www.elsevier.com/locate/cplett

Chemical Physics Letters 396 (2004) 226–231

Water: cavity size distribution and hydrogen bonds

Giuseppe Graziano *

Department of Biological and Environmental Sciences, University of Sannio, Via Port�Arsa 11, 82100 Benevento, Italy

Received 7 June 2004; in final form 30 July 2004

Abstract

There are two sizes for water molecules: (a) the distance of closest approach between two hydrogen bonded molecules, 2.8 A; (b)

the distance of closest approach between two non-hydrogen bonded molecules, 3.2 A, corresponding to the van der Waals diameter

of the oxygen atom. This fact is due to the bunching up effect of hydrogen bonds. Correspondingly, when the hydrogen bonding

potential is turned off in computer models of water, the effective size of water molecules increases. It is shown by means of scaled

particle theory calculations that this basic point has profound effects on the cavity size distribution if the number density is kept

constant. The recognition of the bunching up effect of hydrogen bonds is a key factor in order to address the role played by hydro-

gen bonds in the partitioning of void volume in liquid water.

� 2004 Elsevier B.V. All rights reserved.

1. Introduction

Liquids are characterized by the presence of a signif-

icant fraction of void volume. By performing the ratio of

the van der Waals volume occupied by a mole of liquid

molecules to the molar volume of the liquid itself, one

obtains the so-called volume packing density n. Forcommon organic liquids at room temperature, n is

around 0.5, indicating that about half of the molar vol-

ume is not occupied [1]. However, the total void volume

is not the relevant quantity to shed light on solvation

phenomena [2,3]. What is really important is the manner

in which the void volume is partitioned. In other words,

it would be necessary to know the cavity size distribu-

tion of the liquid. Unfortunately, the determination ofthe cavity size distribution cannot be accomplished by

experimental measurements and has to be performed

by means of computer simulations.

In 1990, Pohorille and Pratt [4,5] were the first to re-

port the cavity size distribution of several liquids,

0009-2614/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.cplett.2004.07.126

* Fax: +39-0824-23013.

E-mail address: graziano@unisannio.it.

including water and n-hexane, determined from a de-

tailed analysis of a large number of configurations gen-

erated by means of molecular dynamics simulations. It

resulted that the most part of the void volume is parti-

tioned in very small cavities, less than 1 A diameter,

and that the number of large cavities is exceedingly

small. In particular, the number of molecular-sized(2–5 A diameter) cavities proved to be markedly smaller

in water than in n-hexane or other non-polar solvents.

From a microscopic point of view, this should be the

origin of the poor solubility of non-polar compounds

in water [2,4,5].

Obviously, a molecular level explanation of the

Pohorille and Pratt results has to be provided. We have

shown that the cavity size distribution of water deter-mined by Pohorille and Pratt can be satisfactorily repro-

duced by means of scaled particle theory [6], SPT,

assigning to water molecules an effective diameter of

2.8 A [7]. This means that the void volume in water is

partitioned in small cavities because the size of water

molecules, the liquid length scale, is small [8]. In this re-

spect, it is worth noting that the effective diameter of

water molecules is markedly smaller than that of com-mon organic solvents [9].

G. Graziano / Chemical Physics Letters 396 (2004) 226–231 227

Recently, Sanchez and co-workers [10,11] determined

the cavity size distribution analysing liquid configura-

tions generated by means of Monte Carlo simulations

in the canonical ensemble, using a newly developed algo-

rithm. They considered three different fluid types: (a) a

hard sphere, HS, fluid with no attractions between par-ticles; (b) a Lennard-Jones, LJ, fluid whose particles

interact according to the Lennard–Jones potential; (c)

the extended simple point charge [12], SPC/E, model

of water that treats explicitly the presence of H-bonds

by means of electrostatic interactions between partial

charges. It resulted that, at densities corresponding to

the liquid state, the cavity size distribution of HS fluid

is very similar to that of LJ fluid (i.e., the two fluids havethe same number density and their particles have the

same size), consistent with the notion that the structure

of simple liquids is mainly determined by repulsive inter-

actions, the excluded volume effect [13,14].

