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A comparison of theoretical models for anomalous water
A Comparison of Theoretical Models for Anomalous Water 1 LELAND
C. ALLEN
AND PETER
A. KOLLMAN
~
Department oJ Chemistry, Princeton University, Princeton, New Jersey 08540 Received July 20, 1970; accepted November 18, 1970 This paper is primarily concerned with a comparison of quantum mechanical results and essentially all of this work has been directed toward the relative stability and geometry of (H20)~. The three different levels of approximate quantum mechanical theory currently in use are critically examined in the light of their successes and failures in quantitatively describing typical covalent and conventional hydrogen bonds. Criteria found from this analysis are employed in a comparison of proposed theoretical models for anomalous water as (H~O)~. New computational results are presented for some structural models that have been heretofore only suggested qualitatively. Two of the models that have been proposed are modifications of qualitative normal liquid theories, and in these cases spectroscopic evidence must be employed for their evaluation.
INTRODUCTION There have been many theoretical studies on possible (H20)~ structures for anomalous water. Some of these have employed quantum mechanical molecular orbital methods, and others, intuitive physical arguments. The purpose of this paper is twofold: first, to examine the usefulness of the molecular orbital methods as a predictive tool in light of the successes and failures of these methods in treating conventional (covalent and hydrogen) bonding problems; and second, to assess which of the structures proposed for anomalous water are most nearly correct from a quantum mechanical viewpoint. THEORETICAL
METHODS
There have been three different levels of quantum mechanical calculations used in studying anomalous water structures: nonempirical (ab initio) molecular orbital methods (i-3), the all-valence electron semiempirica] (CNDO or INDO) methods (4-9) which neglect differential overlap and i Research supported in part by the Directorate of Chemical Sciences, Air Force Office of Scientific Research. 2 NSF Predoctoral Fellow 1966-70.
include only t w o - c e n t e r electron r e p u l s i o n terms, a n d one-electron schemes (e.g., ext e n d e d HiickeI theory) (10, 11) which o m i t
all consideration of electron repulsion and employ intuitively based formulas for diagonal and off-diagonal Hamiltonian matrix elements. In all three methods one-electron molecular orbitals for the system are determined as linear combinations of the atomic orbitals. The ab initio method uses the exact nonrelativistic Hamiltonian with clamped nuclei (Born-Oppenheimer approximation) and solves the time-independent SehrSdinger equation w~thin the space spanned by a particular orbital basis. Minimization of the single determinant wave function energy (Hartree-Fock approximation) then determines the weighting of the atomic basis orbitals in the resultant molecular orbitals (12). The accuracy of ab initio calculations (13) varies significantly depending on the quality of the atomic basis used in the computation, but experience shows that typical calculations are able to ascertain molecular geometry to about 2%, whereas dipole moments and force constants are systematically overestimated (i0 %-25 %), and dissociation energies are greatly underestimated
Journal of Colloidand Interface~cience,"Voh36, No. 4, August1971 461
462
ALLEN AND KOLLMAN
(the well-known "correlation error"). In certain other areas, such as inversion barriers (14), rotational barriers (15), and hydrogen-bond energies and geometries (16), ab initio methods often give excellent agreement with experiment. At the present stage in the development of quantum mechanical techniques, it is only from ab initio wave functions with a fairly high quality basis set that confidence-producing a priori results can be expected. There exist a number of two-electron term containing, all-valence electron semiempirical schemes, and the most widely used are CNDO/2 (17) and INDO (18). These schemes neglect all three- and four-center repulsion integrals as well as approximating and neglecting certain of the one- and twocenter repulsion integrals. Some matrix elements are found from experimental spectra and others are fit to experimental geometries and dipole moments. CNDO systematically underestimates bond distances and overestimates dissociation energies, although in some systems it yields reasonable dimerization energies (19) and qualitatively correct rotational barriers (17). Extended Hiickel theory (EHT) uses experimentally referenced, valence-state ionization potentials as diagonal matrix elements and off-diagonal elements proportional to an overlap weighted average of the diagonal elements. There are several modifications of this one-electron theory which assume slightly different forms for the matrix elements, such as the scheme developed by Cusachs (20) which adds a term proportional to S2 to the off-diagonal matrix elements, and the iterative version introduced by Rein et al. (21). These schemes are generally less successful in reproducing experiments than CNDO/2, although there are exceptions. Studies on the water dimer by EHT (19) and iterative EHT (22) and on methanol by EHT (23) have shown the method to be much poorer in treating hydrogen bonding than CNDO/2. Iterative EHT produces no water dimer stabilization at all, and Rein and Harris (22) attribute this to a basic deficiency in the method. H.~O+OH- is predicted to be more stable than (H~O)~, and the shapes of the proton potential functions Jour*~al of Colloid and Interface Science, VoI. 36, No. 4, AugusL 1971
in (H20)s are vastly different from those found by ab initio or CNDO/2 methods (24). In general one can say that to be completely successful for a molecular system with little precedent, semiempirieal methods need to be referenced to representative ab initio calculations. GEOMETRY OF (H20)~ If anomalous water is (H20)~, what is its geometry? Starting from the IR spectrum result (25) that the hydrogen is in a significantly different environment than in normal water, Allen and Kollman (6) examined possible geometries by CNDO/2 (Fig. 1). Of the structures with symmetric hydrogen bonds only a cyclic hexamer is more stable than six isolated H20 molecules. This search included the structure proposed by O'Konski (26) (a diborane-like arrangement). The latter structure has an energy less stable than isolated water molecules by approximately 100 kcal/H20. Morakuma (4) employed C N D O / 2 to compute linear and condensed negative ion arrays of one to five hexagonal rings. These structures follow the polyelectrolyte sheet model proposed by Lippincott et al. (25) and this model fails both on physical grounds and on the basis of the further quantum mechanical calculation discussed below and listed in Table I. Pederson (5), using INDO, has made calculations for several planar arrays of one to three hexagons along with their corresponding open structures. He has employed bond orders as a criterion of stability, but at best, these yield only a rough qualitative picture for water polymers. Ab initio calculations show that his INDO results grossly overestimate overlap populations, and this leads him to the unreasonable prediction of a 300 % greater bond energy for symmetric hydrogen bonds over those in ordinary water, whereas total energies show the symmetrically bonded structures to be less stable. Azman et al. (7), using a different CNDO parameterization than Morakuma or Allen and Kollman, carried out a few calculations on a single hexamer, and they also concluded that symmetric structures are far more stable than asymmetric. The results of these calculations are clearly in-
463
M O D E L S FOI~ A N O M A L O U S W A T E R
H
(a)
(b)
,c,
H
H
Stability relative to isolated water molecules
.
2
~
H-~-- ~
+64. k cal
.~.~, ~', ~ ' ~ ~ ~: ~ . H
H
H.
>"
IH.
H
(d)
H
-14.3
.
H
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H
-397. H
H
-68.
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(e)
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~A,~,~..,~,~ 14,
-22. H
14,
H,
(f) (g)
-24. H
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-(~H-'~
H
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--56.
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Hk )H 14,, .,H .
(h)
H-q.~__
,~.
