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Molecular theory of second-order sumdashfrequency generation
ELSEVIER
Physica B 222 (1996) 191-208
Molecular theory of second-order sum-frequency generation S.H. Lin a'b'*, M. Hayashi a'b, R. Islampour "'c, J. Yu a, D.Y. Yang ~, George Y.C. Wu d aInstitute of Atomic and Molecular Sciences, Academia Sinica, Taipei, Taiwan ~Department of Chemistry and Center for the Study of Early Events in Photosynthesis, Arizona State University, Tempe, AZ 85287-1604, USA Department of Chemistry, University for Teacher Education, 49 Mofateh Ave., Tehran, lran d Department of Chemistry, University of the Chinese Culture, Taipei, Taiwan Received 23 December 1994; revised 21 November 1995
Abstract
Recently the second-order sum-frequency generation (SFG) has become a powerful surface - spectroscopic technique. In this paper we present a molecular theory for SFG. To systematically treat this process, we have classified SFG into the resonance-resonance case, resonance off-resonance case, off-resonance-resonance case and off-resonance-offresonance case. Among these cases we may have IR-UV SFG and UV-UV SFG. Some of these SFG's are reported in this paper. In each SFG case, we have shown how to calculate the strength (or intensity) and band-shape function.
I. Introduction Second-order nonlinear optical signal generation is dipole-forbidden in media with inversion symmetry. However, such signals have been generated at the interfaces of isotropic media since the earliest days of nonlinear optics [1,2]. An important application of surface nonlinear optical measurements is the determination of adsorbate spectra through the resonant enhancement of the second-order nonlinear susceptibility )(2~. Early measurements exploited electronic resonances to record adsorbate electronic spectra [3,4]. Recently the emphasis has been placed on the application of infrared (IR) + visible or UV sum-frequency generation (SFG) to obtain adsorbate vibrational spectra [5-7]. The vibrational
* Corresponding author.
spectra of a number of adsorbates have now been reported: the methyl modes of alkyl thiol self-assembled monolayers on gold [6]; a number of alcohols on silica [2]; Langmuir - Blodgett films on silica [8,9]; C O on some single crystal metal surfaces [10,11]; methoxy on N i ( l l l ) [11]; acetonitrile on ZrO2 [12] etc. (see, e.g., Ref. [13]). A main purpose of this paper is to give theoretical treatments for various cases of S F G like resonance-resonance, resonance-off-resonance, off-resonance-resonance, and off-resonance-off-resonance cases. We shall also show how to use the B o r n - O p p e n h e i m e r approximation to obtain the expressions for the four cases of S F G for molecular systems. In previous papers [14], the molecular theory of I R - U V S F G has been reported. For the purpose of assisting the future developments in SFG, in this paper we report the molecular theory of other S F G cases.
0921-4526/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 2 6 ( 9 6 ) 0 0 0 0 8 - 7
S.H. Lin et at/Physica B 222 (1996) 191-208
192
2. General consideration
Integration of Eq. (2.4) yields
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(~/~) = Tr (p~3),
(2.1)
where ¢3 denotes the density matrix of the system. The diagonal elements of ~ describe the population of the system, while the off-diagonal elements of describe the phase (or coherence) of the system. Notice that the equation of motion for the density matrix fi(t) of the system is given by the stochastic Liouville equation [15] d~ = _ iLop -/~,~ - iL'(t)~ dt
d~ L'(r)~(r),
where ~i denotes the value of ~(t) at t = ti. Regarding E'(t) as a perturbation and using the perturbation method, we find ~(°)(t) = ~i, ~(1)(t)
= -ill
(2.7a) dr L'(z) ai,
(2.7b)
i
where #(°)(t) and ~(1)(t) represent the zeroth-order and first-order approximations of ~(t), respectively. For the second-order processes, we need the second-order approximation ~(2)(t), dr
dz'L'(z) L'(f)ai.
(2.7c)
The second-order polarization P(g)(t) is given by (2.2)
• ' (2) P(2)ft) = Tr(fi(2)~) = ~--~,~e-'t'uStkOkt (t) + c.c. k
where iL' = iLo + F, and Lo and L'(t) denote the Liouville operators corresponding to the zerothorder Hamiltonian and the Hamiltonian for the interaction 17 between the system and radiation field, respectively./~ represents the damping operator describing the relaxation and dephasing of the system. To evaluate P, it is necessary to solve Eq. (2.2). For this purpose, for the weak field case the perturbation method is often used. This can be done by regarding L'(t) in Eq. (2.2) as a perturbation and solving Eq. (2.2) directly. However, it is often more convenient to eliminate the L~ term in Eq. (2.2) first before solving the equation. For this purpose, we let /~(t) = e-i'L~(t)
(2.6)
Jt~
~(2)(t) = ( - i ) 2
= - iL'op - iL'(t)~ i = - i L ~ p - ~ [,I7",t~],
~(t) = ~, - i
(2.3)
l
(2.8) Using the property of L' [,15] and substituting Eq. (2.7c) into Eq. (2.8) yields P(2)(t) = (--i)2 h ; dr f f d~'EEEe-',°,k,t~,~ i
i
k
k'
I
× { 17(r)kk , I-L'(r') ffi3~,, - [L'(z')#i]kk" ff(r)k't} + C.C.,
(2.9)
where for example rT(Z)kk, = e i~';'r V(r)kk,. Using the relation 1 [L'(r')tTi]k,Z = -~ ¢ (Z')k,l[(tri)tt -- (~i)k'k']
(2.10)
we obtain
to obtain d~ -- = - iL'(t) t~, dt
p(2)(t ) = -(-i)2f) -hT(2.4)
i
k
k'
l
× (rT(~)kk' lT(rt)k '' ['(~i). -- (~i)~'~']
where
L'(t) = ei'L'°L'(t) e-itL'o.
dr f( dz'EEEe-i"°k'p,k i
(2.5)
- 17(~')~, 17(r)k,, [,(a0k,~, - (a3~d} + c.c. (2.11)
S.H. Lin et al. / Physica B 222 (1996) 191 208
i#j
In the dipole approximation, we can write l?(t) as [16]
193
(l~k't"Ei(09~))(l~kk'"Ej(09j))
×2Z'
1 ~
2
i(09~,,,- o9,) + T-~
r/ (t) = --~F.~l ~" Ei(°~i091)e-lt .... /i(t), lj (t) =
e-(',- 0/T,,
(2.12)
lhkAa(kk, k'k')
+
1
1
i(09~,, - 09, - co2)+ ~,~--+ :T--
where/1 represents the dipole operator, a~ takes the values of 1 or - 1 and lj(t) denotes the laser pulseshape function with pulse duration T l. The summation over i in Eq. (2.12) describes the situation that in SFG two lasers are usually involved. Using Eq. (2.12), Eq. (2.11) becomes
i # j E' (lllkk' "El (09i))(/ak't "Ej (09j)) i
j
i(09~k' -
o)i)
1
-[- - -
Ti
( - i) 2
e'2'(tp) -
EYYEYJ,,
k l k' iai jctj
(( x
Aa(ll, k'k')(l~k,,'Ei(e,091))(lal,k,'Ej(~j09j))) -------1--~[.-/---/-, ...... ----1 1
i(09'k't--lXi09i) +~-i)~l(09kl--lXi091--O~j09j)
+(
+ - ~ - i + ~-jj
--- ~Aa(kk'k'k')(lakk" ei(°ci09i))(lak"'Ej(~J09J))-il ~j) i(09~k, - o~,09,)+~ 1(09kt-- O~i(-Di-- ~jO,)j) -~" -~i -~-
x exp [-itp(~i09i
+ ej09j)] +
(2.13)
c.c.,
where, for example, Aa(ll, k'k') (~i)/| (~i)k'k" representing the initial distributions of the system. Eq. (2.13) is general and can be applied to study difference-frequency and sum-frequency generations. For SFG, for convenience, Eq. (2.13) can be written as =
[
- -
+
#lkda(ll, k'k')
L
1
' ~ J Z' ×E
(~'~" ' E , ( - ~ , ) ) (~,~, • & ( - 09j))
i
i(~;,,t + 093 + - -
j
p(2I(tp) = P(2)(09 1 + co2)ex p [-itp(09, + 09z)]
+ c.c. + .-.,
1
Ti
(2.14)
#tkAa(kk, k'k')
+
1
1
i(09~,, + 09~ + 09i) + l,v- + aT--
where
i¢j
P(2)(091 + 092) _
1
i(09~, + c01 + c02) + ~ - + ~-/1
(__ i)2 V ~ ~ {
l~,kAa(ll,k'k') 1
1
~
i(09~,, - 09, -- 092)+ ~ + T--~
J
1 i(o~;,k, + m~) + - -
]*}
Ti
(2.15)
S.H. Lin et al. / Physica B 222 (1996) 191-208
194
In other words, in S F G we are looking for the coefficient of exp[-itv(03 ~ +co;)]. Here in the double summations over i and j, the i = j terms are to be excluded. We shall consider the following energy level diagram for S F G (Fig. 1). In Fig. 1, g,m and k denote the initial, intermediate and final state manifolds, and cox and 032 represent the frequencies of the two lasers used in SFG experiments. According to the definition of the second-order SFG susceptibility ; ~ (COl + 032),
(m,,(/~),~.(y)) +/~k,(a) (i(~0m~+ (D1)+ r;.,) (i (~o~,+ o~ + c~2)+ G,)
+ (i(c0,.~ + co2) + F~,~)(i(cok0 + co~ + ~o2) + F ~
(,~,(/~)p,~(y)) -
P(~2)(e)a + o)2)=
~,,
#ink(a)
(i(coo,.
-- ~0,) + F;.~)(i(~o,~ - o~ - o~2) + r ~ )
Z(~)~(03,+ 032)E,~(c~),)E2~(032), (2.16/ + (i(c%. - co2)+ r~,~) (i (co,. - (~)1 (m.(~),~.(~))
we find
+ #~"(~)
(i(coa., + ¢ol) + r~,g)(i(coa~ + col + ¢o2) + F~,.)
(2.17)
where
("g"(~')u"*(/~))
+ (i(%. + o~) + r~,,)(i(co~. + ~o~ + ~2) +Gm
i2 (2) (01 -F (/)2)1 = ~ ~ E E AtT(Oa'mm) ~a/gy i*
g
" i]* t(2.18)
k m
x ~ak(a) (i(o~ma -- ~01) + F'o)(i(O~k 9 -- OJ1 -- 0)2) + F~ 9)
k
o2
1
(~,~.(/~)m~(y))
.(2) ((2) 1 _[_ 03.02) = Z,,(2) . . + ~ .(2) Za/~v . ~ ~[ .a . + . (~2)1 ~ (03.01 _~. 032) 2
+ ~~,(2) p ~ [, l • + 032)3,
- - (222) ~ - r k m )
o 1 +o 2
m
o!
g Fig. 1. Energy level diagram for SFG.
~,(21t , + 0 9 2 ) 3 c a n be obA.aflyt~l tained from X~)y(col + co2)1 by performing the exchanges k ~ m and k ~ g , respectively. For convenience, their explicit expressions are given in Appendix A. In Eq. (2.18) we have F,~g = F,.g + (l/T,), F,~ = Frog + (l/T2) and F~ = Fkg + (l/T1) + (l/T2). As discussed in a previous paper [16], there are four cases of SFG. They are the resonance-resonance, resonance-off-resonance, off-resonance-resonance and off-resonance-off-resonance cases. For example, the off-resonance-resonance case means that the h031 photon is not in resonance with the g -~ m transition, while the h03 2 photon is in resonance with the k state. To show the application, we consider the resonance-off-resonance case. The energy level diagram of this SFG is shown in Fig. 2. This figure shows that h03, is in resonance with the g ~ m transition. From Eq. (2.16), we can see that due to the resonance of the g ~ m transition with co,, we only need to retain the terms like i(03mg -- CO,) + F,~g or i(03gm + coo + F,~g in X~)~(03, + co2). From ,(2) (031 -[- 092)2
and
S.H. Lin et al. /Physica B 222 (1996) 191-208
195
F o r the off-resonance-resonance case, we have ,,(2) l
k
Z,p~(2)(091 + 092) = z,p~'(2)(091 + 092)1 + ~ W ) l
+ 092)3, (2.21)
m2
where
Ol + m 2
i2 ,[2) i" , Aefl7 tt~'l "}'}'092'1 : h-2 i~7' ~= '"~
a
Ad'((Jg'mm)[l"ok(°O
(09k. - 091 - 092) + F~.
m [g,,O(fl)Pk"(Y) ml
/~,,0(7)gk"(fl)~
(2.22)
and .(2) I A~y ~,t-Ul +
i2 = # Zg Z,.2
a a(kk, mm)#gk(oO . . . . . .
