Rearrangements and tunneling splittings of protonated water trimer

JOURNAL OF CHEMICAL PHYSICS VOLUME, NUMBER 8 8 NOVEMBER 999 Rearrangements and tunneling splittings of protonated water trimer David J. Wales University Chemical Laboratories, Lensfield Road, Cambridge CB2 EW, United Kingdom Received 3 April 999; accepted August 999 Stationary points and rearrangement mechanisms are characterized for protonated water trimer using a variety of basis sets and density functional theory to describe electron correlation. For the largest basis sets there are three distinct low-lying minima separated in energy by only a few wave numbers. Ten distinct transition states were found with barriers spanning nearly three orders of magnitude. Several of these mechanisms should produce observable tunneling splittings. 999 American Institute of Physics. S002-9606993533-6 I. INTRODUCTION The promise of new experimental results for protonated water clusters from far-infrared vibration rotation tunneling FIR VRT spectroscopy 2 6 seems likely to spark a new wave of research in this field, as it did for neutral water clusters. The present work, and a preceding paper on H 5 O 2, 7 are intended to assist in assignment of these new spectra by characterizing rearrangement pathways and the associated tunneling splitting patterns. The magnitude of the splittings presents a difficult problem in quantum nuclear dynamics, and in the present work we only infer the likely feasibility of different rearrangements from the calculated barriers and path lengths. Although simplistic, this approach served to explain the observed splittings of water trimer 8 and predict those recently found for water pentamer. 9,0 The theoretical framework is identical to the procedure described in previous accounts of rearrangements and tunneling splittings in neutral 9, 6 and protonated 7 water clusters. Several experiments have previously probed the infrared spectrum of H 7 O 3 in the OH stretching region. 7 20 In particular, Okumura et al. confirmed the assignment by Schwarz of a band at 2670 cm to H 7 O 3. 20 Comparisons of the present results with previous ab initio calculations 2 30 are made in Sec. III. To deduce the molecular symmetry MS group 3 for a nonrigid molecule we must characterize the reaction graph for permutational isomerization, i.e., the transition states and pathways which connect the minima in question. As before, we adopt the notation of Bone et al. 32 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 that are directly connected by a single transition state are said to be adjacent with respect to the corresponding mechanism, and rearrangements which produce observable tunneling splittings are termed feasible. 3 The largest tunneling splittings are expected for degenerate rearrangements 33 which link permutational isomers of the same structure via a single transition state, i.e., adjacent versions. For the two largest basis sets three minima were found for H 7 O 3 in the present work, the lowest pair separated by a very small energy difference on the order of a few wave numbers, with two nondegenerate rearrangements linking them. The remaining eight transition states all mediate degenerate rearrangements, seven of which are symmetric with the two sides of the path related by a symmetry operation. 34 Throughout this paper we follow Murrell and Laidler s definition of a transition state as a stationary point with a single negative Hessian eigenvalue. 35 The reaction pathways are then defined by steepest-descent paths from the transition states which are, of course, independent of mass, temperature and coordinate system within the Born Oppenheimer approximation when defined properly in terms of covariant derivatives. 