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Model calculations of the vibrations of bonded water molecules
&wnul of MolecularStwcture, 48 (1978) 417430 @EXsevier Scientific Publishing Company, Amsterdam -
MODEL CALCULATIONS MQLECULES
Printed in The Netherlands
OF THE VIRRATIONS OF BONDED WA’IZR
A. ERIKSSON and J. LTNDGREN Institute of Chemistry,
University
crf Uppsala, Box 531, S-751 21 Uppsalu (Sweden)
(First received 26 September 1977; in revised form 13 January 1978)
ABSTRACT Model calculations have been made of the vibrational frequencies and normal modes of a water molecule vibrating in a combined internal and external field. A constant internal force field has been used together with an external central force field from four or three nearest-neighbour atoms to the water molecule. These neighbour atoms have been arranged either tetrahedrally or trigonally around the water molecule. The external force field has been further restricted by the use of five possible site symmetries for the water molecule, C lv, C,, Cs(Q %,.), C,(CF~~)and C,. A series of calculations have been made where the external force constants have been varied within the range l-80 Nrn-. The nine calculated normal modes can be divided into three groups: intra-molecular, rotational and translational vibrations. Among the rotational vibrations it is found that, in the tetrahedral environment, the rocking mode occurs at lower frequencies than the twisting and wagging modes, whereas the opposite occurs for the trigonal environment. Frequency ratios have been calculated using the isotopic species H,O, D,O, HDO and H,“O. The twisting and wagging modes have the V~&PD,Q ratio in the range 1.35-1.41 and the rocking mode in the rauge 1.26-1.41. XNTRODUCTION
In a recent paper [l] a model calculation has been made of the vibratory motions of a water molecule in a condensed phase, A normal coordinate analysis was made using a combined internal and isotropic external force field. In the present work, we have extended this calculation to include the effect on the vibrations of different anisotropic external force fields. We have thus tried to simulate more closely the typical environment of a water molecule in a crystalline hydrate. Additional complications arise on the introduction of anisotropic external force fields; in the isotropic case the same numerical value can be used for all external force constants. The main problem is to find a systematic way to represent the numerous situations possible. In the following, we will assume that the external fo;cce field of the water molecule is produced mainly through interactions between the water molecule and its closest neighbour atoms. The long-range electrostatic interactions between the water molecule and more distant atoms are assumed to be smaller and to contribute isotropically to the external force field.
418
Xn a review [2] of the structures of crystalline hydrates, it was found that, in about 95% of the 683 hydrates considered, the water molecule is surrounded by either four or three closest neighbour atoms. These are arranged in a tetrahedral or planar trigonal manner around the water molecule. Two neighbour atoms are involved in hydrogen bonds with the hydrogen atoms of the water molecule, and the remaining atoms are involved in interactions with the oxygen atom. These latter interactions can be of the hydrogen bond or ionic type. We concentrate our attention, therefore, on these two cases and let the external potential energy be represented by a central force field consisting of atom-atom interactions between an atom in the water molecule and one of the four (or three) nearest-neighbour atoms. The treatment is further systematized by including the different site symmetries possible for the water molecule. Besides the CzVsite symmetry, there are four additional lower symmetries C1, C, (ox=), C, (a,=) and C1. The two symbols ~~~ and uyz s&nit-r mirror planes in and perpendicular to the water molecule plane, respectively. A systematic variation of the external force constants is thus made using the force field as described above and the symmetry restrictions required by the site symmetries_ In the calculations, the internal force field is held constant since our purpose is to study the effect of the external force field on the vibrations of the water molecule. NORMAL
COORDINATE
ANALYSIS
Calculation procedure A computer program has been used which is a modified version of that originally written by Gwinn [3]. In the present version, the force constants and the normal modes can be defined using three types of displacement coordinates; cartesian coordinates, valence coordinates (changes in distances and angles) and rigid body coordinates (Eckart coordinates). The procedure used in the present calculations differs from that used by Pedersen [l] chiefly in that we have also introduced point atoms to represent the environment of the water molecule. With a few exceptions discussed below these atoms are not allowed to vibrate, however, but are used only to define an external force field in which the water molecule vibrates. The procedure used to suppress the vibrational motion of the environment is as follows. Let x bc a column matrix of Cartesiandisplacement coordinates, where the first nine components refer to the atoms of the water molecule. The valence coordin&es are contained in a column matrix r. The relation between xandris r=B’x where the transformation matrix B’ is obtained as described by Gwinn [ 31. The to&Mpotential energy Vtot for both the water molecule and the
419
environment can now be written 2 J&t =fF,r=
t ti’ F, B’ x
where F, is a force constant matrix. Since the environment will not be allowed to move, we want to remove from this expression the contributions associated with displacements of the atoms in the environment. The remaining potential energy for the water molecule alone is wherefx,lo)i=Xj,Bii=Bi:-forj=l, 2, . . ..9a11dFn.o=gF,B.Thatisto say XH,~ contains the first nine elements of x and B contains the first nine columns of B’_ The potential energy for the water molecule is now expressed in cartesian displacement coordinates, and the normal coordinate calculations then follow the procedure given by Gwirm [33. Selection of the force field parameters The internal force field of the water molecule has been chosen to give reasonable stretching and bending frequencies. This force field, which has remained fixed in all calculations, has stretching force constants f,, and fr2 equal to 650 Nm-’ and bending force constant f, equal to 70 Nm rad-‘. A fixed water geometry has been used with rl = r2 = 1.00 ft and ar= 106.3”. The Cartesianpositional coordinates for the water molecule and the surrounding atoms are found in Table 1. The cartesian coordinate system is defined in Fig. 1, The external force field consists of stretching force constants fij between an atom in the water molecule and an atom in the environment, where i and j are the numbers of the atoms. This numbering for the tetrahedral and trigonal environments is found in Fig. 1. The force constant matrix F, is thus a diagonal matrix with force constants f,,, fp,, f, and the ffj’s on the diagonal, The different site symmetries are obtained by using the constraints on the force constants found in Table 2. For the case of the planar trigonal TABLE 1 Positional Cartesian coordinates of atoms in the water molecule and the surrounding atoms in the tetrahedral and triaonal environment. The cartesian axes are defined in Trigonal
Tetrahedral Atom 1 2 3 4 5 6 7
;
y
2
0.0 0.8 -0.8 2.2 -2.2 0.0 0.0
0.0 0.0 0.0 0.0 0.0 2.3 -2.3
z-6” 0:6 1.6 1.6 -1.6 -1.6
Atom
f 3 4 5 6
x 0.0 0.8 -0.8 2.2 -2.2 0.0
Y z-0” 0:O 0.0 0.0 0.0
z
z-: 0:6 1.6 1.6 -2.8
420
‘t.6 4’ /
I
rr7
b
a Fig. 1. Water mclecule
in tetrahedral
(a) and trigonal (b) environment.
TAESLE 2 Force constants (Nm-I) in the different equal are constrained by symmetry
environments
and site symmetries.
