Ionic dissociation of hydrogen bromide in water clusters: a computational study

19 February 1999

Chemical Physics Letters 301 Ž1999. 29–36

Ionic dissociation of hydrogen bromide in water clusters: a computational study Clinton Conley, Fu-Ming Tao

)

Department of Chemistry and Biochemistry, California State UniÕersity, Fullerton, CA 92834, USA Received 14 September 1998; in final form 28 December 1998

Abstract The ionic dissociation of hydrobromic acid in an aqueous environment is studied by density functional theory calculations on the water clusters HBrŽH 2 O. n , n s 1–4. The equilibrium structure, binding energies, vibrational frequencies, and dipole moments of the clusters are calculated in order to understand the mechanism of aqueous acid dissociation. These calculations indicate that HBr does not dissociate until at least three, preferably four, water molecules are positioned around the acid molecule. The Br–H bond grows progressively longer as more water molecules are added in the cluster, causing significant red-shift in its stretching frequency, along with drastic increase in the IR intensity. q 1999 Elsevier Science B.V. All rights reserved.

1. Introduction Hydrogen bromide, with water clusters of various sizes, can be used as a model to study the mechanism of acid dissociation in an aqueous environment. This dissociation occurs at the molecular level, and experimental data conclusively pointing towards the mechanism of dissociation are rare w1x. Consequently, quantum chemical calculations on the clusters involved might be more appropriate for the study of the protonation of HBr in water w2–4x. Chipot et al. w2x performed an early computational study of acid dissociation by means of the self-consistent reaction field approach. It studied the dissociation of HCl and HF in clusters with up to two water molecules, in the attempt to understand the effects of ) Corresponding author. Fax: q1 714 278 5316; e-mail: [email protected]

the environment on the reaction. It showed that, in a polar solvent such as water, the HCl P H 2 O exhibits two minima, one ionically dissociated. The ionic minimum, however, disappeared upon the inclusion of dispersion forces. In a moderately polar solvent Ž ´ s 10.0., HClŽH 2 O. 2 once again showed two minima, one dissociated. The study, however, did not explicitly show the effect of the solvent at the molecular level because it used a continuum model of the solution. The study also predicted that the hydronium ion ŽH 3 Oq. did not take part in the dissociation process, although many subsequent studies showed the existence of the hydronium ion. Lee et al. w3x extended the study of acid dissociation by searching for larger stable clusters of water and dissociated acid without employing a continuum model of the solution. The acids considered in the study were HF, HCl, and H 2 S. Their investigation led to a cluster with four water molecules supporting

0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 0 1 8 - 4

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C. Conley, F.-M. Tao r Chemical Physics Letters 301 (1999) 29–36

the dissociated acid molecule. Three water molecules served to stabilize the dissociation, and the fourth served as a proton acceptor, converting into the hydronium ion. They also showed that the non-dissociated form of the same cluster was, in fact, unstable for HCl, removing all doubt that the proton was indeed transferred in that cluster. Following the strategy, Re et al. w4x examined HCl in water clusters, HClŽH 2 O. n , n s 1–5. In their study, they concluded that the proton remains nontransferred in clusters with n s 1–3. The n s 4 cluster showed partial protonation, whereas the n s 5 cluster exhibits complete proton transfer. HBr is a stronger acid than HCl and a comparison between the clusters may prove very useful in determining the cause of hydrohalic dissociation. Here, we present a computational study of the hydrogen bromide–water clusters HBrŽH 2 O. n , n s 1–4. We wish to focus on the electrostatic interactions present in the clusters. The hydrogen-bonded network contains significant dipole interaction. We consider a thorough study of the varying dipole moments crucial to an understanding of the dissociation mechanism. However, the dipole moments are not the only consideration in dealing with electric fields. Special examination must be given to the individual roles of the polar bonds in the cluster, and equating the complex electrostatic interaction with the simple vector sum of dipole moments is an approximation at best. We also present the vibrational frequencies valuable for future experimental studies and provide an analysis of these frequencies for a further understanding of the acid dissociation mechanism.

