Exploring the exohedral reactivity
and selective encapsulation of
fullerene compounds
Treball Final de Màster
Màster Interuniversitari en Química Teòrica i Computacional
Supervisors: Dr. Josep Maria Luis i Luis i Prof. Miquel Solà i Puig
Marc Garcia Borràs
Institut de Química Computacional and Departament de Química
Universitat de Girona
Girona, Setembre del 2011
I can accept failure, everyone fails at something.
But I can't accept not trying.
Michael Jordan
Als meus avis,
A mis abuelos,
PREFACE
Since fullerene discovering in 1985 by Kroto, Smalley and Curl,1 the interest on this new type of molecules has only
grown because of their potential applications in medicine (as a drug transporter, for example) and technological
(interesting electronic properties) fields. But until now, about 25 years after their discovering, reactivity of fullerene
compounds and their behavior are still largely unknown. The aim of this Master’s Thesis is twofold. First, to shed
some light on the reactivity of fullerene compounds by the computational study of the Diels-Alder cycloaddition
reaction on the Ti2C2@-D3h-C78 metallofullerene. Second, to perform a computational study about host-guest
interactions between recently synthesized 3D nanostructures2 and C60 fullerene to discuss whether these
metalloporphyrin-based nanocages are able to selective encapsulate fullerene molecules.
These studies represent an advance in the knowledge of the fullerene compounds chemistry. Only when we
understand fullerene behavior, we will be able to convert the potential applications of this family of compounds
into real ones.
RESUM DEL TREBALL
Des del descobriment dels ful·lerens per part de Kroto, Smalley i Curl al 1985,1 l’interès en aquestes noves
molècules només ha fet que créixer degut a les seves potencials aplicacions en els camps de la medicina (com a
transportadors de fàrmacs, per exemple) o la tecnologia (interessants propietats electròniques). Però ara, més de
25 anys després del seu descobriment, la reactivitat dels compostos ful·lerènics és encara bastant desconeguda.
L’objectiu d’aquest Treball Final de Màster és avançar en la comprensió de la reactivitat dels compostos ful·lerènics
mitjançant l’estudi computacional de la reacció de cicloaddició Diels-Alder entre el endoful·lerè metàl·lic Ti2C2@D3hC78 i el butadiè. També es duu a terme un estudi de les interaccions “hoste-amfitrió” entre nanoestructures
metal·loporfiríniques 3D recentment sintetitzades2 i la molècula del ful·lerè C60 amb l’objectiu final de ser capaços
d’encapsular selectivament aquest ful·lerè a l’interior de nanocapsules.
Aquests estudis representen un avenç en el coneixement sobre la química dels compostos ful·lerènics. Únicament
quan entenguem el comportament dels ful·lerens, serem capaços de convertir les seves aplicacions potencials en
aplicacions reals.
CONTENTS
CONTENTS
CHAPTER I. GENERAL INTRODUCTION
9
1. FULLERENES
INTRODUCTION
THE BEGINNINGS: DISCOVERING THE C60
CHARACTERIZATION OF C60: STRUCTURAL DESCRIPTION
THE IPR RULE
IPR FULLERENE ISOMERS
FULLERENE PROPERTIES
Carbon-Carbon bond types
Electronic Structure
Aromaticity
2. ENDOHEDRAL METALLOFULLERENES
INTRODUCTION
METALLIC CARBIDE ENDOHEDRAL FULLERENES
Ti2C2@C78 versus Ti2@C80: the stability of metallic carbide
THE MAXIMUM PENTAGON SEPARATION RULE
3. EXOHEDRAL REACTIVITY OF FULLERENE COMPOUNDS
CYCLOADDITION REACTIONS: DIELS-ALDER
The Diels-Alder cycloaddition on endohedral metallofullerene compounds.
4. SUPRAMOLECULAR CHEMISTRY
INTRODUCTION
SELF-ASSEMBLY
HOST-GUEST CHEMISTRY
MOLECULAR RECOGNITION BASED ON SUPRAMOLECULAR CHEMISTRY
Purification and isolation of fullerenes based on host-guest interactions
9
9
9
10
10
12
14
14
15
15
17
17
19
20
22
25
26
28
31
31
32
33
34
35
CHAPTER II. COMPUTATIONAL METHODOLOGIES
39
1. THE HARTREE-FOCK APPROXIMATION
2. THE DENSITY FUNCTIONAL THEORY
THE KOHN-SHAM FORMULATION
THE EXPRESSION FOR THE EXCHANGE-CORRELATION FUNCTIONAL
BASIS FUNCTIONS
CORE ELECTRONS TREATMENT
RELATIVISTIC EFFECTS: ZORA APPROXIMATION
CLASSICAL CORRECTIONS: DISPERSION ENERGY
ENERGY DECOMPOSITION ANALYSIS
41
45
46
48
52
53
53
54
55
7
MASTER THESIS
3. FULLERENE COMPOUNDS USING THE COMPUTATIONAL CHEMISTRY TOOLS
REACTION, ACTIVATION, DEFORMATION AND INTERACTION ENERGIES IN FULLERENE REACTIVITY
PREDICTING THE FULLERENE CHEMISTRY
57
58
60
CHAPTER III. OBJECTIVES
63
CHAPTER IV. RESULTS
65
1. EXOHEDRAL REACTIVITY OF TI2C2@C78 ENDOFULLERENE: DIELS-ALDER CYCLOADDITION ON ALL NON67
67
67
68
71
74
EQUIVALENT BONDS
INTRODUCTION
COMPUTATIONAL DETAILS
STRUCTURAL CHARACTERIZATION OF TI2C2@D3H-C78 ENDOFULLERENE
REACTION ENERGIES FOR THE DIELS-ALDER REACTION ON TI2C2@D3H-C78
ENERGY BARRIERS FOR THE DIELS-ALDER REACTION ON TI2C2@D3H-C78
Effects on the reaction barrier due to the presence of Ti2C2 inside the D3h-C78 fullerene cage.