Constants
put
Tetrahedral
C zv
10
= f,,
40
=
IO
=
=
20 20
30
20
f,,
5
CT
5
25
f*e 35
c, (us=) q.(~TXL)
= fz’II = fzs
10 5
= 10
20 40
20 =f,6
20 20
5 3
40 40
30 = ft,
=
5
10 5
= f,, lo
20 20
10 8
30 20
5 3
30 30
10 12
f,, f,,
f,,
=
= f,,
= f,,
= f14
= f,, = f2.s
= f,, = f,,
10
20
= f,,
=fis
= f,,
30
5
30
30
10 10
fzs
= f,, = f,,
Trigonal
% Cs (axz)
environment (Fig. l), it is not possible to define out-of-plane force constants. We have therefore introduced additional force constants fI, fi and f3 for the Cartesiany displacement coordinates of the oxygen and hydrogen atoms in the water molecule. These constants can be said to represent interactions to more distant atoms above and below the water molecule plane. In so doing, the site symmetries CzV, C, and C, (a,,=) become indistinguishable and only two different cases remain for the planar trigonal case, Cpvand C, (cJ,,). A large number of calculations have been performed in which the force constants fii (and fl, f2 and f3) have been assigned various values in the range l-80 Nm-I. These values have been further restricted, however, to give
421
frequencies in the range 400-900 cm-’ for the mainly rotational vibrations of the water molecule_ Similarly, the predom~~tly translational vibrations are rest&ted to have frequencies in the range 50-350 cm-‘, The ranges have been chosen to correspond to experimentally observed frequencies. In some calculations the positional coordinates of the atoms representing the environment have also been changed from the values given in Table 1. This will result in non-linear hydrogen bonds. fn order to have a basis for a discussion, we have chosen to give the results of the normal coordinate analysis for one representative case of each symmetry and environment. The force constants are given in Table 2 end the frequencies, symmetry types and normal modes in Figs. 2 and 3. The potential energy distributions are given in Table 3. To investigate the situation when the closest neighbour atoms are no longer stationary but are allowed to vibrate, we have also performed a limited number of calculations using the following extended expression for the potential energy 2 ‘Vtot=tB’F,B’x+ii:F‘;x where F: is a diagonal force cons~t matrix. The first nine eIements on the diagonal are equal to zero and the remaining elements are equal to f Nm-‘. F: can thus be said to represent the effect of the next-nearest neighbour atoms on the nearest neighbour atoms to the water molecule. The masses of the neighbour atoms and the force constant f have been varied in these calculations. c2
:rcquancyl-
km-'I
mode
km-‘1
symmewY _
169(B)
.‘Yes_
---
169 iA’)
222 (A’)
222 IB) 278 (A]
708 {A]~
‘I 4
I654
!i :~ Frequency
Normal
symma,ry -_
T-
IA I
3501 (A 1 3550 18 1
291 (A’)
764 IA’)
1659 (A‘1 3509b’l
3558 iA”
i Fig. 2. Frequencies, symmetry types and normal modes for the tetrahedral environment.
422
Normst moda
Normal
mods
Fig, 3. Frequencies, symmetry types and normal modes for the trigonal environment. RESULTS
AND DISCUSSION
The discussion of the results will be based on the examples of calculations presented in Table 3 and Figs. 2 and 3. The potential energy distributions found in Table 3 give the contributions to the frequencies from each force constant. These potential energy distibutions can also be used, however, to predict the change in frequency of a particular mode when one or several force constants are changed. The empty positions in Table 3 correspond to zeros or values which are very small. These values remained small throughout the calculations. The rotational vibrations are usually assumed to occur about the principal inertial axes [4,5f (passing through the centre of gravity) of the water molecule. Our results show that this is not generally true, however. The rotational modes, as found in the calculations, are linear combinations of rotations about the inertial axes, and also contain some translational motion. Nevertheless, each of these modes can, in many cases, be approximately described as rotations about one inertial axis. In such cases, these modes may be described equally well as rotations about one of the axes x, y or z as defined in Fig. I. We will refer in the following to the rotational vibrations as twisting (rotation about the z-axis), wagging (rotation about the x-axis) and rocking (rotation about the y-axis). fn those cases where twisting, wagging or rocking do not properly describe the rnodes, we shall use such expressions as “a mixture of twisting and wagging”_
423
Czp C,, C, fa,,)
and 42,{a,,)
site symmetries
In Figs. 2 and 3, ah vibrations occur either witbin the plane of the water molecule @z-plane) or perpendicular to it. This is not generally true, however. We can have, for example, for the case of C, (oYz) symmetry a rather strong mixture of rocking and twisting. In this case, the atoms move simultaneously in and perpendicular to the plane within a single normal mode. For this situation to occur, the force constants fz6 and fs6 must have values which are very different from f2, and f3,. A difference of almost a factor three is needed to produce any appreciable mixture. Tetrahedral environment
For all symmetries the rocking vibration always occurs at a lower frequency than the other rotational modes. In the CpVsymmetry, the twisting and wagging modes are degenerate. This degeneracy is lifted when the symmetry is lowered. Two different situations occur: in the C2 and C, (Ok=)symmetries, the modes can be described as twisting and wagging; but in the C, (ozz) symmetry, a strong mixture is obtained and the modes become, as seen in Fig. 2, a movement of one or the other of the hydrogen atoms out of the xz-plane. The two modes can have very different frequencies. The isotopic ratios Yn,o/~n,~ for the wagging and twisting modes are 1.41. This ratio varies between 1.25 and 1.41 for the rocking mode, and is mainly due to changes in the force constants fis and f 34. SmalI values of the constants give a low ratio and vice versa. A low ratio results from a translational component arising in the x direction of the rocking mode. When only one hydrogen is substituted with deuterium to produce an HDO molecule, it is found that the twisting and wagging modes become strongly mixed. Two modes are produced with, in the first one, a hydrogen and in the second one a deuterium motion out of the molecular plane. The frequency of the hydrogen mode is between those of wag and twist of HZO, and the frequency of the deuterium mode between those of wag and twist of DzO. If the hydrogen mode frequency is closer to that of wag of H20, the deuterium mode frequency is closer to that of twist of D20 and vice versa. The frequency ratio for the rocking modes vn,o/~nn~ is found to fall in the range 1.15-1.22. The translational vibrations are, in all cases, rather well described as vibrations along the Cartesianaxes. The translational mode along the y-axis is interesting since it can alternatively be described as a rotational vibration about an axis through the two hydrogen atoms (see Figs. 2 and 3). The frequency ratios trH,O/~n,o are close to 1.05 and 1.08 for translations along the z- and x-axes, respectively, and 1.00 for the translation along the y-axis. The E&i0 VH, 1 8&q 260 is about 1.05 for all translational modes. Trigonal environment
The most interesting point here is the possibility for the rocking mode to have the highest frequency of the three rotational modes. This is so
1 2 3 4 6 6 7 8 9
1 2 3 4 6 6 7 8 9
ca (QJ
2 3 4 6 6 7 8 9
1
Mode number
c2
1V
c
Symmetry
