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Specialist Periodical Reports
Chemical Modelling
Applications and Theory
Volume 9
Edited by M Springborg
Chemical Modelling
Applications and Theory Volume 9
Springborg
Now Indexed

in Scopus


Chemical Modelling
Applications and Theory
Volume 9

A Specialist Periodical Report
Chemical Modelling
Applications and Theory
Volume 9
A Review of Recent Literature
Editor
M. Springborg, University of Saarland, Saarbru¨cken, Germany
Authors
Bartolomeo Civalleri, University of Torino, Italy
Roberto Dovesi, University of Torino, Italy
Wu Guozhong, Chinese Academy of Sciences, Shanghai, China
Barbara Kirchner, Leipzig University, Germany
Sha Maolin, Chinese Academy of Sciences, Shanghai, China
George Maroulis, University of Patras, Greece
Sougata Pal, Gourbanga University, Malda, India
Raja Paul, Indian Association for the Cultivation of Science, Kolkata, India
Davide Presti, University of Torino, Italy
Dou Qiang, Chinese Academy of Sciences, Shanghai, China
Sudip Roy, National Chemical Laboratory, Pune, India
Supriya Saha, Gourbanga University, Malda, India
Pranab Sarkar, Visva-Bharati University, Santiniketan, India
Sunandan Sarkar, Gourbanga University, Malda, India
Andreas Savin, CNRS and UPMC Sorbonne University, Paris, France
Stefan Zahn, Monash University, Australia

ISBN: 978-1-84973-412-7
ISSN: 0584-8555
DOI: 10.1039/9781849734790
A catalogue record for this book is available from the British Library
&The Royal Society of Chemistry 2012
All rights reserved
Apart from fair dealing for the purposes of research or private study for
non-commercial purposes, or for private study, criticism or review, as
permitted under the Copyright, Designs and Patents Act, 1988 and the
Copyright and Related Rights Regulations 2003, this publication may not be
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Thank you.

Preface
Michael Springborg
DOI: 10.1039/9781849734790-FP005
You are holding the 9
th
volume of theSpecialist Periodical Reportson
Chemical Modelling: Applications and Theoryin your hands. With this
volume the series is approaching its 15
th
birthday. Whereas the ﬁrst issues
were published biannually, since some few years a new issue appears every
year. For authors this has the additional advantage that their contributions
are included in various citation indices, which, hopefully, adds to the
attractiveness of the series.
With the present issue, I have decided to put emphasis on two quite
diﬀerent issues. One is that of the geographical distribution of the con-
tributing authors, where the present issue has special emphasis on Asian
contributions. No less than three contributions from Indian colleagues are
included and a further contribution comes from China. I hope that it
demonstrates that interesting science is made also outside the more tradi-
tional scientiﬁc regions of Western Europe and Northern America.
In addition, ionic liquids have gradually become of increasing importance
as interesting solvents, although their discovery is not at all recent and
although their properties are only marginally understood. Two contribu-
tions in the present volume from experts in the properties of ionic liquids
provide a thorough and up-to-date discussion of these systems.
Besides these two issues, the present volume contains also other inter-
esting contributions. These include papers on biological systems, density-
functionals, nanostructures, and others.
It is my hope that I have been able to identify exciting and relevant
subjects in more areas of chemical modelling, but I will, of course, be
grateful for any suggestion, both with respect to authors and with respect to
subjects, for the forthcoming issues.
Physical and Theoretical Chemistry, University of Saarland, 66123, Saarbru¨cken, Germany.
E-mail: [email protected]
Chem. Modell.,2012,9,v–v | v
ﬀcThe Royal Society of Chemistry 2012

CONTENTS
Preface v
Michael Springborg
Uncovering molecular secrets of ionic liquids 1
Stefan Zahn and Barbara Kirchner
1 Introduction 1
2 Choice of a suited computational method 2
3 Functionalizing ionic liquids for a low melting point or low
viscosity
5
4 Ionic liquids and water 10
5 Ionic liquids and carbon dioxide 12
6 Surface of ionic liquids 14
7 Summary 15
Acknowledgements 15
References 15
Interaction-induced electric properties 25
George Maroulis
1 Introduction 25
2 Interaction-induced electric properties from ﬁnite-ﬁeld
calculations: A conventional approach
27
Cover
The icosahedral ‘golden fullerene’
WAu
12reproduced by permission of
Pekka Pyykko¨, Chemistry Department,
University of Helsinki, Finland.
Chem. Modell., 2012,9, vii–ix | vii
ﬀcThe Royal Society of Chemistry 2012

3 Theoretical calculations of interaction-induced electric
moments and (hyper)polarizabilities
29
4 Conclusions and some future challenges 56
References 56
Modeling biological cells 61
Raja Paul
1 Introduction 61
2 Dynamics of cytoskeletal ﬁlaments 64
3 Molecular motors 70
4 Adherent cell shape 72
5 Cell polarization, migration and rupture of adhesions 77
6 Stress ﬁber generation 80
7 The stress intercellular interaction and pattern formation 82
8 Discussion 87
References 89
Particle based multiscale simulation methods and applications 92
Sudip Roy
1 Introduction 92
2 Coarse graining based on potentials derived from all
atomistic molecular dynamics simulations
95
3 Adaptive resolution multiscale simulation in the same
simulation box
117
4 Multiscale simulation by parameter transfer from smaller
to larger scale
123
5 Conclusion and outlook 129
Acknowledgement 130
References 130
Size-dependent electronic structure of semiconductor nanoparticles 135
Sougata Pal, Sunandan Sarkar, Supriya Saha and Pranab Sarkar
1 Introduction 135
2 Why are they diﬀerent? 136
3 Some early theoretical studies 137
4 Electronic structure of bare bulk-like semiconductor
nanoparticles
138
5 Electronic structure of hollow and ring-like clusters 145
6 Electronic structure of passivated semiconductor
nanoparticles
148
viii |Chem. Modell., 2012,9, vii–ix

7 Electronic structure of core/shell semiconductor
nanoparticles
153
8 Electronic structure of alloyed nanoparticles 158
9 Conclusions and outlook 163
Acknowledgements 164
References 164
On choosing the best density functional approximation 168
Bartolomeo Civalleri, Davide Presti, Roberto Dovesi and
Andreas Savin
1 Introduction 168
2 Classical measures 171
3 Robust statistics 173
4 Trends 176
5 Reference data 178
6 Conclusion 180
Appendix 182
Acknowledgment 183
References 184
Molecular dynamics simulation of ionic liquids adsorbed onto a solid
surface and conﬁned in nanospace
186
Maolin Sha, Qiang Dou and Guozhong Wu
1 Introduction 186
2 Basis of MD simulation methods for ionic liquids 187
3 Ionic liquids at the solid surface 193
4 Ionic liquids conﬁned in nanospace 201
5 Summary and outlook 211
Acknowledgement 212
References 212
Chem. Modell.,2012,9, vii–ix | ix

Uncovering molecular secrets of ionic
liquids
Stefan Zahn*
1
and Barbara Kirchner*
2
DOI: 10.1039/9781849734790-00001
1 Introduction
Ionic liquids oﬀer high-potential solutions to an amazingly broad range of
applications.
1,2
The large number of cations and anions which can be
combined to a low melting salt suggests the feasibility to design a required
liquid for every task. The variety of possible ionic liquids even outnumbers
traditional solvents in chemistry.
3,4
Unfortunately, little is known about
general properties of ionic liquids except the obvious fact that they consist
solely of ions. It is the human and scientiﬁc nature that always ﬁnds its way
to characterize the unknown or ﬁnd ways to overcome challenges. One such
challenge is the distillation of ionic liquids which was originally thought
impossible due to the low volatility of known ionic liquids. However, ionic
liquids can be distilled.
5
The history of ionic liquids goes back nearly one hundred years. As early
as 1914, Paul Walden reported the ﬁrst systematic study of ionic liquids.
6
However, the scope of ionic liquids was recognized barely until the devel-
opment of air and water stable imidazolium-based ionic liquids in 1992.
7
Since then, the interest in these compounds has increased greatly leading to
manifold applications of these compounds in natural sciences and indus-
try.
3,4,8–18
For example, ionic liquids are used in battery,
19–21
solar cell,
22–26
fuel cell,
27–30
and lubricant applications.
31,32
Nevertheless, even funda-
mental properties of ionic liquids are far from being understood.
Due to the technical progress and the developments in theoretical
chemistry over the last 20 years, computational methods have become a
powerful tool in chemistry because various approaches can be employed for
an investigation at the molecular scale.
33–50
Observed macroscopic prop-
erties can be assigned to functional parts of a molecule which facilitates a
more task-related design of new compounds. Still, the interplay of nuclei
and electrons is too complex for feasible black box methods of systems
larger than a few atoms. Therefore, a computational chemist should always
choose an approach carefully and verify how the necessary approximations
inﬂuence the results.
Ionic liquids are a special challenge for computational chemistry. Due to
the important role of cooperativity,
51–55
the investigation of large systems is
necessary to obtain reliable results. Unfortunately, only for medium sized
systems are there any computational approaches available which possess
1
Wilhelm Ostwald Institut fu¨r Physikalische und Theoretische Chemie, Universita¨t Leipzig,
Linne´str. 2, 04103 Leipzig, Germany. E-mail: [email protected];
homepage: www.stefan-zahn.com
2
Wilhelm Ostwald Institut fu¨r Physikalische und Theoretische Chemie, Universita¨t Leipzig,
Linne´str. 2, 04103 Leipzig, Germany. E-mail: [email protected];
homepage: www.uni-leipzig.de/Bquant/
Chem. Modell., 2012,9,1–24 | 1
ﬀcThe Royal Society of Chemistry 2012

the required ﬂexibility of electronic structure for an accurateab initio
description of cooperativity. Additionally, not only cooperativity, but also
dispersion forces make the choice of a reliable approach for ionic liquids a
challenging task.
56–62
Nevertheless, carefully selected computational
approaches allow predictions for ionic liquids, which can be conﬁrmed by
experiments if possible. One example is the nanoscale segregation of polar and
nonpolar domains (also called microheterogeneity) in ionic liquids. These
domains were found in corse-grained model
63
and in fully atomistic model
molecular dynamics simulations
64
before they were reported by X-ray
diﬀraction
65
or Raman-induced Kerr eﬀect spectroscopy
66
studies as well.
Not only the prediction of the liquid structure but also the calculation of
thermodynamic data, like the gaseous enthalpy of formation, is feasible.
67
These two discussed examples exemplify that carefully selected computational
approaches are a powerful tool for the investigation of ionic liquids.
68–74
2 Choice of a suited computational method
The investigation of large systems is necessary for ionic liquids due to the
important role of cooperativity.
51–55
Additionally, commonly used ionic
liquids consist of inorganic anions and organic cations with alkyl side chains
and aromatic moieties. Both functional groups of the cation are well-known
for a signiﬁcant contribution of dispersion forces to equilibrium structure
and interaction energy.
75–77
Long alkyl chains of ionic liquids result in
nanoscale segregation,
63–66,78–80
in which the nonpolar domains are domi-
nated by dispersion forces. Furthermore,p-p-stacking of aromatic cations
was also observed.
81–84
Even the interplay of counter ions is inﬂuenced
signiﬁcantly by dispersion forces.
58,60
Thus, reliable computational
approaches for an investigation of ionic liquids do not need only a proper
description of electrostatic and induction forces, also an accurate descrip-
tion of dispersion forces is needed.
Ab initiocorrelated or so-called post Hartree–Fock methods provide a
proper description of dispersion forces. Unfortunately, these methods are
computationally limited to systems with few atoms. Only few CCSD(T)
calculations of very small ionic liquid systems were reported so far.
56,84–91
Second-order Møller–Plesset perturbation theory (MP2) might be also a
suitableab initiomethod to study ionic liquids. Recent developments have
made this approach available for systems with hundreds of atoms.
48,92,93
However, calculations of medium sized ionic liquid systems need still
enormous computational resources.
55
Thus, MP2 and similar approaches
seem to be limited to static quantum chemical calculations and are still too
expensive forab initiomolecular dynamics simulations over an appropriate
system size and time frame.
A feasible compromise of accurate forces and available system size
might be Kohn–Sham density functional theory
94,95
(KS-DFT). Several
approaches, especially for general gradient approximation (GGA) func-
tionals, are known to reduce the computational cost much lower than for
conventional correlation methods.
96–104
Unfortunately, KS-DFT accounts
for electrostatic, exchange and induction forces very well, but fails for the
description of dispersion forces.
105–110
Del Po´poloet al.have attributed
2|Chem. Modell., 2012,9,1–24

observed large errors in calculated equilibrium volumes of ionic liquid
crystal structures to the limitations of KS-DFT in dealing with dispersion
interactions.
111
Several possible solutions were proposed to correct this
shortcoming of Kohn–Sham density functional theory.
112–133
Most com-
putational investigations of ionic liquids still use traditional exchange-
correlation functionals,e.g.B3LYP,
134–136
without a dispersion correction.
However, two studies have shown that energy and structure obtained with
traditional functionals deviates to MP2 references similar like Hartree-Fock
calculations.
59,61
Zahn and Kirchner reported that the empirical dispersion
correction proposed by Grimme in 2006
123
reduces the deviation to the
reference values signiﬁcantly. An empirical dispersion correction by a
dispersion-corrected atom-center dispersion potential
137
(DCACP) was also
recommended. Both approaches increase the computational time margin-
ally and the deviance is within the error of the MP2 reference method.
59
Similar results were obtained also by Izgorodinaet al.
61
Additionally, this
study recommended three recently developed functionals (M05-2X,
138
KMLYP,
139
and M05
138
) for the investigation of ionic liquids. Further-
more, a study reported good matching ionic liquid crystal structure volumes
if the non-empirical dispersion correction of Dionet al.
117
is employed.
140
Non-local van der Waals density functionals
131
as well as the third version
of Grimme’s empirical dispersion correction
130
gave excellent results for
ionic liquids, too.
91
The study of Grimme, Hujo, and Kirchner shows also
that the self interaction error of KS-DFT is only important at large ion
distances.
91
Thus, various feasible dispersion corrected KS-DFT approa-
ches are available for reliable investigations of ionic liquid systems.
Ab initiomolecular dynamics simulations might be the best approach
to study middle-sized ionic liquid systems because dispersion corrected
KS-DFT approaches give excellent results, and cooperativity is also con-
sidered. A recent study has shown that electronic properties like the dipole
moment are strongly localized properties of ionic liquids.
141
Furthermore,
Lynden-Bell observed strong electrostatic screening between solute
charges at distances comparable to the radius of the ﬁrst solvation shell.
142
Therefore, the feasible system size ofab initiomolecular dynamics simula-
tions is suﬃcient to study many body eﬀects or solvent-solute interactions.
Numerousab initiomolecular dynamics simulations were published in
the last years which provide a molecular view on ionic liquids.
54,81,141,143–162
As mentioned above, these simulations are limited to medium system
size and short timescales as well. Ionic liquids are often high viscous
liquids which results in slow dynamic processes on the molecular
scale. Additionally,ab initiomolecular dynamics simulations need large
amounts of computational resources. Therefore, classical molecular
dynamics simulations with a well parametrized force ﬁeld could be a
suited alternative or even better choice to study certain properties of ionic
liquids.
Especially for imidazolium based ionic liquids, several force ﬁelds can be
found in the literature.
163–170
The ﬁrst force ﬁeld of Hanke, Price, and
Lynden-Bell was developed to reproduce experimental crystal structures.
163
Unfortunately, a comparison of the reported liquid structure to later pub-
lishedab initiomolecular dynamics simulations show large deviations.
Chem. Modell., 2012,9,1–24 | 3

