J. Phys. Chem. 1983,87,4312-4313
4312
The Structure of Ice I h by Neutron Diffraction W. F. Kuhs’ and M. S. Lehmann Instirut Laue-Langevin, 156X Centre de TrL 38042 Grenoble, France (Received: August 23, 1982; I n Final Form: October 27, 1982)
High-resolutionneutron diffraction studies of ice Ih have been done at 60, 123, and 223 K. The molecular geometry found in previous studies is confirmed with an 0-Hlength of 1.01 8, and an H-0-H angle of 109.5O. The thermal motion follows a harmonic oscillator behavior. The scattering density in the middle of the hydrogen bond indicates that the hydrogen atom motion extends beyond the midpoint of the oxygen-oxygen bond.
The advent of high-flux, short-wavelength neutron sources as, for example, available at the Institute LaueLangevin has greatly facilitated the analysis of thermal motion and disorder in simple crystalline systems. The main reason for this is that more extensive data can be gathered, but likewise the shorter wavelength helps in reducing the effect of secondary extinction in the strongest reflections,’ so that it becomes manageable with present day theories. In this note we describe a recent high-precision diffraction study of ice Ih at three temperatures, 60, 123, and 223 K, using short-wavelength neutrons. The data were collected with the D9 diffractometer located at the hot source of the high-flux beam reactor with a 0-28 scan technique, and the wavelength was 0.71 A. Details of the data collection and handling are given e l s e ~ h e r e .A ~ ~first ~ calculation of structural parameters was done by use of conventional crystallographic leastsquares refinement techniques. The atoms were assumed to undergo harmonic motions, and the two hydrogen atoms were placed in two positions around the midpoint of the oxygen-oxygen bonds following the model of Pauling“ (the split-atom model). So that a further study of the density distribution of the hydrogen atoms could be performed another description of the hydrogen atom was then made, whereby one hydrogen atom at the time was fixed at the midpoint of the oxygen-oxygen bond. By the introduction of terms allowing for nonharmonic motion with a GramCharlier expansion of the scattering density5 (nonsplit atom model) equally good agreement between observed and calculated amplitudes could be obtained. For hydrogen H1 only one additional term (of fourth order) beyond the parameters for a harmonic motion was needed, and as the atom position was fixed the total number of parameters remained constant. Similarly for hydrogen atom H2 three addition fourth-order terms were needed, but as the hydrogen atom position was fixed, again the number of parameters stayed constant. The split-atom model was used for distance and angle calculations, while the nonsplit model was used for studies of the scattering density. The crystallographic agreement factors are s u m marized in Table I. A “bent hydrogen bond” model has been suggested by Chidambaram6on the basis of refinements of the data of Peterson and Levy.’ This model was tested by introducing modifications of the scattering density corresponding to a sixfold splitting of the hydrogen atoms. The improve~
~~
~~
~
~
(1) Lehmann, M. S.; Schneider,J. R. Acta Crystallogr.,Sect. A 1977, 33, 789. (2) Kuhs, W. F.; Lehmann, M. S. Nature (London) 1981, 294, 432. (3) Kuhs, W. F.; Lehmann, M. S. Acta Crystallogr., in press. (4) Pauling, L. J. J. Am. Chem. SOC.1935, 57, 2680. (5) Kuhs, W. F. Acta Crystallogr., Sect. A, in press. (6) Chidambaram, R. Acta Crystallogr. 1961, 14, 467. (7) Peterson, S. W.; Levy, M. A. Acta Crystallogr. 1957, 10, 70.
TABLE I : Crystallograhic Agreement Factor Rwa 60 K
R w ,% N m
2.52 3.05 317 14
123 K 3.09
2.70 235 14
223 K 3.13 3.03 118 14
a X W(Fob, - F c d c d ) / W F o b d ; F,b, and ,F are structure factor amplitudes and the weight is based o n counting statistic plus an additional term which is typically 1.5% of F o b d . N is the number of observations included in the refinement, m is the number of parameters. The left column is for the split-atom model; the right-hand column is for atom H1 being nonsplit.
TABLE 11: Molecular Geometrv and Thermal Motiona 60 K 0-0’ 0-0” 0-H1 0-H2 H1-H1’ H2-H2’ 0’--0-0” 0”-0-0” H1-0-H2 H2-0-H2’
123 K
223 K
Distances, A 2.750 (1) 2.751 (1) 2.752 (1) 2.753 (1) 1.004 ( 2 ) 1.008 ( 2 ) 1.005 (1) 1.004 (1) 0.741 ( 3 ) 0.735 ( 4 ) 0.741 (1) 0.745 ( 2 )
2.759 ( 2 ) 2.760 (1) 1.008 ( 4 ) 1.004 ( 2 ) 0.744(7) 0.753 ( 3 )
Angles, deg 109.33 ( 2 ) 109.34 ( 2 ) 109.61 ( 2 ) 109.60 (31 109.21 (11) 109.40 ( 1 6 ) 109.73 (11) 109.54 ( 1 6 )
109.35 ( 4 ) 109.60 ( 5 ) 109.44 ( 2 7 ) 109.50 ( 2 7 )
Rmsd, A 0.1179 ( 5 ) 0.1521 ( 7 ) 0.2069 ( 1 4 ) 0.2096 ( 1 0 ) OlC 0.1183 ( 9 ) 0.1534 ( 8 ) 0.1337 ( 1 6 ) 0.1626 ( 2 1 ) 0.2115 ( 4 1 ) H1 Ii a Hllc 0.1784 ( 6 ) 0.1988 (11) 0.2409 ( 2 8 ) H1 I1 0-0” 0.1358 (10) 0.1650 ( 1 5 ) 0.2109 (31) H2 Ii a 0.1799 ( 1 0 ) 0.2005 ( 1 4 ) 0.2421 ( 2 3 ) H21a 0.1762 ( 1 0 ) 0.1984 ( 1 5 ) 0.2407 ( 2 7 ) a The bonds 0 - H , and 0-0’ are along the crystallographic c axis. Thermal motion is indicated as root-meansquare displacement (rmsd) in certain directions along or Orthogonal t o axis a, c, or 0 - 0 ” .