Tetrahedral
1.2 1.8 1.1
0.2 46.8, 47.0
0,2 46.8 47.0
0.2 47.0 47.3
0.2 47,o 47.3
1.2 1.8 1.1
1.2 1.6 0.8
0.2 47.0 47.3
0.2 47.0 47.3
1.2 1.6 0.8
1.4 0.9
x.2
f,,
1.4 0.9
1.2
I’,,
92.4 0.2
0.2
2.9
93.1 0.2
0.2
2.3
93.1 0.2
1.9
f,
1.3 1.3
22.6 6.6 0.1
1.3 1.3
22.9 7.2 0.1
1.3 1.3
6.1 0.1
22.8
fi4
1.3 1.3
22.6 6.6 0.1
1.3 1.3
25.9 7.2 0.1
1.3 1.3
6.1 0.1
22.8
fs5
1.1
13.1 0.4 9.6
0.1 1.1
12,6 0.6 lOa8
1,l
0.4 10.7
12.6
f,,
1.1
13.1 0.4 9.6
0.1 1.1
12.6 0.6 10.8
1.1
0.4 10.7
12.6
fad
11,9 3.9 0.2
12.0 4.3 0.2
3.j 0.3
12.0
f14
f,,
11.9 3.9 0.2
12.0 4.3 0.2
7.4
60.0
8.1
60.0
60.0 3.7 - 13.8 0.3
12.0
fis
7.4
60.0
8.1
50.0
50.0 13.8
fi7
0.6 11.6 12.8 28.6 32.2 1.4 1.0 0.9
0.8 16.7 26.8 20.6 16.3 0.8 0.6 0.6
11.9 19.0 2580 26.0 1.0 0.7 0.7
0.7
fib 0.7
0.7 17.0 26.3 21.4 17.6 0.1 0.7 0.7
0.6 10.1 11.2 29.2 33.6 1.3 0.9 0.8
11.9 19.0 26.0 26.0 1.0 0.7 0.7
f,,
0.7
0.5 11,6 12.8 28.6 32.2 1.4 1.0 0.9
0.6 10.1 11.2 29.2 3386 1.3 0.9 0,8
11.9 19.0 26.0 26.0 1.0 0.7 0.7
f3,
0.7
0*7 17.0 26.3 21.4 17.6 0.1 0.7 0.7
0,8 16.7 26.8 20.6 16,3 0.8 0.6 0.6
11.9 19.0 26,O 26.0 1.0 0.7 0.7
f3,
Potentialenergy distributionsin the tetrahedraland trigonalenvironments.The normal modes are numbered in order of increasing frequency
TABLE 3
c1V
Trigonai
2 3 4 5 6 7 8 9
1
O,%
1.3
1.7 7.8
4.1
1.7 1.7
1,4 2.0
2.0 1.9
1.9 0.7
1.1 2.7
0.3 29.0 28.6 2.8 4.0 13.0
0.9 O,% 0,2 0.2 91,8 68.1 26.1 0.2 26.7 67.4
1.4 1.1
2.0 1.9
28.8 28.8 2.8 8.9 %,9
0.8 0,8 0.2 0,2 91.4 46.6 46.6 0.2 46.9 46.9
1.7 1.7
2.2
9.1 1.1
0.0
6.7 0.7
0.1
9.3 1.1
0.1
39.9 39.9 2.9 2.9 1.2 1.2 1.1 1.1
0.0
0.2
41.3 41.7 2.8 2.9 1.8 0.7 0.6 1.7
7.7 10.2 0.8 2.8 0.0 1.2 5.1 21.2 25.2 23.6
0,l
100.0
100.0
100.0
100.0
50.0 60.0 60.0 60.0
0.7 0.9 0.1
17.1 10.6 60.0
6.1 6.1 17.1 2.6 2.6 29.2 29.2 10.6 50.0 60.0 50.0 1.4 1.4 0.7 0.3 0‘3 0.9 1.6 1.6 0.1
8.4 10.4 10,4 0.7 0.7 0.4 3.6 3.6 20.8 23.7 23.7
6.3 11.6 0.6 0.0
9.1 1.1
8.4 0.4
0.0
0.2
1.7
0.2
1.7
0.3 0.3 50.0 50.0 3.5 14.7 14.7
6.6 l&O
6.3 10.0 0.3
0.3
0.3 0.1 93.6 0.7 1.1 12.0 83.9 0.1 0.4 1.2 0.2 81.4 12.6 2.2 0.2
2,9
26.6 17.6 10.4 16.5
2.3 0.2 0.6 0.4
0.8
1.1
426
because, according to Table 3, the rocking frequency is dependent on the force constants fz6 and fS6, whereas the wagging and twisting frequencies depend on the constants f2 and f3_ The latter force constants would be expected to be smaller since they represent interactions with more distant atoms than the nearest neighbours. In other respects the results are analogous to those for the tetrahedral environment. C, site symmetry
In the absence of symmetry restrictions, it is difficult to form a clear view of the general situation. The three rotational vibrations may in principle be a complete mixture of rocking, wagging and twisting. It is found, however, that a rocking mode is always obtained unless an extremely asymmetric force field is used. The remaining two rotational vibrations can be described as wagging and twisting only in those cases where the force constants (fz6, f2,, fx6 and f3, in the tetrahedral; fz6, fJ6, f2 and f3 in the trigonal case) have values in accordance with the constraints for the C,, Cz or C, (oYz) symmetries, as found in Table 2. Otherwise a mixture of wagging and twisting occurs. The effects
of non-linear hydrogen
bonds
The effects of non-linear hydrogen bonds are conveniently treated in three steps. In the first step, the hydrogen-bond acceptor atoms (4 and 5 in Fig. 1) are placed in the plane of the water molecule. The largest effects are then noted in the rocking mode. Both the frequency and the isotopic ratio ~n,~/~n,~ associated with this mode will increase relative to those for linear hydrogen bonds. The effects are similar to those obtained when the force constants fz5 and fa4 are increased. This is because the motion of the hydrogen atoms in the rocking mode are no longer perpendicular to the hydrogen bonds so that the force constants fz4 and fs5 will alsoaffect the rocking mode. In the second step the acceptor atoms are allowed to be out of the molecular plane but have their projections onto the plane restricted to lie along the O-H directions_ An increased mixture of rocking and twisting or wagging (or both) results from this. Finally, simultaneous removal of both restrictions results, naturally enough, in a combination of all effects observed earlier for the first two cases. The effect of a non-stationary
environment
The inclusion of a non-stationary environment in our model increases the number of vibrational degrees of freedom from 9 to 21 in the tetrahedral and from 9 to 18 in the trigonal case. We have found, however, that it is still possible in most cases to select 9 vibrational modes which are
427
predominantly water vibrations and reminiscent of the modes obtained in the stationary environment. In the C, site symmetry, the degeneracy of wagging and twisting is lifted, with a frequency separation of typically 50 cm-“. An increased splitting of wagging and twisting is observed for other site symmetries. The large difference found earlier between the wagging and twisting frequencies on the one hand and the rocking frequency on the other is retained for the case of the tetrahedral environment, however. The frequency ratios ~n,~/~n,~ are lowered to fall in the range 1.35-1.40 for twisting and wagging and 1.25-1.40 for rocking. These values agree well with experimentally observed values [ 4,5]. The translational vibration along the y-axis which could be alternatively described as a rotation about an axis through the hydrogen atoms can be changed such that the rotation axis moves towards or away from the oxygen atom. The frequency ratio z+ro/~n,o then increases from 1.00 to approximately 1.03. The ratio for the translation along the x-axis is lowered to approximately 1.06, Comparison
with other calculations and experiments
Other calculations have been made of the rotational modes of water molecules [5-73. In these calculations it has been assumed that the modes are pure rotational vibrations about the inertial axes, and no translational or internal modes have been allowed for. In one case 163, potential energy functions of the Lippincott-Schroder and Lennard-Jones type were used together with the electrostatic energy calculated from formal point charges on the atoms, In a model calculation of this type, however, we believe that the use of empirical potential energy functions gives no clear picture of the general vibrational situation. The aim of the present study is in any case to acquire some idea of the form of the normal modes and the relative magnitudes of the frequencies, rather than of the nature of the forces between the atoms. Before any comparison can be made with experiment, the limitations and approximations of the model calculations must be considered. (1) A very simplified quadratic force field has been used since only the nearest neighbour atoms have been involved in defining the force constants. (2) The environment atoms beyond the nearest neighbours are stationary. (3) The harmonic rectilinear approximation is used for the motions of the atoms. (4) Only one water molecule has been used. In cases where strong interactions exist between the water molecule and its nexbnearest neighbours, some of our results may be altered because of the approximations implied by point (I) above. We do not believe that points (2) and (3) can affect the results significantly. Point (4) will have an influence when two or more water molecules are directly bonded to each other, In such a case, a vibrational coupling can occur between the water molecules.
