J. Phys. Chem. 1988, 92, 6869-6871
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An AMI and ab Initio Molecular Orbital Study of Water Dimer J. J. Dannenberg Department of Chemistry, Hunter College and the Graduate School, City University of New York, New York, New York 10021 (Received:April 27, 1988; In Final Form: September 26, 1988)
Several structures for the water dimer, including trifurcated structures similar to the optimized AM1 geometry, have been calculated by using the MP4/6-311G** level of ab initio molecular orbital theory. The relative energies of the structures become quite close at the higher levels of calculation. The best trifurcated is only 0.2 kcal/mol higher than the optimized HF/6-31G* structure and only 0.4 kcal/mol higher than the lowest energy structure found (optimized by using AM1 with the H bond constrained to be linear). It appears likely that the potential surface of the water dimer is extremely flat. The experimental geometry, which corresponds to the minimum on the free energy surface, is likely to be dominated by entropy contributions.
Hydrogen bonding has long been recognized as an important factor in molecular structure. As such, it has received considerable attention in molecular modeling studies using molecular orbital theory, molecular mechanics, and empirical force fields. One of the simplest and most important hydrogen-bonding interactions is that in water dimer. This interaction has been studied at various levels of ab initio molecular orbital theory by several groups.' Its structure and energy were thought to be well established by the combination of high-level ab initio calculations and experimental observations. Recently, Dewar has proposed an improved semiempirical molecular orbital method, AM1: which was specifically designed to give better hydrogen-bonding interactions than previous versions (particularly MNDO). In the original AM1 paper, the water dimer was reported to have an optimized geometry that contains a linear H bond with an enthalpy of interaction of -3.3 kcal/mol. It has recently come to our attention that this report is in error? The correct AM1 interaction enthalpy is -5.5 kcal/mol. The optimized structure, I, has three H-bonding interactions, two equivalent and one of longer distance (Figure 1). These results have been confirmed in our laboratory. The initial reaction was to consider the AM1 geometry artifactual as it disagrees with reported experimental and high-level ab intio results. Nevertheless, this geometry seems somewhat reasonable as it uses all possible H-bonding interactions and takes full advantage of interactions of each end of both of the water dipoles. Furthermore, AM1 has been successful in estimating the energies of hydration of individual water molecules to protonated diamines in the gas phase," solvation of ammonium ion in ammonia-water mixtures? as well as explainingthe hydrogen-bonding interactions of various nitroanilines in crystalsS6 AM1 is also successful in modelling water trimers.3b Upon further inspection of the literature, it became apparent that trifurcated geometries had not been seriously considered in any of the recent high-level ab initio calculations, although some bifurcated geometries similar to the AM1 geometry were considered by Clementi.la~b We have calculated the energies of water dimer at the Hartree-Fock (HF), and several Moeller-Plesset (MP2, MP3, MP4DQ, MP4SDQ) levels' with the 6-311G** basis set.8 (1) The following are representative of calculations using large basis sets and correction for electron correlation: (a) Matsuoka, 0.; Clementi, E.; Yoshimine, M. J. Chem. Phys. 1976, 64, 1351. (b) Clementi, E.;Habitz, P. J . Phys. Chem. 1983,87, 2815. (c) Frisch, M. J.; Pople, J. A,; Del Bene, J. E. J. Phys. Chem. 1985,89, 3664. (d) Frisch, M. J.; Del Bene, J. E.; Binkley, J. S.; Schaefer, H. F. I11 J . Chem. Phys. 1986, 84, 2279. (e) Del Bene, J . Chem. Phys. 1987,86, 21 10. (f) Diercksen, G. H. F.; Kraemer, W. P.; Roos, B. 0.Theor. Chim. Acta 1975, 36, 249. (2) Dewar, M. J. S.; Zoebisch, E. G.; Healy, E. G.; Stewart, J. J. P. J . Am. Chem. SOC.1985, 107, 3902. (3) (a) Evleth, E., personal communication (b) Ventura, 0. N.; Coitino, E. L.;Lledos, A.; Bertran, J. THEOCHEM, in press. (4) Dannenberg, J. J.; Vinson, L. K., J . Phys. Chem 1988, 92, 5635. (5) Galera, S.; Lluch, J. M., Oliva, A.; Bertran, J. THEOCHEM 1988, 40, 101.
(6) Vinson, L. K.; Dannenberg, J. J. J. Am. Chem. SOC,in press.
