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Physikalische Chemie
Fakultät Mathematik und Naturwissenschaften
TU Dresden
D-01062 Dresden
Habilitationsschrift
Electron delocalisation and weak
interactions in molecules and
condensed matter: Impact on structure,
stability and properties
Kurzversion
eingereicht von
Thomas Heine
geboren in Osterburg/Germany
2006
2
Inhaltsverzeichnis
Preface........................................................................................................................................ 5
Acknowledgements .................................................................................................................... 6
1
Introduction ........................................................................................................................ 9
Computational Chemistry at the beginning of the 21st century.................................. 9
1.1
1.1.1
Fundamental methods and concepts of computational chemistry...................... 9
1.1.2
The IT revolution and computational chemistry .............................................. 11
1.1.3
The great quest of computational science: Model vs. method ......................... 13
1.1.4
Pushing the limit of quantum mechanical computations ................................. 14
1.2
Electron delocalisation – an important concept in chemistry .................................. 16
1.2.1
Background ...................................................................................................... 16
1.2.2
Electron delocalisation and the aromatic character of molecules .................... 17
1.2.3
Electron delocalisation and physisorption........................................................ 19
1.2.4
Properties of structures with π electron delocalisation .................................... 19
1.3
List of articles included in this work........................................................................ 21
2
Electron delocalisation – A determinant of molecular stability? ..................................... 25
3
Design of new molecules – Application of descriptors of electron delocalisation .......... 30
4
DFTxTB - Development of a fully quantum mechanical hybrid method ........................ 34
5
Nanostructures – “Better Things for Better Living . . . Through Chemistry.”................. 38
6
Calculation of NMR Shifts and Shielding tensors in non-standard situations ................. 44
3
4
Preface
This habilitation thesis is written in English, because colleagues from many countries, but also
from our group at TU Dresden, prefer to read scientific texts in English and asked me for a
copy of this work in advance. With this choice, I can also safely expect that this work will
receive international attention.
I decided to prepare a cumulative habilitation thesis. This means that most of this work is
already published elsewhere or submitted for publication. After this general introduction, the
five thematic chapters are structured in the following way: An introduction, its length depends
on the contents of the articles in the chapter gives motivation and an overview of the state of
research in the field and places my own contributions in this context. The introduction shall
guide the reader through the remainder of the chapter, which are several self-contained review
and research articles. These articles are preprints of publications which summarise the major
part of my scientific life of the last 3 years, a bibliography is given at the end of the
introduction.
In addition to the complete version there is a short version of this work, where the articles are
substituted by hyperlinks to their DOI (digital object identifier) online version. This version
will be publicly distributed on the internet, and I invite readers without access to some of the
included articles to request reprints.
This work contains large amount of development and implementation of new methodology,
which are now part of the deMon computer code. * The binaries and the source code of the
corresponding subroutines are available through the deMon developers community at
http://www.demon-software.com.
The following general introduction includes my personal view of computational chemistry,
how this work can contribute to the development in this field. It also underlines the
importance of electron delocalisation for many phenomena in chemistry.
*
deMon 1.1 experimental version. A.M. Köster, R. Flores, G. Geudtner, A. Goursot, T. Heine, S. Patchkovskii, J.
Ulises Reveles, A. Vela, D. R. Salahub, deMon, NRC, Ottawa, Canada, 2004.
5
6
Acknowledgements
This work could not have been accomplished without the support of my family, my
colleagues, both in Dresden and all over the world, working in various research networks, and
without the financial support given by various foundations. First, I want to thank my family –
my wife, who had to spare me many times during the difficult time of rising two little
children, and my son Valentin and my daughter Luka, who even bravely accompanied their
father to some scientific meetings.
I thank Prof. Dr. Gotthard Seifert for his continuous support, for providing an excellent
working environment, for keeping my administrative workload as small as possible, and for
his permanent availability for discussions.
This work thoroughly challenged our computer capacities, and Dipl.-Phys. Knut Vietze was
able to create and to maintain excellent facilities, even with tight budget, and to help in
various scientific, computational and programming questions. I thank my present and former
PhD students and postdocs, Dr. Clémence Corminboeuf, Dr. Gabriel Merino and M. Sc.
Lyuben Zhechkov for their encouraged and fruitful work, which entered in large extent in this
project. I enjoyed working with them on various scientific topics, we solved many problems,
and their expertise and help was essential for many projects which are part of this work. Also
collaborations with other members of our group, with Dr. Sibylle Gemming, Dr. Viktoria
Ivanovskaya, Dr. Andrey Enyashin, M. Sc. Agnieshka Kuc, Dr. Sandrine Hazebroucq, Dr.
Mathias Rapacioli, Dipl.-Chem. Robert Barthel and Dipl.-Chem. Johannes Frenzel and Dipl.Chem. Regina Ermrich have been essential. I thank our secretary, Frau Siegrid Hehme, for her
support in many administrative matters.
I enjoyed the collaboration with Dr. Serguei Patchkovskii from NRC Ottawa. Serguei, being a
true multi-level scientist, was able to help in all kinds of situations, including method
development and implementation, discussions on scientific literature up to technical
questions.
In the past 4 years I had a close collaboration with Prof. Dr. Hélio A. Duarte from UFMG
Belo Horizonte, Brasil, and his group. I visited them many times, and Hélio joined us in
Dresden. Beside our primary project, the development of DFTxTB (Chapter 4), we started
many interesting studies. I also wish to thank his colleagues Dipl.-Chem. Augusto Oliveira,
Prof. Dr. Helius F. dos Santos and Dr. Heitor de Abreu.
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The deMon developers, in particular Dr. Tzonka Mineva, Dr. Annick Goursot and Prof. Dr.
Andreas Köster are thanked for sharing their developments, encouraging collaborations and
for a very joyful time in science and beyond.
I had a particularly fruitful collaboration with Prof. Dr. Paul von Ragué-Schleyer, who
suggested that recent developments should be applied to many interesting molecules. These
applications form a whole chapter in this work.
I thank the members of the COST D29 network of the European Community for many
interesting meetings and workshops.
The ZIH Dresden is thanked for providing computer facilities and for technical support using
them.
Financial support, most of all of the Deutsche Forschungsgemeinschaft (DFG), the Deutsche
Akademische Austauschdienst (DAAD) and of the Gesellschaft deutscher Chemiker (GDCh),
was essential for this work, to employ post docs and PhD students, to buy equipment, and to
finance visits to collaboration partners, conferences and workshops.
8
1 Introduction
1.1 Computational Chemistry at the beginning of the 21st century
Computational chemistry, including quantum chemistry and molecular simulations, has made
tremendous progress in the past three decades. Today, it is an established discipline of
chemistry. This progress was only possible as computational chemistry has been accepted by
the general chemical community as an independent research field, which is the harvest of hard
work of generations of theoretical chemists, physicists, mathematicians, computer scientists,
as well as graphics specialists and information technology (IT) engineers. While the
fundamentals of quantum chemistry have been formulated as early as in the 1920’s, its
applicability on a routine basis has been made possible only by the revolutionary development
of information technology (IT) in the past 60 years. It is interesting to notice that the
sustainable improvement for practical applications are both tribute to progress in methodology
of computational sciences (scientific methodology and its implementation to software
products) as well as in the development of faster basic computer hard- and software, as
computer processors, storage media, computer memory and basic software as operating
systems and compilers. I shall not forget to mention that the barrier to apply methods of
quantum chemistry has been lowered substantially by the introduction of user-friendly
software packages. In the first two sections a short historical overview of the development of
computational chemistry, from the viewpoint of method and computer developments, is
given. The third section is devoted to today’s attempts to push the limits of quantum
chemistry by new simulation techniques. In a last section I will give my personal view of the
future of computational chemistry.
1.1.1 Fundamental methods and concepts of computational chemistry
The fundamentals of computational chemistry have been laid out a long time before the
computer has been invented: the Schrödinger equation has been published in 1926, the BornOppenheimer approximation only one year later. In the early days of quantum mechanics, its
creators had many expectations for the applicability of this theory to chemistry, which has
been a rather empirical scientific discipline at this time. Quantum mechanics, as rigorous,
fundamental theory of physics, indeed allowed the straight-forward formulation of chemical
problems by formulating the Schrödinger equation for the actual problem. Once the
Schrödinger equation was solved, the wave function is known and the calculation of all
desired quantities, including binding energy, structure, and properties, was possible. It
appeared that all what was left to be done was to find a reasonable mathematical apparatus for
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solving the fundamental equations, and chemistry could be shifted on large scale from
laboratory to the desk. However, even today, after 80 years of progress, the latter problem
proved to be non-trivial, and in the general case even impossible. Indeed, first calculations on
the structure and properties have been limited to small molecules and – except for few special
cases - empirical force fields, as Lennard-Jones or Morse potentials, have been used to
describe the potential energy surface.