428
Experimental data for comparison purposes can be obtained from IR and Raman spectra and from neutron inelastic scattering. It is very difficult, however, to make unambiguous assignments of the external vibrations of water molecules. Single crystal work is needed and, in the case of IR and Raman spectroscopy, polarized radiation. In making assignments of rotational vibrations using IR data, it is necessary to consider the changes in the dipole moment, since it is these which determine the band intensities. In the case of wagging and rocking modes, the permanent dipole moment of the water molecule is moving, whereas only changes in the induced dipole moment contribute to the intensity of the absorption baud for the twisting mode. For the strongly mixed wagging and twisting modes found for C,(oxt) symmetry, the permanent dipole moment is moving. The conclusion here is that, in cases where the vibrations are well described as wagging, twisting and rocking, we would expect two stronger bands corresponding to wagging and rocking, and one weaker, possibly invisible, band corresponding to twisting. In cases where we have a strong mixture of wagging and twisting, we would expect three bands of appreciable intensjty. A large group of compounds containing trigonally bonded water molecules
is found among transition metal complexes. A normal coordinate analysis has been performed [8] for such complexes containing eight or four water molecules coordinated either octahedrally or in a square planar way to the transition metal ion. It was found that the rocking mode frequency is higher than those for wag and twist. This is in agreement with the present results. Another example of trigonal environment is that occurring in MCI*. 2H20 (M = Co, Fe, Mn) [9]_ Here too the rocking mode frequency appears to be higher than that of the other two rotational modes. An interesting observation from the spectra of these compounds is that the translational vibration in the z direction (in ref. 9 denoted as the M-O stretching vibration) has a higher frequency than the wagging vibration. Since we have restricted the translational vibrations to occur below 350 cm-’ and the rotational vibrations above 400 cm-’ such a situation could not be observed in our calculations.
On the other hand, it can be seen from Table 3
that the wagging and twisting vibration frequencies and the frequency of the translational vibration in the z direction depend on different force constants. The experimentally observed results can thus readily be reproduced in the calculations. Another case of trigonal environment is found in K&uC14 2Hz0 [lo] where rocking is found at the highest wavenumber, thus supporting our general conclusion. The position of the H (or D) out-of-plane mode between twisting and wagging is also in perfect agreement with our results. This fact has been used to estimate wavenumbers for the non-observed twisting vibrations in K,HgCl, - H,O [l.l] wh ere the water molecule is coordinated tetrahedrally. As expected, rocking is found at the lowest wavenumber in this compound. l
429
An interesting question in the light of the model calculations is whether any correlation exists between hydrogen-bond strength and rotational vibration frequencies of water molecules. Hydrogen-bond strength would best be expressed by the force constants fz4 and fS5. These constants, however, make a negligible or zero contribution to the potential energy distributions (Table 3) of the rotational vibrations, since the hydrogen atoms move perpendicular (or nearly so) to the hydrogen bonds. There is thus no direct relation between the quantities in question, but as will be shown, an indirect correlation can exist. The force constants have been varied independently in the model calculations in order to see whether any general conclusions can be drawn regarding the vibrations of the water molecule. However, if a potential energy function of some kind is assumed (e.g. a Lennard-Jones potential combined with some form of electrostatic potential), the force constants become strongly correlated. Strong hydrogen bonds imply that large charges must be used on the hydrogen atoms in a point charge representation of the electrostatic potential_ This will lead to large force constants for all interactions between the hydrogen atoms and atoms in the environment. This will lead to high vibrational frequencies for all intermolecular water vibrations including the rotationaI vibrations. From this reasoning, we expect a positive correlation between hydrogen-bond strength and rotational vibration frequencies of the water molecule. A remarkable result of the calculations is the prediction that the approximate translational vibration in the y direction can contain a larger oxygen than hydrogen motion. To examine this prediction experimentally we have investigated 1121 the IR spectra of K2C204- HZ0 using substitutions with D20 and H2180. A band was found at 97.9 cm-’ which had a ratio Y~,~/v~,~ = 1.012 and z~u,~/vu~~a0 = 1.043. This clearly suggests a larger oxygen-than hydrogen motion. A further discussion of this will appear in a forthcoming publication 112 1. CONCLUSIONS
(1) In a tetrahedral environment, the rocking mode of the water molecule is the lowest lying rotational mode; whereas in a trigonal environment, the rocking mode frequency is higher than those of the other two modes. (2) In the C,, C, and C, (a,,) site symmetries, the water molecule has rotational modes which can be described as twisting and wagging modes. In the C, (ox*) symmetry, a strong mixture of wagging and twisting is obtained. (3) The probability for pure twisting and wagging to occur in C1 symmetry is low. (4) The isotopic ratio vH,o/~HDO for rocking falls in the range 1.15-1.22. Two out-of-plane modes for HDO are produced, one being essentially a hydrogen and the other a deuterium motion. The hydrogen mode has a frequency between those of wag and twist for HZ0 and the deuterium mode a frequency between those of wag and twist for DzO.
430
(5) Model calculations are a great aid in making the assignments of the rotational and translational vibrations of water molecules in crystal hydrates, and provide an insight into the nature of the vibratory motions of such molecules. ACKNOWLEDGEMENTS
We would like to thank Prof. B. Pedersen at the University of Oslo for making his results available to us prior to publication and for valuable discussions. We would also like to thank Drs. B. Berglund, J. Tegenfeldt and J. Thomas for valuable discussions. We are also indebted to Prof. I. Olovsson for the facilities he has placed at our disposal. This work has been supported by grants from the Swedish Natural Science Research Council which are hereby gratefully acknowledged. REFERENCES 1 B. Pedersen, Acta Crystailogr. Sect. B, 31 (1975) 874. 2 M. Faik and 0. Knop, in F. Franks (Ed.), Water, A Comprehensive Treatise, Vol. 2, Plenum Press, New York-London, p_ 80-81. 3 W. D. Gwinn, J. Chem. Phys., 55 (1971) 477. 4 H. D. Lutz, H.-J. Kiiippel, W. Pobitschka and B. Baasner, Z. Naturforsch. Teil. B, 29 (1974) 723. 5 J. van der Elsken and D. W. Robinson, Spectrochim. Acta, 17 (1961) 1249. 6 C. L. Thaper, A. Sequeira, B. A. Dasannacharya and P. K. Iyengar, Phys. Status Solidi, 34 (1969) 279. 7 Y. S. Jain, Solid State Commun., 17 (1975) 605. 8 I. Nakagawa and T. Shimanouchi, Spectrochim. Acta, 20 (1964) 429. 9 K. Ichida, Y. Kuroda, D. Nakamura and M. Kubo, Spectrochim. Acta Part A, 28 (1972) 2433. 10 G. H. Thomas, M. Falk and 0. Knop, Can. J. Chem., 52 (1974) 1029. il M. Falk and 0. Knop, Can. J. Chem., 55 (1977) 1736. 12 A. Eriksson and J. Lindgren, to be published.
