J. Phys. Chem. 1994,98,4271-4282
4271
Ab Initio Study of Cooperativity in Water Chains: Binding Energies and Anharmonic Frequencies Lars Ojamie' and Kersti Hermansson' Institute of Chemistry, Uppsala University, Box 531, S-751 21 Uppsala, Sweden Received: May 11, 1993; In Final Form: December 17, 1993' Many-body interaction energies and anharmonic OH stretching frequencies have been calculated for water in chain formations, in a ring structure, and in a tetrahedral arrangement. The calculations were of ab initio type, with the electron correlation energy included by Mlaller-Plesset perturbation correction to second order (MP2) and the basis-set superposition error corrected by the counterpoise procedure. The maximum chain length was seven water molecules, and the ring was five-membered. The molecules were H-bonded head-to-tail. The twoand many-body energies for the chains and ring are all of the same sign (negative), indicating strong cooperativity. The total nonadditivity contribution to the interaction energy is large, 16% for the longest chain and over 18% for the ring. The interaction energy of an individual chain member with the rest of the chain shows even larger nonadditivity: over 25% for a molecule in the middle of the chain. This quantity should be of relevance for molecular dynamics simulations of liquid water. The OH stretching frequency downshift increases for all members of the chain with increasing chain length and is larger for molecules in the interior of the chain (-357 cm-1 for the middle molecule in the 7-chain) than for terminal water molecules. The frequency converges only slowly for water molecules in the interior but faster for terminal water molecules. "Frequency cooperativity" was investigated by calculating many-body contributions in a manner analogous to the energy calculations. The chain and ring exhibit strong cooperativity. Infrared absorption intensities and charge transfer were investigated.
Introduction The local properties of a water molecule are very dependent on its environment and vary widely with for example its state of aggregation-gas, cluster, liquid, or ice. The discussion of interactionsamong water molecules has advanced not in the least via the numerous theoretical calculations of interaction energies of water oligomers.'-10 Much interest has focused on the cooperative nature of these interactions,1-12 i.e how certain geometricalarrangements enhance or diminish the interaction in such a way as to give rise to large deviations from pair additivity.11 This cooperativity is believed to be important for the structure and propertiesof water in the condensed phases. As an example, this is reflected in the equilibrium O--O distance which in the gaseous water dimer is 2.98 A, in liquid water -2.85 A, and in the different ice phases 2.7-2.9 A.13 One situation where energy cooperativityhas been found to be very important is in sequential hydrogen-bonded water chains, where each molecule accepts one H-bond and donates one.'-5 Theoretically, a chain of water molecules can be regarded as a starting point for a model of the condensed phase. But, there are also examples of real water chains in nature, such as short-lived chains in liqui4, water" or organic solutions,15-17 infinite chains in hydrate crystals,I8*l9and short structural water chains in enzymes.13 The interaction energy is only one property, however, where one can see cooperativity at work. Kleeberg,15 Luck,l6 and H ~ y s k e n s have ' ~ studied cooperative effects on OH stretching frequencies of hydrogen-bonded chains of alcohols or water dissolved in organic solvents using vibrational spectroscopy. Generally,the vibrationalpropertiesof hydrogen-bonded (water) chain8 have not been studied much by theoretical methods. Ab initio calculationsof harmonicvibrationalfrequencieswere carried out by Honegger and Leutwyler,20for short chains in small water clusters, and by Knochenmuss and Leutwyler,ZI for (H20)s and ( H 2 0 ) ~clusters; they found the 0-H stretching force constant to decrease strongly with cluster size. Karpfen and Schuster22 calculated the OH Stretching force constant of a water molecule in an infinite chain. K a r ~ f e nhas ~ ~studied the harmonic H-F Abstract published in Advance ACS Absrracrs, February 1, 1994.
frequency shifts in cyclic (HF),, n = 2-4, complexes. Shivaglal et al." performed CNDO calculationsof the OH stretching force constants for water clusters and chains and explicitly discussed force-constant cooperativity effects. Scheiners discussed the cooperativity of harmonic frequency shifts for (HF)3 and (HCl),. In the present paper we study both sequentialwater chains and a closed ring structure. A hydrogen-bonded ring bears many resemblances to an infinite linear chain: it providm, in a sense, the ultimate example of cooperativity. In nature, rings of hydrogen-bonded water molecules occur for example in crystals, such as ice (6-rings in ice Ih, IC,11, VII, and VIII, 5-rings in ice 11113) and hydrates13J9 and probably in liquid water. In liquid water there are some theoretical indications that fivamembered and six-membered rings are dominant among ning species.xJ7 It has even been suggtsted that five-rings are essential for explaining observed anomalous.properties of water.% One very active area for discussion of cooperatiGtyis the field of Monte Carlo and molecular dynamics simulations.12 The intermolecular potential functions that form the basis of these methods have in the past most often been pair-additive. Several more recent investigations have included explicit polarizationenergy terms,ZP-33 which have been seen to have a considerable effect on many structural, dynamicat, and thermodynamical quantities. In fact, also the connectivity in water solutions is influenced.29 The present paper is a quantum-chemical ab initio study of chains of water molecules ((HzO),, n = 2-7), a ring structure ( n = 3,and a tetrahedral pentamer. We study the convergence of interaction energies and vibrational frequencies (with emphasis on the latter) with increasing chain length and compare them with the results for the ring structure and the pentamer. Our purpose is to quantifythe cooperativityphenomena and distinguish the cases where they are important. Theoretical investigations of frequencycooperativityare scarceIuJ5whereas the cooperativity of interaction energies has bem treated in many earlier studies;1-10 here this is done on a more advanced computational level than customary for large systems. The computational treatment includes both electron correlation and corrections for the basis set superposition error and for the frquencies the important anharmonicity contribution to the OH stretching vibrations. 0 1994 American Chemical Society
4212
The Journal of Physical Chemistry, Vol. 98, No.16, 19'94 A
B
C
D
E
F
G
Figure 1. Chains (a) and the ring (b).