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TABLE I: Optimized Geometries and 6-3116** Relative Energies for WateP optimizn method HF/6-31G HF/6-31G* HF/6-31G** HF/6-311G** MP2/6-31G* AM 1
bond 0.9497 0.9473 0.9430 0.9409 0.9684 0.9613
angle 111.56 105.50 105.99 105.48 104.06 103.53
HF 0.76 0.06 0.01 0.00 0.41 0.57
MP2 1.19 0.21 0.37 0.41 0.24 0.00
MP3 MP4DQ 1.06 0.08 0.21 0.24 0.21 0.00
1.10 0.14 0.28 0.32 0.18 0.00
MP4SDQ 1.14 0.17 0.33 0.37 0.17 0.00
'Energies in kcal/mol, bond lengths in A, angles in degrees. AM1-op timized total energies are -76.263 91 1 (MP2) and -76.271 320 (MP4SDQ) hartrees, respectively.
TABLE II: 6311G** Bonding - Energies - (kcal/mol) for Various Geometries of Water DimeP structure HF MP2 MP3 MP4DO MP4SDO I I1 111 IV
V VI
-0.83 -4.58 -3.76 -3.59 -2.71 -6.57 -6.45 -3.33 -2.51 -5.56
-3.60 -6.44 -5.96 -6.40 -5.92 -6.57 -6.15 -6.66 -6.18 -6.77
-2.91 -5.86 -5.44 -5.57 -5.15 -6.37 -6.21 -5.95 -5.53 -6.35
-2.82 -5.83 -5.47 -5.59 -5.23 -6.18 -5.90 -5.94 -5.58 -6.28
-3.08 -6.02 -5.68 -5.84 -5.50 -6.23 -5.89 -6.14 -5.80 -6.40
OEnergies are (a) relative to twice the lowest energy for water optimized at the AM1 level, which gives the lowest energy at all MP levels of calculation (first value); (b) relative to twice the energy for water optimized by the same procedure (second value). Where both values are the same (I and VI) only one entry is made.
TABLE 111: Relative 6-311G** Energies (kcal/mol) for Water Dimer at Different Geometries structure HF MP2 MP3 MP4DO MP4SDO I 5.74 3.17 3.46 3.47 3.32 I1 I11 IV
V VI
1.99 2.97 0.0 2.23 1.01
0.33 0.37 0.20 0.11 0.0
0.50 0.80 0.0 0.40 0.02
0.45 0.70 0.10 0.34 0.0
0.38 0.55 0.16 0.26 0.0
Various different geometries were considered, including the AM 1 optimized geometry, two geometries partially optimized a t the MP2/6-31G* level9 starting from the AM1 geometry, geometries optimized a t the HF/6-31G* and MP2/6-31G* levels, and a (7) (a) Pople, J. A.; Binkley, J. S.; Seeger, R. Int. J . Quantum Chem. Symp. 1976, 10, 1. (b) Krishnan, K.; Pople, J. A. I n f . J. Quantum. Chem. 1976, 14, 91. (8) Krishnan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A. J . Chem. Phys. 1980, 72, 650. (9) (a) Hehre, W. J.; Ditchfield, R.; Pople, J. A. J. Chem. Phys. 1972,56, 2257. (b) Hariharan, P. C.; Pople, J. A. Mol. Phys. 1974, 27, 209.
0 1988 American Chemical Society
6870 The Journal of Physical Chemistry, Vol. 92, No. 24, 1988
Letters
.ToP .. O L % '
I
I1
IV
I11
V
VI
Figure 1. Structures of the water dimers I-VI.
geometry optimized a t the AM1 level with a constrained linear H bond. For comparison, water was optimized by using various levels of ab initio theory and AM1 and then calculated by using the same procedures. The results of the dimer calculations are compared with twice the energy of water calculated, using, alternatively, the lowest energy of the optimized water molecules and the energy of water optimized with the same procedure used for the respective dimers.l0 No attempt was made to correct for zero-point vibrations or basis set superposition errors for reasons discussed below. The results are presented in Tables 1-111. Several surprising features are immediately apparent. First, the partially optimized structure with three H bonds, 11, is very close in energy to the overall minimum linear structure. Second, both the AM1 optimized water molecule and the AM1 (linearly constrained) water dimer have the lowest respective energies of all related structures.
