Quantum chemical calculation of intramolecular vibrational redistribution and vibrational energy transfer of water clusters

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Quantum chemical calculation of intramolecular vibrational

redistribution and vibrational energy transfer of water clusters

Y.L. Niu

a

_{, R. Pang}

a,b

_{, C.Y. Zhu}

a,⇑

_{, M. Hayashi}

c

_{, Y. Fujimura}

d

_{, S.H. Lin}

a,d,⇑

_{, Y.R. Shen}

e a

Department of Applied Chemistry, Institute of Molecular Science and Center for Interdisciplinary Molecular Science, National Chiao Tung University, Hsinchu, Taiwan, ROC

b

State Key Laboratory of Physical Chemistry of Solid Surfaces, and College of Chemistry and Chemical Engineering and Xiamen University, Xiamen 361005, Fujian, PR China

c

Center for Condensed Matter Sciences, National Taiwan University, Taipei, Taiwan, ROC

d

Institute of Atomic and Molecular Sciences (IAMS), Academia Sinica, Taipei, Taiwan, ROC

e_{Department of Physics, University of California, Berkeley, CA, USA}

a r t i c l e

i n f o

Article history: Received 18 March 2013 In ﬁnal form 5 September 2013 Available online 13 September 2013

a b s t r a c t

In present letter the adiabatic approximation is applied to the intramolecular vibrational redistribution (IVR) of water clusters. The isotope, blocking and cluster-size effects are investigated. This letter also examines the assumption associated with the transition state theory applied to unimolecular reactions; that is, IVR is assumed to be completed before the reaction takes place. For this purpose, we choose to study (H2O)2H+?H2O + H3O+, and (H2O)2?2H2O processes. In molecular clusters, the vibrational exci-tation energy transfer between different normal modes has been observed. This will also be investigated for the deuterated species of (HOD)2H+.

1. Introduction

Recently experimental[1–7]and theoretical[8–16] investiga-tions of the vibrational redistribution dynamics of water and other molecules in condensed phases have attracted a considerable attention. Hynes and co-workers studied the vibrational redistri-bution of OH stretch excitations to bending to librational degrees of freedom in water liquid in linear coupling model[15]. Skinner and co-workers discussed the validity of Förster theory model for the vibrational energy transfer in water liquid[13]. Bowman and co-workers simulated the predissociation of water dimers and dis-cuss the process of energy ﬂow from bond stretching to the bond bending[17]. In comparison the ﬁrst principle calculation of the vibrational redistribution dynamics of isolated molecules has re-ceived much less attention[18–20]. A theoretical approach based on the adiabatic approximation model of vibrational redistribution has been developed[8,12]. In this Letter, we shall present its appli-cation to the vibrational dynamics of water clusters (H2O)nand

(H2O)nH+, where n = 2, 3, 4, and their isotope species. The RRKM

(Rice–Ramsperger–Kassel–Marcus) theory is a very popular and well received theory applied to treat the unimolecular reactions of isolated (i.e., collision-free) molecules and clusters[21–24]. Lin and et al. have recently applied Morse potential model to develop

the anharmonic RRKM theory[25–28]. Fundamentally it is based on the transition state theory which assumes that intramolecular vibrational redistribution (IVR) is much faster than unimolecular reactions so that the vibrational equilibrium is established before the reaction takes place. IVR plays an important role not only in unimolecular reactions but also in photochemistry and photophys-ics. Due to the fact that in the harmonic oscillator approximation when a vibrational mode is excited the excitation energy will be localized in that mode and will not ﬂow into other modes, for IVR to take place, the anharmonic potential energy function which describes the coupling among different modes is needed. This information has become available only recently in the quantum chemistry programs and will be employed to perform the calcula-tion of IVR in this Letter[29,30].

In treating the unimolecular decomposition of molecular clus-ters by using the RRKM theory the anharmonic effect is very important and it has been recently included in the conventional RRKM theory. This has been applied to study the decomposition of (H2O)2, ((H2O)2?2H2O) [27] and (H2O)2H+, ((H2O)2H+?(H

2-O)H+_{+ H}

2O)[25]. For the purpose of the investigation of the effect

of IVR on the RRKM theory, we shall calculate the IVR of (H2O)2and

(H2O)2H+ and compare these IVR rates with the anharmonic

decomposition rates of these clusters. This can then be used to examine the validity of the RRKM theory applied to these clusters. In molecular clusters, the vibrational excitation energy transfer from one mode to another has been observed. This will also be investigated in the deuterated species of (HOD)2H+.

