J. Phys. Chem. 1996, 100, 10875-10881
10875
Ab Initio Study of Nonadditive Effects in the (H2O)2‚‚‚H2 Cluster J. Sadlej* Department of Chemistry, UniVersity of Illinois at Chicago, Box 4348, Chicago, Illinois 60680, and Department of Chemistry, UniVersity of Warsaw, 02-093 Warsaw, Poland
S. M. Cybulski* Department of Chemistry, Miami UniVersity, Oxford, Ohio 45056
M. M. Szcze¸ s´niak* Department of Chemistry, Oakland UniVersity, Rochester, Michigan 48309 ReceiVed: September 20, 1995; In Final Form: March 28, 1996X
The three-body interaction energy in the (H2O)2H2 cluster is evaluated using the supermolecular MøllerPlesset perturbation theory and analyzed in terms of the perturbation theory of intermolecular forces. Four main configurations are investigated in which the H2 molecule is attached to either the O or H atoms of the water dimer. In these structures nonadditivity originates mainly from the induction effect. The exchange and dispersion nonadditivities are of secondary importance. The overall three-body effect can be reasonably well reproduced at the uncorrelated level. The water‚‚‚water hydrogen bond is reinforced when its proton donor moiety acts as the hydrogen bond acceptor with respect to the H2 molecule and when the acceptor moiety acts as a hydrogen bond donor. On the other hand, the water‚‚‚water hydrogen bond is weakened when the donor water molecule behaves as a double donor or the acceptor water as a double acceptor.
I. Introduction In modeling condensed phase systems bound by hydrogen bonding and van der Waals forces, it is frequently assumed that the interactions are pairwise additive, i.e. that the potential energy can be approximated as a sum of interactions between all pairs of constituent molecules. However, deviations from this approximation are often substantial and thus have been a focus of intensive studies during the last decade.1-6 The studies of many-body effects have been particularly stimulated by the recent advances in the high-resolution spectroscopy of small clusters.2,3,6 The present study focuses on a cluster composed of two water molecules and the H2 molecule. A detailed understanding of the interactions between water and molecular hydrogen is needed for investigations of condensed phase systems containing these molecules. Examples of such systems which have been the subject of both recent experimental investigations and computational modeling include H2 adsorbate on icy surfaces7-19 and H2 dissolved in water20-22 and in ice.23 Previous ab initio studies involved the H2O‚‚‚H2 dimer as a model for the pair interactions in such systems. In the next step the three-body interactions should be examined. To this end we chose (H2O)2‚‚‚H2 as a model unit. The nonadditive interactions are likely to be of particular interest in the ice‚‚‚H2 adsorbate system due to the asymmetric molecular environment. The goal of this study is to determine the origin and properties of the threebody interactions in this cluster. The examined properties of three-body interactions will help in the modeling of many-body effects in the ice‚‚‚H2 adsorbate system. Since the work of Clementi et al.,24 it has been argued that the many-body effects in water, although fairly small, cannot be neglected in computer simulations of liquid water25,26 and in the investigation of the structure of ice.26,27 The nonadditive X
Abstract published in AdVance ACS Abstracts, June 1, 1996.
S0022-3654(95)02781-X CCC: $12.00
effects in the trimer and larger clusters of water were the subject of numerous studies.24,27-33 A common conclusion is that the major component of the nonadditivity of the interaction in the water trimer originates from the electric polarization.24,25,29 Recently, the origins of three-body effects in the trimers of HF,34 HCl,34 H2O,29 NH3,35 CH4,36 Ar3,37 Ar2HCl,38 and Ar2H2O39 have been analyzed systematically. These studies were carried out within the framework of the intermolecular Møller-Plesset perturbation theory combined with the supermolecular scheme.40-45 This treatment allows for a dissection of interaction energy into the physically interpretable components, such as exchange, induction, and dispersion nonadditivities. The conclusion of these investigations is that the nonadditive three-body effect does not originate from a single source, but rather depends on the nature of the interacting species.4,5 The present study on the (H2O)2‚‚‚H2 cluster was preceded by the investigation of the interaction potential of H2O‚‚‚H2.7,8,46,47 The bonding in this dimer has been found7,8 to be dominated by the electrostatic interaction followed by the dispersion energy. Owing to the fact that the water molecule can act as both a proton donor and an acceptor, two stable configurations of H2O‚‚‚H2 were found. In the “zz” configuration, H2 lies on the water bisector and points to the O atom of water; in the “ohx” configuration, one of the H atoms of H2O points to the H2 center in a T-shaped configuration (with the H2 axis perpendicular to the water plane).7 Investigation of the (H2O)2H2 system thus offers an interesting opportunity to study nonadditive effects in four different bonding configurations, with H2 attached to the O atom of either the proton donor or proton acceptor H2O of the water dimer and with H2 attached to the H atoms of either the proton donor or the proton acceptor water molecule. The two-body and three-body terms will be evaluated for each geometry, and each term will be dissected into its fundamental components. The anisotropy of the three-body components will © 1996 American Chemical Society
10876 J. Phys. Chem., Vol. 100, No. 26, 1996
Sadlej et al.
