Theory of Time-Resolved Sum-Frequency Generation and Its Applications to Vibrational
Dynamics of Water
Michitoshi Hayashi,*,†,‡Ying-Jen Shiu,§ Kuo Kan Liang,|Sheng Hsien Lin,⊥and Yuan Ron Shen#
Center for Condensed Matter Sciences, National Taiwan UniVersity, Taipei 106, Taiwan, Institute for Molecular Sciences, Okazaki, 444-8585, Japan, Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 106, Taiwan, Research Center for Applied Sciences, Academia Sinica, Taipei 115, Taiwan, Department of Applied Chemistry, National Chaio-Tung UniVersity, Hsin-Chu 300, Taiwan, and Department of Physics, UniVersity of California, Berkeley, California 94720
ReceiVed: January 31, 2007; In Final Form: June 27, 2007
A molecular theory of time-resolved sum-frequency generation (SFG) has been developed. The theoretical framework is constructed using the coupled-oscillator model in the adiabatic approximation. This theory can treat not only the vibrational spectroscopy but also vibrational dynamics. An application of this theory is also provided for estimation of the time constants of the intermolecular vibrational energy transfer between water molecules. This approach can be used for molecular analysis of the experimental results of Shen at al. on the SFG studies of vibrational dynamics of water.
1. Introduction
Molecular theory for various kinds of sum-frequency genera-tion (SFG) has been developed. However, few theoretical treatments for time-resolved SFG have been reported. Recently, Shen et al.1 have reported the femtosecond time-resolved vibrational IR-UV SFG for studying the vibrational dynamics of water surface molecules. In this paper, we shall report a molecular theory of femtosecond time-resolved IR-UV SFG. Although numerous MD calculations have been performed to study the vibrational spectroscopy and dynamics of water,2-6 it still seems desirable to develop a theoretical model that can be used to analyze the experimental data of spectroscopy and dynamics of water. For this purpose, in this paper we divide the whole water system into the vibron (intramolecular vibra-tions) system and the phonon (librations and intermolecular vibrations) bath. The interactions among different vibrational modes are described by anharmonic couplings and the adiabatic approximation is introduced as a basis set. In vibrational spectro-scopy, we show how to calculate the spectral shift, intensity of phonon-side bands, temperature effect, and vibration band-shape functions. In vibrational dynamics, we show how to calculate the vibrational excitation energy transfer between different water molecules and the vibrational relaxation due to one-vibron processes, two-vibron processes, and three-vibronic processes. The present paper is organized as follows. In section 2 we briefly present the molecular theory of time-resolved SFG, which will be followed by the description of a coupled oscillator model applied to vibrational spectroscopy and vibrational dynamics (section 3). The applications of the theory to estimate the time constants for the intermolecular energy transfer are given in section 4.
2. General Theory
Let us consider a model for the pump-probe vibrational IR-UV sum-frequency generation experiments in which an IR-pump laser beam is applied to a sample and, with time delay∆t, other IR and UV laser beams are sent to the sample to generate IR-UV SFG. In this case, we start with7-10
where PBpumpSFG(∆t) represents the nonlinear polarization vector due to the probing lasers in the presence of the pump laser,
〈‚‚‚〉denotes the average over the heat-bath modes and orienta-tion configuraorienta-tions of the molecules, and eˆSFG represents the polarization unit vector for the vibrational IR-UV SFG signal detection. On the basis of density matrix method,11the nonlinear polarizability for pump-probe vibrational IR-UV can be expressed as
where PBpumpSFG(l,∆t) stands for the nonlinear polarization vector on the molecule at the position RBl and∆kB ) kBs - kB1- kB2. Substituting eq 2-2 into eq 2-1 and ignoring the incoherent scattering, that is, l ) l′, yield
where
* Corresponding author. E-mail: [email protected]. †National Taiwan University.
‡Institute for Molecular Sciences.
§Institute of Atomic and Molecular Sciences, Academia Sinica.
|Research Center for Applied Sciences, Academia Sinica.
⊥National Chaio-Tung University. #University of California. I(∆t)∼〈|eˆSFG‚PBpump SFG (∆t)|2〉 (2-1) P B pump SFG (∆t) )
∑
l P B pump SFG (l,∆t)ei∆ kB‚RBl (2-2) I(∆t)∝∑
l*l′ 〈{eˆSFG‚PBpump SFG (l′,∆t)}*{eˆSFG‚PBpump SFG (l,∆t)}ei∆ kB‚RBll′〉 (2-3) P B pump SFG (l,∆t) ) Tr[(eˆSFG‚µb)σˆ (4) (t)] )∑
nn′ (eˆSFG‚µbnn′)σˆn′n (4) (t) ) (-i)4∫
ti t dτ1∫
t i τ1 dτ2∫
t i τ2 dτ3∫
t i τ3 dτ4 ×∑
nn′ (eˆSFG‚µbnn′)〈〈n′n|L˜′(τ1) L˜′(τ2) L˜′(τ4) L˜′(τ3)|σˆ(ti)〉〉 (2-4) 10.1021/jp0708300 CCC: $37.00 © 2007 American Chemical Societywhereb denotes the electric dipole moment operator, σˆµ (4)(t) describes the fourth-order reduced density operator with respect to the interaction between radiation field and the molecules, andσˆ(ti) is the initial condition for the reduced density operator. Here L˜′(τ1) is the so-called Liouvillian operator in the interaction representation and〈〈n′n| ‚‚‚ |σˆ(ti)〉〉 denotes the double-index Liouville space matrix element.
