Ab initio calculations of nonadditive effects in the trimers (H2O)2⋯XY,XY=N2 , BF, CS

31 May 2002

Chemical Physics Letters 358 (2002) 237–249 www.elsevier.com/locate/cplett

Ab initio calculations of nonadditive effects in the trimers ðH2OÞ2    XY; XY ¼ N2, BF, CS Michal F. Rode 1, Joanna Sadlej

*

Department of Chemistry, University of Warsaw, Pasteur 1 Str., 02-093 Warsaw, Poland Received 12 March 2002; in final form 8 April 2002

Abstract Optimal structures, interaction energies and harmonic vibrational frequencies of the ðH2 OÞ2    XY, XY ¼ N2 , BF, CS ternary complex have been determined from the supermolecular (SM) calculations with the aug-cc-pVDZ and the aug-cc-pVTZ basis sets. Energetic properties of the complex have been calculated at the MP4 level. We located three low-energy configurations corresponding to two isomeric H-bonded cyclic complexes and one linear structure. Nonadditive interactions play an important role for the ðH2 OÞ2    BF and ðH2 OÞ2    CS trimers. The contribution of the three-body term represents as much as 17% and 13% of the total MP4 interaction energy. Partitioning of the three-body energy was performed in terms of the intermolecular perturbation theory. The nonadditivity originates mainly from the induction effect. The calculations of the vibrational frequencies and infrared intensities for these complexes are presented to facilitate the frequency assignments of future experimental spectra. Ó 2002 Published by Elsevier Science B.V.

1. Introduction Clusters of H2 O with gaseous atoms and molecules received considerable attention, both from experimentalists and from theorists [1–3]. The long term goal of the investigation of such clusters is to gain insight into interactions of small molecules with water in condensed phase systems. By studying the small clusters one can gain considerable insight into exact information on nonadditivity.

*

1

Corresponding author. E-mail address: [email protected] (J. Sadlej). Also corresponding author.

Nonadditive interactions have been the subject of intensive investigations since the work of Axilrod and Teller [4]. The water trimer is perhaps the most frequently studied of all (for a relatively recent review see [2,3]). It is frequently assumed that the interactions are pairwise additive, i.e., that the potential energy can be approximated as the sum of interaction between all pairs of constituent molecules in modeling condensed phase systems bound by hydrogen bonding and van der Waals interactions. However deviations from this approximation are often substantial and have been a focus of intensive studies during the last decade [2,6]. It has been argued that the many body effects in water cannot be neglected in computer simulation of liquid water. The nonadditive effects in the

0009-2614/02/$ - see front matter Ó 2002 Published by Elsevier Science B.V. PII: S 0 0 0 9 - 2 6 1 4 ( 0 2 ) 0 0 5 8 7 - 0

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trimer and larger clusters of water were the subject of numerous studies, as well as for other clusters. Among them the results for the trimer ðH2 OÞ2    CO was already published by us [5]. Recently the origin of three-body effects was investigated in many trimers [2,6]. This treatment allows for a dissection of the interaction energy into the physical interpretable components, such as induction, exchange and dispersion nonadditive three-body effect. In the present Letter the study of the nonadditive effects is extended to include a spectrum of interacting molecules XY with water dimer, which differ by the multipole moments and polarizability. The molecules chosen for this study (N2 , CO, CS) are important in atmospheric chemistry [7]. Nitrogen and carbon oxide interacting with a water molecule has been a subject of various theoretical and experimental studies [8–20]. These molecules are also important in the investigation of weak bonding of gaseous adsorbates to the ice surface [21]. BF molecule was added to have a series of molecules with increasing dipole moment. The purpose of this Letter is to present a comprehensive study of the structure and dynamics of the trimers ðH2 OÞ2    XY, XY ¼ BF, CS, N2 specially: (i) to find the energy minima and the saddle points on the intermolecular potential energy surface (IPES), (ii) to describe the low-energy tunneling pathways, (iii) to characterize the minima including the prediction of spectroscopically observable features, such as vibrational frequency shifts, (iv) to determine the origin and properties of three-body interaction in these trimers within framework of the intermolecular perturbation theory combined with the supermolecular (SM) scheme.

2. Methods of calculations Optimal molecular structures and harmonic vibrational frequencies of the ðH2 OÞ2    XY, XY ¼ N2 , BF, CS were determined by means of second-order Møller–Plesset perturbation theory (MP2) with the aug-cc-pVDZ and the aug-ccpVTZ basis sets at the frozen core approximation. The structures corresponding to the transition