2. Theoretical method The equilibrium geometries of the clusters HBrŽH 2 O. n , n s 1–4, were calculated using density functional theory with Becke’s three-parameter exchange potential and the Lee, Yang, and Parr correlation functional ŽB3LYP. w5–7x. This method was considered to be relatively efficient and reliable in previous studies on similar systems w4,8x. The equilibrium geometries of the clusters were also calculated at the second-order Møller–Plesset perturbation approximation level ŽMP2. w9,10x. The dipole moments and rotational constants were also obtained at

both the B3LYP and MP2 levels. Binding energies and vibrational frequencies of the clusters were calculated with the B3LYP method. All computations used the extended basis set 6-311 q q GŽd,p.. Diffuse functions were included in the basis set to give accurate accounts of molecular orbitals in regions distant from the nuclei. In order to confirm the basis set convergence of the calculations, additional B3LYP calculations are carried out using the basis sets 6-311 q GŽd. and 6-311 q q GŽ2d,p.. The GAUSSIAN 94 program package w11x was used for all computations.

3. Results and discussion Table 1 gives the calculated and the experimental geometrical parameters and dipole moments of the monomers involved in this study. It is clear that both the B3LYP and MP2 methods give geometries that are in good agreement with the experimental values. The calculations, however, have overestimated the dipole moments, compared to the experimental values. This is typical of the level of theory and basis set used, as the dipole moment is very sensitive to small inaccuracies in both charge distribution and relative position. The equilibrium structures for the stable clusters of HBrŽH 2 O. n are shown in Fig. 1, along with the corresponding binding energies. There are two types of these clusters: the almost planar ‘cyclic’ type, consisting of HBr and H 2 O arranged as if on vertices

Table 1 ˚ ., bond angle Ždegrees., and dipole moment Ž m . of Bond length ŽA HBr, H 2 O, and H 3 Oq Monomer

Parameter

HBr

r ŽBrH. m

H 2O

H 3 Oq

a

B3LYP

MP2

Expt.a

1.428 1.083

1.412 1.103

1.414 0.827

r ŽOH. /ŽHOH. m

0.962 105.1 2.159

0.959 103.5 2.187

0.958 104.5 1.854

r ŽOH. /ŽHOH. m

0.980 113.6 1.432

0.978 112.1 1.572

Experimental values are from Ref. w13x.

– – –

C. Conley, F.-M. Tao r Chemical Physics Letters 301 (1999) 29–36

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Fig. 1. The equilibrium structures of the clusters HBrŽH 2 O. n , n s 1–4. The numbers alongside the structures are the binding energies of the clusters, expressed in kcal moly1 . The black atoms are bromine, the dark gray are oxygen, and the light gray are hydrogen. The binding energy was calculated as the difference between the total energy of the cluster and the sum of the energies of the initial HBr and H 2 O monomers. All computations were performed at the B3LYP level.

of a regular polygon, and the ‘non-cyclic’ type, arrangements in multiple planes, or more complex than a simple cycle. This distinction is important in understanding the conditions under which hydrogen bromide dissociates. From the equilibrium structures in Fig. 1 and key geometrical parameters in Table 2, it appears that the first stable cluster containing the ionized species is

cluster 3b of HBrŽH 2 O. 3 . However, this is 2 kcal moly1 less stable than 3a, constituting approximately 3% of all the clusters with three water molecules at room temperature as estimated from the Boltzmann distribution. It does nevertheless show that HBr begins dissociation when surrounded by three water molecules. With four water molecules, the dissociation seems almost certain. The two most stable con-

C. Conley, F.-M. Tao r Chemical Physics Letters 301 (1999) 29–36

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Table 2 ˚ ., bond angles Ždegrees., rotational constants Ž A, B, C: GHz., and dipole moment Ž D . of clustersa Selected bond lengths ŽA Cluster

Theory

X r ŽBrH .

X r ŽOH .

X /ŽBrH O.

A

B

C

m

ns1

B3LYP MP2 b B3LYP MP2 B3LYP MP2 B3LYP MP2 B3LYP MP2 B3LYP MP2 B3LYP MP2

1.447 1.425 1.476 1.440 1.521 1.464 1.980 1.941 2.262 2.264 2.052 2.009 1.792 1.741

1.904 1.958 1.734 1.831 1.542 1.671 1.061 1.058 0.992 0.985 1.036 1.032 1.140 1.151

179.1 179.9 165.3 162.8 175.1 174.1 155.2 155.9 157.1 155.6 157.4 158.7 177.9 177.4

361.704 373.827 7.176 6.976 3.389 3.368 2.769 2.798 1.943 1.942 1.757 1.745 1.946 1.943