At the phase matching direction, only the molecular pair in the bracket in eq 2-3 at a large distance can contribute to SFG signals. In this case, we find
To grasp the essence, we assume that the IR pump-pulse and SFG pulses can be approximated by rectangular pulses with the respective pulse durations Tpumpand TSFG. In the rectangular pulse approximation, the integrals can be divided into two time ranges in eq 2-4 if no overlap between the SFG pulses and pumping pulse is assumed and they are separated by∆t. In this case, we find (see Appendices S1 and S2 in Supporting Information)
where ∆ ) Tpump + ∆t and σˆ(ti) )∑
g P(g)|gg〉〉, with P(g) being the Boltzmann distribution function. In eq 2-6 the time integrals overτ1andτ2lead to SFG responses and those over τ3andτ4give the density matrix for the pumping prepared states. We assume that the IR pumping process with a 130 fs laser pulse cannot generate coherences between vibrationally excited states of any high-frequency vibrational mode at 2800-3800 cm-1. In other words, the pumping process generates populations in the vibrationally excited state g′or in the vibrationally ground state g. In this case, we find
where, for example,
and
Here Fgg(∆t) and Fg′g′(∆t) denote the time development of the population in the vibrational ground state g and that in the vibrational ground state g′, respectively, and Fgg(Tpump) is associated with the pumping condition (see Appendix S3 in the Supporting Information).σegrggSFG (TSFG) in eq 2-8, for example, can be approximately given by (see Appendices S4, S5, and
S6 in the Supporting Information)
The terms S12and S22in eq 2-7 are given in Appendix S5 in the Supporting Information.
In this section, we have derived the time-resolved SFG for studying the population dynamics of surface species. In this paper, we are mainly concerned with the time-resolved IR-UV SFG applied to studying the vibrational energy relaxation (VER) of surface water. In this case, Fgg(or Fg′g′) in eqs 2-8 and 2-9 describes the time-dependent population of vibrational levels. In the next section, we propose a model for vibrational energy transfer, VER, and vibrational spectroscopy for liquid water. Using the Born-Oppenheimer approximation, the vibrational IR-UV SFG has been well treated (see refs 7-10).
3. Coupled Oscillator Model
Liquid water can usually be regarded as consisting of ice-like structures (or clusters) and liquid-ice-like structure (or clusters) which have, in addition to the bending band, 1640 cm-1 and theνO-Hvibrational band of 3200 cm-1, respectively. For this liquid-vapor interface, there exists the additional dangling-bond structures (or clusters) that have a vibrational band around 3700
cm-1.1,12,13To treat the vibrational spectroscopy and dynamics
of water, we shall employ the coupled oscillator model; this is reasonable because in the above-mentioned structures (or clusters) the hydrogen-bonding network is formed between water molecules, and the motion between cluster is slow compared with the vibrational dynamics under consideration.