36,37 II. GEOMETRY OPTIMIZATION AND PATHWAYS As in our previous studies of water clusters all the stationary points were located using eigenvector-following 38 43 in Cartesian coordinates and the scheme described previously for water pentamer. 9 Analytic first and second derivatives of the energy were used at every step with no symmetry constraints and were generated with the CADPAC program. 44 Three basis sets were considered, as for the previous study of H 5 O 2. 7 The first, DZPdiff, is based upon a double- 45,46 plus polarization DZP basis, with polarization functions consisting of a single set of p orbitals on each hydrogen atom exponent.0 and a set of six d orbitals on each oxygen atom exponent 0.9. To these functions were added a diffuse s orbital on each hydrogen atom exponent 0.044 and diffuse s and p orbitals on each oxygen atom exponents 0.0823 and 0.065 for s and p, respectively, 47 giving a total of 02 functions for H 7 O 3. We also employed the standard aug-cc-pvdz and aug-cc-pvtz basis sets, 48,49 with a total of 38 and 340 basis functions, respectively. Correlation corrections were obtained through density functional theory DFT, since MP2 second derivative calculations were impossible due to disk space limitations. We employed the Becke nonlocal exchange functional 50 and the Lee Yang Parr correlation functional 5 together referred to as BLYP. Derivatives of the grid weights were not included and the core electrons were not frozen. Numerical integration of the BLYP functionals was performed using the CADPAC HIGH option. Calculations were deemed to be converged when the root-mean-square gradient fell below 20 6 atomic units. Since derivatives of the grid weights were not included the zero frequencies were sometimes as large as 50 002-9606/99/(8)/8429/9/$5.00 8429 999 American Institute of Physics

8430 J. Chem. Phys., Vol., No. 8, 8 November 999 David J. Wales cm. Pathways were calculated at the DZPdiff/BLYP and aug-cc-pvdz/blyp levels of theory and all thirteen stationary points were also optimized at the aug-cc-pvtz/blyp level. One refinement to the optimization scheme used for H 5 O 2 was necessary. It was impossible to calculate analytic second derivatives for the largest basis set due to program limitations. The aug-cc-pvtz/blyp geometries were therefore relaxed from those obtained at the aug-cc-pvdz/blyp level using the Murtagh Sargent Hessian update scheme 52 and an initial Hessian from the aug-cc-pvtz/blyp calculation. The convergence criteria were met within four or five eigenvector-following steps using the approximate Hessian, and the Hessian index did not change. 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 m Q m, 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, 2 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, 53, 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. If every atom undergoes the same displacement then, while if only one atom moves then n. III. REARRANGEMENTS OF PROTONATED WATER TRIMER Three minima and ten transition states were identified for the two largest basis sets Fig. and Table I. However, the higher-lying minimum, min3, and the transition state that links it to min were not located with the DZPdiff basis. The rotational constants of the three minima are given in Table II. The point group symmetries and Hessian indices are the same for equivalent stationary points with all three basis sets. Harmonic frequencies for the minima and transition states at the aug-cc-pvdz/blyp level are collected in Tables III and IV. Since the low frequency vibrations, in particular, are expected to be rather anharmonic, the harmonic frequencies and zero-point energies tabulated here should be interpreted with caution. Path lengths and cooperativity indices for each rearrangement are given in Table V. Pathways were first calculated by taking small displacements of order 0.05a 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. 