states were optimized at the MP2/aug-cc-pVDZ level. The geometry optimizations have been performed for the isolated subunits and for the entire complex at the MP2 level of theory without taking into account the basis set superposition error in the optimization procedure. As was shown by Hobza et al. [22] for the water dimer, the CPcorrected intermolecular distances are systematically longer than the values evaluated by the standard gradient optimization. But this difference
). The is small for the large basis set (ca 0.02 A interaction energies have been evaluated for the MP2 optimal structures using the MP4 method. The SM calculations have been done with the GA U S S I A N 94 program [23]. SM The SM interaction energy of the complex, Eint , has been obtained by subtraction the energies of the monomers from the energies of the complex (the superscript ‘SM’ denotes the method used in the actual calculation). Here, the geometries of the monomers correspond to those in the complex and are deformed from their equilibrium values. In the calculations of the interaction energies we employed the full basis of the trimer, i.e., we corrected the computed interaction energies for the basis set superposition error using the prescription of Boys and Bernardi [24]. This is a rather commonly accepted procedure to obtain reliable interaction energies [25], despite some contradictory arguments reported in the literature [26]. The dissociation energies DSM were obtained from the 0 interaction energies by adding the correction for the zero-point vibrational motion. The zero-point correction D(ZPE) have been calculated at the MP2 level within the harmonic approximation. There is a number of considerations in the selection of a basis set suitable to describe intermolecular interactions. First, it is necessary to reproduce various electric properties of the monomers as accurately as possible. This will ensure a correct description of the long-range interactions in the complex. Second, the sensitivity of these electric properties to the approach of the partner’s orbitals should be minimized in order to diminish the basis set extension effects. We decided to use the aug-cc-pVDZ and the aug-cc-pVTZ atomic basis sets [27]. As shown [28,29] generally these basis sets accurately reproduce the geome-

M.F. Rode, J. Sadlej / Chemical Physics Letters 358 (2002) 237–249

ð3Þ

tries, the frequencies and the electric properties of the isolated molecules and their complexes. Partitioning of the three-body interaction energy was performed in terms of the intermolecular Møller–Plesset perturbation theory (I-MPPT) as SCF discussed in [2,6]. The nonadditivity of EintNA encompasses nonadditivities of the exchange HL exch SCF and deformation DEdef contributions (three-body SCF SCF Eint in [2,6] we denote as EintNA to stress nonadditive character of this parameter) SCF SCF EintNA ¼ HF exch þ DEdef :

The nonadditivity of the three-body term EintNA contains three parts: the third-order dispersion, exchange and deformation ð3Þ

ð2Þ

ð2Þ

ð3Þ

ð3Þ

ð3Þ

ð30Þ

The disp is the third-order dispersion term, the ð3Þ DEexch is the third-order intracorrelation correction ð3Þ to the exchange effects and the DEdef is the thirdorder deformation correlation energy. It was SCF shown [30] that the DEdef could be approximated ðn;0Þ by the sum of induction components ind;r (r deð20Þ notes response term). The three-body terms ind;r ð30Þ and ind;r terms are calculated using coupled Hartee–Fock theory. The third-order induction nonadditivity may be interpreted as an electrostatic interaction between pairs of moments on A and B induced by the field of the third monomer C. All the terms ðijÞ have been derived within the basis set of the entire complex. The partitioning of the energy was carried out using TR U R L package [31]. Before presenting our results for the clusters, we begin with a comparison of the properties of the isolated subunits calculations with the basis sets used in this Letter. The data are reported in Table 1. As one can see, the basis sets considered here appear to offer a reasonable framework for studying the clusters. Geometries of the subunits agree favorably with the experimental values as do

The electrostatic energy is additive. Physically, the three-body HF exch term includes effects due to single exchanges within pairs of monomers and due to triple exchanges involving all three monoSCF mers simultaneously. The DEdef is determined asymptotically by the classic induction effects due to the multipole electrostatic polarization. The ð2Þ EintNA nonadditivity is dissected into the exchange and the deformation part. The second-order deð2Þ formation correlation, DEdef describes the intramonomer correlation correction to the SCF deformation term. The second-order exchange ð2Þ term, DEexch includes the exchange counterpart of two additive effects: the second-order electrostatic correlation and the second-order dispersion ð2Þ

ð30Þ

EintNA ¼ disp þ DEdef þ DEexch :

ð1Þ

EintNA ¼ DEdef þ DEexch :

239

ð2Þ

Table 1 The multipole moments for the monomers N2 , CO, BF and CS [34,35] MP2/basis set

N2

CO

BF

CS

DZ TZ Expt.

1.132 1.114 1.0977

1.150 1.139 1.1283

1.304 1.271 1.2628

1.561 1.543 1.5349

Frequency mXY (cm1 )

DZ TZ Expt.

2157.0 2186.8 2358.6

2072.0 2109.7 2169.8

1263.3 1395.5 1402.1

1282.6 1296.3 1285.1

Dipole moment lXY (D)

DZ TZ Expt.

0.00 0.00 0.00

0.24 0.25 0.122

0.81 0.95 0.5b

2.15 2.23 1.958

Quadrupole moment HXY , ea20

DZ TZ Expt.

)1.126 )1.163 )1.09

)1.528 )1.489 )1.44

)2.649 )2.704

)2.000 )2.038

Polarizability aXY , a30

DZ TZ Expt.