Deformation energies and Molecular Orbitals.
76
EFFECT OF THE CLUSTER NATURE AND FULLERENE STRAIN ON THE EXOHEDRAL REACTIVITY OF D3H-C78
ENDOFULLERENE COMPOUNDS. THE REACTIVITY OF TI2C2@D3H-C78 AND XN3@D3H-C78 (X = SC, Y)
METALLOFULLERENES.
80
INCLUSION OF DISPERSION CORRECTIONS: CHANGES IN THE ENERGY PROFILE OF DIELS-ALDER REACTION
84
FINAL REMARKS AND CONCLUSIONS
86
2. FINE-TUNABLE METALLOPORPHYRINIC NANOCAGES AS HOSTS FOR FULLERENE ENCAPSULATION
89
INTRODUCTION
89
COMPUTATIONAL DETAILS
90
EXPERIMENTAL CRYSTALLOGRAPHIC STRUCTURES AND DFT STRUCTURES
90
HOST-GUEST STUDIES: EFFECTS OF THE FULLERENE ORIENTATION AND PORPHYRIN METAL IONS
93
Modeling the system: porphyrinic dimer
95
Study of the fullerene orientation and porphyrin metal effects
97
Energy Decomposition Analysis applied to the study of host-guest interactions
100
FINAL REMARKS AND CONCLUSIONS
104
CHAPTER V. CONCLUSIONS
107
CHAPTER VI. ACKNOWLEDGEMENTS
109
AGRAÏMENTS
109
CHAPTER VII. BIBLIOGRAPHY
112
CHAPTER VIII. SUPPORTING INFORMATION
120
8
GENERAL INTRODUCTION
CHAPTER I.
GENERAL INTRODUCTION
1. Fullerenes
Introduction
Fullerenes are molecules composed by an even number of carbon atoms that are
tricoordinated and located at vertices of polyhedra with only pentagonal and hexagonal faces
forming a hollow sphere. They are represented by the formula Cn, where n is the number of
carbon atoms. Pentagonal rings give the curvature to these molecules, as opposed to graphite
structure formed only by hexagonal rings. Fullerenes and graphite are both carbon allotrope,
like diamond and carbon nanotubes (CNT). The main difference between the different
allotrope type is the carbon hybridization. For example, in the diamond, carbon atoms have an
sp3 hybridization and tetrahedral coordination, but for the graphite carbons have an sp2
hybridization. Finally, in fullerenes and carbon nanotubes we have an intermediate situation.
The buckminsterfullerene, C60, is the most popular and studied fullerene structure, and takes
its name from the famous geodesic dome designed by the architect Richard Buckminster
Fuller, which has similar geometric shape. The buckminsterfullerene was the first fullerene
structure discovered and the acronym “fullerene” was adopted to mention related structures
of the family.
The beginnings: Discovering the C60
In 1996, Kroto, Smalley and Curl were awarded with the Nobel Prize of Chemistry for the
discovering buckminsterfullerene. But the possibility of the existence of carbon clusters had
been discussed before their publication in 1985.1
Kroto, Smalley, Curl and co-workers were trying to understand the mechanisms of the
formation of long-chain carbon molecules are formed in interstellar space and circumstellar
shells, vaporizing graphite by laser (Nd:YAG) irradiation. Under these conditions, they observed
a 720 mass peak detected by a time-of-flight mass spectroscopy. That peak corresponded to a
9
MASTER THESIS
C60 structure. They also observed clusters up to 190 carbon atoms, and noted that for clusters
of more than 40 atoms, only those containing an even number of atoms were detected. In
order to satisfy all sp2 valences, they proposed a spheroidal structure based on Buckminster
Fuller’s studies which has icosahedral symmetry for the C60 cluster. They also noticed that the
inner cavity of C60 (about 7 Å of diameter) is able of holding a variety of atoms.
Depending on the vaporization conditions, the formed clusters abundance distribution
changes, but C60 is always the most abundant species. The second carbon cluster in these
distributions is C70, but far from C60 values. Then they concluded that these two structures are
the major constituent of circumstellar shells with high carbon content. As we know nowadays,
they were not far wrong.
Characterization of C60: Structural description
Since Kroto, Smalley and Curl published their discovery, many structural studies of
buckminsterfullerene have been carried out, as well as different studies about properties,
stability, aromaticity and reactivity.
In 1990, Krätschmer et al.3 developed a methodology to get macroscopic quantities of
buckminsterfullerene using pure graphitic carbon soot produced by evaporating graphite
electrodes in an atmosphere of about 100 torr of helium. Then, the resulting product was
dispersed in benzene. The solution had a red-wine to brown color, depending on the C60
concentration. The liquid was then separated from the soot and dried, leaving a residue of
dark brown to black crystalline material.
All structure experimental analysis showed evidences of the presence of C60 in macroscopic
quantities, as mass spectra that had a strong peak at 720 a.m.u., or electron and X-ray
diffraction experiments on synthesized crystals. 13C-NMR measurements carried out by Taylor
et al.4 provided the definitive proof that all the carbons were equivalent, supporting the
buckminsterfullerene structure.