54 The pathways obtained by this procedure have been compared to steepest-descent paths and pathways FIG.. Side views of the three minima found at the aug-cc-pvdz/blyp and aug-cc-pvtz/blyp levels of theory. that incorporate a kinetic metric 37 in previous work the mechanism is usually found to be represented correctly. 55,56 To test this result for the present system all the pathways were recalculated using the quadratic steepest-descent algorithm of Page and McIver. 57 The mechanisms were unchanged, except for the bifurcation mediated by tsb, described below, where a small perturbation changes one of the minima from min to min2 for the DZPdiff basis, and from min to min3 for the aug-cc-pvdz basis. min3 does not appear to exist for the smaller basis set. Aside from this result the values of the distance and cooperativity indices, D and, are practically identical for the eigenvector-following and steepest-descent paths. The integrated path length, however, is always somewhat shorter for the steepest-descent paths, typically by 0% 20%, and in one case by 40%. In view of these results all the tabulated data is for the steepestdescent paths, and all the illustrations are for the larger augcc-pvdz basis. In several previous studies a C 2v symmetry chain structure has been classified as a minimum. 2,23,24,29 In the present work this structure was found to be a transition state tsf2, in agreement with the MP2 calculations of Lee and Dyke. 26 Lee and Dyke also characterize a C s minimum which corresponds to min2 or min3. Two other studies have reported single minima which appear to have C symmetry. 27,30 Geissler et al. 30 also report two transition states for the Stillinger David potential 58 which correspond to tsrc and tsrc2 in the present study, and sampled dynamical transition paths for the corresponding mechanisms. They also report rotational exchange of protons on the terminal water monomers but did not characterize these processes further. The facile monomer inversion process corresponding to tsinv is shown in Fig. 2 and connects min and min2. The

J. Chem. Phys., Vol., No. 8, 8 November 999 Protonated water trimer 843 TABLE I. Energies/hartree Ref. 54 and point groups of the protonated water trimer minima and transition states for three different basis sets using the BLYP functional. Zero-point energies of the minima are also given, along with the energy separation of each stationary point from min at the same level of theory, E the value in brackets includes harmonic zero-point terms; the aug-cc-pvdz frequencies were used for the unknown aug-cc-pvtz zero-point energies. Group DZPdiff aug-cc-pvdz aug-cc-pvtz min C 229.674 703 229.645 58 229.73 497 ZPE/cm 7 622 7 499 7 499 min2 C s 229.674 672 229.645 502 229.73 486 E/cm 7(65) 4(39) 2(38) min3 C s 229.645 27 229.73 205 E/cm 54(39) 64(29) tsinv: monomer inversion C 229.674 652 229.645 42 229.73 367 E/cm (90) 23(46) 29(5) tsrot: internal rotation C 229.673 838 229.644 846 229.72 877 E/cm 90(32) 48(59) 36(48) tsf: flip C s 229.673 783 229.644 520 229.72 227 E/cm 202(242) 29(274) 279(333) tsf2: flip 2 C 2v 229.673 775 229.644 547 229.72 247 E/cm 204(392) 23(83) 274(244) tsf3: flip 3 C 2v 229.644 299 229.7 974 E/cm 268(55) 334(2) tsrc: ring closure C 2 229.664 640 229.635 435 229.703 632 E/cm 2208(999) 223(2004) 265(956) tsrc2: ring closure 2 C s 229.663 262 229.634 283 229.702 47 E/cm 25(2725) 2466(249) 2420(2445) tsb: bifurcation C 229.662 572 229.633 605 229.70 545 E/cm 2662(249) 265(2393) 2623(2402) tsb2: double bifurcation C s 229.645 50 229.67 20 229.685 252 E/cm 6409(6740) 625(6463) 699(6447) tsb3: bifurcation C s 229.64 025 229.62 807 229.680 52 E/cm 739(757) 779(74) 7237(7200) barriers are even smaller than for H 5 O 2. 7 Only one internal rotation mechanism was found in the present work, in contrast to two for H 5 O 2. 