11.69 11.67 11.74

13.23 13.28 13.08

19.81 19.98

28.24 28.42

Monomer Distance rXY


) (A

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the multipole moments. Typical overestimation can be noted when the calculated vibrational frequencies are compared with the experimental values.

3. Results and discussion 3.1. Structure and energetics Similar as for the trimer ðH2 OÞ2    CO [5] three stationary points have been found on the potential energy surface for the trimers ðH2 OÞ2    XY complexes, where XY ¼ N2 , BF and CS. The global minimum in each case corresponds to the cyclic triangular hydrogen-bonded structure (Fig. 1a, c–X–a, for the explanation of the symbols, see the figure caption) with the X atom of the XY

(b)

(a)

(c) Fig. 1. Geometry of the global minimum c–X–a (a) and the minima c–Y–a (b) and l–X–d (c) of the trimer ðH2 OÞ2    XY, XY ¼ BF, CS, N2 (the symbols X or Y mean the X or Y atoms bonded to the proton acceptor (a) or proton donor (d) water molecule in the cyclic (c) or linear (l) complex).

molecule bonded to the O2 atom of the proton acceptor water molecule H2 OðaÞ . The hydrogen atoms not involved in the hydrogen bonds lie alternatively above and below the plane of the heavy atom ring (denoted ‘up’ and ‘down’, or u and d). This structure has no spatial symmetry, but there are two equivalent minima corresponding to the up–down (ud) and down–up (du) enantiomeric structures. In Table 2 we report the interaction energies Eint with respect to the monomers in the same geometries as they have in the clusters and the dissociation energies D0 using the aug-cc-pVDZ and augcc-pVTZ basis sets. We include the results for (H2 OÞ2    CO trimer [5] in this table for comparison. Inclusion of the correlation effect at the MP2 level is important and increases the interaction energies. The MP4 calculations change slightly this result. As was found previously for (H2 OÞ2    CO trimer [5] the convergence of the Møller–Plesset expansion for the interaction energy appears to be good, the MP4 approximation providing over 98% of the CCSD(T) result. The inspection of this table MP2 MP4 shows that the Eint and Eint for global cyclic minima (MG) (the absolute values) is found to decrease in the order X ¼ CðCSÞ > BðBFÞ > CðCOÞ > NðN2 Þ. The interaction energy of linear minimum structures l–X–d was found to decrease in the same order X ¼ CðCSÞ > BðBFÞ > CðCOÞ > NðN2 Þ. The bigger the XY dipole moment is, the deeper is the cyclic global minimum c–X–a as well as the deeper is the linear local minimum l–X–d. Inclusion of the correlation effect at the MP2 level is important and increases the interaction energies. For all the complexes presented in Table 2 the convergence of the Møller– Plesset expansion for the interaction energy appears to be good. One may note that the results obtained with the larger aug-cc-pVTZ basis set for the trimer, listed in Table 2 suggest an average of ca. 10% increase in the interaction energy from the corresponding ones obtained with the smaller aug-cc-pVDZ set. It should be emphasized that the analysis with the aug-cc-pVTZ set was performed at the aug-ccpVTZ optimal geometries. The experimental dissociation energy of this complex is not known. Extensive calculations of the binding energies of

M.F. Rode, J. Sadlej / Chemical Physics Letters 358 (2002) 237–249

241

Table 2 The interaction energies Eint and the dissociation energies D0 (kcal/mol) for the three conformers: global (MG) and two local (ML) minima of the ðH2 OÞ2    XY, XY ¼ N2 , CO, BF, CS trimers calculated at the aug-cc-pVDZ and aug-cc-pVTZ basis sets XY N2 Kind of minimum MG

N2 ML

CO MG

CO ML

CO ML

BF MG

BF ML

BF ML

CS MG

CS ML

c–N–a

l–N–d

c–C–a

c–O–a

l–C–d

c–B–a

c–F–a

l–B–d

c–C-a

l–C–d

aug-cc-pVDZ Interaction energies, Eint SCF Eint )3.336 ð2Þ Eint )2.803 MP2 Eint )6.139 ð3Þ Eint 0.603 MP3 Eint )5.536 MP4 Eint )5.958

)3.563 )1.750 )5.313 0.316 )4.997 )5.234

)3.958 )3.041 )6.999 0.600 )6.398 )6.946

)4.813 )0.988 )5.800 )0.141 )5.942 )5.963

)3.775 )1.907 )5.681 0.310 )5.371 )5.662

)5.670 )3.291 )8.960 0.544 )8.417 )8.562

)4.712 )1.449 )6.162 )0.149 )6.311 )6.053

)4.327 )1.942 )6.269 0.229 )6.039 )6.173

)3.292 )6.424 )9.717 1.839 )7.877 )9.089

)4.304 )2.600 )6.904 0.605 )6.299 )6.813

Dissociation energies, D0 ZPE 2.913 DSCF )0.423 0 )3.226 DMP2 0 DMP3 )2.623 0 DMP4 )3.045 0