~ i(09k9 -
091 -
g Fig. 2. Energy level diagram for the resonance-off-resonance case of SFG.
Eq. (2.18) we find ~(2) ,~,(Z) It., "at- 092)1 a/~y (091 "}- (D2) = Lafl~,~,t~l
i
= -~ ~, Y', ~, Ao(go, mm) g k ,. ]Aok(OO]Amg(fl)lAk,.(])) X
(i(09mo __ 091) -[- /'~nO)(09kg - - 091 - - 0)2)
~Akm(O~)~A,.g(fl)]Agk(~))
jJ
( - i ( % m + 090 + r;,g)(09~" +091 + 092 (2.19) which is obtained by collecting the terms that contain the resonance terms 09,.o ~ 091. Similarly for the resonance-resonance case, we collect the terms that contain the resonance terms 09,.0 - 091 and 09ko -- CO1 + 092 to obtain
+ 09 ): 25,(091 + 09 )1
/$,(09l
i2
= ~ Z £ Z Aa(gg, ram) g k ,. × (i(co,.
o -
o91) + r ; . . ) ( i ( c o k .
-
o~1 -
~2)
+ rg)
(2.20)
X [ "~km(J)~-~"g('-'~) "}" "k"(]l)"mo(fl)] L ('Ok,. -- 091 09kin -- 092 J
.
092) +
Fko (2.23)
In most cases, the Boltzmann factors Aa(kk, mm) ,(2) tQD 1 + 092)3 is are negligible. That is, in this case z~a~ negligible. It should be noted that for the off-resonanceoff-resonance case, all the contributions in Z(2) I • ~(2) ~r., ,(2) t • ~ t ~ , l + 602)1' ],.~fly kU'~l + 092)2 and Aafly ~UJl + c02)3 should be considered. F r o m the above theoretical derivation of the mathematical expressions for SFG, we can see that numerous types of S F G experiments can be designed. This is because not only we have four S F G cases discussed above but also we can use IR or UV for 091 and 092 in SFG.
3. Molecular theory of SFG In this section, we shall treat various cases of S F G for molecular systems by introducing the B o r n - O p p e n h e i m e r ( B - O ) approximation. By using the B - O approximation, the S F G expressions obtained will be expressed in terms of molecular properties and the relation between a given S F G and other optical process can be found. The results presented in this section will be useful to experimentalists for designing S F G experiments and analyzing experimental results.
S.H. Lin et aL / Physica B 222 (1996) 191-208
196
3.1. Resonance-off-resonance case The general expression for this SFG case is given by Eq. (2.19) which can be rewritten as Z(2) ~ ,
ku
tO2
i g
k
tO 1 +60 2
m
X i ( 0 ) m g ~ - ~ l ) + Frog \COkg - - 0)1 -- 0)2
gv'
q 0 ) ~ T ~ - 1 1 - + ~ 2 } ' (3.1) For a randomly oriented system (i.e., isotropic system), it is necessary to carry out the spatial average Of /,~#~ ,~(2) (0)1 + 092) over the entire space; in this case Eq. (3.1) will vanish because of the appearance of odd-power #~g(fl)/~gk(~)#km(Y) in /(,afl),~t'~"l ~2) t , + (DE). This property makes S F G a surface specific experimental method. It can also easily be shown that for a centro-symmetric system, r.h.s, of Eq. (3.1) vanishes. For example, if the g state has a gerade symmetry, then because of #mg(fl) and #k,,(~') (or #kin(a)) the m state must have an ungerade symmetry and k should possess a gerade symmetry. However, in this case #gk(Ct) and #0r(Y) will vanish because both g and k have gerade symmetry. From Eq. (3.1) we can see that Zt2) for the resonance-off-resonance case consists of the resonance part and off-resonance part. The above expression can be applied to I R - U V S F G and U V - U V SFG. Here we have used the notation that for I R - U V SFG, IR is for 0)a and UV for ~2. To distinguish these two SFG's, it is necessary to introduce the B - O approximation. For the I R - U V SFG, we have 9 ~
9v,
m ~ gv',
k ~ ku.
The vibronic energy level diagram for the I R - U V SFG is shown in Fig. 3. Here g and k denote the electronic states, while v, v' and u represent the vibrational states. Due to the fact that in I R - U V SFG, the first transition associated with 0)1 is in the IR range, it must be due to the vibrational transition and is described by gv ~ gv'. Since the laser 0)2 is in the UV-visible range, excited electronic states need to be considered; we use ku to
031
gv Fig. 3. Vibronic energy level diagram for I R - U V SFG. tot is in resonance while ~2 is off-resonance.
denote the excited vibronic manifold. In terms of the new notations, Eq. (3.1) becomes i v
x
u"
ku
i(0)gv,,ov - 0)0 + rgv,,g.
×(
(7_)
\~ku.ov
--
091
--
092
]
0)ku,gv"+ 0)1 + 692/
(3.2) In the B - O approximation, the molecular wave function, for example, %0 can be written as a product of electronic wave function @g and vibrational wave function Ogv, i.e., qggv = ~g(qQ)Ogv(Q),
(3.3)
where q and Q denote the sets of electronic and vibrational coordinates, respectively. In this case, we have tZav.ku(O0 = = -- < O g v l # g k ( = ) l O k . > ,
(3.4)
where lZgk(Ct) = <~g(qQ) l#(ct)l~k(qO)>
(3.5)
197
S.H. Lin et al. 1 Physica B 222 (1996) 191-208
which represents the electronic transition moment for the g+-+k transition. In Eq. (3.4), to evaluate the matrix element of dipole moment P~“,~~(cI),it is necessary to carry out the integrations over both electronic and vibrational coordinates. For this purpose we first perform the integration over electronic coordinates to obtain the electronic transition moment pLgk(tl) which in general depends on vibrational coordinates. In Eq. (3.2), the off-resonance part contains the terms like wk,,+ - o1 - o2 and c&g”’ + o1 + o2 in which the vibrational energies are smaller than electronic energies, and we can use the Placzek approximation, i.e., ok”,go
=
Co ku, gu’
mkg,
= wkg.
(3.6)
The quantities pgg(fi) and txgg(ay) can be expanded in the power series of QI, vibrational coordinate,
/.dB)=&(P)+C
(3.10) 1
and
“gg(“Y)= $g(aY) + c
Qr + ... .
Substituting yields
x
Eqs. (3.10) and (3.11) into Eq. (3.8)
Ct8i)gv.gv -
(Bi)gv’,gv’l
l(@g,IQ~l@,,~)12 X. 1(og”‘,g” - 4 + r;“‘,g”’
In this case, Eq. (3.2) becomes
(3.11)
I
(3.12) ’
where X.
(@,“~IPgg(P)I@g”) l(~g”‘.go - 4 + r;“‘,g”
x
(@gd&k@)~@ku) 0) kg -
[ +
Eqs. (3.12) and (3.13) indicate that for a particular vibrational mode QI to be observed in SFG,
<@ku~~kg(Y)~@gv’) w1
-
(32
1.
(@kubkg(CoI@gu’)
(@gvI~gkb)I@ku)
mkg + 01
+ m2
%7,(P) #
(3.7)
-
(6i)gv’,gv’l
(@g”(agg(cry)(@g”,)) X. (@~~~~~%?(~)~@PJ) mgo’,g” - 4 + r;“‘.g” (3.8) where ~~~(rxy)denotes the electronic polarizability of the ground electronic state, “ggbY)
= 2 k
pgk@)pkg(Y) mkg -
wl
-
+ w2
aQl
(3.14)
0
0
and
Applying the closure relation C ul@k,) (@,,I = 1 to Eq. (3.7) yields
Xzy(Wl+ 02) = $1 C C(ai)gv,gv ” c’
( ) a~,,by)
( 1#O. aQ*
(3.15)
0
Eq. (3.14) corresponds to the IR selection rule, while Eq. (3.15) corresponds to the Raman selection rule. In other words, for the IR-UV SFG, the SFG active mode should be both IR active and Raman active. Some preliminary results of the IR-UV SFG have been reported in previous papers [14]. From Eq. (3.12) we can see that one can introduce the band-shape function for a particular mode in the IR-UV SFG
~gk(Y)~kg(CI) Okg +
wl
+
02
)
Fi(wlw2)
=
(3.9) which in general depends on vibrational coordinates. pJ/?) represents the dipole moment of the ground electronic state.
CC
” 0’
C(6i)gv,gv -
l<@,~1Q~I@,,~)I*
X i bgd,
=
Flt”l~2)r
(Bi)gd.gdl
+
gu -
Fl(“lo2)i
01)
+
3
r;“‘,g”’
(3.16)
198
S.H. Lin et al. / Physica B 222 (1996) 191-208
where
Using the Plazcek approximation to simplify the off-resonance part, Eq. (3.19) becomes
F / ( ° ) l ° ) 2 ) r ~- Z E
[((~i).v, gv -- (~i)gv',gv']
z5' (091 + 092)= i
, Fo,,,,ovK o,,~IQ, IOo,,,)I 2 x (09o,,',g,, - 091) 2 + (F'ov',o,,) 2
(3.17) ×
( Og~,l ~xg,,(091 + 092),,~10,,w) (Om~[/~mg(fl)lOg~) 09,,w, ov - col - iF~, o
and
(3.20) FI (091,092)i -~
E E [(ai)gv, ov - (ai)av,,ov,]
x(091 - o~ov. o~)l(Oo,,IQ, lOo.,)l 2 ' ( 0 9 o . ' . ~ - ~°i) 2 + ( G . ' . , o ) 2
where (3.18)
That is, the band-shape function Fz (09~092) can be separated into the real part Fl(0)1092) r and imaginary part F1(091092)i. The real part Fz (091092)r and the imaginary part F1(091092)i of the SFG band-shape function Fz (091c02) are related to each other by the so-called Kronig-Kramers transforms (for the detailed discussion of the Kronig-Kramers transforms, see Ref. [17]). The real part Ft(091~O2)r is exactly the same as the band-shape function of IR spectra. This means that the SFG band-shape function can be determined if the IR band-shape function is known. We shall consider the UV-UV SFG of the resonance-off-resonance case next. Its vibronic energy level diagram is shown in Fig. 4. Due to the fact that the UV or visible lasers are involved in both 09~ and 092, electronic excited states are to be considered in the intermediate and final manifolds. The location of arrows in the figure can distinguish between the resonance and off-resonance situations. Notice that in this case, Eq. (2.19) takes the following form: ,~a#v
09, -
+
(2) 1 - -
092)2
09kin -F 091 -F 092
(3.21) Again both a~.,(091 + (-D2)a~, and/z,,g(fl) depend on vibrational coordinates Ql and can be expanded as the power series in Qt. If we keep only the dominant terms, in this case it corresponds to the terms without Qt; this is usually called the Condon approximation, In this case, Eq. (3.20)
(k} -= (ku}
(°2
o)l +o)2
{m} =- {row}
_ 091) + Fmo
EZZ¥ ¥ 2j (3.19)
Here (ai). . . . .