7 It also connects min and min2 and involves low barriers Fig. 3. Profiles for these two paths are shown in Fig. 4. Symmetric degenerate rearrangements corresponding to flipping of the unbound proton on the central oxygen were found for all three minima Fig. 5. These processes have relatively low barriers. The five remaining pathways all involve barriers that are an order of magnitude larger. The two mechanisms involving ring closure first found by Geissler et al. 30 are symmetric degenerate rearrangements of min in the present calculations Fig. 6. The rearrangements mediated by tsb, tsb2, and tsb3 all involve bifurcations, two in the case of tsb2. The barriers involved for tsb are similar to those for ring TABLE II. Rotational constants/cm of the three minima for the three different basis sets. DZPdiff aug-cc-pvdz aug-cc-pvtz C 0.0835 0.0834 0.0837 min B 0.0924 0.0925 0.093 A 0.695 0.6775 0.6672 C 0.085 0.0856 0.0859 min2 B 0.0950 0.0962 0.0968 A 0.6482 0.624 0.628 C 0.080 0.08 min3 B 0.0885 0.0887 A 0.7589 0.7503 closure, while the others are significantly larger. All three processes are shown in Fig. 7. There are a total of 23!7!60 480 distinct versions of the H 7 O 3 or D 5 O 2 ) C global minimum, where the first TABLE III. Harmonic frequencies/cm of min, min2, and min3 at the aug-cc-pvdz/blyp level of theory. Intensities in km/mol are given in square brackets. min min2 min3 87 87 0 86 6 06 8 90 2 87 7 26 2 28 49 4 7 259 64 257 43 65 92 299 24 267 408 204 36 338 37 335 50 335 28 376 78 372 96 373 5 385 248 387 38 384 270 44 7 409 8 404 3 455 5 457 7 45 3 603 3 606 4 575 6 079 55 069 5 09 62 94 89 202 85 76 20 530 27 55 8 527 24 548 3 552 569 4 608 35 605 3 577 2 628 22 625 66 625 5 2232 476 2228 450 2275 4260 242 089 244 20 2453 983 3632 62 3635 02 364 7 3636 62 3636 6 364 76 3654 55 366 60 3752 37 3726 43 3730 9 3738 9 373 85 373 320 3738 326

8432 J. Chem. Phys., Vol., No. 8, 8 November 999 David J. Wales TABLE IV. Harmonic frequencies/cm of the H 7 O 3 transition states at the aug-cc-pvdz/blyp level of theory. tsinv tsrot tsf tsf2 tsf3 tsrc tsrc2 tsb tsb2 tsb3 44i 90i 253i 27i 238i 73i 88i 370i 557i 64i 89 88 95 94 92 99 20 75 06 75 0 37 22 0 23 245 207 6 0 24 3 295 47 42 48 26 254 26 8 93 283 326 245 257 57 348 37 224 220 246 337 344 289 267 99 407 383 304 273 263 374 365 359 370 349 456 432 328 282 348 383 374 386 378 384 469 466 368 323 369 403 435 397 407 390 486 55 423 334 404 45 478 47 44 40 603 56 459 370 422 595 700 6 66 594 645 622 482 394 58 068 940 068 076 05 740 755 798 53 99 93 78 65 64 57 929 899 267 32 56 523 529 476 449 493 279 70 485 408 508 569 586 55 535 520 524 572 539 559 527 577 620 597 592 574 60 587 596 602 584 625 676 65 66 68 623 605 625 609 628 2254 22 2236 2229 2272 73 759 905 2809 3079 2442 2409 2395 2385 2425 3432 3434 3509 3523 356 3634 3627 3634 3636 3642 3459 3463 357 3608 3278 3648 3632 3637 3637 3642 367 367 3623 3638 3555 3659 3649 370 376 3706 3644 3655 3638 364 3646 3728 377 3729 3732 3739 3648 3659 376 3726 3693 3748 3726 3733 3732 3740 3699 3699 3724 3727 374 factor accounts for the inversion operation and there are 3! and 7! permutations of the oxygen and hydrogen or deuterium atoms, respectively. For min2 and min3 we must divide by two to account for the order of the point group, 32 and there are only 30 240 distinct versions. The reaction graph for one set of labeled minima involving the most likely feasible mechanisms, i.e. those mediated by monomer inversion, internal rotation and the two flips tsinv, tsrot, tsf and tsf2, is shown in Fig. 8. min3 is not included to simplify the analysis and because it lies somewhat higher in energy than min and min2. The presence of two low energy minima complicates the molecular symmetry group analysis. One way to treat such cases is to use effective generators which relate permuta- TABLE V. Properties of the steepest-descent pathways found for H 7 O 3 with the two smaller basis sets. S, D, and are all defined in Sec. II. Parameter DZPdiff aug-cc-pvdz DZPdiff aug-cc-pvdz tsinv: monomer inversion tsrot: internal rotation S 0.8. 