THEORY OF DOUBLY RESONANT INFRARED-VISIBLE SUM-FREQUENCY AND DIFFERENCE-FREQUENCY-GENERATION FROM ADSORBED MOLECULES

Theory

of

doubly

resonant infrared-visible

sum-frequency

and difFerence-frequency

generation

from

adsorbed molecules

Jung

Y.

Huang

Institute

_of

Electro Op-tical Engineering, Chiao Tung University, Hsinchu, Taiwan, Republic ofChina

Y.

R.

Shen

Department

of

Physics, University

of

California atBerkeley, Berkeley, California 94720 and Materials Sciences Division, Lawrence Berkeley Laboratory, Berkeley, California 94720

(Received 20 December 1993)

We theoretically analyze doubly resonant infrared-visible sum-frequency (DR IVSFG)and diference-frequency generation (DR IVDFG) from a monolayer ofadsorbates at an interface. Our calculated re-sults with a model molecule indicate that the resonant amplitude ofnonlinear optical susceptibility of DR IVDFG and DRIVSFGprocesses conveys information about the electron-vibration coupling in ad-sorbates. Moreover, owing to a dephasing-rephasing procedure involved, DR IVDFG can also be developed into asensitive probe forinvestigating the coherent phase relaxation process in adsorbed mol-ecules without the complication ofinhomogeneous broadening.

PACSnumber(s): 42.50.Md, 68.35.Ja

I.

INTRODUCTION

Molecules at an interface often behave differently than they do in bulk. This may be ascribed to the fact that

molecules at surfaces are better oriented

[1]

and their

electronic structures are modified by the interaction with substrate

[2].

Therefore probing the orientation and

elec-tronic structures

of

molecules at an interface becomes an essential step for better understanding

of

the surface dy-namics and chemistry

of

adsorbates.

Recently, second-order nonlinear optical effects have been demonstrated

to

be an effective and versatile probe

of

the _{static [3,4]} and dynamic [5,6] properties

of

mole-cules and surfaces. During this investigation, various mechanisms

of

resonance enhancement were used to im-prove the spectroscopic capability

of

these techniques. In fact, fairly good agreement between the theoretical

calcu-lation and experimental result

of

resonant second-harmonic generation (SHG) from molecular systems has been achieved

[7].

Unfortunately, the line shape

of

reso-nant SHG is often too broad to be sensitive enough for

identifying molecular species. This, however, has been shown to be remedied by a newly developed technique called infrared-visible sum-frequency generation (IVSFG)

[3].

IVSFG

isalso a second-order nonlinear optical process in which two input laser beams, one at the infrared fre-quency ~& and the other at the visible frequency co2, in-teract and generate an output at the sum-frequency

~,

=co,

+co2 in the visible spectrum

of

light. The

IVSFG

signal from an interface can be resonantly enhanced

if

co&

approaches a surface vibrational resonance. The

enhanced

IVSFG

signai thus carries the vibrational

spec-troscopic information about the interface.

It

is noted

that the signal strength

of IVSFG

can be further

in-creased by an electronic resonance when either the input visible frequency (co@)

or

output sum frequency isnear an

electronic transition. Because the electronic and vibra-tional transitions are excited simultaneously in this dou-bly resonant process,

IVSFG

near double resonance is ideal for investigating electron-vibration coupling in

ad-sorbates. Such acoupling is known

to

play an important role in the coherent evolution

of

a vibrational wave pack-etin anelectronic state.

The ability toprobe the coherent evolution

of

asurface

excitation is crucial for understanding the dynamics and surface reaction

of

adsorbed molecules

[8].

In the past,

this was conducted by probing the phase relaxation pro-cesses through the measurement

of

the excitational linewidth

[9].

Theoretical efFort to understand such

sur-face relaxation isthen focused on relaxation models that can produce the measured spectral line shape

[10].

How-ever, the width

of

aspectral line is often dominated by in-homogeneous broadening.

