Using JCP format - Wales Group

JOURNAL OF CHEMICAL PHYSICS

VOLUME 111, NUMBER 18

8 NOVEMBER 1999

Rearrangements and tunneling splittings of protonated water trimer David J. Wales University Chemical Laboratories, Lensfield Road, Cambridge CB2 1EW, United Kingdom

共Received 13 April 1999; accepted 11 August 1999兲 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. © 1999 American Institute of Physics. 关S0021-9606共99兲31533-6兴 I. INTRODUCTION

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

The promise of new experimental results for protonated water clusters1 from far-infrared vibration–rotation tunneling 共FIR–VRT兲 spectroscopy2–6 seems likely to spark a new wave of research in this field, as it did for neutral water 7 clusters. The present work, and a preceding paper on H5O⫹ 2, 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 trimer8 and predict those recently found for water pentamer.9,10 The theoretical framework is identical to the procedure described in previous accounts of rearrangements and tunneling splittings in neutral9,11–16 and protonated7 water clusters. Several experiments have previously probed the infrared 17–20 spectrum of H7O⫹ In par3 in the OH stretching region. ticular, Okumura et al. confirmed the assignment by Schwarz 20 Comparisons of the of a band at 2670 cm⫺1 to H7O⫹ 3. present results with previous ab initio calculations21–30 are made in Sec. III. To deduce the molecular symmetry 共MS兲 group31 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.’’31 The largest tunneling splittings are expected for degenerate rearrangements33 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 H7O⫹ 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 0021-9606/99/111(18)/8429/9/$15.00

II. GEOMETRY OPTIMIZATION AND PATHWAYS

As in our previous studies of water clusters all the stationary points were located using eigenvector-following38–43 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 7 study of H5O⫹ 2 . The first, DZP⫹diff, is based upon a 45,46 plus polarization 共DZP兲 basis, with polarization double-␨ functions consisting of a single set of p orbitals on each hydrogen atom 共exponent 1.0兲 and a set of six d orbitals on each oxygen atom 共exponent 0.9兲. To these functions were added a diffuse s orbital on each hydrogen atom 共exponent 0.0441兲 and diffuse s and p orbitals on each oxygen atom 共exponents 0.0823 and 0.0651 for s and p, respectively兲,47 giving a total of 102 functions for H7O⫹ 3 . We also employed the standard aug-cc-pVDZ and aug-cc-pVTZ basis sets,48,49 with a total of 138 and 340 basis functions, respectively. Correlation corrections were obtained through density functional theory 共DFT兲, since MP2 second derivative calculations were impossible due to disk space limitations. We employed the Becke nonlocal exchange functional50 and the Lee–Yang–Parr correlation functional51 共together referred to as BLYP兲. 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 2⫻10⫺6 atomic units. Since derivatives of the grid weights were not included the zero frequencies were sometimes as large as 50 8429

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cm⫺1 . Pathways were calculated at the DZP⫹diff/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 H5O⫹ 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 scheme52 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⬇

兺m 兩 Qm⫹1 ⫺Qm兩 ,

共1兲

where Qm 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: 共2兲 D⫽ 兩 Q共 s 兲 ⫺Q共 f 兲 兩 , 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 兺 i 关 Qi 共 s 兲 ⫺Qi 共 f 兲兴 4 ␥⫽ , 共3兲 共 兺 i 关 Qi 共 s 兲 ⫺Qi 共 f 兲兴 2 兲 2 where Qi (s) is the position vector of atom i in starting minimum s, etc. If every atom undergoes the same displacement then ␥ ⫽1, 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 共Fig. 1 and Table I兲. However, the higher-lying minimum, min3, and the transition state that links it to min1 were not located with the DZP⫹diff 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. 1. 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 metric37 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 ts共b1兲, described below, where a small perturbation changes one of the minima from min1 to min2 for the DZP⫹diff basis, and from min1 to min3 for the aug-cc-pVDZ basis. 共min3 does not appear to exist for the smaller basis set.兲 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 10%–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 2 v symmetry chain structure has been classified as a minimum.21,23,24,29 In the present work this structure was found to be a transition state 共ts共f2兲兲, 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 1 symmetry.27,30 Geissler et al.30 also report two transition states for the Stillinger–David potential58 which correspond to ts共rc1兲 and ts共rc2兲 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 ts共inv兲 is shown in Fig. 2 and connects min1 and min2. The