--
(a,).o,.vumw.ov(3) , . . . .
.
~,(z) (09, + c02) = ~ ~ i ( 0 9 m . , g
LO)kg
has been neglected.
{g} -= {gv} Fig. 4. Vibronic energy level diagram for U V - U V SFG of the resonance-off-resonance case.
S.H. Lin et al. I Physica B 222 (1996) 191 208
199
becomes
{k} -- {ku}
= K' E Z m,)....~ v w
I(O~olO,.~512 •
!
,
(3.22)
032
O)mw,g v -- O) 1 -- 1Frog
031 +0~ 2
where 1
{m} ~- {mw}
(3.23)
and I(OgvIO,,~)] 2 denotes the Franck-Condon factor• Eq. (3.23) indicates that for the case in which K' is relatively insensitive to the variation of co~, the measurement of SFG will provide the information of the absorption spectral band-shape for the electronic transition g ~ m. In other words, one can obtain the one-photon absorption spectra from the SFG of adsorbed molecules and vice versa. From the above discussion, we can see that for the IR-UV SFG, the intensity (or strength) is determined by the product of (~l~gg(fl)l~Ql) and (~c%(c~7)/~Qz), while for the UV-UV SFG, the intensity (or strength) is determined by the product of #,,0(fl) and eg,,(e)l + co2). This information is important when one applies molecular orbital calculations to designing molecules with particular properties of X~)~(~ol + (D2).
031
~---~v} Fig. 5. Vibronic energylevel diagram for UV-UV SFG of the off-resonance-off-resonancecase.
where, for example,
1~[ 7- ,7,. 7,,•"(')"''(') ""(')''m(') I (3.25) and
3.2. Off-resonance-off-resonance case Next we shall consider the application of the off-resonance-off-resonance case to the UV-UV SFG. The corresponding vibronic energy level diagram is shown in Fig. 5. In the Plazcek approximation, we obtain
Z (2) t~ "
=-h~v~m(~i)gv, gv~ov
(.Omg--(d)l
+ v.=(/s)~,=(=7) + ~=.(7)=.=(~/7) O)mg -~- 0.) 1
+
~,.,--77~,--7
O,)mg -- 0,) 2
i
'
(3.24)
o7g=(~7) lk~ f #k,(')#,.k(7) h COka + ~ol + 092
+ v_~.(j)c<___.,<(~_).]• O)kra --
(01 -- 0)2~
(3.26) This type of SFG is relatively insensitive to temperature. This can be seen by expanding #mg(fl), ~9,,(~7), ~g,,(~7) etc. in terms of normal coordinates Q~. We will see that the dominant term of Xt2) is independent of temperatures. From Eq. (3.24) we can see that the molecular orbital calculation of the intensity (or strength) of this particular SFG would be extremely difficult.
200
S.H. Lin et al, / P h y s i c a B 2 2 2 (1996) 1 9 1 - 2 0 8
3.3. Off-resonance-resonance case
where
Next we consider the application to the off-resonance-resonance case. We present its vibronic energy level diagram in Fig. 6. Here we are considering the U V - U V SFG. In this case, we retain the terms like i(09kg -- 091 -- 092) + F~g in Eq. (2.17) and from Eq. (2.21) we obtain
1 m~ ~'Ukm ~ m(7)/~"W(fl) g~-~l
IAkm(fl)~Amg(')~ 09rag - - 0 ) 2
_J
(3.29) In the Condon approximation we obtain 1
~(2) ((/)1 + 0)2) ~go, ~. (~)
1 =
.
u
v mw
X
,
09ku, gv - - 0)1 - - 0 ) 2 - - I F k g
X [ ["lraw" gv (J~) ~'lk..~ ~' raw ( ~)) _} ~'lmw, gv ( ~)) /Llku" mw ( fl ) ] . h
09,.,.,.~., -
09~
09,,..,.~o -
092
(3.27) Here due to the Boltzmann distributions, other terms in Eqs. (2.17) and (2.18) can be neglected. Using the Plazcek approximation, Eq. (3.27) becomes
Z~p~(091 + 092)
V V L,~/.,,~
(~)g~'g° I(Og~ I@k")- ~12 . . . . . . --I
.
Eq. (3.30) indicates that when ~kg((l)1092)f17 is relatively insensitive to the variation of 09~ and 092, the SFG can provide the band-shape function of the absorption spectra for the electronic transition g~k. Notice that Eq. (3.30) can be written as a product of the electronic part and vibrational part: Z(2) (091 + ('02)
1 = -h ] l g k ( O O O t k g ( 0 9 1 0 9 2 ) f r F g k ( 0 9 1
1 Z Z ((~,)go,go h u v 09ku,gv
- - 0)1 - - 092 - -
iFko
x (Ok~l~kg(091,092)pr IO~>,
(3.28)
(3.30)
"
u v 09ku, gv -- 091 -- 092 -- IFkg
+
(3.31)
0)2) ,
where the vibrational part Fok(09a + 092) determines the band-shape function of SFG,
Fgk(091 + 092) {k} ~
(~i)gv.gol(Ogol Oku)[ 2
{ku}
O-)ku, gv - - 091 - - 092 - -
~2
(3.32)
IFkg
and the electronic part #gk(Ct)~kg(091092)pr determines the strength of this particular type of SFG. The band-shape function Fgk(091 + 092) can again be separated into the real part Fgk(091 + 092)r and imaginary part Fgk(09~ + 092)i,
CoI +O~2
{m} - {row}
coI
Fgk(091 + 092) -~ F g k ( 0 9 1 "q-
092)r + iFgk(09~ + 092)i,
(3.33)
where
Fgk(091 + 092)r ~,..
Fig. 6. Vibronic energy level diagram for UV-UV SFG of the off-resonance-resonance case.
u
v
,
(09ku.go -
o91 - - 0 9 2 ) [ ( O g o [ O k , ) [
~09ku, gv - - t'~l - -
2)
T
2 kg
(3.34)
S.H. Lin et al. / Physica B 222 (1996) 191-208
201
tion. In the Condon approximation, Eq. (3.36) becomes
and Fgk(0)l + 0)2)i :
(3.35)
z
0)-; -
1 ~2 #gk(OO]Xkm(~})~mg(~) E E 2 Po v v u w
That is, Fgk(0)~ + 0)2)i denotes the band-shape function of the one-photon UV absorption spectra and Fgk(0)l + 0)2)~ is again related to Fgk(0)~ + 0)2)i by the Kronig-Kramers transforms.
(3.37) which can be rewritten as
3.4. Resonance-Resonance case Finally we consider the application of the resonance-resonance case of SFG corresponding to U V - U V transitions (see Fig. 7). Notice that in this case Eq. (2.20) becomes ~(2) [ •
= -
dt e-
itAokg
×
w
Po~/.to~,ku(e)#k . . . . (Y)#,.w,..(fl) [i(o)k.,o~ - c o l - o32) + r~g] [i(o).,w,o~ -- eh) + r~.g]
(3.36) because other terms are unimportant. Here (8~)o~,o~ = Pgv representing the Boltzmann distribu-
{k}
{k,,}
(3,38)
where iA0)ko = i(0)kg -- 0)1
--
(/)2)
-F /~ko'
iA0),, o = i(0), o - o)1) + F~.o
(3.39)
and 1
0)2
fo~ dz e -~a~-.
v u w
i2 O u
X
~Or
Z(2) (0)1 + 0 ) 2 ) K " f o
K" = ~7 #ok(~)#km(Y)#,,o(fl).
(3.40)
In this SFG case, the intensity (or strength) is determined by #0k(a)#k,.(?)/tmg(fl) which can easily be obtained by molecular orbital calculations. Using the closure relations ~ , [ O k . ) ( O k , I = 1 and Y~wlO,,w) (Omwl = 1, Eq. (3.38) becomes
0)1 +0)2
{m} ~ {row} ~t~vt (2) /0) 1 + 6o2) = - K , , f o -~A
d t e -itA°~. fo~ d.c e-i~aw.g i~
i
x ~Pov (Oovle " ke-~a~e-~(t+~)/LIOov). (3.41) 0) 1
Fig. 7. Vibronic energy level diagram for UV-UV SFG of the resonance-resonance case.
12
This indicates that the calculation of double resonance U V - U V S F G ,%~ ~(2) t~,l ~ , + co2) involves the calculation of two-time correlation function. Next we shall consider an important case; that is, the potential energy surfaces of the k and m electronic states
S.H. Lin et al. / Physica B 222 (1996) 191-208
202
are identical. In this case, Eq. (3.37) reduces to t'.., ,#,~,1 Z(2)
+ co2) = - - K " 2 Z
Substituting Eq. (3.47) into Eq. (3.36) yields ~(2) /" ~
Z, a t f l T t t . ~ l
v u
+
Po~l(Oa~l Ok~>l2
[i(coku,g~- oga -- 092)+ r~g] [i(~o.,,~,g~- oh) + F~,g]
xexp
dte-itdo~k.
0)2) = -- K"
-S
{
(3.42)
h0) h0) coth~---.-.~-csch-ZK1 2kT
which can be rewritten as ~(2) ((.O 1
dze -i~'°-.
(3.48)
0)2)
where S = ½fld2, usually called the Huang-Rhys factor or the coupling constant. Eq. (3.48) can be integrated by changing the variables x = t + z and y = t - z; the result is given by lfOo=lOk.)l z e - g"(~ k.- E, . ) e - g'(~ - . - ~,o ,
xZZpo, u
Z(2) ((.01 + (02)
(3.43)
=_K,,e-S(2n+z)~ ~
where
m=O n=O
id0)ko = i(0)ko -- 0)1
--
0)2) 71- / ' k o '
iA0),.0 = i(0),,0 - 0)1) + Frog.
1 - -
(3.44)
[ S ( ~ + 1)]" (S~) m
n!m!
[iA0)r,g + i0)(n - m)] [iA0)kg + i0)(n -- m)]"
For the single-mode case, we obtain
(3.49) At T --- 0, Eq. (3.49) reduces to dte - it~'',
= --K"
dz e-i~a°', G(t,z),
(2)
Z~py(0)l + 0)2) (3.45)
(1/n!)S"
where
= - K " e - S .=o [iA0)~g + in0)] [iA0)kg + in0)] "
G(t, z)
(3.50)
= 2 sinh " - - ~ ~ e-a(v+~)e-U(u+~)l(Og ~l Ok,)12 2kT.
~
(3.46) It follows that [18] G(t, z) = exp
Ifld2{h0)
-- --~ coth 2 ~
h0) - csch 2 k T
xcosh(2-~TT-i0)(t + z ) ) } l , (3.47) where Q' = Q + d and fl = 0)/h. Here d denotes the normal coordinate displacement between the two electronic states.
The types of expressions for SFG given by Eqs. (3.49) and (3.50) are useful because they are expressed in terms of molecular properties like electronic energy gap, coupling constants (or Huang-Rhys factors) etc. They show how one can make good use of the SFG band-shapes to determine molecular properties. The same set of molecular properties are also involved in one-photon absorption coefficients. In other words, through these molecular properties one can relate absorption spectroscopy and SFG together. In concluding this section, we can see that due to the use of the B - O approximation, the molecular expressions for SFG can usually be separated into
203
S.H. Lin et al. 1 Physica B 222 (1996) 191-208
the electronic part which determines the intensity or strength of SFG and the vibrational part which determines the band-shape function of SFG. The strength of SFG is determined by electronic properties like dipole moment, transition moment, polarizability etc, while the band-shape function of SFG is determined by the potential surfaces of different electronic states involved in SFG; harmonic potential surfaces are usually used.