For example, the preferred position of the chloride anion in the study of
Hankeet al.is above and below the imidazolium plane and nearly no anion
can be found in front of the most acidic hydrogen atom of the imidazolium
ring. The force ﬁeld proposed by Morrow and Maginn in 2002 was devel-
oped using a combination ofab initiocalculations and the CHARMM22
force ﬁeld.
164
Most interestingly, the magnitude of used ion charges was less
than one which is in agreement with later published results of large ion
clusters.
51,52
Also, the agreement between experimental and computed
values of the volume expansivity, isothermal compressibility and molar
volumes were good for thisab initioparametrized force ﬁeld. The model of
Liu, Huang, and Wang published 2004 is based on the AMBER force ﬁeld
in which missing parameters were taken from static quantum chemistry
calculations.
165
Additionally, parameters were adjusted to match vibra-
tional frequencies. Furthermore, the van-der-Waals diameter of the most
acidic proton of the imidazolium ring was optimized. A good agreement
between experimental and calculated density was obtained for several ionic
liquids. Also, the obtained liquid structure matches the results from later
publishedab initiomolecular dynamics simulations. In 2004, Canongia
Lopeset al.proposed a force ﬁeld for several imidazolium based ionic
liquids, too.
166
Missing parameters, like torsion energy proﬁles and charge
distribution, were obtained from static quantum chemical calculations.
Crystallographic data and density of fourteen ionic liquids were reproduced
well with this force ﬁeld. Later, Canongia Lopes and Pa´dua published
extensions of their force ﬁeld, which provides parameters for the triﬂate,
bistriﬂymide, dicyanamide, alkylsulfonate, and alkylsulfate anions as well as
for phosphonium, pyridinium, trialkylimidazolium, and alkoxycarbonyl-
imidazolium cations.
167,168,171
Thus, their force ﬁeld facilitate to study a
broad range of commonly used ionic liquids. In 2007, Ko¨ddermann,
Paschek, and Ludwig published a revised version of the force ﬁeld of
Canongia Lopes and Pa´dua for imidazolium based ionic liquids, in which
the Lennard-Jones parameters were adjusted to reproduce experimental
measurements.
169
The ﬁnally obtained model for imidazolium bistriﬂymide
ionic liquids shows improved results for self-diﬀusion coeﬃcients. Addi-
tionally, a good agreement of experimental and calculated heats of vapor-
ization, shear viscosity, and NMR rotational correlation times was found.
Bhargava and Balasubramanian reported also a reﬁned model for imida-
zolium ionic liquids, based on the force ﬁeld of Canongia Lopes and
Pa´dua.
170
They changed Lennard-Jones parameters of few atoms and scaled
the charges of the ions by 0.8 which is close to the value found in large ion
clusters.
51,52
A similar reparametrization was reported by Zhaoet al.as
well.
172
These reﬁned models show an improved agreement of the liquid
structure compared toab initiomolecular dynamics simulations. Especially,
the hydrogen bond structure could be reproduced very well. Additionally,
diﬀusion coeﬃcients and surface tension show a good agreement to
experimental measurements. An investigation of Youngs and Hardacre
showed that the best agreement of classical molecular dynamics simulations
andab initiomolecular dynamics simulations is achieved if the partial
charges obtained from isolated ions are scaled between 0.7 and 0.8 in the
force ﬁeld models.
173
Thus, the charge scaling seems to be a good choice to
4|Chem. Modell., 2012,9,1–24

consider charge transfer between cation and anion and avoid more time
consuming polarisable force ﬁelds. Already in 2004 Yanet al.investigated
the inﬂuence of polarisability by comparing a nonpolarisable with a
polarisable force ﬁeld.
174
A faster dynamics and better agreement to
experimental values was observed for the polarisable model. Similar results
were reported for various polarisable force ﬁeld models.
175–179
These results
highlight that the too slow dynamics of most nonpolarisable force ﬁelds
might be a result of neglecting polarisation and charge transfer between the
ions. Of special interest of the polarisable models might be the the quantum
chemistry-based polarisable force ﬁeld of Borodin because it is available for
various ionic liquids and gives good results for liquid structure, ion self-
diﬀusion coeﬃcients, conductivity, and viscosity.
175
We will come back to
this point in the following chapter.
In summary, various approaches were developed and validated for ionic
liquids over the last years. It was shown that dispersion corrected KS-DFT
approaches allow reliable results for ionic liquids.
59,61,91,111,140
Recently, a
comparison of trajectories obtained fromab initiomolecular dynamics
simulations with and without a dispersion correction revealed that the
dynamics of the system is more accurately described for the dispersion
corrected one,
180
which highlights the necessity of dispersion corrected
approaches inab initiomolecular dynamics simulations. Polarisable force
ﬁelds were developed which allow the investigation of various ionic
liquids.
175
Cheaper nonpolarisable force ﬁelds are also available for a broad
range of ionic liquids.
164–171
However, the latter force ﬁelds tend to show too
slow dynamical properties due to the negligence of polarisability. A feasible
correction for them might be the usage of reduced total ion charges,
164,170,173
which were observed in large scale ionic liquid cluster calculations, too.
51,52
3 Functionalizing ionic liquids for a low melting point or low viscosity
One focus of computational investigations is the characterization of mole-
cular features contributing to a low viscosity or a low melting point of ionic
liquids. A low melting point increases the usable temperature window of the
liquid while a high viscosity inﬂuences several properties undesirably, such
as electrical conductivity or even reaction rates. Most ionic liquids consist of
inorganic anions and organic cations. Unfortunately, the cations are com-
monly too complex to identify easily molecular features contributing to a
low melting point. Thus, investigations with simple model systems might be
reasonable to characterize general molecular features contributing to
desired macroscopic properties.
Two excellent studies of Spohr and Patey employ simple spherical model
systems to characterize the inﬂuence of ion size disparity
181
and charge
asymmetry
182
on macroscopic properties, like viscosity and electrical con-
ductivity. Their ﬁrst study,
181
published 2008, showed that the electrical
conductivity increases from an ion size disparity from 1:1 to 2:1 (cation:
anion). The conductivity is then nearly unaﬀected up to a size ratio of 3:1.
Larger size disparities decrease the electrical conductivity such that the size
ratios 1:1 and 5:1 have nearly the same conductivities. This behavior was
traced back to the competing impact of ion diﬀusion (increasing) and ion
Chem. Modell., 2012,9,1–24 | 5

density (decreasing). Additionally, the inﬂuence of ion size disparity on
shear viscosity is strongest up to a size ratio of 3:1 then decreases and seems
to reach a limiting value at a size disparity of 5:1.
181
Spohr and Patey
reported that charge asymmetry increases electrical conductivity and
decreases viscosity.
182,183
However, this trend is reversed if charge asym-
metry exceeds a critical value. The sharp decrease of conductivity and rapid
increase of viscosity was attributed to the formation of directional long
living ion pairs.
182
This result seems to be in contradiction to the concepts
proposed by Ludwig and our groups who suggested that strong, directional
charge asymmetry induced by hydrogen bonds ﬂuidize ionic liquids.
184–186
However, while Spohr and Patey did not consider a particular class of ionic
liquids and neglected the possibility of hydrogen bonding, the conclusions
of Fuminoet al.and of our investigations are based solely on the obser-
vation of ionic liquids with diverse imidazolium cations. These cations form
hydrogen bonds with anions.
60,144,155,185,187,188
Furthermore, classical
molecular dynamics studies reported a short time correlated motion of
cations and anions for imidazolium ionic liquids but the existence of long
living ion pairs was excluded.
189–191
Overall, the travelled distance of most
ion associates is smaller than the ion-ion distance. Therefore, no signiﬁcant
displacement is reached before they dissociate.
190
In classical molecular
dynamics simulations of molten salts solely short time correlated motions of
ions were found, too.
192
The deviation from the Nernst-Einstein relation of
ionic liquids and molten salts was attributed to the short time correlated
movement.
189–192
Furthermore, stable ion aggregates could be excluded for
imidazolium and pyrrolidinium ionic liquids based on dielectric spectro-
scopy measurements.
193,194
Thus, the term ‘‘ion association’’ seems to be
more suitable for most ionic liquids because distinct ion pairs could not be
observed. Therefore, ionic liquids are discussed also as extremely dis-
sociating solvents because no preferential Coulombic attraction between
any particular cation-anion combination can be observed due to the large
number of undistinguishable neighbors.
195
Distinct long-living ion pairs
might only exist in ionic liquids which consist of highly asymmetric ions,
like in the model systems of Spohr and Patey.
182
All-atom force ﬁeld molecular dynamics simulations have proven as a
powerful tool to investigate the inﬂuence of local charge and charge dis-
tribution on molecular and macroscopic properties, too. A force ﬁeld allows
the user to change easily the charge distribution without changing any other
property of the investigated system. Already ﬁrst classical molecular
dynamics simulations of Morrow and Maginn employed a charge dis-
tribution in which the total charge of the cation isþ0.904 and that of the
anion isﬃ0.904.
164
Thus, this force ﬁeld considers already signiﬁcant
charge transfer between cation and anion which weakens the Coulomb
interaction of them. A signiﬁcant charge transfer was observed in ﬁrst
ab initiomolecular dynamic simulation studies of an ionic liquid where a
charge betweenﬃ0.7 andﬃ0.8 was reported for the chloride anion.
143
In
2006, a systematic quantum chemistry study reported a partial charge
ofﬃ0.735 for the anion of one ion pair while the anion possess on average a
charge ofﬃ0.823 in an inﬁnite ion pair chain.
51
Based on these results,
Koßmannet al.suggested to employ partial charges in force ﬁelds of ionic
6|Chem. Modell., 2012,9,1–24

liquids where the absolute value of the ion charge is smaller than one.
Additionally, Schmidtet al.reported recently reduced ion charges obtained
from 100 snapshots of a CPMD simulation.
52
Therefore, the usage of
reduced ion charges in classical force ﬁelds seems reasonable to consider
many-body eﬀects of ionic liquids. This was suggested by Bhargava and
Balasubramanian who observed a signiﬁcant deviation to experimental
values if the absolute value of the ion charge is set to1.
170
Additionally,
Zhaoet al.suggested a charge scaling in ionic liquid force ﬁelds in 2007.
172
Please note, that few classical molecular dynamics studies
196–198
already
employed reduced partial charges for ionic liquids before 2007 which was
based on the study of Morrow and Maginn in 2002.
164
In 2008, Youngs and
Hardacre investigated the inﬂuence of partial charges scaled between 0.5
and 1.0 on structure and dynamical data.
173
They observed that the mean
square displacement increase with decreased scaling factor. Faster dyna-
mical properties for stronger scaled charges was also observed by Kohagen
et al.
186
Therefore, one can conclude that charge transfer between cations
and anions ﬂuidize ionic liquids.
As one can imagine, not only the total charge of the ions, also the charge
distribution within them inﬂuences properties signiﬁcantly on the molecular
as well as the macroscopic scale. Schro¨der and Steinhauser reported for two
diﬀerent charge distributions used in classical molecular dynamics simula-
tions a signiﬁcant diﬀerence in dynamical properties while the structure
seems to be less aﬀected.
199
Lynden-Bell and Youngs found that equilibrium
properties like energetics or liquid structure depend strongly on the charge
distribution within the ions.
200
Similar results were reported by Kohagen
et al.who compared three partial charge distributions obtained by diﬀerent
partial charge methods.
186,201
Commonly, a delocalized charge is proposed
to lower viscosity and melting point of ionic liquids.
3,87,200,202,203
However,
it was also proposed that a strong localized but well screened charge should
have a similar eﬀect.
202
Polarisation inﬂuences less structural properties than charge distribu-
tion.
200
However, polarisation speed up signiﬁcantly dynamical properties
of ionic liquids,
174–179,200
which is in contradiction to nonionic com-
pounds.
176
Borodin highlighted that the cation polarisation of imidazolium
based ionic liquids has a stronger inﬂuence on dynamical properties than
the anion polarisation.
175
This is consistent with the higher polarisability of
the cation.
175
Therefore, it seems that a large polarisability of the ions
enhance a low viscosity of ionic liquids.
First suggestions attributed the low solid-liquid transition temperature of
ionic liquids to the large size of the ions which should result in a weak
interaction energy.
9
According to the coulomb force, given by:
FC¼
Z
þ
Z
ﬃ
4pe0r
2
ð1Þ
a large distance between charged species results in a weak attractive
force. Because most ionic liquids possess large ions, one may expect a
weak coulomb interaction due to the larger distance of the charge centers
of each ion. However, static quantum chemical calculations show that
the dissociation energy of most isolated ionic liquids ions exceeds
Chem. Modell., 2012,9,1–24 | 7

300 kJ/mol,
56,58,185,204,205
which is signiﬁcant larger than for common
liquids. For example, two water molecules have a dissociation energy of
approximately 20 kJ/mol.
206
Turneret al.found a correlation of the melting
point and the interaction energy for selected compounds of imidazolium
based ionic liquids.
204
Nevertheless, the diﬀerence between successive
melting points is much greater than the diﬀerence between successive
energies and a general correlation can be excluded.
56,58,204,205
Therefore, it
seems questionable whether the investigation of interaction energies leads to
an understanding of the low melting point.
A promising model to discuss the melting point of ionic liquids might be
the energy landscape paradigm. Goldstein stated in 1969: ‘‘...that portion
of the potential energy surface that represents the liquid or glassy region
has, unlike the portion associated with the crystalline solid, a large number
of minima, of varying depths.’’
207
Asymmetrical ions are recognized as one
molecular feature for a low-melting point of ionic liquids.
4,9,203,208–214
These
ions increase the number of minima on the potential energy surface com-
pared to symmetrical ions. Additionally, the energy landscape paradigm
stated that the most stable conformational states of a high melting com-
pound are located in deep and steep parts of the energy potential surface.
Thus, a high activation energy is required for a conformational change of
the compound which is necessary for melting. On the other hand, a low
melting compound possess shallow energy potentials and low transition
states between conformational states, resulting in a gliding movement of the
molecules. A complementary model to the energy landscape paradigm
might be the Lindemann melting rule.
215,216
Lindemann stated that particles
exceed a critical displacement at the solid-liquid transition, which was
shown by Martin and O’Connorin.
217
A probable hypothesis considers the
activation energy for the critical displacement. A low barrier of this dis-
placement would be reﬂected in a low melting temperature which is the
connection to the energy landscape paradigm. Goldstein’s seminal ideas
have been applied especially for glasses and supercooled liquids,
218
which
show similarities to ionic liquids.
219–221
Thus, it seems reasonable to use the
energy landscape paradigm to discuss melting points of ionic liquids.
The substitution of the most acidic proton at C2 of 1,3-dialkylimidazo-
lium based ionic liquids with a methyl group is one example to show the
validity of Goldstein’s model for ionic liquids.
60,222
This substitution
replaces an attractive interaction with a repulsive one and, therefore, a
lower melting temperature and viscosity was expected. However, the
opposite was observed.
203
Hunt attributed the increased melting tempera-
ture and higher viscosity for the 1,2,3-trialkyl species to a decreased free
rotation of the butyl-side chain at C1.
223
A recent NMR-study excluded this
possibility because a similar rotational correlation time of the butyl side
chain in 1-butyl-3-methylimidazolium bromide and 1-butyl-2,3-dimethyli-
midazolium bromide was observed.
224
Another possibility might be the
inﬂuence of the hydrogen bond on the mobility of the anion. The two
stablest conformations of 1,3-dialkylimidazolium ionic liquids possess
nearly the same interaction energy.
56,60,185,222,223
In the ﬁrst one the anion is
located on top of the imidazolium ring while the anion is in front of the
most acidic proton in the second one. The latter minimum structure is
8|Chem. Modell., 2012,9,1–24