Water Monomer The optimized geometries" and relative energies of the monomeric water structures obtained by using different calculational procedures are listed in Table I. With the criterion of lowest energy at MP4SDQ/6-311G**, AM1 gives the best optimized geometry, followed closely by MP2/6-31G*. All of the H F ab initio calculations give structures with obviously large bond angles. Nevertheless, the HF/6-3 1G*-optimized water gives a credible energy a t the MP4/6-311G** level, despite the significant structural differences between it and the other two low-energy structures. Water Dimer
Several points on the water dimer surface are listed in Tables I1 and 111. Structures I and I1 both have trifurcated H bonds. The former is the AMl-optimized structure, while the latter, 11, was partially optimized (a minimum was not reached but the energy changed little from one iteration to the next) at the MP2/6-31G* level, starting from I. Structure 111, which is 1.0 kcal/mol lower than I1 at the MP2/6-31G* level, is the result (10) This is possible as the calculations are size consistent at the levels used. See for example: (a) Gaw, J. F.; Yamaguchi, Y.; Vincent, M. A.; Schaefer, H. F., 111 J . Am. Chem. SOC.1984, 106, 3133. (b) Michael, D. W.; Dykstra, C. E.;Lisy, J. M. J . Chem. Phys. 1984,81, 5998. (c) Frisch, M. J.; Del &ne, J. E.; Binkley, J. S.;Schaefer, H. F. 111 J . Chem. Phys. 1986, 84, 2279. (1 1) Unless indicated, optimizations were performed without any geometrical constraints (including symmetry constraints).
of a continuation of the optimization that led to 11. Structure IV is the best structure we could obtain at the HF/6-3 lG* level using the 0-0distance of 2.964 8, given in ref ICand optimizing the rest of the geometry. It is slightly lower in energy (0.03 kcal/mol) than the value reported in ref I C and has one negative eigenvalue. The H bond is slightly bent (163.8'). Structure V is the completely optimized structure at the MP2/6-3 lG* level obtained by starting with structure IV. Structure VI is obtained by using AM1 with the H bond constrained to be linear. It represents the lowest energy for the M P calculations a t virtually all levels, but is 1.01 kcal/mol higher than IV at the HF level (see Table 111). Clearly, structures with very different geometries have nearly identical energies at the MP4/6-3 11G** level. In particular, the energy difference between the best trifurcated and linear (I1 and VI) structures is minimal. It is likely that the surface is very flat with many extremely shallow wells and low cols connecting them.12 Due to the flatness of the surface, complete optimization at a particular level seems to be less crucial than for structures with well-defined energy minima. It is, perhaps, more relevant to examine reasonable structures with different geometries. Most of the points indicated in the table were not true minima at the MP2/6-31G* level. The structure obtained from AM1 with the H bond constrained to be linear, V, gives the overall lowest energy at the MP4/311G** level. Due to the flatness of the surface, its dependence on basis set, and corrections for electron correlation, no attempt was made to calculate corrections for zero-point vibrations or basis set superposition error. The latter has been shown to lengthen the apparent 0-0distance for optimized structures involving correlated wave function^.'^ The relative energies of the geometries that involve multiple H bonds and/or have been optimized at the MP2/6-31G* level (1-111, and V) are quite sensitive to the electron correlation error. For example, structures, I, 11,111, and V change in binding energy by 2.57, 1.86, 2.81, and 3.33 kcal/mol, respectively, upon MP2 correction, while structures IV and VI change by 0.00 and 1.21 kcal/mol. Consequently, optimization a t the H F level can lead (12) A referee has suggested that the higher relative energy of I should be attributed to an overly close approach of the two waters. While this may very well be the case, one should also consider the relative orientations of the waters as well as the rather large improvement in the relative energy of I upon inclusion of (only partial) electron correlation corrections. (13) Newton, M. D.; Kestner, N. R. Chem. Phys. Lett. 1983, 94, 198. Kestner, N. R.; Newton, M. D.; Mathers, T. L.Int. J . Quantum Chem. Symp. 1983, 17, 431.
