  k' 'k  i  1 Ti




(1-7)

i j

 k 'l  Ei  i   kk '  E j   j 
 lk  lk '


'
i  k' 'l  i  1 Ti
 i  kl  i   j  1 T1  1 T2 i j


*



i j
 k 'k  Ei  i   k 'l  E j   j   
 lk  kk'

 

i  kl'  i   j  1 T1  1 T2 i j
i  k' 'k  i  1 Ti
 
2
 2 

 i


P 1   2 
 2

































where, for example,  lk '  ˆ i ll  ˆ i k 'k ' representing the initial distributions of the system. In other words,
in SFG the coefficient of exp  it p 1  2  or exp it p 1  2  should be needed. Here in the double summations
over i and j, the i = j terms are to be excluded.
One can consider an energy level diagram for SFG as shown in Fig. 1, where g, m and k denote the initial,
intermediate and final state manifolds, and 1 and 2 represent the frequencies of the two lasers used in
SFG experiments.
k
2
1  2
m
1
g
Fig. 1.
Energy level diagram for SFG.
(2)
(1  2 ) ,
According to the definition of the second-order SFG susceptibility 

2 
1  2 E1 1 E2 2 
P2  1  2    

(1-8)

where ,  and  indicate molecular coordinates x, y and/or z, it is found
2 
2 
2 
2 
1  2   
1  2 1  
1  2 2  
1  2 3

where
(1-9)
2 
1  2 1 

i2
2
  gm 
g
k
m


 mg   km  

  gk  
'
'

 i mg  1   mg i kg  1  2   kg



i
mg


 mg   km  
"
ikg  1  2   kg' 
 2   mg

 mg   km  
  kg  
'
'
 i mg  1   mg i kg  1  2   kg


i
mg


 mg   km  
"
' 
 2   mg i kg  1  2   kg 

*

 gm   kg  
  mk  
'
'
 i  gm  1   mg i km  1  2   km






 gm   kg  
igm  2   mg" ikm  1  2   km' 

 gm   kg  
  km  
'
'
 i  gm  1   mg i km  1  2   km



 gm   kg  

igm  2   mg" ikm  1  2   km' 
*

(1-10)





(2)
(2)
(2)

(1  2 )2 and 
(1  2 )3 can be obtained from 
(1  2 )1 by performing the exchanges
(2)
(2)
(1  2 )2 and 
(1  2 )3 are negligible and
k  m and k  g , respectively. The expressions for 
'
 mg  1 T1 ,
will not be produced here. Note that in Eq. (1-10), mg
"
mg
 mg  1 T2 , and
kg'  kg  1 T1  1 T2 .
As discussed in previous papers, there are four cases of SFG.
They are resonance–off-resonance,
off-resonance–resonance, resonance–resonance, and off-resonance–off-resonance cases.
2. Electronic Sum-Frequency Generation
A. Resonance–Off-Resonance Case
kw
2
1  2
mu
1
gv
Fig. 2.
Energy level diagram for resonance–off-resonance SFG.
In this case, from Eq. (1-10) we obtain
2 
1  2  

i
2
  gm
g
k
m
 mg  

'
i mg  1   mg
(2-1)
  gk   km    km   gk   



 kg  1  2 km  1  2 
For electronic SFG, we have to make the following changes due to the use of the Born-Oppenheimer
approximation,
g  gv , m  mu , k  kw
where gv, mu and kw denote the vibronic states. It follows that
2 
1  2  

i
2
  gv, mu
v
kw mu
(2-2)
  gv,kw   kw,mu    kw,mu   gv,kw  
 mu, gv  




'
i mu , gv  1   mg  kw, gv  1  2
kw,mu  1  2 
By using the Placzek approximation and the closure relation,
 kw
kw  1 , we obtain,
w
2 
1  2  

i
2
  gv, mu 
v
kw mu
 mu  mg    gv   mu  gk   km    gv
 mu  gk   km    gv



'
i mu , gv  1   mg
kw, gv  1  2
kw,mu  1  2


or




(2-3)
2 
1  2  


i
2
  gv, mu 
v
kw mu
(2-4)
 mu  mg    gv  gv  gm    mu
'
i  mu , gv  1   mg
and
 2
 2
 2

1  2   Re  
1  2   i Im  
1  2 
(2-5)
where the real part is
1
 2
Re  
1  2   2

   gv, mu 

v
u
 1  mu mg     gv
mu , gv

 gv  gm   mu
(2-6)
 1    '2mu , gv
2
mu , gv
and the imaginary part is
1
 2
Im  
1  2   2


   gv, mu 
v
'
mu , gv
u
mu mg     gv

 gv  gm   mu
(2-7)
 1    '2mu , gv
2
mu , gv
(2)
(1  2 ) is closely related to the electronic absorption
Eq. (2-7) indicates that the imaginary part of 
spectra.
In the Condon approximation we obtain,
Re   1  2   
 2
1
2
mg     gm

     gv, mu 

mu , gv
v
u
 1   gv mu
 1   
2
mu , gv
'2
mu , gv
2
(2-8)
and
Im   1  2   
 2
1
2
mg     gm      gv, mu 
v
u
'mu , gv  gv mu