2.756 )0.807 )2.557 )2.241 )2.478

3.298 )0.660 )3.701 )3.100 )3.648

2.790 )2.023 )3.010 )3.152 )3.173

2.931 )0.843 )2.750 )2.440 )2.730

3.593 )2.076 )5.367 )4.824 )4.969

2.810 )1.902 )3.351 )3.500 )3.243

3.086 )1.240 )3.182 )2.953 )3.087

3.420 0.127 )6.297 )4.458 )5.669

2.992 )1.312 )3.912 )3.307 )3.821

aug-cc-pVTZ Interaction energies, Eint MP2 Eint )6.736

)5.719

)7.756

)6.302

)6.172

)10.161

)6.571

)6.916

)10.927

)7.609

Dissociation energies, D0 ZPE 3.004 DMP2 )3.732 0

2.788 )2.931

3.333 )4.423

2.811 )3.491

2.942 )3.230

3.623 )6.538

2.817 )3.754

3.002 )3.913

3.436 )7.491

3.011 )4.598

van der Waals complexes [25] indicate that all values in the aug-cc-pVTZ basis set are underestimated, mainly due to an underestimation of the dispersion component of the interaction energy, by ca. 5–10%. To continue this paragraph let us compare the geometric parameters of the trimers with the geometry of the respective dimers. The calculated distances between the oxygen and X atoms in the trimers are presented in Table 3 and are compared with the respective distances in the dimers. One may note the shortening of the distances RðO    XÞ between the heavy atoms in the trimers in comparison with the distances in the respective dimers. The O    O distance is also shorter than the corresponding distance in the water dimer,
(in the same basis set). This shortening 2.907 A imposes strong strains in the hydrogen bonds. The heavy atoms form a deformed equilateral triangle, with the angles given in Table 3. One may note here that the equilibrium structures of the trimers

are quite similar to the structure of the water trimer [28]. The second minimum was located for the XY molecule approaching the hydrogen atom of the acceptor water molecule H2 OðaÞ from the side of the Y atom (configuration c–Y–a, Fig. 1b). We found such a local minimum for the XY ¼ BF and previously for CO [5]. All sides and angles in O1 O2 F4 triangle are very similar to their counterparts in the global minimum (see Table 3):
, RðO1    F4 Þ ¼ 2:944 A
, RðO2    F4 Þ ¼ 3:187 A
RðO1    O2 Þ ¼ 2:819 A, \O1 O2 F4 ¼ 58:0°; \O2 F4 O1 ¼ 55:4°; \F4 O1 O2 ¼ 58:0°. In addition to two cyclic structures of the trimers we considered the linear configuration: linear structure l–X–d is a minimum, presented in Fig. 1c. In these linear complexes l–X–d characterize by the smaller interaction energies, the O1    O2 distances (presented in Table 3) are longer than this value in the water dimer (contrary to the cyclic structure c–X–a).

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Table 3
) and the angles (deg) between heavy atoms in the trimers in comparison with the distances in the respective dimers. The distances (A All values calculated at the MP2/aug-cc-pVTZ basis set Distances

Cyclic trimer, c–X–a rðO1    X3 Þ

X ¼ NðN2 Þ C(CO) B(BF) C(CS) ðH2 OÞ3

3.160 3.109 2.784 3.426

Dimer rðO2    X3 Þ

rðO1    O2 Þ

3.132 3.152 3.148 2.988

2.873 2.856 2.821 2.870 2.787

Angles

ðX3    O1    O2 Þ

ðX3    O2    O1 Þ

O1    X3    O2 Þ

X ¼ NðN2 Þ C(CO) B(BF) C(CS)

62.3 63.6 28.3 55.8

63.3 62.1 55.3 71.5

54.3 54.3 56.4 52.6

Distances

Linear trimer, l–X–d rðO2    X3 Þ

rðO1    O2 Þ

rðO1    X3 Þ X ¼ N(N2 ) C(CO) B(BF) C(CS)

5.419 5.410 5.454 5.342

rðO    XÞ 3.240 3.286 3.344 3.158 2.907

Dimer

3.302 3.335 3.392 3.199

3.2. The nonadditivity and the nature of the threebody effects The decomposition of the interaction energy into three-body nonadditive contributions is reported in Table 4 for the structures of the trimer described above. The inspection of this table shows that the three-body nonadditive contribution strongly depends on the nature of interacting systems and on the conformation. The nonadditive effect leads to the stabilization of the trimer energies in the cyclic structures and to the destabilization in the linear structures. This result means that the cyclic character of the complex can be characterized by the sign and the magnitude of the three-body nonadditive term as follows: Eint;NA ðcyclicÞ < 0 < Eint;NA (linear). As in the other polar trimers [32,33] total nonadditivity is quite a large percentage: it varies from 17% for BF molecule, 13% for CS and is the smallest one for N2 molecule. This means that the structure and properties of the ðH2 OÞ2    XY, XY ¼ BF, CS, N2 could not be described by assuming the pairwise additivity of the interaction potential. This resembles the situation in the water trimer (where