(@,“I@,,) (@kul@g”*) xc.l(Wku,gv Wl
u
-
02)
+
rku,gv
(4.4)
Here we have used the Condon approximation 4. Discussion
for
C(kg and
Various IR-UV and UV-UV SFG’s for molecular systems have been treated in the previous section. It should be noted that we have not exhausted all the possibilities of SFG processes. For example, another important SFG will be IR-UV SFG of the resonance-resonance case; its corresponding vibronic energy level diagram is shown in Fig. 8. Here o1 denotes the IR laser frequency while w2 represents the UV laser frequency. In this case, we have
Ql+...
(4.5)
From Eq. (4.4) we can see that this SFG case is very similar to resonance Raman (RR) scattering. The square of the part involving the summation over u in Eq. (4.4) corresponds to the band-shape function of the RR excitation profile [lS].
(4.1)
and using the B-O approximation m --* gv’,
g-+st’,
k + ku
(4.2)
we obtain
(@,,I&k(~)i@ku)
(~k”(~kg(Y)I”g”‘)(og”‘I~gg(P)~og~)
’ [i(wgv’,gv - 01) + rgo’,gvl
or approximately
i2 = h’
pgk@)pkg(?d
To demonstrate the application of the theoretical results presented in this paper we shall show the numerical calculation of the off-resonance-resonante UV-UV SFG. Separating x$!,(oi + w2) into
bbku,go
-
ml
-
O2)
+
rku,gvl
(4.3)
the real part x$,(oi + mZ)r and the imaginary part x$\(o~ + wz)i of Eq. (3.30) we find
1
c ”
c’
da(gv,
g”;
gv’,
d)
S.H. Lin et al. / Physica B 222 (1996) 191-208
204
Notice that the SFG signal is proportional
(ku)
to
Ix$zs),~~ + w2)12, i.e.,
ti
lx:;; h
+ w2)l 2
= IX$J~I
+ ~2)r12 + Ix$‘,(+
+ ~2),12.
(4.10)
For the purpose of numerical calculations, we shall rewrite Eq. (4.6) as
s cc
=
K”’
dt
0
~~(~i)gv.gvl(~gvl~k,)12 ”
”
x exp[I-t(ibku,gv
-01-02)+r.&}].
(4.11)
For the case in which 0,” and @kuare products of harmonic oscillator wave functions, i.e., Fig. 8. Vibronic energy level diagram for IR-UV SFG of the resonance-resonance case.
where
(4.12)
and
X$$lwl + u2)i
=
K”’ 11 t8i)gv,gu ”
@ku
x(~ku,gv
- 01 - O&)2+ (r&)2 ’
(4.7)
Xb24,t”1 + W2)r = K”‘CC(Bi)gv,go ” 11 Oku,g”
(ok",
gv -
-
u2)l<@,"I@ku)12
co1 -
u2)2 + (l-Q2
01
=
(4.13)
nxk&?f)
I
II
G,I(@,“I@kU>12
X(
@go = n Xgui(Qi)
if the oscillators of Qf and Qi are only displaced, we have [19]
- t{i(u& - ml - 02) + rig} (4.8)
and (4.14) (4.9) Notice that &k(a) (see Eq. (3.5)) denotes the a-component of the electronic transition moment, and can be calculated once the electronic wave functions @kand Qs are known. @kand @gare usually obtained by carrying out the molecular orbital calculations. As can be seen from Eqs. (4.7) and (4.8), x$,(wl + ~0~)~and x$L(ol + 02)i are related to each other by the Kronig-Kramers transforms, and x$\(ol + Oz)i is related to one-photon UV absorption spectra for the electronic transition g -+ k.
where Si = (Wi/2ti) dQf, the Huang-Rhys factor. We shall now apply the above theoretical results to the electronic transition So -+ S2 of rhodamine 6G adsorbed on fused silica. No experimental SFG spectra for this system have been reported. However, the SHG (second-harmonic generation) spectra of this system for the off-resonance-resonance case have been reported by Heinz et al. [3], and analyzed by Lin et al. [16]. For the purpose of illustration, the SFG spectra of this system will be calculated in this paper. Notice that for this system,
S.H. Lin et al. / Physica B 222 (1996) 191-208
we have 1-16]
1.0
~Ok9= 28 500 cm- 1, ~oi = 1130 cm -1,
205
Si = 0.23,
Fk9 = 650cm -1.
(4.15)
The calculated absorption spectra are shown in Fig. 9. The parameters given in Eq. (4.15) have been chosen to reproduce the observed absorption spectra. Using these parameters, we can calculate the band-shape function for the SFG of rhodamine 6G adsorbed on fused silica (see Figs. 10-12). From these figures, we can see that the band-shape of SFG spectra is broader than that of absorption spectra; it is mainly due to the existence of the real part of Fko(O) 1 "-b fOE). The real part and imaginary part of Fko(eh + O)z) are shown in Figs. 10 and 11, respectively. The SFG spectra are shown in Fig. 12. For comparison, the experimental SHG spectra obtained by Heinz et al. [3] are shown in Fig. 13. Next we shall examine the effect of inhomogeneities on SFG. For this purpose, we shall assume that
d
v
'~
0.5
== N
Z
0.0 300
320 SPG
wavelength
340 /
360 nm
Fig. 10. Imaginary part of Fkg(~ol + ~o2).
1.0
i
I
I
i
__
0.5
v
v
o
o O.fi
0.0
N
tq
7
o Z
--0.5 0.0
i
300
i
320 wavelength
/
i
I
340
360
nm
w
300
320 SFG
Fig. 9. Calculated absorption spectra of rhodamine 6G adsorbed on fused silica.
wavelength
340 /
360 am
Fig. 11. Real part of Fk~(~ol + co2).
S,H. Lin et al. / Physica B 222 (1996) 191-208
206 I
I
I
I
1.0
this effect appears in the electronic energy gap Ogkg, i.e.,
~kg = tfka + NAco
(4.16)
and PN = m e'
r. t/)
exp --
.
(4.17)
In other words, we assume the distribution of inhomogeneities to be Gaussian. The Gaussian distribution for inhomogeneities is well known. It is used here to demonstrate its effect on the S F G bandshape. In this case, Eq. (4.14) becomes
0.5
0~ e~
(Fko(COl + ~O2)) = f : d t e x p [ - ~ D 2 t 2 A c o 2 -- it(ff~kg -- col -- O~2) -- tF~,9 -- ~i Si{coth ~ T 0.0
e 300
t 320 SFG
t 340
wavelength
/
360
I0 I10
•
6 G
Rhodomine
cosh ~ 2 ~
{4.18)
nm
Fig. 12. Calculated SFG spectra of rhodamine 6G adsorbed on fused silica.
0 Rhodomine
-- csch
O3
As can be seen from Eq. (4.18), the inhomogeneity effect is to smooth and broaden both absorption and S F G spectra. F r o m Eq. (4.18) we can see that the existence of Gaussian inhomogeneities introduces an additional broadening described by exp(-¼D2t2Aco 2) in Eq. (4.18). In concluding the paper, we have presented the theoretical treatment of various S F G cases for molecular systems. It is hoped that this will stimulate experimentalists in the surface science area for designing new experiments. Although the I R - U V S F G has become a very powerful tool as a surface vibrational spectroscopy, in this paper we have given a detailed investigation of off-resonance-resonance U V - U V S F G for molecular systems and performed numerical calculations using rhodamine 6G adsorbed on fused silica as an example. It is hoped that the organic dyes can be used to probe solid or liquid surfaces by using the U V - U V S F G technique.
Acknowledgements 0
I 300
~
I 320
I
SH Wavelength
I 34.0
I 360
(nm)
Fig. 13. Experimental SHG spectra of rhodamine 6G and rhodamine 110 adsorbed on fused silica.
We wish to thank the National Science Council of R O C and US N S F for the financial support of this work, and the referee for useful comments and suggestions.
S.H. Lin et al. / Physica B 222 (1996) 191-208
207
Appendix A The expressions for /I,~,(2) I • 1 "t- 0)2) 2 afly~ua X~)~(0)~ + co2)3 are given in the following:
((2) ( ,
,~,t~,~ + 0)2)2
and
i2 ~ZZZAa(99,kk)
=
0 k m
[
(~*~(/~)~*(T)) x /~m(Z) (i(0)~o -- cO~) + F~)(i(0)mo -- ~ -- 0)2) + F~n~) (~(~)~(/~)) ] + (i(~k~ -- co2) + r;~)(i(~.~ - ~ , - ~:) + r;.~j (~(/~)~(v)) + pmg(~) (i(0)ko + 0)1) -F F~o)(i(0)mo + 0)~ + co2) + r~.~)
+ (i(0)~o + ~2) +
F~'~)(i(0)mo + (01 -F (/)2) Jv
fling )
[ ~*(/~)~(~) -/Xk~(G) L(i(~k -- 0)~) + r ~ . ) ( i ( ~ , - 0)~ - ~:) + rg,)
(i(%~ - ~:) + r L~)((Ti ()o" ~~ ( ~ )- ~ [
- ~:) + r ; ~ i]
"~(/~)~(~)
- M~k(~)L(i(Ogk
+ ~i) + r£o)(i(o~ + ~ + ~ ) + rg~)
&k(7)~,.o(fl) )]*} -~ (i(%k + co2) + rkg~ti't0),.k + o~1 + co2) + C;,k and i2 ~,~fl~, t ('01 -t- (2)2) 3 =
d g
k
a(kk, ram)
m
[ X #kg(~)
(i(0)m k __ ~ 1 ) A-
~(~)~m(~) r~O(i(%, - ~, - ~) + r~)
+ (i(0)mk - - 0)2) -b F~k) (i(0)gk - - C01 -- 0 2 ) + /~kg)
+ flgk(a) (i(mmk + 0)1) + r ~ o ( i ( % , + ~ + ~ ) + r~)
(A J)
S.H. Lin et al. / Physica B 222 (1996) 191-208
208
]2mk(~),Ugm(fl)
1 :'k
+ (i((0..k + co2) + r.~k)(i((00k + (01 + (02) + r~,i - #mg(~)
i((0km - - 0)1) + F m k ) ( i ( w g ~
- - (01 - - (02) + F~,g)
]'lkm(~)]Jgk(fl)
]
+ (i((0~m - o)2) + r;;~)(i((00.. - (01 - (02) + r ; , ) - mm(~) (i((0~m + (01) + r ; ~ ) ( i ( ( 0 . . + o)1 + (02) + r;..)
mm(~)/~ok(')
]*}
-~ (.i((0km + 0)2) + Fmk)(i((0gm -/- o)1 + (02) + Fmo)
References
[1] F. Brown, R.E. Parks and A.M. Sleeper, Phys. Rev. Lett. (1965) 1029. [2] S.R. Meech, in: Advances in Multiphoton Processes and Spectroscopy, Vol. 8, eds. S.H. Lin, A.A. Villaeys and Y. Fujimura (World Scientific, Singapore, 1993) pp. 281-341. [3] T.F. Heinz, C.K. Chen, D. Ricard and Y.R. Shen, Phys. Rev. Lett. 48 (1982) 478. [4] D.C. Nguyen, R.E. Muenchausen, R.E. Keller and N.S. Nogar, Opt. Commun. 60 (1986) 111. [5] J.H. Hunt, P. Guyot-Sionnest and Y.R. Shen, Chem. Phys. Lett. 133 (1987) 189. [6] A.L. Harris, C.E.D. Chidsey, N.J. Levins and D.N. Loiacono, Chem. Phys. Lett. 141 (1989) 350. I-7] R. Superfine, J.Y. Huang and Y.R. Shen, Phys. Rev. Lett. 66 (1991) 1066. 1-8] N. Akamatsu, K. Domen, C. Hirose, T. Onishi, H. Shimiza and K. Masutani, Chem. Phys. Lett. 181 (1991) 175. [9] N. Akamatsu, K. Domen and C. Hirose, J. Phys. Chem. 97 (1993) 10070.