2
'2
mu , gv  1    mu , gv
2
(2-9)
Notice that
  gk  km  
 gm     
k
The
 kg  1  2

 gk  km   
0 
   gm     gm  
kg   gm  1  2 
(2-10)
g  m transition intensity in electronic SFG is determined by the electronic matrix element
 2
mg   gm   rather than  mg in the ordinary absorption spectra.
B. Off-Resonance–Resonance Case
kw
1  2
2
mu
1
gv
Fig. 3.
Energy level diagram for off-resonance–resonance SFG.
We next consider the off-resonance–resonance case. From Eq. (1-10) we find
2 
1  2  

i
2
  gm i
g
k
m
kg
  mg   km    mg   km   
 gk  

 (2-11)
' 
 1   2   kg   mg  1
mg  2 
In terms of vibronic states, we find (with the Placzek approximation mu , gv  mg )
2 
1  2  

i
2
  gv, mu i
v
w
m
 gv  gk    kw
kw , gv
'
 1  2   kw
, gv
  kw  km   mg    kw
 kw  km   mg    gv


mg  1
mg  2


(2-12)




or
2 
1  2  

i
2
  gv, mu
v
w
 gv  gk    kw  kw  kg    gv
'
i kw, gv  1  2   kw
, gv
(2-13)
where
  km   mg  
 kg     
m

 mg  1

 km   mg  

 mg   2 
(2-14)
In the Condon approximation we find that
 2
Re  
1  2  
and
1
2
 gk

         gv, mu 

kw, gv
kg
v
w
 1  2  kw  gv
'2
kw, gv  1  2    kw, gv
2
2
(2-15)
 2
Im  
1  2  
1
 gk    kg       gv, mu 
2
v
w
 'kw, gv kw  gv
2
kw, gv  1  2   '2kw, gv
2
(2-16)
 2
Im 
1  2  shows that it is closely related to the absorption spectra for the electronic transition g  k
with frequency 1  2 .
C. Double Resonance Case
kw
1  2
2
mu
1
gv
Fig. 4.
Energy level diagram for double resonance SFG.
Next we consider the double resonance case. From Eq. (1-10) we obtain
2 
1  2  

i2
2
  gv, mu
v
gv,kw
 
mu kw
(2-17)
 mu , gv   kw,mu  

imu,gv  1   mu' ,gv ikw,gv  1  2   kw' ,gv 
or
2 
1  2  


i2
  gv, mu
 2 v mu kw
 gv  gk    kw  mu  mg    gv  kw  km    mu
i
mu , gv

'
'
 1   mu
, gv i kw , gv  1  2   kw , gv
(2-18)

In the Condon approximation,
2 
1  2  


Notice that
i2
 gk   mg   km     gv, mu 
2
v mu kw
i
 gv  kw  mu  gv  kw  mu
mu , gv

'
'
 1   mu
, gv i  kw , gv  1   2   kw , gv
(2-19)

i2
 2
Re  
1  2   2  gk   mg    km      gv, mu 
v
u
w
   mu , gv  1 kw, gv  1  2    'mu , gv  'kw, gv 

(2-20)
 gv kw kw mu  mu  gv
2
2
'2
'2
 


 mu , gv  1    mu , gv  kw, gv  1  2    kw, gv 
and
i2
 2


Im   1  2    2  gk   mg     km      gv, mu 
v
u
w
  'mu , gv kw, gv  1  2    'kw, gv mu , gv  1  

(2-21)
 gv kw kw mu  mu  gv
2
2
'2
 
  








  '2kw, gv 


mu
,
gv
1
mu
,
gv
kw
,
gv
1
2

 

3. Vibrational Sum-Frequency Generation
A. Resonace–Off-Resonance Case
ku
2
1  2
gv’
1
gv
Fig. 5.
Energy level diagram for resonance–off-resonance SFG.
In this case, from Eq. (1-10) we obtain
2 
1  2  

i
2
  gm
g
k
m
 mg  

'
i mg  1   mg
  gk   km    km   gk   



 kg  1  2 km  1  2 
(3-1)
For vibrational SFG, we have to make the following changes due to the use of the Born-Oppenheimer
approximation,
g  gv, m  gv' , k  ku
It follows that
2 
1  2  

  gv, gv'
i
2
v
v'
ku
 gv',gv  

i  gv',gv  1   gv' ',gv
  gv,ku   ku , gv'    ku , gv'   gv,ku  



ku ,gv'  1  2 
 ku , gv  1  2
(3-2)
Using the Placzek approximation, we obtain
2 
1  2  


i
2
  gv, gv'
v
v'
 gv'  gg    gv
i  gv', gv  1   gv' ', gv
  gk   kg    gk   kg   
 gv  

  gv'










k 

kg
1
2
kg
1
2

(3-3)
or
2 
1  2  

i
2
 gv'  gg    gv
  gv, gv' i
v
v'
gv ', gv
 1   gv' ', gv
 gv  gg    gv'
(3-4)
where  gg   denotes the polarizability of the g electronic state,
  gk   kg  
 gg     
k
 kg  1  2

 gk   kg   

kg  1  2 
(3-5)
Both  gg   and  gg   should be expanded in terms of normal coordinates Ql,
  gg   
 Ql  
 Ql 0
0
    
 gg     gg
l
(3-6)
and
  gg   
 Ql  
 Ql 0
0
    
 gg     gg
l
(3-7)
Substituting Eqs. (3-6) and (3-7) into Eq. (3-4) yields
2 
1  2   i Kl 1 , 2  gv, gv'

v
v'
l
 gv Ql  gv'
2
'
i  gv', gv  1   gv
', gv
(3-8)
where
Kl 1 , 2  
1   gg      gg   



 2  Ql 0  Ql 0
This indicates that for this vibrational SFG to be non-zero, we should have
(3-9)
  gg   

  0
 Ql 0
(3-10)
  gg   

  0
 Ql 0
(3-11)
and
That is, the Ql mode should be both IR and Raman active.
2 
The phase of the SFG can now be determined; in this case 
1  2  should be written in terms of the real
part and the imaginary part. That is,
Re   1  2     Kl 1 , 2   gv, gv ' 
 2
v
v'
l

gv ', gv
 1   gv Ql  gv '
2
(3-12)
gv ', gv  1   '2gv ', gv
2
and
Im   1  2     Kl 1 , 2   gv, gv '
 2
v
v'
l
'gv ', gv  gv Ql  gv '