The IPR rule
As we have seen, fullerene structures are polyhedron where atoms are situated in vertices,
bonds in edges and rings in faces. Taking into account the Euler Theorem, the relationship
between vertices (v), edges (e) and faces (f) is:
𝑣+𝑓 = 𝑒+2
10
(1)
GENERAL INTRODUCTION
And the restriction of three σ-bonds per carbon atom, relates the number of vertices (v) and
the number of edges (e) as gives the equation (2).
2𝑒 = 3𝑣
(2)
If fn is defined as the number of n-sided faces (rings), then the following relationship is
accomplished:
2𝑒 = ∑ 𝑛𝑓𝑛
(3)
𝑛
Now, if we consider a fullerene Ck, the number of vertices (i.e. atoms) is equal to k, and by the
relation in the equation (2) we can obtain the number of edges:
𝑒 = 3𝑘⁄2
(4)
And remembering the Euler Theorem given by the equation (1), the number of faces is given
by the equation (5).
𝑓 = 𝑘 ⁄2 + 2
(5)
As the number of faces can also be expressed as:
𝑓 = ∑ 𝑓𝑛
(6)
𝑛
And making use of the expression given by equation (5), we can obtain equation (7):
∑ 𝑓𝑛 = 𝑘⁄2 + 2
(7)
𝑛
If only fullerenes with pentagonal and hexagonal rings are considered, we can expand the
equation (7) into:
11
MASTER THESIS
𝑓5 + 𝑓6 = 𝑘⁄2 + 2
(8)
And considering equation (3) and (4) (and considering only pentagonal and hexagonal rings)
we have:
5𝑓5 + 6𝑓6 = 𝑘⁄2 + 2
(9)
Then a linear system is obtained, whose solutions indicate that Ck fullerenes must have:
𝑓5 = 12
𝑘
𝑓6 = − 10
2
(10)
that is, 12 pentagonal rings and 𝑘⁄2 − 10 hexagonal rings.5
Based on this theorem, the smallest possible fullerene is C20 which has 0 hexagonal rings
(20⁄2 − 10). When the number of hexagonal rings increase, a wide range of fullerenes can be
generated, and different isomers can be obtained.
The Isolated Pentagon Rule (IPR) proposed by Kroto in 1987,6 states that most stable fullerenes
are those where the 12 pentagons are isolated. When 2 pentagon rings are abutted a high
steric tension is produced, in addition to the destabilizing effect of the π structure impedes the
fullerene formation. These two facts support the IPR rule, and tell us that the final stability of
the fullerene is given by an equilibrium of these two aspects.
When we talk about the destabilizing effect of the π structure, we are referring to the Hückel
4𝑛 + 2/4𝑛 rule. As a general rule, π-electron stabilization is greatest for six membered rings,
somewhat less for sizes 5 and 7, and dramatically less for sizes 4 and 8. Smaller rings (3 atoms)
are unstable because of the σ-strain. In the case of having fused five-membered rings, we have
an eight-cycle in the periphery of these two rings, and according to the Hückel’s rule (8 π
electrons) it has a destabilizing effect over π electronic structure.
IPR fullerene isomers
Taking into account the IPR rule, the final number of isomeric possibilities is substantially
reduced. Now we will focus on the C60 and C78 fullerenes, the two cages which this work is
based on.
As we have seen in the previous sections, the first discovered and characterized fullerene was
the buckminsterfullerene. The 13C-RMN results showed that all the C atoms presents in the C60
12
GENERAL INTRODUCTION
structure were equivalent, which means that the structure synthesized corresponds to the IhC60 isomer. The latter C60 isomer is the only one obeying the IPR rule of all the 1812 possible
isomers for C60. Until very recently, it was the unique buckminsterfullerene isomer identified
and isolate, but in 2008 Tan and coworkers presented chlorinated species of the first non-IPR
C2v and Cs-C60 symmetries represented in Figure 1.7
Figure 1. Geometries of the three C60 isomers obtained from the Tan et al. paper. Carbon atoms along pentagonpentagon fusion are coloured red. The dashed lines indicate the transformations from C s-C60 to C2v-C60 and
eventually to Ih-C60.
There are 5 different IPR structures for C78 cage, which are candidates for its ground state and
have symmetry C2v’, D3, C2v, D3h’, and D3h (see Figure 2). The relative stabilities of the isomers,
calculated at HF level with 6-31G* basis set (in brackets) are: C2v’-C78: 3 (0 kcal/mol) > D3-C78: 1
(3 kcal/mol) > C2v-C78: 2 (4 kcal/mol) > D3h’-C78: 5 (7 kcal/mol) > D3h-C78: 4 (20 kcal/mol).8 As we
will see in next chapters, the stability of the different isomers can be modified by the
encapsulation of different metallic clusters inside. As for example, the Ti2C2@C78 has D3h’
symmetry although the D3h’-C78 free cage is not the most favored. In addition to these 5 IPR
cages, there are 24105 more isomers that do not obey the IPR rule.
Figure 2. Representation of the 5 IPR isomers of C78 fullerene cage.8
13
MASTER THESIS
Fullerene properties
Carbon-Carbon bond types
Because fullerenes are constituted by hexagonal and pentagonal faces, they have two different
bond types. [6,6] bonds are those situated between 2 hexagonal rings, and [5,6] between an
hexagonal and a pentagonal ring (see Figure 3).
Figure 3. Representation of the different bond types [5,5], [5,6], and [6,6] that might be present in any fullerene
structure.