.,~H
Large negative
H~::)---H--Q,H IH~)--H-'O~ H FIG. 1. Possible geometries for anomalous water.
TABLE I PROTONATION OF ~APHTI-IALENE STRUCTURE O10Ht9Protonation At At At In In
c e n t e r oxygen a l p h a oxygen b e t a oxygen center of one ring center of one ring
j~ ( r ( O - - H ) = 1.03A) ( r ( O - - H ) = 1.03A) (r(O--H) = 1.03~) [A (out of plane) = 0.0~x] [A (out of plane) = 1.03A]
--199.01 --199.89 --199.85 --198.60 --198.58
E(O10Hlg-) = --198.24 E(10 isolated H~O) = -198.91 Journal of Colloid and Interface Science,
VoL 36, No. 4, August 1971
464
ALLEN AND KOLLMAN
correct, and this is due to either or both of the following reasons: First, it has not been demonstrated that the parameterization they have used produces reasonable results for conventional hydrogen bonds, and second, they did not systematically optimize the geometry for both asymmetric and symmetric cycles. Goel et al. (8) carried out CNDO/2 calculations on sp 2 and sp 3 hybridized water dimers, trimers, and hexamers and found additional stabilization of sp 2 hydrogen-bonded species over sp 3. An asymmetric, cyclic hexamer is found to be unbound and a symmetric, cyclic hexamer strongly bound--again this result is incorrect as shown unequivocally by the ab initio calculations to be discussed in the next section. G o d et al. have referenced their sp ~ dimer and polymer calculations to isolated sp ~ water monomers without specifying the energy needed to promote the isolated monomers from the normal geometry [0 (HOH) = 105°] to sp2[0 (HOH) = 120°]. Messmer (9), on the basis of C N D O / 2 calculations, believes that p~ delocalization through hydrogen may greatly stabilize symmetric, cyclic structures. Allen and Kollman (6), however, have shown from ab initio calculations on HF2-- and H502+ with a symmetric H bond that this cannot be true. Messmer's numbers result from the inherent nature of CNDO and his particular use of the scheme. On the basis of his crystallographic experience, Donohue (27) has proposed an interesting space-filling structure based on a symmetrically bonded rhombic dodeca~ hedron unit, (H20)1~. We have carried out C N D O / 2 calculations on this unit (28) and also for a 24-molecule cubic oetahedron ( 0 . . . 0 distance equals 2.32A) model. Our results are: Unit
Rhom. dodee. Cubic octah.
Et (z[ )
n (H~O)
Instability (local) per H~O
--277.83413 - 476.4841
-278.4754 - 477.3864
27.1 23.5.
The reason for these high energies is the close approach of symmetric hydrogens (each bearing a charge of -ff.34) necessitated by the
cagelike structures. Another qualitativelyproposed model, the tetramer of Bollander, Kassner, and Zung (29), has been evaluated with ab initio computations by Del Bene and Pople (30), and they find it less stable than isolated water molecules. Two models of a quite different type, both based on modifications of models for the normal liquid, have been suggested to explain anomalous water. One of these, a bistructural model of ordinary water employed by Fabuss (31), derives from the theory of Vdovenko, Gurikov, and Legin (32). The presence in their theory of a nonhydrogen-bonded dense component in addition to the lighter icelike component has led Fabuss to postulate a differential stabilization by the quartz surface of the nonhydrogen-bonded component. This model must be rejected because it would display a normal OH I R stretch and a nonshifted nmr spectrum--both in contradiction to the experimental evidence (25, 33). Erlander (34) has suggested that anomalous water consists of ice II-type clusters in analogy to an ice I-type cluster model for normal water. However, since ice II has a large stretching band absorption at a frequency somewhat greater than 3,000 cm-1, an ice II anomalous water must be expected to have a reasonably close-by absorption, but this again contradicts the I R evidence. The precedents cited by Erlander for the possible disappearance of such stretching frequencies are all based orl high-concentration salt, acid, and base solutions where there is a significant concentration of H2,, -- iO~+, H2~ q- 1On+, and H O H - . . A - . These species would be expected to have very different O-H stretching regions from water, ice I, and ice II. The presence of these species means that there would be less normally bonded water, thus accounting for the observed decrease in intensity of the 3,400 cm-1 (normal water O - - H stretch) band. The presence of H502+ would cause absorptions in the 1,500-2,000 cm-~ range and thus would add to the intensity of the normal water bond (~-4,600 em -1) .
The polyelectrolyte model proposed by Lippincott et aI. (25) would have spectral properties similar to those observed experimentally and similar to those predicted by
Journal of Golloid and Interface Science, VoL 36, No. 4, August 1971
MODELS FOg ANOMALOUS WATER ~he neutral model of Allen and Kollman (6). However, CNDO/2 calculations (Table I) indicate that by far the best place of attachment for a proton to a typical sheet fragment, naphthalene-like H19010-, is above one of the two central oxygens. Thus, any negatively charged ring would automatically pick up a proton and have the characteristics of a neutral structure. Polyeleetrolytes may also be eliminated on physical grounds (6, 34). Therefore, one may conclude that if anomalous water is (H~O)n, it must exist as a neutral network of symmetric, cyclic hexamers. Using iterative extended Hiickel theory (21), Minton (10) has made computations for several of the structures in Fig. 1 as well as the Donohue (27) and Bollander (29) models. In view of the fact that the scheme he has chosen has been shown by its authors (21, 22) to lead to an unstable solution for the conventional hydrogen-bonded water dimer owing to a defect in the method it is neither surprising nor meaningful that Minton finds an unstable solution for the cyclic, symmetric hexamer. Aldrich et al. (11) have applied the one-electron, EHT-]ike scheme of Cusachs (20) and they also find an unstable cyclic, symmetric hexamer. Since the same scheme yields a 32 kcal/mole binding energy for the conventional hydrogenbonded water dimeP 5 (experimental 5-7 kcal/mole), one can have little confidence in their result. :\lore generally, the credibility of one-electron theories is very low because, of necessity, they must base stability predictions on orbital energy sums and/or overlap population, and these criteria are suspect because: (1) In asymmetrically bonded (HF)6, which is experimentally known to be a cyclic hexamer with a Hbond energy of 7 keal/H-bond, ab initio LCAO calculations (3) yield a stabilization energy of 5.3 kcal/H-bond but possess a total overlap population less than that of six isolated H F molecules (36). (2) In many ab initio calculations on hydrogen- and lithium-bonded complexes (37), the sum of the one-electron orbital energies is higher than that for the isolated species, even though the difference in total energies yields a stabilization in agreement with experiment.