We consider a total system consisting of vibrons (intramo-lecular vibrational modes) and phonons (external low-frequency modes including librations). The Hamiltonian of the total system can be expressed as
where Hˆsand Hˆbdenote the Hamiltonian for the vibron system and the heat bath, respectively
and Hˆ′is the interaction between the system and the heat bath. For Hˆ′we shall use the anharmonic
I(∆t)∼
∑
l*l′ l arg e〈{eˆSFG‚PBpumpSFG(l′,∆t)}*〉〈{eˆSFG‚PBpumpSFG(l,∆t)}〉ei∆ kB‚RBll′ (2-5)
〈{eˆSFG‚PBpumpSFG(l,∆t)}〉)
∑
nn′ (eˆSFG‚µbnn′)×∑
mm′ (-i)2∫
0 TSFG dτ1∫
0 τ1 dτ2〈〈n′n|L˜′(τ1+∆) L˜′(τ2+∆)|mm′〉〉× (-i)2∫
0 Tpump dτ3∫
0 τ3 dτ4〈〈mm′|L˜′(τ3) L˜′(τ4)|σˆ(ti)〉〉 (2-6) 〈{eˆSFG‚PBpumpSFG(l,∆t)}〉) S11+ S12+ S21+ S22 (2-7) S11)∑
e(eˆSFG‚µbeg)σegrggSFG (TSFG) Fgg(∆t) Fgg(Tpump)
(2-8) S21)
∑
e (eˆSFG‚µbeg′)σeg′rg′g′ SFG (TSFG) Fg′g′(∆t) Fg′g′(Tpump) (2-9) σegrggSFG (TSFG) ) (-i/p)2∑
g′′ (µbeg′′‚EBUV)(µbg′′g‚EBIR)× 1 i(ω′g′′g- ωIR) 1 i(ω′eg- ωIR- ωUV) (2-10) Hˆ ) Hˆs+ Hˆb+ Hˆ′ (3-1) Hˆs)∑
l(
1 2 Pˆl2+1 2ωl 2Qˆ l 2)
(3-2) Hˆb)∑
j(
1 2pˆj 2+1 2ωj 2 qˆj2)
) Tˆq+1 2∑
j ωj 2 qˆj2 (3-3) Hˆ′) 1 3!∑
l∑
j,k(
∂3V ∂Ql∂qj∂qk)
0Qlqjqk + 1 3!×∑
l,l′∑
j(
∂3V ∂QlQl′∂qj)
0QlQl′qj + 1 3!l,l∑
′,l′′(
∂3V ∂Ql∂Ql′∂Ql′′)
0QlQl′Ql′′ + 1 4!∑
l∑
i,j,k(
∂4V ∂Ql∂qi∂qj∂qk)
0Qlqiqjqk + 1 4!×∑
ll′∑
j,k(
∂4V ∂Ql∂Ql′∂qj∂qk)
0QlQl′qjqk+ ‚‚‚ (3-4)We shall show that different terms in eq 3-4 play different roles in vibrational spectroscopy and dynamics.
For basis sets, we shall employ the adiabatic approximation,14 i.e.,
and
The basis sets will be limited only to harmonic approximation for the vibron system and phonon bath
and
where
and
Here, we use the following notations, for example,
3.1. Intermolecular Vibrational Energy Transfer. We first consider the vibrational energy transfer between two molecules
D and A. In the dipole-dipole interaction15
where and 0denote the dielectric constant of the media and the permittivity of a vacuum, respectively, and RDArepresents the distance between the donor and acceptor molecules. Here
µ
b
D
l
andbµAl′ are given by
Using the Fermi Golden rule
for (nDl ) 1, nAl′ ) 0) f (nDl ) 0, nAl′ ) 1), we obtain
The simplest case will be resonance energy transfer, i.e.,ωl)
ωl′, that is, the vibrational energy transfer of the same vibrational mode between two molecules. In this case, eq 3-19 reduces to
whereΩDAdescribes the relative orientation between D and A, and
Hereγ10denotes the dephasing rate constant associated with the population decay and the pure dephasing. As in the electronic energy transfer case, the vibrational energy transfer rate can be expressed in terms of spectral overlap. This can be accomplished as follows. For simplicity, we replace the Lorenzian D(ωl′
-ωl+ ωV′V) by the delta functionδ(ωl′- ωl+ ωV′V)
and assume that |〈ΘnV|Θn′V′〉|2 ) |〈ΘnV|Θn′V′〉|D2|〈ΘnV|Θn′V′〉|A2 and FnV(b)) [FnV(b)]D[FnV
(b)]