also be studied. A careful analysis of each term will lead to an improved understanding of nonadditivity on a fundamental level. In section II we discuss the method and the definitions. In section III results and discussion are presented. In section IV we discuss cooperative effects. Section V is a summary of the present study.
order of S-MPPT ∆E
SCF
∆E(2)
II. Methods and Definitions A. Partitioning of the Three-Body Interaction. In the supermolecular approach the total energy of a cluster ABC can be decomposed as34 (i) EABC )
TABLE 1: Summary of Nonadditive Contributions Arising in S-MPPT
(i) ∆EXY + EX(i) + ∑ ∆EX(i) + ∑ ∑ X)A,B,C X>Y)A,B,C X)A,B,C (i) ∆EABC
(1)
where (i) denotes a particular level of theory, e.g. HF, an order of Møller-Plesset perturbation theory, or any other sizeconsistent treatment of correlation effects, such as coupled cluster (CC) theory. The second, third and fourth terms describe, respectively, the one-, two-, and three-body contributions. The one-body term describes the effects of the geometry relaxation of subsystem X in the trimer. A two-body term (i) describes the pairwise interaction between two mono∑∆EXY (i) mers, and the ∆EABC term represents the three-body contribution arising between the relaxed-geometry monomers arranged in the same way as they occur in the complex. A major drawback of the supermolecular approach is that it does not provide us with a means of partitioning of nonadditivity into physically-sound contributions, making modeling of supermolecular terms particularly difficult. Such insights are provided by the perturbation theory of intermolecular forces. The Rayleigh-Schro¨dinger (RS) perturbation theory (PT) formalism predicts that there are three fundamental long range interaction energy components, electrostatic, induction, and dispersion, and that two of them, induction and dispersion, are nonadditive. The three-body induction terms appear in the second and higher orders of RSPT. The three-body dispersion terms appear in the third and higher orders. However, in the third order the induction and dispersion effects begin to couple, and the dispersion-induction threebody term can be distinguished. In the intermediate and short range the exchange effects (neglected by the RS formalism) appear, and they are also nonadditive. The intermolecular perturbation theory which includes the exchange effects in the most complete fashion is the symmetry-adapted perturbation theory (SAPT) also known as intermolecular Møller-Plesset perturbation theory (I-MPPT).43-45 Inclusion of exchange effects introduces the nonadditive exchange corrections of the following types: (i) the exchange counterparts of the RS nonadditive contributions and (ii) exchange counterparts of the additive RS terms. Thus in the first approximation, the perturbation theory predicts nonadditivities of the three basic types: the induction, the dispersion, and the exchange. However, it should be clear that within each category there might be effects of varying origins which require separate modeling. In order to understand the contents of supermolecular interaction energies in terms of perturbation theory predictions, a connection has been proposed between the perturbation theory of intermolecular forces, I-MPPT, which utilizes the MøllerPlesset partitioning of the Hamiltonian,40-42 and the supermolecular Møller-Plesset perturbation theory (S-MPPT), which is based on the same partitioning. In I-MPPT43-45 the interaction energy is expanded in terms of two perturbations, the interaction operator (V) and the intramonomer correlation (W).
∆E(3)
nonadditive I-MPPT term
n-body nonadditivity
HL exch SCF ∆Edef
3, 4, ..., n-body
(2) ∆Edef
3, 4, ..., n-body
(2) ∆Eexch
3, 4, ..., n-body
(30) disp (3) ∆Edef (3) ∆Eexch
3-body
3, 4, ..., n-body
3, 4, ..., n-body 3, 4, ..., n-body
The I-MPPT (ij) corrections are of the ith order in V and of the jth order in W (see ref 45 for more details). The relationship between the two theories is shown in Table 1. The nonadditivity of ∆ESCF encompasses nonadditivities of SCF contributions. the exchange HL exch and deformation ∆Edef HL Physically, the three-body exch term includes effects due to single exchanges (SE) within pairs of monomers and due to triple exchanges (TE) involving all three monomers simultaneously.48 The SE term was found important in Ar2-chromophore clusters,6,48,49 but in the interactions of three nonspherical systems it was found of secondary importance.50,51 For this reason no partitioning of HL exch into SE and TE will be SCF term is determined asymptotically attempted here. The ∆Edef by the classic induction effects due to the multipole electrostatic SCF is polarization. In the absence of exchange effects ∆Edef 52 shown to be a sum of I-MPPT induction components, ∞ (n0) ind,r . However, when the exchange