(A.2)
[10] J.C. Owrutsky, J.P. Culver, M. Li, Y.R. Kim, M.J. Sarisky, M.S. Yaganeh, A.H. Yodh and R.M. Hochstrasser, J. Chem. Phys. 97 (1992) 4421. [11] J. Miraglitta, R.S. Polizzotti, R. Rabinowitz, S.D. Cameron and R.B. Hall, Chem. Phys. 143 (1990) 123. [12] S.R. Hatch, R.S. Polizzotti, S. Dougal and P. Rabinowitz, Chem. Phys. Lett. 196 (1992) 97. [13] Y.R. Shen, Solid State Commun. 84 (1992) 171. [143 S.H. Lin, M. Hayashi, C.H. Lin, J. Yu, A.A. Villaeys and G.Y.C. Wu, Mol. Phys. 84 (1995) 453; S.H. Lin and A.A. Villaeys, Phys. Rev. A50 (1994) 5134. [15] S.H. Lin, R.G. Alden, R. Islampour, H. Ma and A.A. Villaeys, Density Matrix Method and Femtosecond Processes (World Scientific, Singapore, 1991) ch. 1. [16] S.H. Lin, R.G. Alden, A.A. Villaeys and V. Pflumio, Phys. Rev. A48 (1993) 3137. [17] M.D. Levenson, Introduction to Nonlinear Laser Spectroscopy (Academic Press, New York, 1982) ch. 1. [18] R. Islampour, M. Hayashi and S.H. Lin, Chem. Phys. Lett. 234 (1995) 7. [19] S.H. Lin, J. Chem. Phys. 44 (1966) 3759.
Physica B 222 (1996) 191-208
Molecular theory of second-order sum-frequency generation S.H. Lin a'b'*, M. Hayashi a'b, R. Islampour "'c, J. Yu a, D.Y. Yang ~, George Y.C. Wu d aInstitute of Atomic and Molecular Sciences, Academia Sinica, Taipei, Taiwan ~Department of Chemistry and Center for the Study of Early Events in Photosynthesis, Arizona State University, Tempe, AZ 85287-1604, USA Department of Chemistry, University for Teacher Education, 49 Mofateh Ave., Tehran, lran d Department of Chemistry, University of the Chinese Culture, Taipei, Taiwan Received 23 December 1994; revised 21 November 1995
Abstract
Recently the second-order sum-frequency generation (SFG) has become a powerful surface - spectroscopic technique. In this paper we present a molecular theory for SFG. To systematically treat this process, we have classified SFG into the resonance-resonance case, resonance off-resonance case, off-resonance-resonance case and off-resonance-offresonance case. Among these cases we may have IR-UV SFG and UV-UV SFG. Some of these SFG's are reported in this paper. In each SFG case, we have shown how to calculate the strength (or intensity) and band-shape function.
I. Introduction Second-order nonlinear optical signal generation is dipole-forbidden in media with inversion symmetry. However, such signals have been generated at the interfaces of isotropic media since the earliest days of nonlinear optics [1,2]. An important application of surface nonlinear optical measurements is the determination of adsorbate spectra through the resonant enhancement of the second-order nonlinear susceptibility )(2~. Early measurements exploited electronic resonances to record adsorbate electronic spectra [3,4]. Recently the emphasis has been placed on the application of infrared (IR) + visible or UV sum-frequency generation (SFG) to obtain adsorbate vibrational spectra [5-7]. The vibrational
* Corresponding author.
spectra of a number of adsorbates have now been reported: the methyl modes of alkyl thiol self-assembled monolayers on gold [6]; a number of alcohols on silica [2]; Langmuir - Blodgett films on silica [8,9]; C O on some single crystal metal surfaces [10,11]; methoxy on N i ( l l l ) [11]; acetonitrile on ZrO2 [12] etc. (see, e.g., Ref. [13]). A main purpose of this paper is to give theoretical treatments for various cases of S F G like resonance-resonance, resonance-off-resonance, off-resonance-resonance, and off-resonance-off-resonance cases. We shall also show how to use the B o r n - O p p e n h e i m e r approximation to obtain the expressions for the four cases of S F G for molecular systems. In previous papers [14], the molecular theory of I R - U V S F G has been reported. For the purpose of assisting the future developments in SFG, in this paper we report the molecular theory of other S F G cases.
0921-4526/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 2 6 ( 9 6 ) 0 0 0 0 8 - 7
S.H. Lin et at/Physica B 222 (1996) 191-208
192
2. General consideration
Integration of Eq. (2.4) yields
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(~/~) = Tr (p~3),
(2.1)
where ¢3 denotes the density matrix of the system. The diagonal elements of ~ describe the population of the system, while the off-diagonal elements of describe the phase (or coherence) of the system. Notice that the equation of motion for the density matrix fi(t) of the system is given by the stochastic Liouville equation [15] d~ = _ iLop -/~,~ - iL'(t)~ dt
d~ L'(r)~(r),
where ~i denotes the value of ~(t) at t = ti. Regarding E'(t) as a perturbation and using the perturbation method, we find ~(°)(t) = ~i, ~(1)(t)
= -ill
(2.7a) dr L'(z) ai,
(2.7b)
i
where #(°)(t) and ~(1)(t) represent the zeroth-order and first-order approximations of ~(t), respectively. For the second-order processes, we need the second-order approximation ~(2)(t), dr
dz'L'(z) L'(f)ai.
(2.7c)
The second-order polarization P(g)(t) is given by (2.2)
• ' (2) P(2)ft) = Tr(fi(2)~) = ~--~,~e-'t'uStkOkt (t) + c.c. k
where iL' = iLo + F, and Lo and L'(t) denote the Liouville operators corresponding to the zerothorder Hamiltonian and the Hamiltonian for the interaction 17 between the system and radiation field, respectively./~ represents the damping operator describing the relaxation and dephasing of the system. To evaluate P, it is necessary to solve Eq. (2.2). For this purpose, for the weak field case the perturbation method is often used. This can be done by regarding L'(t) in Eq. (2.2) as a perturbation and solving Eq. (2.2) directly. However, it is often more convenient to eliminate the L~ term in Eq. (2.2) first before solving the equation. For this purpose, we let /~(t) = e-i'L~(t)
(2.6)
Jt~
~(2)(t) = ( - i ) 2
= - iL'op - iL'(t)~ i = - i L ~ p - ~ [,I7",t~],
~(t) = ~, - i
(2.3)
l
(2.8) Using the property of L' [,15] and substituting Eq. (2.7c) into Eq. (2.8) yields P(2)(t) = (--i)2 h ; dr f f d~'EEEe-',°,k,t~,~ i
i
k
k'
I
× { 17(r)kk , I-L'(r') ffi3~,, - [L'(z')#i]kk" ff(r)k't} + C.C.,
(2.9)
where for example rT(Z)kk, = e i~';'r V(r)kk,. Using the relation 1 [L'(r')tTi]k,Z = -~ ¢ (Z')k,l[(tri)tt -- (~i)k'k']
(2.10)
we obtain
to obtain d~ -- = - iL'(t) t~, dt
p(2)(t ) = -(-i)2f) -hT(2.4)
i
k
k'
l
× (rT(~)kk' lT(rt)k '' ['(~i). -- (~i)~'~']
where
L'(t) = ei'L'°L'(t) e-itL'o.
dr f( dz'EEEe-i"°k'p,k i
(2.5)
- 17(~')~, 17(r)k,, [,(a0k,~, - (a3~d} + c.c. (2.11)
S.H. Lin et al. / Physica B 222 (1996) 191 208
i#j
In the dipole approximation, we can write l?(t) as [16]
193
(l~k't"Ei(09~))(l~kk'"Ej(09j))
×2Z'
1 ~
2
i(09~,,,- o9,) + T-~
r/ (t) = --~F.~l ~" Ei(°~i091)e-lt .... /i(t), lj (t) =
e-(',- 0/T,,
(2.12)
lhkAa(kk, k'k')
+
1
1
i(09~,, - 09, - co2)+ ~,~--+ :T--
where/1 represents the dipole operator, a~ takes the values of 1 or - 1 and lj(t) denotes the laser pulseshape function with pulse duration T l. The summation over i in Eq. (2.12) describes the situation that in SFG two lasers are usually involved. Using Eq. (2.12), Eq. (2.11) becomes
i # j E' (lllkk' "El (09i))(/ak't "Ej (09j)) i
j
i(09~k' -
o)i)
1
-[- - -
Ti
( - i) 2
e'2'(tp) -
EYYEYJ,,
k l k' iai jctj
(( x
Aa(ll, k'k')(l~k,,'Ei(e,091))(lal,k,'Ej(~j09j))) -------1--~[.-/---/-, ...... ----1 1
i(09'k't--lXi09i) +~-i)~l(09kl--lXi091--O~j09j)
+(
+ - ~ - i + ~-jj
--- ~Aa(kk'k'k')(lakk" ei(°ci09i))(lak"'Ej(~J09J))-il ~j) i(09~k, - o~,09,)+~ 1(09kt-- O~i(-Di-- ~jO,)j) -~" -~i -~-
x exp [-itp(~i09i
+ ej09j)] +
(2.13)
c.c.,
where, for example, Aa(ll, k'k') (~i)/| (~i)k'k" representing the initial distributions of the system. Eq. (2.13) is general and can be applied to study difference-frequency and sum-frequency generations. For SFG, for convenience, Eq. (2.13) can be written as =
[
- -
+
#lkda(ll, k'k')
L
1
' ~ J Z' ×E
(~'~" ' E , ( - ~ , ) ) (~,~, • & ( - 09j))
i
i(~;,,t + 093 + - -
j
p(2I(tp) = P(2)(09 1 + co2)ex p [-itp(09, + 09z)]
+ c.c. + .-.,
1
Ti
(2.14)
#tkAa(kk, k'k')
+
1
1
i(09~,, + 09~ + 09i) + l,v- + aT--
where
i¢j
P(2)(091 + 092) _
1
i(09~, + c01 + c02) + ~ - + ~-/1
(__ i)2 V ~ ~ {
l~,kAa(ll,k'k') 1
1
~
i(09~,, - 09, -- 092)+ ~ + T--~
J
1 i(o~;,k, + m~) + - -
]*}
Ti
(2.15)
S.H. Lin et al. / Physica B 222 (1996) 191-208
194
In other words, in S F G we are looking for the coefficient of exp[-itv(03 ~ +co;)]. Here in the double summations over i and j, the i = j terms are to be excluded. We shall consider the following energy level diagram for S F G (Fig. 1). In Fig. 1, g,m and k denote the initial, intermediate and final state manifolds, and cox and 032 represent the frequencies of the two lasers used in SFG experiments. According to the definition of the second-order SFG susceptibility ; ~ (COl + 032),
(m,,(/~),~.(y)) +/~k,(a) (i(~0m~+ (D1)+ r;.,) (i (~o~,+ o~ + c~2)+ G,)
+ (i(c0,.~ + co2) + F~,~)(i(cok0 + co~ + ~o2) + F ~
(,~,(/~)p,~(y)) -
P(~2)(e)a + o)2)=
~,,
#ink(a)
(i(coo,.
-- ~0,) + F;.~)(i(~o,~ - o~ - o~2) + r ~ )
Z(~)~(03,+ 032)E,~(c~),)E2~(032), (2.16/ + (i(c%. - co2)+ r~,~) (i (co,. - (~)1 (m.(~),~.(~))
we find
+ #~"(~)
(i(coa., + ¢ol) + r~,g)(i(coa~ + col + ¢o2) + F~,.)