2.952 2.896 2.059 2.039 1.337 1.311 1.953 1.963 1.187 1.224 1.292 1.332 0.983 1.011

2.935 2.880 1.610 1.586 0.987 0.951 1.337 1.356 1.155 1.175 0.864 0.872 0.667 0.690

4.021 4.031 2.917 2.848 3.061 2.790 4.167 4.559 4.762 5.099 4.488 4.753 5.439 5.249

ns2 n s 3a n s 3b n s 4a n s 4b n s 4c a b

X

H is the proton involved in proton transfer. O is the oxygen intended to receive the proton. The MP2 dipole moments listed were actually calculated at the Hartree–Fock level.

figurations Ž4a, 4b. show the ionized species and the third, non-dissociated, cluster Ž4c. has a relative energy of 5.33 kcal moly1 above the most stable one. The structure of cluster 4a was similar to that of HClŽH 2 O.4 and other acid–water clusters previously studied, and was shown to indeed induce protonation of a variety of acids w3x. Further exploring the varying geometries of the water molecules, we note an interesting trend in the /ŽBrHX O. angle. Here, HX is the transferring proton originating from the HBr molecule, and O is the oxygen atom of the water molecule to accept the proton. Intuitively, it would seem that the larger the angle, or the more linear the alignment of the oxygen with the Br–HX bond, the more conducive the cluster is to proton transfer. However, the proton-transferred clusters Ž3b, 4a, and 4b. have the three lowest values for the angle. This shows that the action of the single water molecule accepting the proton does not determine the outcome of the interaction. Rather, the group of water molecules must be treated as a whole in order to understand the protonation mechanism. Also, there is no precise ‘breaking length’ of the Br–H bond indicated by the series of clusters. It ˚ judgseems to be somewhere between 1.8 and 2 A, ing by clusters 3b and 4c. However, cluster 4c seems extremely close to proton transfer, judging by its short BrH PPP O hydrogen bond length. Conse-

quently, The Br–H bond seems to reach its maxi˚ mum length at approximately 1.8 A. Both HBr and H 2 O are polar molecules. In HBr, the bromine is electronegative, causing the dipole to ‘point’ towards the hydrogen. Likewise, water’s dipole points from its oxygen to the point halfway between its two hydrogens. For the HBr P H 2 O cluster, the dipole–dipole interaction between H 2 O and HBr gives rise to a configuration of H 2 O PPP HBr in which a hydrogen bond forms between the water’s oxygen atom and HBr Žit is not planar due to the alignment of the oxygen atom’s lone pairs.. For the most part, these hydrogen bonds are the dominant intermolecular force present in the clusters. Theoretically, whichever isomer contains the most hydrogen bonds is the most stable. However, when the acid begins to dissociate, the net dipole moment of the water molecules becomes important. For hydrobromic acid, once the bromide anion and hydronium cation form, the overall dipole of the water molecules serves to stabilize the separation of ions. It is interesting to examine the electric field surrounding the HBr molecule created by the total dipole moment of the water molecules in the clusters. Generally, the total dipole moment of the cluster, shown in Table 2, seems to be greater in the dissociated molecules. However, this is not precisely accurate. Cluster 4c has the greatest dipole moment of the

C. Conley, F.-M. Tao r Chemical Physics Letters 301 (1999) 29–36

n s 4 series, yet it is the only one in which proton transfer does not occur! This is because the HBr molecule, or the Hq PPP Bry ion pair, exerts its own influence on the dipole of the cluster. Thus, it would be more accurate to examine only the portion of the dipole moment generated by the water molecules. We believe that the HBr ionic dissociation results from the existence of an electric field generated by the water molecules, which is strong enough to support the separation of the ionic pair Hq and Bry. In the ‘cyclic’ structures, the electric field acting on HBr contributed from the dipole moments of the water molecules is greatly reduced. The circular arrangement of the water molecules allows the individual water dipoles to negate each other. As a result, the electric field around HBr is too weak to stabilize the Hq PPP Bry ion pair. The ‘non-cyclic’ clusters, on the other hand, tend to have a stronger electric field enveloping the acid molecule, due to the unique orientation of the water molecules. At first glance, the dipole moments might not seem sufficiently aligned to merit dissociation of the acid molecule. In cluster 4a, for example, the three water molecules surrounding the Bry anion seem to have dipole moments facing outward from the axis of symmetry. This is because the molecules have a finite space to occupy, and so there is a competition between space and dipole alignment. Consequently, the entire separation of charge in the water molecule is not exploited. Rather, one polar bond from each water molecule is aligned with the

