Ab Initio Path Integral Simulations for the Fluoride IonâWater Clusters

Article pubs.acs.org/JPCA

Ab Initio Path Integral Simulations for the Fluoride Ion−Water Clusters: Competitive Nuclear Quantum Effect Between F−−Water and Water−Water Hydrogen Bonds Yukio Kawashima,† Kimichi Suzuki,† and Masanori Tachikawa* Quantum Chemistry Division, Graduate School of Science, Graduate School of Nanobioscience, Yokohama City University, Yokohama 236-0027, Japan ABSTRACT: Small hydrated fluoride ion complexes, F−(H2O)n (n = 1−3), have been studied by ab initio hybrid Monte Carlo (HMC) and ab initio path integral hybrid Monte Carlo (PIHMC) simulations. Because of the quantum effect, our simulation shows that the average hydrogen-bonded F−···HO distance in the quantum F−(H2O) is shorter than that in the classical one, while the relation inverts at the three water molecular F−(H2O)3 cluster. In the case of F−(H2O)3, we have found that the nuclear quantum effect enhances the formation of hydrogen bonds between two water molecules. In F−(H2O)2 and F−(H2O)3, the nuclear quantum effect on two different kinds of hydrogen bonds, F−−water and water−water hydrogen bonds, competes against each other. In F−(H2O)3, thus, the nuclear quantum effect on the water−water hydrogen bond leads to the elongation of hydrogen-bonded F−···HO distance, which we suggest this as the possible origin of the structural inversion from F−(H2O) to F−(H2O)3. fluctuations. Studies based on the vibrational self-consistent field method7 and discrete variable representation8 have shown that anharmonic correction to vibrational energy levels with respect to the hydrogen-bonded OH stretching mode has a large nuclear quantum effect.7,8 Recently, we have reported a theoretical study on the hydrogen-bonded structures of F−(H2O) by path integral hybrid Monte Carlo (PIHMC) simulation based on fourth-order Trotter expansion, which can evaluate the thermal and quantum nuclear fluctuations.14 The average distance of OH* in water molecule by PIHMC simulation is longer than that by conventional MD one, while the distance of F−···H*O in PIHMC is shorter than that in conventional MD. For quantitative discussion based on accurate hydrogen-bonded structures, we have confirmed that the thermal and nuclear quantum fluctuation effect is indispensable. The above two fluctuations will allow F−(H2O)n clusters to form various structures; thus the increase of water molecules may lead to different character in the hydrogen bonds in these systems. Moreover, additional water molecules introduce possibilities of formation of water−water hydrogen bonds, where the nuclear quantum effect which is not seen in F−(H2O), may affect the F−− water hydrogen bonds. However, a systematic study on the hydrogen-bonded structures by increasing the coordination number of water molecules, especially the nuclear quantum effect on both F−···H*O and O−H···O hydrogen bonds in F−(H2O)n clusters, taking both thermal and nuclear quantum structural fluctuations into account, is not reported yet. In this paper, thus, we would like to report the dependence of coordination number of water molecules on the hydrogenbonded structures in fluoride ion−water clusters (F−(H2O)n, n =

1. INTRODUCTION Fluoride ion−water clusters F−(H2O)n have attracted considerable attention because of their strong interaction compared to typical hydrogen bonds.1−14 Some reports have suggested that the ion−water interaction is over 25 kcal/mol.5,7,9−11 To elucidate the interaction mechanism of the fluoride ion−water clusters, understanding hydrogen-bonded structures is indispensable. IR spectroscopy has been employed to study these clusters to seek the hydrogen-bond nature and found several interesting characteristics: the vibrational frequency of fluoride ionic hydrogen-bonded stretching mode has been observed to be further red-shifted than other halogen ion−water clusters3, the frequency of ionic hydrogen-bonded stretching mode blue-shifts as the coordination number of water molecules increases, and the second solvation shell emerges at coordination number of five.4,5 Several ab initio molecular orbital (MO) calculations have been performed with respect to the equilibrium structures, harmonic vibrations, and some properties to understand the nature of hydrogen bonds in ion−water clusters.5,6 Their reports have shown that binding energies of fluoride ion−water clusters are stronger than those of other halogen ion−water clusters. Consequently, the elongation of distance for OH* (H* represents hydrogen atoms in fluoride ion−water hydrogenbonds) of F−···HO hydrogen bonds in water molecule has been caused by strong ion−water interaction. They have also reported that the OH* distance becomes shorter as the coordination number of water molecules increases, while the distance of F−···H*O (between fluoride and oxygen atoms) becomes longer. It has been qualitatively shown that the interaction between fluoride ion and hydrogen-bonded hydrogen weaken as the coordination number of water molecules increases. We address here that it is well-known that the ionic hydrogenbonded structure is often strongly affected by highly anharmonic and low frequency modes due to thermal and quantum structural © XXXX American Chemical Society