465
Finally, two articles (38, 39) have applied the nonpalred spin orbital (double-quartet) schematic model (40) to structures for anomalous water. Neither of these papers contains quantitative results, but they provide a novel, pictorial representation of the bonding to be expected if anomalous (H20)~ is a symmetrically bonded, tetrahedrally hybridized, three-dimensional zineblende or wurtzite structure (38) or cyclic, symmetrically bonded sheets arranged threedimensionally in a graphite lattice (6, 41). The key bonding unit in Linnett's tetrahedrally coordinated, symmetric hydrogen structure 38 can be modeled by either a tetrahedrally coordinated pentamer or by a cyclic cyclohexane-like puckered ring, and energy optimized results for both 6 show a higher energy than our planar hexagons. Comparing planar and puckered rings it is easy to see the physical origin of the high energy for Linnett's structure: Energy optimized O - - H - - O lengths for planar and puckered hexagons are very similar (2.32 and 2.36 A respectively), and consequently, across-the-ring 0 - - 0 distances are significantly smaller for the puckered conformation. Thus the strongly negative oxygens give rise to a relatively large puckered conformation repulsion energy. Building a larger structure leads to an even more unfavorable situation: Extending the sheet from one to two puckered rings increases the instability relative to two planar rings by an additional 31 kcal/mole. STABILITY
OF (H20)~
It is important to attempt to assess the absolute stability of cyclic structures, and it is clear from the previous sections of this paper that this can be accomplished only through ab initio calculations. Sabin et al. (1), Pople (2), and Allen and Kollman have carried out ab initio calculations on cyclic hexamers. In some of their calculations Sabin et al. find that structures with both symmetric in-ring hydrogens and asymmetric in-ring hydrogens are more stable than isolated water monomers and actually obtained the symmetric structure more stable than the asymmetric, but this latter result is due to the fact. that they have examined only an asymmetric structure with
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466
ALLEN AND KOLLMAN
R ( 0 . . . 0 ) = 2.76~. Pople also finds both structures more stable than isolated monomers, but with the asymmetric structure (R = 2.44A) slightly more stable than the symmetric (R = 2.3A). (This is exactly what is found in CNDO/2 calculations.) Allen and Kollman carried out an ab initio LCAO calculation on the planar, symmetric hex~mer with a basis set superior to these two calculations and found an energy (R (0...0) ~2.32A) slightly less stable than six isolated water molecules. To obtain as realistic an estimate as possible for the absolute stability, one must add correlation and zero point energy corrections to both symmetric and asymmetric structures. When this is done, the results indicate that cyclic hexamers may be slightly more energetically favorable than six isolated water molecules but clearly less stable than liquid water. It should be noted also that there is no barrier (local minimum) in a symmetric hexamer structure to prevent the structure from reverting to an asymmetric (normal water-like) structure. Thus if anomalous water is (H20)~, two other physical effects must come into play: (1) There must be an unusually long-range and strong surface binding force which selectively lowers the energy of symmetric structures below that of the asymmetric. (2) The structures formed must be a threedimensional aggregate, because after its remoral from the surface of formation, there must be a barrier hindering its decomposition into normal water., In another article in this issue, "What Can Theory S~y About the Existence and Properties of Anomalous Water?" we show that neither of these criteria are fulfilled and thus we do not now believe in the existence of anomalous water. SUMMARY AND CONCLUSIONS
This result implies the need for a differentially selective, large, and unprecedentedly long-range force on water-adsorbent surfaces if anomalous water is to exist as (H20).. Within the framework of current digital computer technology, it is not possible to fully assay the potential bonding characteristics and properties of (H20)~ polymers by ab initio calculations. Therefore it is necessary to employ semiempirical methods. For greatest confidence and effectiveness, these methods must be referenced to and calibrated against ab initio calculations that are as nearly as possible representative of the bonding situation under consideration. Several cases have been cited in this paper where failure to do this has limited the usefulness of that work. In particular it was shown that one-electron, extended Hfickcl theorylike models are not likely to be very effective in elucidating possible (H20)~ structures. With the use of semiempirical methods (primarily CNDO/2), it was found that the most favorable symmetrically bonded (H20), structure consists of hexagonally patterned sheets, three-dimensionally arranged in a graphite-like structure (6). Finally, we have shown in another article in this issue, "What Can Theory Say About the Existence and Properties of ArLomalous Water?", that the necessary conditions required for stabilizing cyclic, symmetric (H20)n do not exist. Therefore since this article has sho~n that the most satisfactory geometry of all those proposed is the cyclic, symmetric array described above, we believe that this constitutes very strong evidence against the existence of anomalous water. Of course, this does not rule out some complex combination of substances not considered here, but if such is the case, it almost entirely eliminates the high interest and promise aroused by a simple new water atlotrope.
Of the three levels of quantum mechanical approximation currently in use, it is clear that only ab initio calculations can provide quantitative insight into the absolute stability of the potentially new form of bonding proposed to explain anomalous water. Such calculations show that cyclic, symmetric hydrogen bonding is considerably less stable than the conventional asymmetric form (3).
Note 1: Recently, Ageno [Theor. chim. Acta, 17, 334 (1970)] has proposed a dibora~le-like s t r u c t u r e for anomalous w a t e r a n d has shown t h a t t h e Ii% s p e c t r u m expected for this s t r u c t u r e is consistent w i t h t h a t observed (ref. 25). C i g n i t t i a n d Paolini (Theor. chim. Acta, in press) used C N D O / 2 to assess t h e feasibility of this s t r u c t u r e a n d came to similar conclusions as references 6 a n d 26.
o
Two Notes Added in Proof
Journa$ of Golloidand Interface Science, Vol. 36, No. 4, August; 1971
MODELS FOR ANOMALOUS WATER Note 2: Barclay Kamb has recently published a theoretical paper [Science, 172, 231 (1971)] that presents qualitative bonding arguments based largely on the experimental data which exists for asymmetric hydrogen bonded systems. Several points bear directly on the work reported here: (a) His intuition leads him to favor tetrahedrally hybridized, puckered rings over other structures and feels that lack of ~rbonding in the planar rings implies tetrahedral coordination (p 233, col. 1 and 3; p 234, col. 1 and 2; p 235, col. 1 and 2; p 236, col. 2). We have shown quantitatively that this is incorrect [reference 6 and last paragraph of geometry of (H~.O)n]. (b) Kamb gives a number of hypothesized potential energy curves for symmetric and asymmetric bonds, the data for which is obtained from known asymmetric bonds. He discusses a variety of barrier situations that might permit symmetric bonding, and the one that comes closest to our original ideas 6 is the cooperative hypothesis (p 237, col. 1 and 2), and he abstracts a reasonable per bond destabilization energy (14 keal/mole). (c) The observed existence of stable HLO~+ with its short symmetric bonds leads Kamb to suggest this ion as an activation mechanism that would eliminate the kinetic barrier between symmetric and asymmetric bonds. But the key feature that gave rise to the possibility of the kinetic barrier was the three-dimensional array of neutral cyclic units, and H502+ does not possess any of these properties. Throughout his article Hamb is saying that cyclic symmetric hydrogen bonds do not constitute a straightforward extrapolation of well-established data on asymmetric bonds. With this we have never had any disagreement, but his refutation of their existence on these grounds does not represent a strong case against anomalous water and does not imply errors in the quantum mechanical calculations. REFERENCES 1. SABIN, J. R., H2~RRIS, R. E., ARCItIBALD, T. W., KOLLMAN, P. A., AND ALLEN, L. C., Theor. Chim. Acta, 18, 235 (1970). 2. POPLE, J. A., Private communication. Some calculations are reported in ]:)el Bene, J., and Pople, J. A., J. Chem. Phys. 52, 4858 (1970). 3. ALLEN, L. C., AND KOLLMAN, P. A., ] . A m e r . Chem. Soc. 92, 4108 (1970). 4. MORA~:U~A, K., Chem. Phys. Lett. 4, 358 (1969).