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TABLE I. Energies/hartree 共Ref. 54兲 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 min1 at the same level of theory, ⌬E 共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兲.

min1 ZPE/cm⫺1 min2 ⌬E/cm⫺1 min3 ⌬E/cm⫺1 ts共inv兲: monomer inversion ⌬E/cm⫺1 ts共rot兲: internal rotation ⌬E/cm⫺1 ts共f1兲: flip 1 ⌬E/cm⫺1 ts共f2兲: flip 2 ⌬E/cm⫺1 ts共f3兲: flip 3 ⌬E/cm⫺1 ts共rc1兲: ring closure 1 ⌬E/cm⫺1 ts共rc2兲: ring closure 2 ⌬E/cm⫺1 ts共b1兲: bifurcation ⌬E/cm⫺1 ts共b2兲: double bifurcation ⌬E/cm⫺1 ts共b3兲: bifurcation ⌬E/cm⫺1

Group

DZP⫹diff

aug-cc-pVDZ

aug-cc-pVTZ

C1

⫺229.674 703 17 622 ⫺229.674 672 7(65)

⫺229.645 518 17 499 ⫺229.645 502 4(39) ⫺229.645 271 54(⫺39) ⫺229.645 412 23(46) ⫺229.644 846 148(59) ⫺229.644 520 219(274) ⫺229.644 547 213(183) ⫺229.644 299 268(⫺55) ⫺229.635 435 2213(2004) ⫺229.634 283 2466(2491) ⫺229.633 605 2615(2393) ⫺229.617 201 6215(6463) ⫺229.612 807 7179(7141)

⫺229.713 497 17 499 ⫺229.713 486 2(38) ⫺229.713 205 64(⫺29) ⫺229.713 367 29(51) ⫺229.712 877 136(48) ⫺229.712 227 279(333) ⫺229.712 247 274(244) ⫺229.711 974 334(12) ⫺229.703 632 2165(1956) ⫺229.702 471 2420(2445) ⫺229.701 545 2623(2402) ⫺229.685 252 6199(6447) ⫺229.680 521 7237(7200)

Cs Cs C1 C1 Cs C 2v

⫺229.674 652 11(90) ⫺229.673 838 190(132) ⫺229.673 783 202(242) ⫺229.673 775 204(392)

C 2v C2 Cs C1 Cs Cs

⫺229.664 640 2208(1999) ⫺229.663 262 2511(2725) ⫺229.662 572 2662(2419) ⫺229.645 501 6409(6740) ⫺229.641 025 7391(7517)

7 barriers are even smaller than for H5O⫹ 2 . Only one internal rotation mechanism was found in the present work, in con7 trast to two for H5O⫹ 2 . It also connects min1 and min2 and involves low barriers 共Fig. 3兲. 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 共Fig. 5兲. 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 min1 in the present calculations 共Fig. 6兲. The rearrangements mediated by ts共b1兲, ts共b2兲, and ts共b3兲 all involve bifurcations, two in the case of ts共b2兲. The barriers involved for ts共b1兲 are similar to those for ring

TABLE II. Rotational constants/cm⫺1 of the three minima for the three different basis sets. DZP⫹diff

aug-cc-pVDZ

aug-cc-pVTZ

C min1 B A

0.0835 0.0924 0.6915

0.0834 0.0925 0.6775

0.0837 0.0931 0.6672

C min2 B A

0.0851 0.0950 0.6482

0.0856 0.0962 0.6214

0.0859 0.0968 0.6128

0.0810 0.0885 0.7589

0.0811 0.0887 0.7503

C min3 B A

closure, while the others are significantly larger. All three processes are shown in Fig. 7. There are a total of 2⫻3!⫻7!⫽60 480 distinct versions ⫹ of the H7O⫹ 3 共or D5O2 ) C 1 global minimum, where the first TABLE III. Harmonic frequencies/cm⫺1 of min1, min2, and min3 at the aug-cc-pVDZ/BLYP level of theory. Intensities in km/mol are given in square brackets. min1 87 106 126 259 299 338 376 385 414 455 603 1079 1194 1530 1548 1608 1628 2232 2421 3632 3636 3654 3726 3731