effects are present ∑n)2 the series diverges. Nevertheless, for the purposes of modeling the induction nonadditivity, it is very interesting to examine these induction corrections. Recently Cybulski proposed a (20) (30) and ind,r terms using method of calculating the three-body ind,r the coupled Hartree-Fock intermolecular perturbation theory (r denotes response terms).53 Aware of the divergent nature of induction expansion, Cybulski53 included only the so called two(30) electron part of the ind,r nonadditivity (see ref 54 for more discussion). In the present study the approximation of SCF by these corrections will be tested. ∆Edef The ∆E(2) nonadditivity may be dissected into an exchange part and a deformation part. The second-order deformation (2) correlation, ∆Edef , describes the intramonomer correlation correction to the SCF deformation contribution. In the language of perturbation theory of intermolecular forces, it encompasses the induction correlation energy which allows for exchange effects. The second-order exchange correlation term, ∆E(2) exch, includes the exchange counterparts of two additive effects: the second-order electrostatic correlation, and the second-order dispersion. The nonadditivity of ∆E(3) contains three parts: (30) dispersion, exchange, and deformation. ind,r is the third-order dispersion nonadditivity of the UCHF type, ∆E(3) exch is the thirdorder intracorrelation correction to the exchange effects, and (3) ∆Edef is the third-order deformation correlation effects. All the terms (ij) have been derived within the basis set of the entire complex (i.e. trimer-centered basis sets, TCBS), and all the supermolecular ∆E(i) quantities have been evaluated via the counterpoise procedure,55 i.e. corrected for the basis set superposition error (BSSE).56 All the calculations were carried out using the Gaussian 92 program57 and the intermolecular perturbation theory package Trurl.58 B. Basis Set and Geometries. The calculations were carried out in the medium-polarized basis set of Sadlej,59 which consists
Nonadditive Effects in (H2O)2‚‚‚H2
J. Phys. Chem., Vol. 100, No. 26, 1996 10877
Figure 1. Geometrical configurations of the (H2O)2H2 cluster considered in this work.
of (10s6p4d/6s4p) primitives contracted to [5s3p2d/3s2p]. This basis set was designed to accurately reproduce electric properties of molecules and was shown to be very reliable in the calculations of the intermolecular forces.45 In the H2O‚‚‚H2 cluster, according to refs 7 and 8, the two potential wells were found: zz (H2 approaches the O atom collinearly with the C2 axis of H2O) and ohx (H2 forming a T configuration with the OH bond, perpendicular to the H2O plane). For (H2O)2H2 we considered two main types of configurations: type zz, in which H2 approaches, in the analogous manner, the O atoms of either the proton donor (D) or the acceptor (A), and type ohx, in which H2 approaches the direction of one of the OH bonds of either the proton donor or the acceptor. Overall the four main configurations are thus distinguished: zzD, zzA, ohxD, and ohxA. These structures are presented in Figure 1. The geometry of the water dimer part of the cluster was kept rigid with O‚‚‚O separation equal to 2.98 Å and the angle between the plane of proton acceptor and the O‚‚‚O axis equal to 120°. In the calculations the monomers were kept rigid: r(OH) ) 0.9572 Å, R(HOH) ) 104.52°, and r(HH) ) 0.7408 Å. For each configuration the intersystem distance R(O‚‚‚H) between the water dimer and the hydrogen molecule was taken from ref 7. The anisotropy of the twoand three-body contributions to the trimer energy was investigated by varying the selected orientation angles as defined in Figure 1. III. Results and Discussion A. Nature of the Three-Body Effects. We varied the inplane angles (see Figure 1) θd and θa for the zzD and zzA configurations, respectively, in the range from 70° to 150°. The angular dependence of the three-body term and its components is presented in Figure 2a,b. The equilibrium values for θd ) 127° and θa ) 127° for zzD and zzA configurations, respectively, are shown in Tables 2 and 3. We begin with discussing the two zz configurations presented in Tables 2 and 3. It must be noted that the two-body water‚‚‚ water interactions should be the same in all the configurations considered here (Tables 2-5) because the geometry of the water dimer does not change. The small deviations shown in Tables 2-5 (the second column) reflect the fact that the TCBS basis set which is used to calculate this interaction changes according to the position of H2. In the water‚‚‚water two-body interaction the major attractive (10) component is the electrostatic es energy. The exchange
Figure 2. Angular dependence of the three-body components of interaction energy: (a) the zzD configuration; (b) the zzA configuration.