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axis of symmetry. The alignment allows for a direct electrostatic field to be formed, stabilizing the separation of the Bry and Hq ions. In addition, the close proximity of the bonds to the axis allows for a greater influence on the ions. The hydrogen atoms protruding radially from the cluster have, in fact, little influence on the cluster itself, despite the fact that they have a positive charge. To see this, imagine a cluster similar to 4a, but with the three stabilizing water molecules replaced by HF molecules. It is clear that the HF molecules would be aligned similarly to the the inner O–H bonds of the water molecules. The dipole moments of the molecules are important, but they must be considered in the context of the cluster. It is more important to thoroughly examine the the individual charges present and the role they play towards the stabilization of the ion pair. This is a major problem with the continuum model of solution. It represents solvent molecules solely by their dipole moments, and aligns them as such. Returning to cluster 4a, it would be ludicrous indeed for the water molecules to perfectly align their dipole moments with the ion pair! There simply is not enough space for this to happen, leading to water– water repulsion forces. It is therefore undesirable to represent the electrostatic interaction among water molecules solely as the vector sum of their respective dipole moments. It should be pointed out that the results in Table 2 and Fig. 1 are well converged with respect to the

Table 3 Selected cluster properties from B3LYP calculations with different basis sets a Cluster

Basis set

X r ŽBrH .

X r ŽOH .

A

B

C

m

De

3a

6-311 q GŽd. 6-311 q q GŽd,p. 6-311 q q GŽ2d,p.

1.562 1.521 1.533

1.461 1.542 1.506

3.465 3.389 3.401

1.407 1.337 1.406

1.010 0.987 1.002

3.503 3.061 3.007

y31.75 y25.69 y24.76

3b

6-311 q GŽd. 6-311 q q GŽd,p. 6-311 q q GŽ2d,p.

2.015 1.980 2.022

1.050 1.061 1.050

2.756 2.769 2.840

1.939 1.953 1.952

1.321 1.337 1.358

4.384 4.167 4.003

y29.88 y23.58 y23.42

4a

6-311 q GŽd. 6-311 q q GŽd,p. 6-311 q q GŽ2d,p.

2.302 2.262 2.271

0.988 0.992 0.993

1.941 1.943 1.933

1.169 1.187 1.193

1.144 1.155 1.160

4.309 4.763 4.131

y50.00 y40.93 y40.35

X H is the proton involved in proton transfer. O is the oxygen intended to receive the proton. The dissociation energy De Žin eV. is with respect to isolated water and HBr molecules. a

C. Conley, F.-M. Tao r Chemical Physics Letters 301 (1999) 29–36

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basis set. Table 3 compares some of the key properties of clusters 3a, 3b, and 4a calculated using the B3LYP method with three basis sets, 6-311 q GŽd., 6-311 q q GŽd,p., and 6-311 q q GŽ2d,p.. The values of the bond lengths, rotational constants, and dipole moments show reasonable consistency among the three basis sets, indicating the convergence of the results with the 6-311q q GŽd,p. basis set. The values of the binding energies Ž De . in the table exhibit a similar pattern of convergence among the three clusters. It should be noted that the Ž De . values do not include the effect of the basis set superposition error ŽBSSE.. The BSSE effect contributes to the overestimation of binding energy while such an effect decreases with the improvement of basis set. The De values with the 6-311 q GŽd. basis set show large deviations from those with the larger basis sets, suggesting that they may be severely overestimated due to the BSSE effect. On the other hand, the De values with the two larger basis sets are very close, suggesting that the De values are expected to be stable with respect to further increase of the basis set. Table 4 shows the calculated harmonic frequencies of the monomers considered in this study. They are in reasonable agreement with their experimental counterparts. These frequencies are expected to change with the formation of clusters with water molecules. Our calculated intramolecular harmonic frequencies, corresponding to the frequencies of the monomers, will not only serve as a valuable guide

Table 4 Vibrational frequencies Žcmy1 . and IR intensities ŽkM moly1 . of monomers Description

Expt. freq.a

Freq.

