J. phys. Chem. 1983, 87, 4287-4272
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transformation when nucleation is heterogeneous. As a matter of fact, considering the slight fraction of vitreous phase, it can be expected that the nuclei appear at ice (matrix)-vitreous islets boundaries where thermodynamic conditions are more propitious to nucleation.
0.1
/
0.2
0.3 0.4
4
4.1
u
43
ti4
4.5
46
-
Application of Johnson-MehCAvrami equation. Crystalllzed fraction of vitreous phase as a function of the annealing time at 173 K (after an annealing time at 163 K). A straight line is obtained and its slope gives the value of the parameter n in the Johnson-Mehl-Avrami equation: y = 1 - exp(-kt)" The value of n thus obtained is close to 1. According to Christian18 such a value could correspond to a phase Figwe 8.
Concluding Remarks In order to understand the crystallization of the vitreous phase in an ethylene glycol-water mixture, we have studied the transformation under isothermal and isochronal conditions. During the isothermal annealing treatments carried out at various temperatures from 169to 205 K, the crystallization kinetics have been determined and curves with sigmoidal shapes have been recorded suggesting a phase transformation process occurring in two steps: nucleation then growth. From these results, a time-temperature-transformation (Tl") diagram has been drawn. A simple inspection of such a diagram permits the determination of the domain of existence of the vitreous phase. Other alcohols such as glycerol are now under studv. Registry No. Ethylene glycol, 107-21-1; water, 7732-18-5. (18)Christian, P. "The Theory of Transformation in Metals and Alloys"; Pergamon Press: Oxford, 1965.
Ab Initio Molecular Orbital Estimates of Charge Partitioning between Bjerrum and Ionic Defects in I c e Steve Schelner Department of ChemrSW and BkhsmisW, Southern Iliinols University, Carbon&le, Illinois 62901
and John F. Nagle Departments of phvsics and Biological Sciences, Camegba-h49ilon University, Plttsbwgh, Pennsylvanle 15213 (Received: August 23, 1982; I n Final Form: December 3, 1982)
Ab initio molecular orbital methods are used to determine the fraction eB/e of the full protonic charge carried by the Bjerrum defect in ice. This can be accomplished by calculating the change in dipole moment after a defect has been transported. Dipole moments are calculated for a series of linear chains of water molecules and the resulta extrapolated to infinite chain length. Known deficiencies of the two basis sets used are accounted for by scaling the results against the experimentally determined dipole moment of water. Also, the effect of the three-dimensional hydrogen-bonded network of ice is estimated by including additional water molecules as branches of the chain. These extra bonds, whether present as terminal branches or as closed rings, result in a decrease of only several percent in eB,the charge transmitted by OH bond rotations. Other variations in the calculations include elongation of the OH bond length in ice from 0.97 to 1.01 A, which produces a net increase in eB of 6%, and reduction of the HOH bond angle in each water molecule to 104.5O,which results in a smaller increase. Taking into account the various sources of error and uncertainties as to the precise geometry leads to a value of eB = 0.36 f 0.03,which is in satisfactory agreement with the values obtained in a completely different way from dielectric measurements.
I. Introduction Charge transport in ice is usually explained in terms of bonding (Bjerrum) & f e d , which carry a charge &eB,and ionic defects, which carry a charge feI.1s2 Successive (1) L. Onsager in "Physics and Chemistry of Ice", E. Whalley, S. J. Jones, and L. W. Gold, Ed., Royal Society of Canada, Ottawa, 1973,pp 7-12. 0022-365418312087-4267501.50/ 0
transport of a bonding defect and an ionic defect of the same Sign along the Same Path results in transpod of a full protonic charge with no other changes. Therefore, eB + eI = e; this describes the partitioning of the elementary charge e into the charges of the two kinds of carriers in (2)H.Grhicher, C. Jaccard, P. Schemer, and A. Steinemann, Di8cus. Faraday SOC.,23, 50 (1957).
