MOLECULAR PHYSICS, 1999, VOL. 96, NO. 11, 1659± 1665
The e€ ect of pressure on hydrogen bonding in water: IR study of
m OD HDO at pressures of up to 1500 bar
YU. E. GORBATY*, G. V. BONDARENKO, A. G. KALINICHEV and
A. V. OKHULKOV
Institute of Experimental Mineralogy, Russian Academy of Sciences,
Chernogolovka Moscow Region, 142432 Russia
(Received 5 August 1998; revised version accepted 16 November 1998)
A seeming contradiction has been revealed between the behaviour of the shortest intermolecular separation rOO and that of the frequency of the OH vibration of H2 O under compression in liquid water. Decrease of rOO with increasing pressure up to 2 kbar seems to be
evidence of strengthening of the hydrogen bonding, but the increase in the frequency of t OH
H2 O seems contradictory. To clear up this question, measurements of the integrated intensity
of IR absorption for t OD HDO have been made which de® nitely show that the energy of the
hydrogen bonds decreases under compression at least up to 1.5 kbar. The decrease in the
intensity is of about 20% , i.e., essentially more than the estimated accuracy of the measurements. Monte Carlo simulations of the compression of liquid water show that the behaviour of
rOO, t OH, and the integrated intensity can be explained by the bending of hydrogen bonds.
1. Introduction
Studies of the e€ ect of pressure on the hydrogen
bonding in liquid water can give important information
on the properties of this so far enigmatic liquid, and on
the very nature of the phenomenon of hydrogen
bonding. Since the pioneering experiments of the
Neilson group [1, 2] and the outstanding work of Wu
et al. [3] there have not been many experimental studies
of the in¯ uence of pressure on hydrogen bonding in
water [4± 9]. What is worse, often the data on the pressure e€ ect are contradictory: perhaps not surprising,
taking into account the great di culty of conducting
high pressure experiments and the small values of the
observed changes. Thus far it is even hard to predict a
general trend in the pressure e€ ect on hydrogen
bonding. It has been shown, for example, that pressure
a€ ects the pair correlation functions of water in the
same direction as temperature, decreasing, in particular,
the degree of tetrahedral ordering [4]. So, we could
expect the weakening of hydrogen bonds. However,
the shortest separation rOO between hydrogen bonded
water molecules has been found to decrease with pressure [4, 5] up to about 2000 bar, and this fact seems to be
evidence of strengthening of hydrogen bonds in view of
the well known correlation between the shortest separation and the energy of hydrogen bond [10± 13]. Regret-
* Author for correspondence. e-mail: [email protected] u
tably, even the very fact of rOO decreasing with pressure
remains disputable.
In this paper we would like to draw attention to the
ostensible contradiction between X-ray scattering data
[4, 5] and results of high pressure Raman experiments [8,
9]. Using infrared absorption data and Monte Carlo
simulations, we show that there is no contradiction
and the observed phenomena can be explained in the
framework of the earlier suggested model [5].
2. Experimental
The purpose of the experiment was to obtain the
pressure dependence of the integrated intensity for the
OD stretching band t OD of HDO. The spectra of t OD
was obtained using the technique developed by us [14,
15]. The method is based on the use of high pressure
high temperature cells with changeable path lengths. It
allows us to get rid of the most common measuring
errors arising, for example, from distortions of optimal
geometry, light losses due to absorption and re¯ ection
by sapphire windows, overlapping with the spectra of
water vapour and CO2 from atmosphere, the `lens’
e€ ect, etc.
A new custom made cell (to be described in detail
elsewhere) has been used in the experiment. One of the
sapphire windows can be moved with a driving
mechanism attached to the cell. It consists of a lead
screw and a system of levers. The driving system
allows one to change the path length smoothly between
0 and 0.8 mm so that one full revolution of the screw
Molecular Physics ISSN 0026± 8976 print/ISSN 1362± 3028 online Ñ 1999 Taylor & Francis Ltd
http://www.tandf.co.uk/JNLS/mph.htm
http://www.taylorandfrancis.com/JNLS/mph.htm
1660
Yu. E. Gorbaty et al.
head corresponds approximately to a window shift of
40m m.
To obtain a spectrum of a pure substance free from
the aforementioned errors it is su cient to record two
spectra at di€ erent path lengths in two successive runs.
It is not necessary to know the values of path lengths.