Received: April 3, 2013 Revised: May 11, 2013

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dx.doi.org/10.1021/jp403295h | J. Phys. Chem. A XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry A

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1−3). The emphasis is on the fluoride ion−water, F−···H*O, and water−water, O−H···O, hydrogen-bonded structure with thermal and quantum nuclear fluctuation effects. Computational details are described in the next section, the computational results and their analysis are discussed in Section 3, and this study is concluded in the final section.

2. COMPUTATIONAL DETAILS Ab initio path integral hybrid Monte Carlo (PIHMC) simulation based on fourth-order Trotter expansion at 300 K has been carried out for fluoride ion−water clusters (F−(H2O)n, n = 1−3) with P = 16 and Δt = 2.0 (fs), where the notations of P and Δt are number of imaginary time slices and time step size, respectively, as in our previous work.14 We have used Nmc = 60 000 steps and Nmd/mc = 2 steps, Nmc = 70 000 steps and Nmd/mc = 2 steps, and Nmc = 50 000 steps and Nmd/mc = 1 steps for F−(H2O), F−(H2O)2, and F−(H2O)3, respectively. The notations of Nmc and Nmd/mc are the total number of steps and number of MD steps in one MC cycle, respectively. For comparison, ab initio HMC simulation was carried out using Δt = 2.0 (fs), Nmc = 200 000 steps, Nmd/mc = 2 steps, and Δt = 2.0 (fs), Nmc = 250 000 steps, Nmd/mc = 2 steps, and Δt = 1.8 (fs), Nmc = 250 000 steps, Nmd/mc = 1 steps for F−(H2O), F−(H2O)2, and F−(H2O)3, respectively. Potential energies and its gradients are evaluated at each step with RIMP215,16/aug-cc-pVDZ level using the TURBOMOLE package.17 We later denote the simulations assumed to be quantum (PIHMC) and classical (HMC) nuclei as “quantum” and “classical” simulations. The acceptance ratio was 71%, 81%, 55%, 71%, 44%, and 58% for quantum F−(H2O), classical F−(H2O), quantum F−(H2O)2, classical F−(H2O)2, quantum F−(H2O)3, and classical F−(H2O)3, respectively. For convenience, protons hydrogen bonded with fluoride ion are labeled as “H*”.

Figure 1. Equilibrium structures of (a) F−(H2O), (b) F−(H2O)2, and (c) F−(H2O)3.

Table 1 clearly shows that ROH* shortens as the coordination number of water molecules increases, while RFH* and RFO elongate. The angle of θH*FO enlarges as the number of water molecules increases. Such shift of ROH* with increase of coordination number was also found in the previous ab initio MO calculations.5,6 The difference of these values by adding one water molecule, shown in parentheses in Table 1, decreases as the number of water molecules increases. This is consistent with the tendency found in the interaction energies between F− or F−(H2O)n and additional water molecule. This is also consistent with the harmonic vibrational frequency values in Table 2, where the frequency of O−H* stretching mode becomes higher while those of F −···O intermolecular stretching and F···H*O intermolecular bending modes become smaller as the number of water molecules increases. 3-B. Classical and Quantum Simulations. Table 1 lists the average values of OH* bond length ⟨ROH*⟩, F−···H* length ⟨RFH*⟩, F−···O length ⟨RFO⟩, and H*FO angle ⟨θH*FO⟩ for F−(H2O)n (n = 1−3) with classical and quantum simulations, respectively. For the classical simulation, the average values of each bond length for each cluster are longer, and the average hydrogen-bonded angles ⟨θH*FO⟩ are larger than those of the equilibrium structures due to the thermal structural fluctuations under the anharmonic potential. For all simulations, the average value of the covalent ⟨ROH*⟩ shortens as the coordination number of water molecules increases, while those of ⟨RFH*⟩ and ⟨RFO⟩ elongate. Average values ⟨θH*FO⟩ enlarge as the number of water molecules increases. The difference of these values by adding one water molecule decreases as the number of water molecules increases. These tendencies agree with the results obtained from static MO calculation. We further examine the hydrogen-bonded structural variation with respect to the coordination number of water molecules. Figure 2 shows the one-dimensional distribution of ROH* for classical and quantum F−(H2O)n (n = 1−3). The peak positions are shifted to the shorter bond length region and the distribution becomes more localized, as the number of water molecules increases for each case. Figures 3 and 4 describe a one-