0 1983 American Chemical Society
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three dimensionality of the ice lattice and the effect of uncertainties in the proton position. Finally, the new results are summarized and compared to previous results in section V. ri1
HI
H I
k l A
11. Formulation
Consider a chain of n water molecules hydrogen bonded to one another. As shown in Figure 1,the overall direction r d i of the chain is defined as the z axis. An extra proton is Flgure 1. Schematic diagram of a hydrogen-bonded chain of water assumed to have made its way to the ith molecule of the molecules in which an excess proton has migrated to the middle of chain with which is associated a formal electric charge of the chain. l+. The ith molecule is often called an “ionic defect” or “ionic fault”. Note that the passage of the proton along ice. The actual value of e B / e or e I / e depends upon the the chain has produced a situation wherein for each water molecular details of ice; in particular, it is expected to be to the left of the ith molecule the hydrogen-bonding H quite different for other hydrogen-bonded crystals, such atom is located to the left of its 0 atom. The situation is as KH2POe3 reversed to the right of the ith molecule since the proton In hexagonal or cubic ice eB is simply related4 to the has not yet reached these waters. A more schematic repdipole moment p by resentation of our chain is the following: HO- -HO- -HO- -HO- -HO- -HOH- -OH- -OH- -OH- -0HeB = p ( 3 ) ’ I 2 / R (1) -OH where the hydrogen bond length R = 2.76 A. Values of Of course the presence of the extra proton on the ith p have been obtained from a combination of measurements molecule is expected to have a perturbing influence on the and theories of the dielectric con~tant.’,~?~ However, there local structure of the chain and the excess charge will be are different theories which give different values6 of p. distributed over a number of neighboring molecules. In(Also, while there is agreement among measurements for termolecular distances might be somewhat contracted and the dielectric constant along the a axis, there has been a hydrogen bond angles altered to some degree. We expect 10-15% disagreement along the c axis, although Johari and such perturbations to extend from the ith water by several Jones7 have apparently resolved this.) Clearly, indemolecules. For purposes of illustration, let us assume the pendent methods of estimating eB are desirable, not the influence of the positive charge extends as far as two least because of the constraints that they would impose molecules on either side of it. This “perturbed region”, upon the dielectric theory. encompassing waters i - 2 through i + 2, is denoted by An experimental method for obtaining e B / e Ihas been dotted lines as follows: described by Hubmann.* This method involves measuring HO--HO--HO-.-HO--HO- -HOH- -OH- -OH- i-OH- -OH the ratio of the dielectric constant in the limit of no doping --OH to its value in the limit of heavy doping with NH3. This We emphasize that the extent of the region, here chosen method is independent of the precise value of p, of sysas a total of 5 molecules, is not an important feature of our tematic uncertainties in measurements of to, and of any particular quantitative theory of the dielectric c ~ n s t a n t . ~ model. Our arguments apply unchanged to any size region provided that it is extensive enough to contain the bulk It does depend upon the general theory that Bjerrum of the positive protonic charge. defects (ionic defects) are responsible for the dielectric We now consider the situation when the excess proton relaxation in the limit of no doping (heavy doping), rehas hopped from molecule i to i 1. The “charged” or spectively, and that the values of eB and eI do not change perturbed region has translated by one molecule along with with doping. While we have no good reason to doubt either the proton and now includes waters i - 1 to i + 3. of these assumptions, again independent estimates are desirable. HO- -HO- -HO- -HO-.-HO--HO- -HOH--OH- -oH-.