In the 1950’s and 1960’s, the invention of computers provided new motivation for finding
reasonable approximations to allow the application of the Schrödinger equation to practical
problems of chemistry and physics. Roothaan and Hall introduced a numerical procedure that
allowed solving the Schrödinger equation within the Hartree-Fock approximation on
computers, automatically, for the first time. They formulated Hartree-Fock equations, often
called Roothaan or Roothaan-Hall equations, by introducing atomic orbitals as basis
functions, and expressing the molecular orbitals (MOs) by a linear combination of atomic
orbitals (LCAO). With this ansatz, they could express the Hartree-Fock equations as a
standard problem of linear algebra, a coupled system of linear equations which could be
solved iteratively. The mathematical concept of the numerical solution of an equation by the
fix-point technique was employed. In computational chemistry and physics, this technique is
known as self-consistent field (SCF) method: the MO coefficients are solutions of the HartreeFock equations. At the same time, they are used to construct the Fock matrix (at this point,
they are a guess). At convergence, the MO coefficients of the guess and of the solution
become equal - within some error bars. Today we still use the same concept for quantummechanical calculations of molecules, liquids and materials - not only in Hartree-Fock theory.
The Hartree-Fock approximation has been further simplified by Gaspar and Slater in the
1950’s, whose Thomas-Fermi and Hartree-Fock-Slater approximations lead to the Xα method,
formulated by Schwarz and Slater in 1974. This simplified formulation has later been adopted
for the efficient formulation of Kohn-Sham (KS) density-functional theory (DFT), whose
basic concept was published in 1964 and 1965.
In quantum chemistry, the main goal was to control the approximations to the Schrödinger
equation and to achieve chemical accuracy. 2 The rigorous treatment of this goal involved the
reduction of numerical errors in the two main approximations to the Schrödinger equation
applied to molecules: the choice of basis functions, and the inclusion of electron correlation,
2
As chemical accuracy it is mostly understood that energies are calculated with an error of less than 1 kcal/mol,
even though in many cases such accuracy is hard to be checked experimentally. However, in general sense the
term chemical accuracy should be used to provide comparable error bars in computation and related experiment.
10
which is neglected within the HF method. This lead to the development of various basis sets
and post-HF methods, as configuration interaction (CI) and coupled cluster theory (CC) and
variants of Rayleigh-Schrödinger perturbation theory, including the prominent versions
suggested by Møller and Plesset. On the other hand, for large-scale simulations of more
practical relevance, semi-empirical methods based on the neglection of differential overlap
(NDO) approximation have been developed, among them the common incremental NDO
(INDO) Hamiltonians MNDO, AM1, PM3 and SINDO.
In physics, applicability to condensed matter, in particular to solids, required the development
of scalable first-principle methods: they should not contain empirical parameters,
computations should be reasonably fast, and although accuracy was an issue, the model
character of the calculation was accepted. Accuracy was less crucial than in chemistry. The
Hartree-Fock-Slater method has been applied to solids already in the 1950’s, and Kohn-Sham
density-functional theory, which could be formulated within the same algebra, has been
readily accepted by the physical community. DFT does not – today I have to write necessarily
- include empirical parameters, and with new generations of exchange-correlation functionals
the method was capable to achieve chemical accuracy for most molecules and materials
which are interesting in chemistry. In consequence, DFT became attractive to chemists in the
mid 1990’s.
Simulation techniques, such as molecular dynamics (MD), were developed already in the
1950’s, at the same time first main frame computers have been introduced to science.
Computer simulations of chemical reactions, spectroscopy, diffusion processes etc. have been
developed in the 1980’s, motivated to large extent by the introduction of the Car-Parrinello
technique, and development still continues today. The necessity to simulate realistic systems,
including nanostructures, large biological molecules, solvated molecules, solids and surfaces
motivated the development of multi-level hybrid methods and of multi-scale simulations.
Such developments are also part of this work.
1.1.2 The IT revolution and computational chemistry
Though some scholars might deny the fact, many younger computational chemists will agree
that the principal factor of progress of quantum chemistry was the IT revolution: First of all,
the complexity and performance of computer processors has increased exponentially in time –
this is at least as fast as the scaling of any method of quantum chemistry, usually much faster.
This trend has been proposed more than 30 years ago, holds since then, is believed to hold for
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at least another 10 years and is generally known as Moore’s law. 3 Computers became not only
significantly faster, but also, quite importantly, cheaper and hence more common. The
historical milestone for this development was the introduction of the personal computer (PC).
Indeed, any personal computer, designed for office use, for controlling experiments or even
for computer games, is today an appropriate tool to apply the methods of computational
chemistry.
Cray-1
Mac Mini Core duo
introduction
1976
2006
price
8,86 Mio US$
~800 US$
weight
~5.5 tons
1.31 kg
floating point performance
~250 Mflops peak
~ 6 Gflops
power consumption
115 kW
~100 W
Table 1: Some suitable computers for computational chemistry.
Also the usability of computers has been improved, most importantly by the introduction of
graphical user interfaces: today it is almost inconceivable to create a molecular structure file
without the aid of graphical software. The same holds for the implementation of new methods
and algorithms without graphical development tools. Progress from other scientific disciplines
shall not be neglected. Mathematical algorithms as computer algebra and Fourier
transformations are available in hardware-optimised software libraries, and even fully
numerical three-dimensional integration is today a feasible task. These developments were
necessary to allow the powerful machinery of today’s quantum chemistry. A next challenge
3
The official Intel™ announcement of Moore’s law: “Gordon Moore made his famous observation in 1965, just
four years after the first planar integrated circuit was discovered. The press called it "Moore's Law" and the name
has stuck. In his original paper, Moore observed an exponential growth in the number of transistors per
integrated circuit and predicted that this trend would continue. Through Intel's relentless technology advances,
Moore's Law, the doubling of transistors every couple of years, has been maintained, and still holds true today.
Intel expects that it will continue at least through the end of this decade.”
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for scientific computation will be the efficient use of the newly developed parallel machines
(multi-CPU computers and computer networks).
1.1.3 The great quest of computational science: Model vs. method
The first important step to solve a scientific problem is to make an appropriate model of the
system of interest. This is a crucial point for the introduction of – often most severe –
approximations. In the case of computational chemistry these approximations are, for
example, the neglection of solvent effects, a cluster approximation for a solid or a surface,
neglection of entropy, temperature and relativistic effects. This list could be continued. The
model system which will be used in the calculation and which is subject to these
approximations shall be called structure model in the following. 4 Sometimes only a small part
of the original system is reassembled in the final structure model. However, in the chemical
literature of the last two decades, the inherent assumptions and neglections of these models
often remain undiscussed. At this point it needs to be stressed that the attitude of
computational chemist’s is about to change, and correction schemes for the most severe
approximations have been introduced and become more and more popular. Among the most
important corrections schemes which shall be mentioned here are the calculation of
temperature and entropy effects through the harmonic approximation, the treatment of solvent
effects by continuum models, and the mechanical and electrostatic embedding of a system in
an environment described by a classical theory, for example in hybrid methods, and
convergence tests for cluster models, including periodic boundary conditions.
It might be astonishing, in particular for a physicist, that the other side of the quest, namely
which method shall be applied to solve the problems for the models made above – often
referred as “method of choice” – has been debated much more intensively. It appears to be a
common agreement among quantum chemists that the electronic structure and the properties
of the model have to be solved as accurately as possible, independent on the approximations
involved in the structure model. To calculate energy, forces and properties of the structure
model four principal groups of atomistic methods are available in quantum chemistry. PostHartree-Fock ab initio methods, allowing an extrapolation to correlation and basis set limits,
belong to the first group of most accurate, but also computationally most demanding,
methods. If no symmetry can be applied to the system, their application is restricted to few,
typically about 20, atoms. These methods include configuration interaction and coupled
cluster variants, but also several levels of many-particle perturbation theory. The second
4
The inclusion of temperature and entropy are, on the second glance, also effects on the structure of the system,
as they involve the atomic motions of the system.
13
group are methods employing density-functional theory with its various approximations for
the exchange-correlation potential and basis set representation. DFT scales much better than
the ab initio methods, typically with O(N3), 5 and today one is capable to compute energies
and gradient of some 100 atoms with this method. Semi-empirical methods and approximate
DFT allow the treatment of systems containing up to some 1000 atoms quantum
mechanically, while classical mechanics can treat systems up to the μm scale on today’s
computers. The demand to solve the system of interest with the method of highest accuracy
which is possible for this task motivates, despite the development of computer technology, the
development of faster computational methods.