467
5. PEDERSON, L., Chem. Phys. Letters 4, 280 (1969). 6. ALLEN, L. C., AND KOLLM3-N, P. A., Science 167, 1443 (1970). 7. AZM.~.N,A., HOLLER, J., AND HADZI, D., Chem. Phys. Lett. 5, 157 (1970). 8. GOEL, A., MURTHY, A. S. N., AND RAO, C. N. R., Chem. Commun., 423 (1970). 9. MESSIER, R., Science 167,479 (1970). 10. MINTON, A. P., Nature 226, 151 (1970). 11. ALDRICH, H. S., LADER, H. J., GARY, L. P., CORRINGTON, J. H., AND CUSACHS, L . C., Submitted to Chem. Phys. Lett. 12. PILAR, F., "Elementary Quantum Chemistry." McGraw-Hill, New York, 1968. 13. See ALLEN, L. C., An. Rev. Phys. Chem. 20, 315 (1969) and KRAUSS, M., "Compendium of Ab Initio Calculations of Molecular Energies and Properties," Nat. Bur. Std. (U.S.) Tech. News Bull., p 438 (1967). 14. RAUK, A., ALLEN, L. C., AND CLEMENTI, E., J. Chem. Phys. 52, 4133 (1970). 15. ALLEN, L. C., Chem. Phys. Lett. 2,597 (1968). 16. HOLLMAN,P. A., AND ALLEN, L. C., J. Chem. Phys. 51, 3286 (1969); ibid. 52, 5085 (1970). 17. POPLE, J. A., AND SEGAL, G. A., J. Chem. Phys. 44, 3289 (1966). 18. POPLE, J. A., BEVERIDGE, D. L., AND DOBOStI, P. A., J. Chem. Phys. 47, 2026 (1967). 19. MVRTHY, A. S. N., A~D R.~o, C. N. R., Chem. Phys. Lett. 2,123 (1968). 20. CUSACHS: L. C., AND CUSACHS, B. B., J. Phys. Chem. 71, 1060 (1967). 21. REIN, R., FUKUDA, N., WIN, H., CLARKE, G. A., AND HARRIS, F. E., J. Chem. Phys. 45, 4743 (1966). 22. REIN, R., AND HARRIS, F., J. Mol. Struet. 2,103 (1968). 23. MVRTHY, A. S. N., DAVIS, R., AND RAO, C. N. R., Theor. Chim. Acta 13, 81 (1968). 24. P. KOLLMAN AND L. C. ALLEN (submitted to J. Amer. Chem. Soc.) have compared the proton potential well in NH~-HF by ab initio and CNDO/2 methods and find no double well at the minimum energy N-F distance (although one appears at larger N-F distances). This is in contrast to EHT results on (H~O):. 25. LIPPINCOTT, E., STROMBER, R., GRANT, W., .AND CESSAC, G., Science 164, 1482 (1969). 26. O'HONSKI, C., Science 168, ]089 (1970). O'Konski and Levine (this issue) have carried out a CNDO/2 geometry optimization of the diborane-tike symmetric bonded tretramer and found it unstable by 148 kcal (our unoptimized value was 397 kcal) relative to four water molecules. They have Mso performed ab initio calculations that again
Journal of Colloid and Interface Science, Vol. 36, No. 4, August 1971
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ALLEN AND KOLLMAN
lead to a high energy, unequivocally eliminating this as a possible structure. 27. DONAHUE, J., Science 166, 1000 (1969). 28. We have calculated an energy for this model using R(0...0) = 2.44~_ and symmetric hydrogens. Although energy optimization was not done, the structure is so unfavorable that optimization would not affect the conclusions. 29. BOLLANDER, R. W., KASSNER, J. L., AND ZUNG, J. T., Nature 29.1, 1233 (1969). 30. DEL BENE, J., AND POPLE, J., J. Chem. Phys.
59., 4858 (1970). 31. FABvss, B., "Thermodynamic Aspects of Anomalous Water," J. Colloid Interface Sci., this issue. 32. VDOVENI~O, V. M., GvnlXOV, Y r . V., AND LEGIN, YE. K., Zhu. Strukt. Khim. 8, 18, 403 (1967); ibid. 9, 599, 819 (1968). 33. PETSKO, G., Science 167, 171 (1970); PAGE,
37. 38.
39. 40.
41.
T. F., JR., JACOBSEN, ~:~.J., AND LIPPINCOTT, E. R., ibid., p 51. 34. ERL±NDER, S., Phys. Rev. Lett. 22,177 (1969); Phys. Rev. A1,868 (1970). 35. ALm~ICI~, H. S., GARY, L. P., CORRINGTON,
J. H., AND CUSACI~S, L. C., 159th ACS Meeting, Houston, Texas, February, 1970; Gary, L. P., Private communication. 36. With the use of the wave functions reported in Appendix I, the total overlap for (HF)~ sym = 2.529; for (HF) 6 asym. = 2.933; and for 6HF = 3.086. (Both polymer cMeulations were carried out at R ( F . . . F ) = 2.5/~.) The sum of overlap populations criterion for stability is also basis set depend-
Journal of Colloid and Interface Science,
ent. For example, Kollman and Allen (ref. 16) find the total overlap population in the water dimer to be less than that for two monomers; Hankins, Moskowitz, and Stillinger [J. Chem. Phys. 53, 4544 (1970)], using a larger basis set, find a similar geometry and energy of dimer formation, but a total dimer overlap population greater than that of 2 monomers. KOLLMAN,P. A., LIEBMAN, J. F., AND ALLEN, L. C., J. Amer. Chem. Soe. 92, 1142 (1970). LINNETT, J. W., Science 167, 1719 (1970). ALLEN, L. C., Nature 227 (July 25, 1970). LINNETT: J. W., J. Amer. Chem. Soc. 88, 2643 (1961) ; "Electronic Structure of Molecules." Methuen, London, 1964. The schematic drawing and pictorial description given in reference 39 is just a qualitative "double-quartet" rationalization of the stacked planar hexagons developed in reference 6 (unfortunately the printers omitted reference 8 in that article which was a reference to J. W. Linnett's structure [reference 37 here], and the last reference given should have been numbered 9). APPENDIX I
T h e enerogy results for a s y m m e t r i c (HF)6 a t R = 2.5A, s y m m e t r i c (HF)6 a t R = 2.5A, a n d s y m m e t r i c (H20)6 a t R = 2.35A are g i v e n in reference 3. T h e t a b l e s of w a v e functions have been deposited with the ADI Auxiliary Publications Project, Photoduplic a t i o n Service, L i b r a r y of Congress, W a s h i n g t o n 25, D . C .