absent in the methylated compounds.
60,185,222,223
As a result, the anion is
highly ﬂexible in front of the C2-atom for the 1,3-dialkylimidazolium based
ionic liquids in contrast to the C2 methylated compounds, see Fig. 1.
Therefore, the C2-methylsubstitution decreases the mobility of the anion,
resulting in a higher activation barrier to reach the critical displacement for
melting, and thus the C2-methylsubstitution of 1,3-dialkylimidazolium
based ionic liquids supports the validity of the energy landscape paradigm
for ionic liquids.
60,185,222
Further evidence for the validity of Goldstein’s model for ionic liquids
can be found in molecular dynamics simulations of Urahata and Ribeiro.
225
Their results indicate an enhanced delocalization of the chloride anion with
increasing side chain of 1-alkyl-3-methylimidazolium chlorides from methyl
to the butyl group, while the melting point of the corresponding compounds
decreases. Furthermore, the delocalization of the anion of 1-butyl-3-
methylimidazolium chloride and 1-octyl-3-methylimidazolium chloride is
similar which is also in agreement with the melting point. Moreover,
Canongia Lopeset al.have shown that the bis(triﬂuoro-methanesulfonyl)
amide anion, known to form low melting ionic liquids, is highly ﬂexible.
226
Therefore, the usage of the energy landscape paradigm seems reasonable to
determine molecular features responsible for a low melting point of ionic
liquids.
One open question is which role do the intermolecular forces play in
depressing the melting point. It has been inferred that other forces beside
pure coulomb interactions must play a role in ionic liquids.
56
Force ﬁeld
investigations suggested an important role of dispersion forces which even
might be important as Coulomb forces.
57,227
Symmetry-adapted perturba-
tion theory
35
(SAPT) calculations of ion pairs have shown that the main
contribution to the total interaction energy stems from the electrostatic
interaction.
58,60
However, induction as well as dispersion forces inﬂuence
the equilibrium energy and structure signiﬁcantly. A comparison of the
potential energy proﬁles of several ionic liquid model compounds and a
typical salt obtained by SAPT calculations reveal two important diﬀerences:
ﬁrstly, dispersion forces are negligible in a typical salt in opposite to an ionic
(a) (b)
Fig. 1Spatial distribution function (SDF) of chloride for [C2mim][Cl] (a) and [C2C1mim][Cl]
(b). The isosurface is plotted at 30 times (yellow outer surface) and 20 times (pink interior
surface) of the average anion density less than the corresponding largest local anion density in
each simulation.
Chem. Modell., 2012,9,1–24 | 9

liquid. Secondly, ionic liquids possess a shallower interaction potential than
typical salts as a result of the intermolecular forces interplay.
58,60
Therefore,
ionic liquid counter ions can change their respective distances more easily
than typical ionic solid ones, which contributes to the low melting point of
ionic liquids according to the energy landscape paradigm. An investigation
of Bernardet al.observes for some ionic liquids that the ratio of total ion
pair binding energy to its dispersion component correlates with the melting
point whereas the dispersion component of the ion pair binding energy
correlates with conductivity and viscosity.
62
However, only few ionic liquids
were investigated, and the ﬂattening of the dissociation energy proﬁle due to
the composition of intermolecular forces is only one contribution to the low
melting point of ionic liquids.
In summary, designing low melting, low viscosity ionic liquids is a chal-
lenging task because several molecular features contribute. Additionally
some molecular features, like ion pairing, enhance the mobility of the ions
only over a selected range before their inﬂuence show a reversed eﬀect.
Therefore, semiemperical and quantitative structure-property relationship
(QSPR) approaches seem to be a good choice to estimate melting points or
viscosity of unknown ionic liquids in an appreciable time frame.
212,213,228–230
Also quantum chemical calculations using the Born–Fajans–Haber
cycle might be suitable to get a fast estimate of the melting point.
231
Unfortunately, the latter approach seems to be limited to nonprotic ionic
liquids.
232
4 Ionic liquids and water
Water is one of the typical impurities in ionic liquids, because these low
melting salts tend to absorb water from their environment.
233–235
One might
be anxious that the hygroscopic nature of ionic liquids could limit the usage
of ionic liquids as solvents for water sensitive reactions. However, the
interplay of ionic liquids and water might reduce the reactivity of water with
other solutes signiﬁcantly. For instance, PCl
3can be stored in ionic liquids
with air contact without any hydrolysis.
236
Several further examples of
an altered water reactivity in ionic liquids compared to that in organic
solvents were reported in the literature.
237–241
Additionally, even small
amounts of water can change macroscopic properties of ionic liquids
signiﬁcantly.
233,242,243
A well-known eﬀect of water impurities in ionic liquids is the viscosity
decrease.
233,244–249
However, also ionic liquids were reported, in which
water impurities induces gelation.
250
Spohr and Patey have shown with
ionic liquid model systems that water tends to replace the counter ions from
the ion solvation shell in ionic liquids with small ion size disparity, leading
to a faster diﬀusion of the lighter ion-water clusters.
251
However, water can
increase viscosity of ionic liquids if the ion size disparity is too large, or if
strong directional ion pairs are found. Spohr and Patey attributed this
behavior to extended water-anion chains and strongly bound water-anion-
cation clusters.
251
A classical molecular dynamics study by Raju and
Balasubramanian observed that the anion diﬀuse faster than the cation in
water ionic liquid mixtures in contrast to neat ionic liquids.
252
The larger
10 |Chem. Modell., 2012,9,1–24

diﬀusion coeﬃcient of the anions was related to the faster dynamics of the
water molecules in the hydration layer.
Experimental studies on water imidazolium ionic liquid mixtures reported
anions as preferred sites of interaction for water molecules.
234
This is sup-
ported by classical molecular dynamics studies of water imidazolium ionic
liquid mixtures which highlighted strong water-anion hydrogen bonds but
did not ﬁnd a speciﬁc interaction of water with the cations.
253,254
In
agreement with this, a recent study showed that the dipole moment of the
imidazolium cation is nearly unaﬀected by the presence of water while the
anion is signiﬁcantly stronger polarized with increasing water content.
255
A study by Jianget al.has highlighted that the structural organization of
the water-anion network and nano scale segregation possess a turnover
point after which only loose micelle structure exists due to the dominant
water-water interactions.
256
A competition between the water-water and
water-anion interactions was suggested as cause for the water network
turnover point while the turnover of nano scale segregation was attributed
to the competition between the hydrophobic interactions of the nonpolar
ionic liquid groups and the breakup of the charged polar network with
increasing water content. Spickermannet al.observed a good incooperation
of an imidazolium chloride ion pair in the hydrogen bond network of water
due to the hydrophilic anion solvation and hydrophobic cation solvation.
153
This might give a hint for the longer ion pair life time of ionic liquids in
water compared to typical salts. Bhargava and Klein investigated a series of
long chain 1-n-alkyl-3-methylimidazolium bromides aqueous solutions in
which quasi-spherical micelles were found.
257
In agreement with an
experimental study,
258
increasing aggregation numbers of cations were
observed with increasing alkyl side chain length. Additionally, long alkyl
chains enhance parallel ordering of them to each other.
257
A recentab initio
molecular dynamics study investigated neat 1-ethyl-3-methylimidazolium
acetate and two mixtures of this liquid with water.
255
With increase of water
content, the water structure gets more evident. This is in agreement with
previous studies, which reported that water molecules tend to be isolated
from each other in low water concentrations.
259
Furthermore, a large
ﬂuctuation of the C-H bond of the most acidic proton in the imidazolium
cation was observed in the neat ionic liquid which decreases with increasing
water content. Thus, water prevents the formation of carbenes from the
cation.
As shown by a recent study, the water solvation in nonprotic ionic liquids
diﬀer from protic ones.
158
Cations as well as anions form strong hydrogen
bonds with water in protic ionic liquids. This was also concluded by Herfort
and Schneider based on an experimental study.
260
The tetrahedral solvation
structure in protic ionic liquids suggest a highly polarized water molecule.
However, the opposite is observed: water is depolarized.
158
This might
contribute to the observation that water-protic ionic liquid mixtures create
dielectric environments, which resemble those of neat ionic liquids over an
appreciable composition range,
261
because neat water has a larger relative
dielectric permittivity than neat protic ionic liquids. Kelkar, Shi, and
Maginn suggested a smaller dipole moment of water in nonprotic imida-
zolium ionic liquids than in neat water as well.
262
An agreement of their
Chem. Modell.,2012,9,1–24 | 11

classical molecular dynamics simulations and experimental measurements
could only be observed if the dipole moment of water is decreased compared
to force ﬁelds developed for neat water. A recentab initiomolecular
dynamics study of 1-ethyl-3-methylimidazolium acetate water mixtures
conﬁrms the decreased dipole moment of water in an imidazolium ionic
liquid environment (x
water=0.75).
255
Furthermore, an opposed inﬂuence of
anion and cation on nearby water molecules was observed: anions polarize,
while cations depolarize.
In summary, water interacts stronger with the anions in nonprotic ionic
liquids,
234,253–255
while a similar important role of cations and anions is
observed in protic ionic liquids.
158,260
Interestingly, nonprotic as well as
protic ionic liquids depolarize solvated water molecules.
158,255,262
5 Ionic liquids and carbon dioxide
In 1999 Blanchardet al.reported a good solubility of carbon dioxide in
1-butyl-3-methylimidazolium hexaﬂuorophosphate at high pressures, while
the ionic liquid did not dissolve in carbon dioxide.
263
Therefore, super-
critical carbon dioxide is suited to extract organic solutes from ionic liquids,
and also continuous ﬂow homogeneous catalysis in ionic liquids carbon
dioxide systems is possible.
263–267
First spectroscopic studies show that the
anion dominates the interactions with carbon dioxide by Lewis
acid-base interactions.
268
However, the strength of carbon dioxide anion
interactions did not correlate with carbon dioxide solubility.
268
Thus, strong
anion-carbon dioxide interactions were excluded as major cause for the
carbon dioxide solubility in ionic liquids. Instead, a correlation of carbon
dioxide solubility and the ionic liquid molar volume was observed.
269
Additionally, a signiﬁcant volume decrease of dissolved carbon dioxide was
reported.
263,269,270
Based on these observations a space-ﬁlling mechanism
was suggested.
268,269
This is supported by classical molecular dynamics
simulation studies in which the ionic liquid structure is nearly unperturbed
by the addition of carbon dioxide.
270–272
Only at high carbon dioxide
concentrations (70 mol %) is a signiﬁcant structure change in the radial pair
distribution functions visible.
273
Furthermore,ab initiomolecular dynamics
simulations highlighted the octahedral voids around the hexaﬂuoropho-
sphate anion as most preferred locations of carbon dioxide in 1-butyl-
3-methylimidazolium hexaﬂuorophosphate.
149
On average, four out of
eight of these interaction sites are free for carbon dioxide. Kazarian, Briscoe
and Welton suggested that cation anion interactions aﬀect the space-ﬁlling
mechanism.
268
Weak interacting ions should possess a higher available free
volume for carbon dioxide. This was conﬁrmed by theoretical investigations
which found a correlation of gas phase ion pair bond strength and carbon
dioxide solubility.
274
Additionally to physisorption, chemisorption can contribute to the
solubility of carbon dioxide in ionic liquids. One example is 1-alkyl-3-
methylimidazolium acetate.
275–277
The basicity of the anion allows the
abstraction of the most acidic proton of the cation, forming a carbene and
acetic acid. Subsequently, the former cation is carboxylated. The avail-
ability of carbenes in 1-alkyl-3-methylimidazolium acetate was used already
12 |Chem. Modell., 2012,9,1–24

in synthesis.
278
Furthermore, the stability of carbenes formed from 1-alkyl-
3-methylimidazolium acetate can be observed by photoelectron spectro-
scopy. Instead of ion pairs, carbene acetic acid complexes dominate the
vapor of neat 1-ethyl-3-methylimidazolium acetate.
279
Thus, a similar che-
misorption of carbon dioxide can be expected for imidazolium ionic liquids
combined with basic anions. While the carbon dioxide chemisorption of
1-alkyl-3-methylimidazolium acetate was unintended, several ionic liquids
functionalized for carbon dioxide uptake were proposed as well.
280–284
The
ﬁrst task-related ionic liquids for carbon dioxide uptake used an amine
group attached to the alkyl chain of imidazolium ionic liquids.
280
Approximately one carbon dioxide molecule per two functionalized cations
is absorbed because one amine group forms a carbamic acid (equ. 2) while
in a second step an ammonium ion is formed (equ. 3).
RﬃNH 2þCO 2
ﬃ!
ﬃRﬃNHCO
2H ð2Þ
RﬃNHCO
2HþRﬃNH 2
ﬃ!
ﬃRﬃNHCO
ﬃ
2
þRﬃNH
þ
3
ð3Þ
Zhanget al.reported a similar chemisorption rate for amino acid ionic
liquids. Higher absorption rates were attributed to physisorption.
281
As
later shown by theoretical and experimental investigations, amino acid ionic
liquids favor a 1:1 chemisorption rate because the second reaction step
(equ. 3) will form an anion with two negative charges in close proximity.
283
Thus, the formation of the carbamic acid is favored over the carbamate.
Therefore, the reported absorption rate of 1:1 for dual amino functionalized
ionic liquids might result mainly from the anion.
282
Unfortunately, all ionic
liquids functionalized for carbon dioxide uptake by using an amine group
share one disadvantage: A high viscosity in the neat and carbon dioxide
reacted ionic liquid. Classical molecular dynamics simulations of neat
1-aminoalkyl-3-methylimidazolium ionic liquids revealed the amine group
as new cation anion interaction site consisting of strong hydrogen bonds.
285
Dual amino functionalized ionic liquids possess hydrogen bonds between
the anions, too.
286
Investigations of the carbon dioxide reacted compounds
reveal a salt bridge network which increase the rotational relaxation time by
2–3 orders of magnitude and decrease translational dynamics as well.
287
As
a result, a gel-like network is formed. Based on these results, Gurkanet al.
developed low viscosity ionic liquids for carbon dioxide uptake.
284
Static
quantum chemical calculations identiﬁed pyrrolide and pyrazolide anions
suitable for carboxylation. Subsequent viscosity measurements of the syn-
thesized compounds showed a nearly unaﬀected viscosity by carbon dioxide
uptake.
284
In agreement with this, translational and rotational dynamics are
similar in the neat and carbon dioxide reacted ionic liquid due to the
absence of a hydrogen-bond network.
288
In summary, physisorption of carbon dioxide is mainly driven by a space-
ﬁlling mechanism.
149,268,269
Weak cation-anion interactions seem to
enhance the solubility of carbon dioxide and it seems that the anion is more
important for a large physisorption.
268,274
Especially the tris(penta-
ﬂuoroethyl)triﬂuorophosphate anion can be recommended if a large phy-
sisorption of carbon dioxide is desired.
289
Chemisorption is possible in
Chem. Modell.,2012,9,1–24 | 13