3. RESULTS AND DISCUSSION 3-A. Static MO Calculation. Figure 1 illustrates the optimized equilibrium structures for F−(H2O)n (n = 1−3). The optimized equilibrium structures obtained from our static MO calculations agree with the global minimum structure based on total energy with BSSE correction obtained by Kim et al.6 Our structures for F−(H2O), F−(H2O)2, and F−(H2O)3 correspond to 1(Cs), 2(C2), and 3(C3) structures in their article. The interaction energies between F− or F−(H2O)n and one additional water molecule for F−(H2O), F−(H2O)2, and F−(H2O)3 are 26.8 kcal/mol, 21.2 kcal/mol, and 18.8 kcal/mol with RIMP2/aug-cc-pVDZ level, respectively. Such tendency shows that the charge-induced dipole interaction between fluoride ion and water molecule weakens as the coordination number of water molecules increases. We next focus on the variation of the hydrogen-bonded structures with respect to the coordination number of water molecules. Table 1 lists the values of equilibrium structures for OH* bond length ROH*, F−···H* length RFH*, F−···O length RFO, and H*FO angle θH*FO in F−(H2O)n (n = 1−3), respectively, obtained from static MO calculation. The difference of these values between F−(H2O)n and F−(H2O)n−1 are also shown in parentheses. Table 2 shows the harmonic vibrational frequency values for O−H* stretching, O−H stretching, F−···O intermolecular stretching, F···H*O intermolecular bending, and O···F···O intermolecular bending modes. B



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Table 1. Average Distance and Those Statistical Errors of ROH*, RFH*, RFO, θH*FO for Equilibrium Structure, Classical F−(H2O)n and Quantum F−(H2O)n and Difference of Average Valuesa equilibrium F−(H2O) F−(H2O)2 F−(H2O)3 F−(H2O) F−(H2O)2 F−(H2O)3 F−(H2O) F−(H2O)2 F−(H2O)3 F−(H2O) F−(H2O)2 F−(H2O)3

⟨ROH*⟩

⟨RFH*⟩

⟨RFO⟩

⟨θH*FO⟩

a

1.056 1.019 1.000 1.414 1.529 1.632 2.469 2.544 2.613 1.3 2.5 5.3

classical 1.063(0) 1.022(0) 1.005(1) 1.431(2) 1.566(3) 1.654(6) 2.482(2) 2.569(2) 2.634(6) 4.2(0) 4.8(1) 5.2(1)

(−0.037) (−0.019) (0.115) (0.104) (0.075) (0.070)

(−0.042) (−0.017) (0.135) (0.088) (0.087) (0.065)

(−0.066) (−0.022) (0.175) (0.118) (0.101) (0.088)

Difference between F−(H2O)n and F−(H2O)n−1 are written in parentheses.

where the superscripts (eq), (cl), and (qm) stand for equilibrium geometry, classical and quantum simulations, respectively. The relation is due to the quantum nuclear fluctuation under the anharmonic potential. The OH* distributions in the quantum simulation are highly delocalized by the quantum effect as shown in Figure 2(a). These results indicate that the nuclear quantum effect delocalizes the hydrogen atom within the hydrogen bond. The relation for ⟨θH*FO⟩ is found to be

Table 2. Harmonic Vibrational Frequencies for O−H* Stretching, O−H stretching, F−···O Intermolecular Stretching, F···H*O Intermolecular Bending, and O···F···O Intermolecular Bending Modes for F−(H2O)na

F−(H2O) F−(H2O)2 F−(H2O)3

a

quantum 1.114(1) 1.048(1) 1.026(1) 1.377(4) 1.552(7) 1.67(1) 2.465(2) 2.566(5) 2.654(8) 6.5(0) 6.6(1) 7.4(3)

O−H* str.