5.7 5.9 D 0.7 0.9 4.4 4.5 3.8 3.9 3.5 3.5 tsf: flip tsf2: flip 2 S 2.5 2.9 2.0 2.0 D.9 2..4.5 2.2 2.2 2.6 2.7 tsf: flip 3 tsrc: ring closure S 2.7 3. 2.8 D 2. 0.7 0.6.8.4.4 tsrc2: ring closure 2 tsb: bifurcation S 5. 5.2 0. 9.8 D.2. 7.7 7.4.3.3.9 2.0 tsb2: double bifurcation tsb3: bifurcation 3 S 3.6 3.5 8.5 8.6 D 5.9 6.0 6.2 6. 2.6 2.5 3.6 3.5

J. Chem. Phys., Vol., No. 8, 8 November 999 Protonated water trimer 8433 FIG. 4. Energy profiles of the aug-cc-pvdz/blyp steepest-descent paths for monomer inversion tsinv and internal rotation tsrot. terms of the effective tunneling matrix elements 2 and f2, and assuming a Hückel-type approximation, the splitting pattern is quite simple: FIG. 2. Two views of the monomer inversion pathway for H 7 O 3 calculated at the aug-cc-pvdz/blyp level. tional isomers of the same structure. 59 For min2 suitable generators are AC35627* reflection symmetry, 2 tsinv followed by tsrot, or vice versa and AC72635 the flip, tsf2, and the versions of min2 are then linked in sets of eight. The resulting MS group is the G(6) group first characterized by Dyke for neutral water dimer, 60 which is also appropriate for H 5 O 2. 7 The character table and nuclear spin weights are given in Table VI. In 2 2 f2 2 2 f2 f2 f2 2 2 f2 A B E E B 2 2 2 f2 A 2. FIG. 3. Two views of the internal rotation pathway of H 7 O 3 calculated at the aug-cc-pvdz/blyp level. FIG. 5. Flip rearrangements of H 7 O 3 corresponding to tsf, tsf2, and tsf3 calculated at the aug-cc-pvdz/blyp level.

8434 J. Chem. Phys., Vol., No. 8, 8 November 999 David J. Wales FIG. 6. Ring closure rearrangements of H 7 O 3 corresponding to tsrc left and tsrc2 right calculated at the aug-cc-pvdz/blyp level. FIG. 7. Pathways of H 7 O 3 involving bifurcated transition states calculated at the aug-cc-pvdz/blyp level. If f2 is smaller than 2 the result is a triplet of doublets where the doublets are split by 2 f2 and the components of the triplet are split by 2 2. For min five generators are required to specify all the likely connections, namely 267* for the flip tsf and AC35672*, AC35726*, AC35726* and AC35627* for the indirect processes. The versions of min are then linked in sets of 6 and the appropriate MS group is again G(6). An accidental degeneracy arises in the tunneling spectrum which complicates the symmetry assignments. However, it can be removed by including the generator operation AC35627 as a small perturbation. The resulting spectrum includes a set of levels which match those found for min2, above, and eight more levels with wavefunctions that are not delocalized over the min2 isomers. Since min and min2 are so close in energy we will present a full analysis of the combined spectrum rather than focusing further on the results for the different minima. If min2 lies at an energy above min, and we adopt a Hückel-type approximation for the secular determinant, then an analytic form can be found for the energy levels: 2 8 inv rot 2 f f2 2 2 f f2 2 28 inv rot 2 f f2 2 2 f f2 2 28 2 inv 2 rot f f2 2 2 f f2 2 28 2 inv 2 rot f f2 2 2 f f2 2 28 inv rot 2 f f2 2 2 f f2 2 28 inv rot 2 f f2 2 2 f f2 2 f f A B E E B 2 A 2 A 2,B,E A,B 2,E

J. Chem. Phys., Vol., No. 8, 8 November 999 Protonated water trimer 8435 2 8 inv rot 2 f f2 2 2 f f2 2 B 2 2 8 inv rot 2 f f2 2 2 f f2 2 A 2 2 8 2 inv 2 rot f f2 2 2 f f2 2 E 2 8 2 inv 2 rot f f2 2 2 f f2 2 E 2 8 inv rot 2 f f2 2 2 f f2 2 A 2 8 inv rot 2 f f2 2 2 f f2 2 B. FIG. 8. Reaction graph for H 7 O 3 min and min2 including pathways mediated by monomer inversion, internal rotation and the two flips.