A. Then we can rewrite eq 3-19 as Hˆ ψnV(Q,q) ) EnVψnV(Q,q) (3-5) ψnV(Q,q) )Φn(Q,q)ΘnV(q) (3-6)
(
Hˆs+1 2∑
j ωj2qj2+ Hˆ′)
Φn(Q,q) ) Un(q)Φn(Q,q) (3-7) (Tˆq+ Un(q))ΘnV(q) ) EnVΘnV(q) (3-8) Φn(Q,q) )∏
l Χnl(Ql) ΘnV(q) )∏
j χnVj(Qj(n)) (3-9) Un(q) )∑
l(
nl+1 2)
pωl +1 2∑
j {ωj(n)}2{qj(n)}2-∑
j 1 2{ωj(n)} 2(
∑
l Vllj(
V l+ 1 2)
p ωl)
2 + Hˆ′′ (3-10) {ωj(n)}2) ωj2+ 2∑
l Vlljj(
Vl+1 2)
p ωl (3-11) {qj(n)}2) qj2+ 1 ωj2∑
l Vllj(
nl+1 2)
p ωl (3-12) Hˆ′′)∑
l∑
j*k VljkQlqjqk+∑
l*l′∑
j Vll′jQlQl′qj+∑
l,l′,l′′ Vll′l′′QlQl′Ql′′+∑
l∑
i,j,k VlijkQlqiqjqk+∑
l∑
j*k VlijkQl2qjqk+∑
l*l′∑
j,k Vll′jkQlQl′qjqk+ ‚‚‚ (3-13) EnV)∑
l(
nl+1 2)
pωl +∑
j(
V j+ 1 2)
pωj (n)-∑
j 1 2{ωj(n)}2[
∑
l Vllj(
Vl+1 2)
p ωl]
(3-14) Vljk) 1 3!(
∂3V ∂Ql∂qj∂qk)
0 (3-15) Hˆ′VET) 1 4π0RDA 3[
(µbD l‚µb A l′ ) -3(µ b D l‚RB DA)(bµA l′‚RB DA) RDA2]
) |µbD l||µb A l′| 4π0RDA3ΩDA (3-16) µ b D l )(
∂µb ∂Ql)
Ql bµA l′ )(
∂µb′ ∂Ql′)
Ql′ (3-17) WVET)2π p∑
V∑
V′ F nV (b)|〈 nV|Hˆ′VET|n′V′〉|2D(En′V′- EnV) (3-18) WVET) 2π p2 1 (4π0) 2 RDA6|
(
∂µb ∂Ql)
0|
2|
(
∂µb′ ∂Ql′)
0|
2 |ΩDA| 2(
1 4βlβl′)
×∑
V∑
V′ F nV (b)|〈 ΘnV|Θn′V′〉|2D(ωl′- ωl- ωV′V) (3-19) WVET) 2π p2 1 (4π0) 2 RDA6|
(
∂µb ∂Ql)
0|
2|
(
∂µb′ ∂Ql′)
0|
2 |ΩDA| 2(
1 4βlβl′)
× D(ωl′- ωl) (3-20) D(ωl′- ωl) ) 1 π γ10 γ102+ (ωl′- ωl)2 (3-21) δ(ωl′- ωl+ ωV′V) )∫
dω [δ(ωl+ ωV′V- ω)]D[δ(ω + ωV′V- ωl′)]A (3-22) WVET) 2π p2 1 (4π0) 2 RDA6|
(
∂µb ∂Ql)
0|
2|
(
∂µb′ ∂Ql′)
0|
2 |ΩDA| 2(
1 4βlβl′)
×∫
dω[
∑
V∑
V′ F nV (b)|〈 ΘnV|Θn′V′〉|2δ(ωl+ ωV′V- ω)]
D ×[
∑
V∑
V′ F nV (b)|〈 ΘnV|Θn′V′〉|2δ(ω + ωV′V- ωl′)]
A (3-23)which should be compared with the band-shape function
3.2. One-Vibron Relaxation. In this case, the vibrational relaxation can be described by nl) 1 f nl) 0; that is, the excitation energy of a vibron is directly relaxed into the phonon bath. Using the Hˆ′′given by
we obtain
or
This represents the case in which the one quantum excitation energy of the vibron mode Qlrelaxes into the phonon modes qj and qkwith one quantum each and the rest of all phonon modes whose the energy matches∆ ) pωj +pωk -pωl. Here the summations can be replaced by an integral with the density of state F(Ej)
3.3. Two-Vibron Relaxation. Next we consider the case of relaxation (nl) 1, nl′) 0) f (nl) 0, nl′) 1). In this case, we can use
to obtain the rate of vibrational relaxation as
or
3.4. Three-Vibron Relaxation. This corresponds to the process (nl) 1, nl′) 0) f (nl) 0, nl′) 2). Using
we obtain
or
It should be noted that eq 3-33 can also be expressed in terms of spectral overlap.
Another possibility will be due to
This case can be treated similarly and will not be discussed here.
3.5. Vibrational Spectroscopy. The quantum mechanical expression for the absorption coefficient of vibrational spectra has been given by eq 3-24. Notice that
and that the spectral band-shape function can be expressed as
where
where njidenotes the phonon distribution function.