2.918 2.927 2.942 2.953

rðO    XÞ 3.240 3.286 3.344 3.158

nonadditive effect is ca. 18% [3]) and many other trimers i.e., ðH2 OÞ2    HCl [32,33]. In cyclic trimer ðH2 OÞ2    XY, the water  water hydrogen bond is reinforced when its proton donor moiety acts as the acceptor with respect to the XY molecule and when the acceptor moiety acts as a hydrogen bond donor to XY. On the other hand, the water  water hydrogen bond is weakened in linear trimer, when the water molecule behaves as double donor of proton. This cooperative effects is as much as in the water trimer for XY ¼ BF, CS molecules, but is much smaller in the case of XY ¼ CO and N2 molecules. What is the nature of the nonadditivity in the cyclic and the linear trimers? The perturbation analysis offers a unique opportunity to elucidate the physical origin of the nonadditive effects. To achieve better understanding of the nature of the interaction in the cyclic and linear trimers the three-body contributions to the interaction energy for the global and local minima of the trimers are shown in Table 4. We begin with discussing the global minima. The total three-body effects at the MP3 level are substantial. All the components ð30Þ except disp for the cyclic minima are attractive.

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Table 4 The SM nonadditive energies EintNA and MBPT nonadditive three-body corrections ði;jÞ (kcal/mol) for the three conformers: global (MG) and two local (ML) minima of the ðH2 OÞ2    XY; XY ¼ N2 , CO, BF, CS trimers calculated at the aug-cc-pvDZ basis set XY Kind of minimum

N2 MG

N2 ML

CO MG

CO ML

CO ML

BF MG

BF ML

BF ML

CS MG

CS ML

Eint

c–N–a

l–N–d

c–C–a

c–O–a

l–C–d

c–B–a

c–F–a

l–B–d

c–C–a

l–C–d

Nonadditive energies, EintNA SCF EintNA )0.363 ð2Þ EintNA )0.005 MP2 EintNA )0.368 ð3Þ EintNA 0.033 MP3 EintNA )0.335 MP4 EintNA )0.345

0.154 0.006 0.160 )0.008 0.152 0.151

)0.590 )0.051 )0.641 0.076 )0.565 )0.600

)0.506 0.152 )0.354 )0.036 )0.390 )0.349

0.199 0.051 0.250 )0.032 0.217 0.233

)1.713 0.135 )1.578 0.090 )1.488 )1.478

)0.638 0.068 )0.570 0.003 )0.567 )0.545

0.396 )0.008 0.388 )0.025 0.363 0.361

)1.173 )0.137 )1.310 0.100 )1.210 )1.215

0.377 0.066 0.443 )0.039 0.404 0.417

MP4 MP4 /Eint ,% EintNA

)2.89

8.63

5.85

)4.12

17.26

9.00

)5.85

13.37

)6.12

)0.452 )0.053 )0.300 )0.095 0.019

0.167 0.032 0.102 0.034 )0.003

)1.405 )0.308 )0.862 )0.299 0.077

)0.629 )0.009 )0.318 )0.122 0.020

0.348 0.048 0.275 0.084 )0.002

)1.125 )0.048 )0.779 )0.244 0.037

0.332 0.045 0.227 0.069 )0.005

)5.942

)5.371

)8.417

)6.311

)6.039

)7.877

)6.299

5.79

ði;jÞ

Three-body nonadditive MBPT energy corrections,  SCF DEdef )0.320 0.129 )0.509 HL )0.043 0.025 )0.081 exch ð20Þ ind;r )0.210 0.071 )0.367 ð30Þ ind;r )0.071 0.022 )0.111 ð30Þ disp 0.028 )0.004 0.038 MP3 Eint

)5.536

)4.997

)6.398

The largest contributions come from the attractive SCF EintNA terms. The common feature for the global ð2Þ ð3Þ cyclic trimers are very small EintNA and EintNA ð3Þ terms. The EintNA terms are destabilizing mainly due to the dispersion terms, as this term includes the exchange correlation and deformation effects in addition to the three-body dispersion term. This means that the correlation effects play a secondary role in the total nonadditivity of these structures. The three-body SCF contributions are very close SCF to the three-body DEdef terms. The SCF deformation terms thus determine the entire nonadditivity in the global minima of the trimers. In every cyclic structure the sign and the magSCF nitude of the DEdef effect are determined by the sum of the induction nonadditivity in the second ð20Þ ð30Þ ind;r and in the third-order ind;r of the perturbation theory. The three-body induction effects in the second-order may be viewed as the energy of any monomer in the combined field of the two other. This effect is stabilizing in the cyclic global trimers. The XY molecules act as a ‘hydrogen bond acceptor’ from an H atom of H2 OðaÞ in these structures. They may be viewed as having hydrogen bonds arranged in a concerted fashion as follows:

O ! O ! XY or O ! O ! YX (where arrows describe hydrogen bonds). The third-order induction nonadditivity may be interpreted as an electrostatic interaction between pairs of moments on A and B induced by the field of the third monomer C. As in the case of the second-order these terms are attractive. Contrary to the cyclic trimers, for the linear structures the MP3 three-body contributions MP3 EintNA are destabilizing. The largest contributions come from the SCF term, which may be dissected into the exchange components and the deformation SCF terms. The attractive dispersion terms are negligible. In these structures, as in the cyclic SCF trimers, the sign and the magnitude of the DEdef are determined by the sum of the induction nonadditivities in the second and third-order perturbation theory. The XY molecules act as a ‘hydrogen bond acceptor’ from H atom of H2 OðaÞ , but the destabilizing three-body effect may be viewed as having O ! O XY branched-out of hydrogen bonds. To conclude, in trimers involving a molecule bound to hydrogen-bonded water dimer induction effects are expected to dominate.