O−H str.

F−···O str.

F···H*O bend.

2258 2774 2975 3164 3166 3321

3878 3874 3882 3838 3840 3840

384 310 347 249 249 338

562 511 529 478 479 514

O···F···O bend. 31

(qm) (cl) θH(eq) * FO < ⟨θH * FO⟩ < ⟨θH * FO⟩

59 59 71

which indicates that nuclear quantum effect tends to bend the fluoride ion−water hydrogen-bonded structure. The relations among the average values for the F−···H* and − F ···O lengths shown in Table 1 are found to be

Units in cm−1.

(qm) (eq) (cl) ⟨RFH * ⟩ < R FH * < ⟨R FH *⟩

dimensional distribution as a function of RFH* and RFO, respectively. The peak positions are linearly shifted to the longer bond length region and the distribution becomes more delocalized as the coordination number of water molecules increases. Consequently, these results, again, reflect the results of static ab initio MO calculations as shown in Tables 1 and 2. 3-B-1. F−(H2O). The relation among the average values for OH* bond length shown in Table 1 is found to be (eq) (qm) (cl) R OH * < ⟨R OH *⟩ < ⟨R OH * ⟩

(2)

(3)

and (qm) (eq) (cl) ⟨RFO ⟩ < RFO < ⟨RFO ⟩

(4)

respectively. The tendency of these lengths among static calculation and simulations is the opposite seen in the OH* covalent bond of eq 1. The F−···H* distributions in the quantum simulation are highly delocalized while those of F−···O show not much difference, as shown in Figures 3(a) and 4(a). This

(1)

Figure 2. One-dimensional distributions of OH* distance for (a) F−(H2O)3, (b) F−(H2O)2, and (c) F−(H2O)3 with classical (black) and quantum (red) simulations. Vertical lines correspond to the equilibrium values. C



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Figure 3. One-dimensional distributions of F−···H* distance for (a) F−(H2O), (b) F−(H2O)2, and (c) F−(H2O)3 with classical (black) and quantum (red) simulations. Vertical lines correspond to the equilibrium values.

Figure 4. One-dimensional distributions of F−···O distance for (a) F−(H2O), (b) F−(H2O)2, and (c) F−(H2O)3 with classical (black) and quantum (red) simulations. Vertical lines correspond to the equilibrium values.

Figure 5. One-dimensional distributions of O···O distance for (a) F−(H2O)2 and (b) F−(H2O)3 with classical (black) and quantum (red) simulations. Vertical lines correspond to the equilibrium values.

indicates the effect of nuclear quantum fluctuation mainly arises in bond lengths including hydrogen atoms. 3-B-2. F−(H2O)2. The relation among the average values for the OH* bond length of two water cluster, F−(H2O)2, shown in Table 1 is found to be

(eq) (qm) (cl) R OH * < ⟨R OH *⟩ < ⟨R OH * ⟩

(5)

respectively. This relation and the large delocalization of the OH* distribution found in Figure 2(b) follows the tendency as shown in F−(H2O). The relation for ⟨θH*FO⟩, D