2
'2
gv ', gv  1    gv ', gv
2
(3-13)
 2
 2
Notice that Re  
1  2  and Im  
1  2  satisfy the Kramers-Kroning relation.
B. Double-Resonance Case
ku
1  2
2
gv’
1
gv
Fig. 6.
Energy level diagram for double resonance SFG.
In this case, from Eq. (1-10) we find
2 
1  2  

i2
  gm
2 g m k
 gk   mg   km  

img  1   mg' ikg  1  2   kg' 
In the adiabatic approximation, Eq. (3-15) can be written as
(3-14)
2 
1  2  

i2
2
  gv, gv'
v
v'
u
 gv,ku   gv',gv   ku , gv'  

igv',gv  1   gv' ',gv iku ,gv  1  2   ku' ,gv 
(3-15)
or
2 
1  2  

 gv'  gg    gv
i2




gv
,
gv
'

 2 v v'
i  gv',gv  1   gv' ',gv

 gv  gk    ku  ku  kg    gv'
 
i ku , gv  1  2   ku' , gv
u





(3-16)
This implies that this type of SFG consists of the Raman and resonance Raman scattering,
  gg   

  gv' Ql  gv
2

Q
i
l
0
2 
1  2   2   gv, gv' 

 v v' l
i  gv',gv  1   gv' ',gv

 gv  gk    ku  ku  kg    gv'
 
i ku , gv  1  2   ku' , gv
u

(3-17)




or in the Condon approximation,
  gg   

  gv' Ql  gv
Ql 0
i

2 
0
0
 1  2   2  gk   kg    gv, gv'

i  gv',gv  1   gv' ',gv
v
v'
l
2
(3-18)

 gv  ku  ku  gv' 
 

'
 u i ku , gv  1  2   ku , gv 
4. Surface-enhanced Sum-Frequency Generation
Similar to SERS (surface-enhanced Raman Scattering), its chemical enhancement can be due to the
appearance of charge-transfer bands which originate from the interaction between the adsorbed molecules and
metal (like Ag and Au) (See Figs. 4 and 6).
5. Orientation and Euler Angle Transform
Till now we focused on the second-order susceptibility, (2), of a single molecule. To correlate these
molecular properties to experimental observables, one has to include information of molecular orientation
distribution as well as beam angles defined under laboratory frames.
Given the molecular coordinates, xyz,
laboratory frames, XYZ, and the rotation angles, , in the z-y-z convention (Fig. 7), the Euler rotation matrix
between the two coordinate systems is
Z
z
y
x


Y

X
Fig. 7.
Cartesian coordinates of laboratory frames (XYZ) and molecular axes (xyz) and rotation angles ()
between the two coordinate systems.
 RXx

R   RYx
 RZx

RXz 

RYy RYz 
RZy RZz 
 cos  cos cos  sin sin
 sin  cos cos  cos sin

 sin  cos
RXy
 cos cos sin  sin cos
 sin cos sin  cos cos
sin sin
cos sin 
sin sin 
cos 
(5-1)
Now we consider the projection of molecular second-order susceptibility onto the laboratory frame. With an
orientation average over a group of molecules, the projection is
(2)
(2)
(2)
ˆ )(ˆ  nˆ )  N  lmn

 N  lmn
(ˆ  lˆ)( ˆ  m
RIl RJm RKn
lmn
(5-2)
lmn
where ,  and  indicate laboratory frames X, Y and/or Z, l, m and n refer to molecular coordinates x, y and/or z,
N is the density of molecules at the interface, and
is the orientation average.
(2)
has 27 elements but

most of them vanish supposing the molecules have an azimuthally isotropic distribution on the XY plane. For
example, on the (0001) surface of ice Ih crystal only seven elements survive and some of them are mutually
dependent after the orientation average:
(2)
 ZZZ
,
(2)
(2)
 XXZ
 YYZ
,

(2)
ZXX

(2)
ZXZ
(5-3)