IR int

2577

6

2649

1603 3818 3923

67 9 57

1595 3657 3756

763 1673 3582 3678

518 121 40 499

79

H Br n 1 str. H 2O n 1 bend n 2 sym. str. n 3 antisym. str. H 3 Oq n 1 sym. deform. n 2 deg. deform. n 3 sym. str. n4 deg. str. a

Experimental frequencies are from Ref. w13x.

– – – –

Table 5 Vibrational frequencies Žcmy1 . and IR intensities ŽkM moly1 . of H 79 Br in clusters Cluster

n 1 str. freq.

ns 0 Žmonomer. ns1 ns 2 ns 3a ns 4c

2577 2367 2093 1565 1241

IR int 6 596 1248 1510 3283

for spectroscopic detection of the species, but will also reveal the changing strength of covalent bonds in the presence of water molecules in the clusters. Selected vibrational frequencies for the clusters are shown in Tables 5 and 6. Water shows a characteristic constancy in its frequencies throughout all of the clusters, so the corresponding frequencies are not listed. The HBr stretching frequency in Table 4 shows a large progressive decrease Žred-shift. along with a large increase in IR intensity. This agrees with previous calculations of other strong acids w12,13x. The red-shift indicates the weakening of the Br–H bond in the presence of an increasing number of water molecules in the clusters. This suggests that the water molecules serve to not only stabilize the ionized form of the cluster, but also induce the dissociation directly. The water molecules generate an electric field exerting a force on the acid molecule to cause ionic dissociation, overcoming the Coulombic attraction between the resulting positive and negative ions. In clusters 3b, 4a, and 4b, the HBr stretching frequency with its distinctly large IR intensity disappears, suggesting that the H–Br bond is broken. These clusters possess frequencies corresponding well with those of H 3 Oq, indicative of the formation of the hydronium ion as the product of proton transfer from the acid molecule. It should be noted, however, that there are large deviations between the isolated H 3 Oq frequencies and those assigned for H 3 Oq in the clusters. The primary reason for this observation lies with the strong interaction among H 3 Oq and neighboring H 2 O molecules. As an example, the degenerate stretching mode Ž n4 ., directly proportional to the covalent bond strength between the oxygen and one

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Table 6 Vibrational frequencies Žcmy1 . and IR intensities ŽkM moly1 . of H 3 Oq in clustersa Cluster

n 1 sym. deform.

n 2 deg. deform.

n 3 sym. str.

H 3 Oq Žpure. n s 3b

763 1428

518 711

3582 3118

40 517

n s 4a

1477

558

2927

922

n s 4b

1387

426

1673 1675 1716 1780 1797 1732 1761

3108

712

a

121 35 64 33 42 91 51

n4 deg. str. 3678 2239 3028 2630 2685 2475 2673

499 1974 1313 1383 1276 2310 1537

Vibrational frequencies are in upright font, IR intensities are italicized.

hydrogen, undergoes significant red-shift along with the enhancement in the IR intensity. This is understandable if we consider the possibility of proton transfer, or partial transfer, to a neighboring water molecule. The frequency would be reduced as the covalent bond is weakened, much like the phenomenon exhibited by the original HBr molecule. Because of the extensive proton-sharing between water molecules, the hydronium cation should more q correctly be identified as H 5 Oq 2 or even H 7 O 3 .

frequency calculations support all of these findings and reveal the structure of the hydronium ion in aqueous environments.

Acknowledgements This work was supported by the American Chemical Society Petroleum Research Fund ŽACS–PRF Grant a30399-GB6., The Research Corporation ŽCottrell College Science Award. and CSUF School of Science and Mathematics.

4. Conclusion We have studied the ionic dissociation of hydrogen bromide in water by calculating the equilibrium geometries, binding energies, dipole moments, and harmonic frequencies of the clusters HBrŽH 2 O. n , n s 1–4. The calculations have revealed that hydrogen bromide begins ionic dissociation in clusters with three water molecules, and is almost fully dissociated in clusters with four water molecules. Apparently, the interactions of all water molecules of the cluster contribute to the dissociation process. The bond length of HBr gradually increases with each successive water molecule added. The arrangement of the molecules is important as well; the non-cyclic structures are more conducive to protonation than are the cyclic. This results from the interaction among dipole moments of the water molecules. The ionic structure is formed and stabilized by the electric field generated by this interaction. However, the electrostatic interaction that develops is dependent not only on the dipole moments, but on the entire space-alignment competition that develops in the cluster. The

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