To show an application, we shall consider the theoretical treatments of sidebands in vibrational spectroscopy. A typical sideband in water can be described by (nl) 0, Vj) 0) f (nl) 1, Vj) 1), that is a combination band;16,17its transition moment is given by 〈(nl ) 0, Vj ) 0)|(∂µb/∂Ql)0Ql|(nl ) 1, Vj ) 1)〉. Conventionally, it can be treated as follows. Due to the anharmonic coupling, the state (nl) 0, Vj) 0) can couple with the state (nl) 0, Vj) 1), and the state (nl) 1, Vj) 1) can R(ω) )4π 2 ω 3pc
|
(
∂µb ∂Ql)
0|
2(
1 2βl)
∑
V∑
V′ F nV (b)|〈 ΘnV|Θn′V′〉|2× δ(ωn′V′,nV- ω) (3-24) Hˆ′′)∑
l∑
j*k VljkQlqjqk (3-25) WVET) 2π p2∑
j,k |Vljk| 2(Vjj+ 1)(Vjk+ 1) 8βlβjβk ×∑
{Vm}∑
{Vm′}∏
m m*j,k FV m (b)|〈 χV′ m|χVm〉| 2 D(-ωl+ ωj+ ωk+ ωV′V) (3-26) WVET) 1 p2∑
j,k |Vljk| 2(V j j+ 1)(Vjk+ 1) 8βlβjβk∫
-∞ ∞ dt× exp[it(-ωl+ ωj+ ωk) -γ10|t|] × exp{
∑
m m*j,k Sm[-(2njm+ 1) + (njm+ 1)eitωm+ nj me -itωm]}
(3-27)∑
j f∫
dEjF(Ej) (3-28) Hˆ′′)∑
j Vll′jQlQl′qj (3-29) WVET)2π p2∑
j |Vll′j| 2(V j j+ 1) 8βlβl′βj ×∑
{Vm}{∑
Vm′}∏
m m*j F Vm (b)|〈 χV′m|χVm〉| 2 D(ωl′- ωl+ ωj+ ωV′V) (3-30) WVET) 1 p2∑
j |Vll′j| 2(V j j+ 1) 8βlβl′βj∫
-∞ ∞ dt× exp[it(ωl′- ωl+ ωj) -γ10|t|] × exp{
∑
m m*j Sm[-(2njm+ 1) + (njm+ 1)eitωm+ nj me -itωm]}
(3-31)TABLE 1: Optimized Geometry of Single Water Molecule
atom xa ya za O 0.000000 0.000000 0.117041 H 0.000000 0.763487 -0.468165 H 0.000000 -0.763487 -0.468165 aUnit of Å. Hˆ′′) Vll′l′QlQl′2 (3-32) WVET) 1 p2 |Vll′l′| 2 1 4βlβl′ 2×
∑
V∑
V′ FV(b)|〈 Θn′V′|ΘnV〉| 2 D(2ωl′- ωl+ ωn′V′,nV) (3-33) WVET) 1 p2 |Vll′l′| 2 1 4βlβl′ 2∫
-∞ ∞ dt exp[it(2ωl′- ωl) -γ10|t|] × exp{
∑
j Sj[-(2njj+ 1) + (njj+ 1)eitωj+ nj je -itωj]}
(3-34) Hˆ′′)∑
j Vll′l′jQlQl′2qj (3-35) |〈ΘnV|Θn′V′〉|2)∏
i |〈χnVi|χn′V′i〉| 2 (3-36) R(ω) )2πω 3pc|
(
∂µb ∂Ql)
0|
2(
1 2βl)
∫
-∞ ∞ dt× exp[it(ωn′n- ω) - γn′n|t|]∏
i Gi(t) (3-37) Gi(t) ) exp[Si{-(2nji+ 1) + (nji+ 1)eitωi+ nj ie -itωi}] (3-38)couple with (nl) 0, Vj) 1). We find
The transition moment for the fundamental transition is given by
The intensity ratio between these two bands is given by
In liquid water, a sideband at 2150 cm-1 has been observed and it has been attributed toδOH+ γLwhereδOH(bending) ) 1640 cm-1 andγL(librational mode) ) 505 cm-1. From the above discussion we can estimate the relative band-intensity of this sideband to theδOHband.
Another well-known sideband is observed at 2530 cm-1, which has been attributed toδOH- γL, whereδOH) 3250 cm-1 and γL ) 720 cm-1. This sideband is a hot band and the vibrational transition can be described by (VOH) 0, VL) 1) f (VOH) 1, VL) 0). From eq 3-24 we can see that this band can be described by Fn,VL)1|〈Φn′)1|Ql|Φn)0〉|2|〈Θn′)1,V′)0|Θn)0,V)1〉|2. Due to Fn)0,V)1, this sideband intensity will be weaker compared with the type discussed in the previous paragraph.
Similar to the electronic spectroscopy,|〈Θn′υ′|Θnυ〉|2 in eq
3-27 denotes the Franck-Condon factor. For example, for the transition (nl) 0, υj) 0) f (nl) 1, υj) 1), the intensity ratio between this transition and the transition (nl) 0, υj) 0) f (nl ) 1, υj) 0) is given by
If we ignore the frequency change between the vibron states, i.e., the displaced oscillator case, R takes the form
Using eq 3-12 for calculating∆qj(n) we can see that eq 3-43 reduces to eq 3-41. This shows that eq 3-24 can be used to calculate vibration spectra of molecules in dense media. 4. Discussion
Vibrational spectroscopic and dynamic properties require some of the fundamental properties of molecular systems, as
shown in the previous sections. For example, normal modes, dipole moment derivatives, and anharmonic couplings are needed to estimate rate constants or model spectroscopic features. Quantum chemistry approaches could provide these properties: however, it is practically impossible to treat a whole water system. Here we shall treat water clusters and obtain vibrational properties and anharmonic couplings of water clusters (H2O)n with n ) 1, 2, ..., 8. We perform quantum chemistry calculations by using Gaussian 0318 DFT at the B3LYP level and with the 6-311++G** basis set.