MODEL CALCULATIONS MQLECULES
Printed in The Netherlands
OF THE VIRRATIONS OF BONDED WA’IZR
A. ERIKSSON and J. LTNDGREN Institute of Chemistry,
University
crf Uppsala, Box 531, S-751 21 Uppsalu (Sweden)
(First received 26 September 1977; in revised form 13 January 1978)
ABSTRACT Model calculations have been made of the vibrational frequencies and normal modes of a water molecule vibrating in a combined internal and external field. A constant internal force field has been used together with an external central force field from four or three nearest-neighbour atoms to the water molecule. These neighbour atoms have been arranged either tetrahedrally or trigonally around the water molecule. The external force field has been further restricted by the use of five possible site symmetries for the water molecule, C lv, C,, Cs(Q %,.), C,(CF~~)and C,. A series of calculations have been made where the external force constants have been varied within the range l-80 Nrn-. The nine calculated normal modes can be divided into three groups: intra-molecular, rotational and translational vibrations. Among the rotational vibrations it is found that, in the tetrahedral environment, the rocking mode occurs at lower frequencies than the twisting and wagging modes, whereas the opposite occurs for the trigonal environment. Frequency ratios have been calculated using the isotopic species H,O, D,O, HDO and H,“O. The twisting and wagging modes have the V~&PD,Q ratio in the range 1.35-1.41 and the rocking mode in the rauge 1.26-1.41. XNTRODUCTION
In a recent paper [l] a model calculation has been made of the vibratory motions of a water molecule in a condensed phase, A normal coordinate analysis was made using a combined internal and isotropic external force field. In the present work, we have extended this calculation to include the effect on the vibrations of different anisotropic external force fields. We have thus tried to simulate more closely the typical environment of a water molecule in a crystalline hydrate. Additional complications arise on the introduction of anisotropic external force fields; in the isotropic case the same numerical value can be used for all external force constants. The main problem is to find a systematic way to represent the numerous situations possible. In the following, we will assume that the external fo;cce field of the water molecule is produced mainly through interactions between the water molecule and its closest neighbour atoms. The long-range electrostatic interactions between the water molecule and more distant atoms are assumed to be smaller and to contribute isotropically to the external force field.
418
Xn a review [2] of the structures of crystalline hydrates, it was found that, in about 95% of the 683 hydrates considered, the water molecule is surrounded by either four or three closest neighbour atoms. These are arranged in a tetrahedral or planar trigonal manner around the water molecule. Two neighbour atoms are involved in hydrogen bonds with the hydrogen atoms of the water molecule, and the remaining atoms are involved in interactions with the oxygen atom. These latter interactions can be of the hydrogen bond or ionic type. We concentrate our attention, therefore, on these two cases and let the external potential energy be represented by a central force field consisting of atom-atom interactions between an atom in the water molecule and one of the four (or three) nearest-neighbour atoms. The treatment is further systematized by including the different site symmetries possible for the water molecule. Besides the CzVsite symmetry, there are four additional lower symmetries C1, C, (ox=), C, (a,=) and C1. The two symbols ~~~ and uyz s&nit-r mirror planes in and perpendicular to the water molecule plane, respectively. A systematic variation of the external force constants is thus made using the force field as described above and the symmetry restrictions required by the site symmetries_ In the calculations, the internal force field is held constant since our purpose is to study the effect of the external force field on the vibrations of the water molecule. NORMAL
COORDINATE
ANALYSIS
Calculation procedure A computer program has been used which is a modified version of that originally written by Gwinn [3]. In the present version, the force constants and the normal modes can be defined using three types of displacement coordinates; cartesian coordinates, valence coordinates (changes in distances and angles) and rigid body coordinates (Eckart coordinates). The procedure used in the present calculations differs from that used by Pedersen [l] chiefly in that we have also introduced point atoms to represent the environment of the water molecule. With a few exceptions discussed below these atoms are not allowed to vibrate, however, but are used only to define an external force field in which the water molecule vibrates. The procedure used to suppress the vibrational motion of the environment is as follows. Let x bc a column matrix of Cartesiandisplacement coordinates, where the first nine components refer to the atoms of the water molecule. The valence coordin&es are contained in a column matrix r. The relation between xandris r=B’x where the transformation matrix B’ is obtained as described by Gwinn [ 31. The to&Mpotential energy Vtot for both the water molecule and the
419
environment can now be written 2 J&t =fF,r=
t ti’ F, B’ x
where F, is a force constant matrix. Since the environment will not be allowed to move, we want to remove from this expression the contributions associated with displacements of the atoms in the environment. The remaining potential energy for the water molecule alone is wherefx,lo)i=Xj,Bii=Bi:-forj=l, 2, . . ..9a11dFn.o=gF,B.Thatisto say XH,~ contains the first nine elements of x and B contains the first nine columns of B’_ The potential energy for the water molecule is now expressed in cartesian displacement coordinates, and the normal coordinate calculations then follow the procedure given by Gwirm [33. Selection of the force field parameters The internal force field of the water molecule has been chosen to give reasonable stretching and bending frequencies. This force field, which has remained fixed in all calculations, has stretching force constants f,, and fr2 equal to 650 Nm-’ and bending force constant f, equal to 70 Nm rad-‘. A fixed water geometry has been used with rl = r2 = 1.00 ft and ar= 106.3”. The Cartesianpositional coordinates for the water molecule and the surrounding atoms are found in Table 1. The cartesian coordinate system is defined in Fig. 1, The external force field consists of stretching force constants fij between an atom in the water molecule and an atom in the environment, where i and j are the numbers of the atoms. This numbering for the tetrahedral and trigonal environments is found in Fig. 1. The force constant matrix F, is thus a diagonal matrix with force constants f,,, fp,, f, and the ffj’s on the diagonal, The different site symmetries are obtained by using the constraints on the force constants found in Table 2. For the case of the planar trigonal TABLE 1 Positional Cartesian coordinates of atoms in the water molecule and the surrounding atoms in the tetrahedral and triaonal environment. The cartesian axes are defined in Trigonal
Tetrahedral Atom 1 2 3 4 5 6 7
;
y
2
0.0 0.8 -0.8 2.2 -2.2 0.0 0.0
0.0 0.0 0.0 0.0 0.0 2.3 -2.3
z-6” 0:6 1.6 1.6 -1.6 -1.6
Atom
f 3 4 5 6
x 0.0 0.8 -0.8 2.2 -2.2 0.0
Y z-0” 0:O 0.0 0.0 0.0
z
z-: 0:6 1.6 1.6 -2.8
420
‘t.6 4’ /
I
rr7
b
a Fig. 1. Water mclecule
in tetrahedral
(a) and trigonal (b) environment.
TAESLE 2 Force constants (Nm-I) in the different equal are constrained by symmetry
environments
and site symmetries.