1.1.4 Pushing the limit of quantum mechanical computations
Computational science is now in the position that direct simulation of chemical and physical
processes, from the molecular to the nanometer scale, are just feasible if approximate methods
are applied. A striking example, the simulation of the tensile stress tests of WS2 nanotubes, is
presented in Chapter 5 (Figure 1).
Such and similar simulations have been made possible with an efficient vectorising parallel
implementation of the density-functional based tight-binding method in the deMon code,
which makes efficient use of the features of modern CPUs. In collaboration with the deMon
developers, implementations which aim at large-scale computations, as for example a
QM/MM methodologies, are presently continued.
However, more flexibility in the computational protocol, especially if the whole structure
model shall be treated quantum mechanically, is required to allow a reasonable accuracy and
the simulation of properties which are comparable with experiment. Therefore, we are
currently working on the development of a new, fully quantum-mechanical hybrid scheme on
the basis of density-functional theory, DFTxTB, which will be outlined in Chapter 4.
The application of quantum chemistry to simulate processes gives new challenges to
computational chemists. Today’s interest is in the simulation of processes on the nanometer
and nanosecond scales. For this purpose, in my point of view, there will be no straight,
automatic solution available within the next decade. These systems are too complex, too
large, and processes are running on too long time scales. We need to use our creativity to
study these systems. A relatively new way to approach this goal is the concept of multi-scale
simulations. This is a general concept for computer simulations, and impressive examples can
be found on the web page of the Earth Simulator Centre, Yokohama, Japan. The concept of
multi-scale simulations involves a multi-level approach to simulate parts of the problem on
5
O(N3) states that computation time increases with the 3rd oder of the number of basis functions N.
14
different size- and time scales. The methodology has been introduced for computer
simulations in environmental sciences, for general meteorology, climate forecasts etc.
Figure 1: Left: Tensile stress test of an inorganic WS2 nanotube using an atomic force
microscope. Right: Simulation of the rupture process of (22,0 - left) and (14,14 - right)
MoS2 nanotubes.
Also in this work several problems are solved using the multi-scale technique: The simulation
of hydrogen storage in Chapter 5 starts from high-level quantum-chemical ab initio
simulations, which are used to establish a force field for the interaction of H2 with
nanostructures. Approximate DFT calculations have been used to determine the geometry of
the nanostructure, while force field calculations are employed to generate the H2nanostructure interaction potential on a grid. Finally, the free energy of an H2 molecule in the
potential of a nanostructure has been calculated quantum-mechanically. Another example is
the calculation of shielding tensors of the van-der-Waals crystals biphenyl and benzene,
which involve the geometry calculation of individual molecules, the establishment of a
summation scheme for the NMR calculation and its application to thousands of molecules
contributing to the 1H NMR shielding tensor of the systems (Chapter 6).
15
Figure 2: Sketch of a multi-scale simulation, for details see chapter 5. From left to right:
Ab initio calculations of the interaction energy (y-axis, in kJ/mol) of H2 with aromatic
molecules (H2-benzene distance in Å on x-axis). Calculation of the H2-nanostructure
interaction potential with an empirical force field parameterised with the ab initio
calculations. Lowest-energy states of the probability density of H2 in a carbon bilayer.
Interaction free energy ΔF vs. bilayer distance c and equilibrium constant Keq vs.
temperature, calculated from the energy levels of the H2 states.
1.2 Electron delocalisation – an important concept in chemistry
The role of electron delocalisation in chemistry is much more important than it is presently
acknowledged in text books which are addressing the chemical bond. It is only used to
rationalise the metal bond. The importance of electron delocalisation is also discussed when
aromatic molecules are introduced in organic chemistry, but their influence to the stability and
topology of nanostructures, to properties of van-der-Waals crystals, macromolecules,
supramolecular structures and to systems of biological and biochemical interest is less often
discussed. Systems containing delocalised electrons make up a large part of this work. The
implications of electron delocalisation to the stability of small molecules, related with the
term “aromaticity”, on the magnetic properties of van-der-Waals crystals, on the structure and
stability of nanostructures, in particular of systems based on sp2 carbon, and on the capability
of these systems to physisorb gases, are discussed. This part of the introduction gives only a
brief overview on the topic, in order to motivate the reader to go into the details in the
following chapters. Much more detailed information, including several review articles, is
reprinted there.
1.2.1 Background
When Erich Hückel studied benzene and other aromatic molecules, he discovered that πelectron delocalisation was responsible for the special stability and high-symmetry structures
of this class of molecules (see Figure 3).
16
Figure 3: Benzene. Left: The two Kekulé-forms contain single- and double C-C bonds,
leading to two distinct bonds, and are not in agreement with the D6h symmetry of the
molecule. The four representations of the right-hand side are based on the MO picture,
correctly assembling (from left to right) the electron density of the bonds, the pz orbitals of
the carbons (assuming the molecule is located in the z=0 plane) and the resulting clouds of
delocalised π electrons. The schematic representation on the right-hand side is in
agreement with the structural properties of the molecule. Source: http://de.wikipedia.org.
Working on this problem, he introduced quantum mechanics to organic chemistry, and
perhaps he even initiated the field of computational chemistry with the introduction of his
method. Only a rationalisation on the basis of quantum mechanics could explain the
properties of aromatic molecules. Hückel theory has been extended by London, Dewar, Pople
and others to semiempicical methods, which are capable to study structure, stability, optical
and magnetic properties, and which are not restricted to aromatic molecules. Chapter 2
contains two review articles which discuss this topic in much greater detail.
1.2.2 Electron delocalisation and the aromatic character of molecules
Electron delocalisation has been the rationale for the enhanced stability of aromatic
molecules. Further, aromatic molecules exhibit special magnetic properties, as exalted
magnetic susceptibilities and anomalous proton chemical shifts. London and Pople explained
these special magnetic properties with a ring current model: An external magnetic field,
applied perpendicular to the molecular plane, induces a ring current in the clouds of
delocalised π electrons. This current induces a magnetic response field, which is shielding the
external field in the ring centre, leading to exalted magnetic susceptibilities, and enforcing the
external field at the proton positions, leading to characteristic proton NMR signals (Figure 4).
17
Figure 4: Refined ring current
model for benzene: An external
|Bindz|
magnetic field Bext induces a ring
current in the π cloud, indicated by
Figure 5: Induced magnetic field of
the black circle inside the molecule.
benzene. The plot shows the induced
At the position of the proton (H),
magnetic field if an external field of
the magnetic field is modified by
Bext=1T
the induced field Bind, represented
perpendicular
by the red circle. A 2nd order
isosurfaces of the z-component of the
contribution to Bind, dBind, induced
induced field of (Bind)z=5μT (red) and
at the opposite ring site, has been
(Bind)z=-5μT (blue) are shown. The
suggested
and
induced field lines |Bind|=const. in
coworkers. Source: Pelloni, S.;
planes through the molecular centre are
Ligabue, A.; Lazzeretti, P. Org.
drawn in the faces of the box. The
Lett. 2004, 6, 4451.
colour code is in units of μT.
by
Lazzeretti
strength
to
is
the
applied
ring.
The
The proposition that the special magnetic properties, conveniently explained by Pople’s ring
current model, are the essence of aromaticity is not astonishing. Indeed, many suggestions to
quantify the aromatic character of a molecule by indexes based on ring currents and chemical
shifts have been proposed. They are discussed in the review and research articles reprinted in
Chapters 2 and 3. To discuss the applicability and the limitations of these indexes we
calculated directly the induced magnetic field of molecules (see Figure 5.)
Indeed, if these indexes are applied to molecules, their stability and structure can be
rationalised. Within this work this technique some new, unconventional molecules have been
studied, such as planar Al4 anions, planar tetracoordinate carbon and inorganic Sin2- and
(BH)n2- clusters (Chapter 3). We also calculated the reactivity of aromatic molecules. With a
18
new method to compute orbitally resolved reactivity indexes we could show that the frontier
orbitals of aromatic systems exhibit negative Fukui functions, thus rationalising the low
reactivity of these species (Chapter 4).
1.2.3 Electron delocalisation and physisorption
Due to the delocalisation of the π electrons aromatic molecules show high polarisabilities.
According to the dispersion relation proposed by F. London, physisorption energies are
directly proportional to the polarisabilities of the interacting particles - strong polarisabilities
imply strong physisorption. The empirical London formula is a good, quantitative
approximation for the physisorption energy, and its asymptotic behaviour is easily
implemented in empirical force fields. On the other hand, semi-empirical and densityfunctional based methods do not include dispersion, and hence are not suitable to describe
physisorption. For the density-functional based tight binding method it is, however, straightforward to include this interaction, as shall be lined out in Chapter 4.