Method
Geometry. Three types of topology were considered for the water clusters: water chains up to a heptamer, a pentamer ring, and a tetrahedron. In the chains the water molecules were c o ~ e c t e in d a sequential"head-to-tail" fashion; i.e. each molecule inside the chain accepts one and donates one linear hydrogen bond (Figure la). The0-Odistancebetweenneighboringwater molecules was taken to be 2.84 A. The angle between the O.-O direction and the bisector of the H-bond acceptingwater molecule is 57O. Thisisclose to thegeometricalarrangement in thegaseous water dimer,34 except for the "liquid-like" 0.-0distance chosen here. The 0-H distances are 0.957 A, and the H-CLH angle is 104.52'. Each hydrogen-bonded pair is of C, symmetry. The same geometry for each H-bonded pair was used in the 5-membered ring structure (Figure 1b),obtained by adding water molecules to the beginning and end of the 3-chain so as to close the chain by an additional hydrogen bond. With the dimer geometry used here, a ring then results where "the last" 0-0 distance is 3.1 1 A instead of 2.84 A, and the O-H-*O angle is 173.6O. Knochenmuss and Leuwyler21 and Burkeet al.35 recently optimized the (H20)5 ring structure at the SCF level using a 6-31G* basis and a 6-311G** basis, respectively. They found an almost planar ring to be the global minimum, with 0 - a 0 distances in the ranges 2.814-2.830 and 2.774-2.777 A, respectively. Our tetrahedron was built from the dimers in such a manner that the tetrahedron is of CZ, symmetry. The geometry of a head-to-tail hydrogen-bonded 3-chain is the same in all three structures. We would like to add a few comments on our choice of cluster geometries. Going from a dimer to successively larger clusters, there are several viable strategies open with respect to the choice of geometries in the theoretical calculations, for example: (i) a geometry optimization is carried out for each n-mer,m.21 or (ii) the largest system is optimized and the smaller clusters are taken as fragments of this one, or (iii) some "representative" structure is selected for the largest system and the smaller clusters are taken as fragmentsof this. The advantageof (i) is a self-consistent description of q11 geometries at the computational level chosen. The disadvantage is that the effect of cluster size on the property of interest (in our case, interaction energy and OH frequency)
Ojamae and Hermansson arises both from the change in geometry and from changes in the electronic structure due to the cooperative intermolecular interactions. By keeping the geometry fixed when studying cooperativity as a function of cluster size (ii and iii) we "filter out" the geometrical effect. A further comment is that a system consisting of several moleculesgives rise to an extensiveset of (very different) structures with local energy minima not high above the global minimum. This was recently demonstrated for six-membered water clusters by Franken et ~ 1 At. elevated ~ ~ temperatures in a liquid this situation is even more true. The choice of one appropriate geometry then becomes less straightforward. Here we have chosen to study "ideal" water chains, and the 0.-0distance between neighboring water molecules (2.84A) was selectedwith the notion of being a representative0-0distance between water molecules in solution. Basis Sets and Electron Correlation. The computations were of ab initio restricted Hartree-Fock (RHF) type with correlation energy included by the second-order Maller-Plesset (MP2) correcti~n.~'The basis set used was the contracted-Gaussian double-c plus polarization (DZP) basis set. This is the Dunning and Hay3*contraction of Huzinaga'G9basis set, augmented with polarization functions having exponents of 0.85 for oxygen and 1.OO for hydrogen. Tight convergence criteria were used for the integrals and the SCF convergence. The program used was HONDO8.4 The basis set superposition error (BSSE) was compensated for by using the counterpoise (CP) techniq~e;~l i.e. the energies for the monomers, dimers, and so forth were calculated in the basis set of the heptamer (or pentamer in the ring and tetrahedron cases). These energies were used in the calculation of the manybody energies. The CP technique is an established, albeit not undisputable,4r43method for dealingwith the BSSE of interaction energies, but the situation is less clear where frequencies are concerned. In analogy with the energy case, one can consider a basis set superposition error on calculated frequency shifrs for interacting molecules relative to a noninteractingmolecule." One must then correct the interaction energy for the BSSE by the CP techniqueand use this energy when deriving the frequency. When we do this for a hydrogen-bonded water dimer, we get a correction of the OH stretching frequency shijir by about +20 cm-1. Rather unexpectedly, about two-thirds of this shift can be attributed to the BSSE of the H-bond acceptor and only one-third to the BSSE of the H-bond donor. However, since one is often concerned with absolute frequencies rather than shifts, it is not obvious that the CP technique should be applied. In the present study we have chosen to report BSSE-corrected energies but not to report BSSEcorrected frequencies in the tables. Calculation of Vibrational Frequencies. The uncoupled OH stretching frequency was calculated for each hydrogen-bonded OH bond in the clusters, as well as for the terminal OH bond in the chains. This was done by deriving the OH potential curve from pointwise energy calculations for different 0-H distances. The harmonic OH frequency w was calculated from the second derivativeat the minimum of the OH potential curve. To get the anharmonic OH frequency Y, the Schrbdinger equation for the OH oscillator motion in the above potential was solved variat i ~ n a l l y .Since ~ ~ the OD fragment was assumed rigid, a reduced mass of 0.954 59 was used for the H oscillator in the solution of the vibrational problem. In the calculation of the energy points for the potential curve only the H atom was moved while the rest of the molecule was held fixed. Thus the OH vibration was uncoupled from the vibrations of the other bonds in the chain. It is therefore appropriate to compare the calculated results with experimental results from uncoupled HDO oscillators in a deuterium-subtituted environment. This is in fact the way spectroscopic measurements of OH vibrations in solution are usually performed, i.e. by way of the isotope-isolation technique.lg
The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 4273
Ab Initio Study of Cooperativity in Water Chains
0.20 I
ReSUltS 1. Many-Body Energies. For a cluster of molecules, the total interaction energy (the negative of the binding energy) is
0.15 N
I I
-
where ETOTis the total energy of the cluster and E ( i ) is the energy of each of the isolated subunits (molecules). The interaction energy depends upon the geometricalarrangement of the subunits, i.e. AETOT= A E ~ o ~ ( r ~ , r 2 , . . .One , r ~ )can . always write the interaction energy as an expansion over the subunits:
AETm=c N-L
N
N-2 N-1
A E u + c MYk+
1-1 j='+l
1-1 j=t+l
+ ..
AEiJk'
- &'MY
+ &MY
+ &MY
+ ...
,
(2)
where A E M - w y is the M-body interaction energy for the total cluster, M = 2, 3, ...,N . The AE'j, etc., in (2) are
A E =~~ ( i j-] [E(i) + EO]] Muk= E(ijk) - [E(i) + E o ]
+ E@)] -
cv p > v
AEVp
v,pr Iijkl
= E(# ) - [E(i) + Eo]
+ E(k)] - [AEV + AEik + Ad7
= E(ijk ) + E(i) + E O + E(k) - [E(O] + E(ik) + E(ik)l AEV'
= E(ijkl) - [E(I]+ EO] + E(k) + E(/)] -
etc., where E(ij) is the energy for the complex consisting of molecules i andj, E(ijk) the energy for the complex consisting of i, j , and k, etc. When the total cluster consists of only two molecules, AE'j = AE2-wY,for a cluster of three molecules = AE3-wy. and so on. In the general case, Mi], AE'jk, etc., are contributions to the total AEz-wyand AE3-wy, etc., energies. If AETm is strictly equal to W w y , the interaction energy of the system is said to be pair-additive, otherwise nonadditive. So we define the additiveand nonadditive parts of the interaction energy as
For example, the interaction energy of a system of point charges will be completely pair-additive, and a system of nonpolarizable molecules nearly so. The results of applying these formulas to the chains, the ring, and the tetrahedron are listed in Tables 1-3. I . I . The Chains and the Ring. The many-body interaction energies for the chains in Figure l a are given in Table la. The energies for the subsystems created when one or more molecules are removed from nonterminal positions in a chain are also included in the table. The interaction energy for the smallest cluster, the dimer, is -19 kJ/mol if the molecules are hydrogenbonded neighborsand diminishesrapidly the more empty positions there are between the molecules. Although equivalent in geometry, there is actually a small difference in energy between different connected H-bonded dimers; this is due to the basis set superposition error correction using the counterpoise technique and is further discussed below (in the table, not all counterpoise-
chain length Figure 2. Ratio between nonadditive and total interactionenergy plotted against chain length (solid line). The dashed line represents the ring and its one-, two-, three-, and four-membered connected fragments.
corrected energies are shown for each geometry). Turning to the trimers, one sees that the threebody contribution to the interaction energy is substantial in the connected H-bonded chain. The nonadditive part of the interaction energy is about 9%of the total interaction energy. If the chain is broken (e.g. AB-D) the nonadditive part is usually less than about 1%. In the connected tetramer the nonadditive contribution has increased to 12.4%. The largest part of this is due to the threebody term, which constitutes 12% of the interaction energy. The four-body energy in the connected chain is about 0.27 kJ/mol and is much smaller for the broken chains. For configurations such as AB-DE the four-body term can be up to about 0.06 kJ/mol, whereas for A B L E it is smaller, slightly over 0.01 kJ/ mol. The ratio between nonadditive and total interaction energies increases with chain length and is 16% for the heptamer chain (Figure 2). The three-body part is 15%of the interaction energy and the four-body part about 1%. The many-body interaction etnergies for the 5-ring (Figure 1b) are listed in Table lb. The nonadditiveratio is much larger than that for the chain pentamer, 18.6% us 14.1%, and even larger than that for the chain heptamer (Figure 2). One factor underlying the strongly enhanced nonadditivity in the ring as compared to the chain is that in the ring all molecules are polarized both from their H-bond acceptor and donor ends, whereas this is not so for the end molecules in the chain. Another factor can be inferred from Figure 2: already for the connected tetramer fragment of the ring, the nonadditivityis slightly larger than that for the four-membered chain. This is due to the terminal water dipoles becoming more favorably aligned when the chain is bent to form a ring, thereby increasingthe polarization. Note that for the dashed curve in Figure 2 the ring closing occurs between the values 4 and 5 on the abscissa. The total two-body interaction energiesof the 5-chain and the 5-ring differ by 25 kJ/mol (Table la,b). Out of the difference, 17 kJ/mol originates from the ring closing and 8 kJ/mol from the favorable alignment of the AD and BE dimers. The total three-body energy is 14 kJ/mol for the 5-chain and 24 kJ/mol for the ring. Most of this difference comes from the fact that there are three connected trimer fragments within the chain, whereas in the ring there are five. The many-body energies in the connected chains and the ring are always negative. This reflects that when the chain is built up by adding new subunits, the bonding within the chain will be enhanced. This is a sign of cooperativity. The many-body energies normalized to the number of hydrogen bonds for the connected chain and the ring are listed in Table 2 and displayed in Figure 3. The total interaction and many-body energies per H-bond increase with increasing chain length215but is largest for the ring. The extreme stability of the 5-ring was
4274 The Journal of Physical Chemistry, Vol. 98, No. 16, I994
Ojamiie and Hermansson
TABLE 1 (a)
A A A A A A A A A A A A A A A A A
A A A A A A A A A A A A A A A A A A A A A A A
A A
A A A A A A A A A A A A A A A A A A A A A A A
A A A
B B . .