However, the di€ erence between them, that is, the shift
of movable window ¢ , must be measured with the
highest possible accuracy. For pure water at a density q
kH2 O =
M log [T 2 ( t ) / T 1 ( t
q ¢1
)],
( 1)
where kH2 O is the absorption coe cient, M is the molecular weight of H2 O, T 1 ( t ) is the transmission spectrum obtained with the larger path length, while T 2 ( t )
is obtained with a shorter path length. Deriving this
simple equation, it is easy to see that corrections for
all the errors mentioned above are cancelled out.
To obtain the absorption coe cient kHDO ( t ) from the
spectra of the HDO solution in H2 O we again have to
measure two transmission spectra of the solution: T 3 ( t )
(with a longer path length) and T 4 ( t ) (with a shorter
path length) with ¢2 the di€ erence in path lengths. Then
log [T 4 ( t ) / T 3 ( t
¢2
)]
= kH2O( t ) cH2O + kHDO( t ) cHDO, ( 2)
where cH2 O and cHDO are the concentrations in mol cm­
of H2 O and HDO in the solution at density q . Solving
equations (1) and (2) simultaneously, we have:
3
kHDO( t
)
log [T 4 ( t ) / T 3 ( t
=
cHDO ¢2
)]­
McH2 O log [T 2 (t ) / T 1 ( t
q cHDO ¢ 1
)].
( 3)
(In this case, the spectra T 1 ( t ) and T 2 ( t ) should be
measured in the same spectral range as the spectra
T 3 ( t ) and T 4 ( t )) .
To measure ¢1 and ¢ 2 a measuring microscope with
spiral scale was used that provided an accuracy of
0.5 m m. It is possible, however, to avoid measuring
¢ 2 using the condition
M log [T 2 ( t ) / T 1 ( t
q ¢1
)]
=
log [T 4 ( t ) / T 3 ( t
cH2 O¢2
)]
( 4)
that is valid for a spectral range over which no absorption from HDO occurs. Taking, x¢ 2 = ¢ 1 , we can calculate x as
x=
McH2 O log [T 2 ( t ) / T 1 ( t
q log [T 4 ( t ) / T 3 ( t )]
)].
( 5)
Because of the noise in the experimental spectra, it is
better to calculate an average value of x using a su cient number of points i
Figure 1. Illustration of the method for obtaining a spectrum
of t OD HDO from four spectra: ( a) spectra of pure water;
( b) spectra of HDO in H2 O solution; and ( c) resulting
spectrum of t OD HDO at a pressure of 1200 bar.
x = ( i2 ­ i1 ) ­
1
i2
i1
McH2 O log [T 2 ( t i ) / T 1 ( t i )]
.
q log [T 4 ( t i ) / T 3 ( t i )]
( 6)
In the experiment described i1 corresponds to
1
1
1
2750 cm­ , i2 to 2850 cm­ , and i2 ­ i1 = 100 cm­ .
Then equation (3) may be rewritten as
kHDO( t
)=
log [T 4 ( t ) / T 3 ( t
cHDO x¢ 1
)]­
McH2 O log [T 2 ( t ) / T 1 ( t
q cHDO ¢ 1
)].
( 7)
Figure 1 demonstrates an example of applying the
algorithm. Note the increasing noise at low wavenumbers due to the very low transmittance of sapphire
1
in this spectral range. The same occurs near 2900 cm­ ,
where the absorption of H2 O increases strongly. Note
also an overcompensation of the CO2 band near
2300 cm­ 1 . It is di cult to allow accurate compensation
for this band since the content of CO2 in the atmosphere
tends to ¯ uctuate.
The spectra of t OD HDO were obtained in the spectral
1
1
range 1900± 2900 cm­ with a step of 1 cm­ using a
Perkin-Elmer 983 ratio recording spectrophotometer.
Regrettably, we can use only a cell designed to work
at high temperatures (up to 500 ë C) in the pressure
E€ ect of pressure on hydrogen bonding in HDO
range restricted by 1500 bar. Therefore, the measurements at room temperature have been made in this
pressure range with a step of 100 bar and with an accuracy of 5 bar.
The integrated intensity of absorption A in the region
of the OD vibration mode of HDO was calculated
according to
A=
t 2 =2800
t 1 =2100
kHDO ( t
) dt
.
( 8)
The solution of 4 mol% HDO in H2 O was prepared
using 99.8 wt% D2 O delivered by the `Isotope V/O’
company (Russia) .
3.