To

find the homogeneous linewidth,

i.

e, the inverse

of

the dephasing time, it is necessary to resort

to

spectroscopic techniques capable

of

suppressing inhomogeneous broadening

[11]

or

directly measure the phase relaxation rate in time domain with coherent optical processes such as photon echoes

[12].

A one-to-one correspondence between each wave-mixing process capable

of

producing spectra with re-duced inhomogeneous broadening and photon-echo

pro-cesses was established

[13].

In a recent study, Shen

fur-ther pointed out that doubly resonant difference-frequency generation

(DR

DFG)

in the time domain with properly time-ordered input pulses can yield an output in the form

of

arephased echo

[14].

This immediately sug-gests that doubly resonant infrared-visible difference-frequency generation

(DR IVDFG)

be used to probe the coherent phase relaxation

of

surface vibrations without resorting tohigher-order nonlinear optical processes

[25].

In this paper, we consider this aspect

of

DR

IVDFG

and

its use as the probe

of

the coupling between electronic state and vibrational modes

of

adsorbed molecuies at

sur-faces.

II.

THEORY

A. Doubly resonant infrared-visible sum-frequency generation

The general theory

of

sum-frequency generation (SFG)

in reQection from the surface

of

a nonlinear medium has been described in detail elsewhere

[15].

As shown in

Fig.

1,we explore surface

SFG

in refiection

_(E,

) from an in-terface between linear media 1 and 2 with a dielectric

constant

of

c.&and c2,respectively.

We consider the case in which the visible frequency

(co@)in the

SFG

process is close to a vibronic transition from the ground electronic state manifold So

[gb]

to the first excited state

Si

[evj and cubi in resonance with cob,

[see

Fig.

2(a)]. By applying the diagrammatic technique

of

Yeeand Gustafson

[13,

16] to

Fig.

2(b}, the doubly res-onant nonlinear polarizability canbe expressed as

(gb~rk ~ga )

(ga

~r, ~eu )(ever~ ~gb

)

(roi rogbg

+iyb,

) (&0 _rue„g

+iy

_g)

Here

oi„,

=(E„Eg,

)

—

If&isthe frequency

of

the transition ~ev

)~~ga

).

A common Lorentzian linewidth is taken for

each Franck-Condon (FC}transition (i.e.,

y,

_„g,

=y,

_g). The above equation isconveniently rewritten by introducing the integral representation

of

energy denominators: (a

+ib)

'=(

i)

f

—

o"

e"'+'

'dt _{for b}

)

₀

_[17].

_{After applying} the

clo-sure property

of

the vibrational manifolds,

Eq.

(1}becomes

(t)

(

.

)

I

~ '~st Yegt+~~&~ Yba&

(( '~Hg q s ge( ₎ ~~He q s eg( _}

—_i&'Hg(q)ls

p(

}

It'H _(q)lfi)~d

$2 p

Here,

H

(q) and

H,

(q) are the vibrational Hamiltonians

of

molecules in the electronic ground and excited states.

erg'(q) d—enotes the ith component

of

the electronic transition moment from the ground to excited state, and

erP(q)

de—notes the kth component

of

the electric di-pole moment

of

the ground state. The double bracket

in-dicates the thermal average

of

the operators involved.

The 3N

—

6 normal modes

of

a polyatomic molecule

can be separated into two groups: The first group is comprised

of

the totally symmetric normal modes that

are strongly Franck-Condon active in an optical

transi-tion; and the second is formed by the partially symmetric normal modes that are either

FC

inactive or weakly

ac-tive in the optical transition.

For

atotally symmetric vi-bration, the excited-state potentials can be displaced in the equilibrium configuration relative

to

the ground-state

surface. The displaced excited-state surface exerts a net

force on the initially prepared wave packet, causing its mean position

to

move, hence the dynamic process

of

the wave packet is an important factor in the resonant

excita-tion

of

a totally symmetric vibration.

viS qo

(a)

lev~ Iev& Igb& ga& &gal Ks Igb& &gal to 2 Medium 1 Vi

P,

Interfacial Layer jE Medium 2 IR Iga&

(b)

&gal

FICx. 1. _geometry of sum-frequency

(I(,

) and

difference-frequency (Kd) generation from an interface in the reflected direction. The z direction is taken as zero at the interface be-tween the media and ispositive going into medium 2.