Constants
put
Tetrahedral
C zv
10
= f,,
40
=
IO
=
=
20 20
30
20
f,,
5
CT
5
25
f*e 35
c, (us=) q.(~TXL)
= fz’II = fzs
10 5
= 10
20 40
20 =f,6
20 20
5 3
40 40
30 = ft,
=
5
10 5
= f,, lo
20 20
10 8
30 20
5 3
30 30
10 12
f,, f,,
f,,
=
= f,,
= f,,
= f14
= f,, = f2.s
= f,, = f,,
10
20
= f,,
=fis
= f,,
30
5
30
30
10 10
fzs
= f,, = f,,
Trigonal
% Cs (axz)
environment (Fig. l), it is not possible to define out-of-plane force constants. We have therefore introduced additional force constants fI, fi and f3 for the Cartesiany displacement coordinates of the oxygen and hydrogen atoms in the water molecule. These constants can be said to represent interactions to more distant atoms above and below the water molecule plane. In so doing, the site symmetries CzV, C, and C, (a,,=) become indistinguishable and only two different cases remain for the planar trigonal case, Cpvand C, (cJ,,). A large number of calculations have been performed in which the force constants fii (and fl, f2 and f3) have been assigned various values in the range l-80 Nm-I. These values have been further restricted, however, to give
421
frequencies in the range 400-900 cm-’ for the mainly rotational vibrations of the water molecule_ Similarly, the predom~~tly translational vibrations are rest&ted to have frequencies in the range 50-350 cm-‘, The ranges have been chosen to correspond to experimentally observed frequencies. In some calculations the positional coordinates of the atoms representing the environment have also been changed from the values given in Table 1. This will result in non-linear hydrogen bonds. fn order to have a basis for a discussion, we have chosen to give the results of the normal coordinate analysis for one representative case of each symmetry and environment. The force constants are given in Table 2 end the frequencies, symmetry types and normal modes in Figs. 2 and 3. The potential energy distributions are given in Table 3. To investigate the situation when the closest neighbour atoms are no longer stationary but are allowed to vibrate, we have also performed a limited number of calculations using the following extended expression for the potential energy 2 ‘Vtot=tB’F,B’x+ii:F‘;x where F: is a diagonal force cons~t matrix. The first nine eIements on the diagonal are equal to zero and the remaining elements are equal to f Nm-‘. F: can thus be said to represent the effect of the next-nearest neighbour atoms on the nearest neighbour atoms to the water molecule. The masses of the neighbour atoms and the force constant f have been varied in these calculations. c2
:rcquancyl-
km-'I
mode
km-‘1
symmewY _
169(B)
.‘Yes_
---
169 iA’)
222 (A’)
222 IB) 278 (A]
708 {A]~
‘I 4
I654
!i :~ Frequency
Normal
symma,ry -_
T-
IA I
3501 (A 1 3550 18 1
291 (A’)
764 IA’)
1659 (A‘1 3509b’l
3558 iA”
i Fig. 2. Frequencies, symmetry types and normal modes for the tetrahedral environment.
422
Normst moda
Normal
mods
Fig, 3. Frequencies, symmetry types and normal modes for the trigonal environment. RESULTS
AND DISCUSSION
The discussion of the results will be based on the examples of calculations presented in Table 3 and Figs. 2 and 3. The potential energy distributions found in Table 3 give the contributions to the frequencies from each force constant. These potential energy distibutions can also be used, however, to predict the change in frequency of a particular mode when one or several force constants are changed. The empty positions in Table 3 correspond to zeros or values which are very small. These values remained small throughout the calculations. The rotational vibrations are usually assumed to occur about the principal inertial axes [4,5f (passing through the centre of gravity) of the water molecule. Our results show that this is not generally true, however. The rotational modes, as found in the calculations, are linear combinations of rotations about the inertial axes, and also contain some translational motion. Nevertheless, each of these modes can, in many cases, be approximately described as rotations about one inertial axis. In such cases, these modes may be described equally well as rotations about one of the axes x, y or z as defined in Fig. I. We will refer in the following to the rotational vibrations as twisting (rotation about the z-axis), wagging (rotation about the x-axis) and rocking (rotation about the y-axis). fn those cases where twisting, wagging or rocking do not properly describe the rnodes, we shall use such expressions as “a mixture of twisting and wagging”_
423
Czp C,, C, fa,,)
and 42,{a,,)
site symmetries
In Figs. 2 and 3, ah vibrations occur either witbin the plane of the water molecule @z-plane) or perpendicular to it. This is not generally true, however. We can have, for example, for the case of C, (oYz) symmetry a rather strong mixture of rocking and twisting. In this case, the atoms move simultaneously in and perpendicular to the plane within a single normal mode. For this situation to occur, the force constants fz6 and fs6 must have values which are very different from f2, and f3,. A difference of almost a factor three is needed to produce any appreciable mixture. Tetrahedral environment
For all symmetries the rocking vibration always occurs at a lower frequency than the other rotational modes. In the CpVsymmetry, the twisting and wagging modes are degenerate. This degeneracy is lifted when the symmetry is lowered. Two different situations occur: in the C2 and C, (Ok=)symmetries, the modes can be described as twisting and wagging; but in the C, (ozz) symmetry, a strong mixture is obtained and the modes become, as seen in Fig. 2, a movement of one or the other of the hydrogen atoms out of the xz-plane. The two modes can have very different frequencies. The isotopic ratios Yn,o/~n,~ for the wagging and twisting modes are 1.41. This ratio varies between 1.25 and 1.41 for the rocking mode, and is mainly due to changes in the force constants fis and f 34. SmalI values of the constants give a low ratio and vice versa. A low ratio results from a translational component arising in the x direction of the rocking mode. When only one hydrogen is substituted with deuterium to produce an HDO molecule, it is found that the twisting and wagging modes become strongly mixed. Two modes are produced with, in the first one, a hydrogen and in the second one a deuterium motion out of the molecular plane. The frequency of the hydrogen mode is between those of wag and twist of HZO, and the frequency of the deuterium mode between those of wag and twist of DzO. If the hydrogen mode frequency is closer to that of wag of H20, the deuterium mode frequency is closer to that of twist of D20 and vice versa. The frequency ratio for the rocking modes vn,o/~nn~ is found to fall in the range 1.15-1.22. The translational vibrations are, in all cases, rather well described as vibrations along the Cartesianaxes. The translational mode along the y-axis is interesting since it can alternatively be described as a rotational vibration about an axis through the two hydrogen atoms (see Figs. 2 and 3). The frequency ratios trH,O/~n,o are close to 1.05 and 1.08 for translations along the z- and x-axes, respectively, and 1.00 for the translation along the y-axis. The E&i0 VH, 1 8&q 260 is about 1.05 for all translational modes. Trigonal environment
The most interesting point here is the possibility for the rocking mode to have the highest frequency of the three rotational modes. This is so
1 2 3 4 6 6 7 8 9
1 2 3 4 6 6 7 8 9
ca (QJ
2 3 4 6 6 7 8 9
1
Mode number
c2
1V
c
Symmetry
