Corrections to the OH bond lengths and HOH angles of the water molecules in crystalline hydrates. Application to Ba(ClO3)2.H2O and K2C2O4.H2O

Journal of Molecular Structure, 52 (1979) 107-112 o Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

CORRECTIONS TO THE O-H BOND LENGTHS AND H--O-H ANGLES OF THE WATER MOLECULES IN CRYSTALLINE HYDRATES. APPLICATION TO Ba(C10J1.H20 AND KzC104.Hz0

A. ERIKSSON, B. BERGLUND, Institute

of Chemistry,

University

J. TEGENFELDT

and J. LINDGREN

of Uppsala, Box 531, S-751 21 Uppsala (Sweden)

(Received 15 August 1978)

ABSTRACT Corrections for systematic errors in the O-H bond lengths and H-O-H angles obtained by diffraction methods have been made for the water molecules in Ba(ClO,),.H,O and K,C,O,.H,O. The corrections are made for rigid-body librational motion and anharmonic O-H stretching motion of the water molecules from spectroscopically-determined data. The corrections to the O-H distances are 0.034 and 0.009 A and to the H-O-H angles -2.2 and -1.1” for the water molecules in Ba(ClO,),.H,O and K,C,O,.H,O, respectively. For the water molecule in K,C,O,.H,O, the equilibrium O-H distance is found to-be elongated 0.015( 3) A from the gas phase value.

INTRODUCTION

Bond distances and angles in molecules determined by X-ray or neutron diffraction are known [ 1,2] to be influenced by systematic errors caused by the thermal motion of the atoms. In most crystal structure determinations, it is usual to assume a model of thermal motion which corresponds to the threedimensional harmonic rectilinear motion of each atom. Two types of motion which do not fulfill this assumption are the librational motion of a molecule causing the atoms to move on arcs and the anharmonic motion of atoms along chemical bonds. The effect on the nuclear scattering density of an atom undergoing anharmonic stretching and librational motion are illustrated in Fig. la and b, respectively. In structure determinations assuming harmonic rectilinear motion, ellipsoids are fitted to the actual scattering density distributions. The apparent position of the atom will therefore lie to the right of the equilibrium position in Fig. la and to the left in Fig. lb. Both of these effects are particularly significant for water molecules. The small moments of inertia about the principal inertial axes of the water molecule lead to large amplitude librational motion. Furthermore, the anharmonicity of the O-H stretching motion is large due to the small mass of the hydrogen atoms. In the free water molecule the anharmonicity is already appreciable and is expected to increase further for a bonded molecule.

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Fig. 1. Scattering density distributions motion, and (b) librational motion.

for an atom undergoing

(a) anharmonic

stretching

The estimated standard deviations for an O-H bond length obtained in a neutron diffraction study can be as small as 0.002 A. Systematic errors of about 0.04 A, which are not reflected in the standard deviations, can however occur. With a reasonable model for the types of motion mentioned, some of these errors can be eliminated by correcting for the effect of this motion on the determined bond lengths. A procedure for such correction to atomic coordinates influenced by rigid-body librational motions has been described by Schomaker and Trueblood [2] . In the following, we will calculate vibrational corrections to the internal geometry of the water molecules in the two hydrates Ba(C1O&.HzO and KzCz04.Hz0. We will thereby make use of the results from normal coordinate analyses of the water molecules in these hydrates. These analyses were based on spectroscopic vibrational data and the water molecules were allowed to vibrate in an effective force-field due to the stationary surrounding atoms. The shape and ,vibrational wavenumbers obtained for the different normal modes are shown in Fig. 3 of ref. 3. Neutron diffraction studies of both hydrates at -300 K have been reported [4, 51. CORRECTIONS

FOR LIBRATIONAL

AND BENDING

VIBRATIONS

Corrections for rigid-body librational motion to the atomic positions are given in terms of the librational tensor L of a molecule in ref. 2. The components of this tensor are the mean-square angular displacements (WiWj), where wi’ and wj are rotations about any one of three orthogonal axes. These mean-square values for the water molecules in Ba(C103)2.H10 and KzC104.Hz0 are given in table 1 of ref. 3. Using these values and the formula in ref. 2, a correction to the coordinates of the atoms of the water molecule has been obtained. In order to see the influence of each normal mode on the water geometry, the corrections to the O-H distances and HUH angles have been calculated separately for each mode and the results are shown in Table 1. The effect of the water bending vibrations (mode number 7 in Table 1) on the O-H distances requires special treatment since these are intramolecular

109 TABLE 1 Contributions to the corrections to the O-H distances (A ) and the H-C-H angles (” ) in Ba(ClO,),.H,O and K,C,O,.H,O from each normal mode, and equilibrium values of the O-H distances and H-C-H angles. The modes are ordered in increasing wavenumber. The shapes and vibrational wavenumbers of the different modes are given in ref. 3, fig. 3 Mode