FIG. 2. (a) Schematic for doubly resonant infrared-visible sum-frequency generation process between the potential energy surfaces ofthe electronic ground state [So]and the first elec-tronic excited state

[S,

],

and (b) the corresponding double Feynman diagram.

H,

(q)

=Hs(q)+i

g

LI(a/

a/)+—

fico,

f

(3)

Here,

_Lf

denotes the linear electron-phonon coupling coefficient, %co, is the electronic transition energy ap-propriate to the ground electronic state equilibrium configuration

(q=O),

and a& and

aI

are phonon

annihila-tion and creation operators. We also have Hs(q)

gf

Acof(a&taf

+

g).

Equation (3)can be conveniently converted to the fol-lowing form

[19]:

The random perturbation

of

the partially symmetric normal modes on the periodic motion

of

an optically

ac-tive vibration is embedded in the radiative damping

con-stant

of

the

FC

active vibration

[18].

Similarly to the

case observed with resonance Raman scattering (RRS)

[17,19],

the electron-vibration coupling in each optically active mode will also contribute

to

DR

IVSFG.

In the approximation

of

the Born-Oppenheimer separation

of

electronic and nuclear motions, a single-photon allowed vibronic transition is found

to

be described very well by a linearly coupled vibronic system. The linearly displaced vibrational Hamiltonian

of

the excited state in this model has the form

[19]

tion frequency. The electronic transition moment func-tion e—_{r's(q) for a totally symmetric normal mode can} be replaced by

a

constant which isthe value evaluated at

the equilibrium configuration (q=O)

of

the ground

elec-tronic state,

i.e.

,the Condon approximation

er—

'

(q)

= —

er's—

(q

=0)

.

(sa)

rss

=rss(0)+i

_g

(a

—

a

gq

f

f

This is a good approximation for strongly

FC

active modes that are known

to

be significantly affected by the dynamics

of

wave packets. Vibronic coupling between two excited electronic states can lead

to

linear terms in the expansion

of

er—

's(q) H.owever, this does not be-come a restriction in our molecular model

of

DR

IVSFG

because linear non-Condon terms can be included without much difficulty in calculating Eq. (2)

[19].

The

ground-state dipole moment

of

a molecule resonantly

ex-cited by an infrared photon, however, is modulated by

the vibrations

of

the molecule: drss

rs

_(q)=

r(0}+

—

_g

_qI

Bgf

H,

(q)=e

H

(q)e

_+%cia,

(4)

where

P

=i

_gI

_g/(a/+a~),

with

g&=L//(hcoI).

The

quantity ttPs

=to,

—

_g&gIto& is the zero-phonon

transi-I

=r

_{(s0s)[l+D ]}

.

Substituting Eqs. (4)and (5) into

Eq.

(2),we obtain

(5b)

3

a,

'

k( to„'F02,co—

i)=

_r,

_{"s(0)rjs'(0)rp(0)}

f

_e

'

"

'

f

_{(t, t')dt}

dt',

0

=((e

'"e

_[1+D

₍

—

_t')]&)

_.

To

evaluate

f

(t,

t'),

the function

f

(A;t,

t')

~

~a,

(—~')

'.

'

=((e

'"e

_e s _{)) is defined and then reduced to a}

product

of

c

numbers through the use

of

boson algebra identities

[19,20].

If

we expand exp[ADs(

—

t')

]to aterm linear in A, we can then set

A=1

to recover the thermal average

f

(t,

t').

Thus

_a,

' k in finite temperatures can be

found tohave the expression

E(N CO )f

ct(co)=i

f

(ala(t)&e'

"

""

"dt,

0

(ala

(t) )

=exp

_g

_g

[(nI+1)(e

—

1)

f

+nI(e

I

1)]—

(8) 3

S/;

„=

(Ir, '(0)r'

(0)b,

rP(0)

(7)

««

l~

(t)

&

=exp[iH,

(q)t]la

)

describes the evolution

of

the vibrational wave packet in the excited electronic state, initially prepared by an optical transition from the ground state

lga).