On the other hand, by recognizing that SPC/E water

is an LJ fluid with added electrostatic interactions to

account for the presence of H-bonds, Sanchez and co-

workers [10,11] tried to address the role played by H-bonds on the cavity size distribution by deleting the

partial charges of the SPC/E model. In this manner, they

generated an LJ fluid having the same number density of

water at 25 �C, but with the H-bonding potential turned

off (it will be indicated as the corresponding LJ fluid

hereafter). The cavity size distributions of the two liq-

uids were both unimodal, but that of SPC/E water

proved to be broader and peaked at a larger size thanthat of the corresponding LJ fluid [10,11]. The two distri-

butions are shown in Fig. 1 that corresponds to Fig. 5 in

[10] and Fig. 11 in [11]. The maximum occurs at �0.7 Ærin SPC/E water and at �0.6 Ær in the LJ fluid, where r is

the van der Waals diameter of the oxygen atom, 3.16 A

0,0 0,5 1,0 1,50,0

0,5

1,0

1,5

2,0

2,5

water

LJ

prob

abili

ty d

ensi

ty

cavity size in units of σ

Fig. 1. Cavity size distributions determined by Sanchez and co-

workers in SPC/E water and in the corresponding LJ fluid at 25 �C (see

Fig. 5 in [10] and Fig. 11 in [11]). The corresponding LJ fluid was

obtained by turning off the electrostatic interactions in SPC/E water

and keeping fixed the number density at the experimental value of

water at 25 �C.

in the SPC/E model. The maximum is higher in the LJ

fluid than in SPC/E water and the latter liquid has a

greater number of molecular sized cavities.

Sanchez and co-workers [10,11] rationalized their re-

sults by stating that strongly attractive interactions such

as the H-bonds in water promote �clustering� phenomenathat lead to the existence of larger cavities with respect

to the corresponding LJ fluid. In other words, even

though the two liquids have the same number density,

the void volume should be distributed in larger packets

in the liquid characterized by stronger interactions. In

fact, they wrote [11]: �Hydrogen bonding causes water

to aggregate into clusters that produce a few large cavi-

ties rather than many smaller cavities� and �This resultindicates that applying SPT, which completely ignores

clustering effects, to describe the solvation properties

of water is questionable. The only way to compensate

for clustering in SPT is to use a smaller effective diame-

ter for water that makes water appear to be low density.�The finding that the cavity size distribution in SPC/E

water is broader, smaller and shifted toward larger cav-

ities with respect to that of the corresponding LJ fluidseems to contrast with the ability of SPT to reproduce

the cavity size distribution in TIP4P water determined

by Pohorille and Pratt [4,5] and Graziano [7]. The SPT

success would imply that SPC/E water and a properly

selected HS fluid should have similar cavity size distribu-

tions. We would like to show that this is indeed true. On

this basis, an entirely different explanation of the results

obtained by Sanchez and co-workers can be provided.The fundamental argument is that the role of the

H-bonding potential in determining the cavity size dis-

tribution in water cannot be addressed by simply delet-

ing the partial charges in the SPC/E water model and

keeping fixed the number density.

2. Methods

SPT provides two analytical relationships to calculate

the cavity size distribution pmax(rc), which is the proba-

bility density of the radius of the largest cavity that can

be successfully inserted into a liquid [6,7]. Simple geo-

metric arguments lead to the following exact relation-

ship valid over the size range �r16rc60:

pmaxð�r1 6 rc 6 0Þ ¼ 4p � q1 � ðrc þ r1Þ2; ð1Þwhere q1 = NAv/v1 is the number density of the solvent

and v1 its molar volume; r1 is the radius of the solvent

molecules and rc is the radius of the cavity (i.e., the ra-dius of the spherical region from which any part of

any solvent molecules is excluded). In this respect, it is

worth noting that negative rc values are physically

meaningful, but rc cannot be smaller than �r1 [6]. The

second SPT relationship is an approximate formula

valid over the size range rcP 0 [6,7]:

228 G. Graziano / Chemical Physics Letters 396 (2004) 226–231

pmaxðrc P 0Þ ¼ 2½ðK1=r1Þ þ ð2K2=r21Þrc þ ð3K3=r

31Þr2

c �� expð�DGc=RT Þ; ð2Þ

where K1 = u = 3n/(1�n); K2 = u(u + 2)/2; K3 = n ÆP Æv1/RT; and DGc is the work of cavity creation as expressed

by means of SPT [15]. In these relations R is the gas con-

stant; n is the volume packing density of pure solvent,

which is defined as the ratio of the physical volume of

a mole of solvent molecules over the molar volume of

the solvent ði:e:; n ¼ p � r31 � NAv=6 � v1Þ; rc and r1 are

the hard sphere diameter of the cavity and of the solvent

molecules, respectively; and P is the pressure. In all SPT

calculations, the experimental density of liquid water at

25 �C is used [16], and the pressure is fixed at 1 atm

[15,17,18].