Vol. 36, No. 4, August 1971
C. ALLEN
AND PETER
A. KOLLMAN
~
Department oJ Chemistry, Princeton University, Princeton, New Jersey 08540 Received July 20, 1970; accepted November 18, 1970 This paper is primarily concerned with a comparison of quantum mechanical results and essentially all of this work has been directed toward the relative stability and geometry of (H20)~. The three different levels of approximate quantum mechanical theory currently in use are critically examined in the light of their successes and failures in quantitatively describing typical covalent and conventional hydrogen bonds. Criteria found from this analysis are employed in a comparison of proposed theoretical models for anomalous water as (H~O)~. New computational results are presented for some structural models that have been heretofore only suggested qualitatively. Two of the models that have been proposed are modifications of qualitative normal liquid theories, and in these cases spectroscopic evidence must be employed for their evaluation.
INTRODUCTION There have been many theoretical studies on possible (H20)~ structures for anomalous water. Some of these have employed quantum mechanical molecular orbital methods, and others, intuitive physical arguments. The purpose of this paper is twofold: first, to examine the usefulness of the molecular orbital methods as a predictive tool in light of the successes and failures of these methods in treating conventional (covalent and hydrogen) bonding problems; and second, to assess which of the structures proposed for anomalous water are most nearly correct from a quantum mechanical viewpoint. THEORETICAL
METHODS
There have been three different levels of quantum mechanical calculations used in studying anomalous water structures: nonempirical (ab initio) molecular orbital methods (i-3), the all-valence electron semiempirica] (CNDO or INDO) methods (4-9) which neglect differential overlap and i Research supported in part by the Directorate of Chemical Sciences, Air Force Office of Scientific Research. 2 NSF Predoctoral Fellow 1966-70.
include only t w o - c e n t e r electron r e p u l s i o n terms, a n d one-electron schemes (e.g., ext e n d e d HiickeI theory) (10, 11) which o m i t
all consideration of electron repulsion and employ intuitively based formulas for diagonal and off-diagonal Hamiltonian matrix elements. In all three methods one-electron molecular orbitals for the system are determined as linear combinations of the atomic orbitals. The ab initio method uses the exact nonrelativistic Hamiltonian with clamped nuclei (Born-Oppenheimer approximation) and solves the time-independent SehrSdinger equation w~thin the space spanned by a particular orbital basis. Minimization of the single determinant wave function energy (Hartree-Fock approximation) then determines the weighting of the atomic basis orbitals in the resultant molecular orbitals (12). The accuracy of ab initio calculations (13) varies significantly depending on the quality of the atomic basis used in the computation, but experience shows that typical calculations are able to ascertain molecular geometry to about 2%, whereas dipole moments and force constants are systematically overestimated (i0 %-25 %), and dissociation energies are greatly underestimated
Journal of Colloidand Interface~cience,"Voh36, No. 4, August1971 461
462
ALLEN AND KOLLMAN
(the well-known "correlation error"). In certain other areas, such as inversion barriers (14), rotational barriers (15), and hydrogen-bond energies and geometries (16), ab initio methods often give excellent agreement with experiment. At the present stage in the development of quantum mechanical techniques, it is only from ab initio wave functions with a fairly high quality basis set that confidence-producing a priori results can be expected. There exist a number of two-electron term containing, all-valence electron semiempirical schemes, and the most widely used are CNDO/2 (17) and INDO (18). These schemes neglect all three- and four-center repulsion integrals as well as approximating and neglecting certain of the one- and twocenter repulsion integrals. Some matrix elements are found from experimental spectra and others are fit to experimental geometries and dipole moments. CNDO systematically underestimates bond distances and overestimates dissociation energies, although in some systems it yields reasonable dimerization energies (19) and qualitatively correct rotational barriers (17). Extended Hiickel theory (EHT) uses experimentally referenced, valence-state ionization potentials as diagonal matrix elements and off-diagonal elements proportional to an overlap weighted average of the diagonal elements. There are several modifications of this one-electron theory which assume slightly different forms for the matrix elements, such as the scheme developed by Cusachs (20) which adds a term proportional to S2 to the off-diagonal matrix elements, and the iterative version introduced by Rein et al. (21). These schemes are generally less successful in reproducing experiments than CNDO/2, although there are exceptions. Studies on the water dimer by EHT (19) and iterative EHT (22) and on methanol by EHT (23) have shown the method to be much poorer in treating hydrogen bonding than CNDO/2. Iterative EHT produces no water dimer stabilization at all, and Rein and Harris (22) attribute this to a basic deficiency in the method. H.~O+OH- is predicted to be more stable than (H~O)~, and the shapes of the proton potential functions Jour*~al of Colloid and Interface Science, VoI. 36, No. 4, AugusL 1971
in (H20)s are vastly different from those found by ab initio or CNDO/2 methods (24). In general one can say that to be completely successful for a molecular system with little precedent, semiempirieal methods need to be referenced to representative ab initio calculations. GEOMETRY OF (H20)~ If anomalous water is (H20)~, what is its geometry? Starting from the IR spectrum result (25) that the hydrogen is in a significantly different environment than in normal water, Allen and Kollman (6) examined possible geometries by CNDO/2 (Fig. 1). Of the structures with symmetric hydrogen bonds only a cyclic hexamer is more stable than six isolated H20 molecules. This search included the structure proposed by O'Konski (26) (a diborane-like arrangement). The latter structure has an energy less stable than isolated water molecules by approximately 100 kcal/H20. Morakuma (4) employed C N D O / 2 to compute linear and condensed negative ion arrays of one to five hexagonal rings. These structures follow the polyelectrolyte sheet model proposed by Lippincott et al. (25) and this model fails both on physical grounds and on the basis of the further quantum mechanical calculation discussed below and listed in Table I. Pederson (5), using INDO, has made calculations for several planar arrays of one to three hexagons along with their corresponding open structures. He has employed bond orders as a criterion of stability, but at best, these yield only a rough qualitative picture for water polymers. Ab initio calculations show that his INDO results grossly overestimate overlap populations, and this leads him to the unreasonable prediction of a 300 % greater bond energy for symmetric hydrogen bonds over those in ordinary water, whereas total energies show the symmetrically bonded structures to be less stable. Azman et al. (7), using a different CNDO parameterization than Morakuma or Allen and Kollman, carried out a few calculations on a single hexamer, and they also concluded that symmetric structures are far more stable than asymmetric. The results of these calculations are clearly in-
463
M O D E L S FOI~ A N O M A L O U S W A T E R
H
(a)
(b)
,c,
H
H
Stability relative to isolated water molecules
.
2
~
H-~-- ~
+64. k cal
.~.~, ~', ~ ' ~ ~ ~: ~ . H
H
H.
>"
IH.
H
(d)
H
-14.3
.
H
H
H
-397. H
H
-68.
H
H
(e)
H
~A,~,~..,~,~ 14,
-22. H
14,
H,
(f) (g)
-24. H
H
-(~H-'~
H
H--'~ H-~
H
H
H--(~'- H - " ~ H ~ ) -
H
--56.
H
Hk )H 14,, .,H .
(h)
H-q.~__
,~.