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3.3. Frequency analysis The vibrational harmonic frequencies and IR intensities of three conformers of the trimer were calculated at the MP2/aug-cc-pVTZ level for the respective optimized geometries. The results are presented in Table 5. This table lists also the changes in the parameters caused by formation of the H-bonded complex: the shifts of the harmonic frequencies and the ratio of the intensities in comparison with their monomer values. Fig. 2 illustrates the changes of the frequencies and the intensities of the trimers. Let us discuss the intramolecular modes of the trimer. The H2 O monomer frequencies in the augcc-pVTZ are 1628, 3822, 3948 cm1 . Two stretching modes of water – donor and acceptor, are shifted to the red. The biggest red-shift is observed for the proton donor water molecule in all trimers. The water excitations are delocalized on the complex. One can find the coupling of the proton donor’s and proton acceptor’s OH stretching vibrational modes. This coupling is revealed in the calculated spectrum by the overlapping of the respective frequency values (Fig. 2). The biggest effect is observed for the cyclic global minima (c–X– a); it is smaller for the cyclic local minima (c–Y–a). For the linear local minima (l–X–d) there is no overlapping of the frequencies. The biggest redshift one can notice for the trimer ðH2 OÞ2    CS. The strong intensity enhancement of the symmetric stretching mode of the donor and the acceptor water molecules agrees with the earlier calculations of such systems with the hydrogen bond. The HOH bending frequency is blue-shifted from the monomer value, one (the donor water) with the small intensity changes, second (the acceptor water) with much bigger lowering of intensity. In contrast to the water molecules, the complex mode associated with the intramolecular mode of the XY ¼ BF, CS, N2 is changed little from its monomer value (see Table 5 and Fig. 2). We have observed the blue-shift of the XY stretching frequency when the monomer XY ¼ (CS, BF) is Xapproaching the water molecule and the red-shift when XY is Y-approaching. The small blue-shift has been also observed in the ðH2 OÞ2    N2 complex.

In Table 5 we report the intermolecular frequencies also. Our estimate for the lowest harmonic vibrational frequency, which is interpreted as the intermolecular vibration–rotation-tunneling band is 51 for CS, 41 for BF and 47 cm1 for N2 . The intermolecular stretching modes YX    H–O are 183 for CS, 180 for BF and 173 cm1 for N2 . Because of the difficulty in the detection of bands in the far-IR region, the calculations can play a particularly important role in the prediction and the interpretation of the spectrum. 3.4. Tunneling motion of the molecules in the complex Let us discuss now the internal motion of the molecules in the complex. First we discuss the tunneling motion of the water molecules. Later, we examine the tunneling motion of XY molecules. Both motions are discussed on the basis of aug-ccpVDZ results. To connect the structures corresponding to the equivalent global minima c–X–a (ud) and (ud), we considered two low-energy pathways. Schematic representations of these pathways are reported in Fig. 3a,b. The first, called the flipping pathway (Fig. 3a) connect the enantiomeric forms ud and ud via a flipping transition state (a saddle point on the IPES). The flipping motion is the rotation of the water monomers about their hydrogen bonds. The flipping transition state ppX (see Fig. 3 for the explanation of the symbols), which corresponds to the structure with two free protons in the plane of the ring, is lying 0.37, 0.42, 0.39 kcal/mol (MP2/aug-ccpVDZ) over the minimum level for XY ¼ N2 , BF, CS, respectively. The other pathway, called bifurcated tunneling (Fig. 3b), involves the saddle point (bdX ) with donor water molecule hydrogen-bonded forming the bifurcated hydrogen bond with the acceptor water. It is 1.55, 1.80, 1.50 kcal/mol (MP2/aug-ccpVDZ) above the global minimum (TS, bdX ) for XY ¼ N2 , BF, CS, respectively. During the rotation of the bifurcated water molecule described above, flipping of the free hydrogen of the other water molecule takes place. The O1    O2 and

Table 5 The calculated frequencies (cm1 ), the IR intensities (kM/mol), for the stable conformers of the ðH2 OÞ2 XY; XY ¼ N2 , BF, CS calculated at MP2/aug-cc-pVTZ Vibration No.