TABLE 2: The Two- and Three-Body Contributions to the Interaction Energy for the zzD Structure of the (H2O)2H2 Cluster (in µhartrees), θd ) 127.74°, R(O‚‚‚H) ) 2.7296 Å energy
(H2O)d‚‚‚(H2O)a
(H2O)d‚‚‚H2
(H2O)a‚‚‚H2
3-body
∆ESCF ∆E(2) ∆E(3) ∆E(2) ∆E(3)
-5667.52 -1174.05 69.95 -6841.58 -6771.62
-345.22 -415.42 -12.46 -760.64 -733.10
-68.60 -13.20 0.27 -81.81 -81.53
-145.44 7.85 -4.54 -137.58 -142.12
(10) es HL exch SCF ∆Edef
-11635.94
-1072.23
-66.28
0.00
8816.95
1016.78
0.26
15.10
-2848.52
-289.68
-2.58
-160.54
(20) ind,r
-3711.63
-306.98
-2.56
-97.11
(30) ind,r (20) disp (12) es,r (30) disp
-4468.18
-301.50
-0.01
-38.76
-3010.35
-715.45
-19.60
10.17
69.47
5.97
0.0 0.0 -1.75
(20) repulsion HL exch is partially compensated by the attractive disp SCF and ∆Edef . In the H2O-H2 interaction the following can be noticed: in the interaction between H2 and the nearer H2O molecule, the exchange repulsion and electrostatic attraction nearly cancel one another and the net attraction is provided by (20) disp . In the interaction between H2 and the distant H2O, the exchange repulsion is near zero and the net interaction originates from electrostatics. The total three-body effect at the MP3 level is substantial and amounts to -142.12 µhartrees (-31 cm-1) for zzD and
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TABLE 3: The Two- and Three-Body Contributions to the Interaction Energy for the zzA Structure of the (H2O)2H2 Cluster (in µhartrees), θa ) 127.74°, R(O‚‚‚H) ) 2.7296 Å (H2O)d‚‚‚(H2O)a
(H2O)d‚‚‚H2
(H2O)a‚‚‚H2
3-body
∆E ∆E(2) ∆E(3) ∆E(2) ∆E(3)
-5662.5 -1180.3 68.4 -68442.7 -6774.4
1233.7 -720.1 -99.5 513.5 414.0
-348.0 -422.1 -14.3 -770.2 -784.5
78.8 5.7 41.6 84.5 126.0
(10) es HL exch SCF ∆Edef
-11632.2
-668.3
-1075.7
0.0
energy SCF
8817.1
2220.3
1017.4
-99.9
-2847.5
-318.3
-289.7
178.7
(20) ind,r
-3712.8
-528.1
-306.1
123.4
(30) ind,r (20) disp (12) es,r (30) disp
-4472.3
-528.2
-299.3
21.6
-3013.6
-1061.7
-725.8
0.0
7.0
-98.6
71.1
0.0 36.3
+126.05 µhartrees (27 cm-1) for zzA structures. In order to calculate the percentage contribution of three-body terms, one should consider the three-body effects evaluated at the oneorder higher level than the two-body interactions.4,37,60 A reasonable approach in the present case is to consider a ratio of three-body terms at the MP3 level and the two-body terms at the MP2 level. The percentage contributions of three-body terms for zzD is 1.8 % and for zzA is -1.8 %. In the case of the zzD configuration, the largest contribution of -145.44 µhartrees comes from the attractive ∆ESCF term. The threebody ∆ESCF contribution is very close to the three-body SCF term which amounts to -160.54 µhartrees. The SCF ∆Edef deformation term thus determines the entire nonadditivity in the zzD structure. For the zzA configuration the MP3 three-body contribution is destabilizing and amounts to +126.05 µhartrees. The largest repulsive contribution of 78.78 µhartrees comes from the SCF term which may be dissected into the -99 µhartrees due to the SCF . exchange component and 178.72 µhartrees due to ∆Edef Second in importance for zzA is the three-body ∆E(3) (41.58 µhartrees). This term includes the exchange correlation and deformation correlation effects in addition to the three-body dispersion term (see Table 1). However, the close agreement (30) between disp and ∆E(3) indicates that the exchange and deformation components of ∆E(3) are either small or cancel one another in this geometry. In conclusion, the dominant contributions to the nonadditivity in the zzA configuration of (H2O)2H2 are three-body SCF deformation and HL exchange followed by dispersion. We varied the angles (see Figure 1) φd and φa for the ohxD and ohxA configurations, respectively, in the range from 60° to 150°. The angular dependence of the three-body term and its components is presented in Figure 3a,b. The equilibrium values for φd ) 104° and φa ) 104° for ohxD and ohxA configurations, respectively, are shown in Tables 4 and 5. The total nonadditivity at the MP3 level is destabilizing for the ohxD configuration, while it is stabilizing for ohxA. In both cases, the SCF contribution dominates the entire nonadditivity. The SCF term in SCF nonadditivity, in turn, is dominated by the ∆Edef both ohxD and ohxA configurations. The configurations ohxA and ohxD differ by the signs of the SCF deformation and exchange nonadditivity terms. The common features for both structures are the negligible ∆E(2) and ∆E(3) terms. Overall, the correlation effects play a secondary role in the total nonadditivity of ohxD and ohxA structures. Here, the nonad-
Figure 3. Angular dependence of the three-body components of interaction energy: (a) the ohxD configuration; (b) the ohxA configuration.