imidazolium ionic liquids with basic anions,
275–277
ionic liquids with amine
groups,
280–283
pyrrolide anions,
284
or pyrazolide anions.
284
Of special
interest might be the last two because the viscosity is nearly unaﬀected by
carbon dioxide uptake.
284
6 Surface of ionic liquids
Interfaces of ionic liquids are studied intensively by molecular dynamics
simulations. While some studies are more from a fundamental point of view,
others investigate already used unique ionic liquid interface features. For
example, selected ionic liquids stabilize nanoparticles without additives.
290–293
The ﬁrst computational study of an ionic liquid interface was about the gas-
liquid interface of neat 1,3-dimethylimidazolium chloride and mixtures of
this ionic liquid with water.
294
Lynden-Bell observed that the imidazolium
plane arranges perpendicular to the surface in the neat ionic liquid and in
the mixtures. A similar arrangement of the imidazolium plane was proposed
based on ion recoil spectrometry
295,296
measurements on neat imidazolium
ionic liquids before. However, experimental work suggested by comparing
experimental atomic F/C and H/C ratios that the most acidic hydrogen
atom points into or outside the liquid (Fig. 2a and 2b) while a parallel
arrangement of the C-H vector to the surface was observed in the classical
molecular dynamics simulations, see Fig. 2c. In opposite to these studies,
sum frequency generation vibrational spectroscopy reported a parallel
arrangement of imidazolium ring and surface plane, see Fig. 2d.
297,298
The perpendicular orientations of the imidazolium plane to the surface
was excluded for neat ionic liquids due to missing signals in the spectra.
Furthermore, water impurities reorient the cations by realigning the most
acidic proton into the liquid earlier in hydrophobic ionic liquids than in
hydrophilic ones.
298
A simulation study of Bhargava and Balasubramanian
does observe only a slight preference of a vertical arrangement of the
imidazolium ring and the surface and recommended to avoid molecular
cartoons of the surface.
299
Further simulation studies reported diverse
preferred orientation of the imidazolium ring at the surface of neat ionic
liquids, which might strongly depend on the used force ﬁeld or on the
investigated ionic liquid.
300–303
However, experimental and computational
investigations of imidazolium based ionic liquids agree in one point:
Nonpolar alkyl side chains point away from the surface into the gas
phase
299–307
which is also the case in mixtures with water.
308,309
The
hydrophobic surface can be disrupted if a hydroxyl group is attached to the
alkyl chain.
307
Overall, the surface structure of ionic liquids, especially the
(a) (b) (c) (d)
Fig. 2Suggested orientations of the imidazolium ring on the ionic liquid surface. The black
line illustrate the surface plane a, b: ion recoil spectrometry;
295,296
ﬁrst classical molecular
dynamics simulation;
294
d: sum frequency generation vibrational spectroscopy
297,298
14 |Chem. Modell., 2012,9,1–24

orientation of the imidazolium ring, seems to be very sensitve to the selected
force ﬁeld and highlights limits of classical molecular dynamics simulations.
7 Summary
The ionic liquid research area arises at the same time as classical molecular
dynamics simulations of large systems andab initiomolecular dynamics
simulations of medium sized systems become feasible due to technical
progress. Several unique ionic liquid features were uncovered by compu-
tational investigations and later conﬁrmed by experimental investigations
which illustrates the steady increasing role of theoretical chemistry over the
last years. Nonetheless, ionic liquids are a challenge for computational
chemistry, because induction as well as dispersion forces play a signiﬁcant
role for interaction energy and equilibrium structure. Force ﬁelds and
Kohn-Sham density functional methods were developed and validated for
ionic liquids which allow reliable multi-scale investigations of ionic liquids
today.
Acknowledgements
This work was supported by the DFG, in particular by the projects KI-768/
5-2 and KI-768/5-3 from the SPP-IL program and the projects projects KI-
768/8-1. We would like to thank Annegret Stark, Alfonso Pensado, Oldamur
Hollo´czki, Ekaterina Izgorodina and Douglas Macfarlane for helpful
discussion.
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24 |Chem. Modell., 2012,9,1–24

Interaction-induced electric properties
George Maroulis*
DOI: 10.1039/9781849734790-00025
1 Introduction
The basic aspects of the theory of electric polarizability are widely recog-
nized as of fundamental importance to the rational approach and inter-
pretation of large classes of phenomena.
1
In particular, these properties are
of fundamental importance to intermolecular interaction studies,
2
nonlinear
optics,
3–5
scattering
6
and the simulation of ﬂuids.
7,8
Electric properties are
of importance to the modeling of solvation properties.
9
General molecular
characteristics as hardness,
10
softness,
11
hypersoftness,
12
stiﬀness
13
and
compressibility
14
are now systematically linked to electric polarizability and
hyperpolarizability. Understandably, polarizability is also linked to reac-
tivity.
15
The dipole moment, polarizability and hyperpolarizability are
powerful molecular descriptors that ﬁnd use in quantitative structure-
property (QSPR) or structure activity (QSAR) relationships.
16
Interest
extends also to electric moments higher than the dipole, as evidenced by a
recent study on the toxicity of polychlorinated of dibenzofuranes.
17
Con-
siderable progress and interest in electric polarizability is now obvious in
studies focused on the understanding of pharmacological activity.
18–21
The term interaction-induced polarizability, sometimes also called excess
polarizability, signiﬁes the eﬀect caused by the interaction of two molecular
systems. This property is of central importance to the analysis and inter-
pretation of spectroscopic observations incollision- and interaction-induced
spectroscopy. A particularly rich introduction to early eﬀorts in ﬁeld is
available to reader in two ﬁne, classic collections of papers,Phenomena
Induced by Intermolecular Interactions, edited by Birnbaum
22
andCollision-
and Interaction-Induced Spectroscopy, edited by Tabisz and Neuman.
23
We
also recommend two magisterial texts,Collision-induced absorption in gases
by Frommhold
24
andCollisional Eﬀects on Molecular Spectra: Laboratory
experiments and models, consequences for applicationsby Hartmannet al.
25
The aim of this review is to collect signiﬁcant, recent the most, con-
tributions to the calculation of interaction-induced electric properties.
Explicit presentation of theoretical work and experimental eﬀorts is rather
beyond the scope of this text. Nevertheless, we present in this Introduction a
selection of papers that brings forth important and practical aspects of the
interplay of collision-induced spectroscopy and the theory and calculation
of interaction-induced electric properties.
Hunt reported a model that predicts long range, dipoles, quadrupoles
and hyperpolarizabilities for interacting inert-gas atoms.
26
Among other
examples Hunt calculates the long-range quadrupole moment of homo-
nuclear atom pairs as
Department of Chemistry, University of Patras, GR-26500 Patras, Greece.
E-mail: [email protected]
Chem. Modell.,2012,9,25–60 | 25
ﬀcThe Royal Society of Chemistry 2012

Y
AB
zz
¼
5
2
BC6
a
ﬀﬃ
R
ﬃ6
þ...
whereais the dipole polarizability, B the dipole-dipole-hyperpolarizability
and C
6the van der Waals coeﬃcient. Again, Hunt calculates the mean pair
contribution to the second hyperpolarizability as
D
g¼
1
5
Dg
aabb¼
152
2
a
2
gﬃ
7
15
jC6
a
ﬀﬃ
R
ﬃ6
þ...
wherejj
zzzzzzis a tensor of the sixth rank.
Buckinghamet al.
27
reported a theoretical treatment of the hyperpolar-
izability of interacting atoms. They calculate the long range static hyper-
polarizability as
g
ð0Þ
ðRÞ¼2g
ð0Þ
1þ
76
5
að0Þ
4pe
0R
3
ﬀﬃ
2
þ...
"#
which reproduces the classical pair hyperpolarizability obtained by Hunt in
the equation presented above.
Moon and Oxtoby
28
presented a general theory for collision-induced
absorption, which occurs in the near- and far-infrared region of the spec-
trum, in molecules. Speciﬁc results were presented for the case of symmetric
linear (D
1h) and tetrahedral (Td) molecules. The authors subsequently
applied their nonasymptotic theory of the pair dipole moment to the cal-
culation of binary spectral integrals and the far-infrared spectrum for
dinitrogen.
29
The authors also evaluate the contributions to the second-
order multipole model (including the anisotropy of the polarizability, the
hexadecapole moment and the dipole-octopole polarizability).
Bancewiczet al.
30,31
reported an in-depth examination of the determi-
nation of multipolar polarizabilities from interaction-induced Raman
scattering. The authors focused on the Rayleigh wings of isotropic mole-
cules, as tetrahedral CF
4and octahedral SF6and linear molecules, as N2
and CO
2. They also investigated the wings of the vibrational Raman bands
of CH
4,CF
4and SF
6.
Relying on a spectral theory of far-infrared permanent and induced
dipole absorption of diatomic molecules in rare-gas ﬂuids,
32
Rocoet al.
have been able to extract experimental estimates of the electric multipole
moments of carbon monoxide
33
and dinitrogen.
34
Hohm
35,36
extracted dipole-quadrupole (A) and dipole-octopole (E)
polarizabilities for important small tetrahedral molecules, as P
4and CCl4,
from measurements of collision-induced light scattering spectra. Subse-
quently, Hohm and Maroulis reported experimental and theoretical inves-
tigations ofAandEfor molecules as adamantane,
37
osmium tetroxide
(OsO
4),
38
titanium, zirconium and hafnium tertrachloride (TiCl4, ZrCl4,
HfCl
4)
39
and tetramethyl germanium (Ge(CH3)4)
40
.
El-Kaderet al.
41
reported a study of the binary isotropic and anisotropic
collision-induced light scattering spectra of gaseous methane. An analysis of
26 |Chem. Modell., 2012,9,25–60

the high frequency wings based on the collision-induced rotational Raman
eﬀect allowed the estimation of the dipole-quadrupole and dipole-octopole
polarizability of CH
4. In an interesting paper, El-Kader
42
was able to
estimate the contribution of the quadrupole polarizability to the long-range
polarizability of interacting rare gas atoms or methane molecules. The mean
and anisotropy models were ﬁtted to the intensity of the isotropic and
anisotropic interaction-induced light scattering spectra. A reasonable
agreement was found with ab initio calculations of quadrupole polariz-
ability values.
Verzhbitskiyet al.
43
reported a collision-induced Raman band by mix-
tures of sulfur hexaﬂuoride and dinitrogen. A systematic analysis brings
forth evidence of double incoherent scattering (DRS) by this mixture with
both molecules undergoing two Raman-allowed transitions. The band is
found to be almost fully depolarized. In a subsequent paper Chrysos and
Verzhbitskiy
44
extracted the isotropic spectrum of the above-mentioned
band.
Chrysos, Kouzov and co-workers have recently published signiﬁcant
work on the spectroscopy of carbon dioxide. In a theoretical paper, Kouzov
et al.
45
presented a general treatment of collision-induced spectroscopy with
long-range intermolecular interactions. Subsequently, their ideas were
applied to a joint theoretical/experimental collision-induced Raman
scattering (CIRS) study
46
of the Raman-forbiddenn 3band of CO2.A
further paper, treated the collision-induced absorption by CO
2in the far
infrared.
47
Last, Chrysoset al.
48
provided exact and analytic expressions for
the integrated intensity and the width of collision-induced absorption and
collision-induced scattering bands by centrosymmetric linear molecules.
Applied to the CO
2-CO2system, this new treatment brings forth valuable
insights about the spectroscopy of this system, highly important for
atmospheric science.
Hartman and co-workers published recently a series of three papers
reporting molecular simulations for carbon dioxide spectra.
49–51
In the ﬁrst
paper, they treated line broadening and the far wing of then
3infrared band.
In the second one, they investigated the far infrared collision-induced band.
In the third one, they extended their investigation to permanent and colli-
sion-induced tensors contributions to light absorption and scattering. This
is the ﬁrst treatment of gaseous CO
2with a simultaneous treatment of both
permanent and collision-induced contributions without any adjusted
parameters.
2 Interaction-induced electric properties from ﬁnite-ﬁeld calculations:
A conventional approach
An easily accessible approach to the calculation of the interaction-induced
electric properties of a supermoleculeABorAB, consisting of two
interacting molecular entities A and B, is readily available in most electronic
structure programs. We give here just a brief exposition.
We follow the classic papers by Buckingham
52
and McLean and
Yoshimine
53
in all matters pertaining to terminology and notation,
including the basic philosophy underlying their approach.
Chem. Modell.,2012,9,25–60 | 27

The energy (E
p
) of an uncharged molecule interacting with a weak, static
electric ﬁeld can be written as an expansion
E
p
E
p
ðFa;Fab;Fabg;Fabgd;...Þ
¼E
0
ﬃl
aFaﬃ?1=3ÞH abFabﬃ?1=15ÞX abcFabgﬃ?1=105ÞU abcdFabgdþ...
ﬃ?1=2Þa
abFaFbﬃ?1=3ÞA a;bcFaFbgﬃ?1=6ÞC ab;cdFabFgd
ﬃ?1=15ÞE a;bcdFaFbgdþ...
ﬃ?1=6Þb
abcFaFbFgﬃ?1=6ÞB ab;cdFaFbFgdþ...
ﬃ?1=24Þc
abcdFaFbFgFdþ...
ð1Þ
where the variables Fa,Fab,Fabg,etc., are the ﬁeld, ﬁeld gradient,etc. at the
origin of the molecule. The termsin boldare the permanent properties of the
system: energy (E
0
), multipole moments (l
a,H
ab,O
abc,U
abcd), polariz-
abilities (a
ab,A
a,bc,C
ab,cd,E
a,bcd) and hyperpolarizabilities (b
abc,B
ab,cd,
c
abcd). The subscripts denote Cartesian components and a repeated subscript
implies summation over x, y and z. The number of independent components
needed to specify the above tensors is strictly regulated by symmetry.
In practice, the calculation involves three steps: the calculation of the
property for three systemsAB,AandB. If the desired goal is the calcu-
lation of the interaction-induced mean dipole polarizability
a)orsecond
dipole hyperpolarizability (g), one must ﬁrst obtain the Cartesian components
of both tensors for all three systems. The general deﬁnition of the mean is
a¼ð1=3Þa aaandg¼ð1=5Þg
aabb ð2Þ
The interaction induced mean valueaintorgintPintis then calculated as
PintðABÞ¼PðAB?ﬃPðA?ﬃPðBÞð 3Þ
If a non-saturated basis set is used in the calculations, the resulting basis
set superposition error (BSSE) is removed, usually by the widely applied
counterpoise-correction (CP) method,
54
as
PintðABÞ¼PðAB?ﬃPðAX?ﬃPðXBÞð 4Þ
where P(AX) denote calculation of the property for subsystem A in the
presence of the ghost orbitals of subsystem B,etc. It is easy to show that as
the basis set approaches saturation a nearly ideal limit is reached,
P(AX)EP(A) and P(XB)EP(B).
Several computational aspects of the calculation of interaction-induced
electric properties must be considered in theoretical investigations. These
concern the choice of basis set and method. In general, the symmetry of the
AB supersystem might be basically diﬀerent form that of the interacting
subsystems A and B. In the simple case where A and B are spherical atoms,
AB is of D
Nhor CNusymmetry. Basis set dependence and method sensi-
tivity must be carefully examined. The choice of suitable basis sets is known
to be a highly non-trivial matter in hyperpolarizability calculations.
55
One
expects the degree of diﬃculty to increase in interaction-induced hyperpo-
larizability calculations. Another problem of some complexity is the choice
28 |Chem. Modell., 2012,9,25–60