(2)
ZYY

(2)
YZY
.
Next we take beam angles into account.
Given 1 and 2 the incident beam angles and SFG the output beam
angle with respect to the surface/interface normal, the effective susceptibilities at different combination of beam
polarizations (s or p) areError! Bookmark not defined.
(2)
(2)
 eff
( ssp)  sin 1 LYY (SFG ) LYY (2 ) LZZ (1 )  XXZ
(5-4)
(2)
(2)
 eff
( sps)  sin  2 LYY (SFG ) LZZ (2 ) LYY (1 )  XZX
(5-5)
(2)
(2)
 eff
( ppp)  sin SFG sin  2 sin 1 LZZ (SFG ) LZZ (2 ) LZZ (1 )  ZZZ
(2)
 cos SFG cos  2 sin 1 LXX (SFG ) LXX (2 ) LZZ (1 )  XXZ
(5-6)
(2)
 sin SFG cos  2 cos 1 LZZ (SFG ) LXX (2 ) LXX (1 )  XZX
(2)
 cos SFG sin  2 cos 1 LXX (SFG ) LZZ (2 ) LXX (1 )  XZ
X
where the sequence of beam polarization noted here is SFG, 2 and 1.
In above equations L’s refer to
Fresnel factors which are functions of refractive indices and wavelengths.Error! Bookmark not defined.,
III. Applications
1. Vibrational SFG Studies
A. Early achievements
The SFG technique has become available on analyzing surface species since the works conducted by
Shen’s group in late 1987. With an optical parametric amplifier tuned around 2800 to 3100 cm -1 with a
frequency-doubled Nd:YAG laser, they have successfully recorded IR-Vis SFG spectra of methanol and
pentadecanoic acid (PDA) adsorbed on glass and on water surface.
By tuning the IR frequency, different
vibrational resonance were obtained, and with various combinations of incident beam polarizations (s and/or p),
different C-H stretching modes were revealed.
They have further analyzed C-H vibration signals of PDA at
different adsorption densities, finding the long alkane chains nearly oriented straightly along the normal to the
water surface in the liquid-condensed phase at high density.Error! Bookmark not defined.
Following these pioneering works, numerous applications of vibrational SFG spectroscopy emerged.
One major category of experiments focused on the orientations of functional groups of organic polymers
adsorbed on solid surfaces. Aliphatic and aromatic C-H stretching modes from different chemical species could
be distinguished by resonance frequencies, and their orientations could be deduced from signal intensities
recorded under different combinations of incident/output beam polarizations.
The results showed some
substrate-dependence of the adsorbate orientation. For example, the phenyl groups in polystyrene have an
average tilt angle of ~20° at the polystyrene-air interface when the polymer is adsorbed on sapphire, and ~57°
on oxidized silicon substrate.Error! Bookmark not defined.,Error! Bookmark not defined.
The SFG technique
could identify both chemical species and orientations at the same time, which combines the advantages of other
surface-detecting techniques such as Raman scattering, atomic force microscopy, and X-ray spectroscopy.
B. Water surfaces (water/vapor interfaces)
The configuration and related properties of water surface has long been a crucial topic but it lacks even a
generally acceptable conclusion until the infrared-visible SFG experiments directed by Shen’s group in 1990s.
In a simple model that described  / Q along the OH bond and  / Q with a cylindrical symmetry, only
two components of hyperpolarizabilities were non-vanishing. The tilt angle of dangling OH on the pure water
(2)
(2)
surface could then be deduced from the ratio of  XZX
and  XXZ
which were measured with ssp and sps
polarization combinations, respectively.
It led to a maximal tilt angle of 38°, and furthermore, a minimal
coverage of 20% of dangling OH over the water monolayer.Error! Bookmark not defined.
They have further
recorded ssp, sps, and ppp signals for dangling OHs and given a more comprehensive analysis how the
maximal tilt angle would affect the signal intensity. Richmond’s group, on the other hand, investigated SFG
spectra of not only pure H2O but also pure D2O and their mixture.
In addition to liquid water, Shen’s group paid attention to the surface of ice which, in principle, has a more
regular structure and could yield more solid intepretation.Error! Bookmark not defined., The surface structure
and the melting phenomenon of the (0001) face of ice Ih were studied.
It was supposed each layer of water
molecules in the ice crystal were bonded by the tetrahedral hydrogen bonding, and the topmost layer had free
(dangling) OH bonds pointing outward with a tilt angle of .
For this ice crystal structure, the non-vanishing
elements of the second-order susceptibility reduced to three [Eq. (5-3)], and all three elements could be
determined from SFG signals with ssp, sps, and ppp polarization combinations [Eqs. (5-4) to (5-6)].
In the
temperature-dependence measurements from 173K up to the bulk melting point 273K, the free OH peak at
3695 cm–1 gradually diminished while the broad bonded OH band red-shifted from ~3150 cm–1 to ~3250 cm–1.
The decrease of the free OH peak intensity represented a larger tilt angle and a more disordered structure as the
temperature increased.
Assuming a simple flat orientation distribution at each , it was found the maximum
tilt angle to be 0° at low temperature while ~60° near the melting point. A more surprising founding was that
around 273K, the supercooled liquid water surface was even more ordered than ice (cf. Fig. 9 in Ref. Error!
Bookmark not defined.). This gave a remark on the semi-ordered, ice-like structure on the water surface,
which has raised debate over decades and has not come to consensus yet.
C. Phase-sensitive detections
In early IR-Vis/UV SFG experiments only  (2)
2
could be obtained from signal intensities. Neither
Re[  (2) ] nor Im[  (2) ] could be measured directly; instead, they were fitted from  (2)
2
profiles and might
yield significantly different interpretation (Fig. 8). As early as in 1990 Superfine et al. tried to determine the
“absolute phase” of second-order susceptibility by investigating the relative sign of dipole moment derivative
and polarizability derivative with respect to the vibrational normal mode. This technique became mature in
the 2000s and was applied on the water interface geometry.Error! Bookmark not defined.,
Fig. 8.
A demonstration of quite different fittings to the same  (2)
2
spectrum.
Adopted from Ref. Error!
Bookmark not defined..
In vibrational SFG spectroscopy, the sign of Im[  (2) ] is determined by
 