The single-photon absorption spec-trum

of

a molecule, o(co)/co, is equal to the imaginary

part

of 4(to)

by

[17]

0'(Co) ~ I I i(t0 t0 )t—

r

—~t(

(t2ltt

(t) )e

's 'g _dt

X

[(nI+1}@(co,

—

co&)

—

4(tt),

)],

=1m[A(a)

—

aP,

_)]

.

₍₉₎

where n&

=1/[exp(ficoI/kit

T)

1],

and

h—

rP(0}

—

=

_[drp(q}/dq]

0 is the ground-state dipole moment

derivative. N(to) is the Fourier transform

of

the overlap function

of

the time-evolved wave packet

[21]

Note that all the information about molecular electronic

structure has been absorbed into the time evolution behavior

of

the wave packet. This formalism

of DR

IVSFG

allows us to skip the difficulty

of

the

sum-over-states approach and focus on key physical parameters. Equations (7)

—

(9) provide aconnection between the ab-sorption spectrum and the

SFG

excitation profiles

(Sf

vs co2)

of

normal modes

of

adsorbates.

[(nf+1)4(co,

—

_cof)

—

_4(co,

_)],

_{which appears in the resonant amplitude}

_of

DR IVSFG,

is similar to that found in the

time-correlator formulation

of

resonance Raman scattering

[18].

In Eq.(7),_{

_gf,

cof,

yf,

y,

,and co, I are used to

de-scribe

DR IVSFG.

These parameters are also indispens-able for elucidating the dynamical properties

of

mole-cules. We will show that these parameters can be deter-mined from the fit

of

the experimental

DR

IVSFG

excita-tion profiles to

Eq.

(7)in the discussion section.

B.

Doubly resonant infrared-visible difference-frequency generation

In addition to the

SFG

beam, a difference-frequency signal (Kd) also emerges from the interface, as shown in

Fig.

1.

The doubly resonant difference-frequency genera-tion isespecially interesting since it can suppress inhomo-geneous broadening, as pointed out by Shen

[14].

To

fur-ther reveal its potential applications, we analyze the case in which the visible frequency in the

DR

IVDFG

process is close to a transition from the electronic ground-state manifold So

{ga]

tothe first excited state

S

1{ev] [seeFig. 3(a)]. The resonant nonlinear polarizability,

ad,

'_k,

corre-sponding to

Fig.

3(b) can be written as

(gbIr, ~ev

)

_(evIrjIga )(gc2Irk gb )

ad,

jk(

~'d ~2

~1)

₂

_gP

(~1 b 1Yb ) I.

(2

~1)

b+ir„]

(10)

Following the same procedure as detailed in Sec.

II

A,we

can readily obtain

(2) f,ij k

d,;jk(

—

d'

»

i)

=

X

(coi coj

1)'f

3

Df,

_,

k

=

gf

rs'(0)r,

'

(0)b,

rp(0)

assuming a single dominant inhomogeneous broadening channel. By substituting

Eq.

(8) into Eq.

(11),

we can rewrite the nonlinear susceptibility

of DR IVDFG

as

U

X

[4(cod+cof

)

—

4(cod

)](nf

+

I) .

Note that at zero temperature the intensity profile

of

first-order Raman scattering from the

fth

vibrational mode, If(co2), is proportional to I&(co2)

—

4(co2 cof)I',

which is essentially the same as the absolute square

of

Df

'jk

of

Eq. (1 1). Thus the excitation profiles

of DR

IVDFG

carry dynamical information about wave packets

which is similar to what is widely recognized in reso-nance Raman scattering.

C. Inhomogeneous broadening

The resonant frequencies

of

a molecular system often depend on its local environment.