-oHThe goal of the present paper is to calculate the charge ’ -OH partitioning in ice by using the completely independent We may consider the total effect on the chain as the sum ab initio molecular orbital method. Our approach is to of two simultaneous but conceptually separate processes. perform calculations on small clusters of hydrogen-bonded The first is the motion of the charged region to the right water molecules and to extrapolate the results to larger by a distance d equal to the projection of the R(O0) sepsystems. In section I1 the connection between eBand the aration (in the unperturbed region) onto the chain or z dipole moment of water clusters is established. In section direction (see Figure 1). If we make the reasonable asI11 the molecular orbital methods are discussed, results are sumption that the perturbed region of the chain with net given for small linear clusters, extrapolation to infinite charge e+ is structurally unaffected by its translocation linear clusters are performed, and corrections are made for to the right by d , the change in dipole moment associated deficiencies in the basis sets. In section IV a number of with the first process is simply de+. Comparison of the refinements are introduced to estimate the effect of the chain before and after the proton hop from i to i + 1 has occurred shows a change in the numbers of water molecules to the left and right of the charged region. Translation (3) L. Onsager in “Physical Principles of Biological Membranes“, F. of the charged region to the right results in the appearance Snell, Ed., Gordon and Breach, New York, 1970, pp 137-91. (4) L. Onsager and M. Dupuis in ‘Electrolytes: Proceedings of an of an additional water to its left and a corresponding International Symposium on Electrolytes, Trieste, June, 1959”,B. Peace, disappearance of one water on the right. The second Ed., Pergamon Press: New York, 1962, pp 27-46. process may therefore be considered as the net translation (5)L. Oneager and M. Dupuis, Rend. Sc. Int. Fis. ‘Enrico Fermi”,No. 10, 294-315 (1960). of one water molecule to the left by the distance required (6) J. F. Nagle J. Glaciol., 21, 73 (1978). to traverse the charged region. Since each water molecule (7) G. P. Johari and S. J. Jones, J. Glaciol. 21, 259 (1978). in the uncharged region is electrically neutral, this trans(8) M. Hubmann, 2.Phys. B, 32, 127 (1979). location produces zero net motion of charge. However, an (9) J. F. Nagle, Chem. Phys., 43, 317 (1979). 1
1
I
1
+
The Journal of Physical Chemistry, Vol. 87, No. 21, 1983 4289
Charge Partitioning between Bjerrum and Ionic Defects in Ice
U
U
V
,' \
Iz H
H
Flgure 2. Change in the orientation of the dipole moment of water resulting from the rotation of the molecule.
additional feature of the chain mentioned above is that those waters to the left of the charged region have their hydrogen-bonding proton to the left of the 0 atom; i.e., they are in the HO configuration or "point to the left", whereas molecules to the right of the charged region point to the right. Therefore, associated with the translocation of the neutral water from the right side of the charged region to its left is a concomitant rotation of its hydrogen bonding proton from right to left. Figure 2 illustrates the dipole moment of water and the change resulting from its rotation. The total change of dipole moment associated with the second process is therefore -2pz where p, is the component of the dipole moment of water in the chain or z direction. Since eI is the charge transferred along the chain by the proton-hopping process, the total dipole change resulting from the full transport of a proton along a chain of total length D is eID. We focus here on a single hop of distance d for which the moment change is eId. Equating this expression to the sum of the two aforementioned concurrent processes, viz., translocation of the charged region and rotation of one water, we arrive at eq 2. Equation
eId = +de - 2p2 or eI = e - 2 p 2 / d
(2)
3 is, of course, equivalent to eq 1. Our problem thus reeB = -2p2/d
(3)
duces to evaluation of the z component of the dipole moment of one neutral water molecule in an unperturbed icelike environment.
111. Basic Calculation To this task we apply quantum chemical methods. Specifically, ab initio molecular orbital calculations were performed with the Gaussian-701° package of computer codes. The minimal STO-3G1' and split valence-shell12 4-31G basis sets were used within the restricted HartreeFock formalism. These two basis sets were chosen in part to examine the basis set dependence of our results. In addition, dipole moments computed with the STO-3G and 4-31G basis sets for the water monomer respectively underestimate and overestimate the experimental moment.I3 Use of these two approaches should thus allow interpolation of our computed results for the dipole moment to a reasonable estimate of the experimental value. A water molecule within an ice lattice exists within an infinite three-dimensional network of hydrogen-bonded (10)W. J. Hehre, W. A. Lathan, R. Ditchfield, M. D. Newton, and J. A. Pople, QCPE,No. 236 (1974). (11)W. J. Hehre, R. F. Stewart, and J. A. Pople, J. Chem. Phys., 51, 2657 (1969). (12)R. Ditchfield, W. J. Hehre, and J. A. Pople, J. Chem. Phys., 54, 724 (1971). (13)H. Umeyama and K. Morokuma, J. Am. Chem. SOC., 99, 1316 (1975).