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50.0

0.2

0.4

0.6

0.8

1.0

1.2

SPT-2.8SPT-3.2

p max

(r c)

(ang

stro

m-1

)

rc ( angstrom )

Fig. 2. Comparison of the cavity size distribution determined by

Pohorille and Pratt [4,5] in TIP4P water at 300 K (empty squares), and

those calculated by means of SPT using r = 2.8 A (blue line), and 3.2 A

(black line). On increasing r, the SPT distribution becomes sharper

and peaked at smaller sizes. In all calculations, the experimental

density of water at 25 �C is used, and the pressure is fixed at 1 atm.

(For interpretation of the references to colour in this figure legend, the

reader is referred to the web version of this article.)

3. Results

The starting point is the recognition that the size of

molecules depends on the interactions in which the mol-

ecules are involved. For water there are two relevant

sizes: (a) the distance of closest approach between two

H-bonded water molecules, r@2.8 A, on the basis of

the location of the first peak in the oxygen–oxygen ra-

dial distribution function, rdf, of water [19–21]; (b) thedistance of closest approach between two non H-bonded

water molecules, corresponding to the van der Waals

diameter of an oxygen atom, r@3.2 A [22]. Madan

and Lee [23] determined, by means of Monte Carlo

simulations, that when the H-bonding potential between

water molecules in the TIP4P model [22] is turned off

and the density is kept constant, the distance of closest

approach increases from 2.8 to 3.2 A. Practically, theyfound that the first peak of the oxygen–oxygen rdf

passes from 2.8 to 3.2 A upon turning off the H-bonding

potential [23]. This is because H-bonds are so strong to

bunch up water molecules beyond their van der Waals

size. Since in the TIP4P model the van der Waals diam-

eter of the oxygen atom is fixed at 3.15 A, when the

H-bonding potential is turned off, the distance of closest

approach becomes close to this number. Exactly thesame thing should occur in the SPC/E model, where

the size of the oxygen atom is fixed at 3.16 A, when

the partial charges are stripped and the density is kept

constant.

It is simple to grasp why an LJ liquid obtained by

simply turning off the H-bonding potential in TIP4P

or SPC/E model cannot be the sole reference liquid to

assess the role played by H-bonds in determining thepartitioning of void volume in water. If the number den-

sity is kept constant at the value of water at 25 �C, butthe effective size of the molecules increases from 2.8 to

3.2 A, the void volume existing in the liquid necessarily

decreases. Such a decrease of void volume manifests it-

self in a marked increase of the volume packing density.

Specifically, at 25 �C, n = 0.383 for r = 2.8 A versus

n = 0.572 for r = 3.2 A, using the experimental molar

volume of water, v = 18.07 cm3 mol�1 [16]. The increase

in volume packing density, in turn, causes a sharpening

of the cavity size distribution and an increase of the

magnitude of the work of cavity creation. This reason-ing has a general validity because it is grounded on sim-

ple geometric and physical principles [24].

It can be illustrated by means of SPT calculations.

We have calculated pmax(rc) by means of Eqs. (1) and

(2), using the two different values for the diameter of

water molecules: r = 2.8 and 3.2 A. The SPT pmax(rc)

functions are shown in Fig. 2 together with the distribu-

tion determined by Pohorille and Pratt [4,5] in TIP4Pwater at room temperature. The latter should be consid-

ered the �experimental� cavity size distribution in water,

because it was determined by means of a �brute force�procedure [4,5].

The SPT cavity size distribution calculated for r = 2.8

A is in satisfactory agreement with the �experimental�one, whereas that calculated for r = 3.2 A does not

agree with the �experimental� one. Since the cavity sizedistribution of water can be reproduced by means of

SPT, using r = 2.8 A, the effective size of water mole-

cules, the latter has to be the principal factor in the par-

titioning of void volume in water, when the density is

kept fixed, as already pointed out [7]. On increasing

the diameter of water molecules, the calculated pmax(rc)

distribution becomes sharper and the maximum shifts

to smaller rc values, as anticipated.The two SPT distributions are similar to those deter-

mined by Sanchez and co-workers [10,11] for SPC/E

G. Graziano / Chemical Physics Letters 396 (2004) 226–231 229

water and the corresponding LJ fluid, respectively. The

SPT functions are shown in Fig. 3 versus the cavity size

expressed in units of r = 3.2 A by means of the relation

(rc + 1.6)/3.2. The comparison suggests that the corre-

sponding LJ fluid consists of molecules having an effec-

tive diameter larger than that of molecules in SPC/Ewater. Notwithstanding the claim by Sanchez and co-

workers, SPT calculations reproduce their results, pro-

viding an explanation grounded on basic physical ideas.