Carbon nanostructures include a system of delocalised π electrons and hence belong to the
materials with the highest capability to physisorb molecules. Indeed, the possibility to store
gases is directly proportional to sp2 content and to the active surface of the material. We
studied the interaction of H2 with aromatic molecules and carbon nanostructures, and
determined the theoretical threshold for these materials to store molecular hydrogen, using a
new method to assess the thermodynamic properties of the guest-host system quantum
mechanically (Chapter 5).
1.2.4 Properties of structures with π electron delocalisation
Molecules and nanostructures with a framework of σ bonds and a delocalised π electron
system show interesting properties. These properties can be used for their characterisation,
and they often imply that these molecules and materials are suitable for various applications.
A nice example is the metallic character of graphite and of some carbon nanostructures such
as carbon nanotubes. It is even more interesting if the special electronic and mechanical
properties are related to each other. In this context, we studied the change of conductivity in
carbon nanotubes under tensile stress (Chapter 5). The high mechanical stability of
nanostructures is reason enough for many applications, for example as lubricants, material
coatings or super hard materials with low mass density. We have investigated two families of
such nanostructures: Inorganic WS2 and MoS2 nanotubes show an extraordinary tensile
strength, and are reversibly stable up to an elongation of 15-20% of their length (Chapter 5).
Good candidates for application as light-weight super hard material are solids formed from
19
minor fullerene cages. Such fullerenes below the C60 threshold are instable as molecules, but
show stable derivatives and fullerene solids. We discuss solids based on C20, C28 and C50 in
Chapter 5 (Figure 6).
C50H10
D4 (C100H16)
Tr8
Te6 (C200H28)
xy plane
P4 (C250H34)
xz plane
H4 (C300H40)
yz plane
Figure 6: Left: Oligomers formed from D5h C50H10. Right: Hypothetical solid-state
structure and density-of-states of a fullerene solid based on C20.
Larger
fullerenes
encapsulate
may
endohedral
clusters. If the endohedral
moiety is electron donating or
accepting,
the
surrounding
cage can be charged, and in
the production process special
cage topologies are preferred.
A family of such compounds
δTMS [ppm]
Figure 7. Left: Snapshots of a MD simulation of Sc3NC80.
Right, top: Simulated static
13
C NMR spectrum of
Sc3NC80. The long-time average (0.7 ns) of the 13C NMR
spectrum (Right, bottom, black lines) simulates the
experimental signals (blue) in position and intensity. The
are Sc3N@Cn fullerenes. As
the disc-shaped endohedral
cluster is mobile and subject
to
strong
rotations
and
tumbling motions inside the
cage,
it
is
difficult
to
characterise these species with
C60 reference is given as red dotted line.
vibrational spectroscopy.
On the other hand, 13C NMR has been proven to allow the characterisation of these molecules.
We could simulate, and hence unambiguously confirm, the characterisation of Sc3N@C68 and
20
Sc3N@C80 on grounds of
13
C NMR simulations, based on long-time molecular dynamics
trajectories (Figure 7).
Finally, we were able to determine the 1H NMR shielding tensors of biphenyl and of benzene.
We developed a summation scheme for the efficient computation of these quantities and
benchmarked it on experiment with biphenyl. The same approach has been applied for the
prediction of the 1H NMR shielding tensor of solid benzene (Chapter 6).
1.3 List of articles included in this work
The following articles contain the scientific work which represents this habilitation thesis.
They are listed and numbered as the sections in the respective chapters. The citation is linked
with the digital object identifier (DOI) to allow a direct download of the electronic version of
the article. In case you experience difficulties to access an article please request a reprint per
electronic mail (mailto:[email protected]).
Chapter 2: Electron delocalisation – A determinant of molecular
stability?
1. Description of Electron Delocalization via the Analysis of Molecular Fields
G. Merino, A. Vela, and T. Heine
Chem. Rev. 105 (2005) 3812-3841.
2. The magnetic shielding function of molecules and pi electron delocalization
T. Heine, C. Corminboeuf, G. Seifert
Chem. Rev. 105 (2005) 3889-3910.
3. Aromaticity indices from magnetic shieldings
Z. Chen, T. Heine, P. v. R. Schleyer, and D. Sundholm
in: 'Calculation of NMR and EPR parameters: Theory and Applications', M. Kaupp,
M. Bühl, and V. G. Malkin Eds., Wiley-VCH (2004) pp. 395-408.
4. Evaluation of aromaticity: A new dissected NICS model based on canonical
orbitals
C. Corminboeuf, T. Heine, and J. Weber
Phys. Chem. Chem. Phys 5 (2003) 246.
5. Analysis of Aromatic Delocalization: Individual Molecular Orbital Contributions
to Nucleus-Independent Chemical Shifts
T. Heine, P. v. R. Schleyer, C. Corminboeuf, G. Seifert, R. Reviakine, and J. Weber
J. Phys. Chem. A 107 (2003) 6470.
6. Induced magnetic fields in aromatic [n]-annulenes - interpretation of NICS
tensor components
C. Corminboeuf, T. Heine, P. v. R. Schleyer, and G. Seifert
Phys. Chem. Chem. Phys. 6 (2004) 273-276.
21
7. The Induced Magnetic Field in Cyclic Molecules
G. Merino, T. Heine and G. Seifert
Chemistry – Eur. J. 10 (2004) 4367-4371.
8. Sigma and pi contributions to the induced magnetic field: indicators for the
mobility of electrons in molecules
T. Heine, R. Islas, and G. Merino
Submitted to Journal of Computational Chemistry.
Chapter 3: Design of new molecules – Application of descriptors of
electron delocalisation
1. Do All-Metal Antiaromatic Clusters Exist?
Z. Chen, C. Corminboeuf, T. Heine, J. Bohman, and P. v. R. Schleyer
J. Am. Chem. Soc. 125 (2003) 13930-13931.
2. Antiaromaticity in Bare Deltahedral Silicon Clusters Satisfying Wade's and
Hirsch's Rules: An Apparent Correlation of Antiaromaticity with High
Symmetry
B. King, C. Corminboeuf, T. Heine, and P. v. R. Schleyer
J. Am. Chem. Soc. 126 (2004) 430-431.
3. Aromaticity of four-membered-ring 6 pi-electron systems: N2S2 and Li2C4H4
Y. Jung, T. Heine, P. v. R. Schleyer, and M. Head-Gordon
J. Am. Chem. Soc. 126 (2004) 3132-3138.
4. Design of Molecules Containing a Planar Tetracoordinate [C(C4)] Skeleton
G. Merino, M. A. Mendez-Rojas, H. L. Beltran, C. Corminboeuf, T. Heine, A. Vela
J. Am. Chem. Soc. 126 (2004) 16160-16169.
5. Planar Tetracoordinate Carbons in Cyclic Hydrocarbons
N. Perez, T. Heine, R. Barthel, G. Seifert, A. Vela, M. A. Mendez-Rojas, G. Merino
Org. Letters 7 (2005) 1509-1512.
Chapter 4: DFTxTB - Development of a fully quantum mechanical hybrid
method
1. Efficient computation of orbitally resolved hardness and softness within density
functional theory
T. Mineva and T. Heine
J. Phys. Chem. A 108 (2004) 11086-11091.
2. Orbital hardness tensors from Hydrogen through Xenon from Kohn-Sham
perturbed orbitals
T. Mineva and T. Heine
Int. J. Quantum Chem. 106 (2006) 1396-1405.
3. DFTxTB - a unified quantum-mechanical hybrid method
H. A. Duarte, T. Heine, G. Seifert
Theor. Chem. Acc. 114 (2005) 68-75.
22
4. An Efficient A Posteriori Treatment for Dispersion Interaction in Density
Functional-Based Tight Binding
L. Zhechkov, T. Heine, S. Patchkovskii, G. Seifert, and H. A. Duarte
J. Chem. Theory Comp. 1 (2005) 841-847.
Chapter 5: Nanostructures – “Better Things for Better Living . . . Through
Chemistry.”
1. The Smallest Fullerence C20: From Monomer to Oligomers and Solid States
Z. Chen, T. Heine, H. Jiao, W. Thiel, and P. v. R. Schleyer
Chemistry Eur. J., 10 (2004) 963-970.
2. Hyperdiamond and hyperlonsdaleit : Possible crystalline phases of fullerene C28.
G. Seifert, A. N. Enyashin, and T. Heine
Phys. Rev. B 72 (2005) 012102.
3. C28 fullerites – structure, electronic properties and intercalates
A. Enyashin, S. Gemming, T. Heine, G. Seifert, L. Zhechkov
Phys. Chem. Chem. Phys. in press.
4. D5h C50 fullerene: A building block for oligomers and solids?
L. Zhechkov, T. Heine, and G. Seifert
J. Phys. Chem. A 108 (2004) 11733-11739.
5. Molecular Dynamics Study of the Mechanical and Electronic Properties of
Carbon Nanotubes
V. Ivanovskaya, N. Ranjan, T. Heine, G. Merino, G. Seifert
Small 1 (2005) 399-402.