Figure 3. Linear chain of water molecules in ice. All bond angles are tetrahedral. The hydrogen bonds are all linear; B(OH0) = 180'. Non-hydrogen-bonding protons alternate above and below the plane of the paper.
TABLE I: Average and Incremental Changes in Dipole Moment for Various Chain Lengths of (H,O), [rJz(n + 1) - rJz(n)l, D rJz, D
n
STO-3G
4-31G
STO-3G
4-31G
1 2 3 4 5
1.186 1.461 1.578 1.641 1.681
1.786 2.071 2.200 2.273 2.320
1.736 1.812 1.830 1.841
2.356 2.458 2.492 2.508
m
1.883
2.498
1.882
2.565
molecules. We restrict our attention first to an infinitely long one-dimensional chain of water molecules. The effects on the results of considering additional water molecules and thereby extending the model to three dimensions will be addressed below. The chain of water molecules on which we first focus our attention is illustrated in Figure 3. So that the situation in ice is reproduced,14 all bond angles to each 0 atom are tetrahedral, i.e., 109.47O. All r(0H) bond lengths are 0.97 8, and the distance between 0 atoms is 2.76 A. The direction of the chain is defined as the z axis. H atoms not involved in hydrogen bonds along the chain are alternately above and below the OH..OH.-OH-OH plane as shown in the figqre. Since a direct calculation of an infinitely long chain is not possible, we have designed our procedure to approach this limit asymptotically. We begin by calculating the z component of the dipole moment ( p z ) of a single water molecule ( n = 1) and successively increase n by adding additional molecules to our chain. For each chain length n, the average p, attributed to each individual molecule is defined as p2 = p J n . The electronic distributions of terminal molecules may of course be significantly different than those more deeply ensconsed in the chain but the proportionate contributions of the former molecules, i.e., end effects, may be expected to be reduced dramatically as n increases. Average dipole moments, calculated via both the STO3G and 4-31G basis sets, are presented in Table I for chain lengths up to the pentamer. The results indicate that each successive addition of a water molecule to the end of the chain produces a further increase in the average dipole moment. In other words, the dipole moments are not simply additive but instead each addition of a water molecule to the chain acts to further polarize the electronic distribution within the chain. This "cooperative" effect (14)S. J. LaPlaca, W. C. Hamilton, B. Kamb, and A. Prakash, J . Chem. Phys., 58,567 (1973);W. C. Hamilton, B. Kamb, S. J. LaPlaca, and A. Prakash, ibid., 55,1934(1971);I. Olovsson and P.-G. Jonsson in 'The Hydrogen Bond-Recent Developments in Theory and Experiments", P. Schuster, G. Zundel, and C. Sandorfy, Ed., NorthHolland, Amsterdam, 1976,pp 393-456;E. Whalley, ibid.,pp 1425-70; E. Whalley, Mol. Phys., 28,1105(1974);B. Kamb, Acta Crystallog., 17, 1437 (1964).
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Scheiner and Nagle
0.8 1 -
k=t
p T 0 - 3 G
I
1
0
Flgm 4. Least-squares fits of the reciprocal of p, as a linear function of the reciprocal of the chain length n (solM lines). Dashed lines represent fits of reciprocal of [p,(n 1) - p , ( n ) ] .