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(qm) (cl) θH(eq) * FO < ⟨θH * FO⟩ < ⟨θH * FO⟩

while those of F−···O show not much difference as shown in Figures 3(c) and 4(c). Strictly speaking, it is interesting that the distributions longer than 3.0 Å in F−···O distance are found in the quantum simulation in Figure 5(a). To discuss the origin of the inversion in the relations for ⟨RFO⟩ and ⟨RFH*⟩ with respect to the coordination number of water molecules, or in other words, the difference in the relations for ⟨RFO⟩ and ⟨RFH*⟩ between F−(H2O) and F−(H2O)3, we show the distribution of ROO between two water molecules in F−(H2O)3 in Figure 5(b). In Figure 5(b), the classical simulation of F−(H2O)3 shows a broad distribution in a wide range of ROO distance, while the quantum simulation shows localized distribution, as found in F−(H2O)2 in Figure 5(a). In the case of F−(H2O)3 cluster, the distribution of the quantum simulation shows one sharp peak within the range of 2.8−3.7 Å around the equilibrium position of 3.25 Å. Especially, the distribution of the quantum simulation below 3.0 Å is greater than that of the classical simulation, which indicates that the nuclear quantum effect enhances the formation of water−water hydrogen bonds. To make the difference between classical and quantum simulations clear, the snapshot structures of the two simulations are illustrated in Figure 6. Hydrogen atoms of water, which bind

(6)

also follows the tendency seen in F−(H2O), although the difference of ⟨θH*FO⟩ among the two simulations are smaller compared to F−(H2O). The relations among the average values for the F−···H* and − F ···O lengths shown in Table 1 are found to be (eq) (qm) (cl) RFH * < ⟨R FH * ⟩ < ⟨R FH *⟩

(7)

and (eq) (qm) (cl) RFO < ⟨RFO ⟩ < ⟨RFO ⟩

(8)

respectively. These intermolecular bond lengths show the same tendency; however, the difference between two simulations is very small compared to that in F−(H2O). The F−···H* distributions in the quantum simulation are highly delocalized, while those of F−···O do not show much difference as shown in Figures 3(b) and 4(b). Despite the difference found in the average bond length values, the nuclear quantum fluctuation of hydrogen atoms are similar as found in F−(H2O), that is, the distributions show similar tendency. We next analyzed the hydrogen-bond length between two oxygen atoms, ROO, in each water molecule, to seek the water− water interaction for F−(H2O)2, which is not present in F−(H2O). Figure 5 shows the one-dimensional distribution as a function of ROO with the quantum and classical simulations. For F−(H2O)2 in Figure 5(a), the classical simulation shows a broad distribution from 3.0 Å to 5.5 Å, while the distributions of the quantum simulation are much more localized (where the equilibrium position is 3.64 Å). This result implies that the fluctuation of hydrogen-bonded water molecules in F−(H2O)2 is somewhat suppressed by the nuclear quantum effect. However, we should note here that most configurations do not form a hydrogen bond in the two water molecular case of F−(H2O)2, since few distributions below 3.0 Å are found in Figure 5(a). 3-B-3. F−(H2O)3. The relation among the average values for the OH* bond length of the three water cluster, F−(H2O)3, shown in Table 1 is found to be (eq) (qm) (cl) R OH * < ⟨R OH *⟩ < ⟨R OH * ⟩

Figure 6. Snapshot structures of F−(H2O)3 obtained from classical and quantum simulations: (a) top-view and (b) side-view of snapshot structure from the classical simulation, and (c) top-view and (d) sideview of snapshot structure of the centroids for each atom from the quantum simulations.

(9)

This relation and the large delocalization of OH* distribution found in Figure 2(c) follows the tendency as seen in F−(H2O)n and F−(H2O)2. The relation for ⟨θH*FO⟩ is (qm) (cl) θH(eq) * FO ≤ ⟨θH * FO⟩ < ⟨θH * FO⟩

with F−, point toward the ion in both simulations. On the other hand, other hydrogen atoms do not interact with other water molecules for the classical simulation. In the quantum simulation, however, some hydrogen atoms point toward an oxygen atom of water, as shown in Figure 6(c) and (d). Water molecules in the classical simulation have negligible interaction with each other and fluctuate largely, while water molecules with water−water interaction in the quantum simulation develop certain structures. This results in a localized distribution for ROO in the quantum simulation. The emergence of such nuclear quantum effect on water− water hydrogen bonds leads to a decrease of the nuclear quantum effect on F−−water hydrogen bonds, which results in elongation of ⟨RFO⟩ and ⟨RFH*⟩ as shown in Figures 3(c) and 4(c). The hydrogen-bonded oxygen atom to fluoride ion is drawn toward the oxygen atom of the other water molecule because of the nuclear quantum effect on water−water hydrogen bonds. As found in the case of F−(H2O), the nuclear quantum effect on the fluoride ion−water hydrogen bond draws the oxygen atom toward F−. The nuclear quantum effect on the different hydrogen bonds, F−−water and water−water hydrogen bonds, causes them