H-C-H

C-H distances (A ) Ba( CIO,),.H,O

K,C,O,.H,O

Ba(ClO,),.H,O

1 2 3

0.0030 0.0022 0.0000

0.0037 0.0000 0.0003

-0.45 -0.29 0.00

4 5 6 7 8 9

0.0162 0.0116 0.0114 0.0029 0.0000 0.0000

0.0075 0.0066 0.0059 0.0027. 0.0000 0.0000

-2.39 1.00 -0.07

Total librational correction Anharmonicity correction

0.0473

0.0267

-2.20

Diffraction value Equilibrium value

-0.0134

-0.0179

angles (” )

0.00 0.00 0.00

-

0.926( 10)

0.963( 3)

110.7(1.4)

0.960

0.972

108.43

K,C,O,.H,O -0.56 0.00 -0.03 -0.09 -0.93 0.52 0.00 0.00 0.00 -1.09 107.62( 33) 106.53

vibrations. If we consider a displacement (Yof the water angle from its equilibrium value, this corresponds to a displacement of (r/2 of each O-H bond. The effect of the bending vibration can therefore be simulated by a simple librational tensor with a mean-square angular displacement in the water molecular plane of (cY*>/~. CORRECTIONS

FOR ANHARMONIC

VIBRATIONS

Anharmonic O-H stretching vibrations will cause the mean O-H distance to be longer than re, the distance at equilibrium. This elongation of the O-H distances in the free water molecule has been calculated by Kuchitsu and Bartell [6]. These calculations were based on an extensive expression for the potential energy of the water molecule. The values obtained for Hz0 and D20 were 0.014 and 0.011 8, respectively. In the present case, where our vibrational data are naturally not of the same accuracy as those for the free water molecule [7] , we have tried a much simpler expression for the potential energy. We have assumed that the hydrogen atoms vibrate independently against a stationary oxygen atom in a Morse potential [8]. Furthermore, we have assumed that only the lowest vibrational state is populated at 300 K, since it was found in ref. 6 that, for the free

110

water molecule, the contributions to the corrections from excited vibrational states were negligible. We have used the expression for the probability density given by Ibers [9] : p(z)

=

(&,Q)+ [r(x;l

-

I)]-’

e-ZZ(l-xe)/xe

where 2 = zc;’ exp [-(2x,(11)+ (r - r,)] and (Y= 4n2mo,c/h We have tested this function on the free water molecule using the values 3887 and 180 cm-’ for we and 20,3c, as given in ref. 10. The elongations obtained for Hz0 and D20 were 0.015 and 0.011 A. We consider this to be in sufficiently good agreement with the more accurate values of ref. 6. The w, and 2w,3c, values used for the hydrates under consideration are those given in ref. 10 for isotopically-dilute HDO molecules. We believe, however, that the correction to the G-H distances will be overestimated if calculated from the mean displacements relative to re using the probability density described above. We have therefore tried to simulate the result which would be obtained from a crystallographic least-squares refinement by fitting a Gaussian probability density to that obtained from the Morse potential. The corrections calculated in this way are somewhat smaller and are given in Table 1. For a further discussion on this point see below. DISCUSSION

From Table 1, it can be seen that the librational correction is positive (the O-H distance observed by diffraction methods is shorter than re). The correction for anharmonic stretching motion is negative. The absolute value of the librational correction is larger in the two cases studied but it would seem possible for the reverse to occur, for strongly-bonded water molecules in particular. Generally, therefore, we can expect to observe by diffraction O-H distances in hydrates which are shorter or longer than the re values. In the appendix of a recent review article [ 1 l] on X-ray and neutron diffraction studies of hydrogen-bonded systems O-H distances determined by neutron diffraction are listed. If we limit ourselves to studies on hydrates where the standard deviation is equal to or less than 0.005 A, the O-H distances fall in the range 0.92-1.00 A. In a theoretical study on the free water molecule and water molecules influenced by crystal fields [12] , a maximum elongation of an G-H distance at equilibrium was 0.025 A, accompanied by a shift in the harmonic stretching wavenumber of 415 cm-‘. Since this corresponds to rather strong hydrogen bonding, the elongation of the G-H distance (0.025 A) would be close to the upper limit, at least for the type of hydrates considered here. The substantially wider range of O-H distances observed must therefore arise from systematic errors in the diffraction studies.