. . .
B B B B B . . .
. .
. .
.
C C .
. .
. C . .
.
C C C C
.
. .
.
. .
. .
B B B B B B B B B B . .
C C
C C
.
. .
. . .
D
E
F
G
.
.
.
.
.
. . .
.
.
,
E
D
.
. .
D .
.
. D . .
.
D D D
. ,
.
,
. .
. .
.
.
B B B B B B B B B B
C
,
C
. , . . B B B B B
C C
C C C .
.
. C
C C
.
C C C C
.
.
C
B
C
B B
E .
.
E
. .
F
.
.
F
.
. .
. .
F
F
.
G
. . G . G G
.
. .
F
. .
. .
E E
F
E E E
.
E E E E
.
E E E E
C
.
G
F F
,
D D
-19.222
. . G . .
.
E E
.
D
-112.609
.
G G G
F
.
-132.897
.
G F
E E
.
,
. . G
D D D
D D D D
G G G G
-13.990 -9.461 -9.233 -5.326 -4.781 -4.916 -5.394 -0.984 -0.613 -4.951 -9.461 -5.033 -0.909 -4.783 -9.332
. F
F
D D D
-89.032 -69.327 -65.961 -66.447 -50.215 -63.438 -66.440 -46.109 -46.459 -63.460 -69.721 -50.437 -46.688 -50.814 -66.576
G
.
,
-103.643 -79.077 -75.471 -71.916 -55.022 -68.441 -71.970 47.163 47.101 -68,501 -79.466 -55.492 -47.672 -55.615 -76.193
.
.
E
D D D . .
.
E . , . F . . . G E F . E . G . F G
E
.
-8.975 -4.656 -4.523 -4.394 -0.698 -0,333 -0.254 -0.501 -0.163 -0.196 -4.641 -0,410 -0.297 -0.257 -0.010 -0.211 -4.602 -0.433 -0,184 -4.423
.
F
D D D
.
-65.514 -46.258 -42.924 -43.054 -42.753 -26.546 -22.576 40.037 -23.201 -39.923 -46.635 -26.328 -23.406 -26.945 -11.157 -23.779 -43.307 -23.420 -23.169 -43.667
.
. .
E E
C
.
-74.752 -50.928 -47.455 -47.451 -43.513 -26.882 -22.833 40.591 -23.368 40.136 -51.283 -26.740 -23.704 -27.203 -11.166 -23.993 47.918 -23.853 -23.357 -48.095
. . F .
F . F
,
-0.010 -0.005 -0.183 -0.016 -0.006 -0.092 -0.005 -0.006
E
,
E .
.
-42.499 -22.622 -19.738 -19.903 -19.428 -23.005 -7.247 -3.312 -3.885 -20.111 -3.302 0.223 -20.291 -3.899 -20.020
G
D D D . .
. .
-46.789 -22.850 -19.870 -19.991 -19.438 -23.154 -7.246 -3.322 -3.890 -20.294 -3.319 0317 -20.383 -3.904 -20.025
.
. .
. E E
C C C C C
-19.367 -3.377 0.131 -0.482 -0.046 -0.015
.
. . .
AEGbody
F
. . E .
D D D
,
AEK, -19.367 -3.377 0.131 -0.482 -0.046 -0,015
. . .
D
. . .
.
.
F F F
.
F F F F
.
F
F F F F
D
G
. G G . G G G ,
D E B
B B
AEpEwY AEcw
AEsbodY
AEGw
AEl-bOdy
-4.290 -0.228 -0.131 -0.088 -0.010 -0,148
C
D E
-19.761 -3.446 -4.068 -16.665 -19.759 -3.443 -4.073
1.O
0.7 0.4 0.0 0.6 0.0 0.3 0.1 0.9 0.5 -2.9 0.4 0.1 0.0
O.Oo0
-19.761 -3.446 -4.068 -16.665 -19.759 -3.443 -4.073
lOO(AE-/AE.mr)
9.2
-0,263 -0.014 -0.007 -0.004 -0.062 -0.003 -0.002 -0.053 -0,004 -0.016 -0.007 -0.001 -0.001 -0,002 0.000
12.4 9.2 9.5 9.3 1.7 1.3 1.1 1.4 0.7 0.5 9.1 1.5 1.3 1.o 0.1 0.9 9.6 1.8 0.8 9.2
-0.004
-0.009 0.000 -0.004 -0.005 -0.612 -0.289 -0.277 -0,141 -0.026 -0.085 -0.134 -0.070 -0,028 -0.088 -0.284 -0.022 -0.075 -0.017 -0.284 -1.039 -0.650 -0.448 -0.249 -0.445 -0.650 -1.510
4.010 4.001
14.1 12.3 12.6 7.6 8.7 7.3 7.7 2.2 1.4 7.4 12.3 9.1 2.1 8.6 12.6
O.Oo0
-0.002 O.Oo0
-0.002 -0.002 O.Oo0
4.001 -0.002 -0.001 O.Oo0 O.Oo0 O.Oo0 O.OO0
-0.025
-0.001
G -108.049 -92.881 -14.507 -0,010 O.OO0 G -100.349 -89.733 -10.162 4.006 O.OO0 G -100.680 -90.362 -10.060 -0.008 O.OO0 G -100.461 -89.755 -10.255 -0.006 0.OOO G -108.667 -93.482 -14.523 -0.012 O.Oo0 G -162.480 -136.417 -24.503 -0.047 -0.002 4.001 (b) Many-BodyEnergies for Water Molecules in the 5-Ring Geometry (See Text)* E AEZ-wY AE3-wY AEcwY L\EsmY
C
A
Many-BodyEnergies for the Water Chains'
15.3 14.0 10.6 10.2 10.7 14.0 16.0 1 0 0 ( ~ ~ l E w r )
Ab Initio Study of Cooperativity in Water Chains
The Journal of Physical Chemistry, Vol. 98, No. 16, I994 4215
TABLE 1 (Continued) C C
D D
A A A A A A
B B B
C D E C C
B B
A
D D D
C C
B
A A A A
E E
D D D
c
B B B
C C
B B
C C
D D D D
c
E E
E E E E E E E E
-19.760 -3.457 -19.874
-19.760 -3.457 -19.874
-47.359 -28.175 -43.809 -28.234 -24.285 -43.545 -47.358 -28.199 -28.346 -47.492 -8 1.366 -76.869 -76.41 1 -76.632 -8 1.509 -140.416
-42.967 -27.272 -40.499 -27.274 -23.569 -40.607 -42.963 -27.290 -27.390 -43.091 -70.238 -67.162 -67.884 -67.271 -70.367 -1 14.307
-4.392 -0.903 -3.310 -0.959 4.717 -2.938 -4.395 -0.909 -0.956 -4.401 -10.650 -9.328 -8.106 -9.015 -10.661 -23.880
9.3 3.2 7.6 3.4 3.0 6.7 9.3 3.2 3.4 9.3 13.7 12.6 11.2 12.2 13.7
-0.479 -0.379 4.421 -0.346 -0.481 -2.106
-0.123
18.6
The molecules are labeled according to Figure la. The exact geometries are given in the text. MP2(DZP) calculations. Energies in kilojoules per mole. Note that only the 'geometry-unique" configurationsare shown but not all 'counterpoise-unique" configurations. MP2(DZP) calculations. Energies in kilojoules per mole. TABLE 2 Many-Body Energies per Number of H-Bondsfor the Connected chains, the Ring, and the Connected Tetrahedron 0
Frrnmeatsf'
chains A B A B A B A B A B A B 5-ring tetrahedron H A A . . . H A H A
. .