Results and discussion
3.1. Background
Studying the e€ ect of pressure on the structure of
liquid water with the energy-dispersive X-ray di€ raction
technique (EDXD), Okhulkov et al. [5] have found that
the average separation between nearest molecules rOO
exhibits peculiar behaviour. It might be expected intuitively that rOO should decrease as a result of compression. Indeed, it decreases with the pressure rise up
to 2 kbar, as ® gure 2 shows, but at higher pressures it
begins to grow, virtually reaching the initial value at
4± 5 kbar. Some authors [6], however, reckoned this
result as debatable. Indeed, in some previous experiments [1± 3] a very small contraction of rOO was found.
Also computer simulation of the e€ ect of pressure did
not predict an essential decrease in rOO with increasing
pressure [16, 17]. We should note, however, that this
rather subtle e€ ect would be di cult to reveal explicitly
in neutron di€ raction or usual angular scanning X-ray
di€ raction experiments.
The energy-dispersive X-ray di€ raction technique
used by Okhulkov et al. has certain advantages over
the classical X-ray di€ raction method because of the
high statistical accuracy achieved with shorter exposure
times and due to the absence of any contribution from
the container to the scattering intensity. This presents
the possibility of using the long range part of the structure function si( s) (where s is the scattering parameter or
wavefactor and i( s) is the structure factor) to obtain the
value of rOO, using a large number of experimental
points. The procedure is based on the fact that the
remote part of the structure function for certain values
of s is virtually the Fourier transform of the ® rst peak of
the radial distribution function 4p rq 0 g( r) , where q 0 is
the bulk density at given pressure and g( r) is the molecular pair correlation function. The procedure gives an
additional advantage over the usual way to obtain rOO,
from the position of the ® rst peak of the radial distribution function. The ® rst peak of g( r) is overlapped
1661
strongly with the more broad peak at 3.2± 3.3 A [4]
and this results in a shift of the ® rst peak to larger r.
However, if the structure factor i( s) or structure function si( s) is considered, these two distributions do not
overlap in the remote part of inverse space for s higher
1
than 8± 9 A­ . It is well known that the position of the
® rst peak depends on the method of calculation of g( r) .
In particular, modi® cation functions used frequently to
force the structure functions to approach zero within the
experimental range of s inevitably lead to decreasing of
g( r) resolution and, consequently, to the additional shift
of rOO. Therefore, there is not much sense in comparing
absolute values of rOO obtained in a variety of experimental studies. Usually, they di€ er within 0. 05 A, i.e.,
larger than the e€ ect under discussion. More important
is the relative change of rOO observed with variation of
external conditions. Incidentally, Bellissent-Funel and
Bosio [7] seem to con® rm qualitatively the behaviour
of rOO observed by Okhulkov et al. [5].
We ought to mention that the values of r obtained in
[5] are not exactly the separations between the nearest
oxygen atoms. They correspond to the separations
between the centres of electron distributions in the
nearest molecules. However, as may be seen from the
behaviour of the form factors derived by Narten and
Levy [18], the electron distribution of the water molecule
is nearly spherical.
Okhulkov et al. have explained the speci® c behaviour
of rOO in terms of `preferential’ structure ¯ uctuations
taking the form of deformed local structures of the ice
high-pressure polymorphs (similar ideas have been
discussed by other authors [19, 20]). This suggestion
originated from the fact of increasing rOO in the series
ice I ! ice VIII [13, 21]. It was shown [5] with the
analysis of the pair correlation functions that at low
pressures the preferential structure ¯ uctuations of the
type of ice I and ice III prevail while at higher pressure
the short lived molecular con® gurations typical of
distorted structures of higher pressure ice polymorphs
dominate. This presents a reasonable explanation of the
minimum in the pressure dependence of rOO presented in
® gure 2.
The experimentally observed behaviour rOO leads one
to assume that the strength of hydrogen bonds in compressed water increases ® rst and then after 2± 3 kbar
decreases. Such a notion is based on the correlation
between the energy of hydrogen bonding and the OÐ
O separation [11± 13]. On the other hand, this correlation is not well de® ned. If a large number of hydrogen
bonded systems is considered, the scatter of points in the
correlation is very large, so that the correlation between
rOO and the energy of bonding can only be guessed
rather than seen clearly. This fact witnesses that besides
rOO a large number of factors in¯ uence the energy of
1662
Yu. E. Gorbaty et al.
Figure 2. Pressure dependence of the shortest intermolecular
separation rOO between hydrogen bonded water molecules. The line is a guide to the eye.
hydrogen bonds. In this sense there is no direct relationship between rOO and the energy of hydrogen bonds.