6. Direct Tensile Tests of Individual WS2 Nanotubes
I. Kaplan-Ashiri, S. R. Cohen, K. Gartsman, R. Rosentsveig, V. Ivanovskaya, T. Heine,
G. Seifert, H. D. Wagner, and R. Tenne
Proc. Natl. Acad. Sci. 103 (2006) 523-528.
7. Hydrogen storage by physisorption on nanostructured graphite platelets
T. Heine, L. Zhechkov, and G. Seifert
Phys. Chem. Chem. Phys. 6 (2004) 980-984.
8. Graphene nanostructures as tunable storage media for molecular hydrogen
S. Patchkovskii, J. Tse, S. Yurchenko, L. Zhechkov, T. Heine, G. Seifert
Proc. Natl. Acad. Sci. USA 102 (2005) 10439-10444.
Chapter 6: Calculation of Nuclear Magnetic Shifts and Shielding tensors
in non-standard situations
1. Fullerenes
T. Heine
in: 'Calculation of NMR and EPR parameters: Theory and Applications', M. Kaupp,
M. Bühl, and V. G. Malkin Eds., Wiley-VCH (2004) pp. 409-421.
23
13
2.
C NMR fingerprint characterises time-averaged structure of Sc3N@C80
endohedral fullerene
T. Heine, K. Vietze, and G. Seifert
Magn. Res. Chem. 42 (2004) 199-201.
3.
13
C NMR pattern of Sc3N@C68 - structural assignment of the first fullerene with
adjacent pentagons.
U. R. Reveles, T. Heine, and A. M. Köster
J. Phys. Chem. A 109 (2005) 7068-7072.
4. Influence of Dynamics on the Structure and NMR Chemical Shift of a Zeolite
Precursor
S. Krishnamurty, T. Heine, and A. Goursot
J. Phys. Chem. B 107 (2003) 5728.
5. The proton nuclear magnetic shielding tensors in biphenyl: Experiment and
theory
F. Schönborn , H. Schmitt , H. Zimmermann , U. Haeberlen , C. Corminboeuf , G.
Großmann and T. Heine
J. Magn. Res. 175 (2005) 52-64.
6. Proton magnetic shielding tensors in benzene – from individual molecule to the
crystal
T. Heine, C. Corminboeuf, G. Grossmann, U. Haeberlen
Submitted to Angew. Chem.
24
2 Electron delocalisation – A determinant of molecular
stability?
In the following two chapters I will demonstrate how electron delocalisation and related
quantities, including molecular fields based on the electronic density and on magnetic
properties, can be used as a concept to study the stability of molecules. In this context I will
use the phrase stability synonymously with possibility of existence, as hypothetical and new
molecules, partially with very short life times, will be discussed. In nature, stability is only
related to energetic properties, such as relative (free) energy and the shape of the potential
energy surface around the minimum structure of the molecule. Clearly, the principle of lowest
(free) energy is the underlying concept of nature and has been applied in countless studies of
relative energies of isomeric structures of molecules, clusters, and solids. In practical
application this approach is limited by the complex character of nature: it is only meaningful
to compare energies for isomeric systems, which are reasonably isolated or have well defined
boundaries, such as for example gas-phase molecules or perfect solids or surfaces. However,
the energy of a single system is a meaningless quantity. Only energy differences with respect
to reference structures are meaningful, they allow the interpretation of the (relative) stability
of a system. It is, of course, always possible to define references, for example free atoms,
diatomic molecules, perfect solids etc. Calculation of the relative energy to the reference
might, in practice, be complicated or even impossible by various technical or even
methodological difficulties, as for example the approximate treatment of the environment, the
impossibility to calculate the reference with the method of choice or inaccurate results of the
reference energies, or special treatments of the quantum mechanical boundaries which prevent
to compare energies (e.g. saturation of cluster boundaries).
Chemical bonding is related to redistribution of electrons in the molecular fragments. This
redistribution can, for example, be “dislocation”, as for the anion/cation example, or
perturbation like polarisation for a solvated molecule. In many cases, electron delocalisation
is a meaningful concept to address chemical bonding. This concept is used for the traditional
explanation of metal bonding (cf. the Jellium model), and also for the explanation of the
stabilisation of aromatic molecules. To describe aromatic stabilisation, isolated localised p
electrons are in a similar state as in the atomic reference, and delocalisation allows them to
interact and to lower their energy. Applied to cyclic organic molecules, this behaviour is the
essence of aromaticity. In classical chemical understanding, aromaticity is related to systems
of delocalised π electrons in planar molecules. However, there is no rigorous definition of the
term aromaticity. In fact, it has been related to molecules with certain structural elements
25
(equal or nearly equal bond lengths), relative energies (aromatic stabilisation energies, ASE),
special reactivity and exalted magnetic properties as magnetic susceptibilities or nucleusindependent chemical shifts (NICS). Two special issues of Chemical Reviews have been
devoted to this topic. The latter one, which has appeared in October 2005, drives the focus to
electron delocalisation, and contains two articles which are also part of this chapter.
Not without controversy, the term aromaticity has been extended to assess stabilisation of
molecular structures by delocalised mobile electrons. Hirsch introduced homoaromaticity, an
extension of aromaticity to non-planar three-dimensional clusters exhibiting high symmetry.
In analogy to classical Hückel theory, he proposed the 2(N+1)2 π electron rule for the stability
of homoaromatic clusters Cn, (BN)n, Sin etc. Also, delta-aromaticity, a concept in which metal
d-electrons delocalise and lead to particularly stable clusters, falls into this category. One can
argue that it is more misleading than helpful to stress such systems to be aromatic, and that
even more striking analogies of such systems can be found when they are compared with
metal clusters or Zintl ions. However, present terminology of chemical literature prefers the
term aromaticity, as it is used in several of the articles summarised in this chapter of this
work, while electron delocalisation is preferred by others.
Several descriptors are suitable to discuss electron localisation and electron delocalisation.
They include the topological analysis of the electron density, which is useful when the
bonding framework of a structure needs to be determined, the electron localisation function
(ELF), the Laplacian of the electron density, the Fermi hole function and others. A detailed
review, which appeared in the October 2005 issue of Chemical Reviews, summarises and
discusses those descriptors and is the first section of this chapter.
The second section is another article of the October 2005 issue of Chemical Reviews and
deals with the tensorial magnetic shielding function. NICS are directly related to this quantity,
they are the negative isotropic values of the tensorial shielding function at selected points in
space. A historical overview on the usage of chemical shieldings at positions different from
nuclei is given, and the special magnetic properties of aromatic molecules are reviewed. The
results obtained by the shielding function are compared with those of methods which are
based on current densities and the topological analysis of the current density. Comparison for
organic and inorganic examples, including cyclobutadiene, benzene and Al4n- anions are
completing this section.
Delocalised electrons show exalted magnetic properties. As they are delocalised, these
properties are not restricted to the vicinity of an atom or of a bond, but present inside and
around the molecule. Therefore, Schleyer and co-workers introduced nucleus-independent
26
chemical shifts (NICS), an empirical index to estimate the degree of electron delocalisation,
or, in the best sense of the authors, aromaticity. The method has been used numerous times to
an enormous range of applications. In the third section of this chapter a review on extensions
of the NICS index and their applications to various molecules containing delocalised electrons
presented.
In most cases, only a subset of electrons of a system are delocalised. For example, the core
electrons are always well localised. The same holds usually for σ electrons which participate
in a covalent bonding framework. If a magnetic index, as NICS, is employed to study electron
delocalisation or aromaticity, one is usually interested to study the influence of the π systems.
Within the IGLO method, this can be achieved by separate localisation of groups of molecular
orbitals. Such an approach, called NICScπ, is presented in Section 4.
The disadvantage of the NICScπ approach is its restriction to the IGLO method to compute
shielding contributions. The consequent continuation of the development is to compute MO
contributions to NICS, and to discuss those contributions which are relevant in the case one is
studying. Beside others, the GIAO method allows the computation of MO contributions to the
shielding tensor, and is applied in the MO-NICS analysis for a series of test systems, which
are discussed in Section 5.
Shielding
Region
H
H
Deshielding
Region
H
H
Deshielding
Region
H
H
Bext
Induced Magnetic Field, Bind
Shielding
Region
Scheme 1.