. C C C C C
.
.
.
D D D D
. E E E
. . F F
B B B B B B
. C C C C C
.
. . . G -28.0833
I I . I
-19.3666 -23.3945 -24.9 173 -25.9109 -26.5793 -27.0800 -22.8614
-19.3666 -2 1.2496 -21.8379 -22.2581 -22.5218 -22.7362 -4.7759
-2.1448 -2.99 18 -3.4974 -3.8444 -4.0839 -0.421 3
-1 5.3346 -23.4784 -16.3917 -21.9782 -22.2861 -23.2697
-16.4136 -21.3896 -1 7.6492 -19.9993 -20.3582 -20.4207
1.0791 -2.0888 1.2575 -2.0360 -1.9276 -2.9360
-0.0876 -0.1529 -0.2078 -0,2516 -0.0246
0.0571 4.0003 0.0836
-0.0024 -0.0051 -0,0079
-0.0002 -0.0003
9.2 12.4 14.1 15.3 16.0
-0.0001 18.6
-7.0 8.9 -7.7 9.0 8.7 12.2
0.0034
0 In the chain the number of H-bonds,NHB,is one less than the chain length, whereas in the ring NHBquals the number of molecules. Energies in kilojoules per mole. TABLE 3: Many-Body Energies for Water Molecules in the TebabedralGeometry'
A
B
C
A
B B
C
A A
H
H
C
i
C A A A A A A A A 0
I
B B B
B B B
H C C C C C C C C
H
i I
H H
I I
H
I
MTOT
MtWY
U 3 - M Y
-18.998 -20.392 5.168 -3.389 5.486 -30.669 -46.957 -32.783 -1.521 -1.235 -65.935 -66.858 -2.616 -93.079
-18.998 -20.392 5.168 -3.389 5.486 -32.827 -42.779 -35.298 -1.610 -1.292 -59.998 -61.075 -2.903 -8 1.683
2.158 -4.178 2.515 0.089 0.057 -6.108 -5.783 0.293 -1 1.744
MCWY
MEWY
100(bEdf&yrOr)
-7.0 8.9 -1.7 -5.9
-4.7 0.171 -0.001 -0.006 0.334
0.014
9.0 8.7 -11.0 12.2
MPZ(DZP) calculations. Energies in kilojoules per mole.
also noted by Del Bene and Pople in their minimal-basis SCF calculations on water oligomers.* Consecutive many-body energies per hydrogen bond differ by about an order of magnitude (Figure 3). The seven-bodyenergy (4.001 kJ/mol) falls outside the general pattern, but this energy is the one most affected by round-off, geometry, and convergenceerrors, etc.,and this energy is therefore likely not to be significant. Generally speaking, caution is needed when quantities are calculated as differences between much larger numbers, and
moreover when several additional arithmetic manipulations are needed, as is the case for the higher-order many-body energies here. An indication that the calculated six-body energies are in fact significant is the regularity of these seen in Table l a for the hexamers. I .2. The Tetrahedron. In the above structures not all possible hydrogen bonds to and from a molecule are saturated. The simplest example where this is the case is the tetrahedron pentamer (Figure 4). The many-body energies for the tetrahedron and its
4276 The Journal of Physical Chemistry, Vol. 98, No. 16, 1994
chain length Figure 3. Many-body energies in the chains (solid lines) plotted against chain length. The crosses represent the energies for the five-membered ring.
C
I
Figure 4. The water cluster in tetrahedral geometry.
fragments are collected in Table 3 and the energies per H-bond for the connected fragments in Table 2. From Table 3 we see that the trimer with the largest (by magnitude) three-body term is the head-to-tail trimer ABC (or “acceptor-donor” trimer). For the other types of trimer (one water accepts two H-bonds, HAB, or donates two, BCI) the three-body term is smaller in magnitude and positive, making the nonadditivity ratio negative. These results are ~ e l l - k n o w n and ~ . ~ are ~ ~ included here to facilitate comparison. In the tetrahedron tetramers, the four-body energies are smaller than in the sequential 4-chain and not always of the same sign. The “acceptor-acceptor-donor” tetramer HABC has the largest negative three-body contribution because it consists of two (symmetry-related) acceptor-donor trimers with negative three-body terms but only one double acceptor with a (smaller) positive three-body term.’ In the tetrahedron pentamer the magnitudes of the interaction energies are smaller than in the pentamer chain. The smaller size of the three-body term stems from the positive contribution of the double acceptor and double donor trimers. Both the four-body and the five-body contributions are positive. The tetrahedron pentamer nonadditivity ratio is smaller than for the chain pentamer, resulting both from the smaller size of the three-body energy and from the positive sign of the four- and five-body energies. I .3. Correlation Energy and BSSE Effects. The energies were all calculated at the MP2(DZP) level and with the full basis set of the heptamer (for the chain) or pentamer (for the ring/ tetrahedron) in the evaluation of the energies of the fragments. First the effect of electron correlation (from MP2) for the chains was investigated. The correlationenergy contributionto the dimer interaction energy calculated in the heptamer basis set was -2.17 kJ/mol for the AB dimer, -2.40 kJ/mol for the BC, CD, DE, and EF dimers, and -2.46 kJ/mol for the FG dimer. If instead the dimer interaction energy is calculated in the dimer basis set, the correlation energy is -6.45 kJ/mol. For the heptamer, with the fragment calculated in the heptamer basis set, the correlation energy contributions to the total, and the two-, three-, four-, five-, six-, and seven-body energies (per H-bond) were -2.79, -2.49,
Ojamae and Hermansson -0.25, -0.043, -0.0020, -0.0002, and -0.00006 kJ/mol, respectively (cf. Table 2). Second, the effect of the BSSE on thechain interaction energies was examined. For the dimers the effect of the counterpoise correctionson theinteractionenergy at theRHFleve1were +2.52, +2.36, and +2.22 kJ/mol for AB, CD, and FG, respectively, i.e. about 11%, and at the MP2 level +6.80, +6.41, and +6.21 kJ/ mol, respectively. For the total, two-, three-, four-, five-, six-, and seven-body energies in the chain the counterpoise corrections on the RHF level were +2.38, +2.47, -0.12, +0.023, +0.0056, +0.0003, and -0.00005 kJ/mol, respectively, and on the MP2 level +6.60, +6.54, +0.009, +0.043, +0.012, -0.0005, and +0.00009 kJ/mol. It is interesting to note that the trends for the higher-order (16) many-body energies discernible in Table l a for the BSSEcorrected MP2 energies could also be seen in the BSSE-corrected and non-BSSE corrected RHF energies, but not in the non-BSSE corrected MP2 energies (not shown). It thus appears to be essential to carry out counterpoise corrections for MP2 energies. Since the effects of the correlation energy and the counterpoise correction are most often of opposite signs, they will counteract each other. The differences between the counterpoise-corrected MP2 and the noncounterpoise RHF interaction energies, i.e. between our “highest-level” and “lowest-level”results, are for the AB, CD, and FG dimers as small as +0.35, -0.04, and -0.24 kJ/mol and for the total energy and two-, ...,seven-bodyenergies per H-bondofthe heptamer-0.41,-0.022,-0.37,-0.020,+0.0036, +0.00012, and -0.0001 1 kJ/mol. The dimer interaction energies thus do not differ much between the highest-level and lowestlevel theories. The same could be said for the many-body energies, although the magnitude of the three-body term seems to be underestimated at the noncounterpoise RHF level. 2. Individual Binding Energies. The many-body energiesabove tell us about the energetics of a cluster of molecules. If we focus our attention on the role of a particular molecule rather than the cluster, such energies may be less appropriate, however. The quantity of interest will then be the interaction energy of the moleculewith the rest of its surroundings. Consider,for example, molecule i in the pentamer cluster consisting of molecules h, i, j, k , and 1. We define the “individual” binding energy as195
AEi
+
E(hijkl) - [ E ( h j k l ) E ( i ) ]
(6)
where E(h-jkl) is the energy of the cluster consisting of molecules h, j , k, and 1 and the underscore indicates that the position if molecule i is vacant. This can also be written AEi = AEmT(hi$l)
=
- AEE,dh-jkl)
V+l
...