To

account for this possible inhomogeneous broadening, the nonlinear sus-ceptibility should be convolved with a Gaussian distribu-tion funcdistribu-tion

of

the resonant frequencies, giving

viS qo lev& Igb& gQ& cgbl 1ev&

yd(2)(cod=co2

—

co,)

=X

—

—

e ~ Io'

_[ad

(2)(cod,

rj)]drI,

STY/0 (12) Vis 0 Iga& &gbl t0 2

where _g represents a set

of

local parameters. The reso-nant frequencies

of

adsorbates in different local

environ-ments can be expressed as [22] Iga& &gal IR

=Q,

+b,

co,

(g)

=0,

+greco,

g,

cof cof

+

Ibcof(71 ) cof

+

cjkcof

(13) FIG.3. Schematic for (a)the energy level and (b)the

corre-sponding double Feynman diagram of doubly resonant

3

Xd,~'Jjk'( ~d ~1 ~2)' ' _g22

Xkf

'

j

1

~

P

x

_g

n,=O nN=0

+oo ], (q/go) dn

& 7T'gp (N) cof ')}Ecof l_y

j

)

N

[co2 o))

+

(cof

+

rjkcof)

(r—

jato,

(t

+

Q,

g}

—

g

n&( o)&

+

rjht0& )

+

i

y,

(t]

Q=1

N

[o)~

—

o),

—

(rjtI),o),

+Q,

e)

—

g

ng(oPg+rjb,

tog)+iy,

]

Q=1

(14)

Byusing the properties

of

the plasma dispersion function, Oudar and Shen [22]have shown that in adouble-resonance

case,

if

the imaginary parts

of

the energy denominators in nonlinear optical susceptibility

(y)

have opposite signs, then

y

scanned over the resonance yields aspectral line with aLorentzian linewidth. Indeed, near the double resonance (i.

e.

,

co)

=o)f

and

o)d=

—

o)f+Qs+

_g&

n&oP& or o))

=o)f

and o)d

=Qe+

_g&

n&co&) Eq. (14) exhibits a Lorentzian line shape with ahalf-width

of

_yf

_+y,

_g. On the contrary, this isnot the case for

DR

IVSFG

where the imaginary parts

of

the denominators ing( '_{have the same sign. Thus y(} '_{exhibits no singularity near the double resonance. Infact, at the}

double resonance, g( '(

—

_o)„'o)),

_o)z)exhibits awidth more than twice the width

of

inhomogeneous broadening.

D. Transient effect

The reduced spectral linewidth via

DR IVDFG

can be better appreciated in the time domain. The coherence be-tween ~ev

)

and (,gb~ [see

Fig.

3(b)],which is set up by

E(co,

) and E(co&), can directly emit the output radiation at

cod=(vz

—

o),

if

transition between ~ev

}

and (gb~ is single-photon allowed. The polarization set up by pulsed excitations

has difFerent phases fordifferent molecules in difFerent local environments. The resulting dephased polarization can be expressed as

&d"(tod=to2

to„t)—

=N

f

n(rj)Tr[

«pd"—

(t, rj)]&rj

=(

e)N

f

n(g)

—

g g

&gb[r[ev

_)pd,

'„(,

(t,

7(I)drj+c.

c.

b u

(15)

where n

(rj}

is the distribution function

of

inhomogeneous broadening and Nthe surface density. Byfollowing the di-agrammatic rules

of

the time-dependent density matrix, we canwrite

(2) e (()g2

—

)g() r(g)—t'[n,g+(tt—1)tdf}(

_r,

_gg+r,g(2—rf((2 &,

)—

pd,gtt(tb ttrj

—

2e

x

zptm(evlrlga

)(g

lrlg _&af ' tte' '

"

'+

"'

(t )'dr"tt

tft'

_e

t'

'

"d(ltt

_{tdt, e} a

(16)

assuming that each level in the vibrational manifolds

of

the excited and ground electronic states isequally affected by the inhomogeneous broadening Here

.