For each bond type, we can found different environments that allow us to classify them into
different subtypes. In the [6,6] bond case, we have: pyracylenic or type A, Type B, and pyrenic
or type C. The first (type A) corresponds to a C‒C bond situated between two pentagonal rings,
they are the shortest bonds and also those having C atoms with the highest pyramidalization
angles. These properties provide a strong double bond character to pyracylenic bonds. Type B
bond is the one situated between a hexagon and a pentagon. And finally, the type C is
localized between two hexagonal rings, and it has the lowest pyramidalization angles which
produces a more planar region of the fullerene structure.
For the [5,6] bonds we can also classify them in two different subtypes: corannulene or type D,
and Type F. Finally, [5,5] bond are pentalene or type E. As type F and type E (pentalene) bonds
have two pentagons abutted, they cannot be found in C60 and D3h-C78 structures which obey
the IPR rule.
14
GENERAL INTRODUCTION
Electronic Structure
The C60 electronic structure results in a HOMO orbital which has bonding π interactions in the
[6,6] bonds, and antibonding π interactions in the [5,6]. This implies the occupation of the
HOMO orbitals leads to cut down the [6,6] bond distance and to extend the [5,6] bond types.
This fact produces a C‒C bond distance alternation which does not allow a fully delocalization
of the π electron over different bonds. The LUMO and LUMO+1 represent an inverse situation,
so including electrons into these orbitals causes an increase of the [6,6] bond distance, and a
decrease of the [5,6], which favor the aromatic character of the molecule because all different
types for C‒C bond have a similar bond distance. These particularities can be extended to all
fullerene compounds family.
Aromaticity
The presence of hexagonal rings in the fullerene structures was originally interpreted in the
sense of C60 was a possible superaromatic molecule. Fullenenes cannot undergo substitutions
reactions, characteristic of the aromatic compounds, because they haven’t hydrogen atoms
that could be substituted. Moreover, the pyramidalization of C atoms, responsible for the
strain, should be taken into account to discuss the fullerene aromaticity. The fullerene
chemical reactivity is more similar to the electron-deficient olefins, and is favored by strain
release.
Centering our attention into the structural point of view, as we have seen previously C60 has
bond length alternation which is a clear difference from the prototypical aromatic molecule
benzene. The 2(𝑁 + 1)2 rule was proposed for the spherical fullerenes9 which is an equivalent
to the 4𝑁 + 2 Hückel rule for planar polycyclic aromatic compounds. For example, according
10+
to this rule, the cationic 𝐶60
molecule is closer to the rule than C60, and indeed the bond
10+
differences in 𝐶60 are smaller than in C60.
There are several studies of the C60 aromaticity10-13 which have shown that six-membered rings
are partially aromatic, while five-membered rings are antiaromatic. There also exists electron
delocalization studies14 that show a delocalized character of the C60 π-system similar to the
aromatic systems such as benzene or naphthalene.
In conclusion, we can consider C60 and fullerene in general to show only modest forms of
aromaticity, and equally for the different fullerene compounds.
15
MASTER THESIS
16
GENERAL INTRODUCTION
2. Endohedral Metallofullerenes
Introduction
When Kroto, Smalley, Curl and co-workers reported the discovery of the buckminsterfullerene
in 1985, they already hypothesized about the fact that the diameter of this molecule was large
enough to hold a variety of atoms inside the carbon cage.1 Fullerenes can encapsulate atoms
(such noble gases), ions, metallic clusters and small molecules (such H2, CO, H2O, NH3, or CH4).
One of the most interesting candidates to be encapsulated inside the fullerene cages are
metals. In fact, metals can be (1) incorporated into the fullerene carbon surface (heterohedral
metallofullerenes) (2) located outside the cage (exohedral metallofullerenes) or (3) trapped
inside the hollow fullerene cage (endohedral metallofullerenes) (Figure 4).
Figure 4. Representation of the different types of metallofullerenes: (1) Heterohedral metallofullerene, (2)
Exohedral metallofullerene, and (3) Endohedral metallofullerene.
Endohedral fullerenes are formed when the metal atom is encapsulated inside the cage. The
first evidence of the existence of this type of metallofullerenes was presented by Heat et al.15
the same year of the discovery of the C60. They found evidences of the formation of a stable
C60La, with the La atom trapped inside the C60 cage.
To describe the endohedral metallofullerene, the most commonly used nomenclature is M@Ck
where the symbol @ indicates that the metal M is trapped inside the cage Ck. Endohedral
metallofullerenes (EMF) can be classified into several classes (see Figure 5):16,17 (a) the socalled classical EMFs of the type M@C2n and M2@C2n; (b) metallic tri-nitride templates (TNT)
EMFs (M3N@C2n); (c) metallic carbide EMFs (M2C2@C2n, M3C2@C2n, M4C2@C2n, M3CH@C2n, and
M3CN@C2n); (d) metallic oxide EMFs (M4O2@C2n and M4O3@C2n); and (e) metallic sulfide
(M2S@C2n).
17
MASTER THESIS
Figure 5. representation of some types of endohedral metallofullerenes: (a) classical (La@C82)18; (b) metallic trinitride template (TNT) (Sc3N@C78)17; (c) metallic carbide (Sc2C2@C84)19; and (d) metallic oxide (Sc4(μ3-O)2@C80).20
The classical metallofullerenes have C2n cages, where 68 ≤ 2𝑛 ≤ 92. The most abundant
classical EMFs family corresponds to M@C82, being La@C82 the fist example synthesized.18,21
In 2001, Shinohara and co-workers reported the first characterization of a metallic carbide
endohedral compound.19 It was a Sc2C2 unit encapsulated inside the D2d-C84 cage. Other
examples of metallic carbide EMFs are Ti2C2@D3h-C78 (which is studied in this work), Y2C2@C82
(isomers Cs, C2v, and C3v), Sc2C2@C2v-C68, or Gd2C2@D3-C92.22
Using the Krätschmer-Huffman arc method with presence of nitrogen, TNT EMFs can be
produced in macroscopic quantities.23 In this group is found the Sc3N@C80, which is the third
most abundant fullerene,24 only exceeded by C60 and C70. Other interesting TNT member is the
non-IPR Sc3N@C68.