hE’vfi+
A@‘+ V
(7)
v p>v V,)Lfl
AE1v’’x+
(8)
v p>v)i>p
v,p,h
= L ~ E +~ u Z~b OYd y + A E ~ +~... Y
(9)
where the individual two- and many-body energies in (9) have been defined in a way analogous to (2). We see from (7) that AE,is the difference in interaction energy of the cluster with and without molecule i present. In the following, these individual binding energieswill be discussed for the members of the connected water chains. Figure 5 shows how the individualbinding energy of a molecule depends on its position in the two-, ...,seven-membered chains. It is seen that the members can be divided into three groups: the two end molecules of each chain constitute one group, the first but one and the last but one form a second group, and the “inner” molecules make up thedast group. The energy splitting between the first and second groups is mostly due to the two-body term, and between the second and third groups it is due to both the twoand three-body terms. The interaction energy has essentially converged with respect to chain length when we reach the 7-chain (Figure 6).
The Journal of Physical Chemistry, Vol. 98, No. 16, I994 4277
Ab Initio Study of Cooperativity in Water Chains IO
6ot
“/ 30
position in chain
position in chain
-4-Body
1
x
position in chain
position in chain
Fipwr 5. Individual binding energies for the connected water chain members. The solid lines connect energy points for members of the same chain. The leftmost point on each solid line is a hydrogen-bond donor and the rightmost an acceptor. The cross denotes the energy for the 5-ring. (a, top left) Total energy; (b, top right) 2-body. (c, bottom left) 3-body, and (d, bottom right) 4-body energies.
IO
50
I
-
f m t but one & last but one
c:
lot
O
’
i
i
i
5
6
i
I
chain length Figure 6. hdividual binding energy for members of the chains plotted against chain length. Only the energies for the first, the first but one, and the middle medum of each chain is shown. In Figure 7 tBe nonadditive ratio is displayed. Whereas the
nonadditivity in the total energy is at most 18.6% for the chains,
the nonadditivity in an individual binding energy can be as large as 25.5%. This occurs for the middle molecules in chains with n 1 5. It can be noted that if the bond lengths had been optimized for each chain length, the nonadditivity effect for the “inner” molecules had been enhanced and for the “outer” molecules decreased. 3. Frequencies and Many-Body Frequency Shifts. 3.1. OH Frequencies in the Ckoins and Ring. The uncoupled OH stretchingfrequenciesfor the H-bonded OH bonds were calculated for the chains, the ring, and the tetrahedron and in the chain also for the non-H-bonded OH of the rightmost molecule in Figure 1. The resulting force constants and harmonic and anharmonic frequencies for the chains and ring are collected in Table 4. For the chains, we find that the frequencies are divided into four major groups; see Figure 8. These correspond to the position of the water molecule along the chain. Group I consists of the rightmost molecule in each chain, group I1 of the leftmost molecule, and group I11 of the leftmost but one, and finally group IV is made up of all the inner molecules. The leftmost molecule in the dimer, 2A, although formally belonging to group 11, constitutes a group of its own. In Figure 9 the frequency shift with respect to the monomer frequency has been plotted against chain length. The frequencies for the members of groups 1-111 appear to have converged, but this is more questionablefor group IV. We also see from Figure 8 that the frequency shift for a
4218
Ojami4e and Hermansson
The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 0.3 I
1
hEnon-add
I i
I1 0.0
I11 I
IV
5-ring
’
I
2A
IV ,
I1 I11
I
position in chain Figure 7. Fraction of nonadditive individual energy out of the total individual energy for each member of the connected chains. The solid linea connected energy points for membersof the samechain. The leftmost point for each chain is the leftmost molecule in Figure la.
cm”
*
’
0
-100
-200
-300
-400
Rgure 8. Frequencies of the OH group participatingin the OH-bonded
network in the chains and in the ring. 400
TABLE 4 OH Stretching Force Constants and Frequencies for the Members of the Wins and the Ring (See Figure 1) (a,H~~rmonic Frequency; Y, Anharmonic Frequency)
h
MP2
ki
RHF (mdyn/ unit Y (cm-1) A) 1A 2A 2B 3A
4045 3954 4032 3893 3B 3880 3C 4030 4A 3887 4B 3811 4C 3872 4D 4029 5A 3882 5B 3802 5C 3801 5D 3867 5E 4029 6A 3880 6B 3796 6C 3793 6D 3795 6E 3863 6F 4027 7A 3879 7B 3793 7C 3788 7D 3787 7E 3792 3862 7F 7G 4028 ring 3770
k3
(mdyn/
AZ)
4.4508(1) -10.402( 1) 4.181 1 (1) -10.158(3) 4.4090(1) -10.269(2) 4.0449(1) -9.988(4) 4.0058(1) -9.881(3) 4.3941(2) -10.269(2) 4.0274(3) -9.952(1) 3.8524(2) -9.688(6) 3.9849(2) -9.834(8) 4.4010(1) -10.249(6) 4.0172(2) -9.941(7) 3.8327(3) -9.662( 11) 3.8307(1) -9.647(3) 3.973l(2) -9.835(6) 4.3999(3) -10.244(6) 4.0099(3) -9.943(1) 3.82O4(2) -9.636(6) 3.8102(4) -9.627(13) 3.8175(4) -9.642(14) 3.9670(3) -9.840(1) 4.3995(2) -10.248(3) 4.0088(3) -9.932(9) 3.8125(2) -9.634(8) 3.7980(1) -9.598(4) 3.7968(3) -9.603(9) 3.8098(2) -9.629(5) 3.9646(3) -9.819(8) 4.3996(2) -10.243(4) 3.7614(1) -9.568(4)
k4
(mdyn/ w v Au A’) (cm-1) (cm-1) (cm-1) 16.18(2) 15.52(3) 15.96(2) 15.21(2) 14.97(1) 15.91(4) 15.18(6) 14.75(3) 14.98(4) 15.93(7) 15.14(4) 14.64(6) 14.54(2) 14.95(3) 15.95(6) 15.20(6) 14.62(3) 14.48(7) 14.58(7) 14.84(6) 15.91(3) 15.09(5) 14.62(4) 14.39(2) 14.48(5) 14.53(3) 14.86(4) 15.84(4) 14.48(2)
3978 3856 3960 3793 3774 3953 3784 3701 3764 3956 3780 3692 3691 3759 3956 3776 3686 3681 3684 3756 3955 3776 3682 3675 3674 3681 3755 3956 3657
3811 3664 3795 3590 3572 3792 3582 3487 3562 3791 3576 3475 3473 3554 3791 3572 3469 3461 3466 3548 3790 3571 3464 3454 3454 3461 3549 3790 3433
0 -147 -16 -22 1 -239 -19 -229 -324 -249 -20 -235 -336 -338 -257 -20 -239 -342 -350 -345 -263 -2 1 -240 -347 -357 -357 -350 -262 -21 -377
IIxI1
’
,
: I
h
-
2
3
~
~
L
4
]
‘f
~
I
I
5
6
1
chain length Figure 9. OH frequenciesagainst chain length. Group I consists of the last (non-hydrogen-bonded) molecule in each chain, group I1 ofthe first hydrogen-bonded molecule, and group I11 of the last but one, and group IV are the inner molecules (cf. Figure 8).
Avi(ijklm)= vi(Oklm) - v(monomer) =
c Av;‘+
Av:~
K
K P>K
K f i
~ , +ip
= AVPwy
+ ec K
Av/‘@+
P > K k p ~ , p ,+il
+ AVFwy + AVfbody + ,,.