[cf. Eq. (3-10)], and
Q Q
hence in the case of water surface, this sign can reflects directions of OH bonds supposing the OH stretching
modes are well decoupled from each other, i.e. performed as local modes. This decoupling is practical since
each OH bond faces somewhat different environment constructed by surrounding water molecules.
As a result,
OH bonds pointing outward from the surface are positive, and those pointing into the bulk are negative, in the
Im[  (2) ] spectra. Tian and Shen have carried out phase-sensitive vibrational SFG spectroscopy experiments
for pure H2O, and later on pure D2O as well as H2O/D2O mixture at different ratios. On the surface of pure H2O,
the dangling OH bonds, as expected, had a sharp positive peak at 3690 cm1. The hydrogen-bonded OH bonds,
however, present one negative band centered at 3450 cm1 and one positive band centered at 3100 cm1 (Fig. 9).
The same features were also observed by Tahara and coworkers.
It should be noticed that the positive band
could not be correctly resolved previously by just fitting real and imaginary parts to the  (2)
2
spectrum.Error!
Bookmark not defined.
Comparing these band frequencies to IR absorption bands of bulk ice and liquid samples, it was concluded
that OHs in the topmost layer DAA, DDA, and DDAA (Fig. 10, where D and A represent hydrogen donor and
hydrogen acceptor, respectively, in the hydrogen-bonding network) molecules loosely donor-bonded to
molecules below contributed to the “liquid-like” negative band, while OHs of tetrahedrally bonded DDAA
molecules in lower layers contributed to the “ice-like” positive band through their donor-bonding to upper DAA
and DDA molecules. This supported the idea that the surface layer of water is relatively disordered, while the
lower layers have some ordered (to a certain extent) ice-like structure.Error! Bookmark not defined.,Error!
Bookmark not defined.
Fig. 9.
Phase-sensitive vibrational SFG spectra of water surface in the OH stretching region.
Adopted from
Ref. Error! Bookmark not defined..
Fig. 10.
Configurations of water molecules categorized by hydrogen donating/accepting in the
hydrogen-bonding network.
D. Theoretical and computational approaches
Case 1: Organic molecules adsorbed on graphene surface
Graphene is a novel material due to its unique geometric structure and hence related properties such as
electric potential, conductivity, etc.
It raises interest as well in the adsorption behavior of this substrate, e.g.
the orientations of organic polymers adsorbed on the graphene surface, and people have initiated SFG
experiments to study this topic. From the theoretical point of view, we have carried out a preliminary SFG
simulation of styrene–graphene adsorption system.
In our approach, the first step was to construct molecular models of ethylbenzene, styrene monomer, and
polymerized styrene oligomers (up to 4 units) adsorbed on a (finite-sized) graphene sheet.
It followed
geometric optimization of styrene molecules by DFT computations and it showed the phenyl groups tended to
“stand” rather than “lay down” on the graphene surface. The adsorption energies were weak but not negligible.
With calculated dipole derivatives and polarizability derivatives the vibrational SFG spectra of phenyl and alkyl
CH stretching modes were simulated with different polarization combinations. These results, some of which
are demonstrated in Figs. 11 and 12, have provided the first theoretical prediction to the graphene-based
adsorption system and is waiting for experimental investigation.
Fig. 11.
Optimized structures and stabilization energies (in kcal/mol) of ethylbenzene–graphene and styrene
oligomer–graphene systems calculated by B3LYP/6-31G(d).
Adopted from Ref. Error! Bookmark not
defined..
8
ssp
sps (5x)
ppp
etb(a)
6
10
sty2(a)
ssp
sps (5x)
ppp
6
4
SFG Intensity (arb. unit)
4
2
0
0
etb(b)
3
ssp
sps (10x)
ppp
10
20
8
sty2(b)
ssp
sps (5x)
ppp
6
4
1
0
0
etb(c)
3
ssp
sps (10x)
ppp
ssp
sps (5x)
ppp
0
8
sty2(c)
ssp
sps (5x)
ppp
sty3(c)
ssp
sps (5x)
ppp
6
2
4
1
0
2800
sty3(b)
2
1
2
0
2900
3000
3100
3200
2800
-1
Wavenumber (cm )
Fig. 12.
ssp
sps (5x)
ppp
0
2
5
2
sty3(a)
10
2
15
30
8
0
2900
3000
3100
-1
Wavenumber (cm )
3200
2800
2900
3000
3100
3200
-1
Wavenumber (cm )
Simulated SFG spectra of ethylbenzene– and styrene-oligomer–graphene systems.
corresponding conformers are shown in Fig. 11.
The
Adopted from Ref. Error! Bookmark not defined..
Case 2: Molecular orientation on the water surface (liquid water/vapor interface)
Since the first report of the water surface SFG experiment, theoreticians started to make efforts on
interpreting and simulating the spectrum.
A main approach is to calculate the second-order susceptibility by a
time correlation function of polarizability tensors and dipole vectors obtained from molecular dynamics (MD)
simulation. The major problem in this approach is that the accuracy of the MD potential field definitely affects
the accuracy of simulated spectra, in which the positive band centered at ~3100 cm1 (Refs. Error! Bookmark
not defined.Error! Bookmark not defined., cf. Fig. 9) in Im[  (2) ] has never been reproduced until very