Hd(t,rj)

=

Ij(gto,g

(rj)t

—

[j(go),g

(rj)t

&+bee&,

(rj}(t

2

t,

)]] denote—s the phase difference

of

the induced nonlinear polarization at the local environment specified by g;and

A;(t;)

is the envelope

of

the exciting pulsed field which has a carrier frequency at o);and peaks at t; As noted

.

from

Eq.

(16),all the molecules

at the surface will generate the maximum coherent output whenever the phase difference vanishes

[13,

14],

i.

e., Od(t,

g}=0,

(17) This indicates that with properly time-ordered input pulses (t2&

_t)

},the transient

DR

IVDFG

will give rise

to a

pho-ton echo

of

frequency o)d at

t

=t,

which is later than both t2 and

t,

[14]

[see

Fig.

4(a)].

The amplitude

of

the

DR

inho-mogeneous broadening. Thus in frequency domain

DR

IVDFG

exhibits a reduced linewidth which contains the

intrin-sicdynamical information about molecules.

Byapplying similar analysis to

DR IVSFG,

we find

e ~(kz+k() r(.&) (—(Q, +U~f)t r—

, (+r,

i,

r—

f(t,

t'(—) iI

[0

—co +(v—1)co jIt

&&

gp',

,

(ever)gb)(gb~r~ga)

I

e

"

'

'

'A2(t)

_)dt's

J

e '

'gi(ti)dtie

a

(18}

where

_8,

(t,

rt)

=

Ibrp,g(rt)t [pro—

,

g(rt)tz ~rambo(rt)( z

t,

)]

j—

.

The time at which 6),

=0

is always less than t2 [see Fig. 4(b}]. Therefore

DR

IVSFG

does not involve a

dephasing-rephasing process

of

photon echoes and its spectral line will be affected by inhomogeneous broaden-ing.

III.

NUMERICAL RESUI.TSAND DISCUSSION

The chemisorption

of

molecules onasubstrate not only

can perturb the electronic states

of

the substrate but also change the potential energy surfaces and vibrational structures

of

adsorbates

[2].

These changes, once they have been measured, could yield valuable information about the static and dynamical behaviors

of

adsorbates. In this section, we will show that such measurements can

be fulfilled effectively by the use

of DR IVSFG

and

DR

IVDF G

processes.

It

isnoted that Eqs.(7) and (9)allow the connection be-tween the absorption spectrum and

DR

IVSFG

ampli-tudes

of

normal modes

of

adsorbates. This relation is similar to the transform theory

of

resonance Raman scattering

[17],

which seeks to determine the Raman exci-tation profile (REP)from the absorption cross section. In an indirect approach

of RRS

model potentials are used to

calculate the time cross correlation function

((b~a(t))

)

which can then be transformed to the frequency domain.

The parameters

of

the potential energy surfaces can be adjusted in order to get a good fit

of

the experimental profile. In this respect, adirect inversion

of

resonant

Ra-Vis rephase

DF dephase = = =, Echo

I

man excitation profiles to yield time domain information is more attractive. Recently, a direct inversion scheme has been proven tobe feasible

[23].

In the case

of

an interfacial system,

DR

IVSFG

and

DR IVDFG

can be used to determine the parameters

of

the vibrational structures and potential energy surfaces

of

adsorbed molecules, which are indicated by

_[gf

cof y

f,

y,

, happ, and rlpkcp

j.

From the experimental point

of

view, this approach can be done by first scanning the in-frared beam (c0)) through each vibrational mode with a

fixed visible frequency. Equation (7}isthen used to fitthe observed resonant nonlinear optical susceptibility. The

fit determines cof,

_yf,

and

_Sf.

In the second step, we let cu&=cof and measure

_Sf

as a function

of

cu2. This

mea-surement yields a set

of

excitation profiles

of

DR

IVSFG.

In an indirect approach, we can fit these

DR IVSFG

profiles to Eq. (7) with the help

of

model potential sur-faces to determine the rest parameters

_(gf

_{y g} 7/pktp _g,

and to, ). We will elucidate this procedure in detail by using a model molecule with linearly displaced harmonic potential surfaces along three

FC

active normal mode

coordinates. The vibrational frequencies and disp1ace-ment parameters

of

these vibrational modes aredescribed in Table

I.