The EMFs are promising materials with a number of potential interesting applications related
to magnetism, superconductivity, and nonlinear optical (NLO) properties.25,26 By varying the
encapsulated metal cluster, the optoelectric properties of EMFs might be tailored without
changing the outer carbon cage. Moreover, the relative inertness of the EMFs carbon structure
makes these compounds ideal for medical applications, for instance as hosts of radioactive
atoms for use in nuclear medicines,27,28 or as effective magnetic resonance imaging (MRI)
contrast agents.29,30 Also, photoinduced charge transfer using EMFs as electron acceptor in
electron donor-acceptor dyads can lead to promising photovoltaic materials to be used in solar
energy conversion/storage systems.31 Finally, it has also been suggested that some EFs with
long spin lifetimes might in the future be used in quantum computing or spintronic devices.32,33
In this work, we study the exohedral reactivity of Ti2C2@C78 EMF, so in the next section we will
give a more detailed description of metallic carbide endohedral complexes.
18
GENERAL INTRODUCTION
Metallic carbide endohedral fullerenes
Metallic carbide EMFs are a special type of fullerenes that encapsulate a metallic carbide
moiety in their interior (see Figure 6).22 The first structural characterization of a metallic
carbide EMF, as it has been mentioned in the latter section, was reported in 2001 by Shinohara
and co-workers.19 They used the 13C-RMN and synchrotron X-ray diffraction to determine the
endohedral character of Sc2C2 unit inside the C84-D2d cage.
Figure 6. Representation of: (left) half section of the equicontour surface of the electron charge density for
Sc2C2@C84; and (right) the structure of Sc2C2@C84.22
The same metallofullerene had previously been isolated by the same research group but they
erroneously assigned it as Sc2@C86. In a similar way, theoretical calculations as well as 13C-NMR
spectroscopy and refined X-ray structural analyses have revealed over the past of the years
other metallic carbide EMFs such as Y2C2@C82 (isomers Cs, C2v and C3v), Ti2C2@D3h-C78 (about
which we will discuss later), Sc2C2@C2v-C68, Sc2C2@C3v-C82, Sc3C2@Ih-C80, and Gd2C2@D3-C92.22 It
is also interesting to remark that the only non-IPR carbon cage of the above described EMFs is
the Sc2C2@C2v-C68.22
A good example of the complexity of determining the true structures of endohedral fullerenes
is the special case of the Sc3C2@C80.17 The metallofullerene Sc3C82 was synthesized by arc
discharge of Sc-impregnated graphite rods in 1992.34,35 Initially, electron paramagnetic
resonance, theoretical studies and synchrotron X-ray powder diffraction studies suggested
that a C82 cage encapsulated three Sc ions. However, single-crystal X-ray structures of a
chemically functionalized fullerene36 and, more recently, high-resolution powder X-ray
diffraction37 revealed the presence of a Sc3C2 unit inside an Ih-C80 cage. The theoretical DFT
results showed that the two Sc3C2@Ih-C80 isomers computed (see Figure 7) are 30.2 and 30.9
kcal mol-1 more stable than the lowest-energy structure of Sc3@C82.36
19
MASTER THESIS
Figure 7. Representation of: (a) the X-ray single-crystal structure of Sc3C2@C80 functionalized by adamantylidene
carbine; and (b) computed structures of Sc3C2@C80 isomers.36
Poblet and co-workers38 studied the electronic structure of metallic carbides EMFs using their
ionic model. They proposed that empty cages with a large (LUMO-3)-(LUMO-2) gap are more
suitable for encapsulation the M2C2 moiety because of the stabilization obtained by the formal
transfer of four electrons from the cluster to the LUMO-1 and LUMO-2 of the carbon cage. A
carbide-containing endohedral fullerene can be seen as (M2C2)4+@(C2n)4- with the C2 unit
considered as an acetylide ion, C22-. That is true for the Sc2C2 case, where only 4 electrons are
transferred. But in the Ti2C2 case, each Titanium transfers one more electron to the carbon
cage than Scandium (6 electrons from the Ti2C2 moiety). So we have to take into account the
(LUMO-4)-(LUMO-3) gap instead of the (LUMO-3)-(LUMO-2) gap. Then, the LUMO-1, LUMO-2,
and LUMO-3 of the carbon cage will be occupied giving the (M2C2)6+@(C2n)6- unit.
Ti2C2@C78 versus Ti2@C80: the stability of metallic carbide
As said before, in the present work we will study the Diels-Alder cycloaddition on the
Ti2C2@C78 EMF. In this section we will describe the discovery of the Ti2C2@C78 EMF and how
this was initially assigned as the Ti2@C80 compound.