... (1 0)
where Av:~= ~((0’) -v(monomer) = Avi(0’)
-
AV;~‘= vi(@) v(monomer) -
Avi” P
molecule in the ring is much larger than even the largest shift for a molecule in the chain. This parallels the behavior of the interaction energies. 3.2. Many-Body Frequency Corrections in the Chains and Ring. We have performed a many-body frequency analysis, analogous to the “individual energy” analysis. Thus, for example, for a certain vibrating OH group in molecule i of a pentamer cluster ijklm we write the frequency shift with respect to the monomer frequency as
PE
AV.V~‘
= vi(gkl)-v(monomer)-
ilk1
cP
AV~”
P€ (ik4
-C
C
AV:@
(11)
P l>,k
P A € w1
The many-body frequency contributions have been cal~culatedto third order for the water molecules in the chains and to fifth order for the ring and the tetrahedron (Table 5). We start by discussing the chains. The dimer calculation results show that the hydrogen-bonded first neighbor has the largest influence on
Ab Initio Study of Cooperativity in Water Chains
The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 4279
TABLE 5 (a) Anharmonic OH Stretching Frequency Shifts and Many-Body Frequency Shifts Relative to the Free HDO Molecule (381 1 cm-I) for the Water Molecules in the Cham& a A A a A A A A a a a a a a A a a a A a a a a A a a a a a A a a a a a a
b b
c
d
e
f
g
C
i b b .
. c d d
c
e
C
B B b .
c C C
b B b b b B b b b b B b b b b b B b b
c c c C c c c C c c c c C c c c c c C c
.
d . d
c
b
c b b
.
e .
D d d d D d d d D d d d d D d d d d d D E d d
d c c
. h e
e e E e e e e E e e e e e
. f f f f f F f
f f f
f
g
F f
e e
Av
AfiWy
-147 -10 -16 -221 -149 -1 1 -1 1 -239 -30 -19 -151 -1 2 -17 -229 -324 -749 -20 -235 -336 -338 -257 -20 -239 -342 -350 -345 -263 -2 1
-147 -10 -16 -157 -146 -8 -9 -164 -26 -17 -148 -10 -1 6 -1 56 -173 -164 -16 -155 -172 -174 -164 -16 -154 -171 -173 -173 -163 -15 -154 -170 -172 -172 -173 -162 -15
. . g -240 g - 3 4 7 g - 3 5 7 g - 3 5 7 3 5 0 -262 6 -2 1
[AfiwY
-64 -3 -3 -2 -76 -4 -2 -3 -1 -1 -70 -143 -8 1 -3 -77 -151 -149 -8 5 -4 -80 -159 -158 -155 -89 -7 -79 -163 -167 -165 -160 -92
+ A+.bodY + ...I
29 2 27 18 32 13 11 2 8 6 32 47 34 20 34 49 49 36 20 36 50 51 50 38 29 36 51 52 52 51 38 29
-3 -8 -3 -1 -3 -1 3 -14 -8
0 -4 -12 -20 -17 -1 1 1
-7 -13 -18 -20 -17 -7 -2
-4
100(Avpol.dd/Awf)
(b) The 5-Ring. Total Frequency Shifts and Many-Body Frequency Shifts Relative to the Free H P O Molecule (381 1 cm-I) for the Water Molecules in the R i n p a a
b
d
C
e
AV
e
-4 -17 -148 -1 1
C b
C C
d
C a a a
b b b
a a a
b b b b
a
C C C
d
C C
d
C C C C C C
d d
e e e e e e e
d d d
-22 -167 -17 -238 -30 -22 1 -273 -39 -251 -327 -377
Av3-Wy -4 -17 -148 -1 1 -2 1 -152 -14 -165 -28 -1 59 -169 -3 1 -162 -176 -180
Av3-WY
-1 -1 5 -2 -73 -2 -63 -89 -6 -80 -138 -157
AfiWY
-1 5 -2 -8 -1 3 -39
AvS-bodY
-3
1OO(AV-/AV,)
5 9 12 31 7 29 38 21 35 46 53
(c) Thd Tetrahedron. Frequency Shifts and Many-Body Frequency Shifts Relative to the Free HDO Molecule (381 1 cm-l) for the Water Molecules in the Tetrahedron.bJ* Frequencies in cm-' a
h
h a a
B
a
B B B B B B B
a h h a
b B B B
a a a h
C
i
C
i C C
i i
C C C
*
i i i
AVOH -17 -148 14 -38 -239 -2 -95 -338 -23 -183 -277
AvoHZ-~"~~ -17 -148 14 -34 -165 -3 -133 -182 -20 -150 -167
AUOH'*
lOO(A~/Avf)
-4 -74
31
1 39 -153 -3 -35 -113
-41 46 13 18 40
11
-
-3 0 2 1
2
MP2(DZP) calculations. Frequencies in cm-'. A capital letter in the first column denotes the vibrating molecule; small letters, the nonvibrating molecules. e The arrow in Figure 4 or Figure 11 indicates the vibrating OH group. a
4280
The Journal of Physical Chemisfry. Vol. 98. No. 16, 1994
Ojamae and Hermansson Also for the ring (Table 5b) the nonadditivity ratio is above
“r
50%. and the sum of four- and higher-body terms is more than
-4Ocm-I. The five-body term is small. Neverthelessring-closing by hydrogen bonding is very important for the frequency. The reason for the large frequency downshift of the ring compared to the chain is mostly found in the four-body term. The fourbody term is much larger for the ring than the 4 5 terms in the 5-chain. The bent abCd tetramer in the ring geometry is downshifted hy 24 cm-l more than the corresponding straight tetramer in the chain geometry, which is probably a result of morefavorablealigneddipoles (this is contrary to what westated in ref 46). This is only one factor that makes the four-body term for the ring larger than for the chain. The water molecule in the ring is also a member of a larger number of connected tetramer fragments than a molecule in the middle of the 5-chain. The polarization of the former molecule is therefore larger. The twoand three-body downshifts are also somewhat larger for the ring than for the 5-chain. 3.3. OH Frequencies in the Tetrahedron. The frequenciesfor the tetrahedron and its fragments are given in Table 5c and displayed in Figure 11. A hydrogen-bonded neighbor on the donor side of the central vibrating molecule induces opposite effects depending on whether it is connected to the vibrating hydrogenor to thenonvibratingatom. In thefirstcasea downshift of about 148 cm-l is induced, in the second w e , an upshift of 14 cm-I. A neighbor on the acceptor side induces a small downshift. Looking at the trimers, one sees that adding a second nearest neighbor on theacceptor side increases thedownshift and theeffect is almost pair-additive. In the head-to-tail arrangement the nonadditivity is large, as we have previously seen for the trimer chain. If the molecule donates two hydrogen bonds, the nonadditivecontributionisalsosubstantial, hut now it isofopposite sign. Adding a neighbor to the nonvibrating H of an already H-bonded molecule diminishes the frequency downshift caused by the first neighbor; the same phenomenon is seen also for the additionofamoleculeatthenonvibratingOHgroupin the trimer and tetramer. For the tetramers and the pentamer the four-body contribution is significantly smaller than for the four-body contribution in the chains. The five-body contribution in the pentamer is insignificant. To conclude: Since the connected chains are short (3) in the tetrahedron, this does not allow significant woperativity beyond the three-body term. For double-donor or double-acceptor configurations the nonadditivity works in the opposite direction (‘antiwoperativity”) as compared to the acceptor4onor wnfigurations. 4. Charge Transfer. In a hydrogen-bonded water dimer, it is well-known that electron-charge transfer takes place from the H-bond accepting molecule to the H-bond donating molecu1e.l In a chain ofwater molecules, ‘multiplechargetransfers” between the molecules will then occur. The situation for chainsof different lengths is displayed in Table 6, where the Mulliken charges on the hydrogen-bondedH-atomandtotalchargesoneachmolecule are listed. The polarization is seen to be largest for molecules in the interior of the chain, as expected. A net electron-charge transfer has occurred from the “acceptor” side of thechain to the “donor” side. The terminal molecules have acquired most of the charge (+0.03e) and the total molecular charge diminishes rapidly when moving toward the middle of the chain (cf. the HF-chain results in ref 3 ) . As a result of the charge transfer, a middle molecule will reside in a charged environment and experiencean electric field from the terminal molecules’ charges. The situation will be the most extreme for a trimer due to the proximity of the terminal charges. For a truly infinite chain all molecules will be uncharged, and the same is true for a ring. This charge transfer might have consequences for practical calculations if a long or infinite chain has to be modeled by a short fragment. One example is when from a crystal structure containing infinite water chains, one selects a water molecule and only includes its nearest neighbors in the ‘supermolecule” to
+
A
B
C
D
E
F
G
position in 7-chain Figure 10. Many-body frequency contributions for the water moleculss in the heptamer chain.