(2.17)
where
("g"(~')u"*(/~))
+ (i(%. + o~) + r~,,)(i(co~. + ~o~ + ~2) +Gm
i2 (2) (01 -F (/)2)1 = ~ ~ E E AtT(Oa'mm) ~a/gy i*
g
" i]* t(2.18)
k m
x ~ak(a) (i(o~ma -- ~01) + F'o)(i(O~k 9 -- OJ1 -- 0)2) + F~ 9)
k
o2
1
(~,~.(/~)m~(y))
.(2) ((2) 1 _[_ 03.02) = Z,,(2) . . + ~ .(2) Za/~v . ~ ~[ .a . + . (~2)1 ~ (03.01 _~. 032) 2
+ ~~,(2) p ~ [, l • + 032)3,
- - (222) ~ - r k m )
o 1 +o 2
m
o!
g Fig. 1. Energy level diagram for SFG.
~,(21t , + 0 9 2 ) 3 c a n be obA.aflyt~l tained from X~)y(col + co2)1 by performing the exchanges k ~ m and k ~ g , respectively. For convenience, their explicit expressions are given in Appendix A. In Eq. (2.18) we have F,~g = F,.g + (l/T,), F,~ = Frog + (l/T2) and F~ = Fkg + (l/T1) + (l/T2). As discussed in a previous paper [16], there are four cases of SFG. They are the resonance-resonance, resonance-off-resonance, off-resonance-resonance and off-resonance-off-resonance cases. For example, the off-resonance-resonance case means that the h031 photon is not in resonance with the g -~ m transition, while the h03 2 photon is in resonance with the k state. To show the application, we consider the resonance-off-resonance case. The energy level diagram of this SFG is shown in Fig. 2. This figure shows that h03, is in resonance with the g ~ m transition. From Eq. (2.16), we can see that due to the resonance of the g ~ m transition with co,, we only need to retain the terms like i(03mg -- CO,) + F,~g or i(03gm + coo + F,~g in X~)~(03, + co2). From ,(2) (031 -[- 092)2
and
S.H. Lin et al. /Physica B 222 (1996) 191-208
195
F o r the off-resonance-resonance case, we have ,,(2) l
k
Z,p~(2)(091 + 092) = z,p~'(2)(091 + 092)1 + ~ W ) l
+ 092)3, (2.21)
m2
where
Ol + m 2
i2 ,[2) i" , Aefl7 tt~'l "}'}'092'1 : h-2 i~7' ~= '"~
a
Ad'((Jg'mm)[l"ok(°O
(09k. - 091 - 092) + F~.
m [g,,O(fl)Pk"(Y) ml
/~,,0(7)gk"(fl)~
(2.22)
and .(2) I A~y ~,t-Ul +
i2 = # Zg Z,.2
a a(kk, mm)#gk(oO . . . . . .
~ i(09k9 -
091 -
g Fig. 2. Energy level diagram for the resonance-off-resonance case of SFG.
Eq. (2.18) we find ~(2) ,~,(Z) It., "at- 092)1 a/~y (091 "}- (D2) = Lafl~,~,t~l
i
= -~ ~, Y', ~, Ao(go, mm) g k ,. ]Aok(OO]Amg(fl)lAk,.(])) X
(i(09mo __ 091) -[- /'~nO)(09kg - - 091 - - 0)2)
~Akm(O~)~A,.g(fl)]Agk(~))
jJ
( - i ( % m + 090 + r;,g)(09~" +091 + 092 (2.19) which is obtained by collecting the terms that contain the resonance terms 09,.o ~ 091. Similarly for the resonance-resonance case, we collect the terms that contain the resonance terms 09,.0 - 091 and 09ko -- CO1 + 092 to obtain
+ 09 ): 25,(091 + 09 )1
/$,(09l
i2
= ~ Z £ Z Aa(gg, ram) g k ,. × (i(co,.
o -
o91) + r ; . . ) ( i ( c o k .
-
o~1 -
~2)
+ rg)
(2.20)
X [ "~km(J)~-~"g('-'~) "}" "k"(]l)"mo(fl)] L ('Ok,. -- 091 09kin -- 092 J
.
092) +
Fko (2.23)
In most cases, the Boltzmann factors Aa(kk, mm) ,(2) tQD 1 + 092)3 is are negligible. That is, in this case z~a~ negligible. It should be noted that for the off-resonanceoff-resonance case, all the contributions in Z(2) I • ~(2) ~r., ,(2) t • ~ t ~ , l + 602)1' ],.~fly kU'~l + 092)2 and Aafly ~UJl + c02)3 should be considered. F r o m the above theoretical derivation of the mathematical expressions for SFG, we can see that numerous types of S F G experiments can be designed. This is because not only we have four S F G cases discussed above but also we can use IR or UV for 091 and 092 in SFG.
3. Molecular theory of SFG In this section, we shall treat various cases of S F G for molecular systems by introducing the B o r n - O p p e n h e i m e r ( B - O ) approximation. By using the B - O approximation, the S F G expressions obtained will be expressed in terms of molecular properties and the relation between a given S F G and other optical process can be found. The results presented in this section will be useful to experimentalists for designing S F G experiments and analyzing experimental results.
S.H. Lin et aL / Physica B 222 (1996) 191-208
196
3.1. Resonance-off-resonance case The general expression for this SFG case is given by Eq. (2.19) which can be rewritten as Z(2) ~ ,
ku
tO2
i g
k
tO 1 +60 2
m
X i ( 0 ) m g ~ - ~ l ) + Frog \COkg - - 0)1 -- 0)2
gv'
q 0 ) ~ T ~ - 1 1 - + ~ 2 } ' (3.1) For a randomly oriented system (i.e., isotropic system), it is necessary to carry out the spatial average Of /,~#~ ,~(2) (0)1 + 092) over the entire space; in this case Eq. (3.1) will vanish because of the appearance of odd-power #~g(fl)/~gk(~)#km(Y) in /(,afl),~t'~"l ~2) t , + (DE). This property makes S F G a surface specific experimental method. It can also easily be shown that for a centro-symmetric system, r.h.s, of Eq. (3.1) vanishes. For example, if the g state has a gerade symmetry, then because of #mg(fl) and #k,,(~') (or #kin(a)) the m state must have an ungerade symmetry and k should possess a gerade symmetry. However, in this case #gk(Ct) and #0r(Y) will vanish because both g and k have gerade symmetry. From Eq. (3.1) we can see that Zt2) for the resonance-off-resonance case consists of the resonance part and off-resonance part. The above expression can be applied to I R - U V S F G and U V - U V SFG. Here we have used the notation that for I R - U V SFG, IR is for 0)a and UV for ~2. To distinguish these two SFG's, it is necessary to introduce the B - O approximation. For the I R - U V SFG, we have 9 ~
9v,
m ~ gv',
k ~ ku.
The vibronic energy level diagram for the I R - U V SFG is shown in Fig. 3. Here g and k denote the electronic states, while v, v' and u represent the vibrational states. Due to the fact that in I R - U V SFG, the first transition associated with 0)1 is in the IR range, it must be due to the vibrational transition and is described by gv ~ gv'. Since the laser 0)2 is in the UV-visible range, excited electronic states need to be considered; we use ku to
031
gv Fig. 3. Vibronic energy level diagram for I R - U V SFG. tot is in resonance while ~2 is off-resonance.
denote the excited vibronic manifold. In terms of the new notations, Eq. (3.1) becomes i v
x
u"
ku
i(0)gv,,ov - 0)0 + rgv,,g.
×(
(7_)
\~ku.ov
--
091
--
092
]
0)ku,gv"+ 0)1 + 692/
(3.2) In the B - O approximation, the molecular wave function, for example, %0 can be written as a product of electronic wave function @g and vibrational wave function Ogv, i.e., qggv = ~g(qQ)Ogv(Q),
(3.3)
where q and Q denote the sets of electronic and vibrational coordinates, respectively. In this case, we have tZav.ku(O0 =
(3.4)
where lZgk(Ct) = <~g(qQ) l#(ct)l~k(qO)>
(3.5)
197
S.H. Lin et al. 1 Physica B 222 (1996) 191-208
which represents the electronic transition moment for the g+-+k transition. In Eq. (3.4), to evaluate the matrix element of dipole moment P~“,~~(cI),it is necessary to carry out the integrations over both electronic and vibrational coordinates. For this purpose we first perform the integration over electronic coordinates to obtain the electronic transition moment pLgk(tl) which in general depends on vibrational coordinates. In Eq. (3.2), the off-resonance part contains the terms like wk,,+ - o1 - o2 and c&g”’ + o1 + o2 in which the vibrational energies are smaller than electronic energies, and we can use the Placzek approximation, i.e., ok”,go
=
Co ku, gu’
mkg,
= wkg.
(3.6)
The quantities pgg(fi) and txgg(ay) can be expanded in the power series of QI, vibrational coordinate,
/.dB)=&(P)+C
(3.10) 1
and
“gg(“Y)= $g(aY) + c
Qr + ... .
Substituting yields
x
Eqs. (3.10) and (3.11) into Eq. (3.8)
Ct8i)gv.gv -
(Bi)gv’,gv’l
l(@g,IQ~l@,,~)12 X. 1(og”‘,g” - 4 + r;“‘,g”’
In this case, Eq. (3.2) becomes
(3.11)
I
(3.12) ’
where X.
(@,“~IPgg(P)I@g”) l(~g”‘.go - 4 + r;“‘,g”
x
(@gd&k@)~@ku) 0) kg -
[ +
Eqs. (3.12) and (3.13) indicate that for a particular vibrational mode QI to be observed in SFG,
<@ku~~kg(Y)~@gv’) w1
-
(32
1.
(@kubkg(CoI@gu’)
(@gvI~gkb)I@ku)
mkg + 01
+ m2
%7,(P) #
(3.7)
-
(6i)gv’,gv’l
(@g”(agg(cry)(@g”,)) X. (@~~~~~%?(~)~@PJ) mgo’,g” - 4 + r;“‘.g” (3.8) where ~~~(rxy)denotes the electronic polarizability of the ground electronic state, “ggbY)
= 2 k
pgk@)pkg(Y) mkg -
wl
-
+ w2
aQl
(3.14)
0
0
and
Applying the closure relation C ul@k,) (@,,I = 1 to Eq. (3.7) yields
Xzy(Wl+ 02) = $1 C C(ai)gv,gv ” c’
( ) a~,,by)
( 1#O. aQ*
(3.15)
0
Eq. (3.14) corresponds to the IR selection rule, while Eq. (3.15) corresponds to the Raman selection rule. In other words, for the IR-UV SFG, the SFG active mode should be both IR active and Raman active. Some preliminary results of the IR-UV SFG have been reported in previous papers [14]. From Eq. (3.12) we can see that one can introduce the band-shape function for a particular mode in the IR-UV SFG
~gk(Y)~kg(CI) Okg +
wl
+
02
)
Fi(wlw2)
=
(3.9) which in general depends on vibrational coordinates. pJ/?) represents the dipole moment of the ground electronic state.