Nevertheless, it seems intuitively that the average
energy of bonding should increase as rOO decreases.
The frequency of OH stretching vibrations is also a
strong function of the hydrogen bond strength. The
higher the energy of hydrogen bonding, the lower is
the frequency t OH . So, one ought to expect the behaviour of t OH to be qualitatively the same as that of rOO.
However, as seen in ® gure 3, the pressure trend of t OH is
opposite to the expected one. Here the Raman data on
the maxima of the band position [8] and centres of
gravity [9] for the OH stretching band of water are
shown. It should be noted, though, that because of the
large scatter of experimental points, Walrafen and
Abebe [8] approximated their data with a linear ® t.
Still, a combination of the data from [8] and [9] presented in ® gure 3 plus the behaviour of the halfwidth
of the OH band contour found in the work of Cavaille et
al. [9] de® nitely indicate a singularity in the pressure
range 2± 3 kbar. This circumstance seems to con® rm
the singularity found for rOO. However, taking t OH
and halfwidth as criteria for the strength of the
hydrogen bond in compressed water, one can only
state that up to 2± 3 kbar hydrogen bonds weaken
while at higher pressures they strengthen. So, the question arises: do we have increasing or decreasing the
strength of hydrogen bonding in water under
compression?
Figure 3. Pressure dependence of the frequency of H2 O
stretching vibrations: circles, position of the band maximum [8]; and squares, centre of gravity of the band [9].
3.2. Spectroscopic results
To answer the above question, we have studied the
behaviour of the integrated intensity of absorption by
OD vibrations of HDO. This characteristic is very sensitive to the energy of hydrogen bonds. The integrated
intensity of t OH decreases approximately by a factor of
25± 30 at the transition of water from the solid state to
the gaseous [10], while the frequency shift is of about
10% only. Because of the aforementioned experimental
limitations, we could check only the left descending
branch of the pressure dependence of rOO, shown in
® gure 2. However, even the limited pressure range
1± 1500 bar is enough to reveal the main trend of the
pressure e€ ect on hydrogen bonding at the beginning
of compression.
Figure 4 shows the pressure dependence of the integrated intensity A for t OD HDO. It demonstrates convincingly that A, and consequently the energy of
hydrogen bonds, decrease with the pressure rise. The
decrease in A over the pressure range explored is
20% . This value is much higher than the estimated
error of the intensity measurements ( 5- 6% ) but it is,
in fact, very small, bearing in mind the full range of the
possible variation of the integrated intensity.
The data for low pressures are in a good agreement
with those reported previously [22± 25]. In general, measurement of intensity in any kind of spectroscopy is
always a much more di cult task than analysis of
band positions, halfwidths, or band shape. However,
E€ ect of pressure on hydrogen bonding in HDO
Figure 4.
t
1663
Pressure dependence of the integrated intensity of
HDO. The lines are guides to the eye.
OD
when studying hydrogen bonded systems, the integrated
intensity provides the most explicit information about
the degree or probability of hydrogen bonding in the
system [10]. We were not able, for example, to use for
this purpose the position of the maximum of the OD
band because the scatter of the data was so great in so
narrow a pressure range.
Thus, we have arrived at the conclusion that at least
up to 1.5± 2 kbar hydrogen bonding weakens with compression of liquid water and shortening rOO. It is for
future experiments to reveal what is going on at pressures higher than 2 kbar. Relying upon the data shown
in ® gure 3, we dare to predict that the integrated intensity and, consequently, the strength of hydrogen bonds
should increase.
3.3. Monte Carlo simulations
To explain the seeming contradiction between the behaviour of rOO and the strength of hydrogen bonds with
compression of liquid water, isothermal± isobaric Monte
Carlo simulations have been performed, using the
TIP4P e€ ective site± site potential.
There have been, of course, other attempts to study
the e€ ect of pressure, using MD and MC calculations
[16, 17, 27± 30], but the present study was aimed at a
comparison of the simulated and experimental pair correlation functions [5] during the step by step compression of water. The simulations were performed in
the pressure range 0.001± 10 kbar for the NPT ensemble
of 216 molecules in a cubic cell with periodic boundary
conditions. A detailed discussion of the results has been
given elsewhere [31]. Here we con® ne ourselves to a few
Figure 5. Results of Monte Carlo simulations: ( a) rOO as a
function of pressure; ( b) energy of hydrogen bond versus
pressure; and ( c) the pressure dependence of the average
angle of hydrogen bonds.
particular results most relevant to the problem. We
could not, of course, anticipate good qualitative agreement of the simulated data with experiment. The TIP4P
potential is too sti€ at short molecular distances and
gives essentially even shorter values of rOO at ambient
pressure. This is a well known drawback of many intermolecular water± water potentials used in computer
simulations, and that may be a reason for the low rate
of contraction usually found in computer experiments.