-,.“**,
I
Figure 11. OH frquency shifts in the tetrahedron and its fragments. The linea indicate those clusters which are fragments of a cluster in the row immediately below.
the frequency downshift (-147 cm-1). The first neighbor on the lone-pair side of the vibrating water molecule induces a much smaller shift (-16 cm-I), slightly larger than the second neighbor on the donor side. Neighbors beyond the second are unimportant for the dimer shift (note that the standard deviation for the frequency is 1 or 2 cm-I). For the trimers in the chain geometry, the three-body contribution is large for the connected chain and very small for the nonconnected trimers. For all the water molecules in the connected tetramer, pentamer, hexamer, and heptamer chains the nonadditive contribution is large, and especiallysofor water moleculesin theinteriorof thechains. The many-body contributions for the members of the 7-chain are shown in Figure 10, and we note that out of the large frequency downshift (-357 cm-I) for molecules in the interior of the chain (i.e. group IV) about 50% is due to the nonadditive three-body contribution. Also the sum of four- and higher-body terms is quite large, -20 cm-I.
Ab Initio Study of Cooperativity in Water Chains TABLE 6 Charge Transfer in the Connected C h a i i as a Function of the Position of the Water Molecule and of the chain Length. chain length A B C D E F G H
0
Hz0
O.OO0 +0.031 +0.034 +0.036 +0.036 +0.036 +0.036
+0.018 +0.046 +0.050 +0.051 +0.051 +0.051
+0.019 +0.047 +0.051 +0.052 +0.052
+0.020 +0.048 +0.021 +0.051 +0.048 +0.021 +0.052 +0.052 +0.048 +0.021
1 2 3 4 5 6 7
o.OO0 -0.040 -0,046 -0.045 -0.046 -0.046 -0.046
-0.014 -0.052 -0.058 -0.057 -0.058 -0.058
-0.017 -0.055 4.061 -0.060 -0.061
-0.016 -0.055 -0.017 -0.060 -0.055 -0.017 -0.060 -0.061 -0.055 -0.017
1 2 3 4 5 6 7
O.OO0 -0.021 -0.027 -0.025 -0.026
+0.001 +0.025 -0.004 +0.004 +0.026
-0.027 -0,004
I
h
e
+0.022 -0.003
I
% ri-
1 2 3 4 5 6 7
-0.027 -0.004
The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 4281
O.OO0 +0.004 +0.026 0.000 O.OO0 +0.005 +0.026 0.000 0.OOO O.OO0 +0.004 +0.026
* Both Mulliken charge shifts relative to the free water molecule (qH = +0.338; qo = -0.676) and the total charge on each water molecule are given. RHF(DZP) calculations. model the short range interactions. Even if more remote neighbors are described as point charges, it will nevertheless be so that an electric field is created by thecharge-transfer charge; the direction of the charge transfer will lead to a frequency upshift as compared to a cluster where no charge transfer has occurred (i.e.the infinite system). Modelingnext- and next-next-nearestneighborsby point charges may then not be sufficient, since the charge transfer within the small cluster will still be there. Some test calculations were performed for the middle molecule in the k h a i n , for which the frequency shift is -338 cm-l. The two terminal molecules were replaced by point charges, +0.40 for H and -0.80 for 0,and the resulting frequency shift was then -302 cm-l. If only the leftmost molecule on the acceptor side is replaced, the frequency shift is -33 1 cm-1, whereas if the rightmost molecule is replaced, the frequency shift is -307 cm-’. Consequently, if one wants to simulate with reasonable accuracy the vibrations of a water molecule in a long chain, it appears to be necessary to include in the supermolecule at least two neighboring water molecules of the water molecule, whereas more distance molecules can be represented by point charges. If, on the other hand, only one neighbor is selected, the frequency shift may be underestimated by -30 cm-l. On the acceptor side it appears to be acceptable to include only one neighbor explicitly and represent the others by point charges. 5. Infrared AbsorptionIntensities. The IR absorptionintensity of each vibrating OH bond in the chains was calculated from the squared dipole moment derivative, obtained by pointwise differentiation with respect to the OH distance.47 The intensity variation with the position in and length of the chain are shown in Figure 12. The intensity increaseswith increasing chain length for all the members of a chain, and inner-chain molecules have higher intensities than terminal molecules. In the same way as for the frequencies, the intensity variation divides the molecules into the four groups I-IV. In fact, theintensity variation parallels the frequency variation. Epilogue We have discussed nonadditivity, both for energies and frequencies, for different geometrical arrangements of water molecules. As demonstrated by Tables 1-3 and Figures 2,3, and 5-7, the energy nonadditivity is considerable for water in chain
4e
position in chain Figure 12. IR absorption intensity for the OH stretching vibration for different positions in the chain and for different chain lengths. The solid linesconnect intensitypointsfor members of thesame chain. Theleftmost point on each solid line is the leftmost molecule on the corresponding chain in Figure la.
and ring formations. The nonadditivity increases with increasing chain length and is largest for molecules in the interior of the chains. OH frequency downshifts are nonadditive. For water clusters where the connected hydrogen-bonded chains of the headto-tail type are not longer than threeunits, an approximate “threebody additivity”prevails. In general, however, for the frequencies not even three-body additivity applies. The largest frequency shift and nonadditive frequency contribution exist for the ring structure. Here, all molecules are polarized in an equal fashion, whereas in the chains the terminal water molecules are less polarized. Strict Cooperativity. Typically, cooperativity is taken to be synonymous with the type of nonadditivity which enhances bonding, and anticooperativity with nonadditivity which diminishes bonding.ll For thechainsand thering we thenundisputably have cooperativity, both in the energies and in the frequencies. We here define a new sort of cooperativity where a stronger condition applies: all many-body terms should be of the same sign as the two-body term. If we call this form strict cooperativity in the case of negative signs, and correspondingly strict anticooperativity for positive signs, the chains and the ring are all of the first type, both with respect to energies and to frequencies. Water molecules in a tetrahedral coordination display cooperativity, but not strict cooperativity. Water in M D Simulations. The calculated nonadditivities in the total interaction energies were high, over 18%in the ring. But we also found that the nonadditive contribution to interaction energy of an individual molecule with the rest of the system could be much higher: over 25% for molecules in the middle in the longer chains. In molecular dynamics simulations of liquid water pair-additive potentials have been used routinely. Although some of these pair-potentials take nonadditivity into account in an effectiveway, others48were constructed from ab initio calculations on isolated water dimers. In liquid water, probably both rings and short chains are present. It is the individual interaction energies rather than the total interaction energy that governs the motion of a certain molecule. Considering that the potential energy a certain molecule experiences can evidently be in error by 25%, it is clear that the inclusion of many-body terms is advisable in simulations. From the present calculations we can see that the major part of the nonadditive energy can be recovered if three-body interactions are included, as has in fact been done in some simulations.31 Globular or Linear? Experimentally, if water dissolved in an organic solvent is studied by IR spectroscopy,l6basically two OH stretching bands are found. This has been taken as a support for the globular cluster model, where the broad lower frequency peak