However, the comparison cannot be quantitative.

Even though the distribution determined by Sanchez

and co-workers should be a volume distribution [10],

giving the probability p(vc)dvc that a cavity has volume

between vc and vc + dvc, we suspect that it is similar to,but not exactly, p(vc)dvc. On the other hand, the distri-

bution determined by Pohorille and Pratt and the SPT

ones provide the probability pmax(rc)drc that a cavity

has radius between rc and rc + drc. Furthermore, com-

parison of the cavity size distribution in SPC/E water

of Sanchez and co-workers with that in TIP4P water

of Pohorille and Pratt indicates that the algorithm de-

vised by Sanchez and co-workers overestimates the cav-ity diameters, as already noted by the authors [10].

The argument that the molecular diameter is the fun-

damental factor controlling the division of void volume

in water, when the density is kept constant, can be fur-

ther validated. Madan and Lee [23] showed that by

adjusting the size of LJ particles at 2.8 A and using

the same number density of water at 25 �C, the DGc

magnitude in such LJ fluid is comparable to that inTIP4P water. Similarly, Pratt and Pohorille [5] found

that the cavity contact correlation function in an LJ

fluid, with particles of the same size of water ones and

having the same number density of water, is similar to

that determined in TIP4P water.

0,0 0,2 0,4 0,6 0,8 1,00,0

0,2

0,4

0,6

0,8

1,0

1,2

SPT-3.2

SPT-2.8

p max

( r c )

(a

ngst

rom

-1 )

cavity size in units of σ

Fig. 3. SPT cavity size distributions calculated using the experimental

density of water at 25 �C and two different diameters for the particles

2.8 and 3.2 A, respectively, reported as a function of cavity size

expressed in units of r = 3.2 A by means of the relation (rc + 1.6)/3.2.

In this manner, the similarity with the distributions of Fig. 1 should be

more evident.

The SPT estimates of the work to create a cavity of 4

A diameter in water at 25 �C are: 26.0 kJ mol�1 for

r = 2.8 A and 61.1 kJ mol�1 for r = 3.2 A. The reliabil-

ity of the SPT value calculated using r = 2.8 A is readily

verified. To create a 4 A diameter cavity: (a) Guillot and

Guissani found DGc = 26 kJ mol�1 in SPC/E water at300 K [25]; (b) van Gunsteren and co-workers calculated

DGc = 27 kJ mol�1 in SPC water at 300 K [26], upon

accounting for the transformation from the cavity ther-

mal diameter to the hard sphere one [27]; (c) Pohorille

and Pratt found DGc = 27 kJ mol�1 in TIP4P water at

300 K [4]. The corresponding cavity insertion probabil-

ity, po = exp(�DGc/RT) = 2.8·10�5 for r = 2.8 A, and

2.0·10�11 for r = 3.2 A. The latter number is nine or-ders of magnitude larger than the SPT estimate,

3·10�20, reported by Sanchez and co-workers [11] to in-

sert a 4 A diameter cavity into an HS fluid with particles

of 3.16 A diameter and having the same number density

of water at 25 �C. The origin of the huge discrepancy is

because Sanchez and co-workers calculated DGc using

the pressure of the HS fluid as determined by SPT [6],

27600 atm. In this respect, several authors [15,17,18]have indicated that SPT calculations of DGc have to

be performed at the experimental pressure of the real liq-

uid, not at the pressure of the HS fluid. For this reason,

we used P = 1 atm in SPT calculations.

4. Discussion

The present analysis indicates that, if the number

density is kept fixed, the cavity size distribution of water

can be reproduced by means of SPT assigning to water

molecules an effective diameter of 2.8 A. The latter num-

ber corresponds to the location of the first peak in the

oxygen–oxygen rdf of water and so to the distance be-

tween two H-bonded water molecules [19–21]. This

means that H-bonds play an indirect role in determiningthe cavity size distribution by reducing the size of water

molecules from 3.2 to 2.8 A.