Next we shall estimate the rate of intermolecular energy transfer. The transfer rate for the resonance case can be expressed as (see eq 3-18)
where the coupling is given by
Let us first consider a dimer cluster. In this example, we assume a model system in which each water molecule has its own normal modes and they are coupled with each other via the interaction defined by eq 4-2.bµDl in eq 4-2, for example, is defined by eq (3-17) so that we make use of the calculated geometry and dipole moment derivatives for the water mo-nomer, and they are listed in Table 1 and Table 2. To estimate the magnitude of eq 4-2, we further use the vacuum permit-tivity 8.854 × 10-12J-1 C2 m-1. The optimized structure of the dimer is given in Table 3, and the distance between the two oxygen atoms is separated by RO-O) RDA) 2.901 76 Å. If we choose a bending mode of 1602 cm-1 and place its dipole moment derivatives on each oxygen atom with the direction given in Table 4, the corresponding orientation factor is then ΩDA ) -1.386 57 and the estimated value for the interaction is given by |H′DA|/2πcp ) 3.549 71 cm-1. 〈(nl) 0, Vj) 0)|
(
∂µb ∂Ql)
0Ql|(nl) 1, Vj) 1)〉) 〈(nl) 0, Vj) 0)|VlljQl2qj|(nl) 0, Vj) 1)〉 -pω j × 〈(nl) 0, Vj) 1)|(
∂µb ∂Ql)
0Ql|(nl) 1, Vj) 1)〉+ 〈(nl) 1, Vj) 0)|VlljQl2qj|(nl) 1, Vj) 1)〉 pωj × 〈(nl) 0, Vj) 0)|(
∂µb ∂Ql)
0Ql|(nl) 1, Vj) 0)〉)(
∂µb ∂Ql)
0 Vllj pωjx
1 4βjβl (3-39) 〈(nl) 0, Vj) 0)|(
∂µb ∂Ql)
0Ql|(nl) 1, Vj) 0)〉)(
∂µb ∂Ql)
0x
1 2βl (3-40) R ) p|Vllj| 2 2ωj 3 ωl 3 (3-41) R )|〈 χ1l1j|χ0l0j〉| 2 |〈χ1l0j|χ0l0j〉| 2 (3-42) R ) Sj)ωj 2p∆qj(n) 2 (3-43)TABLE 2: Normal Modes and Transition Moment Derivativeawith Respect to Each Normal Mode
normal mode frequency/cm-1 ∂µx/∂Qi ∂µy/∂Qi ∂µz/∂Qi
Q1(a1) 3819 9.2× 10-10 4.2× 10-9 -3.0
Q2(a1) 1602 7.0× 10-10 2.4× 10-9 -8.2
Q3(b2) 3924 -1.2 × 10-8 -7.5 1.9× 10-8 aUnit of D/(Å amu)1/2.
TABLE 3: Optimized Geometry of Dimer
atom eq no. xa ya za O 1 -0.006242 1.525404 0.000000 H 1a 0.909757 1.525404 0.000000 H 1b 0.035907 0.556437 0.000000 O 2 -0.006242 -1.376356 0.000000 H 2a -0.422895 -1.782687 0.767083 H 2b -0.422895 -1.782687 -0.767083 aUnit of Å.
TABLE 4: Unit Vectorsaof Dipole Moment Derivatives
Originated at Each O Atom
O no. eˆx eˆy eˆz
1 0.81977 -0.572693 0.000000
2 -0.715913 -0.698177 0.004215
aeˆn) (∂µb
n/∂Ql)/[(∂b/∂Qµ l)‚(∂µb/∂Ql)]1/2, where n specifies the oxygen atom. W )2π p2 |H′DA| 21 π γDA (γDA) 2+ (∆ω)2 (4-1) H′DA) 1 4π0RDA 3
[
(µbD l‚µb A l′ ) -3(µ b D l‚RB DA)(bµA l′‚RB DA) RDA2]
(4-2)Suppose that∆ω ) 0 in eq 4-1,
If the dephasing rate constantγDA ) 1012s-1, we find W ) 2|H′DA/p|2(1/γ
DA) ) 8π2c23.549712(1/1012)∼ 8.954 03 × 1011 s-1 which is 1.116 82 ps.
