Rearrangements and tunneling splittings of protonated water dimer

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1 JOURNAL OF CHEMICAL PHYSICS VOLUME 110, NUMBER 21 1 JUNE 1999 Rearrangements and tunneling splittings of protonated water dimer David J. Wales University Chemical Laboratories, Lensfield Road, Cambridge CB2 1EW, United Kingdom Received 18 December 1998; accepted 25 February 1999 Stationary points and rearrangement mechanisms are characterized for protonated water dimer with a variety of basis sets and both Møller Plesset and density functional theory to describe electron correlation. The results are consistent at each level of theory and suggest that this molecule will exhibit interesting tunneling splitting patterns which should be resolvable experimentally American Institute of Physics. S I. INTRODUCTION Far-infrared vibration-rotation tunneling FIR-VRT spectroscopy 1 5 has sparked renewed interest in the structure and dynamics of neutral water clusters. The promise of new results for protonated water clusters 6 with similar resolution of around 1 MHz (310 5 cm 1 ) will present a further challenge to theoreticians. In the present paper a number of rearrangement pathways are characterized for the protonated water dimer, and the resulting tunneling splitting patterns are predicted. The theoretical framework employed is identical to the procedure described in previous accounts of rearrangements and tunneling splittings in neutral water clusters Previous experimental work 14,15 includes results with a resolution of about 300 MHz (0.01 cm 1 ) 16 for spectra in the OH antisymmetric stretch region around 3700 cm 1. Some effort was made in the latter study to interpret the results in terms of tunneling splittings, but the analysis was severely hampered because the tunneling paths could only be guessed at. There have been a number of previous theoretical studies of this system, 17 31,63 and Valeev and Schaefer have recently characterized the two lowest-lying stationary points using coupled cluster methods based upon a Bruekner reference determinant. 32 Their results serve as a useful benchmark for the present calculations. A complete analysis of the reaction graph in terms of permutational isomers and their interconnections is needed to deduce the effective molecular symmetry MS group. 33 In the present account we adopt the notation of Bone et al. 34 where a structure is understood to mean a particular molecular geometry and a version is a particular labeled permutational isomer of a given structure. Versions which are directly connected by a single transition state are said to be adjacent with respect to the corresponding mechanism. Rearrangements which produce observable tunneling splittings are termed feasible. 33 Significant tunneling splittings are most likely for degenerate rearrangements 35 which link permutational isomers of the same structure via a single transition state, i.e., adjacent versions. Here we follow Murrell and Laidler s definition of a transition state as a stationary point with a single negative Hessian eigenvalue. 36 All the rearrangements found in the present work are symmetric degenerate rearrangements 37 where the two sides of the path are related by a reflection or a C 2 rotation. II. GEOMETRY OPTIMIZATION AND PATHWAYS All the stationary points reported in the following section were located using eigenvector-following using the scheme described previously for water pentamer. 8 Analytic first and second derivatives of the energy were used at every TABLE I. Energies/hartree (1 h J and point groups at various levels of theory for the protonated water dimer global minimum and the four transition states described in the text. The zero-point energy of the minimum is also given, along with the energy separation of each transition state at the same level of theory, E the value in brackets includes harmonic zero-point terms. group DZPdiff DZPdiff aug cc pvdz aug cc pvtz minimum C ZPE/ cm H 2 O inversion C s E/ cm 1 80(42) 95(67) 118(11) 155(4) internal rot 1 C 2h E/ cm 1 284(231) 267(237) 222(172) 212(171) internal rot 2 C 2v E/ cm 1 545(316) 496(335) 409(286) 410(293) bifurcation C s E/ cm (4477) 4682(5140) 4510(4979) 4555(5055) /99/110(21)/10403/7/$ American Institute of Physics