In fact, in the Ti2C80, the two Ti atoms were believed to be encapsulated in a D 5h-C80 or Ih-C80
cage with a total of four electrons transferred to the carbon cages, although both cages prefer
to accept six electrons to attain a closed-shell electronic configuration.39 That was the first
assignment when Ti2C80 was synthesized.40 But later studies showed the possibility of having
the Ti2C2@C78 compound instead of Ti2@C80. In particular, an analysis of ultraviolet
photoelectron spectra (UPS) has shown that this Ti-encapsulated EMF exhibit different
characteristics from those of empty C80 fullerene.41 This suggested that the cage which
encapsulates Ti was not the C80. According to 13C-NMR experiments,42 the most plausible
option is having two Ti atoms and a C2 molecule encapsulated inside a C78 cage. Transmission
20
GENERAL INTRODUCTION
electron microscopy experiments43 also corroborate the existence of Ti2C2@C78 encapsulated
in single-walled carbon nanotubes. However, more details of the structure of this EMF were
not known until computational studies were carried out.39,42,44
Different computational results showed that the most favorable isomer of C78 cage to allow
the Ti2C2 moiety is the D3h-C78 instead of the D3h’ or C2v C78 isomers42 (see Figure 8_(1)). To
accommodate the cluster inside, the cage expands along the C3 rotation axis relative to the
empty C78 cage. In contrast to the Sc2C2@C78 EMF case, the Ti2C2 cluster adopts a linear
conformation inside the C78 cage.44 The encapsulation of Ti2C2 moiety modifies the geometry of
the cage. In particular, it elongates the bond lengths of the hexagon adjacent to the Ti atoms.
Figure 8. (1) Representation of the geometric structures of three isomers of Ti2C2@C78 metallofullerene: (a) D3h;
(b) D3h’; and (c) C2v.42 (2) Schematic representation of the C2 rotation around the dititanium center inside the C78
cage.44
Moreover, theoretical results show that the Ti-cage bonding exists and it’s much stronger than
the Ti-acetylide (Ti-C2). The Ti2C2@C78 can be viewed as Ti28+C22-@C786- with covalent dative
bonding between the Ti4+ cations and the C786- cage, as well as ionic Ti4+-acetylide
interactions.39,44 This fact is manifested with the position of the Ti atoms, which are too close
to the center of hexagonal faces, indicating the existence of dative bonding. The presence of
the C2 between the two titanium cations reduces the Coulomb repulsion energy between the
two positive charges (two Ti4+), which stabilizes the final endofullerene formed.
The rotation of the C2 molecule around the two titanium atoms inside the C78 cage has also
been studied. As can be seen in Figure 8_(2), the rotation of the C2 unit that transforms the
end-on bridging mode of C2 to the side-on one is prevented by an activation energy of 2.11 eV
(about ~48 kcal mol-1) relative to the linear isomer.44
21
MASTER THESIS
In conclusion, the Ti2C2@C78 is an endohedral metallofullerene with a characteristic molecular
and electronic structure that makes it an interesting species to be studied. In this work, its
reactivity will be analyzed and will be compared with the reactivity of the free C78 cage and of
other related EMFs.
The maximum pentagon separation rule
As we have seen in previous sections, Poblet and co-workers tried to explain the selection of a
particular cage isomer to host a specific metal cluster based on the molecular orbital rule. That
is the (LUMO+(n+1))–(LUMO+n) gap, 2n being the number of electrons formally transferred
from the cluster to the fullerenic cage. But although this rule is useful to identify the best cage,
does not provide the complete explanation. More recently, Poblet’s group has shown that the
stability of a particular isomer is related to the separation among pentagons, which can be
measured with the inverse pentagon separation index (IPSI) computed by the formula: 17,45
12 12
𝐼𝑃𝑆𝐼 = ∑ ∑ 1⁄𝑅𝑖𝑗
(11)
𝑖=1 𝑗>𝑖
where 𝑅𝑖𝑗 is the Euclidean distance between the centroids of pentagon i and j. The largest
pyramidalization of the [5,6] carbon-carbon bonds induces higher concentration of the
negative charge on the pentagons. Consequently, the isomers with largest separation among
pentagons (smallest IPSI values) reduce the coulombic repulsion and thus become the most
stable.17,45 As an example, for the hexaanions of the 35 IPR isomers of C88 fullerene they
represented the AM1 relative energies as a function of IPSI (see Figure 9). These results reveal
that there exists a correlation between the separation among the pentagons and the stability
of the isomers. The IPR isomer 35, which is the IPR isomer with a lowest IPSI and the largest
concentration of negative charge on the pentagons, has been identified by X-ray
crystallography as the cage that encapsulates Tb3N and Gd3N. So, those structures with
favorable disposition of pentagons, i.e. lowest IPSI values, not only minimize the steric strain,
but also the Coulomb repulsion, which is the major factor in determining the stability of
charged fullerenes (endohedral metallofullerene cages).
Finally, it’s important to remark that the maximum pentagon separation rule is less likely to be
obeyed when the charge transferred from the metal to the cage is small or when the cage size
is large.
22
GENERAL INTRODUCTION
Figure 9. Representation of: (a) the relative stability of the 35 IPR isomers of C 88 in their hexaanion form with
respect to the Inverse Pentagon Separation Index (IPSI); and (b) charge localized on the pentagons. The most
stable IPR isomer 35 encapsulating the Tb3N and Gd3N guests has been characterized by X-ray crystallography.17,45
23
MASTER THESIS
24
GENERAL INTRODUCTION
3. Exohedral reactivity of fullerene
compounds
We have seen in the previous chapters that fullerenes are usually involved in typical reactions
of π-conjugated electron deficient alkenes, as reductions, cycloadditions, nucleophilic
additions, hydrogenations, radical additions, and halogenations. But these are only a few
possibilities of all the extensive fullerene chemistry, which is represented in Figure 10.46,47
Figure 10. Schematic representation of the different reactivity of C60.47
One of the objectives of the present work is to study and understand the Diels-Alder
cycloaddition between the endohedral metallofullerene Ti2C2@C78 and 1,3-cis-butadiene. That
is the reason why now we will focus our attention to the cycloaddition reactions which
involves EMFs, and the Diels-Alder reactions (DA) in particular.