关1兴 关18兴 关21兴 关164兴 关214兴 关37兴 关178兴 关248兴 关17兴 关5兴 关3兴 关55兴 关189兴 关27兴 关13兴 关35兴 关22兴 关4176兴 关1089兴 关62兴 关62兴 关155兴 关143兴 关185兴

min2 87 90 128 257 267 335 372 387 409 457 606 1069 1202 1515 1552 1605 1625 2228 2414 3635 3636 3661 3730 3731

关0兴 关2兴 关49兴 关43兴 关408兴 关50兴 关196兴 关138兴 关18兴 关7兴 关4兴 关51兴 关185兴 关18兴 关11兴 关3兴 关66兴 关4150兴 关1120兴 关102兴 关16兴 关160兴 关9兴 关320兴

min3 86 87 114 165 204 335 373 384 404 451 575 1091 1176 1527 1569 1577 1625 2275 2453 3641 3641 3752 3738 3738

关16兴 关17兴 关17兴 关92兴 关316兴 关28兴 关151兴 关270兴 关3兴 关3兴 关6兴 关62兴 关201兴 关24兴 关14兴 关21兴 关5兴 关4260兴 关983兴 关71兴 关76兴 关137兴 关9兴 关326兴

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TABLE IV. Harmonic frequencies/cm⫺1 of the H7O⫹ 3 transition states at the aug-cc-pVDZ/BLYP level of theory. ts共inv兲

ts共rot兲

ts共f1兲

ts共f2兲

ts共f3兲

ts共rc1兲

ts共rc2兲

ts共b1兲

ts共b2兲

ts共b3兲

144i 89 110 131 283 337 374 383 403 451 595 1068 1193 1523 1569 1577 1625 2254 2442 3634 3648 3659 3728 3748

90i 88 137 295 326 344 365 374 435 478 700 940 1178 1529 1586 1620 1676 2211 2409 3627 3632 3649 3717 3726

253i 95 122 147 245 289 359 386 397 417 611 1068 1165 1476 1515 1597 1615 2236 2395 3634 3637 3710 3729 3733

271i 94 110 142 257 267 370 378 407 414 616 1076 1164 1449 1535 1592 1616 2229 2385 3636 3637 3716 3732 3732

238i 92 123 148 157 199 349 384 390 410 594 1051 1157 1493 1520 1574 1618 2272 2425 3642 3642 3706 3739 3740

173i 199 245 261 348 407 456 469 486 603 645 740 929 1279 1524 1601 1623 1731 3432 3459 3617 3644 3648 3699

188i 201 207 254 371 383 432 466 551 561 622 755 899 1170 1572 1587 1605 1759 3434 3463 3617 3655 3659 3699

370i 75 116 216 224 304 328 368 423 459 482 798 1267 1485 1539 1596 1625 1905 3509 3517 3623 3638 3716 3724

557i 106 110 181 220 273 282 323 334 370 394 531 1132 1408 1559 1602 1609 2809 3523 3608 3638 3641 3726 3727

641i 75 124 193 246 263 348 369 404 422 581 919 1156 1508 1527 1584 1628 3079 3156 3278 3555 3646 3693 3741

sion, internal rotation and the two flips 关ts共inv兲, ts共rot兲, ts共f1兲 and ts共f2兲兴, is shown in Fig. 8. min3 is not included to simplify the analysis and because it lies somewhat higher in energy than min1 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-

factor accounts for the inversion operation and there are 3! and 7! permutations of the oxygen and hydrogen 共or deuterium兲 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 inver-

TABLE V. Properties of the steepest-descent pathways found for H7O⫹ 3 with the two smaller basis sets. S, D, and ␥ are all defined in Sec. II. Parameter