2 10404 J. Chem. Phys., Vol. 110, No. 21, 1 June 1999 David J. Wales TABLE II. Rotational constants/cm 1 of the C 2 minimum at various levels of theory. TABLE IV. Geometrical parameters of the C s symmetry transition state for monomer inversion. Bond lengths are in Å, angles in degrees. DZPdiff DZPdiff aug cc pvdz aug cc pvtz C B A step and were generated with the CADPAC program. 44 No symmetry constraints were applied. Pathways were calculated by taking small displacements of order 0.01 a 0 away from a transition state parallel or antiparallel to the transition vector, and then employing eigenvector-following energy minimization to find the associated minimum. The atomic unit of distance, a 0, is m. The pathways obtained by this procedure have been compared to steepestdescent paths and pathways that incorporate a kinetic metric 45 in previous work the mechanism is usually found to be represented correctly. 46,47 Three basis sets were considered. The first, DZPdiff, is based upon a double- Refs. 48,49 plus polarization DZP basis, with polarization functions consisting of a single set of p functions on each hydrogen atom exponent 1.0, and a set of six d functions on each oxygen atom exponent 0.9. To these functions were added a diffuse s function on each hydrogen atom exponent and diffuse s and p functions on each oxygen atom exponents and for s and p, respectively, 50 giving a total of 70 functions for H 5 O 2. We also employed the standard aug cc pvdz and aug cc pvtz basis sets, 51,52 with a total of 95 and 235 basis functions, respectively. Correlation corrections were obtained through both second order Møller Plesset theory 53 and density functional theory DFT. The calculations were only possible with the DZPdiff basis due to disk space limitations. In the DFT calculations we employed the Becke nonlocal exchange functional 54 and the Lee Yang Parr correlation functional 55 together referred to as. Derivatives of the grid weights were not included and the core electrons were not frozen. Numerical integration of the functionals was performed using the CADPAC HIGH option. Calculations were deemed to be converged when the root mean-square gradient fell below 10 6 a.u. This tolerance is sufficient to reduce the six zero normal-mode frequencies TABLE III. Geometrical parameters of the C 2 symmetry global minimum see Fig. 1. Bond lengths are in Å, angles in degrees. Parameter DZPdiff DZPdiff aug cc pvdz aug cc pvtz d(o A -H 5 ) (O A -H 5 -O B ) d(h 2 -O A ) d(h 1 -O A ) (H 2 -O A -H 5 ) (H 1 -O A -H 5 ) Parameter DZPdiff DZPdiff aug cc pvdz aug cc pvtz d(o A -H 5 ) d(o B -H 5 ) (O A -H 5 -O B ) d(h 1 -O A ) d(h 2 -O A ) (H 2 -O A -H 5 ) (H 1 -O A -H 5 ) d(h 3 -O B ) (H 3 -O B -H 5 ) to less than 1 cm 1 in the calculations. Since derivatives of the grid weights were not included in the DFT calculations, the zero frequencies were sometimes as large as 50 cm 1. Using the scheme described previously for water pentamer 8 we encountered no numerical problems in the geometry optimizations and pathway calculations. The tighter binding of this protonated water cluster actually made the present investigation significantly easier than previous work on neutral water clusters. Pathways were calculated at the DZPdiff/, DZPdiff/, and aug cc pvdz/ levels of theory. The five stationary points were also optimized at the aug cc pvtz/ level. Three parameters are useful to describe the rearrangement mechanisms. The first is the integrated path length, S, calculated as a sum over eigenvector-following steps, m: S m Q m1 Q m, 1 where Q m is the 3n-dimensional position vector for n nuclei in Cartesian coordinates at step m. The second is the distance between the two minima in nuclear configuration space, D: DQsQ f, TABLE V. Harmonic frequencies/cm 1 for the five stationary points characterized in the present study at the DZPdiff/ level of theory. Minimum H 2 O inversion Internal rot 1 Internal rot 2 Bifurcation i 166i 220i 465i