TABLE 4: The Two- and Three-Body Contributions to the Interaction Energy for the ohxD Structure of the (H2O)2H2 Cluster (in µhartrees), Od ) 104.52°, R(O‚‚‚H) ) 3.2776 Å energy
(H2O)d‚‚‚(H2O)a
(H2O)d‚‚‚H2
(H2O)a‚‚‚H2
3-body
∆ESCF ∆E(2) ∆E(3) ∆E(2) ∆E(3)
-5658.2 -1172.7 70.7 -6830.9 -6760.2
-209.48 -551.94 -21.32 -761.42 -782.73
65.12 -34.61 -3.07 30.51 27.44
127.57 -8.87 -1.96 118.70 116.74
(10) es HL exch SCF ∆Edef
-11629.0
-1092.46
71.72
0.00
8816.6
1285.45
0.71
10.41
-2845.9
-402.48
-7.32
117.16
-3711.4
-427.03
-7.38
83.33
-4473.1
-231.50
-0.04
19.28
-3003.4
-736.76
-27.96
0.0
5.4
19.30
-8.07
0.0
(20) ind,r (30) ind,r (20) disp (12) es,r (30) disp
0.63
ditivity is strongly dominated by the induction effect and is well represented at the SCF level. Further insights into the nonadditive behavior of the (H2O)2H2 cluster can be gained by considering the angular dependence of the individual components of the three-body term. The variations of θd in the zzD geometry and θa in zzA are shown in parts a and b of Figure 2, respectively. Variations of φd and φa for the ohxD and ohxA geometries, respectively, are displayed in parts a and b of Figure 3. In all the trimer
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J. Phys. Chem., Vol. 100, No. 26, 1996 10879
TABLE 5: The Two- and Three-Body Contributions to the Interaction Energy for the ohxA Structure of the (H2O)2H2 Cluster (in µhartrees), Oa ) 104.52°, R(O‚‚‚c.o.m.(H2)) ) 3.2776 Å energy
(H2O)d‚‚‚(H2O)a
(H2O)d‚‚‚H2
(H2O)a‚‚‚H2
3-body
∆E ∆E(2) ∆E(3) ∆E(2) ∆E(3)
-5670.3 -1173.8 69.6 -6844.0 -6774.4
-88.1 -24.1 -0.6 -112.2 -112.8
-208.5 -554.0 -21.2 -762.5 -783.9
(10) es HL exch SCF ∆Edef
-11638.4
-84.7
-1091.1
0.0
8817.1
1.0
1286.6
-17.1
-2848.9
-4.4
-404.0
-135.8
(20) ind,r
-3711.0
-4.4
-427.4
-84.2
(30) ind,r
-4465.1
-0.2
-229.6
-38.4
(20) disp
-3006.6
-33.0
-739.2
0.0
(12) es,r (30) disp
6.7
8.2
19.5
0.0
SCF
-152.8 2.4 7.1 -150.4 -143.0
0.7
configurations the anisotropy of the total three-body interaction originates at the SCF level. Both components of the SCF nonadditivity, the SCF deformation effect and the HL exchange nonadditivity, are very important to reproduce the overall trends in ∆ESCF. In the zzD geometry the total nonadditivity is nearly constant with respect to the θd variations. In the zzA geometry, the repulsive total effect has a shallow minimum around the θa value of 127°. This minimum is determined by the HL exchange effect. In the ohxD geometry the repulsive total effect has a distinct maximum at φd ) 104.5° which coincides with SCF . Finally, in the ohxA the analogous maximum in ∆Edef geometry the total nonadditivity has a pronounced minimum SCF SCF effect. Overall the ∆Edef effect also determined by the ∆Edef HL is the most anisotropic followed by exch as a distant second. The correlation effects are both very small and nearly isotropic except for the dispersion term in the zzA configuration. The effect of the enlargement of the basis set on H2 was also examined. As pointed out previously,7 the quadrupole moment on H2 is difficult to describe without diffused d-type functions. Therefore, we carried out additional calculations for the zzD configuration with the basis set with d functions (exponent 0.18) on the H2 monomer. These calculations confirmed our previous conclusions34 that the three-body effects are less basis set dependent than the two-body effects. This is due to the fact that the most basis set sensitive components of interaction energy, such as the electrostatic and dispersion (second-order) terms, are additive. B. Induction Nonadditivity. As indicated above, the SCF effect plays a prominent role in the overall nonadditiv∆Edef ity for every configuration. In every geometry the sign and SCF effect are determined by the sum the magnitude of the ∆Edef of the induction nonadditivities in the second order and in the third order of perturbation theory. The perturbational analysis offers a unique opportunity to elucidate the physical origin of the induction nonadditivity in the present complex (see Table 6). It is instructive to start from the two-body induction effect. This effect can be interpreted as resulting from the energy lowering of one monomer in the field of the other. Among the pair interactions in the present cluster the strongest energy lowering is experienced by the proton acceptor H2Oa in the field of the proton donor H2Od. By comparison, the lowering of the energy of H2Od in the field of H2Oa is over two times smaller. This effect can be rationalized by the fact that the electric field of a water molecule is maximized along the O-H bond.61 The
TABLE 6: Analysis of the Induction Components in the (H2O)2H2 Clustera 3-body 2-body
second-order zzD
third-order
-32 -16 -49 -97
H2(d), a(d) a(d), d(H2) d(a), H2(d) total
-10 -11 -14 -39
133 -67 57 123
H2(d), a(d) H2(a), d(a) a(d), H2(a) d(a), H2(d) d(H2), a(d) total
37 20 -24 -18 13 22
ohxD H2(d, a) 69 a(d, H2) 13 total 83
H2(d), a(d) total
14 19
ohxA H2(d, a) -59 a(d, H2) -20 total -84
H2(a), a(d) total
-27 -39
a(d) d(a) H2(d) d(H2) total
-2534 -1178 -262 -45 -4021
H2(d, a) a(d, H2) d(a, H2) total
a(d) d(a) H2(d) H2(a) a(H2) total
-2530 -1182 -465 -261 -45 -4547
H2(d, a) a(d, H2) d(a, H2) total
a(d) d(a) H2(d) total
-2531 -1181 -22 -4146
a(d) d(a) H2(a) a(H2) total
-2534 -1177 -405 -22 -4143
zzA
a The symbols x(y) denote the energy of x in the field of y (in µhartrees). Only the values larger or equal to 10 µhartrees are listed.