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to erroneous results. In fact, it is unlikely that any optimization performed so far be definitive as the surface is too flat for calculations with an error range of even 1-2 kcal/mol. Frisch et al. optimized water dimer at many different HF and MP2 calculational levels; however, it does not appear that they explored the surfaces except near to the minima found.ld They reported that correction for electron correlation tended to shorten the 0-0 distance but did not report calculations on structures resembling the trifurcated ones considered here. Clementi et al. reported minima for several water dimer structures including one containing a bifurcated H bond.Ib They seem not to have considered structures similar to I and 11, which have three H-bonding interactions. The binding energies in Table I1 are reported in two ways: (a) relative to twice the lowest energy calculated for water at each level, which corresponds to the AM1-optimized geometry; and (b) relative to twice the energy of water optimized by the same procedure used to optimize the dimer. Using water optimized at the various a b initio levels used would introduce an error of 0.34-2.28 kcal/mol (twice 0.17-1.14) into the energy of two waters at the highest level used here. The common practice of comparing the dimer to waters optimized at the same level also introduces errors of the same nature for the dimer. It is not clear that these errors will always compensate for each other. It is likely that errors of this nature may apply to previously reported ab initio binding energies. The experimental determinations of the s t r ~ c t u r eand ’ ~ binding energyI5 of the water dimer were performed at 350-400 K. On so flat a surface the free energy surface may be largely determined by entropic factors. For example, if one estimates the optimized structure to be 10 eu lower than the linear structure, at 400 K
the linear structure would be favored by 4 kcal/mol from the entropic contribution to the free energy surface. This would be more than enough to render the linear structure lower in energy at the experimental temperatures. Comparison of AMI-calculated energies with experiment poses an unusual problem for the case of water dimer. AM1 is parametrized to give enthalpies of formation at 298 K. The tacit assumption in such a system is that the optimized geometries will not differ significantly upon temperature change. As this is evidently not the case here, it is not clear whether the AM1 bonding energies should be compared with experimental bonding energies or bonding enthalpies (5.4 and 3.7 kcal/mol, respectively13). Curiously, the trifurcated and linear structures have AM1 bonding energies of 5.5 and 3.3 kcal/mol, respectively. Despite the imperfections of the comparisons, it is abundantly clear that AM1 does give interaction energies that are within reason. This is particularly important for more complex systems where other geometrical constraints preclude the possibility of forming three H bonds between water molecules.
Dyke,T. R.; Mack, K. M.;Muenter, J. S . J . Chem. Phys. 1977,66,
Supplementary Material Available: Table of calculated 631 1G** energies (hartrees) (1 page). Ordering information is given on any current masthead page.
(14) 498.
(15) Curtiss, L. A.; Frurip, D. 2703.
J.; Blander, M. J. Chem. Phys. 1979, 71,
Conclusion
Both A M I and high-level ab inito calculations appear to lead to structures of reasonable energy for the water dimer. That the “optimized” structures differ significantly in geometry seems to be more the result of the flatness of the surface than of any fundamental inadequacies of the AM1 methodology. In fact, AM1 optimization leads to the best structures for monomeric water and (constrained) linear water dimer. A structure qualitatively similar to the overall AM1 minimum is negligibly higher in energy than the best dimer considered in this study.
Acknowledgment. This work was supported in part by a PSC-CUNY grant.
Thermalization Distances and Times for Subexcitation Electrons in Solid Water T. Goulet and J.-P. Jay-Gerin* Groupe du Conseil de Recherches Medicales du Canada en Sciences des Radiations et Departement de Medecine Nucldaire et de Radiobiologie, Facult8 de Medecine, UniversitP de Sherbrooke, Sherbrooke, Quebec, Canada J l H SN4 (Received: May 31, 1988; In Final Form: September 7, 1988)
We report the results of our Monte Carlo simulations of the thermalization of subexcitation electrons in solid water. The electron scattering cross sections used in the calculations were those recently determined by Michaud and Sanche from slow-electron-impactexperiments on thin amorphous ice films. We find an average electron thermalization distance of about 13 nm, which is larger than what is usually assumed (-2-7 nm) in models describing diffusion-controlled track reactions s, in good agreement with experimental in irradiated liquid water. Our calculated average thermalization time is observations. The possibility for the subexcitation electrons to undergo a dissociative attachment to water molecules was also considered. We show that this process could explain the unscavengeable initial yield of molecular hydrogen.
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the “histories” of both the incident particles and the numerous
Introduction
since this substance is the chief constituent of biological matter. As proposed by Platzman,’ the succession of events that accompanies the passage of ionizing radiation through a medium may be divided into the physical, the physicochemical, and the chemical stages. For liquid water, the physical stage (S1O-I5 s) is well described by sophisticated Monte Carlo codesz4 which simulate (1) Platzman, R. L. The Vortex 1962, 23, 372.
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(2) Hamm, R. N.; Wright, H. A.; Ritchie, R. H.; Turner, J. E.; Turner, T. P. In Proceedings of rhe Fifrh Symposium on Mierodosimetry, Verbania-Pallama, 22-26Sept. 1975; Report No. EUR 5452; Booz, J., Ebert, H. G., Smith, B. G. R., Eds.; Commission of the European Communitics: Luxembourg, 1976; p 1037. (3) Malbert, M.; Carel, C.; Patau, J. P.; Terrissol, M. In Seventh Symposium on Microdosimetty, Oxford, 8-12Sept. 1980; Report No. EUR 7147; Booz, J., Ebert, H. G., Hartfiel, H. D., Eds.; Harwood Academic: London, 1981; p 359. (4) Zaider, M.; Brenner, D. J.; Wilson, W. E. Radial. Res. 1983, 95, 231.
0 1988 American Chemical Society