Now we estimate the time constants for the intermolecular energy transfer in two octamers (cube and bicyclic) in a similar fashion. Figures 1 and 2 show the optimized structure of cube and bicyclic octamers, respectively, that agree with Ohno’s work.19Table 5 lists the optimized molecular parameters for the cube octamer. For the cube case, the total number of possible two water molecule pair is 28. Table 6 lists the unit vectors of the dipole moment derivatives originated at each O atom. For the case in which RO-O< 3 Å, we find that there are 12 water molecule pairs. Table 7 summarizes RO-O,Ω, and the calculated rate constants, and these values versus H2O pairs are plotted in Figure 3. Here the numbers 1-8 specify the oxygen atoms listed in Table 5. One can see from Figure 3 that there can be three groups for the energy transfer mechanism: the first group consisting of two water molecules with short distances around 2.68 Å and|Ω| values around 1.2, and the second group with longer distances around 2.86 Å and larger|Ω| around 1.38, the last group with similar distances as the second group and smaller |Ω| values ∼0.36. For the first group, the alignment of the two water molecules in each pair is similar to that of the dimer given in Table 1, and for the second group, the two water molecules form a tilted dimer structure, but this alignment is more preferable to have a larger value for|Ω|. In the last group, the dimer structures are quite different from that listed in Table 1. We also estimate the intermolecular energy transfer time constants for molecular pairs whose RO-Ois larger than 3 Å. Table 8 lists the calculated time constants. The shortest time constant in this case is about 60 ps. The molecular pair in this group is located diagonally either in each plane in the cube (e.g., 1 T 3 pair) or in the cube (e.g., 1 T 7 pair).
For the bicyclic octamer case, the time constants for the intermolecular energy transfer taking place in pairs with RO-O
< 3 Å are also calculated and shown in Figure 4. In this case, the time constants for the intermolecular energy transfer show strong|Ω| dependence.
It may be informative to estimate the time constants of the intermolecular energy transfer in the tetrahedral water cluster whose structure is shown in Figure 5. The calculated values are also plotted versus the water pair. If we look at the water
Figure 1. Structure of water cube octamer. The numbers correspond
to the oxygen molecules listed in Table 5.
Figure 2. Structure of water bicyclic octamer. Each number specifies
the oxygen molecule used in Figure 4.
W ) 2 p2
|H′DA|
2 1
γDA (4-3)
TABLE 5: Optimized Geometry of the Cube Octamer
atom eq no. Xa Ya Za O 1 1.566303 1.449495 -1.285755 H 1a 1.642469 0.465981 -1.412737 H 1b 2.219554 1.861437 -1.859035 O 2 1.477277 -1.228116 -1.392212 H 2a 0.547244 -1.424396 -1.608627 H 2b 1.582434 -1.516535 -0.467514 O 3 -1.379417 -1.375764 -1.539816 H 3b -1.515428 -1.502382 -0.562028 H 3b -1.966234 -1.989406 -1.991442 O 4 -1.298728 1.485903 -1.308213 H 4a -0.346812 1.643307 -1.445290 H 4b -1.441147 0.568211 -1.602903 O 5 1.287805 1.214574 1.359794 H 5a 0.340021 1.421122 1.662674 H 5b 1.512659 1.500452 0.661773 O 6 -1.554271 1.386748 1.359794 H 6a -1.571839 1.521934 0.375568 H 6b -2.180900 2.006591 1.744949 O 7 -1.468370 -1.47477 1.132528 H 7a -1.647155 -0.557323 1.406315 H 7b -0.544209 -1.636833 1.39733 O 8 1.366881 -1.458804 1.467499 H 8a 1.944994 -1.880499 2.110213 H 8b 1.444504 -0.475781 1.599667 aUnit of Å.
TABLE 6: Unit Vectorsaof Dipole Moment Derivatives
Originated at Each O Atom in the Cube Octamer
O no. eˆx eˆy eˆz
1 0.636658 -0.472214 -0.609656 2 -0.706978 -0.405819 0.579217 3 -0.629753 -0.645315 0.432411 4 0.679273 -0.638178 -0.362378 5 -0.587233 0.415327 -0.694738 6 -0.565406 0.659135 -0.495839 7 0.644368 0.61664 0.452266 8 0.572913 0.463705 0.675832 aeˆn) (∂µb
n/∂Q)/[(∂µbn/∂Q)‚(∂bµn/∂Q)]-1/2where n specifies the oxygen atom.
TABLE 7: Calculated Time Constants for the
Intermolecular Energy Transfer in the Cube Octamer with RO-O< 3 Å RDA) RO-Oa Ωb T ) 1/(W× 10-12)c 2 T 1 2.68120 -1.18936 0.944596 4 T 1 2.86535 -0.354928 15.8008 5 T 1 2.87400 -1.40118 1.03235 3 T 2 2.86431 -0.377116 13.9658 8 T 2 2.87112 -1.37431 1.06669 4 T 3 2.87216 -1.38538 1.05197 7 T 3 2.67566 -1.19086 0.930579 6 T 4 2.68205 -1.18375 0.955373 6 T 5 2.85468 -0.312932 19.8762 8 T 5 2.67633 -1.18819 0.936167 7 T 6 2.87181 -1.38303 1.05479 8 T 7 2.85501 -0.398997 12.235
aUnit of Å.bCalculated using eq 4-2.cCalculated using eq 4-3 and unit of ps.
molecule 2, there are 4 water pairs that consist of the nearest neighbor water molecules. The respective calculated time constants are 1.26 ps for the 1 T 2 pair, 15.2 ps for the 2 T 5 pair, 27.3 ps for the 2 T 4 pair, and 44.0 ps for the 2 T 3 pair. According to the coupled oscillator model presented in section 3, to treat VER we need to have the information of cubic and quartic anharmonic coupling constants. In our DFT calculations
of water clusters, this information has been obtained up to the cluster of size 8. So, in principle, we should be able to perform the quantum chemistry calculations of the rates of energy transfer (see eqs 3-27, 3-31, and 3-34) and VER. For example, from Figure 1 for the cube octamer we can determine how the cluster is excited initially by the pumping laser (determined by laser wavelength, pulse duration, and laser intensity). Each excited state will evolve (or decay) and its dynamical process can be described by using the coupled oscillator model, which requires the information of cubic and quartic anharmonic couplings.