+
has been noted previously15 for hydrogen bond energies as well as dipole moments. The data in Table I indicate that self-consistency has not yet been achieved at the pentamer level. Instead we see a significant increase in pz beteen n = 4 and n = 5 and expect further enlargement as n is increased beyond 5. It is at first sight unclear how to accurately extrapolate our data to infiite chain length. However, a technique applied by Sheridan et al.'& to determine the average hydrogen bond energy of an infinitely long chain of formamide molecules and by Nagle and Wilkins0nl6to find the infinite limit of T,offers a convenient means of extrapolating our dipole moments to n = m. Confronted with a similar situation, Sheridan et al. noted a good linear correlation between the reciprocals of their average hydrogen bond energies and the number of molecules in the chain. Following this approach, we present in Figure 4 a plot of @J1 as a function of l / n for both the STO-3G and 4-31G dipole moments. The fit of the data to a straight line is excellent in each case; the correlation coefficients are 0.9999 and 0.9995, respectively. By extrapolating these lines to l / n = 0 we obtain the infinite limits of our average dipole moments presented in the last row of Table I. These values correspond to the z component of the dipole moment of a water molecule in a (one-dimensional) unperturbed icelike environment. An alternate method of extracting a bulk property from a series of progressively longer chains is by means of the incremental changes in this property. The last two columns of Table I contain the increases in the z component of the dipole moment resulting from each lengthening of the chain by one water molecule. As in the case of the average dipole moments in the first two columns of the table, this quantity has not converged to a clear asymptote at the pentamer level. However, it is possible once again to extract the infiiite limit by means of a double reciprocal plot. Least-squares fits of the reciprocals of the incremental dipole moment increases to l / n are shown by the (15)(a) R. P. Sheridan, R. H. Lee, N. Peters, and L. C. Allen, Biopolymers, 18,2451 (1979). (b) s. Scheiner in 'Aggregation Processes in Solution", E. Wyn-Jones and J. Cormally, Ed., North-Holland,Amsterdam, in press. (c) D. Hankim, J. W. Moakowitz, and F. H. Stillinger, J. Chem. Phys., 53,4544(1970);69,995 (1973). (d) J. E. Del Bene and J. A. Pople, ibid., 52,4858(1970);68,3605(1973). (e) J. E.Del Bene, ibid., 65,4633(1971). (0 B. R. Lentz and H. A. Scheraga, ibid., 68,5296 (1973); 61, 3493 (1974). (g) A. Johanson, P. Kollman, S. Rothenberg, and J. McKelvey, J.Am. Chem. Soc., 96,3794(1974). (h) L. A. Curth, J.C h m . Phys. 67,1144(1977). (i) J. F.Hinton and R. D. Harpool, J. Am. Chem. Soc., 99,349(1977). 6)S.Scheiner and C. W. Kem, ibid., 99,7042(1977). (16) J. F. Nagle and D. A. Wilkinson, Biophys. J., 23, 159 (1978).
H/OK H Figure 5. Linear dimer of water (H20)a(H10)bincluding "branch" or extemai waters (H,O), and (H20),,. Ail bond angles are tetrahedral and H bonds linear. Ail 00 distances are 2.76 A.
dashed lines in Figure 4. The correlation coefficients for these fits are 0.996 and 0.999 for the STO-3G and 4-31G data, respectively. Most importantly, the intercepts of these dashed lines with ( l / n ) = 0 are clearly quite close to those obtained by using the average dipole moments (solid lines). This near coincidence indicates the results are rather insensitive to the particular method of extrapolation to infinite chain length. If we now substitute the average dipole moments of the infinite chain from the first two columns of Table I into eq 3 we find that eB is calculated to be 0.348 via STO-3G and 0.462 by 4-31G. Clearly these two results are substantially different. This discrepancy is not surprising, however, since the two procedures are commonly known to lead to substantially different dipole moments for various molecules including water. In fact, dipole moments calculated with either of these basis sets are typically significantly different than values obtained by experimental meas~rements.'~It is possible to use the experimental dipole moment of ~ a t e r ' ~in , ' the ~ gas phase (1.85 D) to scale our calculated results and arrive at a more realistic estimate of eB. When the STO-3G and 4-31G basis sets are applied directly to the experimental geometry of the water monomer, the calculated dipole moments are 1.72 and 2.60 D, re~pective1y.l~ Thus, STO-3G underestimates the dipole moment compared to experiment by 7% while the 4-31G value is too high by 41%. Scaling our infinite limits of p, in Table I by the equation (4) we find scaled values of the infinite limit of pz of 2.03 and 1.78 D for the STO-3G and 4-31G basis sets, respectively. Incorporating our adjustments into eq 3, we finally arrive at respective estimates of eB of 0.374 and 0.328. With regard to the treatment of H-bonded systems by the two basis sets, Del Bene and Poplelsd have demonstrated that both lead to satisfactory descriptions of the interaction between water molecules. They further noted that calculations with both basis sets indicate a cooperative effect, i.e., a nonadditivity of H-bond energies in the oligomer. (17)R. D.Nelson, D. R. Lide, and A. A. Maryott, Natl. Stand. Ref. Data. Ser., Natl. Bur. Stand., No. 10 (1967);S.A. Clough, Y. Beers, G. P. Klein, and L. S. Rothman, J. Chem. Phys., 59, 2254 (1973).