of method.Ab initiomethods are in general more reliable than density
functional theory approaches in electric property calculations. The applic-
ability of high-level conventionalab initiois limited to small and middle-
sized molecules while density functional theory oﬀers an economical
alternative to large-scale calculations. Little is known about the relative merit
of the two classes of methods in interaction-induced (hyper)polarizability
calculations. Clearly, systematic comparisons of their performance are
needed.
3 Theoretical calculations of interaction-induced electric moments
and (hyper)polarizabilities
3.1 Rare gas dimers
Several early theoretical attempts for the determination of the induced
dipole moment of rare gas heterodimers have been recorded.
56,57
Dacre
58
reported dipole polarizabilities for rare gas homodiatoms. Dacre and
Frommhold
59
subsequently used the calculated curves to compute collision-
induced Raman spectra for all pairs He
2,Ne2,Ar2,Kr2and Xe2.We
mention also the calculation of dipole moments and polarizabilities for all
rare gas pairs Rg
1-Rg
2(Rg
1,Rg
2=He, Ne, Ar, Kr and Xe) by Pearson
et al.
60
who used the modiﬁed electron gas Drude model. Experimental
values are also available. Bar-Ziv and Weiss
61
deduced the dipole moment
of He-Ar, He-Kr, He-Xe, Ne-Ar, Ne-Kr, Ne-Xe, Ar-Kr and Ar-Xe from
translational absorption data. Ja¨geret al.
62
extracted the dipole moment of
Ne-Xe, Ar-Xe and Kr-Xe from pure rotational spectra measurements. The
dipole moment of Ne-Ar was estimated by Grabowet al.
63
Last, the dipole
moment of Ne-Kr and Ar-Kr has been extracted from measurements of
pure rotational spectra by Xuet al.
64
We are aware of only one paper
reporting an experimental study of the dipole polarizability of rare gas
dimers. This is the pioneering work of Minemotoet al.
65
who obtained the
dipole polarizability anisotropy (Da)ofAr
2,Kr2and Xe2by laser-induced
alignment. Nevertheless, several attempts have been made to construct
empirical or semiempirical models of the dipole polarizability, able to
reproduce correctly this property for a wide range of internuclear separa-
tions. Cecheriniet al.
66
presented an one-parameter analytical model of the
interaction-induced polarizability of He
2,Ne
2,Ar
2,Kr
2and Xe
2.An
empirical model for the mean (
a) and the anisotropy (Da)ofKr
2was
reported by El-Kader.
67
Last, we mention a state-of-the-art semiempirical
model for the interaction-induced dipole polarizability anisotropy of Ar
2
reported by Chrysos and Dixneuf.
68
The model reproduces accurate
experimental data.
Moszynskiet al.
69
calculated the interaction-induced polarizability of the
helium dimer for the internuclear separations 3rR/a
0r10. Their com-
putational approach relied on spin-adapted perturbation theory (SAPT)
calculations with a large basis set of [5s4p3d2f] size. Subsequently, they
determined the polarized and depolarized Raman spectrum of the dimer.
The computed polarized spectrum displays fair agreement with experiment.
Most important, the computed intensities of the depolarized spectrum agree
quite well with the experimental data reported by Proﬃttet al.
70
Chem. Modell.,2012,9,25–60 | 29

Bishop and Dupuis
71
reported calculations of the interaction-induced
polarizability and second hyperpolarizability of the helium dimer. The
respective mean values
aintandgintwere calculated with a large (12s5p3d2f)
primitive basis set of Gaussian-type functions contracted to [6s5p3d2f].
Their self-consistent ﬁeld (SCF), second- (MP2) and fourth-order (MP4)
Møller-Plesset perturbation theory calculations pertain to a range of
internuclear separations deﬁned by 3rR/a
0r13. The authors presented a
lucid discussion of long-range dispersion and dipole-induced-dipole eﬀects.
They also reported determinations of the second dielectric virial coeﬃcient
and the second virial coeﬃcient of the hyperpolarizability.
Ferna´ndezet al.
72
calculated the frequency dependent interaction-
induced second hyperpolarizability of two argon atoms. Subsequently, they
evaluated the dielectric, the refractivity, the Kerr and the hyperpolariz-
ability second virial coeﬃcients. They obtained the interaction-induced
mean and the anisotropy of the dipole polarizability and the mean second
hyperpolarizability of Ar
2in the range of internuclear separations deﬁned
by 5rR/a
0r30 at the CCSD level of theory, keeping the ten innermost
MO frozen. They tested several basis sets at the experimental bond length of
the argon dimer. Their ﬁnal choice for the calculations was a d-aug-cc-
pVTZ-33211 basis.
One of the most interesting contributions to the ﬁeld, an unparalleled
achievement as regards completeness, is a series of three papers reported by
Ha¨ttiget al.,
73
Kochet al.
74
and Rizzoet al.
75
In the ﬁrst were reported
calculations of the interaction-induced (hyper)polarizability for the helium
and argon dimers. The second paper reported dielectric, refractivity, Kerr
and hyperpolarizability second virial coeﬃcients. Last, the third paper
reported quantum statistical calculations of the dielectric second virial
coeﬃcients. The calculation of the interaction properties was performed at
the full conﬁguration interaction (FCI), CCSD and CCSD(T) levels of theory
with very large basis sets. The dependence of the calculated values on the
internuclear separation (R-dependence) of the helium dimer was obtained
at the FCI/d-aug-cc-pVTZ-3321 and CCSD/d-aug-cc-pV5Z-3321 levels of
theory. For the argon dimer the authors obtained the R-dependence at the
CCSD/d-aug-cc-pVQZ-33211 level. For He
2at the internuclear separation of
5.6 a
0they reported reference CCSD/d-aug-cc-pV6Z-3321 values
aint=
ﬃ0.001109 andDa
int=0.059774 e
2
a
0
2E
h
–1and
g
int=ﬃ0.8961 e
4
a
0
4E
h
–3.
Maroulis
76
calculated the interaction-induced dipole polarizability and
hyperpolarizability of the He
2,Ne2,Ar2and Kr2homodiatoms relying on
ﬁnite-ﬁeld Møller-Plesset perturbation theory and coupled cluster calcula-
tions. Special attention was paid to the design of ﬂexible basis sets, suitable
for interaction-induced electric property calculations. Atom-speciﬁc, pre-
pared basis sets were used on all atoms. The construction is completed in
four steps:
1. A strong, reliable substrate is chosen for the He, Ne, Ar and Kr atoms.
2. The substrate is augmented with diﬀuse s- and p-GTFs.
3. Polarization GTFs are added and their exponents are systematically
chosen to maximize the dipole polarizability (p-GTF for He, d-GTF for Ne,
Ar and Kr).
30 |Chem. Modell., 2012,9,25–60

4. More polarization GTFs are added and their exponents are system-
atically chosen to maximize the quadrupole polarizability (d-GTF for He,
f-GTF for Ne, Ar and Kr).
The basis sets used on the He-He interaction are the contracted [6s4p3d],
[6s4p3d1f] and a large uncontracted basis set (13s11p7d3f). The latter was
used to check the convergence of the results at the SCF level of theory. In
Figs. 1 and 2 we show the R-dependence (i.e.the internuclear separation of
the two He centres) of the interaction-induced hyperpolarizability calcu-
lated at the CCSD(T) level of theory. Figure 1 shows clearly that the
interaction is very anisotropic for short distances. The domination of the
longitudinal interaction componentg
zzzzis clearly brought out in Fig. 2: of
all three invariants of the second hyperpolarizability,D
1gis the most
important term for 2rR/a
0r4. It is worth presenting a few important
characteristics of the interaction. At R/a
0=2, the large (13s11p7d3f) basis
set yields SCF values for the (hyper)polarizability invariants
a=ﬃ0.1490
andDa
=0.7526 e
2
a
0
2E
h
–1, g=39.36 e
4
a
0
4E
h
–3. At the same distance, at
the CCSD(T) level of theory basis [6s4p3d1f] yields
a=ﬃ0.1307 and
Da
=0.8361 e
2
a0
2Eh
–1, g=56.35 e
4
a0
4Eh
–3. Thus, electron correlation eﬀects
are of some importance for short distances. The diﬀerences between
the [6s4p3d] and [6s4p3d1f] at R/a
0=2 are not substantial. Around the
estimated experimental equilibrium separation of 5.6 a
0,for the interval
3oR/a
0o8, the
a,Daandginvariants vary as
2345678
–50
0
50
100
150
200
250
He-He
CCSD(T)
Basis set [6s4p3d1f]
He-He γ
int
/e
4
a
0
4
E
h
–3
R/a
0
γ
zzzz
γ
xxxx
γ
xxzz
Fig. 1R-dependence of the Cartesian components of the interaction-induced second hyper-
polarizability of two He atoms.
Chem. Modell.,2012,9,25–60 | 31

ﬀaðR޼ﬃ0:0012þ0:0025ðRﬃR
e?ﬃ0:0039ðRﬃR eÞ
2
þ0:0034ðRﬃR eÞ
3
ﬃ0:0009ðRﬃR eÞ
4
DaðRÞ¼0:0638ﬃ0:0245ðRﬃR eÞþ0:0021ðRﬃR eÞ
2
ﬃ0:0007ðRﬃR eÞ
3
þ0:0005ðRﬃR eÞ
4
ﬀgðR޼ﬃ0:74þ1:91ðRﬃR
e?ﬃ1:36ðRﬃR eÞ
2
þ0:06ðRﬃR eÞ
3
þ0:10ðRﬃR eÞ
4
A [7s5p4d1f] basis set was used in the calculations on the Ne2homo-
diatom. The interaction mean polarizability of this system is systematically
negative for short distances while the anisotropy is positive. The interaction
second hyperpolarizability is given in Fig. 3. The R-dependence of the
invariants
g,D1gandD 2gdisplays the same patterns as in the case of He2.
TheD
1gcomponent is signiﬁcantly large. Very important electron correla-
tion eﬀects are obvious at a short distance as R/a
0=3. The respective
CCSD(T) values are (SCF values in parentheses)
g=14.5 (1.8),D 1g=618.6
(240.6) andD
2g=113.1 (57.1) e
4
a0
4Eh
ﬃ3.
The interaction (hyper)polarizability of two Ar atoms was calculated with
a [8s6p5d4d] basis set at the SCF and MP2 levels of theory. At the shortest
separation of R/a
0=4 the MP2 values of the dipole polarizability invariants
are (SCF values in parentheses) are
a=2.4619 (1.5679) andDa =16.1488
(13.7622) e
2
a0
2Eh
ﬃ1. The interaction mean second hyperpolarizability at the
2345678
0
200
400
600
800
1000
He-He
Basis set [6s4p3d1f]
CCSD(T)
γ
Δ
1
γ
Δ
2
γ
He-He γ
int
/e
4
a
0
4
E
h
–3
R/a
0
Fig. 2R-dependence of the invariants of the interaction-induced second hyperpolarizability of
two He atoms.
32 |Chem. Modell., 2012,9,25–60

same distance isg=8700.3 (5302.8) e
4
a0
4Eh
ﬃ3. In Fig. 4 we show the R-
dependence of the interaction second hyperpolarizability invariants. Very
much as in the case of the He-He and Ne-Ne systems, the longitudinal
componentg
zzzzand consequently theD
1ginvariant are dominant.
2345678910
0
100
200
300
400
500
600
700
Ne-Ne
CCSD(T)
Basis set [7s5p4d1f]
γ
Δ
1
γ
Δ
2
γ
Ne-Ne γ
int
/e
4
a
0
4
E
h
–3
R/a
0
Fig. 3R-dependence of the invariants of the interaction-induced second hyperpolarizability of
two Ne atoms.
24681012
0
20000
40000
60000
80000
100000 Ar-Ar
MP2
Basis set [8s6p5d4f]
γ
Δ
1
γ
Δ
2
γ
Ar-Ar γ
int
/e
4
a
0
4
E
h
–3
R/a
0
Fig. 4R-dependence of the invariants of the interaction-induced second hyperpolarizability of
two Ar atoms.
Chem. Modell.,2012,9,25–60 | 33

The interaction-induced (hyper)polarizability of two Kr atoms was cal-
culated with a [8s7p6d5f] basis set at the SCF and MP2 levels of theory. At
R/a
0=4, the MP2 values of the interaction (hyper)polarizability invariants
are (SCF values in parentheses)a=3.6127 (2.4113) andDa =24.8369
(21.8086) e
2
a
0
2E
h
ﬃ1. The interaction mean second hyperpolarizability at the
same distance is g=17837.0 (11719.3) e
4
a0
4Eh
ﬃ3. The electron correlation
eﬀects are important for all properties at short separations. The R-depen-
dence of the polarizability invariants and the Cartesian components and the
mean of the second hyperpolarizability are shown in Figs. 5 and 6, respec-
tively. In both cases, the strong anisotropic character of the (hyper)polar-
izability for short separations is very much obvious.
A very accurate determination of the interaction-induced polarizability of
He
2at the experimental internuclear separation of 5.6 a
0was reported by
Jaszun´skiet al.
77
The authors used a very large 11s8p6d5f4g3h basis set for
He and high-precision explicitly correlated R12 methods. Their most
accurate results for the mean and the anisotropy polarizability were cal-
culated at the CCSD(T)-R12 level of theory and are are
aint=ﬃ0.00104
andDa
=0.06179 e
2
a0
2Eh
ﬃ1. These values represent reference estimates of
the interaction-induced dipole polarizability of two helium atoms.
Maroulis and Haskopoulos calculated the interaction-induced dipole
moment, polarizability and ﬁrst hyperpolarizability of NeAr.
78
They per-
formed ﬁnite-ﬁeld ab initio and density functional theory calculations with
two ﬂexible basis sets: A
=[7s5p4d1f/8s6p5d3f] and B=[7s5p5d3f/
8s6p5d4f]. The construction of these atom-speciﬁc basis sets has been
reported in earlier work. Their completeness is tested by calculating the
dipole (hyper)polarizability and the quadrupole polarizability of the Ne and
4681012
0
5
10
15
20
25 Kr-Kr
MP2
Basis set [8s7p6d5f]
α
Δα
Kr-Kr α
int
/e
2
a
0
2
E
h
–1
R/a
0
Fig. 5R-dependence of the interaction-induced polarizability invariants of two Kr atoms.
34 |Chem. Modell., 2012,9,25–60

Ar atoms. The comparison of the performance of the two basis sets is very
satisfactory. For the shortest internuclear separation of 4 a
0they obtain
SCF and MP2 values of the interaction-induced mean hyperpolarizabilityb
int=12.7 and 15.1 e
3
a0
3Eh
ﬃ2for both basis sets A and B. With basis A they
calculate CCSD and CCSD(T) values
b
int=15.1 and 14.5 e
3
a
0
3E
h
ﬃ2. Last,
at the B3LYP/B level of theory they report a slightly higher value 15.3
e
3
a0
3Eh
ﬃ2. In Figs. 7 and 8 are shown the R-dependence of the SCF,
CCSD(T) and B3LYP values of the interaction-induced mean and aniso-
tropy of the dipole polarizability. It is obvious that the B3LYP is more
successful for the mean than the anisotropy of the polarizability. In Fig. 9 is
shown the mean and anisotropy of the interaction induced ﬁrst hyperpo-
larizability for R
=4a
0. Again the B3LYP method is in better agreement
with the ab initio values for the mean (
b
int) and less so for the anisotropy
(Db
int). Last, a polynomial ﬁt of the induced dipole moment around the
potential minimum R
e=6.57624644 a0shows that at the SCF/A and
CCSD(T)/A levels of theory this property varies as
mðRÞ¼0:0025ﬃ0:0039ðRﬃR
eÞþ0:0031ðRﬃR eÞ
2
ﬃ0:0022ðRﬃR eÞ
3
þ0:0008ðRﬃR eÞ
4
mðRÞ¼0:0019ﬃ0:0034ðRﬃR eÞþ0:0030ðRﬃR eÞ
2
ﬃ0:0022ðRﬃR eÞ
3
þ0:0008ðRﬃR eÞ
4
The two methods produce similar results.
4681012
0
10000
20000
30000
40000
50000
60000
70000
Kr-Kr
MP2
Basis set [8s7p6d5f]
γ
zzzz
γ
xxxx
γ
xxzz
γ
Kr-Kr γ
int
/e
4
a
0
4
E
h
–3
R/a
0
Fig. 6R-dependence of the Cartesian components and the mean interaction-induced second
hyperpolarizability of two Kr atoms.
Chem. Modell.,2012,9,25–60 | 35