Even though the magnetic indexes of aromaticity have been applied thoroughly to many
aromatic molecules, it has not been possible to use them to explain the classical ring current
picture of benzene. The classical model of an aromatic ring has been given by Pople, who
suggested that a ring current is induced by a field which is normal to the molecule. In return,
this ring current induces a counter field which opposes the applied field, and leads to diatropic
27
NICS values at the ring centre, and in less shielded protons. This model is rationalised by the
special topology of the system: The π electrons are located parallel to the xy plane in two
rings, one on top and one below the molecular cycle. A classical ring current can hence be
induced only parallel to the xy plane, and is related to an applied field normal to the ring. This
situation is not reflected by the isotropic shielding at the ring centre. Instead, the zz
component of the Cartesian shielding tensor is related to the integrated net current in xy
planes. The problem to relate NICS with ring current densities is discussed in Section 6.
The groups of Lazzeretti and Fowler studied current densities to discuss aromaticity of many
molecules in the past decade. The current density, induced by an external magnetic field, has
been used in analogy to Pople’s model to discuss aromatic ring currents. Their approach is,
however, numerically very sensitive. As it is difficult to quantify their results, only pictorial
representations in current density maps have been given.
On the other hand, the induced magnetic field is directly connected to the induced current
density, and it is directly quantifyable. Being a molecular vector field, it offers immediate
visualisation, but also the mathematical apparatus of topological analysis can be applied. To
my surprise, induced magnetic fields of molecules have not been reported in the literature,
even though directly related quantities as current densities have been discussed rather
intensively. Section 7 introduces the computation of the induced magnetic field and compares
the results of aromatic, antiaromatic and non-aromatic molecules.
Figure 1: Induced magnetic field of benzene, with an external field applied normal to the
rings. Field lines are drawn for the contribution of the σ (left) π (centre) and of all
electrons (right).
Finally, in Section 8, we demonstrate by first principles that the Pople model is a very realistic
picture of the induced magnetic field of the π electrons in benzene by direct computation of
the magnetic field induced by all, σ, and π electrons (Figure 1). The field lines, shown in this
plot, quickly lose the shape of the molecular framework and become circles. We also discuss
the role of the σ electrons in benzene and the σ and π contributions to the induced magnetic
28
field for benchmark systems containing a double bond (C2H4), a triple bond (C2H2) and for
antiaromatic cyclobutadiene.
This chapter dealt with the comparison and development of magnetic descriptors of systems
exhibiting strong electron delocalisation. In the next chapter we will apply these methods to a
series of new molecules and clusters.
29
3 Design of new molecules – Application of descriptors of
electron delocalisation
In the previous chapter, descriptors of electron delocalisation and aromaticity have been
introduced, and many examples for their application have been given. In particular the
magnetic quantities have been proven to be sensitive indexes of electron delocalisation, and
therefore special emphasis has been given to NICS and to the induced magnetic field.
In this chapter I want to give some examples where these descriptors are helpful, either for the
design of new molecules, or for explanation why certain molecules express an – at first glance
– unexpected stability. All molecules discussed in this chapter show a common feature:
Electrons, which are not included in the bonding framework, in the physicist’s terminology
“dangling bonds”, are delocalised. Generally, delocalisation of electrons increases their
interaction, and is aligned with lowering the energy of these electrons. In consequence,
additional stability of the molecule is gained. As the degree of delocalisation, or the overlap of
the “dangling electrons”, is maximised in certain topologies, the resulting molecules exhibit a
special, often planar, structure. This is the underlying concept for the stabilisation of aromatic
molecules. Hence, in chemistry the term “aromaticity”, sometimes with the prefix “homo” for
three-dimensional structures, or “delta” if d-electrons are involved, is often used to
characterise such type of stabilisation. In some sections of this chapter we make use of this
terminology.
It is important to note that the term aromaticity has been overused – and perhaps even
misused – in the past. One of the reasons might be that a clear definition is still missing.
Furthermore, a survey of the recent literature reveals that many scientists are so much
concerned on the aromaticity of molecules that this has become the primary point of their
research – ring current maps and NICS points are calculated and intensively discussed, but not
much attention is given to basic, much more relevant quantities, as for example to the
potential energy surface around the minimum. This criticism is often expressed, and I
observed some polarisation of the chemical community: It is often challenged if it makes
sense at all to compute quantities which are related to aromaticity, in particular the ones of
magnetic origin, as they are not directly accessible in experiment. 6 On the contrary, one can
argue that a big manifold of new molecules and bonding types have been proposed in the past
years by using the NICS index: Schleyer and his coworkers use the diatropicity of molecules
6
Related quantities which are experimentally accessible and meaningful indexes for electron delocalisation are
rare, the best example are perhaps 3H NMR chemical shifts of endohedral fullerenes containing He.
30
or parts of molecules as initial criterion for the design of new molecules. Their success in this
field shall be proof enough that those quantities might be useful.
The chapter starts with the discussion
Al4Li3 (-) Isomers:
Cs Singlet
of planar Al4n- clusters (Figure 1):
Cs Triplet
–29.0
Li
–4.8
Li
After the synthesis of these species
Al
Al
by Boldyrev and Wang their special
Al
Al
Al
Al
Al
stability
Al
Li
Li
Li
has
been
controversially.
interpreted
The
term
‘aromaticity’ has been used since the
Li
beginning to discuss these species,
Figure 1: Planar Al44- anion stabilised by three Li+
ions. NICS values denote paratropic (that is,
deshielding an external magnetic field, red) and
diatropic
(shielding,
green)
regions
of
the
molecule. The size of the spheres is proportional to
the magnitude of NICS.
and
analogies
to
aromatic
[n]-
annulenes have always been made.
Several theoretical works have been
devoted to this topic, and it has been
found - in general agreement - that
the stabilisation is mainly due to the
delocalised, metal-like, σ-framework.
Obviously, those molecules also have a π system, which is different in Al42- and Al44-.
Electron counting suggests that Al42- is aromatic while Al44- is antiaromatic. In the first
section we show that more involved calculations reveal that in both cases the σ frameworks
are responsible for the stability of these anions.
In the 2nd section another four-membered ring, N2S2, is discussed. It has been claimed that this
molecule is a diradical, but our highly correlated ab initio calculations, as well as the
evaluation of magnetic indices, find
this molecule rather aromatic, with a π
Si
B
system containing six electrons. The
Si
B
B
B
B
Si
Si
Si
Si
results are compared with Li2C4H4, that
B
1.787Å
2.601Å
B
is, cyclobutadiene stabilised by two Li
B
B
B
B
Si
Si
Si
Si
Si
B
atoms (Figure 2). This stabilised [4]annulene
Si
Figure 2: Isostructural clusters B12H122- and
Si122-. NICS values are given with the same
convention as in Figure 1.
has
also
a
framework and an isoelectronic π
system as N2S2.
Highly
symmetric
dianionic clusters (BH)n231
quadratic
closo-borane
have been
found to exhibit a strong electron delocalisation, and they have been used as standard
examples for homoaromatic clusters. We compare the magnetic properties of these clusters
with those of typical Zintl ions Sin2- in Section 3. As main result we can conclude that the
closo-borane dianions show all diatropic NICS in the cage centres, and indeed these clusters
have been all found experimentally. On the other hand, for the Sin2- ions diatropic as well as
paratropic NICS at cage centres have been found for 5 ≤ n ≤ 12. This is in line with
experimental observation: While, for example, homoaromatic 9-vertex D3h dodecahedra,
single tricapped Si92- trigonal prisms is prevalent in Zintl ion chemistry, no antiaromatic Si122have been detected yet (Figure 3).
Planar tetracoordinate carbon (ptC) is an exciting new topic in organic chemistry. The
possibility to coordinate carbon with four other carbon atoms in a plane opens many new
possibilities to design molecules and solids with special properties. Our computations suggest
that it is possible to coordinate a ptC to four further carbons by forming the C52- dianion. Such
a dianion can be stabilised by metal counterions, and we find that such species are both
energetically and topologically stable. Again, delocalisation of the p orbital of the ptC plays
the central role in the stabilisation process of these species. The 4th section of this chapter is
discussing this topic.
In the final section we show
that ptC in the form of C52can also be incorporated into
organic rings. We studied
hydrocarbon clusters where
five- to eight-membered rings
include a ptC. The resulting
Figure
3:
The
magnetic
response
of
molecular
frameworks containing a ptC fragment. The response is
antiaromatic for odd-membered (left) and aromatic for
even-membered rings. In both cases, three atoms of the C5
molecules are planar, and the
ptC p orbital is incorporated
in the π system of the ring.
The overall π system can be
aromatic
fragment are included in the ring.
and
antiaromatic,
and the (4n+2) π electron rule
also holds for these molecules. As these molecules contain a low amount of H atoms, they
might be present in interstellar carbon clusters. Indeed, an isomer of C7H2 has been detected
in interstellar medium.