The present letter is organized as follows. Following the intro-duction, a brief theory of IVR will be presented in Section2, which

http://dx.doi.org/10.1016/j.cplett.2013.09.019

⇑ Corresponding authors at: Department of Applied Chemistry, Institute of Molecular Science and Center for Interdisciplinary Molecular Science, National Chiao Tung University, Hsinchu, Taiwan, ROC.

E-mail addresses:[email protected](C.Y. Zhu),[email protected]

(S.H. Lin).

Contents lists available atScienceDirect

Chemical Physics Letters

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2. Theory of IVR

We consider the adiabatic approximation model for isolated molecules or clusters, which is similar to the Born–Oppenheimer approximation model for molecules; that is, electronic coordi-nates corresponds to high frequency normal coordicoordi-nates {QI},

nu-clear coordinates corresponds to low frequency normal coordinates {qi}, UV–visible spectra corresponds to IR vibrational

spectra and internal conversion corresponds to IVR. It follows that to solve

H

W

avðQ ; qÞ ¼ Eav

W

avðQ ; qÞ ð1Þ

where

H ¼ TLþ THþ V ¼ TLþ HH ð2Þ

is the total vibrational Hamiltonian, and

HH THþ V ð3Þ

is the high frequency oscillator Hamiltonian including the interac-tion between high frequency modes and low frequency modes. The subscript ‘H’ and ‘L’ represent ‘high frequency modes’ and ‘low frequency modes’ respectively. We ﬁrst solve

HH

U

aðQ ; qÞ ¼ Uað Þq

U

aðQ ; qÞ ð4Þ

to get the ‘potential energy surface (PES)’ of low frequency modes. Then we solve

TLþ Uað Þq

½

H

avð Þ ¼ Eq av

H

avð Þq ð5Þ

to obtain the total wavefunction

W

avðQ ; qÞ ¼

U

aðQ ; qÞ

H

avð Þq ð6Þ

Here semicolon means that q is regarded as a parameter inUa(Q;q)

The performance for the adiabatic approximation has been tested and is shown to be acceptable[31]. In Eqs.(4)–(6), a denotes the quantum state of the high frequency modes Q, while

v

represents the quantum states of the low frequency modes q.

Notice that V Q ; qð Þ ¼ VHð Þ þ VQ Lð Þ þ Vq intðQ ; qÞ ð7Þ VHð Þ ¼Q X I 1 2

x

2 IQ 2 I þ 1 6 X IJK VIJKQIQJQKþ ð8Þ VLð Þ ¼q X i 1 2

x

2 iq2i þ 1 6 X ijk Vijkqiqjqkþ ð9Þ VintðQ ; qÞ ¼ 1 2 X IJi VIJiQIQJqiþ 1 2 X Iij VIijQIqiqjþ ð10Þ

Vijk is the anharmonic expansion coefﬁcients of the PES; for

example, VIJi @3V @QI@QJ@qi ! 0 ð11Þ

Using the same treatment in the internal conversion process

[32], in the adiabatic approximation the IVR rate for a ? b can be expressed as[8,31] where

U

b @ @ql

U

a ¼

U

b@V@ql

U

a D E Uað Þ UQ bð ÞQ

U

0_b @V @ql

U

0_a D E U0a U 0 b ð14Þ

and D(EavEbu) denotes the line-shape function. This IVR theory has recently been applied to compare with the experimental data of IVR for the two N–H vibrational modes of isolated aniline re-ported by Ebata et al.[19,20]. The agreement is acceptable in[33]. 3. Calculated results-(H2O)2H+and (H2O)2

3.1. The isotope effect in IVR of (H2O)2H+

We now consider the application of the adiabatic approximation model of IVR to calculate the IVR rates of the hydrogen-bonded water dimer (H2O)2H+ and its deuterated and tritiated species.

We select the MP2 method and 6-311++G(d, p) basis set to optimize the geometry of all the water clusters usingGAUSSIAN09 program

[34], and then calculate the anharmonic coupling parameters. The optimized structure of (H2O)2H+is shown inFigure 1.

The vibrational modes of (H2O)2H+ are shown inFigure S1 in

supplementary materialand the rates of IVR for the O–H related modes that play the important role for the dissociation of (H2O)2H+to H2O and H3O+will be presented.