Charge Partitioning between Bjerrum and Ionic Defects in Ice ’H
?;
H
\A
77
O 5
0
e-?4
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-z Pi-% H,
d
t-
I_
-,$
The Journal of Physical Chemistry, Vol. 87, No. 21, 1983
_ _ _ _ _ _ _ _ _2
a
H
b
C
Figure 6. Six-membered rings of water. All bond angles are tetrahedral and H bonds linear; R(O0) = 2.76 A, r(OH) = 0.97 A. Each ring is in the “chair” conformation characteristic of cyclohexane. The broken box in a surrounds the linear trimer described in Figure 3.
e
d
Flgure 7. Hexameric rings of water in which the linear trimer (within
the box) acts as both proton donor and acceptor to the external waters.
IV. Refinements As was pointed out above, each water molecule in ice is part of a three-dimensional network of hydrogen bonds. It is therefore important to examine the effects on our results of adding additional waters that branch off our one-dimensional dimer chain. With this objective we took as a starting point our one-dimensional dimer chain and added two waters to form the cluster depicted in Figure 5. 0, and Obare the oxygen atoms of the base dimer while 0, and O d of the branch waters form “external” H bonds to 0,. Specifically, 0, interacts through a hydrogen of 0, while O d forms a H bond via a lone pair of 0,. (The symmetry of the cluster is such that any dipole moments of the two branch waters cannot directly contribute to the z component of the total dipole.) The central water (H20), of our cluster is thus in a situation more representative of ice than in the one-dimensional dimer where the branch waters are absent. Calculations of the z component of the dipole moment of this cluster by STO-3G and 4-31G yield 2.88 and 4.09 D, respectively. These values are quite close to the corresponding dipoles of the dimer (H20),(H20),,obtained when the branch waters are absent; inclusion of these external H bonds reduces the dipole by only 0.04 D. The two branch waters (H20), and (H20)dwere added also to the monomer (H20),. The geometry of this cluster is identical with Figure 5 except that (H20)bis absent. Inclusion of the external H bonds to the n = 1chain results in a similar 0.04-D reduction of p, as was found for n = 2. Incorporating a decrease of 0.04 D into our infinite limits of p, results in a reduction of only 2% in the values of eB reported above. An interesting feature of the three-dimensional network in ice is the presence of what may be considered as “hydrogen bonded rings”. Examples of such rings are illustrated in Figures 6 and 7. Each ring contains six water molecules, each of which is connected to two others in the ring by hydrogen bonds. The rings in Figure 6 have several characteristics in common. First, each contains as a subunit the exact same trimer configuration illustrated in Figure 3 which served as a subject of the calculations on the one-dimensional chain described above. This trimer unit is outlined by the broken-line box in Figure 6a. In all three hexameric rings, the geometry of the three additional waters 4-6 (outside the box) is such that there is no direct contribution possible from these waters to the z component of the total dipole moment. Thus the dipole calculated for each ring, when compared with the linear trimer chain, may be interpreted as the perturbation upon the latter structure by the other three waters as a result of ring closure. For example, the z component of the dipole moment of ring a was calculated by STO-3G to be 4.78 D, as compared to the value of 4.73 D found previously for
Geometries are as described in Figure 6.