32
33
4 DFTxTB - Development of a fully quantum mechanical
hybrid method
This chapter contains methodological developments which are either still in progress
(DFTxTB), or which are used in the following chapters on nanostructures. The central part of
the chapter is devoted to DFTxTB. DFTxTB is a hybrid method, joining full and
approximated density functional theory (DFT). The approximated variant is the densityfunctional based tight-binding (DFTB) method, which has been developed by Seifert and
coworkers in the 1980s and which is the working horse methodology applied in our group. It
is particularly suitable to study nanostructures, large molecules and solids, and for long-time
molecular dynamics simulations. Many investigations on structure and properties of
nanostructures and fullerenes, given in the next two chapters, have been performed employing
this method. DFTB is an approximation to DFT. The approximations have been introduced in
such way that numerically elaborate procedures can be reduced completely. This is achieved
by three major approximations, which, if applied in a smart way, result in a speedup of DFTB
of four to five orders of magnitude with respect to full DFT calculations for systems
containing up to several thousand valence electrons. The three major approximations of
DFTB can be briefly summarised as follows:
1. DFTB uses a minimum valence basis of atomic orbitals, computed for sphericalised
atoms. Gain: Linear algebra costs reduce significantly, both in CPU and in memory
utilisation.
2. The DFTB Hamiltonian contains only two-centre contributions, all three-centre terms
of the Coulomb and of the external potential are neglected. This is motivated by the
screening argument: The nuclei are shielded by the electrons of the atom, and the total
net charge is approximately zero. In terms of contributions to the Hamiltonian,
external and Coulomb potentials cancel to a large extend. This approximation allows
to employ integral tables (Slater-Koster tables) which contain pre-computed
interaction integrals of the Kohn-Sham and of the overlap matrix elements. In such an
implementation, matrix elements can be interpolated instead of computed. Gain: The
computational costs for integrals and their derivatives is – even for small structures of
about 50 atoms – negligible.
3. Double counting terms, necessary to compute the total energy, are merged together
with the nucleus-nucleus repulsion and given in a parameterised, two-centre
expression. Gain: No grid techniques and no three-dimensional numerical integration
are necessary for calculation of total energy and gradients.
34
The idea to work on DFTxTB is rather old, during my PhD I was starting to work on this
method. As always, the devil is in the detail, and it turned out that it is easier to write a new
DFTB program which is working in an “ab inito” way, i.e. computing all necessary data on
the fly, then to combine existing DFTB implementations with a DFT code. Therefore, DFTB
has been implemented into the new release of deMon in two ways, one employing the
traditional parameter tables, and another one, with an on-the-fly generation of parameters.
deMon uses the latter method to calculate a DFT startup density. The mathematical
description of all quantities as basis functions, MO coefficients, and expansion coefficients of
the electron density follow the same protocol as for the deMon LCGTO-DFT (LCGTO:
Linear combination of Gaussian-type orbitals) method, and the formulation of a hybrid
method became much simpler. As DFTB is an approximation to DFT, all integral routines are
available in the DFT code. The implementation is much more straight-forward.
In DFTxTB, the DFTB approximations are only done for a part of the system, the “low level”
part. The remainder is treated as in standard DFT. The LCGTO, however, spans over the
whole system. Therefore, no artificial boundaries between the subsystems are necessary.
Introducing two or more computational levels into one molecule, even only two basis sets of
different size, leads to a change in the chemical potential and consequently to spurious charge
transfer between the subsystems. This charge transfer is not just an artefact of inadequate
population analysis methods, it also occurs in the auxiliary density and is visible in spurious
dipole moments, for example in D∞h diatomics. DFTxTB uses the self-consistent-charge
(SCC) correction to control the charge transfer between DFT and TB regions during the SCF
procedure. As the SCC technique uses the hardness values of reference atomic orbitals to
control charge fluctuations, a reliable method to compute atomic hardness parameters was
necessary. In the first section, the efficient implementation of a method to compute orbitally
resolved hardness tensors is presented. As a by-product, we could proof that there is, contrary
to wide-spread claims in the literature, no dependency on exchange-correlation functionals for
quantities related to Persson’s hardness-softness acid-base (HSAB) principle (hardness,
softness, Fukui function). Those claims have been artefacts of inappropriate implementations
to compute the hardness, in particular methods which unnecessarily include unoccupied
orbitals.
The latter claim becomes even more evident when I demonstrate that our implementation is
not only functional-independent, but also stable and robust: Atomic hardness parameters from
H through Xe are calculated and presented in the second section.
35
The DFTxTB methodology, and specifics for its implementation into the deMon code, are
presented in the third section. As the method is still in development – no gradients are
developed and implemented yet – no real applications have been performed until now.
However, we expect that the method will be applicable in the next year.
In the fourth section, a scheme to treat weak interactions within the DFTB method is
presented. In this method an empirical long-range London dispersion term is added to the
quantum-mechanical interaction energy, and, correspondingly, it also accounts for the longrange dispersion forces. We demonstrate that this method is suitable for van-der-Waals
complexes and van-der-Waals solids. The advantage of the dispersion-corrected DFTB
method is the possibility to study chemical reactions in a system with significant van-derWaals interactions, for example in nanotubes or in graphite.
36
37
5 Nanostructures – “Better Things for Better Living . . .
Through Chemistry.” *
Computer simulations of structural and electronic properties and atomistic processes of
nanostructures take up a large part of this work. The majority of the applications are devoted
to carbon nanostructures. The prefix “nano” became omnipresent in today’s science, in
particular in chemistry, physics and material science. Indeed, the assembly of characteristic,
well-defined structures on the nanometer scale progressed tremendously in the past decade,
and new materials with new and exciting properties are reported on a daily basis.
The design of new nanostructures with new properties is one of the central issues of the
research carried out in our group. In this work, contributions in three major directions are
followed:
1. Design of fullerene solids with interesting mechanical, electric and magnetic
properties.
2. Design of organic and inorganic nanotubes with special mechanical and
electromechanical properties.
3. Design of carbon nanostructures with high sp2 content suitable for the storage of
molecular hydrogen.
It has been known for a long time that fullerenes can be functionalised, that they can form
dimers and polymers. These observations have been made for the larger, stable fullerenes, for
which holds the isolated pentagon rule (IPR). In these cases, the fullerenes do not change their
topology when a solid is formed, and the intercage bonds are relatively weak. However,
interesting properties may arise for those polymers. One of the most striking examples was
the discovery of “magnetic carbon” by Makarova and coworkers in 2001.
Our applications focus stronger on carbon solids made from small fullerenes. Small
fullerenes, those below the C60 threshold, cannot satisfy the IPR by topological reasons. This
means that a certain number of pentagonal rings are fused, and an energy penalty of 80-100
kJ/mol is connected with each pentagon-pentagon adjacency. † The adjacent pentagons,
together with the high curvature of the small cages, imply the extraordinary high chemical
reactivity of small fullerenes. Hence, they readily form fullerene hydrates or oxyhydrates, and
if they reach a substrate, solid films and eventually solids. The first fullerene solid of this
class which has been debated in the literature, C36, was introduced by Piskoti, Yarger and
*
Old chemical company’s slogan, DuPont 1939.
†
An intensive survey on the role of the fullerene topology for the stability of small fullerenes has been carried
out in my PhD thesis, available on the internet at http://theory.chm.tu-dresden.de/~theine/
38
Zettl in 1999. Details of this topic can also be found in my PhD thesis. Many solid state
topologies have been suggested for this solid, but up to today there is no common agreement
on its stability and structure, neither by theoreticians nor by experimentalists. However, it is
has been shown that such C36 solids can be formed, while their characterisation remains to be
a challenging task.
Another recently reported fullerene solid is
based on the smallest fullerene topologically
possible, C20, which has been synthesised in
2002. Unfortunately, experiment only delivered
TEM images, but no rigorous characterisation of
the new solid. The topology of the molecule
Figure 1: C20H8 and C20H12. The C-H
suggests
bonds correspond to possible linking
neighbouring C20 building blocks to form a solid
sites to form C20-based solids.
(Figure 1). In the first section we propose several
8
or
12
connection
sites
with
possible candidate structures of solid C20. The
investigation starts from C20, and then explores structure and stability of its dimers, oligomers
and 2D and 3D polymers. The density of states shows that all our proposed structures are
insulators or semiconductors. Also, our simulations show that the shape of the fullerenes is
probably lost when intercage bonds of a three-dimensional solid are formed.
In the second and third sections we discuss hypothetical solids based on the C28 fullerene. C28
has tetrahedral symmetry (Td), and the cage has at each corner of the tetrahedron three fused
pentagons. This special topology suggests that the linking positions follow the symmetry
pattern of the cage, and solids compatible to those of sp3-hybridised carbon can be formed.