The frequencies of normal modes in (H2O)2H+, (H2O)2D+, and

(H2O)2T+, which are related to the motions of the bridged

hydro-gen, deuterium and tritium atoms, obviously decrease due to mass increase. Overall vibrational redistribution rates for high frequency modes 12–15 of (H2O)2H+, (H2O)2D+and (H2O)2T+are calculated

according to Eq.(13)and listed inTable 1. From this table, we ﬁnd that the IVR rates slow down with the increase of the atomic weight of the bridged atom, that is, the IVR rates of the O–H modes in the (H2O)2H+are faster than those in the (H2O)2D+and (H2O)2T+

species. This phenomenon is often referred to as ‘blocking effect’, for which the bridge hydrogen atom is replaced by deuterium and tritium atom. The decrease of frequency means there must be more quanta of vibrational modes to accept energy, which low-ers the vibrational transition rate. For example, for vibrational mode 13, the IVR rate for the H-species is 1.96 ps, while for the D-species and T-species, the rates are 5.20 and 7.46 ps, respectively.

Next we consider other deuterated species of (H2O)2H+which are

listed in Table 2. Let us ﬁrst consider (H2OHHOD)+ and (H

2-ODHOD)+_{. We can see that the IVR rates of any O–H mode in the}

H bridged species are faster than those in the D bridged species. On the contrary the IVR rate of the O–D mode is faster in the D bridged species. Similarly we compare the IVR rates in (D2OHD2O)+

and (D2ODD2O)+; the IVR rates of the four O–D modes in the

H-spe-cies are faster than those in the D-speH-spe-cies except mode 12. Finally we compare the IVR rates in (HODHHOD)+and (HODDHOD)+ ex-cept mode 12. In conclusion, the IVR rates of the O–H modes in the H-species almost remain to be faster than those in the D-species.

3.2. The vibrational redistribution of water clusters (H2O)n

The optimized structure of water dimer is shown inFigure 2. The point group of water dimer is CS. There are 8 symmetric

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modes and 4 anti-symmetric modes. The IVR rates of water di-mer (H2O)2 are given inTable 3 and Table 4. Table 3 gives the

overall IVR rates of six intramolecular modes which seem to cor-relate with the energy gap to be relaxed into low frequency modes, which calculated according to Eq. (12). Table 4 gives

the important detailed vibrational redistribution rates, that is, the fastest IVR paths of water dimer. From these tables, we know that the fastest vibrational redistribution rate is 1.94 1011_s1 _{for the mode 9. An important feature is that}

the detailed redistribution rates such as W11,8,8, W10,7,7, W9,8,8, Figure 1. Equilibrium geometry of (H2O)2H+, (H2O)3H+and (H2O)4H+.

Table 1

The overall IVR rates of (H2O)2H+, (H2O)2D+and (H2O)2T+.

(H2O)2H+ (H2O)2D+ (H2O)2T+

Mode x(cm1₎ _{Rate (s}1₎ _s_(ps) _x_(cm1₎ _{Rate (s}1₎ _s_(ps) _x_(cm1₎ _{Rate (s}1₎ _s_(ps)

12 3790 1.89 1011 _5.30 ₃₇₈₉ _{7.64 10}10 _13.1 ₃₇₈₉ _{4.55 10}10 _22.0 13 3798 5.11 1011 _1.96 ₃₇₉₈ _{1.92 10}11 _5.20 ₃₇₉₈ _{1.34 10}11 _7.46 14 3893 4.65 1010 21.5 3893 1.34 1010 74.7 3893 9.59 109 104 15 3893 8.87 1010 11.3 3893 1.19 1010 84.0 3893 8.34 109 120 Table 2

IVR rates of deuterated (H2O)2H+, (H2O)3H+and (H2O)4H+.