8436 J. Chem. Phys., Vol., No. 8, 8 November 999 David J. Wales TABLE VI. Character table for the MS group G(6) and nuclear spin statistical weights for H 7 O 3 and D 7 O 3. G(6) E (2)(67) (2) 67 (AC)(35)(726) AC35627 (AC)(35)(7)(26) AC35627 E* (2)(67)* (2)* 67* (AC)(35)(726)* AC35627* (AC)(35)(7)(26)* AC35627* H 7 O 3 D 7 O 3 A 9 339 A 2 26 26 B 9 336 B 2 24 26 E 2 2 0 0 0 2 2 0 0 0 30 540 A 9 339 A 2 26 26 B 9 336 B 2 24 26 E 2 2 0 0 0 2 2 0 0 0 30 540 From the calculated barrier heights and path lengths a reasonable guess for the magnitudes of the tunneling matrix elements in wavenumbers for H 7 O 3 corresponding to tsinv-tsf2 is probably inv : rot : f : f2 0::0.:0.. The energy levels are then arranged in the order given above. Whatever the value of the levels always appear in doublets split by f f2. This doubling may be the smallest observable splitting if none of the other mechanisms corresponding to tsrc-tsb3 are feasible. In the limit 0 the energy levels obey a mirror relation about the energy zero with two triplets of doublets at around 2 inv, assuming that inv is the largest matrix element Fig. 9. The triplet spacing is roughly 2 rot if we also neglect f and f2. When is large compared to the tunneling matrix elements the two triplets of doublets move to energies around zero and Fig. 9 as the wavefunctions become more strongly localized on either min or min2 isomers. The triplet splitting is around 4 inv rot /. The expected intensities of the lines in the triplet of doublets are 9:9:30:30:24:26 or 8:60:50 if the splitting due to the flip is not resolved. For D 7 O 3 the corresponding ratios are 339:336:540:540:26:26 and 675:080:432. If any of the other mechanisms corresponding to the remaining transition states are feasible then the MS group will be enlarged and further splittings of the tunneling states will occur. However, the present results suggest that the effects of ring closure and bifurcation are probably beyond the limit of current experimental resolution, especially for D 7 O 3. Although bifurcation tunneling is resolved in (H 2 O) 3,(D 2 O) 3,2,6 63 and probably (H 2 O) 5, 9,0 the barriers in H 7 O 3 are higher. IV. CONCLUSIONS The present calculations of rearrangement pathways in protonated water trimer reveal a more complicated situation than for the protonated dimer, with two low energy minima interconverting via a variety of rearrangements and a third minimum lying only a little higher in energy before zeropoint effects are included. The nondegenerate monomer inversion and internal rotation processes seem likely to produce observable splittings via indirect tunneling 4 in which both min and min2 are involved. The degenerate flip rearrangements involve higher barriers and should result in an additional doublet splitting if they are feasible. The other mechanisms, including bifurcation processes, involve significantly larger barriers and do not appear to have been observed in previous quantum nuclear dynamics simulations of either H 5 O 2 64 or a hydrated proton in water. 65 Calculations on protonated water clusters are notoriously sensitive to changes in basis set and treatment of electron correlation, 66 and the additional stationary points found with the two largest basis sets provide further evidence of this problem for H 7 O 3. Extensive searches for lower energy pathways connecting min3 to either min or min2 were not conducted. However, the patterns predicted in the present work would probably not change much if such additional paths exist or if the details of the low energy pathways were altered, so long as the connectivity of the reaction graph is unaffected. New results of high resolution spectroscopy in the far infrared should soon test this theory. ACKNOWLEDGMENTS The author gratefully acknowledges the support of the Royal Society of London and the EPSRC. FIG. 9. Splitting pattern for H 7 O 3 calculated using a Hückel-type approximation for 0, 20, and 40 cm. The other parameters were fixed at inv 0, rot.5 and f f2 0.4, all in wave numbers. R. J. Saykally personal communication. 2 R. C. Cohen and R. J. Saykally, J. Phys. Chem. 94, 799 990. 3 N. Pugliano and R. J. Saykally, J. Chem. Phys. 96, 832 992. 4 R. J. Saykally and G. A. Blake, Science 259, 570 993. 5 K. Liu, J. D. Cruzan, and R. J. Saykally, Science 27, 929 996.

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