It should be also noted that the phase diagram of TIP4P
water does not correspond quantitatively to the real
diagram of water. Furthermore, it can hardly be
expected that computer simulations are able to reproduce so a complicated a phenomenon as the preferential
variations of the nearest environment (structure ¯ uctuations [5]) involving a large number of water molecules.
Still, e€ ectively computer simulations can help in
understanding and interpreting experimental data.
Figure 5( a) shows the pressure dependence of the
shortest separation between the oxygen atoms of the
neighbouring hydrogen bonded water molecules. For
the sake of simplicity, only geometrical criteria of
hydrogen bonding [32] were used. Despite the experimentally observed minimum, rOO is not reproduced,
and one can see two distinct pressure ranges with the
1664
Yu. E. Gorbaty et al.
di€ erent contraction rates. This may be considered as a
weak re¯ ection of the peculiarity observed in ® gure 2.
However, an even more important ® nding is that the
average potential energy of hydrogen bonding UHB
decreases with the pressure rise, as may be seen in
® gure 5( b). Undoubtedly, the reason for this e€ ect is
the deviation of hydrogen bonds from their favourable
near-linear arrangement at compression of liquid water.
Figure 5( c) shows a noticeable decrease in the OHO
angle H with increasing pressure.
Thus, we may consider the bending of hydrogen
bonds as the same reason for both the shortening of
rOO and weakening of the hydrogen bonds. This
means that there is no contradiction between the behaviour of rOO and the value of the red shift of the OH(OD)
stretching vibrations in liquid water. It is also clear that
the correlation between rOO and the energy of the
hydrogen bonding is not universal and must be used
cautiously.
4. Conclusion
In conclusion, we would like to summarize the main
inferences drawn from the present study.
A seeming contradiction has been revealed between
the behaviour of the shortest intermolecular separation
rOO and that of the frequency of OH vibration of H2 O
with compression of liquid water. Decrease of rOO with
the pressure rise up to 2 kbar may be considered as
evidence of strengthening of hydrogen bonding, but
the increase in the frequency of t OH H2 O seems contradictory.
To clarify this, measurements of the integrated intensity of IR absorption for t OD HDO have been made
which de® nitely show that the energy of hydrogen
bonds decreases with compression at least up to
1.5 kbar. The decrease in the intensity is about 20% ,
i.e., essentially more than the estimated accuracy of
measurements. On the other hand, the e€ ect, though
quite explicit, is rather small compared with the full
range of possible change in the intensity of absorption.
Therefore, it is not surprising, that the variations of rOO
and t OH are also small, just slightly exceeding the limits
of experimental error.
Monte Carlo simulations of the compression of liquid
water show that the behaviour of rOO, t OH , and the
integrated intensity can be explained by the bending of
hydrogen bonds taking place during the initial stage of
compression. At the same time, MC simulations do not
reproduce the experimentally observed behaviour of the
shortest intermolecular separation above 3± 4 kbar. A
possible reason for this fact is the excessive rigidity of
TIP4P intermolecular potential.
The degree of hydrogen bond bending de® nes the
structures of high pressure solid polymorphs of water
and the P- T regions of their stability [33]. In terms of
preferential structure ¯ uctuations, we may say that in
the range of pressures studied the structure ¯ uctuations
of ice I type dominate. The distortion of the geometry of
hydrogen bonds with further increase in pressure should
lead to the appearance of the short-lived structure variations typical of the distorted structures of `higher’ types
of ice with hydrogen bonds bent even more.
It would be very interesting to obtain the pressure
dependence of the integrated intensity of HDO or
H2 O (D2 O) stretching bands at pressures higher than
those achieved in this experiment. It may be predicted
that there should be a change in the behaviour of the
integrated intensity above 2± 3 kbar.
The authors are grateful to Professor G. E. Walrafen
for kind permission to cite Raman data. This material is
based upon work supported by the US Civilian
Research and Development Foundation under Award
No. RC1-170. Grants from INTAS 96-1989 and from
RFBR (97-05-65956 and 97-03-32587) are greatly
appreciated.