0 I1 c
ments in the agreement were hardly significant (corresponding to a 5% level with the test suggested by Hamilton*), and three additional parameters had to be added. Moreover the scattering density showed no positive modification in the direction of the bend. There is thus no reason to depart from the standard model, although it should be noted that the motion orthogonal to the bond is definitely much larger than along the bond as expected. The main structural result is a confirmation of the somewhat unusual dimensions and angles in the molecule orginally found by Peterson and Levy.7 The values are summarized in Table 11, and are calculated from the positions found in the harmonic split-atom model. It is (8) Hamilton, W. C. Acta Crystallogr. 1965, 18, 502.
0022-3654/83/2087-4312$01.50/00 1983 American Chemlcal Society
Structure of
The Journal of Physical Chemisfry, Vol. 87, No. 21, 1983 4313
Ice I h 60 K
123 K
223 K
...
Figure 1. Probability density maps of H1 in the (1070) plane as described by harmonic and anharmonic terms up to fourth order (equidistant contours, zero contour omitted). The small negative regions are due to minor shortcomings of the chosen mathematical description. Within the limits of error these probability densities are following the expectations for simple temperature dependance assuming BoRzmann statistics.
well-knowngthat this model leads to a foreshortening of calculated bond lengths, so the values given for the oxygen-hydrogen bond are lower limits. A more extensive analysis of the 60-K data2 shows that the mode-mode distance in the O-H bond is around 0.008 A longer. The values observed do not vary significantlywith temperature, and as observed by WhalleylO the distances are considerably longer than the value around 0.97 A normally found for bound water in crystalline compounds. Similarly, the bond angle is larger than 105' which is commonly observed. Opening of the H-O-H bond angle corresponds to an approach of the "lone-pair" orbitals in the nearly sp3-hydridized molecule, and leads to increased interelectronic repulsion. This could be released by rearranging slightly the electron density, and indeed this will happen because of hydrogen bond formation. A relatively strong hydrogen bond is formed leading to a considerable stretch of the 0-H bond length. So although we have not yet any quantitative explanation of the anomalies the two deformations follow a common trend. The thermal motion parameters given in Table I1 show a variation with temperature for the oxygen which is in very good agreement with the behavior of a harmonic oscillator. The slightly larger motions observed for the hydrogen come from intramolecular contributions. These parts are constant within the error limits indicating that the various oscillations are always in the ground state, as (9) Busing, W.R.;Levy, M.A. Acta Crystallogr. 1964,17, 142. (10) Whalley, E. Mol. Phys. 1974,28,1105.
expected. Likewise it is worth noting that the oxygen atom motion is very nearly isotropic, and that the two hydrogen atoms have nearly identical behavior. The main distinction between these two atoms comes from the bonding angles, where the angle Hl-O-H2 is, in general, smaller than H2-O-H2. This corresponds to the angles 0-0-0being larger when two of the oxygen atoms are in the hexagonal plane, leading to a slight compression along the c axis. The axis ratio c / a is therefore 1.6284l19l2rather than the ideal value of 1.633. The oxygen-oxygen distances, on the other hand, are the same in the two cases. One additional curious result of this analysis arises from a careful inspection of the scattering density of the hydrogen atoms. This is obtained from the parameters used in the nonsplit atom models where higher than quadratic terms are included in the description of the scattering density, and it is depicted in Figure 1for atom H1 for the three temperatures. In general, a direct Fourier transform of diffraction data suffers from resolution effects, i.e., the maps will be smeared because of limited spatial resolution and, in addition, ripples might occur from series termination effects and errors in the observations. When data are fitted to a reasonable model before calculation of the density from the model parameters these effects are reduced5 as is obvious from the figure. The interesting feature of this map is the nonnegligible scattering density in the midpoint of the oxygen-oxygen bond which increases with increasing temperature. In a diffraction study we sample over time and space so this map arises from the mean of domains of the crystal where the proton is in an asymmetric potential bound to one of the two oxygen atoms. It does, however, mean that the hydrogen atom is present at the middle of the hydrogen bond for an observable part of the time. Whether this means that the proton moves across the midpoint toward the next oxygen we cannot tell from the diffraction study. This would very much depend on the location of the two hydrogen atoms bound to the oxygen atom in question and the dynamics of the system. Presently we are therefore investigating possible motions of this kind using low-energy, high-resolution, quasi-elastic neutron-scattering techniques. Acknowledgment. The authors thank A. Chaillou (Laboratoire de Glaciologie, Grendole) who kindly provided the single crystals used in this study. Registry No. H20, 7732-18-5. (11) LaPlaca, S. J.; Post, B. Acta Crystallogr. 1960, 13, 503. (12) Brill, R.; Tippe, A. Acta Crystallogr. 1967,23, 343.