c–N–a mc

l–N–d

c–B–a

Dm

Ic =Im

mc

Ic

Intramolecular frequencies Stretch. XY 2191 Bend. d 1629 Bend. a 1652 Stretch. sym. d 3706 Stretch. Sym. a 3800 Stretch. As. D 3913 Stretch. As. A 3924

0.7 84.3 35.4 255.2 52.2 105.3 153.7

4 1 24 )116 )22 )35 )24

1.18 0.49 45.89 9.38 1.40 2.04

2189 1629 1652 3729 3814 3909 3935

0.1 84.1 27.7 255.3 13.6 234.3 95.3

Intermolecular frequencies 8 47 9 58 10 80 11 104 12 116 13 173 14 178 15 199 16 253 17 448 18 614

9.8 2.5 26.1 29.7 48.0 71.9 18.0 75.5 196.9 61.5 100.3

16 37 60 73 83 133 160 196 293 400 627

2.7 0.6 5.0 48.1 4.8 0.9 48.5 229.6 103.9 59.7 84.7

l–B–d

Dm 2 1 24 )93 )8 )39 )13

mc

Ic

Dm

Ic =Im

mc

Ic

Dm

Ic =Im

1.17 0.39 45.91 2.44 3.10 1.26

1434 1629 1651 3642 3681 3896 3901

151.8 64.1 31.1 142.5 404.9 117.7 106.0

38 1 23 )180 )141 )52 )47

0.93 0.89 0.43 25.63 72.81 1.56 1.40

1347 1627 1643 3700 3810 3910 3931

174.5 99.3 23.3 269.3 14.9 116.9 116.3

)49 )1 15 )122 )12 )38 )17

1.07 1.39 0.33 48.43 2.68 1.55 1.54

75 88 124 143 180 193 197 304 346 556 686

8.9 5.8 95.7 23.6 10.0 0.4 80.9 77.2 140.0 107.4 97.6

c–C–a

40.9 57.7 86.4 102.8 127.3 160.1 176.0 201.6 212.0 397.5 632.8

l–C–d

Ic

Dm

Ic =Im

mc

Ic

Dm

Ic =Im

mc

1418 1630 1653 3728 3812 3846 3934

168.3 81.7 17.4 121.1 48.8 414.3 91.6

23 2 25 )94 )10 )102 )14

1.03 1.14 0.24 21.78 8.77 5.49 1.21

1318 1638 1665 3640 3685 3895 3902

19.6 54.6 59.0 254.7 308.3 130.8 86.0

22 10 37 )182 )137 )53 )46

0.48 0.76 0.82 45.81 55.44 1.73 1.14

1314 1630 1656 3728 3809 3835 3933

Intermolecular frequencies 8 20 9 52 10 71 11 77 12 100 13 112 14 156 15 201 16 429 17 439 18 613

5.7 7.9 49.0 9.7 21.9 1.8 14.9 251.8 67.4 48.7 74.5

51 60 92 142 164 183 194 299 363 555 651

14.2 7.2 16.1 90.8 17.4 11.0 79.8 60.5 149.4 124.8 93.9

18 40 61 63 96 101 156 203 465 486 604

Ic

Dm

34.4 18 76.569 2 28.2 28 84.8 )94 108.2 )13 517.3 )113 89.398 )15 9.8 9.4 12.6 49.5 5.6 22.1 9.0 298.9 39.7 93.1 64.3

Ic =Im 0.211 1.0682 0.3939 15.253 19.459 6.8547 1.1846

0.2243 1.0474 7.3796 16.723 55.677 26.905 95.47 174.68 108.83 44.995 95.454

245

mc Stretch. XY Bend. d Bend. a Stretch. sym. d Stretch. Sym. a Stretch. As. D Stretch. As. A

c–F–a

Ic =Im

M.F. Rode, J. Sadlej / Chemical Physics Letters 358 (2002) 237–249

Ic

246

M.F. Rode, J. Sadlej / Chemical Physics Letters 358 (2002) 237–249

TS :

ppðH2 OÞ2 > ppBðBFÞ > ppCðCSÞ ppCðCOÞ > ppNðN2 Þ > ppOðCOÞ þ0:58

þ0:44

þ0:39

þ0:39

þ0:37

þ0:34

DEintNA : þ0:16

þ0:05

þ0:04

þ0:02

þ0:02

DEint :

Second pathway – the bifurcated tunneling gives the following series: TS :

bdðH2 OÞ2 < bdCðCSÞ < bdNðN2 Þ < bdCðCOÞ < bdOðCOÞ < bdBðBFÞ þ1:46

þ1:50

þ1:55

þ1:58

þ1:72

þ1:82

DEintNA : þ0:56

þ0:12

þ0:19

þ0:14

þ0:67

DEint :

Fig. 2. Comparison of the harmonic frequencies and intensities calculated at the MP2/aug-cc-pVTZ level for the ðH2 OÞ2    XY trimer and isolated monomers, XY ¼ BF, CS, N2 .