Following this analogy, we investigated phases isostructural to diamond and lonsdaleite,
which we call hyperdiamond and hyperlonsdaleite. We discuss the overall stability, strainstress curves, and the electronic properties of these materials as well as their endohedral (with
Ti and Zn) and exohedral (with K) intercalates, and the probability of the material to store
molecular hydrogen. Figure 2 shows the interaction potential of hyperlonsdaleit with H2.
39
Figure 2: Structure of hyperlonsdalite and its guest-host interaction potential with H2.
The fourth section is devoted to C50:
This lower fullerene has recently
been synthesised in saturated form
Figure 3: C50H10 – a flying saucer-shaped
fullerene – has an isoelectronic π structure to two
corannulene and 5 C2H6 units.
C50Cl10.
isolated,
The
species
characterised,
has
and
been
its
structure has been determined as the
D5h C50Cl10 isomer. Interestingly, the
isolated cage is not the C50 isomer with the lowest number of pentagon adjacencies. However,
the extraordinary stability of this molecule can be explained by its topology (see Figure3). We
explain the high stability of D5h C50Cl10 and discuss dimers and oligomers based on this cage,
as well as the possibility to form fullerene solids.
Another key application studied by our group is the stability and the electromechanical
properties of organic and inorganic nanotubes. In the fifth section we explore the change of
conductivity when stretching zigzag and armchair nanotubes. We find a rather strong change
of conductivity in the stretching process, and therefore carbon nanotubes might be suitable for
applications as electromechanical switches.
40
Figure 4: Tensile stress tests on a WS2 nanotube using an atomic force microscope.
Inorganic nanotubes, for example WS2 or MoS2, have very interesting mechanical properties
and a high potential for various applications. The most prominent one is probably a lubricant,
NanoLub™, which has been developed recently. Here, WS2 nanoonions are added to grease,
and the friction, even at enormous loads, is reduced strongly. In collaboration with the group
of Reshef Tenne of the Weizmann Institute in Israel we studied the mechanical properties of
MoS2 and WS2 nanotubes. A tensile stress test of a WS2 nanotube with an atomic force
microscope, for which we performed the computer simulation, is shown in Figure 4. In this
work, it was possible to understand the experimental measurements by the computer
simulations, and we could explain the mechanism which is responsible for the strong tensile
stability: The material breaks only after 15-20% of elongation along its principal axis (Section
6).
The final two sections of this chapter are devoted
to the storage of molecular hydrogen in carbon
nanostructures. Our aim is to explore the
possibility to physisorb H2 in and on aromatic
structures, as for example on graphene platelets or
in nanoonions. Several studies on this topic have
been
Figure
5:
interaction
Extrapolation
energy
of
of
H2
the
with
graphene platelets to the graphene
surface. Energies are in kJ mol-1,
distance R in Å.
performed
with
rather
controversial
experimental results. Theoretical investigations are
quite challenging, as the interaction is dominated
by electron correlation, and a reasonable treatment
requires post-Hartree-Fock methods and nearly
converged basis sets. In Section 7 we extrapolate
the H2-graphene interaction on the basis of highly-
correlated ab initio calculations near the basis set limit (see Figure 5).
41
The H2 storage capacity, however, is not
determined only by the interaction energy. As
the guest-host-interaction is weak and H2 is a
light-weight
molecule,
entropy
plays
an
important role, and the free interaction energy
at a given temperature is the determining
quantity. As the potential energy surface is
Figure 6: Isosurfaces of the interaction
potential of H2 with a graphene layer.
Attractive interactions are given in dark
blue, repulsive ones in red.
strongly anharmonic, and as the low mass of
H2 might cause non-classical behaviour, we
calculated the free energy for graphene and
graphene sandwiches quantum mechanically
by solving the Schrödinger equation of a single
particle of the mass of H2 in the external potential of the nanostructure (see Figure 6), as
discussed in Section 8.
42
43
6 Calculation of Nuclear Magnetic Shifts and Shielding
tensors in non-standard situations
The calculation of NMR chemical shifts and shielding tensors has become a standard
application of quantum chemistry. Today, all major commercial and academic chemistrybased electronic structure codes have the possibility to perform NMR calculations at most
levels of theory, including semi-empirical methods, Hartree-Fock, DFT, and post-HF ab initio
methods. Progress, in particular in the past decade, allows the calculation of NMR properties,
even in difficult situations, as for molecules in solution or for complexes with small HOMOLUMO gaps. Numerous reviews on this topic have appeared in the last years, one of them,
concentrating on fullerenes, is given in the first section of this chapter.
In the remainder of this chapter I will show that special effort is necessary if NMR parameters
are computed for systems with strong intermolecular effects, and for systems with high
fluxionality. As NMR theory is highlighted in the 2nd chapter of this work this chapter only
includes five selected applications.
13
In the second section, the
chemical
shifts
of
an
C NMR
endohedral
fullerene are studied. In Sc3N@C80, a
D3h Sc3N moiety is encapsulated in a
C80 cage of Ih symmetry (see Figure 1).
The overall structure must have a lower
symmetry
than
Ih,
and
indeed,
vibrational spectroscopy indicates that
this cage has either C3 or C3v symmetry.
Figure 1: Snapshot of Sc3N@C80, taken from a
molecular dynamics simulation.
However, on the NMR time scale,
which is typically in the order of μs, a
pattern with two signals, denoting Ih
symmetry, was recorded. Our molecular dynamics simulations show that the encapsulated
Sc3N moiety can rotate and tumble nearly without barrier, and therefore the time-averaged
simulated 13C NMR spectrum correctly shows only two signals, corresponding to Ih symmetry
of the cage topology, and the positions are in excellent agreement with experiment.
44
In the third section, the
smaller “sister” Sc3N@C68
(Figure 2) is discussed. This
was the first fullerene with
adjacent pentagons which
could
be
quantities
produced
in
which
allow
kinds
of
Figure 2: Structure of Sc3NC68, the first fullerene with
various
adjacent pentagons (given in red).
spectroscopy, including
13
C
NMR. We show that the
behaviour of this smaller cage is different to Sc3NC80: The Sc3N cluster is fixed in three
“pockets” of the cage and hence does not show fluxional behaviour. However, the motion of
the central N atom breaks the D3 symmetry, and the optimised structure coincides with the
time averaged one, corresponding to the observed
13
C NMR pattern. We simulate the
13
C
NMR pattern, could assign the cage isomer to unambiguously, and furthermore observed an
error in the line assignment of the original experimental work.
The influence of solvent and of fluxionality on
a zeolite precursor is the topic of the 4th
section. Indeed, to simulate the
29
Si NMR
spectrum of polar Si(OH)4, a full shell of water
has to be included in the simulation, and the
vibrations,
coordination
rotations,
to
and
changes
neighbouring
of
solvent
molecules have to be treated correctly. We
found that both polarisation and motion on the
ps time scale attribute to ~3 ppm to the
chemical shift, both additive to lower fields.
In section 5, the intermolecular effects on the
1
H chemical shielding tensors of biphenyl in
the crystal phase has been investigated (Figure
Figure 3: The unit cell of solid biphenyl
3). A combined experimental/theoretical study
contains two crystallographically distinct
was necessary to assign the proton shielding
molecules.
tensors correctly, and we could determine the
influence of the intermolecular contributions,
45
which can be as large as 2.5 ppm. It was evident from the comparison of experiment and gasphase calculation, both performed more than 10 years ago, that there is an appreciable
intermolecular effect on the 1H shielding. It was, however, up to now not clear how this effect
should be evaluated. We proposed a simple superposition scheme for this purpose, in which
individual shielding contributions of all neighbours around the molecule, up to a radius of 72
Å, were calculated. Even though our computation is still a crude model, neglecting molecular
vibrations and electronic intermolecular interaction, the comparison with experiment is
strikingly good.
Finally, in Section 6, we apply the
same method for the shielding tensor
of benzene. It is indeed true that in
2006 the proton shielding tensor of
benzene, one of the most commonly
used compounds in chemistry, is still
unknown. We show that there is
reason
for
this
fact,
as
its
experimental determination is close
to
impossible,
and
theoretical
calculations suffered the fact that
long-range interactions have been
neglected so far. An important result
is
that
the
intermolecular
contributions can have an influence
of up to 5 ppm for individual
Figure 4: 1H NMR shielding tensor of the benzene
components of the shielding tensor,
molecule (top) and benzene in solid state (bottom)
and 1.74 ppm for isotropic shieldings,
in tensorial representation. The difference (centre)
15% of the whole organic proton
is the intermolecular contribution. Principal axes
NMR scale! Therefore, if computed
are indicated, the lengths are proportional to the
proton shielding tensors should ever
magnitude of the shielding tensor component.
be compared to experiment, it is
absolutely necessary to include these
interactions in the calculations of systems containing aromatic rings, for example in most
biological molecules.
46