TABLE 11: Effects of Ring Closure upon the Dipole Moment of the Linear Trimer of Water
STO-3G linear trimer ring a ring b ring c ring d ring ea
p2,D 4.73 4.78 4.53 4.55 4.81 9.33
A,%
1 -4 -4 2 -2
4-31G p2,D 6.60 6.57 6.43 6.42 6.59 12.99
4,%
-0.4 -3 -3 -0.2 -2
a 4 is computed for ring e as the difference between the dipole moment of the ring and twice that of the trimer.
the linear trimer. Thus ring closure by the three “external” waters 4-6 produces a 1% increase in the dipole of the trimer (1-3) and therefore in eB as well. The 4-31G basis set results in a very small decrease of 0.4%. Ring closure via structure a is therefore found to produce a negligible change in the value of eB obtained for the linear chains with no external H bonds. In ring a, the “external” waters interact with the lone electron pairs of the first and third waters of the linear trimer in the box. Structure b is quite similar to a except that the interaction is instead through OH bonds of the latter terminal waters. As may be seen in Table 11, both STO-3G and 4-31G predict small decreases of about 3% for ring b. Structure c is quite similar to b; in both rings the three external waters interact with OH bonds of the trimer. However, in ring b, the external waters 4 and 6 acting as proton acceptors to the trimer 1-3 serve as proton donors to water 5, whereas these same waters are proton acceptors to water 5 in structure c. Nevertheless, as may be seen from Table 11, the calculated results for rings b and c are nearly identical. Finally, the last type of ring requiring consideration is that in which the original linear trimer acts as proton donor at one end and as acceptor at the other end. Rings d and e, illustrated in Figure 7, contain hydrogen bonds via a lone electron pair of O1and an OH bond of 03.18The three external waters in dimer d are positioned such that there is no direct contribution to the dipole moment. Table I1 indicates that the perturbations induced by ring closure in dimer d are rather small; +2% with STO-3G and -0.2% with 4-31G. The situation is different in dimer e in that the OH bonds of the three external waters (04,05,0,) all (18) A slight alteration in the geometry of the linear trimer within the box was necessary in order to construct rings d and e; viz., the non-hydrogen bonding H of O1is in front of the plane of the paper rather than behind it as in Figure 6. However, the dipole moment of this trimer, as computed with both STO-3G and 4-31G, is essentially identical with that of the trimer of Figure 6.
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The Journal of Physical Chemistry, Vol. 87, No. 27, 7983
"point to the right", as do the OH bonds of the original trimer (01, 02,03).In fact, the two sets of trimers, if separated from one another, would have identical dipole moments. The dipole of the hexamer ring e, when compared to twice the dipole of the linear trimer, represents the perturbation induced by ring closure between the two identical trimers. As shown in the last row of Table 11, this type of ring closure leads to a small (2%) reduction in the dipole. The effects of external hydrogen bonds upon N~ of the linear chain may be summarized as follows. External hydrogen bonds without ring closure produce a reduction of p z by 2 % . Closing the ring results in a decrease of 3 or 4% when the additional waters interact with two OH bonds of the chain. H-bond formation with two lone electron pairs of the trimer produce a much smaller change. Small perturbations, varying between +2 and -2 %, are observed when the trimer acts simultaneously as both proton donor and acceptor. In summary, the values of eB calculated for the linear chains may be expected to be reduced by at most several percent by consideration of the three-dimensional network of ice. Another point worthy of consideration is the effect of small variations in the molecular geometry of ice upon eB. In particular, the above calculations have made use of a tetrahedral structure with R(O0) = 2.76 A. The OH bond lengths were taken as 0.97 A. Although the latter value is in agreement with most structural studies of ice,14 Chamberlain et al.19 have suggested a bond length of 1.01 A. To test the sensitivity of e B to small changes in this parameter, we repeated the calculations summarized in Table I and Figure 4 with r(0H) = 1.01 A. Extrapolation of the inverse of the mean g Eto l / n = 0 yielded infinite chain length values of pr of 2.01 (STO-3G) and 2.65 D (4-31G). These dipoles are about 6% higher than those shown in the last row of Table I. We conclude that pz and therefore e B may be increased by this amount if the OH bond lengths in ice are 1.01 A rather than 0.97 A. Chidambar" has pointed out that while the skeletal O(000) angles in ice may be tetrahedral, the internal 8(HOH) bond angle in each water molecule may be closer to the smaller equilibrium angle of the monomer in the gas phase. Calculations were therefore carried out as well for noncyclic systems, as in Figure 3, ranging from the monomer to the pentamer, in which each 8(HOH) bond angle was set at 104.5°.21 The resulting dipole moments were slightly higher than those listed in Table I. Extrapolation to infinite chain length led to values of pz equal to 1.90 (STO-3G) and 2.56 D (4-31G). These dipoles are respec(19) J. S. Chamberlain, F. H. Moore, and N. H. Fletcher in 'Physics and Chemistry of Ice", E. Whalley, et al., Ed., Royal Society of Canada, 1973, pp 283-4. (20) R. Chidambaram, Acto Crystallogr., 14, 467 (1961). (21) Each HOH bond angle was decreased from 109.5' to 104.5' by bending each H atom of the H20 unit in toward the HOH angle bisector by 2.5'; i.e. keeping the C2 rotation axis fixed. These bends lead to slightly nonlinear H bonds in which B(OH--0) = 176".