energy lowering of H2 in the field of either H2Oa or H2Od is, depending on the orientation, 1-2 orders of magnitude smaller. The three-body induction effects in the second order may be viewed as the energy of any monomer in the combined field of the other two. Unlike in the two-body case this energy can be either stabilizing or destabilizing. The energy of H2 in the field of the water dimer represents a dominant (or nearly dominant) effect in all the configurations examined. This effect is stabilizing in the zzD and the ohxA configurations and destabilizing in the zzA and the ohxD configurations. In the zzD the H2 molecule acts as a “hydrogen bond donor” to the O atom of H2Od, and in the ohxA H2 acts as a “hydrogen bond acceptor” from an H atom of H2Oa. Thus both structures may be viewed as having hydrogen bonds arranged in a concerted fashion as follows: zzD as HfOfO and ohxA as OfOfH2 (where arrows describe hydrogen bonds). Using the same interpretation, the configurations with the destabilizing threebody effect, zzA and ohxD, may be viewed as having OfOrH and H2rOfO branched-out chains of hydrogen bonds, respectively. 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. Alternatively, the moments on A and B may also be induced by pairs of A and C or B and C, respectively (in both cases coupling all three monomers). As in the case of the second-order effect, the total third-order nonadditivity is repulsive in the zzA and ohxD configurations, while attractive in the zzD and ohxA ones. In zzA and ohxD the repulsion is dominated by the effect of H2Od inducing moments on both H2Oa and H2 which in turn repel each other. In the remaining configurations the dominant terms involve the pair H2Od and H2Oa inducing moments on H2 and H2Od in zzD, and on H2 and H2Oa in ohxA, respectively. To conclude, in complexes involving a nonpolar molecule (or an atom) bound to a hydrogen-bonded dimer induction effects are expected to dominate. A rough prediction of the favorable three-body induction effects can be based on directions
10880 J. Phys. Chem., Vol. 100, No. 26, 1996 of maximal electric field around the hydrogen bond donor molecule and around the overall hydrogen-bonded dimer. Such favorable induction effects occur in the structures involving hydrogen bonds arranged in a concerted fashion, e.g. HfOfO in zzD or OfOfH2 in ohxA. Finally, it is worthwhile to comment on the validity of SCF by the sum of perturbational induction representing ∆Edef (n0) ∞ (n0) ind,r corrections ind,r. As mentioned above, the sum ∑n)2 SCF converges to ∆Edef only in the absence of the exchange effects.52 The first two terms from the induction series are listed SCF values. It appears that the in Tables 2-5 along with the ∆Edef SCF three-body ∆Edef term can be fairly well described by the sum (20) (30) of ind,r and ind,r , thus the nonexchange approximation is a very reasonable approach. However, the approximation is not as good in the case of two-body contributions. For example, (30) (20) for the H2O-H2O pair interaction, ind,r is larger than ind,r and SCF their sum is larger in magnitude than the respective ∆Edef term. It should be emphasized that the inclusion of only the (30) two-electron part in the evaluation of the ind,r term appears to prevent the divergency problems. This strategy was applied in (30) term, but not in the case of the the case of the three-body ind,r (30) two-body ind,r term. (For more discussion on the artificial charge transfer in induction terms, see ref 54). IV. Cooperative Effects The concept of the cooperative effect among hydrogen bonds was postulated by Frank and Wen.62 The cooperativity implies that a hydrogen bond between a proton donor group, A-H, and a proton acceptor B becomes stronger when another hydrogen bond is formed with a further A-H group, i.e. A-H‚‚‚A-H‚‚‚B.63 Although cooperative effects are usually discussed on the basis of the shifts in the hydrogen bond stretching frequencies,64 it may be useful to carry out a similar analysis on energetic grounds. Such an energetic cooperativity is difficult to define uniquely since it is not a property of the complex itself, but that of the ways of its formation or dissociation. Nevertheless, it is interesting to examine the effect the presence of H2 attached to one of the water molecules has on the interaction between two water molecules. By rearranging eq 1 we find that such an interaction between H2H2O‚‚‚H2O can be expressed in terms of two-body terms and the threebody term as