recently. Morita and Tahara considered the dipoleinduced-dipole interaction in a pair of water molecules and
attributed the low-frequency positive band to the strong hydrogen bonding between the two; however, the
direction of induced dipole is questionable.
Skinner and coworkers have constructed a potential which
explicitly describes three-body interactions and finally obtained the positive band, although the peak positions
still required modification.
We would like to treat this problem with a more quantum chemical approach based on DAA, DDA, and
DDAA configurations. By randomizing orientations and distances of the surrounding three or four molecules,
the explicit, averaged dipole and polarizability derivatives of the center water molecule at each configuration
could be calculated. The overall simulated SFG spectrum could reproduce detailed features quite well, and the
arrangement of report is in the process.
2. Electronic SFG studies
A. Recent experiments
In addition to VSFG, Tahara and co-workers have recently made heterodyne-detected electronic
sum-frequency generation (ESFG) applicable.
In their experiments, two visible/near-infrared beams, one was
795 nm and the other was tunable between 540 nm and 1.2 m, were guided to the sample where the sum
frequency was tuned resonant with an electronic state. By this means they have resolved the orientation of
surfactant-like molecules floating on water surface. Following that they have studied the pH spectrometry of an
air/cationic surfactant/water interface by the same technique, obtaining an insight to the acid-base equilibrium,
and furthermore, to the pH value of water surface, where the value is lower than the bulk by 1.7.
It turns out
that electronic SFG is helpful to solve this long-debated issue; however, it should be noticed that a solid
theoretical framework to interpret these excellent experimental results is still in demand.
B. Recent theoretical approaches
We have chosen 4-methyl-7-hydroxycoumarin (MHC), a pH-indicator surfactant molecule analogous to
which Tahara’s group used in their experiments,Error! Bookmark not defined. to provide a theoretical support
to ESFG spectroscopy.
A loose optimization showed that the MHC molecule tends to float on the water
surface (simulated by a cluster of ~50 water molecules) with its oxygen atoms immersed into water while
methyl group pointing outward (Fig. 13). Under the off-resonance–resonance scheme, the dominant transition
occurred from the ground state (1 1A') to the first excited state (2 1A'). The corresponding transition dipole
moments, two-photon matrix elements, and Franck-Condon integrals were calculated according to Eqs. (2-13)
to (2-16) and the SFG spectrum was then simulated (Fig. 14). This preliminary result is the first theoretical
ESFG spectrum to our knowledge, and it may be further improved by including more detailed factors like
non-Condon contributions, more definite orientation distributions, and explicit solve effects.
Fig. 13.
MHC anion floating on the surface of a small water cluster, where red and dark gray spheres
represent oxygen and carbon atoms, respectively.
0.005
eff,sps
(2)
(arb. unit)
0.004
0.003
0.002
0.001
0.000
Im[
Re[
-0.001
(2)
]
]
(2)
-0.002
-0.003
260
280
300
320
Wavelength (nm)
340
360
Fig. 14.
Simulated electronic SFG spectrum of MHC with the sps polarization.
References
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13. X. Wei, S. C. Hong, A. I. Lvovsky, H. Held, and Y. R. Shen, “Evaluation of Surface vs Bulk Contributions
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16. E. A. Raymond, T. L. Tarbuck, M. G. Brown, and G. L. Richmond, “Hydrogen-Bonding Interactions at the
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Mixtures and Molecular Dynamics Simulations”, J. Phys. Chem. B 107, 546 (2003).
17. Xing Wei, Paulo B. Miranda, and Y. R. Shen, “Surface Vibrational Spectroscopic Study of Surface Melting
of Ice”, Phys. Rev. Lett. 86, 1554 (2001).
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Orientation of Water Molecules at Interfaces”, Chem. Rev. 106, 1140 (2006).
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moment derivatives: an infrared-visible sum frequency generation absolute phase measurement study”,
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generation”, Opt. Lett. 15, 1276 (1990).
21. Victor Ostroverkhov, Glenn A. Waychunas, and Y. R. Shen, “New Information onWater Interfacial
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22. N. Ji, V. Ostroverkhov, C. S. Tian, and Y. R. Shen, “Characterization of Vibrational Resonances
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(2008).
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Sum-Frequency Vibrational Spectroscopy”, J. Am. Chem. Soc. 131, 2790 (2009).
24. S. Nihonyanagi, S. Yamaguchi, and T. Tahara, “Direct evidence for orientational flip-flop of water
molecules at charged interfaces: A heterodyne-detected vibrational sum frequency generation study”, J.
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Study on Structure and Sum-Frequency Generation (SFG) Spectroscopy of Styrene–Graphene Adsorption