25
MASTER THESIS
Cycloaddition reactions: Diels-Alder
Cycloaddition reactions belong to one major class of reactions called pericyclic reactions. They
are a kind of organic reactions which involve two reactants, one identified as an electrophilic
and the other as a nucleophilic species. These are usually rearrangement reactions, wherein
the transition state of the molecule has a cyclic geometry, and the reaction progresses in a
concerted fashion.
The most famous cycloaddition reaction is the Diels-Alder reaction which takes place between
a conjugated diene and an olefin called the dienophile, usually conjugated, to form a
cyclohexene (see Figure 11). This reaction was described by Otto Diels and Kurt Alder in
1928,48 and in 1950 they were awarded the Nobel Prize in Chemistry because of this work.
Figure 11. Representation of the Diels-Alder reaction mechanism, where the transition state has six π delocalized
electrons.
The Diels-Alder reaction goes in a single step. Its mechanism is often described as a rotation of
the electrons round a six-membered ring (as represented in Figure 11), which transition state
has six delocalized electrons giving aromatic character to this structure. Finally, two π bonds
disappear and two σ bonds are formed with four electrons moving smoothly out of the π
system into the σ orbitals.
In the sense of the Woodward-Hoffmann description, the Diels-Alder reaction is often
described as a [4𝜋𝑠 + 2𝜋𝑠 ] reaction. In this notation, numbers followed by 𝜋 denote the
number of 𝜋 electrons implied, so in this case, we have 4 𝜋 electrons from diene and 2 𝜋
electrons from dienophile. The suffix ‘s’ means suprafacial. Suprafacial is a topological concept
that, together with ‘a’ antarafacial, describe the relationship between two simultaneous
chemical bond making and/or bond breaking processes in a reaction center. When both
changes occur at the same face, the suprafacial interaction is produced. For the application of
orbital symmetry to pericyclic reactions, K. Fukui and R. Hoffmann won the Nobel Prize in 1981
(R.B. Woodward died in 1979).
From a Diels-Alder reaction we can obtain stereospecific products depending on the reaction
conditions and the isomeric characteristics of the reactants. If the dienophile is unsymmetrical,
there exist two possible stereochemical orientations with respect to the diene. When the
reference substituent on the dienophile is oriented toward the π system of the diene the
orientation is called endo, whereas it is named exo when the substituent is situated away from
the π system (see Figure 12).
26
GENERAL INTRODUCTION
Figure 12. Representation of the different approaches than can take place in a Diels-Alder reaction.
Because of the great versatility of the Diels-Alder reaction, there exists a big interest in
understanding the mechanism of these cycloadditions. But the real mechanism which takes
place is not a closed issue yet. Huisgen proposed a concerned mechanism that can be or not
asynchronous, whereas Firestone suggested a two-step mechanism.49-53 The majority of the
actual theoretical studies consider that a concerted [4𝜋𝑠 + 2𝜋𝑠 ] mechanism is followed
because the two-step process is less favorable as it is higher in energy than the concerted
one.54-58
In 1967, Fukui developed the Frontier Molecular Orbital (FMO) Theory.59-63 This methodology is
used to understand and predict the reactivity and regioselectivity of cycloaddition reactions,
and for Diels-Alder reactions in particular. From the FMO, one can see that the usual strongest
orbital interaction is produced between the HOMO of the diene and the LUMO of the
dienophile. But there exists a strong electronic substituent effect on the Diels-Alder addition,
in the sense that, if an electron-poor diene and an electron-rich dienophile react, the strongest
orbital interaction which take place is between the HOMO of the dienophile and the LUMO of
the diene. That is called inverse electron demand Diels-Alder reaction. Usually Diels-Alder
reactions in fullerenes follow a normal electronic demand.
In a recent publications,64,65 Ess and Houk have shown that the reactivity for cycloadditions of
1,3-dipoles with alkenes or alkyls is controlled by the energy required to distort the 1,3-dipole
and dipolarophile to the transition state geometry. This model can also be applied on DielsAlder reactions.
The distortion energy (deformation energy)66,67 is defined as the energy required to distort the
reactants into the geometry they present in the transition state, without allowing the
interaction between the fragments. Thus, the activation energy of a reaction can be expressed
27
MASTER THESIS
ⱡ
as the deformation energy (∆𝐸𝑑ⱡ ) and the interaction energy (∆𝐸𝑖𝑛𝑡
) between reactants
involved in the transition state:
ⱡ
∆𝐸 ⱡ = ∆𝐸𝑑ⱡ + ∆𝐸𝑖𝑛𝑡
(12)
A good correlation has been found between activation barriers and distortion energies. A high
reaction barrier implies a higher deformation of the reactants in the transition state, although
this is not always the case.68
In the present work we propose to study the Diels-Alder reaction between 1,3-cis-butadiene
and the Ti2C2@C78 EMF. In the next section we will review some previous and similar studies
carried out on the field of this new project.
The Diels-Alder cycloaddition on endohedral metallofullerene compounds.
In this section we will revise the first Diels-Alder reactions described on endohedral
metallofullerenes, from the experimental field to the computational studies.
The first organic derivative of a metallic nitride EMF was prepared in 2002 by Dorn and coworkers on Sc3N@Ih-C80.69 The crystal structure of the Diels-Alder product between the
mentioned EMF and the 6,7-dimethoxyisochroman-3-one which was reported later,70 showed
that the addition occurs on the corannulene-type [5,6] bonds.