DZP⫹diff

aug-cc-pVDZ

S D ␥

ts共inv兲: monomer inversion 0.8 1.1 0.7 0.9 3.8 3.9

S D ␥

2.5 1.9 2.2

DZP⫹diff

aug-cc-pVDZ

ts共rot兲: internal rotation 5.7 5.9 4.4 4.5 3.5 3.5

ts共f1兲: flip 1

ts共f2兲: flip 2 2.9 2.1 2.2

ts共f1兲: flip 3

2.0 1.4 2.6

2.0 1.5 2.7

S D ␥

2.7 2.1 1.8

ts共rc1兲: ring closure 1 13.1 12.8 10.7 10.6 1.4 1.4

S D ␥

ts共rc2兲: ring closure 2 15.1 15.2 11.2 11.1 1.3 1.3

ts共b1兲: bifurcation 1 10.1 9.8 7.7 7.4 1.9 2.0

S D ␥

ts共b2兲: double bifurcation 13.6 13.5 5.9 6.0 2.6 2.5

ts共b3兲: bifurcation 3 8.5 8.6 6.2 6.1 3.6 3.5

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Protonated water trimer

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FIG. 4. Energy profiles of the aug-cc-pVDZ/BLYP steepest-descent paths for monomer inversion 共ts共inv兲兲 and internal rotation 共ts共rot兲兲.

terms of the effective tunneling matrix elements ␤ 12 and ␤ f2 , and assuming a Hu¨ckel-type approximation, the splitting pattern is quite simple:

FIG. 2. Two views of the monomer inversion pathway for H7O⫹ 3 calculated at the aug-cc-pVDZ/BLYP level.

59

tional isomers of the same structure. For min2 suitable generators are 共AC兲共35兲共16兲共27兲* 共reflection symmetry兲, 共12兲 共ts共inv兲 followed by ts共rot兲, or vice versa兲 and 共AC兲共17兲共26兲共35兲 共the flip, ts共f2兲兲, and the versions of min2 are then linked in sets of eight. The resulting MS group is the G(16) group first characterized by Dyke for neutral water 7 dimer,60 which is also appropriate for H5O⫹ 2 . The character table and nuclear spin weights are given in Table VI. In

FIG. 3. Two views of the internal rotation pathway of H7O⫹ 3 calculated at the aug-cc-pVDZ/BLYP level.

2 ␤ 12⫹ ␤ f2

A⫹ 1

2 ␤ 12⫺ ␤ f2

B⫺ 1

␤ f2

E⫺

⫺ ␤ f2

E⫹

⫺2 ␤ 12⫹ ␤ f2

B⫹ 2

⫺2 ␤ 12⫺ ␤ f2

A⫺ 2 .

FIG. 5. Flip rearrangements of H7O⫹ 3 corresponding to ts共f1兲, ts共f2兲, and ts共f3兲 calculated at the aug-cc-pVDZ/BLYP level.

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FIG. 6. Ring closure rearrangements of H7O⫹ 3 corresponding to ts共rc1兲 共left兲 and ts共rc2兲 共right兲 calculated at the aug-cc-pVDZ/BLYP level.

If ␤ f2 is smaller than ␤ 12 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 ␤ 12 . For min1 five generators are required to specify all the likely connections, namely 共12兲共67兲* for the flip 共ts共f1兲兲 and 共AC兲共35兲共1672兲*, 共AC兲共35兲共1726兲*, 共AC兲共35兲共17兲共26兲* and 共AC兲共35兲共16兲共27兲* for the indirect processes. The versions of min1 are then linked in sets of 16 and the appropriate MS group is again G(16). An accidental degeneracy arises in the tunneling spectrum which complicates the symmetry assignments. However, it can be removed by including the genera-

David J. Wales

FIG. 7. Pathways of H7O⫹ 3 involving bifurcated transition states calculated at the aug-cc-pVDZ/BLYP level.

tor operation 共AC兲共35兲共1627兲 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 min1 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 min1, and we adopt a Hu¨ckel-type approximation for the secular determinant, then an analytic form can be found for the energy levels:

1 2

共 冑8 共 ␤ inv⫹ ␤ rot兲 2 ⫹ 共 ␤ f1⫺ ␤ f2兲 2 ⫹⌬ 2 ⫹ 共 ␤ f1⫹ ␤ f2兲兲 ⫹

⌬ 2

A⫹ 1

1 2

共 冑8 共 ␤ inv⫹ ␤ rot兲 2 ⫹ 共 ␤ f1⫺ ␤ f2兲 2 ⫹⌬ 2 ⫺ 共 ␤ f1⫹ ␤ f2兲兲 ⫹

⌬ 2

B⫺ 1

2 2 ⫹ ␤ rot 兲 ⫹ 共 ␤ f1⫺ ␤ f2兲 2 ⫹⌬ 2 ⫹ 共 ␤ f1⫹ ␤ f2兲兲 ⫹ 共 冑8 共 ␤ inv

⌬ 2

E⫺

2 2 ⫹ ␤ rot 兲 ⫹ 共 ␤ f1⫺ ␤ f2兲 2 ⫹⌬ 2 ⫺ 共 ␤ f1⫹ ␤ f2兲兲 ⫹ 共 冑8 共 ␤ inv

⌬ 2

E⫹

1 2

共 冑8 共 ␤ inv⫺ ␤ rot兲 2 ⫹ 共 ␤ f1⫺ ␤ f2兲 2 ⫹⌬ 2 ⫹ 共 ␤ f1⫹ ␤ f2兲兲 ⫹

⌬ 2

B⫹ 2

1 2

共 冑8 共 ␤ inv⫺ ␤ rot兲 2 ⫹ 共 ␤ f1⫺ ␤ f2兲 2 ⫹⌬ 2 ⫺ 共 ␤ f1⫹ ␤ f2兲兲 ⫹

⌬ 2

A⫺ 2

␤ f1

⫹ ⫺ A⫹ 2 ,B 1 ,E

⫺ ␤ f1

⫺ ⫹ A⫺ 1 ,B 2 ,E

1 2

1 2

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Protonated water trimer

⫺ 12 共 冑8 共 ␤ inv⫺ ␤ rot兲 2 ⫹ 共 ␤ f1⫺ ␤ f2兲 2 ⫹⌬ 2 ⫺ 共 ␤ f1⫹ ␤ f2兲兲 ⫹

⌬ 2

B⫹ 2

⫺ 21 共 冑8 共 ␤ inv⫺ ␤ rot兲 2 ⫹ 共 ␤ f1⫺ ␤ f2兲 2 ⫹⌬ 2 ⫹ 共 ␤ f1⫹ ␤ f2兲兲 ⫹

⌬ 2

A⫺ 2

2 2 ⫺ 21 共 冑8 共 ␤ inv ⫹ ␤ rot 兲 ⫹ 共 ␤ f1⫺ ␤ f2兲 2 ⫹⌬ 2 ⫺ 共 ␤ f1⫹ ␤ f2兲兲 ⫹

⌬ 2

E⫺

2 2 ⫺ 12 共 冑8 共 ␤ inv ⫹ ␤ rot 兲 ⫹ 共 ␤ f1⫺ ␤ f2兲 2 ⫹⌬ 2 ⫹ 共 ␤ f1⫹ ␤ f2兲兲 ⫹

⌬ 2

E⫹

⫺ 21 共 冑8 共 ␤ inv⫹ ␤ rot兲 2 ⫹ 共 ␤ f1⫺ ␤ f2兲 2 ⫹⌬ 2 ⫺ 共 ␤ f1⫹ ␤ f2兲兲 ⫹

⌬ 2

A⫹ 1

⫺ 21 共 冑8 共 ␤ inv⫹ ␤ rot兲 2 ⫹ 共 ␤ f1⫺ ␤ f2兲 2 ⫹⌬ 2 ⫹ 共 ␤ f1⫹ ␤ f2兲兲 ⫹

⌬ 2

B⫺ 1 .

FIG. 8. Reaction graph for H7O⫹ 3 min1 and min2 including pathways mediated by monomer inversion, internal rotation and the two flips.

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David J. Wales

⫹ TABLE VI. Character table for the MS group G(16) and nuclear spin statistical weights for H7O⫹ 3 and D7O3 .