System”, J. Phys. Chem. C 117, 1754 (2013).
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39. H. Watanabe, S. Yamaguchi, S. Sen, A. Morita, and T. Tahara, “‘Half-hydration’ at the air/water interface
revealed by heterodyne-detected electronic sum frequency generation spectroscopy, polarization second
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40. S. Yamaguchi, K. Bhattacharyya, and T. Tahara, “Acid-base equilibrium at an aqueous interface: pH
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(2012)
1.
PUBLICATION LIST (JOURNALS, CONFERENCES, BOOKS, BOOK CHAPTERS, etc.）
Journal paper summary
1.
“Absorption and fluorescence spectra of the neutral and anionic green fluorescent protein
chromophore: Franck-Condon simulation”, Tsung-wei Huang, Ling Yang, Chaoyuan Zhu, Sheng
Hsien Lin, Chemical Physics Letters (ISSN: 0009-2614), 541(10), 110-116 (2012)
2. “Analysis of Initial Reactions of MALDI based on Chemical Properties of Matrices and Excitation
Condition”, YH Lai, CC Wang, CW Chen, BH Liu, SH Lin, YT Lee, YS Wang, J PHYS CHEM B
(ISSN: 1520-6106), 116(32), 9635-9643 (2012)
3. “Ionization/Dissociation Processes of Methyl-Substituted Derivates of Cyalopentanone in Intense
Femtosecond Laser Field”, Q Wang, D Wu, D Zhang, M Jin, F Liu, H Liu, Z Hu, D Ding, YA Dyakov,
H Mineo, Y Teranishi, SD Chao, AM Mebel and SH Lin, Chemical Physics (0301-0104), (2012/12)
4. “Quantums Switching of pi-Electron Rotations in a Nonplanar Chiral Molecule by Linearly Polarized
UV Laser Pulses”, H Mineo, M Yamaki, Y Teranishi, M Hayashi, SH Lin and Y Fujimura, Journal of
the American Chemical Society (ISSN: 0002-7863), 134(35), 14279-14282 (2012)
5. “Localized Emitting State and Energy Transfer Properties of Quadrupolar Chromophores and (Multi)
Branched Derivatives”, LY Yan, XD Chen, QG He, YY Wang, XF Wang, QJ Guo, FL Bai, AD Xia, D
Aumiler, V Silviie, SH Lin, Journal of Physical Chemistry A (ISSN: 1089-5639), 116(34), 8693-8705
(2012)
6. “Semiclassical Instanton Theory of Nonadiabatic Reaction Rate Constant. I. Formulation”, Y
Teranishi, H Nakamura, SH Lin, AIP Conference Proceedings (ISSN: 0094-243X), 1456, 119-130
(2012)
7. “Thermodynamics of protein folding using a modified Wako-Saitô-Muñoz-Eaton model ”, Min-Yeh
Tsai, Jian-Min Yuan, Yoshiaki Teranishi, Sheng Hsien Lin, Journal of Biological Physics (ISSN:
0092-0606), 38(4), 543-571 (2012)
8. “Quantum Chemical Calculation of Intramolecular Vibrational Relaxation and Vibrational Energy
Transfer of Water Clusters Corresponding”, Sheng-Hsien Lin, Ying-Li Niu, Ran Pang, Chao-Yuan
Zhu, Michitoshi Hayashi, Yuichi Fujimura, Yuen-Ron Shen, Chemical Physics Letters,
9. “Visible-light Photocatalytic Activity of Ni-doped TiO2 from ab initio Calculations”, Y.-M. Lin, Z.-Y.
Jiang, C. Zhu, X.-Y Hu, X.-D. Zhang, J. Fan, Mater. Chem. Phys., 133, 746 (2012)
10. “Highly hydrated Nafion/activated carbon hybrids”, H.-C. Chien, L.-D. Tsai, A. Kelarakis, C.-M. Lai,
J.-N. Lin, J. Fang, C. Zhu, F.-C. Chang, Polymer, 53, 4927-4930 (2012)
11. “Enhanced optical absorption and photocatalytic activity of anatase TiO2 through (Si, Ni) codoping”,
Y.-M. Lin, Z.-Y. Jiang, C. Zhu, X.-Y. Hu, X.-D. Zhang, H.-Y. Zhu, and J. Fan, Appl. Phys. Lett.
(ISSN: 0003-6951), 101, 062106 (2012)
12. “Recent developments in radiationless transitions”, Y. Niu, C.-K. Lin, L. Yang, J. Yu, R. He, R.
Pang, C. Zhu, M. Hayashi, and S. H. Lin, Prog. Chem., 26, 87 (2012)
13. “Recent Developments in Theoretical Chemistry”, Y. L. Niu, M. Yamaki, C. Zhu, M. Hayashi, Y.
Fujimura, and S. H. Lin, Bulletin of Association of Asia Pacific Physical Societies (ISSN: 0218-2203),
22, 12-16 (2012) (IF: 1.3)
14. “Coherent ring currents in chiral aromatic molecules induced by linearly polarized UV laser pulses”,
Manabu Kanno, Hirohiko Kono, Hirobumi Mineo, Sheng Hsien Lin, and Yuichi Fujimura, Key
Engineering Materials(Materials and Applications for Sensors and Transducers II-Proceedings of the
2nd International Conference), 543, 381-384 (2013) (ISSN: 1662-9795)
15. “Coherent π-electron dynamics of (P)-2,2_-biphenol induced by ultrashort linearly polarized UV
pulses: Angular momentum and ring current”, H. Mineo, S. H. Lin and Y. Fujimura, J. Chem. Phys.,
138(7), 074304(16) (2013)(ISSN: 0021-9606)
16. “Neutral Fragmentation Paths of Methane Induced by Intense Ultrashort IR Laser Pulses: Ab Initio
Molecular Orbital Approach”, Shiro Koseki, Noriyuki Shimakura, Yoshiaki Teranishi, Sheng Hsien
Lin, and Yuichi Fujimura, J. Phys. Chem. A (ISSN: 1089-5639), 117(2), 333-341, (2013)
17. “Anharmonic Rice-Ramsperger-Kassel-Marcus (RRKM) and product branching ratio calculations for
the partially deuterated protonated water dimmers: Dissociation and isomerization”, Di Song, Hongmei
Su, Fan-ao Kong, and Sheng-Hsien Lin, J. Chem. Phys. 138(10), 104301(7), 2013 (ISSN: 0021-9606)
18. “C/B codoping effect on band gap narrowing and optical performance of TiO2 photocatalyst: a
spin-polarized DFT study”, Y. Lin, Z. Jiang, C. Zhu, X. Hu, X. Zhang, H. Zhu, J. Fan, and SH Lin, J.
Mater. Chem. A (ISSN: 0959-9428), 1, 4516-4524 (2013)