3 J. Chem. Phys., Vol. 110, No. 21, 1 June 1999 David J. Wales TABLE VI. Harmonic frequencies/cm 1 for the five stationary points characterized in the present study at the aug cc pvtz/ level of theory. Intensities in km/mol are given in square brackets for the minimum. Minimum H 2 O inversion Internal rot 1 Internal rot 2 Bifurcation i 155i 182i 507i where Q(s) and Q( f ) are the 3n-dimensional position vectors of the minima at the start and finish of the path. The third is the moment ratio of displacement, 56, which gives a measure of the cooperativity of the rearrangement: n iq i sq i f 4 i Q i sq i f 2, 3 2 where Q i (s) is the position vector of atom i in starting minimum s, etc., and n is the number of atoms. If every atom undergoes the same displacement in one Cartesian component then 1, while if only one atom has one nonzero component then n. III. REARRANGEMENTS OF PROTONATED WATER DIMER One minimum and four transition states were found in the present study, as summarized in Table I. The rotational TABLE VII. Properties of the four pathways found for H 5 O 2 at three different levels of theory. S, D, and are all defined in Sec. II. Parameter DZPdiff DZPdiff aug cc pvdz monomer inversion S D internal rotation 1 S D internal rotation 2 S D bifurcation S D FIG. 1. Monomer inversion pathway for H 5 O 2 calculated at the aug cc pvdz/ level. constants of the C 2 minimum are given in Table II. The point group symmetries, Hessian indices, and rearrangement pathways are consistent at all levels of theory. This is a significant result given the difficulties encountered in previous work for H 5 O 2 see Ref. 32 for a summary. Geometrical parameters and harmonic frequencies for the C 2 minimum and the lower energy C s transition state are presented in Tables III VI for comparison with Valeev and Schaefer. 32 We generally find good agreement with the latter, more accurate results for these two stationary points, although the bond lengths calculated using DFT seem to be systematically larger than the coupled cluster values in Ref. 32. For the present purposes it is the Hessian indices of the stationary points and the permutations corresponding to the rearrangement mechanisms that are of primary importance. Path lengths and cooperativity indices are given in Table VII. There are a total of 22!5!/2240 distinct versions of the H 5 O 2 or D 5 O 2 ) global minimum, where the first factor accounts for the inversion operation and there are 2! and 5! permutations of the oxygen and hydrogen or deuterium atoms, respectively. We must divide by two to account for the order of the point group. 34 The largest possible MS group when disruption of the OH or OD terminal bonds is not feasible has order 22!(2!) 2 16, where the first factor again accounts for the inversion operation, the

4 10406 J. Chem. Phys., Vol. 110, No. 21, 1 June 1999 David J. Wales TABLE VIII. Character table for the group G(8), which is appropriate if the monomer inversion mechanism is feasible, and nuclear spin statistical weights for H 5 O 2 and D 5 O 2. G(8) E (12)(34) AB1423 AB1324 (34)* (12)* AB1324* AB1423* H 5 O 2 D 5 O A B E second factor accounts for permutation of the two oxygen nuclei, and the last term accounts for the permutation of the two hydrogen or deuterium atoms within each monomer. This group is isomorphic to that obtained for neutral water dimer by Dyke 57 and will be denoted G(16). In fact there are a number of analogies between the present system and the neutral water dimer, which was originally investigated by Dyke, Mack, and Muenter. 58 Rearrangements for the latter system have been presented elsewhere 11,12 building upon the characterization of a number of stationary points by Smith et al. 59 For H 5 O 2 and D 5 O 2 the largest number of distinct versions that can be interconverted without breaking covalent bonds is 16/28. The low barrier and short path length found for the monomer inversion pathway suggest that it should produce an observable tunneling splitting of order 1 cm 1 for H 5 O 2 and perhaps one or two orders of magnitude less for D 5 O 2. The pathway is illustrated in Fig. 1 and the corresponding generator permutation-inversion PI operation is (34)*. If this mechanism alone is feasible then versions are linked in sets of four and the resulting MS group, denoted G(8), has order 8. The character table and nuclear spin weights for H 5 O 2 and D 5 O 2 are given in Table VIII. Each version is adjacent to two others and the resulting splitting pattern, in a nearest-neighbor approximation, is isomorphous to the system of cyclobutadiene: FIG. 2. First internal rotation rearrangement pathway of H 5 O 2 calculated at the aug cc pvdz/ level. FIG. 3. Second internal rotation rearrangement pathway of H 5 O 2 calculated at the aug cc pvdz/ level.