.,~H
Large negative
H~::)---H--Q,H IH~)--H-'O~ H FIG. 1. Possible geometries for anomalous water.
TABLE I PROTONATION OF ~APHTI-IALENE STRUCTURE O10Ht9Protonation At At At In In
c e n t e r oxygen a l p h a oxygen b e t a oxygen center of one ring center of one ring
j~ ( r ( O - - H ) = 1.03A) ( r ( O - - H ) = 1.03A) (r(O--H) = 1.03~) [A (out of plane) = 0.0~x] [A (out of plane) = 1.03A]
--199.01 --199.89 --199.85 --198.60 --198.58
E(O10Hlg-) = --198.24 E(10 isolated H~O) = -198.91 Journal of Colloid and Interface Science,
VoL 36, No. 4, August 1971
464
ALLEN AND KOLLMAN
correct, and this is due to either or both of the following reasons: First, it has not been demonstrated that the parameterization they have used produces reasonable results for conventional hydrogen bonds, and second, they did not systematically optimize the geometry for both asymmetric and symmetric cycles. Goel et al. (8) carried out CNDO/2 calculations on sp 2 and sp 3 hybridized water dimers, trimers, and hexamers and found additional stabilization of sp 2 hydrogen-bonded species over sp 3. An asymmetric, cyclic hexamer is found to be unbound and a symmetric, cyclic hexamer strongly bound--again this result is incorrect as shown unequivocally by the ab initio calculations to be discussed in the next section. G o d et al. have referenced their sp ~ dimer and polymer calculations to isolated sp ~ water monomers without specifying the energy needed to promote the isolated monomers from the normal geometry [0 (HOH) = 105°] to sp2[0 (HOH) = 120°]. Messmer (9), on the basis of C N D O / 2 calculations, believes that p~ delocalization through hydrogen may greatly stabilize symmetric, cyclic structures. Allen and Kollman (6), however, have shown from ab initio calculations on HF2-- and H502+ with a symmetric H bond that this cannot be true. Messmer's numbers result from the inherent nature of CNDO and his particular use of the scheme. On the basis of his crystallographic experience, Donohue (27) has proposed an interesting space-filling structure based on a symmetrically bonded rhombic dodeca~ hedron unit, (H20)1~. We have carried out C N D O / 2 calculations on this unit (28) and also for a 24-molecule cubic oetahedron ( 0 . . . 0 distance equals 2.32A) model. Our results are: Unit
Rhom. dodee. Cubic octah.
Et (z[ )
n (H~O)
Instability (local) per H~O
--277.83413 - 476.4841
-278.4754 - 477.3864
27.1 23.5.
The reason for these high energies is the close approach of symmetric hydrogens (each bearing a charge of -ff.34) necessitated by the
cagelike structures. Another qualitativelyproposed model, the tetramer of Bollander, Kassner, and Zung (29), has been evaluated with ab initio computations by Del Bene and Pople (30), and they find it less stable than isolated water molecules. Two models of a quite different type, both based on modifications of models for the normal liquid, have been suggested to explain anomalous water. One of these, a bistructural model of ordinary water employed by Fabuss (31), derives from the theory of Vdovenko, Gurikov, and Legin (32). The presence in their theory of a nonhydrogen-bonded dense component in addition to the lighter icelike component has led Fabuss to postulate a differential stabilization by the quartz surface of the nonhydrogen-bonded component. This model must be rejected because it would display a normal OH I R stretch and a nonshifted nmr spectrum--both in contradiction to the experimental evidence (25, 33). Erlander (34) has suggested that anomalous water consists of ice II-type clusters in analogy to an ice I-type cluster model for normal water. However, since ice II has a large stretching band absorption at a frequency somewhat greater than 3,000 cm-1, an ice II anomalous water must be expected to have a reasonably close-by absorption, but this again contradicts the I R evidence. The precedents cited by Erlander for the possible disappearance of such stretching frequencies are all based orl high-concentration salt, acid, and base solutions where there is a significant concentration of H2,, -- iO~+, H2~ q- 1On+, and H O H - . . A - . These species would be expected to have very different O-H stretching regions from water, ice I, and ice II. The presence of these species means that there would be less normally bonded water, thus accounting for the observed decrease in intensity of the 3,400 cm-1 (normal water O - - H stretch) band. The presence of H502+ would cause absorptions in the 1,500-2,000 cm-~ range and thus would add to the intensity of the normal water bond (~-4,600 em -1) .
The polyelectrolyte model proposed by Lippincott et aI. (25) would have spectral properties similar to those observed experimentally and similar to those predicted by
Journal of Golloid and Interface Science, VoL 36, No. 4, August 1971
MODELS FOg ANOMALOUS WATER ~he neutral model of Allen and Kollman (6). However, CNDO/2 calculations (Table I) indicate that by far the best place of attachment for a proton to a typical sheet fragment, naphthalene-like H19010-, is above one of the two central oxygens. Thus, any negatively charged ring would automatically pick up a proton and have the characteristics of a neutral structure. Polyeleetrolytes may also be eliminated on physical grounds (6, 34). Therefore, one may conclude that if anomalous water is (H~O)n, it must exist as a neutral network of symmetric, cyclic hexamers. Using iterative extended Hiickel theory (21), Minton (10) has made computations for several of the structures in Fig. 1 as well as the Donohue (27) and Bollander (29) models. In view of the fact that the scheme he has chosen has been shown by its authors (21, 22) to lead to an unstable solution for the conventional hydrogen-bonded water dimer owing to a defect in the method it is neither surprising nor meaningful that Minton finds an unstable solution for the cyclic, symmetric hexamer. Aldrich et al. (11) have applied the one-electron, EHT-]ike scheme of Cusachs (20) and they also find an unstable cyclic, symmetric hexamer. Since the same scheme yields a 32 kcal/mole binding energy for the conventional hydrogenbonded water dimeP 5 (experimental 5-7 kcal/mole), one can have little confidence in their result. :\lore generally, the credibility of one-electron theories is very low because, of necessity, they must base stability predictions on orbital energy sums and/or overlap population, and these criteria are suspect because: (1) In asymmetrically bonded (HF)6, which is experimentally known to be a cyclic hexamer with a Hbond energy of 7 keal/H-bond, ab initio LCAO calculations (3) yield a stabilization energy of 5.3 kcal/H-bond but possess a total overlap population less than that of six isolated H F molecules (36). (2) In many ab initio calculations on hydrogen- and lithium-bonded complexes (37), the sum of the one-electron orbital energies is higher than that for the isolated species, even though the difference in total energies yields a stabilization in agreement with experiment.