(10) −

−

which also follows the tendency seen in F (H2O) and F (H2O)2, while the difference among the classical and the quantum simulations is larger than those of F−(H2O) and F−(H2O)2. The relations among the average values for the F−···H* and F−···O lengths shown in Table 1 are found to be (eq) (qm) (cl) RFH * < ⟨R FH *⟩ < ⟨R FH * ⟩

(11)

and (eq) (cl) (qm) RFO < ⟨RFO ⟩ < ⟨RFO ⟩

(12)

respectively. The order of these intermolecular bond lengths shows the opposite tendency found in both F−(H2O) and F−(H2O)2. Average values in the quantum simulation are shorter than those in the classical one for F−(H2O), where those relations are the opposite for F−(H2O)3. The F −···H* distributions in the quantum simulation are highly delocalized, E



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to compete against each other. Our results suggest that the competition of the two different hydrogen bonds is the origin of the inversion of the relation for ⟨RFO⟩ and ⟨RFH*⟩ as the coordination number of water molecules increases from F−(H2O) to F−(H2O)3.

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(1) Jeffrey, G. A. An Introduction to Hydrogen Bonding; Oxford University Press: New York, 1997. (2) Meot-Ner, M. The Ionic Hydrogen Bond. Chem. Rev. 2005, 105, 213−284. (3) Robertson, W. H.; Johnson, M. A. Molecular Aspects of Halide Ion Hydration: The Cluster Approach. Annu. Rev. Phys. Chem. 2003, 54, 173−213. (4) Robertson, W. H.; Diken, E. G.; Price, E. A.; Shin, J.-W.; Johnson, M. A. Spectroscopic Determination of the OH− Solvation Shell in the OH−•(H2O)n Clusters. Science 2003, 299, 1367−1372. (5) Xantheas, S. S.; Dunning, T. H. Structures and Energetics of F−(H2O)n, n = 1−3, Clusters from ab initio Calculations. J. Phys. Chem. 1994, 98, 13489−13497. (6) Kim, J.; Lee, H. M.; Suh, S. B.; Majumdar, D.; Kim, K. S. Comparative ab initio Study of the Structures, Energetics and Spectra of X−(H2O)n=1−4 [X = F,Cl,Br,I] Clusters. J. Chem. Phys. 2000, 113, 5259− 5272. (7) Chaban, G. M.; Xantheas, S. S.; Gerber, R. B. Anharmonic Vibrational Spectroscopy of the F−(H2O)n Complexes, n = 1, 2. J. Phys. Chem. A 2003, 107, 4952−4956. (8) Horvath, S.; McCoy, A. B.; Roscioli, J. R.; Johnson, M. A. Vibrationally Induced Proton Transfer in F−(H2O) and F−(D2O). J. Phys. Chem. A 2008, 112, 12337−12344. (9) Xantheas, S. S.; Dang, L. X. Critical Study of Fluoride−Water Interactions. J. Phys. Chem. 1996, 100, 3989−3955. (10) Xantheas, S. S. On the Importance of the Fragment Relaxation Energy Terms in the Estimation of the Basis Set Superposition Error Correction to the Intermolecular Interaction. J. Chem. Phys. 1996, 104, 8821−8824. (11) Weis, P.; Kemper, P. R.; Bowers, T. M.; Xantheas, S. S. J. Am. Chem. Soc. 1999, 121, 3531−3532. (12) Xantheas, S. S. Quantitative Description of the Hydrogen Bonding in Chloride−Water Clusters. J. Phys. Chem. 1996, 100, 9703− 9713. (13) Cabarcos, O. M.; Weinheimer, C. J.; Lisy, J. M.; Xantheas, S. S. Microscopic hydration of the Fluoride Anion. J. Chem. Phys. 1999, 110, 5−8. (14) Suzuki, K.; Tachikawa, M.; Shiga, M. Efficient ab initio Path Integral Hybrid Monte Carlo Based on the Fourth−Order Trotter Expansion: Application to Fluoride Ion−Water Cluster. J. Chem. Phys. 2010, 132, 144108. (15) Weigend, F.; Häser, M. RI−MP2: First Derivatives and Global Consistency. Theor. Chim. Acta 1997, 97, 331−340. (16) Weigend, F.; Häser, M.; Patzelt, H.; Ahlrichs, R. RI−MP2: Optimized Auxiliary Basis Sets and Demonstration of Efficiency. Chem. Phys. Lett. 1998, 294, 143−152. (17) Ahlrichs, R.; Bär, M.; Häser, M.; Horn, H.; Kölmel, C. The Electronic Structure Calculations on Workstation Computers: The Program System Turbomole. Chem. Phys. Lett. 1989, 162, 165−169. (18) Dorsett, H. E.; Watts, R. O. Probing Temperature Effects on the Hydrogen Bonding Network of the Cl−(H2O)2 Cluster. J. Phys. Chem. A 1999, 103, 3351−3355. (19) Craig, I. R.; Manolopoulos, D. E. Quantum Statistics and Classical Mechanics: Real Time Correlation Functions from Ring Polymer Molecular Dynamics. J. Chem. Phys. 2004, 121, 3368−3373. (20) Shiga, M.; Nakayama, A. Ab initio Path Integral Ring Polymer Molecular Dynamics: Vibrational Spectra of Molecules. Chem. Phys. Lett. 2008, 451, 175−181.