5 J. Chem. Phys., Vol. 110, No. 21, 1 June 1999 David J. Wales TABLE IX. Character table for the group G(4), which is appropriate when internal rotation is feasible, and nuclear spin statistical weights for H 5 O 2 and D 5 O 2. G(4) E AB1423 (12)(34)* AB1324* H 5 O 2 D 5 O mi A 1, 0E, 2 mi B 2, 4 where mi is the tunneling matrix element for monomer inversion. The predicted intensity ratio is 14:12:14 for H 5 O 2 and 81:108:81 for D 5 O 2. Pathways for the mechanisms, denoted internal rotation 1 and 2, are illustrated in Figs. 2 and 3. The corresponding generator PI is (12)(34)* in both cases. These two mechanisms correspond to relative rotation of the terminal H 2 O units in opposite senses. The pathways are not equivalent, but connect the same two versions. If either pathway or both is feasible then the resulting MS group, denoted G(4), has four members and versions are linked in pairs. The characters are give in Table IX. The splitting pattern in the nearest-neighbor approximation is ir A 1, ir A 1, 5 where ir is the tunneling matrix element corresponding to internal rotation. The predicted intensity ratio is 20:20 for H 5 O 2 and 135:135 for D 5 O 2. If the monomer inversion and an internal rotation pathway are both feasible then the MS group is enlarged to G(16) see Table X. This is the largest MS group possible without disrupting the terminal OH bonds. The versions are connected in sets of eight, and the predicted splitting pattern in the nearest-neighbor approximation is: 2 mi ir, 2 mi ir B 2, ir E, ir E, 2 mi ir B 2, 2 mi ir. Each version is adjacent to three others, however the reaction graph still contains no odd-membered rings and the tunneling levels therefore occur in plus minus pairs. 60 Since the barriers for both internal rotation mechanisms are higher than for monomer inversion, and the path length is longer, the resulting tunneling splittings will probably be at least an order of magnitude smaller than for monomer inversion. We therefore expect each line to be split into a triplet of doublets with intensity ratio 2:12:6:6:12:2 for H 5 O 2 and 63:18:54:54:18:63 for D 5 O 2, but the doublet splitting may not be resolvable. The aug cc pvdz/ energy profiles for the above rearrangements are shown in Fig. 4. The final mechanism found in the present study has a significantly higher barrier because it involves a rearrangement of the OH bonds Fig. 5. The transition state is characterized by a bifurcated hydrogen-bonding pattern of the type found in donor tunneling for the neutral water dimer 11,12,59 and bifurcation tunneling in neutral water trimer 7,9 and pentamer. 8 The generator corresponding to Fig. 5is(35)*and the corresponding MS group contains 12 operations Table XI. On its own, this mechanism connects versions in sets of 6 and gives a splitting pattern analogous to the system of benzene: 2 b A, b E, b E, 2 b A, 6 where b is the tunneling matrix element for this bifurcation process. The predicted intensity ratio is 12:8:8:12 for H 5 O 2 and 63:72:72:63 for D 5 O 2. When combined with monomer inversion the MS group is enlarged to 240 elements, and when bifurcation is com- TABLE X. Character table for the group G(16), which is appropriate when both monomer inversion and internal rotation are feasible, and nuclear spin statistical weights for H 5 O 2 and D 5 O 2. G(16) E (12)(34) (12) 34 AB1423 AB1324 AB1423 AB1324 E* (12)(34)* (12)* 34* AB1423* AB1324* AB1423* AB1324* H 5 O 2 D 5 O A 2 B E A 2 B E