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The diffraction study of Ba(C1O&.HZO resulted in rather large standard deviations, making the calculated equilibrium O-H distances correspondingly uncertain. It is not possible to conclude if the latter distance (0.960 A) actually is longer than the gas phase value 0.957 A [7]. The interest of the values for Ba(C1O&.HzO in Table 1 lies rather in the magnitudes of the corrections. For KzCz04.H20, the standard deviations are smaller and the elongation of the O-H distance of 0.015 .& from the gas phase value is certainly real. The distribution of the librational corrections (cf. Table 1) shows that the rotational modes 4, 5 and 6 are responsible for the largest corrections, as expected. Furthermore the corrections are approximately evenly distributed among these modes. The effect of librational motions on the observed HUH angle has not often been considered. Pedersen [ 131 found in a model calculation that, in general, the HUH angle observed is too large. From Table 1 it can be seen that both positive and negative corrections can be obtained, depending on the nature of the particular mode. To investigate this further we define a Cartesian coordinate system as in ref. 3, with the origin at the center of mass and the y-axis as the two-fold symmetry axis of the water molecule. The zand Ix-axes are normal to and in the plane, respectively. In general, for rotational vibrations about the y-axis (twisting) a positive correction to the observed HUH angle is expected, while a negative correction is expected for rotational vibrations about the x-axis (wagging). The third type, rotation about the z-axis, will give no correction at all. These results can be easily visualized if the scattering density in Fig. lb is placed at the positions of the hydrogen atoms and oriented properly for the three types of rotational vibrations. Since the rotational vibrations (modes number 4, 5 and 6) can be described as twisting, wagging and rocking (note that the order is different in Ba(ClO& .fi,O and K&O4 .HzO) the angle corrections are as expected. The total angular correction for Ba(C1O&.HZO was -2.2”. In this compound, the water molecule resides in a planar trigonal environment and the chemical bonding to the water molecule from atoms in the molecular plane is expected to be much stronger than from atoms which lie out of this plane. This will cause the water rotational vibrations out of the plane to occur at low wavenumbers. The force constant for wagging motion in Ba(C103)2.HZ0 was 0.052 as compared to 0.24 mdyn A for rocking motion [3]. We would therefore expect particularly large angular corrections for trigonallycoordinated water molecules. However the equilibrium angle itself in such water molecules is expected to be large (see ref. 14 and references therein). Thus part of the enlargement actually observed [14] might be only apparent enlargement, especially when the water molecule is coordinated to transition metal ions (which lead to trigonal coordination). Aninteresting case in this connection is CuF2.2Hz0. This hydrate has been studied at two temperatures [15,16] and the H--O-H angles were observed as 115.5(4)” at 298 K and llO.l(4)“at 4.2 K. There is a NQelpoint at 10.92 K, but the interatomic distances

112

differ very little above and below this point. The decrease of 5.4” in the water angle is therefore most likely explained as a systematic error resulting from librational motion at 298 K. Even the value at 4.2 K (110.1(4)“) is subjected to systematic errors, mainly resulting from zero-point vibrations and should be further decreased before the equilibrium value comparable with the gas phase value of 104.5” [7] is obtained. A better way of taking into account the influence of librational and anharmanic motions in a least-squares refinement is to incorporate them directly using higher order cumulant expressions [ 171. This would be particularly important in connection with studies of bonding-electron distributions by difference Fourier techniques. We are presently working on this approach for water molecules, using spectroscopically determined contributions to the temperature factors. ACKNOWLEDGEMENTS

The authors wish to thank Prof. Ivar Olovsson for the facilities he has placed at their disposal. This work has been supported by grants from the Swedish Natural Science Research Council which are hereby gratefully acknowledged. REFERENCES 1 D. W. J. Cruickshank, Acta Crystallogr., 9 (1956) 757. 2 V. Schomaker and K. N. Trueblood, Acta Crystailogr., Sect. B, 24 (1968) 63. 3,A. Eriksson, M. A. Hussein, B. Berglund, J. Tegenfeldt and J. Lindgren, J. Mol. Struct., 52 (1979) 95. 4,s. K. Sikka, S. N. Momin, H. Rajagopai and R. Chidambaram, J. Chem. Phys., 48 (1968) 1883. 5 A. Sequeira, S. Srikanta and R. Chidambaram, Acta Crystailogr., Sect. B, 26 (1970) 77. 6 K. Kuchitsu and L. S. Bartell, J. Chem. Phys., 36 (1962) 2460. 7 W. S. Benedict, N. Gailar and E. K. Plyler, J. Chem. Phys., 24 (1956) 1139. 8 P. M. Morse, Phys. Rev., 34 (1929) 57. 9 J. A. Ibers, Acta Crystailogr., 12 (1959) 251. 10 B. Berglund, J. Lindgren and J. Tegenfeldt, J. Mol. Struct., 43 (1978) 169. 11 I. Olovsson and P.-G. Jijnsson, in P. Schuster, G. Zundel and C. Sandorfy (Eds.), The Hydrogen Bond, II, North-Holland, Amsterdam, 1976, p. 393. 12 J. Aimlof. J. Lindgren and J. Tegenfeldt, J. Mol. Struct., 14 (197 2) 427. 13 B. Pedersen, Acta Crystallogr., Sect. B, 31 (1975) 869. 14 M. Falk and 0. Knop, in F. Franks (Ed.), Water - A Comprehensive Treatise, Vol. 2, Plenum Press, New York, 1973, p. 55. 15 S. C. Abrahams and E. Prince, J. Chem. Phys., 36 (1962) 56. 16 S. C. Abrahams, J. Chem. Phys., 36 (1962) 56. 17 B. T. M. Willis and A. W. Pryor, Thermal Vibrations in Crystallography, Cambridge University Press, London, 197 5.