X3    O2 distances and the angles in the triangle do not differ much from the equilibrium values especially for the flipping transition state. One may note that the heights of the barriers separating the enantiomeric global minima are quite low, so it should be possible to observe spectroscopic transitions resulting from the tunneling between the equivalent equilibrium structures via flipping pathways. Considering the tunneling motion of the water dimer protons, can one inquire what is the influence of the monomer XY on the heights of the barriers? Adding the results for the XY ¼ CO [5] and for the water dimer to our data we have found that the height of the tunneling barrier is determined by the cyclic character of the respective transition state (TS) structure. For flipping pathway we have the series:

Fig. 4a presents the rotation of the XY molecule in the trimer from the global minimum configuration c–X–a to the local minimum c–Y– a configuration. The transition state, denoted as baXY , is located at 0.90, 1.66 and 2.74 kcal/ mol (MP4/aug-cc-pVDZ) above the global minimum and contains the XY molecule, which is perpendicular to the ring plane and forms the hydrogen bond with acceptor water molecule, for XY ¼ N2 , CO and BF, respectively. CS molecule do not form the stable trimer ðH2 OÞ2    SC. Looking at the numbers cited above, we can answer the question: what is the influence of the XY monomer on the height of the proton tunneling barrier in the water dimer? For the case of the flipping pathway ppX , when the cyclic character of the complex is maintained along the pathway, the height of the tunnelig barrier of the water dimer protons is lowered when compared with the dimer of water. Contrary to that, for the bifurcated pathway bdX this barrier is higher in comparison with the isolated water dimer.

M.F. Rode, J. Sadlej / Chemical Physics Letters 358 (2002) 237–249

247

Fig. 3. Geometries of the (a) flipping and (b) bifurcated transition states on the potential energy surface of the c–X–a trimer ðH2 OÞ2    XY. The numbers calculated at the MP4/aug-cc-pvDZ level relative to the global minimum are given in kcal/mol (the symbols ppX means two water molecules in the plane bonded by the X atom to the XY; bdX means the donor water molecule H2 OðdÞ bifurcated).

248

M.F. Rode, J. Sadlej / Chemical Physics Letters 358 (2002) 237–249

Fig. 4. Geometries of the transition state corresponding to the rotation of the XY molecule ðH2 OÞ2    XY. The numbers calculated at the MP4/aug-cc-pvDZ level relative to the global minimum are given in kcal/mol (the symbol baXYo means bifurcated configuration to water acceptor H2 OðaÞ , out-of-plane).

4. Conclusions In the present study we have investigated the structural, energetic and spectroscopic properties of the ðH2 OÞ2    XY, XY ¼ N2 , BF, CS trimers. Our results can be summarized as follows: 1. The global minimum on the potential energy surface of the ðH2 OÞ2    XY trimer corresponds to the cyclic triangular hydrogen-bonded structure. The heavy atoms form a slightly deformed equilateral triangle. The hydrogen atoms forming hydrogen bonds lie almost perfectly in the plane of the heavy atoms, while the external protons H1 and H3 are strongly distorted from the hydrogen-bonded ring. This structure has no spatial symmetry, but there are two equivalent minima corresponding to the ud and du enantiomeric structures. 2. SM calculations of the pair and three-body interaction energies show that the structure and properties of the ðH2 OÞ2    XY complex could not be accurately described by assuming the pairwise additivity of the interaction potential in the case of XY ¼ BF, CS molecules, the three-body terms representing as much as 13– 17% of the total MP4 interaction energy of the trimers. On the other hand, the three-body

terms are representing only few % of the total interaction energy of the trimer for XY ¼ N2 . The analysis of the components of the threebody terms shows that the nonadditivity in SCF the trimers is determined by the DEdef , i.e., the induction effect which is restrained by the exchange effect. The three-body induction leads to the additional stabilization in the cyclic trimers, which can be schematically described as O ! O ! XY. The three-body term is destabilizing in the linear trimers which can be described as O ! O XY. The arrows in the same direction correspond to reinforcement of the induction effect, while those in the opposite direction correspond to induction effects which interfere destructively. 3. Despite a fairly weak character of the hydrogen bond in the ðH2 OÞ2    XY trimer, a substantial red-shift and intensification was found for the OH stretch of water. The spectrum of the XY is also changed (blue-shift). 4. A study of the tunneling motion of the water molecules across the plane of the ring results in two low-energy pathways corresponding to the hydrogen bond network rearrangement processes. The flipping of the free hydrogen atoms across the plane ring is the lowest energy rear-

M.F. Rode, J. Sadlej / Chemical Physics Letters 358 (2002) 237–249

rangement pathway. The other pathway involves transition state with one water molecule hydrogen-bonded to the other water in a bifurcated manner. It should be possible to observe spectroscopic transitions resulting from the tunneling between the equivalent structures via these pathways because the heights of the barriers corresponding to these transition states are quite low.

Acknowledgements We are grateful to Drs. S.M. Cybulski and G. Chalasinski for the TR U R L program. This work was supported by the Polish Scientific Research Council (KBN), Grant No. 3 T09A 100 19. We thank ICM in Warsaw University for computer time.

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