Mode Frequency (cm1

) Rate(s1

) Lifetime (ps) Mode Frequency (cm1

) Rate(s1 ) Lifetime (ps) 1. (H2ODH2O)+ 2. (H2OHHOD)+ 12 3789 7.64 1010 13.1 12 2792 4.17 1010 24.0 13 3798 1.92 1011 _5.20 ₁₃ ₃₇₉₄ _{5.64 10}11 _1.77 14 3893 1.34 1010 _74.7 ₁₄ ₃₈₄₉ _{1.04 10}11 _9.64 15 3893 1.19 1010 84.0 15 3893 5.191010 19.3 3. (H2OHD2O)+ 4. (HODHHOD)+ 12 2735 7.17 1011 1.39 12 2789 3.08 1010 32.4 13 2859 8.85 1010 11.3 13 2795 2.69 1011 3.72 14 3794 5.57 1011 1.79 14 3847 3.80 1010 26.3 15 3893 5.25 1010 19.0 15 3851 5.93 1010 16.9 5. (H2ODHOD)+‘ 6. (H2ODD2O)+ 12 2791 1.40 1011 7.17 12 2734 4.27 1011 2.34 13 3793 2.29 1011 _4.37 ₁₃ ₂₈₅₉ _{3.47 10}10 _28.8 14 3849 2.95 1010 _34.0 ₁₄ ₃₇₉₃ _{2.14 10}11 _4.66 15 3893 2.49 109 ₄₀₂ ₁₅ ₃₈₉₃ _{2.17 10}9 ₄₆₀ 7. (HODDHOD)+ 8. (DOHDHOD)+ 12 2788 9.98 1010 10.0 12 2790 1.18 1011 8.44 13 2794 1.10 1011 9.07 13 2798 1.37 1011 7.31 14 3847 5.67 109 176 14 3842 1.82 1010 54.9 15 3851 2.51 1010 40.0 15 3850 3.95 1010 25.3 9. (D2OHD2O)+ 10. (D2ODD2O)+ 12 2730 2.07 1011 4.82 12 2729 2.32 1011 4.32 13 2740 8.89 1011 1.13 13 2740 7.15 1011 1.40 14 2859 6.51 1010 _15.4 ₁₄ ₂₈₅₉ _{2.12 10}10 _47.1 15 2859 1.59 1011 6.28 15 2859 3.74 1010 26.7 11. (H2O)3H+ 12. (H2O)4H+ 20 3844 1.44 1011 6.95 28 3852 2.22 1011 4.51 21 3845 2.31 1011 4.33 29 3852 1.79 1011 5.58 22 3863 2.08 1011 4.80 30 3853 2.88 1011 3.62 23 3951 2.52 1010 39.7 31 3958 3.20 1010 31.2 24 3951 2.61 1010 38.4 32 3958 3.19 1010 31.4 33 3959 2.11 1010 _47.5

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W8,6,6, and W7,6,6play important roles in the vibrational

redistri-bution of (H2O)2, which indicates the low frequency modes tend

to accept two quanta energy from the high frequency modes. The calculated results of IVR for other common water clusters (H2O)nand (H2O)nH+where n = 3 and 4 are also shown inTable 3

andTable 2. As can be seen from these results, the IVR rates in most cases do increase with the size of cluster (maybe slowly). This indicates that it may be possible to regard the liquid water to con-sist of various size of clusters n = 2–4 or 5. For the case of (H2O)3

and (H2O)3H+, the IVR rates of the ﬁrst three O–H modes are much

faster than the remaining ones. On the other hand, for the case of (H2O)4 and (H2O)4H+, while there are four fast OH modes in

(H2O)4. The fastest IVR rates of the OH modes in (H2O)nare usually

much faster than those of the corresponding (H2O)nH+. For the IVR

rates of (H2O)nwith n = 2, 3, 4 shown inTable 3, we compare the

IVR rates of the lowest O–H modes, 3807 cm1 _{for n = 2,}

3688 cm1 _{for n = 3 and 3526 cm}1 _{for n = 4. We can see their}

IVR rates are, respectively, 5.15 ps, 1.91 ps and 0.39 ps. There exists approximately a general tendency, that is, the IVR rates vary with their O–H vibrational frequencies.

From the results of our calculated IVR rates of (H2O)n, we can

suggest that in the bulk water the size of water clusters must be n P 3.

4. Discussion

4.1. Dipole vibrational–vibrational energy transfer

The vibrational energy transfer can also take place in a cluster or between molecules through dipole–dipole interaction.[35] In this Letter only the resonance energy transfer case is considered; the non-resonance case can also be treated[32,35].

The vibrational energy transfer may take place between O–H stretching modes or O–D stretching modes in (HOD)2H+. We

de-ﬁne the vibrational energy transfers from the ‘Donor’ part to the ‘Acceptor’ part. According to Ref.[12,35], the transfer rate by di-pole–dipole interaction can be presented as