Figure 5shows the calculated absorption spectra

of

the model molecule, for which the short-dashed, solid, and dotted curves are the calculated results obtained with the displacement parameters

_[gf

_j taken from sets 1,2, and 3,

respectively. In the calculations, we model the homo-geneous broadening

of

an electronic transition by a

damping constant,

_y,

,and the inhomogeneous broaden-ing by the parameter

of

rtb,to, [see Eq.

(13)].

In Fig. 5, these two parameters are assumed to be (a)

_y,g=300

cm

',

lb,

r,

~=00 cm

',

(b)

_y,g=600

cm

',

tlhcp,

g=0

cm

_{i;and{c) y,g=300cm}

i,

_{n~~,g=300cm-i.}

Asex-pected, the higher vibronic features in the absorption

spectra become more distinctive when the linear electron-phonon coupling is increased

($3=0.

7:

short-Vis _TABLE

_I.

_{The frequencies,}

cd, and displacement parame-ters, _{gf, of} three Franck-Condon active vibrational normal modes ofamodel molecule.

I

t'

2

FIG.

4. Pulse sequences for (a) transient infrared-visible difference-frequency generation, and (b) sum-frequency genera-tion processes. Mode no. _cd (cm

')

900 1200 1500 Set 1 0.3 0.6 0.7

f

Set 2 0.3 0.6 1.0 Set 3 0.3 0.6 1.5

~rt 0 0 t gf 4 0 20 . 10-0 -4000 20-10. 0 -4000 2000 2000 (ug-(ueg (Cm 1₎ 8000 20 10-0 8000-4000

(c)

2000 Mg—Greg (Cm-1 ) 8000

FIG.

5. Calculated absorption spectra for linearly displaced harmonic potential energy surfaces along Franck-Condon active normal mode coordinates. The vibrational frequencies and displacement parameters of this model molecule are described in Table

I.

The calcu-lated spectrum with the displacement parame-ters, _{g~),taken from set 1 in Table

I

is indi-cated by the short-dashed curve; solid curve is from set 2, and dotted curve for set 3. In addi-tion, the electronic damping constant

_y,

_g and inhomogeneous broadening parameter gpss,co,g

are assumed to be (a)

_y,

_g

=300

cm

gphco,_g

=0

cm

';

(b)

_y,

_g

=600

cm 'gpkco _g 0 cm ', and (c) _Z~

=

300 cm gphco,_g

=

300cm (b)mode 2 4 ~IS/ C4 0 -4000 2000 (a)mode 1 8000

FIG.

6. The resonant amplitude,

Sf(

—

co&

—

cof,co&, cof ),of DR IVSFG for each vibrational normal mode is plotted as a func-tion ofthe visible frequency (co&

—

co~). The

absorption spectra ofthe molecule are shown in Fig.5(a). 4-0 -4000 2000 GfS

-

(d_eg (cm

-

1₎ 0 8000-4000 2000 1dg 1Ieg (Cm 1) ~ ~ 8000 (b)mode 2 0 -4000 2000 (a)mode 1 8000

FIG.

7. The resonant amplitude

Sf(

—

co&

—

&of,'co&,

~f)

of DR IVSFG for each vibrational normal mode is plotted as a func-tion ofthe visible frequency (co&

—

co,g). The absorption spectra ofthe molecule are shown inFig.5(b). Q '0 74 4 CO 2-0 -4000 2000 G)P-fdeg ₍Cm 1₎ 0 8000-4000 2000 (dP-(de (crn- ) 8000

(b)mode 2 Q '0 A. E U 0 -4000 6 3 A. E U 0 -4000 I 2000 (a)mode 1 2000 (cm-i ) 8000 0 8000-4000 I 2000 cu2-~,g (Cm-I ) 8000

FIG.

8. The resonant amplitude

Sf

ofDR

IVSFG for each vibrational normal mode is plotted as a function ofthe visible frequency (co&

—

co,gj. The absorption spectra ofthe

mol-ecule are shown in Fig. 5(c).

dashed line;

(3=

1.