465
Finally, two articles (38, 39) have applied the nonpalred spin orbital (double-quartet) schematic model (40) to structures for anomalous water. Neither of these papers contains quantitative results, but they provide a novel, pictorial representation of the bonding to be expected if anomalous (H20)~ is a symmetrically bonded, tetrahedrally hybridized, three-dimensional zineblende or wurtzite structure (38) or cyclic, symmetrically bonded sheets arranged threedimensionally in a graphite lattice (6, 41). The key bonding unit in Linnett's tetrahedrally coordinated, symmetric hydrogen structure 38 can be modeled by either a tetrahedrally coordinated pentamer or by a cyclic cyclohexane-like puckered ring, and energy optimized results for both 6 show a higher energy than our planar hexagons. Comparing planar and puckered rings it is easy to see the physical origin of the high energy for Linnett's structure: Energy optimized O - - H - - O lengths for planar and puckered hexagons are very similar (2.32 and 2.36 A respectively), and consequently, across-the-ring 0 - - 0 distances are significantly smaller for the puckered conformation. Thus the strongly negative oxygens give rise to a relatively large puckered conformation repulsion energy. Building a larger structure leads to an even more unfavorable situation: Extending the sheet from one to two puckered rings increases the instability relative to two planar rings by an additional 31 kcal/mole. STABILITY
OF (H20)~
It is important to attempt to assess the absolute stability of cyclic structures, and it is clear from the previous sections of this paper that this can be accomplished only through ab initio calculations. Sabin et al. (1), Pople (2), and Allen and Kollman have carried out ab initio calculations on cyclic hexamers. In some of their calculations Sabin et al. find that structures with both symmetric in-ring hydrogens and asymmetric in-ring hydrogens are more stable than isolated water monomers and actually obtained the symmetric structure more stable than the asymmetric, but this latter result is due to the fact. that they have examined only an asymmetric structure with
Journal of Colloid and Interface Science, Vol. 36, No. 4, August 1971
466
ALLEN AND KOLLMAN
R ( 0 . . . 0 ) = 2.76~. Pople also finds both structures more stable than isolated monomers, but with the asymmetric structure (R = 2.44A) slightly more stable than the symmetric (R = 2.3A). (This is exactly what is found in CNDO/2 calculations.) Allen and Kollman carried out an ab initio LCAO calculation on the planar, symmetric hex~mer with a basis set superior to these two calculations and found an energy (R (0...0) ~2.32A) slightly less stable than six isolated water molecules. To obtain as realistic an estimate as possible for the absolute stability, one must add correlation and zero point energy corrections to both symmetric and asymmetric structures. When this is done, the results indicate that cyclic hexamers may be slightly more energetically favorable than six isolated water molecules but clearly less stable than liquid water. It should be noted also that there is no barrier (local minimum) in a symmetric hexamer structure to prevent the structure from reverting to an asymmetric (normal water-like) structure. Thus if anomalous water is (H20)~, two other physical effects must come into play: (1) There must be an unusually long-range and strong surface binding force which selectively lowers the energy of symmetric structures below that of the asymmetric. (2) The structures formed must be a threedimensional aggregate, because after its remoral from the surface of formation, there must be a barrier hindering its decomposition into normal water., In another article in this issue, "What Can Theory S~y About the Existence and Properties of Anomalous Water?" we show that neither of these criteria are fulfilled and thus we do not now believe in the existence of anomalous water. SUMMARY AND CONCLUSIONS
This result implies the need for a differentially selective, large, and unprecedentedly long-range force on water-adsorbent surfaces if anomalous water is to exist as (H20).. Within the framework of current digital computer technology, it is not possible to fully assay the potential bonding characteristics and properties of (H20)~ polymers by ab initio calculations. Therefore it is necessary to employ semiempirical methods. For greatest confidence and effectiveness, these methods must be referenced to and calibrated against ab initio calculations that are as nearly as possible representative of the bonding situation under consideration. Several cases have been cited in this paper where failure to do this has limited the usefulness of that work. In particular it was shown that one-electron, extended Hfickcl theorylike models are not likely to be very effective in elucidating possible (H20)~ structures. With the use of semiempirical methods (primarily CNDO/2), it was found that the most favorable symmetrically bonded (H20), structure consists of hexagonally patterned sheets, three-dimensionally arranged in a graphite-like structure (6). Finally, we have shown in another article in this issue, "What Can Theory Say About the Existence and Properties of ArLomalous Water?", that the necessary conditions required for stabilizing cyclic, symmetric (H20)n do not exist. Therefore since this article has sho~n that the most satisfactory geometry of all those proposed is the cyclic, symmetric array described above, we believe that this constitutes very strong evidence against the existence of anomalous water. Of course, this does not rule out some complex combination of substances not considered here, but if such is the case, it almost entirely eliminates the high interest and promise aroused by a simple new water atlotrope.
Of the three levels of quantum mechanical approximation currently in use, it is clear that only ab initio calculations can provide quantitative insight into the absolute stability of the potentially new form of bonding proposed to explain anomalous water. Such calculations show that cyclic, symmetric hydrogen bonding is considerably less stable than the conventional asymmetric form (3).
Note 1: Recently, Ageno [Theor. chim. Acta, 17, 334 (1970)] has proposed a dibora~le-like s t r u c t u r e for anomalous w a t e r a n d has shown t h a t t h e Ii% s p e c t r u m expected for this s t r u c t u r e is consistent w i t h t h a t observed (ref. 25). C i g n i t t i a n d Paolini (Theor. chim. Acta, in press) used C N D O / 2 to assess t h e feasibility of this s t r u c t u r e a n d came to similar conclusions as references 6 a n d 26.
o
Two Notes Added in Proof
Journa$ of Golloidand Interface Science, Vol. 36, No. 4, August; 1971
MODELS FOR ANOMALOUS WATER Note 2: Barclay Kamb has recently published a theoretical paper [Science, 172, 231 (1971)] that presents qualitative bonding arguments based largely on the experimental data which exists for asymmetric hydrogen bonded systems. Several points bear directly on the work reported here: (a) His intuition leads him to favor tetrahedrally hybridized, puckered rings over other structures and feels that lack of ~rbonding in the planar rings implies tetrahedral coordination (p 233, col. 1 and 3; p 234, col. 1 and 2; p 235, col. 1 and 2; p 236, col. 2). We have shown quantitatively that this is incorrect [reference 6 and last paragraph of geometry of (H~.O)n]. (b) Kamb gives a number of hypothesized potential energy curves for symmetric and asymmetric bonds, the data for which is obtained from known asymmetric bonds. He discusses a variety of barrier situations that might permit symmetric bonding, and the one that comes closest to our original ideas 6 is the cooperative hypothesis (p 237, col. 1 and 2), and he abstracts a reasonable per bond destabilization energy (14 keal/mole). (c) The observed existence of stable HLO~+ with its short symmetric bonds leads Kamb to suggest this ion as an activation mechanism that would eliminate the kinetic barrier between symmetric and asymmetric bonds. But the key feature that gave rise to the possibility of the kinetic barrier was the three-dimensional array of neutral cyclic units, and H502+ does not possess any of these properties. Throughout his article Hamb is saying that cyclic symmetric hydrogen bonds do not constitute a straightforward extrapolation of well-established data on asymmetric bonds. With this we have never had any disagreement, but his refutation of their existence on these grounds does not represent a strong case against anomalous water and does not imply errors in the quantum mechanical calculations. REFERENCES 1. SABIN, J. R., H2~RRIS, R. E., ARCItIBALD, T. W., KOLLMAN, P. A., AND ALLEN, L. C., Theor. Chim. Acta, 18, 235 (1970). 2. POPLE, J. A., Private communication. Some calculations are reported in ]:)el Bene, J., and Pople, J. A., J. Chem. Phys. 52, 4858 (1970). 3. ALLEN, L. C., AND KOLLMAN, P. A., ] . A m e r . Chem. Soc. 92, 4108 (1970). 4. MORA~:U~A, K., Chem. Phys. Lett. 4, 358 (1969).