Tetrahedral
1.2 1.8 1.1
0.2 46.8, 47.0
0,2 46.8 47.0
0.2 47.0 47.3
0.2 47,o 47.3
1.2 1.8 1.1
1.2 1.6 0.8
0.2 47.0 47.3
0.2 47.0 47.3
1.2 1.6 0.8
1.4 0.9
x.2
f,,
1.4 0.9
1.2
I’,,
92.4 0.2
0.2
2.9
93.1 0.2
0.2
2.3
93.1 0.2
1.9
f,
1.3 1.3
22.6 6.6 0.1
1.3 1.3
22.9 7.2 0.1
1.3 1.3
6.1 0.1
22.8
fi4
1.3 1.3
22.6 6.6 0.1
1.3 1.3
25.9 7.2 0.1
1.3 1.3
6.1 0.1
22.8
fs5
1.1
13.1 0.4 9.6
0.1 1.1
12,6 0.6 lOa8
1,l
0.4 10.7
12.6
f,,
1.1
13.1 0.4 9.6
0.1 1.1
12.6 0.6 10.8
1.1
0.4 10.7
12.6
fad
11,9 3.9 0.2
12.0 4.3 0.2
3.j 0.3
12.0
f14
f,,
11.9 3.9 0.2
12.0 4.3 0.2
7.4
60.0
8.1
60.0
60.0 3.7 - 13.8 0.3
12.0
fis
7.4
60.0
8.1
50.0
50.0 13.8
fi7
0.6 11.6 12.8 28.6 32.2 1.4 1.0 0.9
0.8 16.7 26.8 20.6 16.3 0.8 0.6 0.6
11.9 19.0 2580 26.0 1.0 0.7 0.7
0.7
fib 0.7
0.7 17.0 26.3 21.4 17.6 0.1 0.7 0.7
0.6 10.1 11.2 29.2 33.6 1.3 0.9 0.8
11.9 19.0 26.0 26.0 1.0 0.7 0.7
f,,
0.7
0.5 11,6 12.8 28.6 32.2 1.4 1.0 0.9
0.6 10.1 11.2 29.2 3386 1.3 0.9 0,8
11.9 19.0 26.0 26.0 1.0 0.7 0.7
f3,
0.7
0*7 17.0 26.3 21.4 17.6 0.1 0.7 0.7
0,8 16.7 26.8 20.6 16,3 0.8 0.6 0.6
11.9 19.0 26,O 26.0 1.0 0.7 0.7
f3,
Potentialenergy distributionsin the tetrahedraland trigonalenvironments.The normal modes are numbered in order of increasing frequency
TABLE 3
c1V
Trigonai
2 3 4 5 6 7 8 9
1
O,%
1.3
1.7 7.8
4.1
1.7 1.7
1,4 2.0
2.0 1.9
1.9 0.7
1.1 2.7
0.3 29.0 28.6 2.8 4.0 13.0
0.9 O,% 0,2 0.2 91,8 68.1 26.1 0.2 26.7 67.4
1.4 1.1
2.0 1.9
28.8 28.8 2.8 8.9 %,9
0.8 0,8 0.2 0,2 91.4 46.6 46.6 0.2 46.9 46.9
1.7 1.7
2.2
9.1 1.1
0.0
6.7 0.7
0.1
9.3 1.1
0.1
39.9 39.9 2.9 2.9 1.2 1.2 1.1 1.1
0.0
0.2
41.3 41.7 2.8 2.9 1.8 0.7 0.6 1.7
7.7 10.2 0.8 2.8 0.0 1.2 5.1 21.2 25.2 23.6
0,l
100.0
100.0
100.0
100.0
50.0 60.0 60.0 60.0
0.7 0.9 0.1
17.1 10.6 60.0
6.1 6.1 17.1 2.6 2.6 29.2 29.2 10.6 50.0 60.0 50.0 1.4 1.4 0.7 0.3 0‘3 0.9 1.6 1.6 0.1
8.4 10.4 10,4 0.7 0.7 0.4 3.6 3.6 20.8 23.7 23.7
6.3 11.6 0.6 0.0
9.1 1.1
8.4 0.4
0.0
0.2
1.7
0.2
1.7
0.3 0.3 50.0 50.0 3.5 14.7 14.7
6.6 l&O
6.3 10.0 0.3
0.3
0.3 0.1 93.6 0.7 1.1 12.0 83.9 0.1 0.4 1.2 0.2 81.4 12.6 2.2 0.2
2,9
26.6 17.6 10.4 16.5
2.3 0.2 0.6 0.4
0.8
1.1
426
because, according to Table 3, the rocking frequency is dependent on the force constants fz6 and fS6, whereas the wagging and twisting frequencies depend on the constants f2 and f3_ The latter force constants would be expected to be smaller since they represent interactions with more distant atoms than the nearest neighbours. In other respects the results are analogous to those for the tetrahedral environment. C, site symmetry
In the absence of symmetry restrictions, it is difficult to form a clear view of the general situation. The three rotational vibrations may in principle be a complete mixture of rocking, wagging and twisting. It is found, however, that a rocking mode is always obtained unless an extremely asymmetric force field is used. The remaining two rotational vibrations can be described as wagging and twisting only in those cases where the force constants (fz6, f2,, fx6 and f3, in the tetrahedral; fz6, fJ6, f2 and f3 in the trigonal case) have values in accordance with the constraints for the C,, Cz or C, (oYz) symmetries, as found in Table 2. Otherwise a mixture of wagging and twisting occurs. The effects
of non-linear hydrogen
bonds
The effects of non-linear hydrogen bonds are conveniently treated in three steps. In the first step, the hydrogen-bond acceptor atoms (4 and 5 in Fig. 1) are placed in the plane of the water molecule. The largest effects are then noted in the rocking mode. Both the frequency and the isotopic ratio ~n,~/~n,~ associated with this mode will increase relative to those for linear hydrogen bonds. The effects are similar to those obtained when the force constants fz5 and fa4 are increased. This is because the motion of the hydrogen atoms in the rocking mode are no longer perpendicular to the hydrogen bonds so that the force constants fz4 and fs5 will alsoaffect the rocking mode. In the second step the acceptor atoms are allowed to be out of the molecular plane but have their projections onto the plane restricted to lie along the O-H directions_ An increased mixture of rocking and twisting or wagging (or both) results from this. Finally, simultaneous removal of both restrictions results, naturally enough, in a combination of all effects observed earlier for the first two cases. The effect of a non-stationary
environment
The inclusion of a non-stationary environment in our model increases the number of vibrational degrees of freedom from 9 to 21 in the tetrahedral and from 9 to 18 in the trigonal case. We have found, however, that it is still possible in most cases to select 9 vibrational modes which are
427
predominantly water vibrations and reminiscent of the modes obtained in the stationary environment. In the C, site symmetry, the degeneracy of wagging and twisting is lifted, with a frequency separation of typically 50 cm-“. An increased splitting of wagging and twisting is observed for other site symmetries. The large difference found earlier between the wagging and twisting frequencies on the one hand and the rocking frequency on the other is retained for the case of the tetrahedral environment, however. The frequency ratios ~n,~/~n,~ are lowered to fall in the range 1.35-1.40 for twisting and wagging and 1.25-1.40 for rocking. These values agree well with experimentally observed values [ 4,5]. The translational vibration along the y-axis which could be alternatively described as a rotation about an axis through the hydrogen atoms can be changed such that the rotation axis moves towards or away from the oxygen atom. The frequency ratio z+ro/~n,o then increases from 1.00 to approximately 1.03. The ratio for the translation along the x-axis is lowered to approximately 1.06, Comparison
with other calculations and experiments
Other calculations have been made of the rotational modes of water molecules [5-73. In these calculations it has been assumed that the modes are pure rotational vibrations about the inertial axes, and no translational or internal modes have been allowed for. In one case 163, potential energy functions of the Lippincott-Schroder and Lennard-Jones type were used together with the electrostatic energy calculated from formal point charges on the atoms, In a model calculation of this type, however, we believe that the use of empirical potential energy functions gives no clear picture of the general vibrational situation. The aim of the present study is in any case to acquire some idea of the form of the normal modes and the relative magnitudes of the frequencies, rather than of the nature of the forces between the atoms. Before any comparison can be made with experiment, the limitations and approximations of the model calculations must be considered. (1) A very simplified quadratic force field has been used since only the nearest neighbour atoms have been involved in defining the force constants. (2) The environment atoms beyond the nearest neighbours are stationary. (3) The harmonic rectilinear approximation is used for the motions of the atoms. (4) Only one water molecule has been used. In cases where strong interactions exist between the water molecule and its nexbnearest neighbours, some of our results may be altered because of the approximations implied by point (I) above. We do not believe that points (2) and (3) can affect the results significantly. Point (4) will have an influence when two or more water molecules are directly bonded to each other, In such a case, a vibrational coupling can occur between the water molecules.