0:

solid line; and

(3=1.

5:

dotted). By

comparing

Fig.

5(b} with 5(c),we note that the separation

of

the line broadening into the homogeneous and inho-mogeneous parts leads to different absorption strength.

But the difference in the absorption line shape between two cases cannot be detected clearly.

The displaced difference operation on

4

gives rise to

the sensitive dependence

of

_Sf

on

_gf.

This can be clearly seen in

Fig.

6, where the resonant amplitude

Sf

'jk( cop cof cog cof)

of

DR

IVSFG

foreach

vibration-al normal mode is plotted as a function

of

the visible fre-quency (co~

—

co,

).

The corresponding molecular absorp-tion spectra are depicted in

Fig.

5(a).

It

is interesting to

note that the position

of

the second peak (

—

1825 cm

')

in

_Fig.

6 coincides with that

of

the first vibronic peak

of

the absorption spectra [see

Fig. 5(a}j.

Furthermore, the frequency difference

of

the first and second spectral peaks in Fig. 6is identical to the corresponding normal mode frequency. In addition, _g3 not only changes the line shape

(S3) of

mode 3 but also affects the profiles

of

the

other two normal modes. Particularly noticeable, the second peak height

of

_Sf

is found to decrease as g3

in-creases. At sufftciently large g3, this peak disappears completely. The molecule with larger broadening exhib-its a similar trend, which is depicted in Figs. 7 and

8.

From the comparison between Figs. 7 and 8,we can find that

_Sf

can resolve the effect

of

the homogeneous and in-homogeneous broadening with higher accuracy than that

with absorption spectrum. Moreover, with the immunity

of

DR

IVDFG to

inhomogeneous broadening, we can in principle measure the homogeneous broadening directly from the spectral profiles

of

DR

IVDFG.

Even by using

DR IVSFG

only, agood fit

of

the excitation profiles pro-vides a very strict test for the parameters used. The

simulations show that

_Sf

asa function

of

co2indeed

accu-rately reflects the electron-vibration coupling in an

adsor-bate.

Recently, the application

of

the resonant third-harmonic generation (THG) technique to investigate the

vibronic structures

of

all-trans P-carotene in solution has been reported by van Beck, Kajzar, and Albrecht [24]. Their results indicate that a suitable fit

of

the nonlinear third-harmonic susceptibility dispersion could be

accom-plished using fewer

FC

active normal modes, but all

FC

active normal modes in an electronic transition are re-quired to correctly fit the observed resonance Raman

ex-citation profiles. The higher sensitivity

of

REP

to the vi-brational structures

of

molecules can be attributed tothe selective excitation

of

molecular vibrations, which ap-pears in

RRS

but not in

THG.

The resonant vibrational transitions induced by the infrared photons in

DR

IVSFG

and

DR IVDFG

processes warranty the sensitivi-ty to the vibronic structures

of

molecules. In addition, both

DR IVSFG

and

DR IVDFG

belong to a

lower-order wave-mixing process with double-resonance enhancement, therefore they are capable

of

generating stronger signal than that via

RRS

and other higher-order wave-mixing processes [25,

26j.

Furthermore,

DR

IVDFG

and

DR

IVSFG

are surface specific, thus are ideally suited for surface studies.

In summary, an analytic expression

of

doubly resonant infrared-visible difference-frequency and sum-frequency susceptibilities in terms

of

the overlap function

of

the wave packet in an excited electronic state has been de-rived. Our results show that these second-order non-linear optical effects can be developed into an effective probe forthe electron-vibration coupling in molecules ad-sorbed at surfaces. In time domain,

DR IVDFG

isfound

to go through a dephasing-rephasing process

of

photon

echoes, therefore it can become asensitive technique for

the investigation

of

coherent dynamical processes appear-ing at surfaces.

ACKNOWLEDGMENT

The author

(J.

Y.

H.

)acknowledges the financial support

from the National Science Council

of

ROC under Grant

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