In other words, the partitioning of void volume in

water is determined by the size of the molecules, but

the latter has to be considered with care. This is due

to the bunching up effect of H-bonds that are so strong

to allow the interacting molecules to come closer than

their van der Waals size [23]. Even though all the stand-ard computer models of water use a van der Waals

diameter for the oxygen atom of 3.1–3.2 A [12,22], the

obtained oxygen–oxygen rdf is always peaked at

2.8 A, in line with X-ray and neutron scattering meas-

urements [19–21]. The bunching up effect of H-bonds

is operative also in liquid methanol and ethanol. For

methanol, the van der Waals diameter is 4.10 A [31],

whereas the effective diameter is 3.69 A [1], or 3.83 A

[32]. For ethanol, the van der Waals diameter is 4.66

230 G. Graziano / Chemical Physics Letters 396 (2004) 226–231

A [31], whereas the effective diameter is 4.34 A [1], or

4.44 A [32].

Therefore, in order to address the role of H-bonds in

determining the cavity size distribution in water, it is

necessary, upon turning off the H-bonding potential,

to reduce the diameter of LJ particles in order to obtainthe same effective size of water molecules and so the

same amount of void volume in the simulated liquid.

This need was pointed out by Madan and Lee [23],

and is now well recognized by scientists working in the

field [5,28–30]. In contrast, the corresponding LJ fluid

of Sanchez and co-workers differs from the SPC/E water

model not only for the absence of the H-bonding poten-

tial, but also for the size of the molecules. This is the ori-gin of the finding that SPC/E water has larger cavities

than the corresponding LJ fluid, a result readily rational-

ized when the molecular size is properly taken into ac-

count, as shown by means of SPT in Figs. 2 and 3.

Since an HS fluid having the same number density of

water and consisting of particles of the same size of

water molecules is characterized by a cavity size distri-

bution close to that of water, the space occupation inwater is largely determined by the effective molecular

size.

It is important to note that the real situation is more

subtle: the two sizes characterizing water molecules

operate simultaneously in water. The average coordina-

tion number in water at room temperature is about 5

[19–21]. There are 4 water molecules at 2.8 A that are

H-bonded to the central one; and there is also an inter-stitial molecule at 3.2 A that is not H-bonded to the cen-

tral one, but occurs in its first coordination shell [33].

This implies that an LJ fluid consisting of molecules

having a well defined diameter could not be able to ex-

actly account for the partitioning of void volume in

water. Similarly, SPT should be considered in a way

unsatisfactory to describe the packing properties of

water because it can make use of only one size at a time.Even though this is strictly true, the computer simula-

tion results of both Pohorille and Pratt [4,5], and Madan

and Lee [23] indicate that an LJ fluid with particles of

2.8 A diameter and the same number density of water

at 25 �C should possess a cavity size distribution close

to that of SPC/E or TIP4P water models. This is further

supported by the finding that SPT, an approximate the-

ory of HS fluids, is able to reproduce in a satisfactorymanner the �experimental� cavity size distribution deter-

mined by Pohorille and Pratt in TIP4P water [7], by

using r = 2.8 A for water molecules. The excluded vol-

ume effect related to the effective size of water molecules,

in part determined by the bunching up effect of

H-bonds, plays the pivotal role in the partitioning of

void volume in liquid water.

Clearly, if the effective size of water molecules wereequal to the van der Waals diameter of the oxygen atom,

3.2 A, water would be �dense� with n = 0.572. The solu-

bility of non-polar compounds in this hypothetical

�dense� water would be even poorer than in water, be-

cause DGc would be larger, as emphasized by the cavity

size distribution determined by Sanchez and co-workers

for the corresponding LJ fluid, and the SPT one calcu-

lated for r = 3.2 A. The H-bonds, by rendering smallerthe water molecules and so less costly the process of cav-

ity creation, should aid the solubility of non-polar com-

pounds in water. Similar arguments were already

discussed by Lee [9,23] and by us [34].

In conclusion, the present analysis points out that if

the H-bonding potential is turned off, the effective size

of water molecules increases from 2.8 to 3.2 A. An LJ

fluid obtained by turning off the H-bonding potentialand keeping fixed the number density has less void vol-

ume than water and cannot be the sole reference fluid to

clarify the role of H-bonds in determining the cavity size

distribution of water. Another LJ fluid should be con-

sidered: after turning off the H-bonding potential, the

particle size has to be reduced to the effective diameter

of water molecules so to have the same void volume of

water. The role played by the diameter assigned to watermolecules in determining the cavity size distribution is

clarified by means of SPT calculations. The latter show

unequivocally that, when the number density is fixed,

the molecular diameter plays the fundamental role

for the partitioning of void volume and the magnitude

of the work of cavity creation in water. The H-bonds

play an indirect role by determining the effective size

of water molecules. The present results support the smallsize of water molecules as the origin of the hydrophobic

effect [7–9,29,34], but do not support the dense packing

effect as proposed by Sanchez and co-workers.