45678910
–0.4
–0.3
–0.2
–0.1
0.0
NeAr
α
int
/e
2
a
0
2
E
h
–1
R/a
0
SCF
CCSD(T)
B3LYP
Fig. 7R-dependence of the mean interaction-induced polarizability of NeAr. SCF and
CCSD(T) values with [7s5p4d1f/8s6p5d3f]. B3LYP with [7s5p5d3f/8s6p5d4f].
45678910
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
NeAr
Δα
int
/e
2
a
0
2
E
h
–1
R/a
0
SCF
CCSD(T
B3LYP
Fig. 8R-dependence of the interaction-induced anisotropy of the polarizability of NeAr. SCF
and CCSD(T) values with [7s5p4d1f/8s6p5d3f]. B3LYP with [7s5p5d3f/8s6p5d4f].
36 |Chem. Modell., 2012,9,25–60

The interaction-induced dipole polarizability and second hyperpolariz-
ability of two neon atoms was reported by Ha¨ttiget al.
79
They subsequently
used the calculated values along with an accurate potential for Ne
2to
estimate the refractivity and hyperpolarizability second virial coeﬃcients of
gaseous neon. The calculation ofaint,Daintandg
intwas performed at the
CCSD level of theory with a d-aug-cc-pVQZ-33211 basis set. The R-
dependence of the interaction-induced electric properties was obtained at a
range of internuclear separations deﬁned by 3rR/a
0r20.
Marouliset al.
80
constructed ﬁve new, carefully optimized, xenon-speciﬁc
gaussian-type basis sets. Their size ranges from B1
=[9s8p7d1f] to
B5
=[9s8p7d5f3g]. The interaction-induced dipole polarizability of the
xenon dimer was calculated for a wide range of internuclear separations
4oR/a
0o50. At the estimated equilibrium of Re=8.5 a0
81and in the region
of 7oR/a
0o10, the anisotropy decreases monotonically as
DaðRÞ¼7:42ﬃ2:24ðRﬃR
eÞþ0:31ðRﬃR eÞ
2
The calculated anisotropy is in essential agreement with the experimen-
tally determined value reported by Minemotoet al.(see Ref. 65).
The interaction-induced dipole moment, polarizability, ﬁrst and second
hyperpolarizability of the HeNe, HeAr and NeAr heterodimers was cal-
culated by Lo´pez Cacheiroet al.
82
The properties reported arem int,
aint,
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
R = 4 a
0
NeAr
CCSD(T)
CCSD
MP2
SCF
β (B3LYP) = 15.3
Δβ (B3LYP) = 11.3
β
int
/e
3
a
0
3
E
h
–2 β
Δβ
Fig. 9Method-dependence of the interaction-induced ﬁrst dipole hyperpolarizability of NeAr
at R
=4a0. Basis set [7s5p4d1f/8s6p5d3f]. B3LYP values with [7s5p5d3f/8s6p5d4f].
Chem. Modell.,2012,9,25–60 | 37

Daint,b
intandg
int. The calculations were performed at the CCSD level of
theory with a d-aug-cc-pVQZ-33211 basis set, for internuclear separations
ranging from 3 to 20 a
0.
Haskopouloset al.
83
reported SCF, MP2 and B3LYP calculations of the
interaction induced dipole moment and (hyper)polarizability of the KrXe
heterodiatom. They employed a [8s7p6d5f/9s9p7d5f] basis set composed of
atom-speciﬁc parts for Kr and Xe. In a small neighborhood around the
experimental bond length of R
e=7.94 a0(see Ref. 62), the interaction
properties vary as
m
intðRÞ=ea 00:0036þ0:0061ðRﬃR e?ﬃ0:0048ðRﬃR eÞ
2
þ0:0018ðRﬃR eÞ
3
ﬀa
intðRÞ=e
2
a0
2Eh
ﬃ10:04þ0:09ðRﬃR e?ﬃ0:04ðRﬃR eÞ
2
DaintðRÞ=e
2
a0
2Eh
ﬃ15:37ﬃ1:58ðRﬃR eÞþ0:27ðRﬃR eÞ
2
ﬃ0:04ðRﬃR eÞ
3
ﬀ
b
intðRÞ=e
3
a0
3Eh
ﬃ28:72þ7:81ðRﬃR e?ﬃ2:23ðRﬃR eÞ
2
þ0:15ðRﬃR eÞ
3
The predicted equilibrium valuem
int(R
e)=ﬃ0.0036 ea
0is quite close to
the experimental estimate of9m
int9=0.003 reported by Ja¨geret al.(Ref. 62).
The R-depenence of the Cartesian components of the induced polarizability
are shown in Fig. 10. The electron correlation eﬀect on the Cartesian
components of the ﬁrst hyperpolarizability is shown in Fig. 11.
0 1020304050
–10
0
10
20
30
40
50
Interaction-induced
polarizability
MP2
[8s7p6d5f/9s8p7d5f]
KrXe
α
αβ
/e
2
a
0
2
E
h
–1
R/a
0
α
zz
α
xx
Fig. 10Longitudinal and transversal component of the induced dipole polarizability of KrXe.
38 |Chem. Modell., 2012,9,25–60

Haskopoulos and Maroulis
84
calculatedab initioand DFT values of
the interaction dipole moment of Rg-Xe (Rg
=He, Ne, Ar, Kr). At
the CCSD(T) level of theory and the respective equilibrium bond
lengths, the interaction dipole moment varies as m
int(Rg-Xe)/
ea
0=ﬃ0.0025(He),ﬃ0.0047(Ne),ﬃ0.0055(Ar), andﬃ0.0037 (Kr). The
R-dependence of the dipole moment was also calculated.
Marouliset al.
85
calculated the interaction-induced dipole moment and
(hyper)polarizability of the HeAr heterodiatoms. The calculations were
performed at the SCF and MP2/[6s4p3d1f/8s6p5d3f] level of theory. The
obtained R-dependence of the interaction properties was subsequently
applied to the calculation of the collision-induced hyper-Rayleigh (CIHR)
spectra both quantum mechanically and semi-classically for the frequency
shifts up to 1200 cm
ﬃ1
. The R-dependence of the mean and the anisotropy
of the interaction-induced ﬁrst hyperpolarizability is given in Figs. 12 and 13.
It is observed that electroc correlation eﬀects are particularly signiﬁcant
for short internuclear separations.
G"azet al.
86
calculated collision-induced hyper-Rayleigh (CIHR) for
HeNe at temperatures of 95 and 295 K. Both quantum mechanical and
classical approached were applied, for frequency shifts up to 1000 cm
ﬃ1
.
The calculations were based onab initiodata obtained at the SCF and MP2
level of theory with atom-speciﬁc basis sets for He
=[6s4p3d1f] and
Ne
=[9s6p5d1f]. These basis sets yield SCF values for the dipole polariz-
abilitya
He=1.322 anda Ne=2.368, to be compared to the numerical
4 6 8 10 12 14 16 18 20 22
–50
–40
–30
–20
–10
0
10
20
Interaction-induced
hyperpolarizability
KrXe
MP2
[8s7p6d5f/9s8p7d5f]
β
αβγ
/e
3
a
0
3
E
h
–2
R/a
0
β
zzz
(SCF)
β
zzz (MP2)
β
zxx
(SCF)
β
zxx
(MP2)
Fig. 11R-dependence of the ﬁrst hyperpolarizability components of KrXe.
Chem. Modell.,2012,9,25–60 | 39

Another Random Document on
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not cross her own. But she had so many private side-paths down
which she was liable to wander, that one never knew for certain
where she would come out next, or how she would act in any given
set of circumstances. But as long as doing a kindness to another did
not interfere with what she desired herself, she was always ready,
even at the cost of trouble and personal exertion, to help her friends
if they approached her in the proper spirit, which implied a good
deal of abasement. She had been in her time a very considerable
political intriguer, and, following her invariable rule of always getting
whatever she wanted, she had built up her husband into the edifice
of the Conservative Government. But the game—for it had never
been more to her than that—had now ceased to amuse her, and she
cared no longer how greatly her poor Ardingly floundered in the
spacious halls of the Admiralty. This he seldom failed to do. She was,
finally, the very antipodes of those women who, because generals
and statesmen tell them things not generally known, consider
themselves, in that they are at the centre of things, as wielding
some vague political influence, and fly about telling all their friends
what everybody has said. Lady Ardingly never flew about; she sat
quite still and gave orders. Why people did as she told them they
never quite knew; it arose, perhaps, from her habit of always being
right.
Ardingly House was a vast and modern erection in Pall Mall. "So
convenient for Ardingly," as his wife used to say in her slow foreign
speech, "now that he is at the Admiralty. He can come home to
lunch, and tell me all the blunders he has made since breakfast. And
there is plenty of time for him to take two steps and make them all
over again before dinner." Not long ago, at the time when Mrs.
Maxwell was house-hunting, she had heard a vague rumour that
there was a possibility of this mansion being in the market, and had
had the temerity to call on Lady Ardingly to know if it was so. She
heard her in silence, not helping her out at difficult points, and then
remarked: "Yes, we are going to sell it, and live at Clapham Junction.
So convenient a train service." This Mrs. Maxwell had rightly
interpreted to be a denial of the rumour, and had quitted the subject

with some precipitation. It was also characteristic of Lady Ardingly
that she did not fly about town, making the place ring with the story.
Here, perhaps, lay one of the secrets of her effectiveness: she never
dissipated her energy.
It was to this lady that Mrs. Brereton decided to carry her doubts
and perplexities. There was only a small dinner-party that night, and
before the men left the dining-room she found herself sitting by her
on a sofa. Lady Ardingly happened to be in an admirable temper,
and the opportunity was golden.
"I have not seen you for very long, dear Mildred," said she. "Tell me
your news. How is Jack Alston? Have you seen him lately?"
This kind of frankness even Mildred found a little embarrassing. Lady
Ardingly, of course, knew everything about everybody, and never,
except when there was something to be got by it, assumed
ignorance.
"Jack Alston? Oh, yes, I constantly meet him, in the way one does
meet in London," she said rather foolishly.
"Yes, dear, I know you are great friends. Who does not? Do you
hope he will get a Government post after the election? Tell me; I am
really asking for news."
"Well, Jack hopes for it, of course. The War Office is what he is
running for."
"The War Office? He knows about rifles and powder, does he not?
Well, there is a feeling just now for having men who know their
work. Ardingly, I find, is reading Nelson despatches. Very nice for
him. What is there of news? Never mind politics; they are dull. Some
scandal."
"They say Mrs. Alington has made a mess of her affairs," said
Mildred. "I always knew she would, dabbling in the mining-market
like that. Her husband is furious."
"Ah! Now, I wonder who can have told you that? I saw Alington only
this evening. It is not so at all. They are the best of friends. What

else?"
"Did you hear about Jim Netson? I am told he was down at Brighton
on Sunday with——"
"Dear Mildred, where can you get these things from?" asked Lady
Ardingly. "Jim Netson was lunching with me on Sunday. What else?"
Mildred found it difficult to bear this sort of thing quite good-
naturedly. Like many other women, she repeated what she heard,
adding a little here and there, not caring particularly about the truth
of a story so long as it amused. But Lady Ardingly contradicted her
flat, and, the worst of it was, she was invariably right. She did not in
the least care for made-up stories, and Mildred, who was by way of
being a well-informed woman on the matter of other people's
backyards, was rather nettled. But she swallowed her pique and
laughed.
"Dear Lady Ardingly," she said, "it is no use my telling you things.
You always know best and most."
Lady Ardingly took some coffee, and as she removed the cup from
the tray, the spoon clattered on the floor.
"Clumsy fool!" she said to the footman, and without a pause: "You
have got something on your mind, Mildred. What is it? Always get
things off your mind, my dear, as soon as possible. It is very
enfeebling to worry. Is it"—and her eye fell on Maud, who was
talking in a group on the other side of the room—"is it about your
daughter? She is getting a big girl. It is time you married her."
Mrs. Brereton gave a little staccato note of admiration.
"You are too wonderful!" she said. "Yes, it is exactly that. Anthony
Maxwell wants to marry her."
"Very nice. The son of the great Mr. Maxwell, you mean?" asked
Lady Ardingly, without the slightest inflection of irony.
"Yes."
Lady Ardingly laughed.

"What a pity we did not sell them this house! Maud would have been
mistress here," she said. "At present she does not wish to marry
him. Is it so? I do not wonder, dear Mildred, at a momentary
hesitation. Do you? But it would be a very good marriage for her."
"So I have told her."
"Then, do not tell her so again. Ah, here come the men! Let us play
Bridge immediately. Only I will not play with your husband, dear
Mildred. I would sooner play with a groom out of the stables. We will
have two tables, and he shall be at the other one. Send Maud here a
moment. I will speak to her."
Mrs. Brereton rose with alacrity.
"Dear Lady Ardingly, you are too kind!" she said with heartfelt
gratitude.
"And do not put your oar in, my dear," said Lady Ardingly
impassively.
Maud, looking very shy and tall, came in obedience to the summons.
"You are too unkind, dear Maud, to an old woman," said Lady
Ardingly. "You have not said a word to me all the evening, and now
we are going to play Bridge. They all insist on playing Bridge. You
would like to play with your father, would you not? We will arrange a
table for you. Yes, that will be very pleasant. You must come and
talk to me one of these days quite quietly. To-morrow—no, to-
morrow will not do. Come to lunch with me on Friday. What a tall girl
you are! and, my dear, do you know you are wonderfully handsome?
Now they want me to play Bridge."