19. "Solvent Effects on Vibrational Normal Modes of Solute Molecule: Franck-Condon Simulation of the
X ¹A_g↔ A¹B_1u Absorption and Fluorescence
Spectra of Perylene in Benzene Solution", Yang, Ling; Zhu, Chaoyuan; Yu, Jian-Guo; Lin, Sheng
Hsien, Journal of Physical Chemistry Letters, Manuscript ID: jz-2013-00024y (2013)
20. “Noncovalent interaction and its influence on excited-state behavior: A theoretical study on the mixed
coaggregates of dicyanonaphthalene and pyrazoline”, Y. Dai, M. Guo, J. Peng, W. Shen, M. Li, R. He,
C. Zhu, and SH Lin, Chem. Phys. Lett. (ISSN: 0009-2614), 556, 29, 230-236 (2013)
21. “Theoretical Study on Structure and Sum- Frequency Generation (SFG) Spectroscopy of
Styrene-Graphene Adsorption System”, C-K. Lin, C-C. Shih, Y. Niu, M-Y. Tsai, Y-J. Shiu, C. Zhu, M.
Hayashi, and SH Lin, J. Phys. Chem. C (ISSN: 1932-7447), 117, 1754-1760 (2013)
22. "Solvent Effects on Vibrational Normal Modes of Solute Molecule: Franck-Condon Simulation of the
X ¹Ag↔ A¹B_1u Absorption and Fluorescence Spectra of
Perylene in Benzene Solution", Yang, Ling; Zhu, Chaoyuan; Yu, Jian-Guo; Lin, Sheng Hsien, Journal
of Physical Chemistry A, Manuscript ID: jp-2013-016382 (2013)
23. "Explore the Role of Varied-Length Spacers in the Charge Transfer: A Theoretical Investigation on the
Pyrimidine-Bridged Porphyrin Dyes", Guo, Meiyuan; Li, Ming; Dai, Yulan; Shen, Wei; Peng,
Jing-dong; Zhu, Chaoyuan; Lin, Sheng Hsien; He, Rongxing, Journal of Physical Chemistry C,
Manuscript ID: jp-2013-01293h (2013)
24. “The Foldon behavior of Cytochrome c as revealed by Molecular Dynamics Simulations and Some
Experimental Evidences”, M. Y. Tsai and S. H. Lin, in preparation
25. “Molecular Dynamics Insight into the Thermodynamics of a Beta-Hairpin Peptide”, M. Y. Tsai, J. M.
Yuan, M. Yamaki and S. H. Lin, in preparation
26. "Theoretical Formulation and Simulation of Electronic Sum-Frequency Generation (SFG)
Spectroscopy", Lin, Chih-Kai; Hayashi, Michitoshi; Lin, Sheng Hsien, Journal of Physical Chemistry
C (Submission, Manuscript ID: jp-2013-07881a) (2013)
Workshop summary
1.
“非平面キラル芳香分子(P)-2,2’BIPHENOL における光誘起Π電子運動の 量子スイッチ ”,
HIROBUMI MINEO, MASAHIRO YAMAKI, YOSHIAKI TERANISHI, MICHITOSHI HAYASHI, SHENG-HSIEN
LIN, AND YUICHI FUJIMURA, 分子科学討論会, 東京大學 (2012/09/18-21)
2. 14. “線形紫外レーザーパルス 誘起非平面キラル分子におけるΠ電子回転 ダイナミクスの
理論的研究: 角運動量 環電流”, HIROBUMI MINEO, SHENG-HSIEN LIN AND YUICHI FUJIMURA,
分子科学討論会, 東京大學 (2012/09/18-21)
“RECENT PROGRESS IN MOLECULAR PHOTOCHEMISTRY”, Y. L. NIU, R. PAUG, L YANG, C. K. LIN, H.
MINEO, C. Y. ZHU, S. D. CHUO, Y. FUJIMURA, H. HAYCSHI AND S. H. LIN*, THE 7TH TAIWAN-JAPAN
BILATERAL SYMPOSIUM ON ARCHITECTURE OF FUNCTIONAL ORGANIC MOLECULES, HSINCHU, NCTU
(2012/10/21-24)
4. “DISCUSSION AND INVESTIGATIONS OF MOLECULAR VIBRATIONAL DYNAMICS”, SHENG-HSIEN LIN,
德國慕尼黑工業大學化學院物理與理論化學研究所(FAKULTÄT FÜR CHEMIE/INSTITUT FÜR
PHYSIKALISCHE UND THEORETISCHE CHEMIE/TECHNISCHE UNIVERSITÄT MÜNCHEN/GERMANY),
(2012/11/18-27)
5. “MOLECULAR THEORY OF SUM-FREQUENCY GENERATION”, SHENG-HSIEN LIN, 理論計算化學研討
會, 台科大化工系, TAIPEI (2013/01/18-19)
6. “THEORY AND APPLICATION OF INTRAMOLECULAR VIBRATIONAL RELAXATION(共同討論及研究分
子振動動態機制)”, SHENG-HSIEN LIN, ARIZONA STATE UNIVERSITY/ PHOENIX AND UNIVERSITY OF
OKLAHOMA/ NORMAN, USA (2013/02/09-22)
7. “”, SHENG-HSIEN LIN, 中國/重慶/西南大學化學化工學院, 廈門/廈門大學固體表面物理化學國
家重點實驗室及北京/中國科學院化學所 (2013/04/25-05/04)
8. “APPLICATION OF INTENSE LASERS TO BIO-ORGANIC MASS SPECTROMETRY”, YUICHI FUJIMURA AND
SHENG-HSIEN LIN, INVITATION TO ATTEND THE SHORT PULSE STRONG FIELD LASER PHYSICS
INTERNATIONAL SYMPOSIUM HONOURING SEE LEANG CHIN, CENTER D’OPTIQUE, PHOTONIQUE ET
LASER, UNIVERSIÉ LAVAL, QUÉBEC, CANADA (2013/05/20-26)
9. “THEORY AND APPLICATIONS OF SUM-FREQUENCY GENERATIONS AND SURFACE-ENHANCED SFG”,
C. K. LIN, M. HAYASHI, L. YANG, C. Y. ZHU, Y. FUJIMURA AND S. H. LIN, 6TH WCTCC, 第六屆 世界
華人理論及計算化學研討會(THE 6TH WORLDWIDE CHINESE THEORETICAL AND COMPUTATIONAL
CHEMISTRY CONFERENCE (WCTCC)), 24-28 JUNE, 2013, TAMKANG UNIVERSITY, TAIWAN
(2013/06/24-28)
10. “NEW DEVELOPMENTS IN BIOLOGICAL PHYSICAL CHEMISTRY AT MOLECULAR LEVEL”,
SHENG-HSIEN LIN, CONFERENCE IN HONOUR OF THE 90TH BIRTHDAY OF RUDOLPH MARCUS
FUNDAMENTALS IN CHEMISTRY & APPLICATIONS (紀念魯道夫•馬庫斯教授在基礎化學及應用的第
90 個生日研討會), 22-24 JULY, 2013, NANYANG TECHNOLOGICAL UNIVERSITY, SINGAPORE
(2013/07/21-25)
Books
1. Wolf Prize in Chemistry: An Epitome of Chemistry History in 21th Century and Beyond, Lou-Sing
Kan and Sheng-Hsien Lin (World Scientific Publishing, 2012)
2. Advances in Multiphoton Processes and Spectroscopy, Vol. 21, S.H. Lin, A.A. Villaeys and Y.
Fujimura (World Scientific Publishing, 2012)
3. “Application of Density Matrix Method to Ultrafast Processes”, YL Niu, CK Lin, CY Zhu, H Mineo,
SD Chao, Y Fujimura, M Hayashi and SH Lin, Progress in Theoretical Chemistry and Physics, SPi
Technologies India Private Ltd., Manatec Towers, (2013)
4. Theory and Applications of Sum-Frequency Generations”, C. K. Lin, L. Yang, M. Hayashi, C.Y.
Zhu, Y. Fujimura, Y. R. Shen, and S.H. Lin∗, JCCS (Journal of the Chinese Chemical Society),
No.:jccs.201300416 (Accepted, in the Book of Advances in Multi-Photon Processes and
Spectroscopy, Vol. 16) (2013/08/15)
3.