References
[1] Neilson, G. W., Page, D. I., and Howells, W. S., 1979,
J. Phys. D, 12, 901.
[2] Gaballa , G. A., and Neilson, G. W., 1983, Molec.
Phys., 50, 97.
[3] Wu , A. Y., Whalley, E., and Dolling , G., 1982,
Molec. Phys., 47, 603.
[4] Gorbaty, Y u . E., and Demianets, Y u . N., 1985, Molec.
Phys., 55, 571.
[5] Okhulkov , A. V., Demianets, Yu . N., and Gorbaty,
Y u . E., 1994, J. chem. Phys., 100, 1578.
[6] Ohtaki, H., R adnai, T., and Y amaguchi, T., 1997,
Chemical Society Reviews, 41.
[7] Bellissent-Funel , M. C., and Bosio, L., 1995, J. chem.
Phys., 102, 3727.
[8] Walrafen, G. E., and A bebe, M., 1978, J. phys. Chem.,
68, 4894.
[9] Cavaille, D., Combes, D., and Zwick , A., 1996, J.
Raman Spectrosc., 27, 853.
[10] Gorbaty, Y u . E., and Kalinichev, A. G., 1995, J. phys.
Chem., 99, 5336.
[11] Pimentel, G. C., and Mc Clellan, A. L., 1960, The
Hydrogen Bond (San Francisco: Freeman).
[12] Sokolov , N. D., 1972, Zhurnal Vsesoyuznogo
Khimicheskoko Obschestva, 17, 299 (in Russian) .
[13] Eisenberg , D., and Kautzman, W., 1969, The Structure
and Properties of Water (Oxford University Press).
[14] Babashov, I. V., Bondarenko, G. V., Gorbaty, Y u . E,
and Epelbaum, M. B., 1970, Pribory I tekhnika eksperimenta, No. 2, 215 (in Russian).
[15] Gorbaty, Y u . E., and Bondarenko, G. V., 1993, Rev.
sci. Instrum., 64, 2346.
[16] Stillinger, F. H., and R ahman, A., 1974, J. chem.
Phys., 61, 4973.
E€ ect of pressure on hydrogen bonding in HDO
[17] Impey, R. W., Klein, M. I., and Mc Donald , I. R.,
1981, J. chem. Phys., 74, 647.
[18] Narten, A. H., and Levy, H. A., 1971, J. chem. Phys.,
55, 2263.
[19] Vedamuthu, M., Singh, S., and R obinson, G. W., 1995,
J. phys. Chem., 99, 9263.
[20] R obinson, G. V., and Zhu , S.-B., 1994, Reaction
Dynamics in Clusters and Condensed Phases, edited by
J. Jortner et al. (Dordrecht: Kluwer) p. 423.
[21] Minceva -Sukarova , B., Sherman, W. F., and
Wilkinson, G. R., 1984, J. Phys. C, 17, 5833.
[22] Swenson, C. A., 1965, Spectrochim. Acta, 21, 986.
[23] Wyss, H. R., and Falk , M., 1970, Can. J. Chem., 48, 607.
[24] Frank , E. U., and R oth, K., 1967, Discuss. Faraday
Soc., 43, 108.
[25] Iogansen, A. V., and R oz enberg, M. Sh., 1978, Opt.
Spectrosc., 44, 49 (in Russian).
1665
[26] Janscoí , G., Bopp, P., and Heiz inger , K., 1984, Chem.
Phys., 85, 377.
[27] Paí linkaí s, G., Bopp, P., Janscoí , G., and Heinzinger,
K., 1984, Z. Naturforsch., 39a, 179.
[28] R eddy , m. R., and Berkowitz , M., 1987, J. chem. Phys.,
87, 6682.
[29] Madura , J. D., Pettitt, B. M., and Calef, D. F., 1988,
Molec. Phys., 64, 325.
[30] Bagchi, K., Balasubramanian, S., and Klein, M. L.,
1997, J. chem. Phys., 107, 8561.
[31] Kalinichev, A. G., Gorbaty, Y u . E., and Okhulkov ,
A. V., 1999, J. molec. L iquids, in press.
[32] Kalinichev, A. G., and Bass, J. D., 1997, J. phys. Chem
A, 101, 9720.
[33] Kamb, B., 1968, Structural Chemistry and Molecular
Biology, edited by A. Rich and N. Davidson (San
Francisco: Freeman) p. 507.