6 10408 J. Chem. Phys., Vol. 110, No. 21, 1 June 1999 David J. Wales FIG. 4. Rearrangement pathways of H 5 O 2 calculated at the aug cc pvdz/ level. The pathway for bifurcation is omitted for clarity. bined with either internal rotation the complete nuclear permutation inversion CNPI group 61 is obtained. Since the bifurcation mechanism is unlikely to be feasible further details of these groups are omitted. In their study, Yeh et al. guessed that the pathways corresponding to monomer inversion and internal rotation might be important. 16 They also correctly discounted mechanisms involving the shared proton. However, they proposed that the largest tunneling splittings would be caused by an effective C 2 rotation of a monomer unit, by analogy with the pathway that leads to the largest splittings in neutral water dimer. In fact, the latter mechanism actually corresponds to a methylamine-type process 62 with inversion of the acceptor monomer and internal rotation. 11,12 Nevertheless, Yeh et al. were still able to employ the G(16) group and tentatively assigned the groups of 12 R branches in their spectrum to two sets of overlapping tunneling levels, each consisting of six lines. 16 Some of their results appear to fit a doublet of triplets pattern better than the triplet of doublets splitting suggested above, but the signal to noise ratio precludes any firm conclusions. It is likely that the barrier heights have not yet converged in the present calculations, and hence the doublet of triplets pattern cannot be ruled out. FIG. 5. Bifurcation pathway of H 5 O 2 calculated at the aug cc pvdz/ level. IV. CONCLUSIONS The present calculations of rearrangement pathways in protonated water dimer suggest that the monomer inversion process should give rise to observable tunneling splittings in both H 5 O 2 and D 5 O 2. Internal rotation may also give rise to observable effects, particularly in H 5 O 2, and we have deduced the corresponding splitting patterns and intensity ratios. The bifurcation tunneling mechanism, which leads to scrambling of the shared proton, is subject to a much higher TABLE XI. Character table for the group G(12), which is appropriate when the bifurcation mechanism is feasible, and nuclear spin statistical weights for H 5 O 2 and D 5 O 2. G(12) E (253) 235 AB1423 AB1435 AB1425 AB14* AB14253* AB14235* (35)* 25* 23* H 5 O 2 D 5 O 2 A B E A B E

7 J. Chem. Phys., Vol. 110, No. 21, 1 June 1999 David J. Wales barrier. The results of high resolution spectroscopy in the far infrared should test these predictions in the near future. ACKNOWLEDGMENTS The author gratefully acknowledges the support of the Royal Society of London and the EPSRC, and fruitful discussions with Professor R. J. Saykally. 1 R. C. Cohen and R. J. Saykally, J. Phys. Chem. 94, N. Pugliano and R. J. Saykally, J. Chem. Phys. 96, R. J. Saykally and G. A. Blake, Science 259, K. Liu, J. D. Cruzan, and R. J. Saykally, Science 271, J. B. Paul, C. P. Collier, R. J. Saykally, J. J. Scherer, and A. O Keefe, J. Phys. Chem. 01, R. J. Saykally private communication. 7 D. J. Wales, J. Am. Chem. Soc. 115, D. J. Wales and T. R. Walsh, J. Chem. Phys. 105, T. R. Walsh and D. J. Wales, J. Chem. Soc., Faraday Trans. 