∆E[H2(H2O)a‚‚‚(H2O)d] ) ∆E[H2‚‚‚(H2O)d] + ∆E[(H2O)a‚‚‚(H2O)d] + ∆E3-body (2) By comparing the value of the interaction energy ∆E[H2(H2O)a‚‚‚(H2O)d] with the two-body term ∆E[(H2O)a‚‚‚(H2O)d)], we can gain insights into the “cooperative” strengthening or weakening of the H2O‚‚‚H2O interaction due to the presence of H2; thus, the difference between these two quantities will be denoted ∆Ecoop. The results of such an analysis for the four selected configurations of the trimer are shown in Table 7. ∆Etrimer represents the total interaction energy of the trimer, that is, the sum of all the two-body interactions and the three-body term. As pointed out earlier, the three-body terms should be evaluated at a level one-order higher than the one used for twobody terms. Thus the trimer interaction energy, ∆Etrimer, shown in Table 7 was evaluated by summing up the ∆E(2) values for the two-body terms and the ∆E(3) value of the three-body term. ∆Ecoop is the difference between the ∆E(H2O‚‚‚H2O) two-body term and ∆E[H2(H2O)‚‚‚(H2O)] defined in eq 2. It should be reiterated (see above) that the small differences in the values of ∆E(H2O‚‚‚H2O) reflect the fact that they are evaluated in
Sadlej et al. TABLE 7: Comparison of the Cooperative Effect of the H2 Moleule on a Water‚‚‚Water Interaction and the Body Term for the Four Configurations of the (H2O)2H2 Clustera ∆Etrimer
∆E(H2O‚‚‚H2O)
-7826.1
-6841.6
-6973.3
∆E(H2OH2‚‚‚H2O)
∆Ecoop
∆E3-body
zzD -7065.5
-223.9
-142.1
-6842.8
zzA -6203.2
639.6
126.0
-7445.1
-6830.9
ohxD -6683.7
147.2
116.7
-7861.8
-6844.0
ohxA -7099.3
-255.2
-143.0
a
All values are in µhartrees; for definitions see the text.
TCBS which changes from structure to structure due to the varying position of H2. It is seen that the attachment of H2 to the O atom of the donor H2O (zzD) results in the reinforcement of the water‚‚‚water interaction and so does the attachment of H2 to the H atom of the acceptor water molecule (ohxA). On the other hand, the presence of H2 near the O atom of the acceptor molecule (zzA), as well as its presence next to the H atom of the donor (ohxD), leads to a weakening of the water‚‚‚water interaction. These trends fully agree with the conclusion of Mo et al.30 stating that a “water dimer will be reinforced when its donor moiety acts as H-bond acceptor with respect to the third molecule or when the acceptor moiety acts as a H-bond donor.” The magnitude of ∆Ecoop is much larger than the three-body effect, although both quantities display the same sign. V. Summary and Conclusions The interaction within the cluster (H2O)2H2 was dissected into two-body and three-body components. The nature of manybody terms was studied by partitioning these terms into their fundamental components: electrostatic, exchange, induction, and dispersion in the case of two-body interactions, and exchange, induction, and dispersion in the case of the three-body term. The analysis of the components of the three-body term shows that the nonadditivity in this cluster is determined by the SCF term, i.e. the induction effect which is restrained by the ∆Edef exchange effect. The examination of the nonexchange apSCF in the form of the classical CHF proximation to ∆Edef induction series indicates that this approximation works satisfactorily in the three-body case. In a trimer involving at least two polar molecules the induction nonadditivity is expected to dominate regardless of whether the third monomer is an atom, a nonpolar molecule, or a polar molecule. In the four configurations of the (H2O)2H2 clusters, zzD, zzA, ohxD, and ohxA, we found that the attractive or repulsive induction effects may be related to the directions of