CC
” 0’
C(6i)gv,gv -
l<@,~1Q~I@,,~)I*
X i bgd,
=
Flt”l~2)r
(Bi)gd.gdl
+
gu -
Fl(“lo2)i
01)
+
3
r;“‘,g”’
(3.16)
198
S.H. Lin et al. / Physica B 222 (1996) 191-208
where
Using the Plazcek approximation to simplify the off-resonance part, Eq. (3.19) becomes
F / ( ° ) l ° ) 2 ) r ~- Z E
[((~i).v, gv -- (~i)gv',gv']
z5' (091 + 092)= i
, Fo,,,,ovK o,,~IQ, IOo,,,)I 2 x (09o,,',g,, - 091) 2 + (F'ov',o,,) 2
(3.17) ×
( Og~,l ~xg,,(091 + 092),,~10,,w) (Om~[/~mg(fl)lOg~) 09,,w, ov - col - iF~, o
and
(3.20) FI (091,092)i -~
E E [(ai)gv, ov - (ai)av,,ov,]
x(091 - o~ov. o~)l(Oo,,IQ, lOo.,)l 2 ' ( 0 9 o . ' . ~ - ~°i) 2 + ( G . ' . , o ) 2
where (3.18)
That is, the band-shape function Fz (09~092) can be separated into the real part Fl(0)1092) r and imaginary part F1(091092)i. The real part Fz (091092)r and the imaginary part F1(091092)i of the SFG band-shape function Fz (091c02) are related to each other by the so-called Kronig-Kramers transforms (for the detailed discussion of the Kronig-Kramers transforms, see Ref. [17]). The real part Ft(091~O2)r is exactly the same as the band-shape function of IR spectra. This means that the SFG band-shape function can be determined if the IR band-shape function is known. We shall consider the UV-UV SFG of the resonance-off-resonance case next. Its vibronic energy level diagram is shown in Fig. 4. Due to the fact that the UV or visible lasers are involved in both 09~ and 092, electronic excited states are to be considered in the intermediate and final manifolds. The location of arrows in the figure can distinguish between the resonance and off-resonance situations. Notice that in this case, Eq. (2.19) takes the following form: ,~a#v
09, -
+
(2) 1 - -
092)2
09kin -F 091 -F 092
(3.21) Again both a~.,(091 + (-D2)a~, and/z,,g(fl) depend on vibrational coordinates Ql and can be expanded as the power series in Qt. If we keep only the dominant terms, in this case it corresponds to the terms without Qt; this is usually called the Condon approximation, In this case, Eq. (3.20)
(k} -= (ku}
(°2
o)l +o)2
{m} =- {row}
_ 091) + Fmo
EZZ¥ ¥ 2j (3.19)
Here (ai). . . . .
--
(a,).o,.vumw.ov(3) , . . . .
.
~,(z) (09, + c02) = ~ ~ i ( 0 9 m . , g
LO)kg
has been neglected.
{g} -= {gv} Fig. 4. Vibronic energy level diagram for U V - U V SFG of the resonance-off-resonance case.
S.H. Lin et al. I Physica B 222 (1996) 191 208
199
becomes
{k} -- {ku}
= K' E Z m,)....~ v w
I(O~olO,.~512 •
!
,
(3.22)
032
O)mw,g v -- O) 1 -- 1Frog
031 +0~ 2
where 1
{m} ~- {mw}
(3.23)
and I(OgvIO,,~)] 2 denotes the Franck-Condon factor• Eq. (3.23) indicates that for the case in which K' is relatively insensitive to the variation of co~, the measurement of SFG will provide the information of the absorption spectral band-shape for the electronic transition g ~ m. In other words, one can obtain the one-photon absorption spectra from the SFG of adsorbed molecules and vice versa. From the above discussion, we can see that for the IR-UV SFG, the intensity (or strength) is determined by the product of (~l~gg(fl)l~Ql) and (~c%(c~7)/~Qz), while for the UV-UV SFG, the intensity (or strength) is determined by the product of #,,0(fl) and eg,,(e)l + co2). This information is important when one applies molecular orbital calculations to designing molecules with particular properties of X~)~(~ol + (D2).
031
~---~v} Fig. 5. Vibronic energylevel diagram for UV-UV SFG of the off-resonance-off-resonancecase.
where, for example,
1~[ 7- ,7,. 7,,•"(')"''(') ""(')''m(') I (3.25) and
3.2. Off-resonance-off-resonance case Next we shall consider the application of the off-resonance-off-resonance case to the UV-UV SFG. The corresponding vibronic energy level diagram is shown in Fig. 5. In the Plazcek approximation, we obtain
Z (2) t~ "
=-h~v~m(~i)gv, gv~ov
(.Omg--(d)l
+ v.=(/s)~,=(=7) + ~=.(7)=.=(~/7) O)mg -~- 0.) 1
+
~,.,--77~,--7
O,)mg -- 0,) 2
i
'
(3.24)
o7g=(~7) lk~ f #k,(')#,.k(7) h COka + ~ol + 092
+ v_~.(j)c<___.,<(~_).]• O)kra --
(01 -- 0)2~
(3.26) This type of SFG is relatively insensitive to temperature. This can be seen by expanding #mg(fl), ~9,,(~7), ~g,,(~7) etc. in terms of normal coordinates Q~. We will see that the dominant term of Xt2) is independent of temperatures. From Eq. (3.24) we can see that the molecular orbital calculation of the intensity (or strength) of this particular SFG would be extremely difficult.
200
S.H. Lin et al, / P h y s i c a B 2 2 2 (1996) 1 9 1 - 2 0 8
3.3. Off-resonance-resonance case
where
Next we consider the application to the off-resonance-resonance case. We present its vibronic energy level diagram in Fig. 6. Here we are considering the U V - U V SFG. In this case, we retain the terms like i(09kg -- 091 -- 092) + F~g in Eq. (2.17) and from Eq. (2.21) we obtain
1 m~ ~'Ukm ~ m(7)/~"W(fl) g~-~l
IAkm(fl)~Amg(')~ 09rag - - 0 ) 2
_J
(3.29) In the Condon approximation we obtain 1
~(2) ((/)1 + 0)2) ~go, ~. (~)
1 =
.
u
v mw
X
,
09ku, gv - - 0)1 - - 0 ) 2 - - I F k g
X [ ["lraw" gv (J~) ~'lk..~ ~' raw ( ~)) _} ~'lmw, gv ( ~)) /Llku" mw ( fl ) ] . h
09,.,.,.~., -
09~
09,,..,.~o -
092
(3.27) Here due to the Boltzmann distributions, other terms in Eqs. (2.17) and (2.18) can be neglected. Using the Plazcek approximation, Eq. (3.27) becomes
Z~p~(091 + 092)
V V L,~/.,,~
(~)g~'g° I(Og~ I@k")- ~12 . . . . . . --I
.
Eq. (3.30) indicates that when ~kg((l)1092)f17 is relatively insensitive to the variation of 09~ and 092, the SFG can provide the band-shape function of the absorption spectra for the electronic transition g~k. Notice that Eq. (3.30) can be written as a product of the electronic part and vibrational part: Z(2) (091 + ('02)
1 = -h ] l g k ( O O O t k g ( 0 9 1 0 9 2 ) f r F g k ( 0 9 1
1 Z Z ((~,)go,go h u v 09ku,gv
- - 0)1 - - 092 - -
iFko
x (Ok~l~kg(091,092)pr IO~>,
(3.28)
(3.30)
"
u v 09ku, gv -- 091 -- 092 -- IFkg
+
(3.31)
0)2) ,
where the vibrational part Fok(09a + 092) determines the band-shape function of SFG,
Fgk(091 + 092) {k} ~
(~i)gv.gol(Ogol Oku)[ 2
{ku}
O-)ku, gv - - 091 - - 092 - -
~2
(3.32)
IFkg
and the electronic part #gk(Ct)~kg(091092)pr determines the strength of this particular type of SFG. The band-shape function Fgk(091 + 092) can again be separated into the real part Fgk(091 + 092)r and imaginary part Fgk(09~ + 092)i,
CoI +O~2
{m} - {row}
coI
Fgk(091 + 092) -~ F g k ( 0 9 1 "q-
092)r + iFgk(09~ + 092)i,
(3.33)
where
Fgk(091 + 092)r ~,..
Fig. 6. Vibronic energy level diagram for UV-UV SFG of the off-resonance-resonance case.
u
v
,
(09ku.go -
o91 - - 0 9 2 ) [ ( O g o [ O k , ) [
~09ku, gv - - t'~l - -
2)
T
2 kg
(3.34)
S.H. Lin et al. / Physica B 222 (1996) 191-208
201
tion. In the Condon approximation, Eq. (3.36) becomes
and Fgk(0)l + 0)2)i :
(3.35)
z
0)-; -
1 ~2 #gk(OO]Xkm(~})~mg(~) E E 2 Po v v u w
That is, Fgk(0)~ + 0)2)i denotes the band-shape function of the one-photon UV absorption spectra and Fgk(0)l + 0)2)~ is again related to Fgk(0)~ + 0)2)i by the Kronig-Kramers transforms.
(3.37) which can be rewritten as
3.4. Resonance-Resonance case Finally we consider the application of the resonance-resonance case of SFG corresponding to U V - U V transitions (see Fig. 7). Notice that in this case Eq. (2.20) becomes ~(2) [ •
= -
dt e-
itAokg
×
w
Po~/.to~,ku(e)#k . . . . (Y)#,.w,..(fl) [i(o)k.,o~ - c o l - o32) + r~g] [i(o).,w,o~ -- eh) + r~.g]
(3.36) because other terms are unimportant. Here (8~)o~,o~ = Pgv representing the Boltzmann distribu-
{k}
{k,,}
(3,38)
where iA0)ko = i(0)kg -- 0)1
--
(/)2)
-F /~ko'
iA0),, o = i(0), o - o)1) + F~.o
(3.39)
and 1
0)2
fo~ dz e -~a~-.
v u w
i2 O u
X
~Or
Z(2) (0)1 + 0 ) 2 ) K " f o
K" = ~7 #ok(~)#km(Y)#,,o(fl).
(3.40)
In this SFG case, the intensity (or strength) is determined by #0k(a)#k,.(?)/tmg(fl) which can easily be obtained by molecular orbital calculations. Using the closure relations ~ , [ O k . ) ( O k , I = 1 and Y~wlO,,w) (Omwl = 1, Eq. (3.38) becomes
0)1 +0)2
{m} ~ {row} ~t~vt (2) /0) 1 + 6o2) = - K , , f o -~A
d t e -itA°~. fo~ d.c e-i~aw.g i~
i
x ~Pov (Oovle " ke-~a~e-~(t+~)/LIOov). (3.41) 0) 1
Fig. 7. Vibronic energy level diagram for UV-UV SFG of the resonance-resonance case.
12
This indicates that the calculation of double resonance U V - U V S F G ,%~ ~(2) t~,l ~ , + co2) involves the calculation of two-time correlation function. Next we shall consider an important case; that is, the potential energy surfaces of the k and m electronic states
S.H. Lin et al. / Physica B 222 (1996) 191-208
202
are identical. In this case, Eq. (3.37) reduces to t'.., ,#,~,1 Z(2)
+ co2) = - - K " 2 Z
Substituting Eq. (3.47) into Eq. (3.36) yields ~(2) /" ~
Z, a t f l T t t . ~ l
v u
+
Po~l(Oa~l Ok~>l2
[i(coku,g~- oga -- 092)+ r~g] [i(~o.,,~,g~- oh) + F~,g]
xexp
dte-itdo~k.
0)2) = -- K"
-S
{
(3.42)
h0) h0) coth~---.-.~-csch-ZK1 2kT
which can be rewritten as ~(2) ((.O 1
dze -i~'°-.
(3.48)
0)2)
where S = ½fld2, usually called the Huang-Rhys factor or the coupling constant. Eq. (3.48) can be integrated by changing the variables x = t + z and y = t - z; the result is given by lfOo=lOk.)l z e - g"(~ k.- E, . ) e - g'(~ - . - ~,o ,
xZZpo, u
Z(2) ((.01 + (02)
(3.43)
=_K,,e-S(2n+z)~ ~
where
m=O n=O
id0)ko = i(0)ko -- 0)1
--
0)2) 71- / ' k o '
iA0),.0 = i(0),,0 - 0)1) + Frog.
1 - -
(3.44)
[ S ( ~ + 1)]" (S~) m
n!m!
[iA0)r,g + i0)(n - m)] [iA0)kg + i0)(n -- m)]"
For the single-mode case, we obtain
(3.49) At T --- 0, Eq. (3.49) reduces to dte - it~'',
= --K"
dz e-i~a°', G(t,z),
(2)
Z~py(0)l + 0)2) (3.45)
(1/n!)S"
where
= - K " e - S .=o [iA0)~g + in0)] [iA0)kg + in0)] "
G(t, z)
(3.50)
= 2 sinh " - - ~ ~ e-a(v+~)e-U(u+~)l(Og ~l Ok,)12 2kT.