G(16)

E

A⫹ 1 A⫹ 2 B⫹ 1 B⫹ 2 E⫹ A⫺ 1 A⫺ 2 B⫺ 1 B⫺ 2 E⫺

1 1 1 1 2 1 1 1 1 2

(12) (AC)(35)(1726) (AC)(35)(17)(26) (12) * (AC)(35)(1726) * (AC)(35)(17)(26) * (12)(67) 共67兲 共AC兲共35兲共1627兲 共AC兲共35兲共16兲共27兲 E * (12)(67) * 共67兲* 共AC兲共35兲共1627兲* 共AC兲共35兲共16共27兲* 1 1 1 1 ⫺2 1 1 1 1 ⫺2

1 ⫺1 1 ⫺1 0 1 ⫺1 1 ⫺1 0

1 1 ⫺1 ⫺1 0 1 1 ⫺1 ⫺1 0

1 ⫺1 ⫺1 1 0 1 ⫺1 ⫺1 1 0

1 1 1 1 2 ⫺1 ⫺1 ⫺1 ⫺1 ⫺2

From the calculated barrier heights and path lengths a reasonable guess for the magnitudes of the tunneling matrix elements in wavenumbers for H7O⫹ 3 corresponding to ts共inv兲-ts共f2兲 is probably ␤ inv : ␤ rot : ␤ f1 : ␤ f2⫽10:1:0.1:0.1. The energy levels are then arranged in the order given above. Whatever the value of ⌬ the levels always appear in doublets split by ␤ f1⫹ ␤ f2 . This doubling may be the smallest observable splitting if none of the other mechanisms corresponding to ts共rc1兲-ts共b3兲 are feasible. In the limit ⌬→0 the energy levels obey a mirror relation about the energy zero with two triplets of doublets at around ⫾ 冑2 ␤ inv , assuming that ␤ inv is the largest matrix element 共Fig. 9兲. The triplet spacing is roughly 冑2 ␤ rot if we also neglect ␤ f1 and ␤ f2 . When ⌬ is large compared to the ␤ tunneling matrix elements the two triplets of doublets move to energies around zero and ⌬ 共Fig. 9兲 as the wavefunctions become more strongly localized on either min1 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 18:60:50 if the splitting due to the flip is not resolved. For D7O⫹ 3 the corresponding ratios are 339:336:540:540:216:216 and 675:1080:432. 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. How-

1 1 1 1 ⫺2 ⫺1 ⫺1 ⫺1 ⫺1 2

1 ⫺1 1 ⫺1 0 ⫺1 1 ⫺1 1 0

1 1 ⫺1 ⫺1 0 ⫺1 ⫺1 1 1 0

1 ⫺1 ⫺1 1 0 ⫺1 1 1 ⫺1 0

⫹ H7O⫹ 3 D7O3

␮

9 26 9 24 30 9 26 9 24 30

339 216 336 216 540 339 216 336 216 540

ever, the present results suggest that the effects of ring closure and bifurcation are probably beyond the limit of current experimental resolution, especially for D7O⫹ 3 . Although bifurcation tunneling is resolved in (H2O) 3, (D2O) 311,12,61–63 and probably (H2O) 5, 9,10 the barriers in H7O⫹ 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 tunneling14 in which both min1 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 or a hydrated proton in water.65 either H5O⫹64 2 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 H7O⫹ 3 . Extensive searches for lower energy pathways connecting min3 to either min1 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.1 ACKNOWLEDGMENTS

The author gratefully acknowledges the support of the Royal Society of London and the EPSRC. R. J. Saykally 共personal communication兲. R. C. Cohen and R. J. Saykally, J. Phys. Chem. 94, 7991 共1990兲. N. Pugliano and R. J. Saykally, J. Chem. Phys. 96, 1832 共1992兲. 4 R. J. Saykally and G. A. Blake, Science 259, 1570 共1993兲. 5 K. Liu, J. D. Cruzan, and R. J. Saykally, Science 271, 929 共1996兲. 1 2

H7O⫹ 3 calculated ⫺1

FIG. 9. Splitting pattern for using a Hu¨ckel-type approximation for ⌬⫽0, 20, and 40 cm . The other parameters were fixed at ␤ inv⫽10, ␤ rot⫽1.5 and ␤ f1⫽ ␤ f2⫽0.4, all in wave numbers.

3

J. Chem. Phys., Vol. 111, No. 18, 8 November 1999 6

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