CHAPTER VI
It was Sunday afternoon, and Riversdale, by reason of the gaiety
gathered there, had eclipsed the gaiety of all other places. Some
dozen people were staying in the house, but the most of them had
come down from London to spend the afternoon and return after
dinner, and the lawns, which the company of blameless fools had
caused to wear their most ravishing appearance, were suitably
crowded. A set of croquet-hoops had been put up on one, and a
game was proceeding in the orthodox Sunday afternoon style; that
is to say, a nervous, palpitating little man, to whom at the moment
croquet seemed of more importance than his eternal salvation, was
busy, with a tea-party of four balls, separating adversaries and
making hoops with intolerable precision, while a long, willowy girl,
his partner, trailed after him in his triumphal progress and gave faint
and languid sounds of sycophantic applause.
"There you see they are separated, Miss Martin," said the zealot at
length, "and now I'll mobilize with you. Then you can make your
hoop next time, and I ought to go out."
"Yes, it's quite too beautiful," said Miss Martin; "but I know I'll miss.
Oh, it's not my turn, is it? Where are they gone?"
"They" at this moment—a Guardsman of the most pronounced type
and a middle-aged woman of the most un-middle-aged type—being
weary with this faultless exhibition, had retired to a seat at the far
end of the garden, and were talking very low and laughing very
loud. They were recalled with difficulty, still lingering on the way, and
the unpromising situation was carefully explained to them by the
palpitating man in a voice in which the endeavour not to appear
jubilant was rather too marked. It being the lady's turn, she chipped
her ball sideways at about right angles to the required direction,
and, without even affecting to look where it had gone, dropped her

mallet in the middle of the lawn, and instantly retired with her
Guardsman again.
Elsewhere other groups were forming and dispersing. In the new
wooden shelter Lady Ardingly had taken up her permanent position
at the Bridge-table, and, while others cut in and out, kept her seat
with tree-like composure, and played rubber after rubber with a
success which appeared monotonous to her adversaries. Anthony
Maxwell occasionally took a hand at her table, and in the intervals
chased Maud Brereton from terrace to terrace with a hunter's
pertinacity, conscious of the approving eye both of his mother and of
Maud's. The fathers of them both would no doubt have viewed his
employment with equal approbation, had they not been deeply
engaged in a secluded corner in trying to rook each other at piquet,
each, however, finding to his indescribable dismay that he had
caught a Tartar. Like many very rich men, they played for very low
stakes, and exhibited an inordinate greed for half-crowns, and even
smaller coins.
Jack Alston and his wife had been among the guests who came
down from the Saturday till Monday, but he had gone over for the
day, rather to Mildred's disgust, to a neighbouring golf-links, and
would not be back till dinner. Marie, however, had been, so Mildred
considered, at her very best all the afternoon, conferring, as she in
some mysterious manner had always the power to do, an air of
distinction and success to the party. Wherever she was there was a
crowd; wherever she was there was more constant laughter, more
animated conversation. She had the gift, rare and inimitable, of
making people play up. Dull folk aroused themselves when she
talked to them, brilliant people coruscated, for there went from her,
an unconscious but pervading emanation, some air of freshness and
vitality, which acted like a breath of wind in a close atmosphere,
reviving and bracing. At present she was talking to Lady Devereux
and Arthur Naseby, who wore a straw hat which was strangely
unsuitable to him and appeared stouter than ever, in the
comparative privacy of the lower lawn.

"Ah yes," she was saying, "that is just the fault with us all now. We
think we can be amused merely by having people to amuse us. It is
not so; being amused depends almost entirely on one's self. Some
days nothing amuses one; on others one is amused by the other sort
of nothing."
"It's always the other sort of nothing with me," said Arthur Naseby.
"And what I like really best of all is the pantomime. You find in the
pantomime exactly what you take there. I take there an invincible
gaiety. That is why I find it there."
"That's what I mean," said Marie. "It is the case with everything. I
love the pantomime, like you. Everything takes place without the
slightest reason. It is so like life; and, like the clowns, we belabour
each other with bladders and throw mud at butter belonging to
other people. But the audience—the part of it like Mr. Naseby and
me—are enormously amused."
"You are horribly unjust, Marie," said Lady Devereux in her sleepy,
drawling voice. "We never belabour you. You are a privileged
person; you go flying over hedges and ditches, while if I, for
instance, as much as look over a hedge, I am supposed to be there
for no good purpose. Is it the consciousness of innocence that gives
you such license! One can acquire almost anything by practice. I
think I shall set about that."
Marie laughed.
"I would, dear. Be innocent for an hour a day, to begin with, and
increase it by degrees."
"Ah, it's not innocence, but the consciousness of it, I want," said
Blanche. "It is a different matter."
"But it leads to absolutely nothing," said Arthur Naseby, in a
discontented voice, "except, perhaps, promotion in the Church; but I
have given up all real thought of that."
"I thought the real way to get on in the Church now was to preach
heretical doctrine," remarked Lady Devereux. "Our parson at Rye

always casts doubt on things like Jonah and the whale, or tries to
explain them by supposing it was not a whale, but an extinct animal
with an enormous gullet, which seems to me just as remarkable.
They tell me he is certain to be made a Bishop. My grandfather was
a Bishop."
"And mine was a draper," said Arthur Naseby. "I am thankful every
day that he was such a successful one. Really, nothing matters
nowadays except money. That is so convenient for the people who
have some. Here is a most convenient person, for instance, just
coming."
Jim Spencer entered the tent with the air of looking for somebody.
He also had the air of having found somebody when he saw Marie,
and sat down in a low chair by her.
"I have been playing croquet," he said; "but I shall never play
again."
"What happened?"
"Nothing happened. I remained in sublime inactivity, except when
other people used me for their own base ends. I never felt so useful
in my life."
"But that, again, is no use," said Arthur—"like the consciousness of
innocence which Lady Devereux means to cultivate. Being simply an
opportunity for other people seems to me the very type of a wasted
life. I am continually being an opportunity for other people, and the
opportunity I give them is to make unkind remarks about me; they
constantly take advantage of it."
"What do they say?" asked Marie.
"They say I am idle, and therefore probably vicious. Now, nothing
was ever less proved than that; it is a perfect fallacy, entirely due to
that pessimistic person who said that Satan finds some mischief still
for idle hands to do. That I am idle is, of course, quite true. For
thirty years I have been very busy doing nothing whatever, and
every day I live I find more nothing to do, if you understand."

"Then, you allow the world doesn't libel you?" said Lady Devereux.
"Certainly it does. It is that to which I so strongly object. People go
about saying all sorts of things about me which are perfectly true.
The greater the truth, the greater the libel."
Marie got up from her chair.
"It is true that the world has a keen grasp of the obvious," she said.
"Why don't you disappoint them, Mr. Naseby, and do something?"
"I am ready to do almost anything in the world," said he, "for a
suitable inducement; but nobody ever induces me."
"Well, I shall go for a stroll," said Marie, "and expect neither
inducement nor companionship unless any one is inclined."
Jim Spencer got up instantly.
"Please let me come," he said.
The two left the tent, but Arthur Naseby and Lady Devereux
continued to sit there. There was a moment's pause, and then in a
shrill whisper, "Yes, the case certainly presents some points of
interest," said he; "and as a consulting doctor, although nobody has
shown the slightest desire to consult me, I don't see why I shouldn't
give my diagnosis. Briefly it is this: This exceeding warm weather
will undoubtedly cause the snowflake to melt; if it does not, it is no
true snowflake. But it must be, for anything but a snowflake would
have melted long ago; in fact, it is proved."
Lady Devereux considered this.
"Marie is a great friend of mine," she said; "but I have one criticism
to make upon her: Her extraordinarily healthy way of looking at
things cannot be genuine; she would not be human if it was. She
gave me a lecture the other day about the vulgarity of lying down to
be trampled on. Now, any one that was human would know that that
is just about the only thing in the world worth doing. Personally, I
consider it an instance of the wonderful self-abandonment and self-
sacrificing character of love."

"And she wouldn't even call it love," said Arthur.
"No; she would use some perfectly antiquated and shocking word.
Now, whatever I am, I am not antique. It is absurd to treat me as if
I was Old Testament history. But Marie is a great dear. She has been
too sweet about the bazaar, and has promised to hold a stall every
day."
"I never can quite make out what people see in her," said Arthur. "Of
course I adore her, simply because one has to—it is unheard of not
to—but is there anything there after all, except—except what one
sees?"
"Yes, of course there is," said Blanche. "There is in her all that you
and I and the rest of us are without. To put it baldly, she is a good
woman. You get force from being good if you are clever as well. Yes,
you may laugh, but it is so true. Now, the rest of us are not good—
neither you, nor I, nor dear Mildred."
"But Andrew is," said Arthur.
"That is why one never knows whether he is in the room or not,"
said Lady Devereux. "He is, or may be, good; but there is nothing
else there whatever. Mere goodness is pretty colourless by itself; but
Marie is everything else, and good as well. She is about five times as
clever as all of us. She has tact, else she would have made rows
long ago; she is a woman of the world, but she is also good."
"I suppose that is probably why I am never quite comfortable with
her," said Arthur in a mild, ruminating voice.
"Very likely. It is also why you are quite wrong in your diagnosis just
now. Oh, there's Lady Ardingly looking for people to make up her
table. She has probably cleaned everybody else out. Come, Arthur,
let us go and be cleaned out too."
They both laughed loudly and went.
Marie and Jim Spencer meantime had strolled away from the crowds
on the lawn towards the meadow and the river. Even though he had
been only a fortnight or so back in England, he had begun clearly to

recognise that his experiment of going away, his self-banishment to
South Africa in order to win back freedom from the spell which she
had cast on him, had been a failure. He had thought that by filling
his mind with other interests, by drugging his soul with the pursuit
of gold, as you can drug an aching body into unconsciousness, he
would still that pain. So, indeed, he had done for the time, but the
opiate, it appeared, was not permanent in its effects; the drowsiness
had passed off, and again at the sight of her his love had awoke. It
seemed, too, to him now that he loved her with a more devout
passion than ever before; all the old longing was there with this
added—that his heightened and matured perception could now
appreciate how fine she was; how different from the jostling race
that swirled round her, who clutched like greedy children with both
hands at the two things they alone thought worthy of effort:
pleasure, at whatever cost or violation; and money, which was worth
any sacrifice except that of pleasure. Like the whole of the rest of
London, he knew the intrigue which Jack had been carrying on for
years, and which was now so stale that it had almost ceased to form
a subject for gossip, and this thought was bitterly poisonous to his
mind. Could it be possible, he wondered, that Marie knew and
condoned it? that she had accepted that for which there was no
remedy but divorce, played gooseberry to her husband, and knew
what were his relations to the woman whose hospitality she was
even now enjoying? That she and Jack had drifted into the apathetic
estrangement which so often is the result of childless marriages, he
did not doubt; but was the reason for it that which was so well
known to everybody else? Again and again during this last fortnight
this unworthy and debasing suspicion had assailed him, and, to do
him justice, he had as often cast it from him, his trust and whole-
hearted belief in her rejecting and strangling it; and as often as it
presented itself, he vowed that he would give it no home. But the
other alternative, the only other possible, though it left her stainless
and unsullied, was hardly less painful; and it was an intolerable
thought to him that she alone should be ignorant of that of which all
his better mind told him she was ignorant. Three-quarters of the
world, no doubt, if they ever gave a thought now to a piece of

scandal which had long outlived its first youth, commended her for
her admirable common-sense in recognising the folly of making a
fruitless public exhibition of her private affairs; the other quarter no
doubt wondered idly how long her blissful ignorance would continue,
and saw material for drama the moment that enlightenment came.
And in this wonder he could not help joining—what would she do if
ever she found out? Her worldly wisdom would assuredly indicate a
direction completely opposite to that in which her moral sense would
point. That there would be a struggle he regarded as inevitable; but
even he, knowing her as well as he did, could form no conjecture as
to which way it would go. Marie accepting what had happened, and
not quarrelling with the irremediable, made a picture unpaintable;
but Marie, living the life of a woman who had separated herself from
her husband, was almost equally outside possibilities. He had a
vague sense of approaching storm and brewing mischief, remote it
might be, but marching inevitably nearer, even as in some spell of
sultry and oppressive days we know that it is only through thunder
and a convulsion of elements that we can get back to cool and dewy
mornings, and again regard sunshine as a friend, not as a thing to
be shunned and shrunk from.
It may have been that the vividness with which he was conscious in
every fibre of threatening disaster was communicated by some
subtle brain-wave to her; in any case, her first words as they walked
down the shady path below the full-fledged elms bore very distinctly
on that which filled his mind.
"How hot it is!" she said. "There will surely be a storm."
The echo made by her audible voice to his inaudible thought startled
him.
"What sort of storm?" he asked quickly, still busy on his own ground.
She laughed.
"So you have been thinking of storms, too," she said. "We often
used to think in harness—do you remember, Jim? What sort of
storm? Well, I too had other storms than thunder in my mind. You

used to dislike real thunder-storms, I remember; but I always loved
them. I expect other sorts of storms affect one similarly. I hate
compromise, you know. If one is absolutely at cross-purposes with
other people, it is much better to have it out fair and square, to
upset the furniture and smash the china if necessary, rather than
concede a little here and have a little conceded there. That always
results in a state of things no better than before, and an added
distaste on both sides to open the subject again."
He did not at once answer; this bore directly on his stifled
questionings, and answered them.
"Was anything particular in your mind?" he asked at length.
"No—I mean yes. I can't lie to any purpose, Jim; it's no good my
trying. Yes, what was partly in my mind was a disagreement I had
with Jack some ten days ago. We patched it up quite beautifully, and
agreed that nothing was worth bothering about. I acquiesced,
though I should personally have preferred to have it out. At least I
am sure of this, that if one differs fundamentally from any one, it is
no use arguing, or, as he says, bothering. And fundamentally Jack
and I are very different."
She paused a moment and glanced suddenly at him.
"And that is why we get on so excellently," she added, with just a
suspicion of hurry in her words.
Jim longed to applaud her quickness; it had been excellently done.
But the most elementary courtesy forbade him to call attention to it.
"Asides" are conventionally observed at other places besides the
theatres.
"I am glad of that," he said in a perfectly even voice.
This was a turning of the tables; his conventionality was as obvious
as hers; she silently noticed it and also passed on.
"Yes, that little patch-up with Jack was in my mind," she said; "but
then, as I told you, we have privately settled to have no storms. No,
the storm which I mean will be a bigger storm than that. On that

subject Jack and I are quite agreed. I mean a national storm, a
general upheaval. My goodness! some high towers and steeples will
be smashed. And here we all go, meantime, dancing in the middle of
the thunder-clouds, with the lightning, so to speak, playing about
us."
They had emerged from the wooded walk on to the edge of the
meadow bordering the river, and as Marie spoke she pointed across
the field to the lawn visible beyond it, filled with gay figures, and
bordered with the bright colour of the flower-beds, and set in the
sombre green of the yew hedges. Jim followed her finger.
"Yes, assuredly we are dancing," he said. "But Sunday afternoon in
the country is an innocuous sort of high-dress dance, isn't it?"
"Certainly; but if we dance all and every day we don't get on with
our work. And in point of fact, Jim, all our dances are not very high-
dress. No; the fact is we are going to the dogs as quick as ever we
can. Money, money, money! That is a perfectly sound and legitimate
cry if the means you adopt are those that increase wealth. But if I
get a tip from a City man and speculate, I am merely snatching at
what I want. Did you go to the Maxwells' the other night? I did,
because we all do. That is what we all have come to; but it does not
spell efficiency. We worship the golden calf, but instead of feeding it
we try to cut little pieces off it."
"And Deborah was a prophetess before the Lord," said Jim.
"Proceed, Deborah."
"I wish I were," said Marie. "Oh, you should hear the truth if I was!
But, Jack, I must tell you, comes pretty near to being a prophet."
"Then you are the prophet's wife. Tell me what he says about it."
"Ah! he is a prophet in all but the one thing needful—I mean the fire
and the burning. The prophet is like the phœnix; he is born from the
ashes of a conflagration. In Jack's case all the message is there, but
it is delivered—I don't know if it will mean anything to you, but
personally I feel it—it is delivered out of cold lips. He needs the
touch of the red-hot coal like Isaiah. Did you hear or read his speech