4. CONCLUSIONS We have performed path integral hybrid Monte Carlo simulation to evaluate the nuclear quantum effect on hydrogen bonds for fluoride ion−water clusters (F−(H2O)n n = 1−3) with respect to the coordination number of water molecules. In our simulation, ROH* shortens as the coordination number of water molecules increases, while RFH* and RFO elongate. The difference of these values by adding one water molecule decreases as the number of water molecules increases. These tendencies agree with the results obtained from static MO calculations. The relation among the average bond lengths of covalent (cl) (qm) bonded ROH* follows in the order of R(eq) OH* < ⟨ROH*⟩ < ⟨ROH* ⟩ for all coordination number of water molecules, because of the quantum fluctuation under the anharmonic potential. In the case of F−(H2O), the average of RFO in the quantum simulation is shorter than that in the classical one, while the relation was opposite in the case of F−(H2O)3. In the case of F−(H2O)3 we have confirmed that the nuclear quantum effect enhances the formation of hydrogen bonds between two water molecules and the oxygen atoms are drawn toward each other. This increases the F−−water hydrogen-bond length. On the other hand, the nuclear quantum effect on F−−water hydrogen bonds shortens the bond length as found in the case of F−(H2O). Thus, the nuclear quantum effect on the different hydrogen bonds, F−− water and water−water hydrogen bonds, causes them to compete against each other. This suggests that that the competition of these hydrogen bonds is the origin of the inversion of the relation for ⟨RFO⟩ and ⟨RFH*⟩ as the coordination number of water molecules increases from F−(H2O) to F−(H2O)3. For further understanding, comparison with observed IR spectra is essential as studied by Dorsett et al. for Cl−(H2O)2.18 Observed IR spectra in a comparable temperature with simulation are available.13 We plan to carry out ring polymer molecular dynamics19 simulations for ion−water cluster systems for this purpose. Ab initio ring polymer molecular dynamics20 allows us to construct IR spectra accurately from the time-correlation function of a real time simulation. Competition due to nuclear quantum effect on the different kinds of hydrogen bonds should be indispensable for the quantitative analysis for systems including different hydrogen-bonds such as ion liquid-water systems, or simulating biomolecules functioning in water such as proton relay in bacteriorhodopsin.

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Author Contributions †

Yukio Kawashima and Kimichi Suzuki contributed equally to this work. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The present study has been supported in part by a Grant-in-Aid for Scientific Research and for the Priority Area by the Ministry of F


Ab Initio Path Integral Simulations for the Fluoride IonâWater Clusters

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Ab Initio Path Integral Simulations for the Fluoride IonâWater Clusters