Two types of VER have been reported.16,17One is the VER of VOH) 3700 cm-1into two quanta of 1650 cm-1. That is, this type of VER belongs to three-vibron processes Vll′l′QlQl′2. There are a number of possibilities (i.e., paths) found in our DFT calculations; the cubic anharmonic coupling constants cover the range 0-90 cm-1(see Appendix S7 in the Supporting Information). Another type of VER is the so-called thermal-ization. In our opinion the most probable processes will be the VER of the 1650 cm-1modes into two quanta of the vibrational modes with frequencies smaller than 1000 cm-1. In this case, the cubic anharmonic coupling constants cover the range 0-102 cm-1(see Appendix S7 in the Supporting Information). From the values of anharmonic coupling constants for VER in comparison with the |H′DA| values for vibrational energy transfer, the time scale of VER can also be estimated. In a future
Figure 3. Calculated time constants for the intermolecular energy transfer in the cube octamer with RO-O< 3 Å.
TABLE 8: Calculated VET Time Constants for the Intermolecular Energy Transfer in the Cube Octamer with
RO-O> 3 Å RDA) RO-Oa Ωb T ) 1/(W× 10-12)c 3 T 1 4.08949 -0.112493 1329.4 4 T 2 3.88319 -0.105825 1506.79 5 T 2 3.84032 -0.842586 66.6621 5 T 3 4.84398 -0.234159 272.646 5 T 4 3.87550 0.229208 234.729 6 T 1 4.09156 0.00584479 337725. 6 T 2 4.85812 -0.86013 63.9108 6 T 3 4.00871 0.213419 272.517 7 T 1 4.85887 -0.878554 60.1967 7 T 2 3.88741 -0.24601 246.614 7 T 4 3.84078 -0.125602 1041.53 7 T 5 3.87507 0.231926 226.547 8 T 1 4.00979 0.0306694 12274.4 8 T 3 4.07345 -0.8504 64.3896 8 T 4 4.84576 0.145168 577.869 8 T 6 4.07945 -0.0778607 2734.42
aUnit of Å.bCalculated using eq 4-2.cCalculated using eq 4-3 and
unit of ps.
Figure 4. Calculated time constants for the intermolecular energy transfer in the bicyclic octamer with RO-O< 3 Å.
Figure 5. Structure and calculated time constants for the intermolecular
paper, we shall report the detailed first-principle calculations of vibrational spectroscopy, VER, and vibrational energy transfer.
5. Conclusion
In this paper, using the density matrix method, we have developed a theory of time-resolved SFG that takes into account the dynamical behaviors of both population and coherence of the surface system. To study the vibrational dynamics of surface waters, we limit ourselves to discuss the case of time-resolved vibrational IR-UV SFG. To treat the VER, vibrational energy transfer, and vibrational spectroscopy of liquid water, we have used the coupled oscillator model in which the basis set chosen is based on the diabatic approximation to separate the system oscillators and bath oscillators. In the harmonic oscillator approximation, vibrational energy transfer, VER, and vibrational spectroscopy can be calculated by including cubic anharmonic couplings that are obtained by DFT calculations. In this way we have demonstrated that it is possible to carry out the first-principle calculations of vibrational spectroscopy and dynamics of clusters.
Acknowledgment. Dedicated to Prof. M. C. Lin on the occasion of his 70th birthday. We thank the financial support by National Science Council of Taiwan and Academia Sinica. M.H. also thanks the NSC for financial support (NSC-95-2113-M-002-027-MY3) and National Taiwan University for financial support (95R0066-23), and the National Center for High-performance Computing in Hsinchu for providing access to the computational resources.
Supporting Information Available: The rectangular pulse treatments and derivation of pumping process are given in Appendices S1 and S2. A detailed derivation of Fgg(Tpump) and Fg′g′(Tpump) is presented in Appendix S3, that of σegrggSFG (TSFG) andσg′′erggSFG (TSFG) is given in Appendix S4, and that of S12and S22is shown in Appendix S5, respectively. Appendix S6 shows a definition of Fg′g′(Tpump) and Fgg(Tpump). Normal modes and anharmonic coupling of cube octamer are given in Appendix
S7. This material is available free of charge via the Internet at http://pubs.acs.org.
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