92, D. J. Wales and T. R. Walsh, J. Chem. Phys. 106, D. J. Wales, in Advances in Molecular Vibrations and Collision Dynamics, edited by J. M. Bowman and Z. Bačić JAI, Stamford, 1998, Vol. 3, pp D. J. Wales, in Theory of Atomic and Molecular Clusters II, edited by J. Jellinek Springer-Verlag, Heidelberg, 1999, pp , available from the Los Alamos preprint server at URL D. J. Wales, in Recent Theoretical and Experimental Advances in Hydrogen-Bonded Clusters, edited by S. Xantheas Kluwer, Dordrecht, in press, available from the Los Alamos preprint server at URL xxx.lanl.gov/abs/physics/ , L. I. Yeh, M. Okumura, J. D. Meyers, J. M. Price, and Y. T. Lee, J. Chem. Phys. 91, M. Okumura, L. I. Yeh, J. D. Myers, and Y. T. Lee, J. Phys. Chem. 94, L. I. Yeh, Y. T. Lee, and J. T. Hougen, J. Mol. Spectrosc. 164, P. A. Kollman and L. C. Allen, J. Am. Chem. Soc. 92, W. P. Kraemer and G. H. F. Diekersen, Chem. Phys. Lett. 5, M. D. Newton and S. Ehrenson, J. Am. Chem. Soc. 93, M. J. Frisch, J. E. Del Bene, J. S. Binkley, and H. F. Schaefer, J. Chem. Phys. 84, G. R. J. Williams, J. Mol. Struct.: THEOCHEM 138, G. E. Scuseria, A. C. Scheiner, T. J. Lee, J. E. Rice, and H. F. Schaefer, J. Chem. Phys. 86, C. Mijoule, Z. Latajka, and D. Borgis, Chem. Phys. Lett. 208, Y. Xie, R. B. Remington, and H. F. Schaefer, J. Chem. Phys. 101, D. Wei and D. R. Salahub, J. Chem. Phys. 101, H.-P. Cheng, R. N. Barnett, and U. Landman, Chem. Phys. Lett. 237, L. Ojamäe, I. Shavitt, and S. J. Singer, Int. J. Quantum Chem., Quantum Chem. Symp. 29, L. Kar and S. Scheiner, Int. J. Quantum Chem., Quantum Chem. Symp. 29, F. F. Muguet, J. Mol. Struct.: THEOCHEM 368, S. Klein, E. Kochanski, A. Strich, and A. Sadlej, J. Phys. Chem. 01, H.-P. Cheng, J. Phys. Chem. 02, E. F. Valeev and H. F. Schaefer, J. Chem. Phys. 108, H. C. Longuet-Higgins, Mol. Phys. 6, R. G. A. Bone, T. W. Rowlands, N. C. Handy, Mol. Phys. 72, R. E. Leone and P. v. R. Schleyer, Angew. Chem. Int. Ed. Engl. 9, J. N. Murrell and K. J. Laidler, Trans. Faraday Soc. 64, J. G. Nourse, J. Am. Chem. Soc. 102, C. J. Cerjan and W. H. Miller, J. Chem. Phys. 75, J. Simons, P. Jorgensen, H. Taylor, and J. Ozment, J. Phys. Chem. 87, D. O Neal, H. Taylor, and J. Simons, J. Phys. Chem. 88, A. Banerjee, N. Adams, J. Simons, and R. Shepard, J. Phys. Chem. 89, J. Baker, J. Comput. Chem. 7, J. Baker, J. Comput. Chem. 8, CADPAC, The Cambridge Analytic Derivatives Package, Issue 6, R. D. Amos, I. L. Alberts, J. S. Andrews, S. M. Colwell, N. C. Handy, D. Jayatilaka, P. J. Knowles, R. Kobayashi, K. E. Laidig, G. Laming, A. M. Lee, P. E. Maslen, C. W. Murray, J. E. Rice, E. D. Simandiras, A. J. Stone, M.-D. Su, and D. J. Tozer, Cambridge, A. Banerjee and N. P. Adams, Int. J. Quantum Chem. 43, T. R. Walsh and D. J. Wales, Z. Phys. D 40, G. N. Merrill and M. S. Gordon, J. Phys. Chem. 02, T. H. Dunning, J. Chem. Phys. 53, S. J. Huzinaga, J. Chem. Phys. 47, J. E. Fowler and H. F. Schaefer, J. Am. Chem. Soc. 117, T. H. Dunning, J. Chem. Phys. 90, R. A. Kendall, T. H. Dunning, and R. J. Harrison, J. Chem. Phys. 96, C. Møller and M. S. Plesset, Phys. Rev. 46, A. D. Becke, Phys. Rev. A 38, C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B 37, F. H. Stillinger and T. A. Weber, Phys. Rev. A 28, T. R. Dyke, J. Chem. Phys. 66, T. R. Dyke, K. M. Mack, and J. S. Muenter, J. Chem. Phys. 66, B. J. Smith, D. J. Swanton, J. A. Pople, H. F. Schaefer, and L. Radom, J. Chem. Phys. 92, C. A. Coulson and S. Rushbrooke, Proc. Cambridge Philos. Soc. 36, P. R. Bunker, Molecular Symmetry and Spectroscopy Academic, New York, M. Tsuboi, A. Y. Hirakawa, T. Ino, T. Sasaki, and K. Tamagake, J. Chem. Phys. 41, E. P. F. Lee and J. M. Dyke, Mol. Phys. 73,