Scheiner and Nagle
tively 1and 2.5% greater than those obtained for tetrahedral HOH bond angles.
V. Conclusions The findings of our calculations may be summarized as follows. The STO-3G and 4-31G basis sets give quite different raw values of f?B. However, when the known errors of these basis seta in calculating the dipole moment of water are accounted for, the scaled results are in much closer agreement; egsn>-3G= 0.37, egP31G = 0.33. Inclusion of the effects of external hydrogen bonds upon the onedimensional chain reduces eBby only a few percent. On the other hand, a more sizable increase of 6% is observed when the OH bond length is increased from 0.97 to 1.01 A; somewhat smaller increases arise from reduction of the internal HOH bond angle by 5'. Incorporating the different basis set results as well as the three-dimensional structure of ice and uncertainity as to the precise geometry, we estimate that e B = 0.36 f 0.03. Previous results for eB/e obtained from interpretation of electrical measurements are 0.43 by Jaccard,220.38 by Onsager and Dupuis; 0.36 by Worz and Cole,230.39 by Johari and Jones,I 0.44 by Camplin et al.,24and 0.38 by H u b m a " The result given by Camplin et al. was obtained by an analysis that did not require e = eI + eB;their ratio eB/el = 0.60 is in good agreement with more recent values. The most recent value eB/e = 0.38 f 0.01 obtained by Hubmann would appear to be reliable and to be a consensus of most of the other values. Our independent theoretical estimate is in agreement with it. We note finally that in addition to our extrapolation of calculated results for finite clusters to chains of infinite length, crystal orbital approaches may be used directly upon the infinitely repeating system with periodic boundary ~ o n d i t i o n s . ~Calculations ~ of chains of HF molecules25dindicate that both approaches converge to similar results. We feel that our cluster technique is probably more appropriate to investigate the effect of side waters and ring formation within the three-dimensional lattice of ice. Acknowledgment. This work was supported by grants to S.S. from the National Institute of General Medical Sciences (GM29391) and the Research Corporation and to J.F.N. by the National Science Foundation (DMR8115979). An allocation of computer time from Southern Illinois University is acknowledged. Registry No. H20, 7732-18-5. (22) C. Jaccard, Helu. Phys. Acta, 32, 89 (1959). (23) 0.WBrz and R. H. Cole, J. Chem. Phys., 51, 1546 (1969). (24) G. C. Camplin, J. W. Glen, and J. G. Paren, J. Claciol., 21, 123 (1978). (25) (a) G. Del Re,J. Ladik, and G.Biczo, Phys. Reu., 155,997 (1967). (b) J. J. Ladik, Adu. Quantum Chem., 7, 397 (1973). (c) H.Chojnacki, Int. J . Quantum Chem., 16, 299 (1979). (d) P. Schuster in "Intermolecular Interactions: From Diatomic8 to Biopolymers", B. Pullman, Ed., Wiley, Chichester, 1978, pp 363-432.