467
5. PEDERSON, L., Chem. Phys. Letters 4, 280 (1969). 6. ALLEN, L. C., AND KOLLM3-N, P. A., Science 167, 1443 (1970). 7. AZM.~.N,A., HOLLER, J., AND HADZI, D., Chem. Phys. Lett. 5, 157 (1970). 8. GOEL, A., MURTHY, A. S. N., AND RAO, C. N. R., Chem. Commun., 423 (1970). 9. MESSIER, R., Science 167,479 (1970). 10. MINTON, A. P., Nature 226, 151 (1970). 11. ALDRICH, H. S., LADER, H. J., GARY, L. P., CORRINGTON, J. H., AND CUSACHS, L . C., Submitted to Chem. Phys. Lett. 12. PILAR, F., "Elementary Quantum Chemistry." McGraw-Hill, New York, 1968. 13. See ALLEN, L. C., An. Rev. Phys. Chem. 20, 315 (1969) and KRAUSS, M., "Compendium of Ab Initio Calculations of Molecular Energies and Properties," Nat. Bur. Std. (U.S.) Tech. News Bull., p 438 (1967). 14. RAUK, A., ALLEN, L. C., AND CLEMENTI, E., J. Chem. Phys. 52, 4133 (1970). 15. ALLEN, L. C., Chem. Phys. Lett. 2,597 (1968). 16. HOLLMAN,P. A., AND ALLEN, L. C., J. Chem. Phys. 51, 3286 (1969); ibid. 52, 5085 (1970). 17. POPLE, J. A., AND SEGAL, G. A., J. Chem. Phys. 44, 3289 (1966). 18. POPLE, J. A., BEVERIDGE, D. L., AND DOBOStI, P. A., J. Chem. Phys. 47, 2026 (1967). 19. MVRTHY, A. S. N., A~D R.~o, C. N. R., Chem. Phys. Lett. 2,123 (1968). 20. CUSACHS: L. C., AND CUSACHS, B. B., J. Phys. Chem. 71, 1060 (1967). 21. REIN, R., FUKUDA, N., WIN, H., CLARKE, G. A., AND HARRIS, F. E., J. Chem. Phys. 45, 4743 (1966). 22. REIN, R., AND HARRIS, F., J. Mol. Struet. 2,103 (1968). 23. MVRTHY, A. S. N., DAVIS, R., AND RAO, C. N. R., Theor. Chim. Acta 13, 81 (1968). 24. P. KOLLMAN AND L. C. ALLEN (submitted to J. Amer. Chem. Soc.) have compared the proton potential well in NH~-HF by ab initio and CNDO/2 methods and find no double well at the minimum energy N-F distance (although one appears at larger N-F distances). This is in contrast to EHT results on (H~O):. 25. LIPPINCOTT, E., STROMBER, R., GRANT, W., .AND CESSAC, G., Science 164, 1482 (1969). 26. O'HONSKI, C., Science 168, ]089 (1970). O'Konski and Levine (this issue) have carried out a CNDO/2 geometry optimization of the diborane-tike symmetric bonded tretramer and found it unstable by 148 kcal (our unoptimized value was 397 kcal) relative to four water molecules. They have Mso performed ab initio calculations that again
Journal of Colloid and Interface Science, Vol. 36, No. 4, August 1971
468
ALLEN AND KOLLMAN
lead to a high energy, unequivocally eliminating this as a possible structure. 27. DONAHUE, J., Science 166, 1000 (1969). 28. We have calculated an energy for this model using R(0...0) = 2.44~_ and symmetric hydrogens. Although energy optimization was not done, the structure is so unfavorable that optimization would not affect the conclusions. 29. BOLLANDER, R. W., KASSNER, J. L., AND ZUNG, J. T., Nature 29.1, 1233 (1969). 30. DEL BENE, J., AND POPLE, J., J. Chem. Phys.
59., 4858 (1970). 31. FABvss, B., "Thermodynamic Aspects of Anomalous Water," J. Colloid Interface Sci., this issue. 32. VDOVENI~O, V. M., GvnlXOV, Y r . V., AND LEGIN, YE. K., Zhu. Strukt. Khim. 8, 18, 403 (1967); ibid. 9, 599, 819 (1968). 33. PETSKO, G., Science 167, 171 (1970); PAGE,
37. 38.
39. 40.
41.
T. F., JR., JACOBSEN, ~:~.J., AND LIPPINCOTT, E. R., ibid., p 51. 34. ERL±NDER, S., Phys. Rev. Lett. 22,177 (1969); Phys. Rev. A1,868 (1970). 35. ALm~ICI~, H. S., GARY, L. P., CORRINGTON,
J. H., AND CUSACI~S, L. C., 159th ACS Meeting, Houston, Texas, February, 1970; Gary, L. P., Private communication. 36. With the use of the wave functions reported in Appendix I, the total overlap for (HF)~ sym = 2.529; for (HF) 6 asym. = 2.933; and for 6HF = 3.086. (Both polymer cMeulations were carried out at R ( F . . . F ) = 2.5/~.) The sum of overlap populations criterion for stability is also basis set depend-
Journal of Colloid and Interface Science,
ent. For example, Kollman and Allen (ref. 16) find the total overlap population in the water dimer to be less than that for two monomers; Hankins, Moskowitz, and Stillinger [J. Chem. Phys. 53, 4544 (1970)], using a larger basis set, find a similar geometry and energy of dimer formation, but a total dimer overlap population greater than that of 2 monomers. KOLLMAN,P. A., LIEBMAN, J. F., AND ALLEN, L. C., J. Amer. Chem. Soe. 92, 1142 (1970). LINNETT, J. W., Science 167, 1719 (1970). ALLEN, L. C., Nature 227 (July 25, 1970). LINNETT: J. W., J. Amer. Chem. Soc. 88, 2643 (1961) ; "Electronic Structure of Molecules." Methuen, London, 1964. The schematic drawing and pictorial description given in reference 39 is just a qualitative "double-quartet" rationalization of the stacked planar hexagons developed in reference 6 (unfortunately the printers omitted reference 8 in that article which was a reference to J. W. Linnett's structure [reference 37 here], and the last reference given should have been numbered 9). APPENDIX I
T h e enerogy results for a s y m m e t r i c (HF)6 a t R = 2.5A, s y m m e t r i c (HF)6 a t R = 2.5A, a n d s y m m e t r i c (H20)6 a t R = 2.35A are g i v e n in reference 3. T h e t a b l e s of w a v e functions have been deposited with the ADI Auxiliary Publications Project, Photoduplic a t i o n Service, L i b r a r y of Congress, W a s h i n g t o n 25, D . C .
Vol. 36, No. 4, August 1971