4282 The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 corresponds to water molecules within the cluster and the higher frequency peak to molecules at the surface of the cluster. This is one possible explanation. Table 4, however, provides an alternative explanation. In principle the higher frequency peak could correspond to the non-hydrogen-bonded OH groups and the lower to hydrogen-bonded OH within the chain. Indeed, in MD simulations of water dissolved in an organic solvent short chains and rings have been seen.49 In an experiment where the water contents in a CH3CN-water mixture was gradually increased, it was found that the uncoupled OH frequency of selfassociated water decreased monotonicallytoward the liquid water value.50 This was interpreted as the water structure developing in a chainlike manner rather than a globular one, since in the latter case the frequency of the inner and the surface water molecules should be constant. Our calculations of increasingly longer chains also show a monotonic increase of the frequency shifts. The uncoupled OH frequency shift for a molecule inside the chain seems to approach a value below -360 cm-l rather than the downshift of -300 cm-1 in liquid water. The calculated value is more comparable to the frequency shift in amorphous ice, -400 cm-1-51 In amorphous ice each water molecule probably forms four H-bonds. Table 5c shows that saturating by H-bonds the two remaining free H-bond positions of a trimer leads to a further downshift of -38 cm-I. If this value is added to the calculated frequency of a molecule in the middle of the long chain, -360 cm-1, one arrives at a value close to -400 cm-l, similar to the amorphous ice frequency. Note also that from Table 5 follows that out of the pentamers considered, the energetically most favorable one is the ring, followed by the chain, and lastly the tetrahedron,2 suggesting that a large part of small water clusters will be in ring or unbranched chain forms. Acknowledgment. We thank Dr. J. Lindgren for invigorating discussions and The Swedish Natural Science Research Council (NFR) for supporting the work. References and Notes (1) Schuster,P. In The Hydrogen Bond-Recent Developmentsin Theory andExperiments;Schuster,P., Zundel, G., Sandorfy, C., Eds.;North-Holland Publishing Co.: Amsterdam, 1976; Vol. I, p 25. (2) Del Bene, J.; Pople, J. A. J. Chem. Phys. 1970, 52, 4858. (3) Kollman, P. A.; Allen, L. C. J. Am. Chem. Soc. 1970, 92, 753. (4) Hankins, D.; Moskowitz, J. W.; Stillinger, F. H. J. Chem. Phys. 1970,53,4544. ( 5 ) Karpfen, A.; Ladik, J.; Russegger, P.; Schuster, P.; Suhai, S.Theor. Chim. Actu 1974, 34, 115. (6) Clementi, E.; Kola, W.; Lie, G. C.; Ranghino, G. Int. J. Quantum Chem. 1980,17, 377. (7) Kistenmacher, H.; Lie, G. C.; Clementi, E. J. Chem. Phys. 1974,61, 546. ( 8 ) Yoon, B. J.; Morokuma, K.; Davidson, E. R. J. Chem. Phys. 1985, 83, 1223. (9) Hermansson, K. J. Chem. Phys. 1988,89,2149. (10) White, J. C.; Davidson, E. R. J. Chem. Phys. l w , 93, 8029. (11) Schuster. P.: Karafen. A.: Bever. A. In Molecular Interactions: Raiajczak, H., Owille-Thdmas; W.J. Ehs.;John Wiley &Sons: New York; 1980; Vol. 1, p 117.
Ojamie and Hermansson (12)W6jcik, M.J. J . Mol. Srrucr. 1988, 189, 89. (13) S&,vage,H. In Wuter Science Reviews; Franks, F., Ed.; Cambridge University Press: Cambridge, U.K., 1986; p 67. (14) Rapaport, D. C. Mol. Phys. 1983,50, 1151. (15 ) Kleeberg,H. In Interactionsof Waterin Ionicand Nonionlc Hydrates; Kleeberg, H., Ed.; Springer-Verlag: Berlin-Heidelberg, 1987; p 89. (16) Luck, W. A. P. In Water,A Comprehensive Treutise;Franks, F., Ed.; Plenum Press: New York, 1973; Vol. 2, Chapter 2. (17) Kuyskens, P. L. In Interactions of Water in Ionic and Nonionic Hydrates; Kleeberg, H., Ed.; Springer-Verlag: Berlin-Heidelberg, 1987; p 113. (18) Wells, A. F. Structural Inorganic Chemistry; Clarendon Press: Oxford, U.K., 1984. (19) Falk, M.; Knop, 0.In Water,A Comprehensive Treatise;Franks, F., Ed.; Plenum Press: New York, 1973; Vol. 2, Chapter 2. (20) Honegger, E.; Lcutwyler, S . J. Chem. Phys. 1988,88, 2582. (21) Knochenmuss, R.; Lcuwyler, S . J. Chem. Phys. 1992,96, 5233. (22) Karpfen, A.; Schuster, P. Can. J. Chem. 1985,63, 809. (23) Karpfen, A. Int. J. Quantum Chem.: Quuntum Chem. Symp. 1990, 24, 129. (24) Shivaglal, M. C.; Brakaspathy, R.; Singh, S . Proc. Indian Acad. Sci. 1988,100,413. (25) Scheiner, S.J. Mol. Strucr. (THEOCHEM) 1989,200, 117; 1989, 202, 177. (26) Rahman, A,; Stillinger, F. H. J . Am. Chem. Soc. 1973, 95, 7943. (27) Sciortino, F.; Corongiu, G. J. Phys. Chem. 1993, 98, 5694. (28) Speedy, R. J. J. Phys. Chem. 1984,88, 3364. (29) Barnes, P.; Fmey, J. L.; Nicholas, J. D.; Quinn, J. E. Nature 1979, 282, 459. (30) Lybrand, T. P.; Kollman, P. A. J. Chem. Phys. 1985,83, 2923. (31) Wojcik, M.; Clementi, E. J. Chem. Phys. 1986,84, 5970. (32) AhhtrGm, P.; Wallqvist, A.; Engatram, S.;Jbnason, B. Mol. Phys. 1989. 68. 563. --(33) Niesar, U.; Corongiu, G.; Clementi, E.; Kneller, G. R.; Bhattacharya, D. K. J. Phys. Chem. 1990,94, 7949. (34) Odutola, J. A.; Dyke, T. R. J. Chem. Phys. 1980, 72, 5062. (35) Burke, L. A.; Jensen, J. 0.; Jensen, J. L.; Krishnan, P. N. Chem. Phys. Lett. 1993, 206, 293. (36) Franken, K. A.; Jalaie, M.; Dykstra, C. E. Chem. Phys. Lett. 1992, 198, 59. (37) Msller, C.;Plesset, M. S.Phys. Rev. 1934,618. (38) Dunning, T. H., Jr.; Hay, P. J. In Modern Theoretical Chemistry: Methods of Electronic Structure Theory; Schaefer, H. F., 111, Ed.; Plenum: New York, 1977; Vol. 3, p 1. (39) Huzinaga, S.J . Chem. Phys. 1965,42, 1293. (40) Dupuis, M.; Rys, J.; King,H.F. J. Chem.Phys. 1976,65,111, Dupuis, M.; Farazdel, A.; Karna, S. P.; Maluendas, S . A. In MOTECC, Modern TechniquesinComputarbnal Chemistry;Clementi, E., Eds.;ESCOM: Lciden, The Netherlands, 1990. (41) BOYS,S.F.; Bemardi, F. Mol. Phvs. 1970. 19, 553. (42) F&h, M. J.; Del Bene, J. E.; Binkley, J. S.;Schaefer, H. F., 111. J . Chem. Phys. 1986,84,2279. (43) Alagona, G.;Ghio, C.; Cammi, R.; Tomasi, J. Int. J. Quantum Chem. 1987, 32, 207. (44) van Duijneveldt-van de Rijdt, J. G. C. M.; van Duijncveldt, F. B. J . Compur. Chem. 1992, 13, 399. (45) W6jcik, M. J.; Lindgnn, J. Chem. Phys. Lett. 1983, 99, 116. (46) Hermansson, K.;Ojamie, L. Int. J. Quantum Chem. 1992,42,1251. (47) Zilles, B. A.; Person, W. B. J . Chem. Phys. 1983, 79,65. (48) Matsuoka, 0.;Clementi, E.; Ycahimine, M. J. Chem. Phys. 1976, 64, 1531. (49) Kovacs, H.; Laaksonen, A. J. Am. Chem. Soc. 1991, 113, 5596. (50) Jamroz, D.; Stangret, J.; Lindgren, J. J. Am. Chem. Soc. 1993,115, 6165. (51) Bergren. M. S.;Schuh, D.; Sceats, M.G.; Rice, S . A. J. Chem. Phys. 1978,69, 3477. ~
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