WVET¼ 2

p

h X v;u

q

ðbÞ nvjhn

v

jH’DAjn’

v

’ij2D Eð n’v’ EnvÞ ð15Þ where

q

ðbÞ

nv is the distribution function, and H’DA¼ ~

l

l D ~

l

k A 4

pee

0R3DA

X

ð16Þ

X

¼ cos hlkDA 3 cos h l Dcos h k A ð17Þ D Eð n’v’ EnvÞ ¼

_p

1 h

c

En’v’ Env ð Þ2þ h2

c

2 ð18Þ The dipoles ~

l

Dand ~

l

kAin Eq.(16)are deﬁned as ~

l

l D¼ @~

l

D @QD;l 0 QD;l; ~

l

lA¼ @~

l

A @QA;k 0 QA;k: ð19Þ

hlk_DAis the angle between ~

l

Dand ~

l

kA;and h l D;h

k

Arepresent the angles

between ~

l

D; ~

l

kA and ~RDA, respectively. For the transition

ðnl

D¼ 1; nkA¼ 0Þ ! ðnlD¼ 0; nkA¼ 1Þ, Eq.(15)can be rewritten as WVET¼ 2

p

h2

X

2 4

pee

0 ð Þ2R6DA @~

l

D @QD;l 0 2 @~

l

A @QA;k 0 2 h 2 4

x

D;l

x

A;k D

x

D;l

x

A;k: ð20Þ

Figure 2. Equilibrium geometry of (H2O)2, (H2O)3and (H2O)4.

Table 3

The overall IVR rates of (H2O)2, (H2O)3and (H2O)4.

Mode Frequency (cm1 ) Rate (s1 ) Lifetime (ps) (H2O)2 7 1639 2.20 109 454 8 1664 3.52 109 ₂₈₄ 9 3807 1.94 1011 _5.15 10 3877 3.16 1010 31.6 11 3974 8.82 109 113 12 3990 1.77 109 566 (H2O)3 16 3688 5.22 1011 1.92 17 3739 1.10 1011 9.05 18 3747 1.59 1011 6.29 19 3961 1.99 1010 _50.14 20 3964 2.14 1010 46.69 21 3965 2.04 1010 49.01 (H2O)4 23 3526 2.55 1012 0.392 24 3608 4.77 1011 2.10 25 3608 4.77 1011 2.10 26 3644 4.13 1011 2.42 27 3953 1.65 1010 _60.6 28 3955 1.64 1010 _61.0 29 3955 1.64 1010 61.0 30 3955 2.15 1010 46.6 23 3526 2.55 1012 0.392 Table 4

Vibrational relaxation paths of (H2O)2. Accepting Energy =xnxlxk:

I l k Rnlk Accpt. energy (cm1) Rate (s1)

7 6 6 0.054 301 2.20 109 8 6 6 0.027 325 3.42 109 9 8 8 0.063 479 1.861011 10 7 7 0.065 599 3.16 1010 11 8 8 0.015 646 6.08 109 12 8 3 0.104 2149 8.51 108

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To calculate the vibrational energy transfer between O–H bond stretching modes and O–D bond stretching modes, we ﬁrst opti-mize the equilibrium structure of (HOD)2H+. Then we calculate

the dipole derivatives of O–H bond stretching mode and O–D bond stretching mode in monomer HOD. The frequencies of OH and OD stretching modes are 3946 cm1_{and 2864 cm}1_{, respectively. We}

use O–O bond length in (HOD)2H+for the distance RDA, which is

2.38 Å. According Eq.(17), XOH and XOD are 0.843 and 0.101

respectively. The resonance energy transfers taking place in (HOD)2H+and the dipole derivatives localizing on each mode in

the dimer (HOD)2H+are shown inFigure 3. The numbers inFigure 3

represent the unit vectors of OH!, OD!and OO!, which is used to cal-culateXOHandXOD. In the resonance case, Ebu EavThen

Lorentz-ian function in Eq.(18)can be approximately derived as

D Eð bu EavÞ ¼

_p

1_h

_c

ð21Þ

Set the dephasing parameter

c

= 1012_s1_{, and according to Eqs.}

(15)–(21), The vibrational energy transfer rates between OH stretching modes and OD stretching modes in (HOD)2H+ are

1.24 1011_s1_{and 7.61 10}8_s1_{, respectively.}

In photochemistry, the electronic excitation energy transfer be-tween the excited donor molecule D⁄_{and un-excited acceptor}

mol-ecule A is usually described by the Förster theory using the dipole– dipole interaction which is in turn expressed in terms of the spec-tral overlap between the normalized ﬂuorescence spectra of D⁄_and

the absorption spectra of A. However, it should be noted that in this case to describe the electronic states of D⁄_{and A no}

conﬁgura-tional interactions have been considered, which resulted in the Coulomb interaction and exchange interaction. Furthermore, the exchange interaction is ignored and the Coulomb interaction is re-placed by the multipole expansion and only ﬁrst non-vanishing di-pole–dipole interaction is retained.