428
Experimental data for comparison purposes can be obtained from IR and Raman spectra and from neutron inelastic scattering. It is very difficult, however, to make unambiguous assignments of the external vibrations of water molecules. Single crystal work is needed and, in the case of IR and Raman spectroscopy, polarized radiation. In making assignments of rotational vibrations using IR data, it is necessary to consider the changes in the dipole moment, since it is these which determine the band intensities. In the case of wagging and rocking modes, the permanent dipole moment of the water molecule is moving, whereas only changes in the induced dipole moment contribute to the intensity of the absorption baud for the twisting mode. For the strongly mixed wagging and twisting modes found for C,(oxt) symmetry, the permanent dipole moment is moving. The conclusion here is that, in cases where the vibrations are well described as wagging, twisting and rocking, we would expect two stronger bands corresponding to wagging and rocking, and one weaker, possibly invisible, band corresponding to twisting. In cases where we have a strong mixture of wagging and twisting, we would expect three bands of appreciable intensjty. A large group of compounds containing trigonally bonded water molecules
is found among transition metal complexes. A normal coordinate analysis has been performed [8] for such complexes containing eight or four water molecules coordinated either octahedrally or in a square planar way to the transition metal ion. It was found that the rocking mode frequency is higher than those for wag and twist. This is in agreement with the present results. Another example of trigonal environment is that occurring in MCI*. 2H20 (M = Co, Fe, Mn) [9]_ Here too the rocking mode frequency appears to be higher than that of the other two rotational modes. An interesting observation from the spectra of these compounds is that the translational vibration in the z direction (in ref. 9 denoted as the M-O stretching vibration) has a higher frequency than the wagging vibration. Since we have restricted the translational vibrations to occur below 350 cm-’ and the rotational vibrations above 400 cm-’ such a situation could not be observed in our calculations.
On the other hand, it can be seen from Table 3
that the wagging and twisting vibration frequencies and the frequency of the translational vibration in the z direction depend on different force constants. The experimentally observed results can thus readily be reproduced in the calculations. Another case of trigonal environment is found in K&uC14 2Hz0 [lo] where rocking is found at the highest wavenumber, thus supporting our general conclusion. The position of the H (or D) out-of-plane mode between twisting and wagging is also in perfect agreement with our results. This fact has been used to estimate wavenumbers for the non-observed twisting vibrations in K,HgCl, - H,O [l.l] wh ere the water molecule is coordinated tetrahedrally. As expected, rocking is found at the lowest wavenumber in this compound. l
429
An interesting question in the light of the model calculations is whether any correlation exists between hydrogen-bond strength and rotational vibration frequencies of water molecules. Hydrogen-bond strength would best be expressed by the force constants fz4 and fS5. These constants, however, make a negligible or zero contribution to the potential energy distributions (Table 3) of the rotational vibrations, since the hydrogen atoms move perpendicular (or nearly so) to the hydrogen bonds. There is thus no direct relation between the quantities in question, but as will be shown, an indirect correlation can exist. The force constants have been varied independently in the model calculations in order to see whether any general conclusions can be drawn regarding the vibrations of the water molecule. However, if a potential energy function of some kind is assumed (e.g. a Lennard-Jones potential combined with some form of electrostatic potential), the force constants become strongly correlated. Strong hydrogen bonds imply that large charges must be used on the hydrogen atoms in a point charge representation of the electrostatic potential_ This will lead to large force constants for all interactions between the hydrogen atoms and atoms in the environment. This will lead to high vibrational frequencies for all intermolecular water vibrations including the rotationaI vibrations. From this reasoning, we expect a positive correlation between hydrogen-bond strength and rotational vibration frequencies of the water molecule. A remarkable result of the calculations is the prediction that the approximate translational vibration in the y direction can contain a larger oxygen than hydrogen motion. To examine this prediction experimentally we have investigated 1121 the IR spectra of K2C204- HZ0 using substitutions with D20 and H2180. A band was found at 97.9 cm-’ which had a ratio Y~,~/v~,~ = 1.012 and z~u,~/vu~~a0 = 1.043. This clearly suggests a larger oxygen-than hydrogen motion. A further discussion of this will appear in a forthcoming publication 112 1. CONCLUSIONS
(1) In a tetrahedral environment, the rocking mode of the water molecule is the lowest lying rotational mode; whereas in a trigonal environment, the rocking mode frequency is higher than those of the other two modes. (2) In the C,, C, and C, (a,,) site symmetries, the water molecule has rotational modes which can be described as twisting and wagging modes. In the C, (ox*) symmetry, a strong mixture of wagging and twisting is obtained. (3) The probability for pure twisting and wagging to occur in C1 symmetry is low. (4) The isotopic ratio vH,o/~HDO for rocking falls in the range 1.15-1.22. Two out-of-plane modes for HDO are produced, one being essentially a hydrogen and the other a deuterium motion. The hydrogen mode has a frequency between those of wag and twist for HZ0 and the deuterium mode a frequency between those of wag and twist for DzO.
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(5) Model calculations are a great aid in making the assignments of the rotational and translational vibrations of water molecules in crystal hydrates, and provide an insight into the nature of the vibratory motions of such molecules. ACKNOWLEDGEMENTS
We would like to thank Prof. B. Pedersen at the University of Oslo for making his results available to us prior to publication and for valuable discussions. We would also like to thank Drs. B. Berglund, J. Tegenfeldt and J. Thomas for valuable discussions. We are also indebted to Prof. I. Olovsson for the facilities he has placed at our disposal. This work has been supported by grants from the Swedish Natural Science Research Council which are hereby gratefully acknowledged. REFERENCES 1 B. Pedersen, Acta Crystailogr. Sect. B, 31 (1975) 874. 2 M. Faik and 0. Knop, in F. Franks (Ed.), Water, A Comprehensive Treatise, Vol. 2, Plenum Press, New York-London, p_ 80-81. 3 W. D. Gwinn, J. Chem. Phys., 55 (1971) 477. 4 H. D. Lutz, H.-J. Kiiippel, W. Pobitschka and B. Baasner, Z. Naturforsch. Teil. B, 29 (1974) 723. 5 J. van der Elsken and D. W. Robinson, Spectrochim. Acta, 17 (1961) 1249. 6 C. L. Thaper, A. Sequeira, B. A. Dasannacharya and P. K. Iyengar, Phys. Status Solidi, 34 (1969) 279. 7 Y. S. Jain, Solid State Commun., 17 (1975) 605. 8 I. Nakagawa and T. Shimanouchi, Spectrochim. Acta, 20 (1964) 429. 9 K. Ichida, Y. Kuroda, D. Nakamura and M. Kubo, Spectrochim. Acta Part A, 28 (1972) 2433. 10 G. H. Thomas, M. Falk and 0. Knop, Can. J. Chem., 52 (1974) 1029. il M. Falk and 0. Knop, Can. J. Chem., 55 (1977) 1736. 12 A. Eriksson and J. Lindgren, to be published.