last week about the Army Estimates? The First Secretary had given
his statement in an apologetic kind of way, apparently wishing to
conciliate the Opposition for the estimates being so high. The usual
bickering over rounds of ammunition followed, and then Jack got up.
Instead of apologizing for their being so high, he fell foul of his own
chief for their being so low. He wanted to know why the autumn
manœuvres had been curtailed; he wanted to know why the
experiments at Lydd had been abandoned in the middle; he wanted
to know why the projected battery at Gibraltar had not been
constructed: was it because of the expense? Why, in fact, they had
not spent twice as much as they had."
"That would hardly be a popular speech from a member of the
Government."
"Popular? No. The prophet is no opportunist, and thank God Jack
cares absolutely not one jot what either his own side or the
Opposition think of him. The press the next morning was worth
reading; he got the most violent abuse from both sides."
"Won't that sort of thing damage him both in and after the next
election? I should not think his party would like it, and I am sure the
Government will not."
"Ah, I disagree with you there," said Marie. "Jack, I think, is getting
a great hold on the people—the masses, if you like. He dislikes
them, and he treats them like dirt; but the masses, as you know, are
profound snobs, and rather enjoy that. They like a lord to behave in
the way they imagine lords do behave. They even like a wicked lord.
On the other hand, they are beginning to see that Jack means
business. He thinks the army is wholly inadequate, and, judging
from the length of the Boer War, would crumple under a great
stress. You see, he considers that the walk-over which was
anticipated has degenerated into a stroll. So, instead of joining in the
hymn of praise to the British Empire which the Government spend
their time in singing, rather out of tune with each other, he stands
apart, and says bluntly that we must set to and put ourselves in a
far greater state of efficiency, otherwise 'Pop goes the Empire.' Now,

that is impalatable, but I think people in general are beginning to
see that it may be medicinal."
"I should say it was lucky he's a hereditary legislator," said Jim. "But
how about a Government post afterwards?"
"Well, I think the Government may see that, too. They know
perfectly well that Jack doesn't care one straw about party
questions. He has said as much. What he does care about is the
Empire—I think he cares for it more than anything else in the world
—and what he knows about is the army. And if this cry for efficiency
—which certainly is getting louder—in the country continues, they
will have a far better chance of remaining in power if Jack is put at
the War Office."
They had come to the far end of the meadow, and Marie paused a
moment, looking at the broad, patient stream. Hundreds of
pleasure-boats were scattered over its surface, and electric launches
and river steamers crowded with roaring Sunday excursionists did
their best to make vile one of the most beautiful rivers in the world.
Each, indeed, seemed a Bedlam let loose and packed tight. Even the
stuffy little cabins were full of feather-hatted girls and amorous
young men, who changed hats with each other, without finding the
brilliancy of this wit grow the least stale even in endless repetition;
took alternate mouthfuls of solid refreshment out of paper bags and
of beer out of the same bottle, with shouts of laughter at slightly
indelicate suggestions. The poor river was flecked with fragments of
bun-bags and floating bottles where trout should have been feeding,
and echoes of the music-halls, with absolutely independent
wheezings of concertinas, owning no suzerainty, by way of
accompaniment, came to Marie and her companion with that curious
sharp distinctness with which sound travels over water. For a
moment they stood there in silence; then Marie turned quickly in the
direction of the lawn behind them, and back to the river again.
"After all—" she said half to herself.

Jim laughed. It was somehow strangely pleasant to him to find
himself, as Marie had said, thinking in harness.
"Yes, but less loudly so," he answered, replying to that which she
had not said.
"So it seems to us. Those good folk on the river don't seem loud to
themselves. But—oh dear me, Jim! what an awkward and
inconvenient thing it is to be different from the people one moves
among!"
He did not feel that he owed her any mercy on this point. She had
refused deliberately the other life he had once offered her.
"Ah, you find that, do you?" he said, his love for her surging up with
bitterness in his throat. "Yet you chose it yourself."
They had begun walking back towards the lawn again, but at his
words Marie suddenly stopped. From one side came the sound of
laughter and talk, from the other, now more remote, fragments of
"D'isy, D'isy."
She well knew what was in his mind, and thanked him silently for
not putting it into words.
"I know I did," she said; "and no doubt the very fact that I am
different to most of my milieu is what makes it so entertaining."
At that moment Jim saw where he stood. He knew that his taunt
that her lot was of her own choosing had been dictated by that
which was bitter within him, and was of the nature of revenge,
however ineffectual. And Revenge is a very smoky lamp wherewith
to guide one's steps in this world, and he had the justice to quench
it without more ado. But he knew also that the void which she might
have filled ached horribly, and by the irony of fate he had now in
abundance that of which the lack years ago had made it impossible
then that she should fill it. She had been but a girl, he but a boy;
and in him, he felt now, that which had subsequently flowered into
this great bloom of love had been but in bud. But the bud, it was
now proved, was authentic, for there was no mistaking the flowers.

Marie also was troubled. She could not but guess something of what
was in his mind, but his taunt seemed to her unworthy of him, and
she did not regret the light finality of her answer. But as they walked
back by the meadow-side, already growing tall with hay, and
redolent with the hundred unprized flowers of English meadows, her
mind changed. He had loved her with an honourable love; she on
her side had liked him, but it had been impossible—so she told
herself rather hurriedly. If she had been free, and he came to her
now—but she dismissed such unprofitable conjectures. Meanwhile
she had been harsh, though perhaps deservedly, to her old friend.
So just as he held the gate into the garden open for her:
"But I am so glad you have come back, Jim!" she said.

CHAPTER VII
Tea—or, rather, the modern substitute for tea, which consistes of
most things except tea, from caviare sandwiches to strawberry ice,
and whisky-and-soda to iced coffee—had just been brought out
when the two returned to the lawn, and Mildred Brereton's guests
had fallen upon it with the most refreshingly healthy appetites, and
were fluttering about the tables like a school of gulls fishing. Every
one, according to the sensible modern plan, foraged privately and
privateerly for himself, and there were no rows of patient women
agonizing for things to eat and drink, until some man languidly
brought them something they did not want, instead of that which
they desired. Nor, on the other hand, were there rows of men
parading slowly up the female line, like sightseers at an exhibition,
with teacups slipping and gliding over the saucers, and buns being
jerked from their plates by neighbouring elbows. Instead, every one
flocked to the tables, seized what he wanted, and retired into
corners to eat it. Anthony Maxwell in particular, who had a wonderful
gift for mimicry, was loading up with great care and solidity.
Something in his air might have reminded an observer of a steamer
coaling for a trip. He had had, in fact, a little conversation with both
Mrs. Brereton and Lady Ardingly during the afternoon.
"Yes, dear Mr. Anthony," Mildred had said. "You received my note,
did you not? And I am delighted you could come here to-day! Of
course, it is a dreadful thing to me to think that my little girl will be
taken away so soon. But that is what every mother has to go
through. Dear me! it seems only yesterday that she came into my
room, a little toddling mite, to announce that when she was grown
up she was going to marry the groom, because then she could
always live among horses."

"Oh, that'll be all right," said Anthony. "She can have plenty of
them."
"How generous of you to say that! You have not—ah—spoken to her
yet?"
"No. I've been trying to all the afternoon, but I couldn't get an
opportunity."
"Dear Maud! She is—how shall I say it? But, anyhow, it is so
characteristic of her."
"She seemed to want to avoid me," said Anthony with a bluntness
that rather distressed Mrs. Brereton.
"Yes, it would seem like it," said she; "but indeed— What I wanted
to say to you was this: You must be patient with her, and I expect
you will need a little perseverance. It is a rare thing, you know, to
come and see and conquer, like Julius Cæsar, or whoever it was.
Dear Maud perhaps scarcely knows her own mind. I am sure I do
not know it. You see, she is young, very young, and I do not think
that hers is a nature that expands very early."
The young man's rather heavy, commonplace face flushed; for the
moment it was lit up, as it were, by a flame from within.
"Oh, I'm not going to be impatient," he said. "And as for
perseverance, why, there's nothing I would not do, nor any number
of years I would not wait, to get her."
Mrs. Brereton looked at him critically for a half-moment. "Why, he's
in love!" she said to herself. Then aloud, "Dear Mr. Anthony! I am
convinced of it," she said. "And bear that in mind when you speak to
Maud. Also bear in mind that there is no marriage which either her
father or I so much desire. Ah, there is the Duchess of Bolton just
come! I must go and speak to her."
His interview with Lady Ardingly had been briefer, but, he felt, more
to the point.

"She will probably refuse you," said that lady. "In that case you had
better wait a month and ask her again. You have everything on your
side and everybody—except, perhaps, the girl. But eventually she
will do what is good for her. Here is a fourth. Let us play Bridge
immediately."
This particular game of Bridge had rather taken it out of Anthony, for
he had been Lady Ardingly's partner, and had had the misfortune to
revoke in playing a sans-à-tout hand. Her remarks to him were
direct.
"You might just as well pick my pocket of twenty pounds," she said
to him, "as do that. Do you not see it so? By your gross carelessness
you have lost us the rubber, a mistake which one intelligent glance
at your hand would have avoided. Come, there are other pursuits,
are there not, in which you wish to be engaged? You will, perhaps,
follow them with better attention."
Then, seeing the young man's discomfiture, her admirable good-
nature returned. "Croquet, for instance," she added. "I hear you are
a great player. Ah! there is Lord Alston. No doubt he will make our
fourth."
Maud, it is true, had spent the hours since lunch in flying before her
admirer, but her reasons, it must be confessed, were not those
which one would be disposed to think natural on the part of a young
girl. There was not, in fact, one atom of shyness or shirking about
her; she had not the least objection to hear impassioned speeches
or blunt declarations, whichever mode Anthony should choose to
adopt, nor did the thought of him in any way fill her with horror. She
had listened very attentively to her mother's advice when they drove
down to Windsor earlier in the week; she had also listened with the
same consideration to Lady Ardingly's far more convincing and
sensible remarks when she had lunched with her on Friday, and her
only reason for refusing Anthony an opportunity all the afternoon
was that she really had not the slightest idea whether she should
say yes or no. She did not, as she had told her mother, love him; she
did not, either, dislike him. He was merely quite indifferent to her, as,

indeed, all men were. Men, in fact, as far as she thought about them
at all, seemed to her to be unattractive people; she could not
conceive what a girl should want with one permanently in the house.
They were for ever either putting tobacco or brandy into their
mouths or letting inane remarks out, and they stared at her in an
uncomfortable and incomprehensible manner. On the other hand,
she knew perfectly well that it was the natural thing for girls to
marry; every one always did it, and they were probably right. She
supposed that she also would ultimately marry, but was this—this
utter absence of any emotion—the correct thing? She was aware
that tremblings and raptures were in the world of printed things
supposed to be the orthodox signals flown by the parties engaged;
she should be a creature of averted eyes and deep blushes. But she
did not feel the least inclined to either; there was nothing in Anthony
that would make her wish to avert her eyes, nor, as far as she knew,
did he ever say things which would make her blush. He was simply
indifferent to her, but so, for that matter, were all men. Was she,
then, to be a spinster? That was equally unthinkable.
There were other things as well. A great friend of hers, with whom
she had been accustomed to spend long days in the saddle, or in the
company of dogs in endless walks over moors, had been married
only a month ago, for no other reason, as far as Maud or Kitty
Danefield herself knew, but the one that every girl married if
possible, that it was the natural thing to do. Maud had seen her
again only two days ago for the first time since her marriage, and
had found quite a different person. Kitty had become a woman,
radiantly happy, with an absorbing interest in life which seemed
quite to have eclipsed the loves of earlier days. She still liked horses,
dogs, great open country, Maud herself; but all these things which
had been the first ingredients of existence had gone into a
secondary place, and the one thing that made life now was her
husband. To Maud this was all perfectly incomprehensible—would
Anthony, if she accepted him, ever fill existence like that? She could
not help feeling that existence would be a much narrower thing if he
did. Kitty, in fact, had just arrived, and had rushed at Maud.

"Darling, I am so pleased to see you!" she said, "and we'll have a
nice long talk. Where's Arthur! Arthur is really too tiresome; he
asked Tom Liscombe to come down with us when I had counted on
a nice quiet empty carriage all to ourselves. He didn't want him, nor
did I; but that is so like Arthur, to do good-natured things from a
sort of vague weakness. He saw Tom, and asked him without
thinking what he was doing. You look rather careworn, Maud. What
is it now?"
"Oh, come for a stroll, Kitty," said the other; "I want to talk."
"Very well; I must say good-bye to Arthur."
Maud laughed.
"Oh, you ridiculous person," she said; "you will be away ten minutes.
Would you like to make your will, too?"
"Well, if it's only ten minutes—oh, he's looking. There!" and she
waved a tiny morsel of a handkerchief to him.
Maud looked at her with grave attention.
"Now, I cannot understand that," she said.
"No, dear, of course not. You're not married. I should have thought it
as ridiculous as you before. By the way, Maud—oh, that's why you
look careworn. Is it true you are going to marry Anthony Maxwell?
Darling, how nice, and simply rolling!"
"You think that is important?" asked Maud.
"Why, of course. It's the only crumpled rose-leaf Arthur and I have.
It makes us quite miserable; there's always that little ghost in the
corner. Can we afford this? Can we spare the money for that? But
you haven't answered me. Is it true?"
"I haven't the slightest idea," said Maud.
Kitty laughed.
"You absurd creature!" she said; "you must know. Has he proposed
to you?"

"No, but he has told mother he wants to. And he has been stalking
me all the afternoon."
Kitty turned quickly back.
"He shall stalk you no longer," she said. "Really, Maud, you are
behaving very unfairly to him. If you are going to marry him, say so;
if not—well, if not, you will be a very foolish person, but still say so.
He has a mother, I know that, but really his mother matters very
much less than the man himself. He's all right, isn't he? Behaves
nicely—I mean, hasn't a vice about him—looks decent?"
"Moderately," said Maud.
"Oh, my dear, what do you want? Every one can't be an Adonis, and,
as the copybooks used to say, human nature is limited. I dare say
he's not a genius; well, no more are you. As for beauty, you've got
enough for two, and he's got money enough for three—baby, as
well, do you see? Oh yes, I am indelicate, I know, but it's far better
than being delicate. Being delicate never pays; on the other hand,
you have to pay for it, and I haven't got enough money for it. You
are lucky, Maud."
"Why? I want to talk to you about it."
"My dear girl, there is nothing to say. You will be a fool if you don't
marry him, as I told you. There is simply nothing else to talk about. I
was in a state of blank indifference about Arthur before I married
him. My mother—and I bless her for it—absolutely obliged me to
accept him. So will yours do if she has any sense, and I am certain
she has heaps. Unless you are a visionary or a fanatic of some kind,
you will be glad to be married. Glad? Good gracious! it is much more
than that."
She turned sharply on her heel, Maud following.
"Then, why are there so many unhappy marriages?" asked the latter.
"Ah, in books, only. They are there because the author does not
know what else to say. 'You can't write about happy marriages,' so

an author assured me. 'They are so dull. Happy people have no
history.'"
Maud was silent a moment.
"You have changed very much, Kitty," she said at length.
"Thank goodness, I have! Oh, Maud, I don't mean to be nasty to
you. Those old days were really dear days. But one can't always
remain a girl, Maud. It is mercifully ordained that girls become
women. And the door by which they enter is marriage."
"It means all that?"
"All. More——"
Maud found herself struggling for utterance. The blush and the
downcast eye which she had thought Anthony could never have
produced in her were hers now.
"You mean a man—the fact of a man?" she said stammeringly.
Kitty laughed the laugh of a newly-married woman, which is as old
as Eve.
"Put it that way if you like," she said. "But there is another—the fact
of a woman."
"But I am content," she said almost piteously. "Why does everybody
—you, mother—want me to marry?"
"You have left out Anthony," remarked Kitty rigorously. "I and your
mother, because we are women; he, because he is a man."
They had come to the populated lawn again, and further intimate
conversation would next moment be impossible. Kitty turned to her
hurriedly.
"Oh, my dear, it is like having a tooth out," she said. "No doubt it is
a shock. But it no longer aches. There is Mr. Anthony; let him ask
you, anyhow. That is bare justice; and remember what I have said."
"I shall not forget it," said Maud.

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