The vibrational energy transfer can also take place in cluster or between molecules. Here again only the dipole–dipole interaction is employed. In this Letter only the resonance energy transfer case is considered; the non-resonance case can also be treated[31,35]. In the non-resonance case the ‘Franck–Condon’ factor |hHbu|Havi|2

of Eq.(15)should be properly treated.

4.2. IVR and the RRKM theory

The RRKM rate constant for an isolated molecule with energy E can be expressed as k Eð Þ ¼1 h W– _{E E}– 0

q

ð ÞE ð22Þ where E–

0 denotes the activation energy, W –_{E E}–

0

;the total num-ber of states of the activated complex and

q

ð Þ; the density of statesE of the reactant. The dissociation rate of (H2O)2H+was calculated by

using the anharmonic RRKM theory[25].

The activation energy of the variational TS is obtained to be 30.95 kcal/mol for the dissociation of (H2O)2H+ in the paper by

Song et al.,[25]which corresponds to 10 825 cm1_{. We calculate}

the vibrational redistribution rate from n = 3 to n = 2 and from n = 2 to n = 1. From the Eq.(12)we can derive that approximately

W n ! n 1ð Þ ¼ nW 1 ! 0ð Þ ð23Þ

From Eq. (23), the IVR rates from n = 3 to n = 2 are 5.66 1011s1_, _{1.54 10}12

s1_, _{1.39 10}11

s1 _and _2.66

1011_s1_{corresponding to modes 9–12, respectively. Now we shall}

examine the validity of the RRKM theory by comparing the IVR rates and the dissociation rate in RRKM method. Song et al. [25]

have applied the anharmonic RRKM theory to calculate the disso-ciation rate of (H2O)2H+. Near the dissociation threshold of

(H2O)2H+, the dissociation rate calculated in microcanonical

ensemble is about 4.34 106_s1_{. This is much slower than the}

IVR rates listed above. Thus in this case it is valid to use the RRKM theory to calculate the decomposition of (H2O)2H+.

Yao et al. have calculated the dissociation rate of water dimer with the anharmonic RRKM method[27]. Comparing the IVR rates (seeTable 3) and the dissociation rate of (H2O)2, 1.95 1011s1

[27], we can see that the transition state theory is not valid. In this case the quantum mechanical vibrational predissociation theory should be employed[36,37].

In conclusion, to check the validity of the RRKM theory for uni-molecular decompositions of these water clusters, we ﬁnd that for the dissociation of (H2O)2, the RRKM theory is not valid to be

em-ployed, and for the case of (H2O)2H+, the RRKM theory holds to a

considerable excitation energy range.

5. Summary

Recently vibration dynamics of liquid water and surface water has attracted considerable attention experimentally. To study this vibration dynamics the anharmonic coupling potentials are re-quired which have become available only recently. In this Letter we use this information to calculate the vibrational redistribution in water clusters (H2O)nand (H2O)nH+with n = 2,3 and 4. The IVR

rate does exhibit its increase with the size of cluster and reaches the values of liquid water at n P 3. For (H2O)3and (H2O)4; their

fastest IVR rates are in the same order of magnitudes as these of bulk and surface water[1]. This indicates that in the bulk water or surface water, the water clusters of n P 3 exist and although their structures may change, the changed rates must be slower than picoseconds.

The isotope effect on IVR is also investigated. We make use of (H2OHH2O)+ to (H2ODH2O)+ and (H2OTH2O)+ to study the

blocking effect of IVR, that is, the IVR rate decreases with H+_{, D}+

and T+_{in the cluster. Making use of (HODHHOD)}+_{we study the}

resonance energy transfer between O–H and O–H, and O–D and O–D in the cluster and ﬁnd the vibrational energy transfer rates to be in the range of 1.24 1011_s1_{–7.61 10}8_s1_{, sensitively}

depending on the relative orientation of the vibrational dipole mo-ments and relative distance.

We make use of the calculated IVR rates in (H2O)2and (H2O)2H+

to check the validity of the use of the RRKM theory for the dissoci-ation of (H2O)2and (H2O)2H+. We ﬁnd that the RRKM theory can be

applied to the case of (H2O)2H+but not to (H2O)2. Figure 3. Vibrational energy transfer between (a) OH-OH and (b) OD-OD in (HOD)2H+.

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Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.cplett.2013. 09.019.

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