electric field around the donor H2O, as well as the overall dimer (H2O)2. The three-body induction leads to the additional stabilization in zzD and in ohxA, which can be schematically described as HfOfO and OfOfH2, respectively (the arrows denote the hydrogen bonds). The three-body term is destabilizing in the zzA and ohxD configurations which can be described as OfOrH and H2rOfO, respectively.65 The direction of arrows can also be viewed as the direction of polarizing effect caused by the hydrogen bonds. The arrows in the same direction correspond to reinforcement of the induction effect, while those in the opposite directions correspond to induction effects which interfere destructively. We also examined the cooperative effects which the presence of H2 has on the water‚‚‚water interaction. Such considerations require the division of the trimer into the interaction between
Nonadditive Effects in (H2O)2‚‚‚H2 the preformed dimer H2H2O with H2O. Although difficult to define uniquely, this concept is useful since it may be related to the observable dissociation channels of a trimer.3 The water‚‚‚water interaction is strengthened when H2 acts as a H-bond donor to the donor moiety of (H2O)2 (zzD) or if it acts as the H-bond acceptor to the acceptor moiety (ohxA). The water‚‚‚water interaction weakens when its donor moiety acts as a double donor (ohxD) or as a double acceptor (zzA). Acknowledgment. The support of the BST-502/20/95 grant is gratefully acknowledged by J.S. J.S. wishes to thank Victoria Buch for suggesting this project and for numerous discussions. Support by the National Science Foundtion (Grant No. CHE9215082) is acknowledged by M.M.S. References and Notes (1) Meath, W. J.; Koulis, M. J. Mol. Struct. 1991, 226, 1. (2) Elrod, M. J.; Saykally, R. J. Chem. ReV. 1994, 94, 1975. (3) Suhm, M. A.; Nesbitt, D. J. Chem. Soc. ReV. 1995, 24, 45. (4) Szcze¸ s´niak, M. M.; Chaleasin´ski, G. J. Mol. Struct. THEOCHEM 1992, 261, 37. (5) Chaleasin´ski, G.; Szcze¸ s´niak, M. M. Chem. ReV. 1994, 94, 1723. (6) Cooper, A. R.; Hutson, J. M. J. Chem. Phys. 1993, 98, 5337. (7) Zhang, Q.; Chenyang, L.; Ma, Y.; Fish, F.; Szcze¸ s´niak, M. M.; Buch, V. J. Chem. Phys. 1992, 96, 6033. (8) Sadlej, J.; Rowland, B.; Devlin, J. P.; Buch, V. J. Chem. Phys. 1995, 102, 4804. (9) Devlin, J. P. In Physics and Chemistry of Ice; Maeno, N., Ed.; Hokkaido University: Sapporo, Japan, 1992. (10) Devlin, J. P.; Silva, S. C.; Buch, V. In Hydrogen Bond Networks; Bellissent-Funel, M. C., Doore, J. C., Eds.; Kluver: Amsterdam, 1994. (11) Zhang, Q.; Buch, V. J. Chem. Phys. 1990, 92, 1512. (12) Rowland, B.; Devlin, J. P. J. Chem. Phys. 1991, 94, 812. (13) Buch, V.; Devlin, J. P. J. Chem. Phys. 1991, 94, 4091. (14) Rowland, B.; Fisher, M.; Devlin, J. P. J. Chem. Phys. 1991, 95, 1378. (15) Rowland, B.; Fisher, M.; Devlin, J. P. J. Chem. Phys. 1993, 97, 2485. (16) Devlin, J. P. J. Chem. Phys. 1992, 96, 6185. (17) Hixson, H. G.; Wojcik, M. J.; Devlin, M. S.; Devlin, J. P.; Buch, V. J. Chem. Phys. 1992, 97, 726. (18) Buch, V.; Devlin, J. P. J. Chem. Phys. 1993, 98, 4195. (19) Buch, V. J. Chem. Phys. 1992, 97, 726. (20) Taylor, D. G.; Strauss, H. L. J. Chem. Phys. 1993, 90, 2265. (21) Hunter, J. E.; Taylor, D. G.; Strauss, H. L. J. Chem. Phys. 1992, 97, 50. (22) Xiao, L.; Coker, D. F. J. Chem. Phys. 1995, 102, 1107. (23) Taylor, D. G.; Strauss, H. L. J. Chem. Phys. 1992, 96, 3367. (24) Clementi, E.; Koleos, W.; Lie, G. C.; Ranghino, G. Int. J. Quantum Chem. 1980, 17, 377. (25) Clementi, G.; Corongiu, G. Int. J. Quantum Chem. 1983, 10, 31. (26) Corongiu, G. Int. J. Quantum Chem. 1992, 42, 1209. (27) Yoon, J.; Morokuma, K.; Davidson, E. R. J. Chem. Phys. 1985, 89, 1223. (28) Van Duijneveldt-van de Rijdt, J. G. C. M.; Van Duijneveldt, F. B. Chem. Phys. 1993, 175, 271. (29) Chaleasin´ski, G.; Szcze¸ s´niak, M. M.; Cieplak, P.; Scheiner, S. J. Chem. Phys. 1991, 94, 2873. (30) Mo, O.; Yan˜ez, M.; Elguero, J. J. Chem. Phys. 1992, 97, 6628. (31) Schu¨tz, M.; Burgi, T.; Leutwyler, S.; Burgi, H. B. J. Chem. Phys. 1992, 99, 5228.
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