~
(3.46) It follows that [18] G(t, z) = exp
Ifld2{h0)
-- --~ coth 2 ~
h0) - csch 2 k T
xcosh(2-~TT-i0)(t + z ) ) } l , (3.47) where Q' = Q + d and fl = 0)/h. Here d denotes the normal coordinate displacement between the two electronic states.
The types of expressions for SFG given by Eqs. (3.49) and (3.50) are useful because they are expressed in terms of molecular properties like electronic energy gap, coupling constants (or Huang-Rhys factors) etc. They show how one can make good use of the SFG band-shapes to determine molecular properties. The same set of molecular properties are also involved in one-photon absorption coefficients. In other words, through these molecular properties one can relate absorption spectroscopy and SFG together. In concluding this section, we can see that due to the use of the B - O approximation, the molecular expressions for SFG can usually be separated into
203
S.H. Lin et al. 1 Physica B 222 (1996) 191-208
the electronic part which determines the intensity or strength of SFG and the vibrational part which determines the band-shape function of SFG. The strength of SFG is determined by electronic properties like dipole moment, transition moment, polarizability etc, while the band-shape function of SFG is determined by the potential surfaces of different electronic states involved in SFG; harmonic potential surfaces are usually used.
(@,“I@,,) (@kul@g”*) xc.l(Wku,gv Wl
u
-
02)
+
rku,gv
(4.4)
Here we have used the Condon approximation 4. Discussion
for
C(kg and
Various IR-UV and UV-UV SFG’s for molecular systems have been treated in the previous section. It should be noted that we have not exhausted all the possibilities of SFG processes. For example, another important SFG will be IR-UV SFG of the resonance-resonance case; its corresponding vibronic energy level diagram is shown in Fig. 8. Here o1 denotes the IR laser frequency while w2 represents the UV laser frequency. In this case, we have
Ql+...
(4.5)
From Eq. (4.4) we can see that this SFG case is very similar to resonance Raman (RR) scattering. The square of the part involving the summation over u in Eq. (4.4) corresponds to the band-shape function of the RR excitation profile [lS].
(4.1)
and using the B-O approximation m --* gv’,
g-+st’,
k + ku
(4.2)
we obtain
(@,,I&k(~)i@ku)
(~k”(~kg(Y)I”g”‘)(og”‘I~gg(P)~og~)
’ [i(wgv’,gv - 01) + rgo’,gvl
or approximately
i2 = h’
pgk@)pkg(?d
To demonstrate the application of the theoretical results presented in this paper we shall show the numerical calculation of the off-resonance-resonante UV-UV SFG. Separating x$!,(oi + w2) into
bbku,go
-
ml
-
O2)
+
rku,gvl
(4.3)
the real part x$,(oi + mZ)r and the imaginary part x$\(o~ + wz)i of Eq. (3.30) we find
1
c ”
c’
da(gv,
g”;
gv’,
d)
S.H. Lin et al. / Physica B 222 (1996) 191-208
204
Notice that the SFG signal is proportional
(ku)
to
Ix$zs),~~ + w2)12, i.e.,
ti
lx:;; h
+ w2)l 2
= IX$J~I
+ ~2)r12 + Ix$‘,(+
+ ~2),12.
(4.10)
For the purpose of numerical calculations, we shall rewrite Eq. (4.6) as
s cc
=
K”’
dt
0
~~(~i)gv.gvl(~gvl~k,)12 ”
”
x exp[I-t(ibku,gv
-01-02)+r.&}].
(4.11)
For the case in which 0,” and @kuare products of harmonic oscillator wave functions, i.e., Fig. 8. Vibronic energy level diagram for IR-UV SFG of the resonance-resonance case.
where
(4.12)
and
X$$lwl + u2)i
=
K”’ 11 t8i)gv,gu ”
@ku
x(~ku,gv
- 01 - O&)2+ (r&)2 ’
(4.7)
Xb24,t”1 + W2)r = K”‘CC(Bi)gv,go ” 11 Oku,g”
(ok",
gv -
-
u2)l<@,"I@ku)12
co1 -
u2)2 + (l-Q2
01
=
(4.13)
nxk&?f)
I
II
G,I(@,“I@kU>12
X(
@go = n Xgui(Qi)
if the oscillators of Qf and Qi are only displaced, we have [19]
- t{i(u& - ml - 02) + rig} (4.8)
and (4.14) (4.9) Notice that &k(a) (see Eq. (3.5)) denotes the a-component of the electronic transition moment, and can be calculated once the electronic wave functions @kand Qs are known. @kand @gare usually obtained by carrying out the molecular orbital calculations. As can be seen from Eqs. (4.7) and (4.8), x$,(wl + ~0~)~and x$L(ol + 02)i are related to each other by the Kronig-Kramers transforms, and x$\(ol + Oz)i is related to one-photon UV absorption spectra for the electronic transition g -+ k.
where Si = (Wi/2ti) dQf, the Huang-Rhys factor. We shall now apply the above theoretical results to the electronic transition So -+ S2 of rhodamine 6G adsorbed on fused silica. No experimental SFG spectra for this system have been reported. However, the SHG (second-harmonic generation) spectra of this system for the off-resonance-resonance case have been reported by Heinz et al. [3], and analyzed by Lin et al. [16]. For the purpose of illustration, the SFG spectra of this system will be calculated in this paper. Notice that for this system,
S.H. Lin et al. / Physica B 222 (1996) 191-208
we have 1-16]
1.0
~Ok9= 28 500 cm- 1, ~oi = 1130 cm -1,
205
Si = 0.23,
Fk9 = 650cm -1.
(4.15)
The calculated absorption spectra are shown in Fig. 9. The parameters given in Eq. (4.15) have been chosen to reproduce the observed absorption spectra. Using these parameters, we can calculate the band-shape function for the SFG of rhodamine 6G adsorbed on fused silica (see Figs. 10-12). From these figures, we can see that the band-shape of SFG spectra is broader than that of absorption spectra; it is mainly due to the existence of the real part of Fko(O) 1 "-b fOE). The real part and imaginary part of Fko(eh + O)z) are shown in Figs. 10 and 11, respectively. The SFG spectra are shown in Fig. 12. For comparison, the experimental SHG spectra obtained by Heinz et al. [3] are shown in Fig. 13. Next we shall examine the effect of inhomogeneities on SFG. For this purpose, we shall assume that
d
v
'~
0.5
== N
Z
0.0 300
320 SPG
wavelength
340 /
360 nm
Fig. 10. Imaginary part of Fkg(~ol + ~o2).
1.0
i
I
I
i
__
0.5
v
v
o
o O.fi
0.0
N
tq
7
o Z
--0.5 0.0
i
300
i
320 wavelength
/
i
I
340
360
nm
w
300
320 SFG
Fig. 9. Calculated absorption spectra of rhodamine 6G adsorbed on fused silica.
wavelength
340 /
360 am
Fig. 11. Real part of Fk~(~ol + co2).
S,H. Lin et al. / Physica B 222 (1996) 191-208
206 I
I
I
I
1.0
this effect appears in the electronic energy gap Ogkg, i.e.,
~kg = tfka + NAco
(4.16)
and PN = m e'
r. t/)
exp --
.
(4.17)
In other words, we assume the distribution of inhomogeneities to be Gaussian. The Gaussian distribution for inhomogeneities is well known. It is used here to demonstrate its effect on the S F G bandshape. In this case, Eq. (4.14) becomes
0.5
0~ e~
(Fko(COl + ~O2)) = f : d t e x p [ - ~ D 2 t 2 A c o 2 -- it(ff~kg -- col -- O~2) -- tF~,9 -- ~i Si{coth ~ T 0.0
e 300
t 320 SFG
t 340
wavelength
/
360
I0 I10
•
6 G
Rhodomine
cosh ~ 2 ~
{4.18)
nm
Fig. 12. Calculated SFG spectra of rhodamine 6G adsorbed on fused silica.
0 Rhodomine
-- csch
O3
As can be seen from Eq. (4.18), the inhomogeneity effect is to smooth and broaden both absorption and S F G spectra. F r o m Eq. (4.18) we can see that the existence of Gaussian inhomogeneities introduces an additional broadening described by exp(-¼D2t2Aco 2) in Eq. (4.18). In concluding the paper, we have presented the theoretical treatment of various S F G cases for molecular systems. It is hoped that this will stimulate experimentalists in the surface science area for designing new experiments. Although the I R - U V S F G has become a very powerful tool as a surface vibrational spectroscopy, in this paper we have given a detailed investigation of off-resonance-resonance U V - U V S F G for molecular systems and performed numerical calculations using rhodamine 6G adsorbed on fused silica as an example. It is hoped that the organic dyes can be used to probe solid or liquid surfaces by using the U V - U V S F G technique.
Acknowledgements 0
I 300
~
I 320
I
SH Wavelength
I 34.0
I 360
(nm)
Fig. 13. Experimental SHG spectra of rhodamine 6G and rhodamine 110 adsorbed on fused silica.
We wish to thank the National Science Council of R O C and US N S F for the financial support of this work, and the referee for useful comments and suggestions.
S.H. Lin et al. / Physica B 222 (1996) 191-208
207
Appendix A The expressions for /I,~,(2) I • 1 "t- 0)2) 2 afly~ua X~)~(0)~ + co2)3 are given in the following:
((2) ( ,
,~,t~,~ + 0)2)2
and
i2 ~ZZZAa(99,kk)
=
0 k m
[
(~*~(/~)~*(T)) x /~m(Z) (i(0)~o -- cO~) + F~)(i(0)mo -- ~ -- 0)2) + F~n~) (~(~)~(/~)) ] + (i(~k~ -- co2) + r;~)(i(~.~ - ~ , - ~:) + r;.~j (~(/~)~(v)) + pmg(~) (i(0)ko + 0)1) -F F~o)(i(0)mo + 0)~ + co2) + r~.~)
+ (i(0)~o + ~2) +
F~'~)(i(0)mo + (01 -F (/)2) Jv
fling )
[ ~*(/~)~(~) -/Xk~(G) L(i(~k -- 0)~) + r ~ . ) ( i ( ~ , - 0)~ - ~:) + rg,)
(i(%~ - ~:) + r L~)((Ti ()o" ~~ ( ~ )- ~ [
- ~:) + r ; ~ i]
"~(/~)~(~)
- M~k(~)L(i(Ogk
+ ~i) + r£o)(i(o~ + ~ + ~ ) + rg~)
&k(7)~,.o(fl) )]*} -~ (i(%k + co2) + rkg~ti't0),.k + o~1 + co2) + C;,k and i2 ~,~fl~, t ('01 -t- (2)2) 3 =
d g
k
a(kk, ram)
m
[ X #kg(~)
(i(0)m k __ ~ 1 ) A-
~(~)~m(~) r~O(i(%, - ~, - ~) + r~)
+ (i(0)mk - - 0)2) -b F~k) (i(0)gk - - C01 -- 0 2 ) + /~kg)
+ flgk(a) (i(mmk + 0)1) + r ~ o ( i ( % , + ~ + ~ ) + r~)
(A J)
S.H. Lin et al. / Physica B 222 (1996) 191-208
208
]2mk(~),Ugm(fl)
1 :'k
+ (i((0..k + co2) + r.~k)(i((00k + (01 + (02) + r~,i - #mg(~)
i((0km - - 0)1) + F m k ) ( i ( w g ~
- - (01 - - (02) + F~,g)
]'lkm(~)]Jgk(fl)
]
+ (i((0~m - o)2) + r;;~)(i((00.. - (01 - (02) + r ; , ) - mm(~) (i((0~m + (01) + r ; ~ ) ( i ( ( 0 . . + o)1 + (02) + r;..)
mm(~)/~ok(')
]*}
-~ (.i((0km + 0)2) + Fmk)(i((0gm -/- o)1 + (02) + Fmo)
References
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