Acknowledgements

I thank Dr. B. Lee (Center for Cancer Research,NCI, NIH, Bethesda, MD) for carefully reading many

earlier drafts of the manuscript. This work is supported

by the COFIN 2002 grant from the Italian Ministry of

Instruction, University, and Research (M.I.U.R.,

Rome).

References

[1] E. Wilhelm, R. Battino, J. Chem. Phys. 55 (1971) 4012.

[2] L.R. Pratt, Annu. Rev. Phys. Chem. 53 (2002) 409.

[3] K.E.S. Tang, V.A. Bloomfield, Biophys. J. 79 (2000) 2222.

[4] A. Pohorille, L.R. Pratt, J. Am. Chem. Soc. 112 (1990) 5066.

[5] L.R. Pratt, A. Pohorille, Proc. Natl. Acad. Sci. USA 89 (1992)

2995.

[6] H. Reiss, Adv. Chem. Phys. 9 (1966) 1.

[7] G. Graziano, Biophys. Chem. 104 (2003) 393.

[8] B. Lee, Biopolymers 31 (1991) 993.

[9] B. Lee, Biopolymers 24 (1985) 813.

G. Graziano / Chemical Physics Letters 396 (2004) 226–231 231

[10] P.J. in�t Veld, M.T. Stone, T.M. Truskett, I.C. Sanchez, J. Phys.

Chem. B 104 (2000) 12028.

[11] M.T. Stone, P.J. in�t Veld, Y. Lu, I.C. Sanchez, Mol. Phys. 100

(2002) 2773.

[12] H.J.C. Berendsen, J.R. Grigera, T.P. Straatsma, J. Phys. Chem.

91 (1987) 6269.

[13] H. Reiss, R.V. Casberg, J. Chem. Phys. 61 (1974) 1107.

[14] D. Chandler, J.D. Weeks, H.C. Andersen, Science 220 (1983) 787.

[15] R.A. Pierotti, Chem. Rev. 76 (1976) 717.

[16] G.S. Kell, J. Chem. Eng. Data 20 (1975) 97.

[17] F.H. Stillinger, J. Solution Chem. 2 (1973) 141.

[18] S. Shimizu, M. Ikeguchi, S. Nakamura, K. Shimizu, J. Chem.

Phys. 110 (1999) 2971.

[19] A.H. Narten, H.A. Levy, Science 165 (1969) 447.

[20] A.K. Soper, F. Bruni, M.A. Ricci, J. Chem. Phys. 106 (1997) 247.

[21] J.M. Sorenson, G. Hura, R.M. Glaeser, T. Head-Gordon,

J. Chem. Phys. 113 (2000) 9149.

[22] W.L. Jorgensen, J. Chandrasekhar, J.D. Madura, R.W. Impey,

M.L. Klein, J. Chem. Phys. 79 (1983) 926.

[23] B. Madan, B. Lee, Biophys. Chem. 51 (1994) 279.

[24] H. Reiss, J. Phys. Chem. 96 (1992) 4736.

[25] B. Guillot, Y. Guissani, J. Chem. Phys. 99 (1993) 8075.

[26] T.C. Beutler, D.R. Beguelin, W.F. van Gunsteren, J. Chem. Phys.

102 (1995) 3787.

[27] F.M. Floris, M. Selmi, A. Tani, J. Tomasi, J. Chem. Phys. 107

(1997) 6353.

[28] S.R. Durell, A. Wallqvist, Biophys. J. 71 (1996) 1695.

[29] M. Ikeguchi, S. Shimizu, S. Nakamura, K. Shimizu, J. Phys.

Chem. B 102 (1998) 5891.

[30] H. Tanaka, Chem. Phys. Lett. 282 (1998) 133.

[31] J.T. Edward, J. Chem. Educ. 47 (1970) 261.

[32] D. Ben-Amotz, K.G. Willis, J. Phys. Chem. 97 (1993) 7736.

[33] K.R. Gallagher, K.A. Sharp, J. Am. Chem. Soc. 125 (2003) 9853.

[34] G. Graziano, J. Phys. Chem. B 106 (2002) 7713.