In 2005, Stevenson et al.71 reported the synthesis of some bisadducts of Gd3N@C80. It was
shown that two o-quinodimethane molecules were attached to the Gd3N@C80 surface forming
the latter bisadduct. This compound is very interesting because of its potential applications as
MRI contrast agent.
The decrease of the DA reactivity between the trimetallic EMFs respect the empty cages and
classical EMFs, was used by Dorn and co-workers for the purification of metallic nitride EMFs
from the extracted fullerenes72,73 as represented in Figure 13.
Figure 13. Schematic representation of the purification of trimetallic nitride template endohedral
metallofullerenes using the Diels Alder reaction proposed by Dorn and co-workers.72,73
28
GENERAL INTRODUCTION
There exist several theoretical studies which try to explain the experimental results. In 2006,
Campanera et al.74 performed a theoretical study that correctly describes the reactive
exohedral sites of Sc3N@C80 for the Diels-Alder reaction. From these studies, they found that
the most reactive bonds were those with high Mayer Order Bond (MOB)75 and high
pyramidalization angles.76,77 In the same line, Osuna, Swart and Solà reported the study of the
Diels-Alder cycloaddition between 1,3-butadiene and free D3h’-C78 , and between butadiene
and the endohedral Sc3N@D3h’-C78 fullerene.78 As it was to expect, they found that when we
have the metallic cluster inside, the final reactivity of the cage decreases respect the free cage.
Then, the authors expanded the study to the Y3N@D3h’-C78 EMF.79 They found that the most
stable regiosiomer for the Y3N compound was the one obtained over a corannulene-type [5,6]
bond, which exhibits the longest bond distance and a large pyramidalization angle. This was
the first reported case where the most stable cycloaddition reaction takes place over one of
the longest C─C bonds in the cage. In contrast to the Sc3N compound, where bonds close to
the scandium atoms were destabilized, in this new case the most reactive bond has one
yttrium atoms in close contact. This preference for reacting with those bonds situated close to
the yttrium atom is due to two different factors. First, the D3h cage is extremely deformed
(especially in the areas close to the cluster atoms) and breaking the C─C bond the cage reduces
his strain energy. Second, in the final adduct, the Y3N cluster gets additional space to adopt a
more planar configuration, since it is pyramidalized in the Y3N@D3h’-C78.
Figure 14. Superposition of the Y3N@D3h-C78 and Sc3N@D3h-C78 structures. The main difference are localized on
the pyramidalization of the nitrogen of the cluster (h = 0.693 Å) and the pyracylene units situated close to the
metals atoms (colored in light green).79
From these previous theoretical studies, we can conclude that although the shortest bonds
and those which have the highest pyramidalization seems to be the most reactive bonds of the
EMF considered, the metal cluster encapsulated inside has an enormous influence on the
reactivity of these compounds and can dramatically change the regiochemistry of the
fullerenic cage.
29
MASTER THESIS
30
GENERAL INTRODUCTION
4. Supramolecular Chemistry
Introduction
Although supramolecular chemistry has a previous origin, it not was until the 60’s when it
focuses the interest of chemists for the synthesis of complex structures. The field of
supramolecular chemistry has been defined by one of its leading proponents, Jean-Marie Lehn,
awarded with Nobel Prize in 1987 for his work in the development of this chemistry area, as
“chemistry beyond the molecule”.80,81
Supramolecular chemistry, instead of traditional chemistry which is normally based in covalent
interactions, is based on weak interactions like van der Waals, π-π, hydrophobic interactions,
etc. This kind of interactions allows a reversible equilibrium between the reactants and
products, where the stabilization of the final structures obtained determines the success of the
reaction. The supramolecular strategy, as represented in Figure 15, makes it possible to build
up complex structures avoiding problems of low yield and inherent difficulties of traditional
chemistry, which is based in the formation of new covalent bonds.2,82,83
Figure 15. Schematic representation of traditional molecular and supramolecular synthesis based on covalent
bonds between atoms and intermolecular non-covalent interactions between molecules, respectively.83
Covalent interactions are usually between 35-105 kcal mol-1 for single bonds. Non-covalent
interactions are considerably weaker that covalent ones: from 1 kcal mol-1 for dispersion
interactions, to 70 kcal mol-1 for ion-ion interaction. But the supramolecular chemistry is an
31
MASTER THESIS
efficient strategy because of the cooperative manner, which is the fact that allows a final
supramolecular complex to exist. As said before, van der Waals forces, hydrophobic effects, ππ stacking, or hydrogen bonding, electrostatics interactions and classical coordination
interactions are considered non-covalent interactions which provide many different tools for
the final stabilization and effective formation of supramolecular structures.84
Supramolecular chemistry can be understood from two different points of view: host-guest
chemistry and self-assembly. These two categories are differentiated by size and shape terms.
If one molecule is significantly larger than another that can be wrapped by the former, then
the large molecule is termed as “host” and the smaller one as “guest”. If the size of the
different two structures is not significant, and any species is acting as host for another, then
the non-covalent interaction between these two molecular building blocks to produce an
aggregate is called self-assembly.82
One of the projects presented in this work uses both techniques: first, it synthesizes some
macromolecular structures by the self-assembly technique which then are used to encapsulate
in their void inner space a fullerene structure using host-guest interactions. As this project
forms part of a collaboration between theoretical (Institut de Química Computacional,
Universitat de Girona) and experimental group (Grup de Química Bioinorgànica i
Supramolecular, Universitat de Girona). But in the work presented in this master thesis we
focus on the theoretical study of the formation of host-guest complexes.
Self-Assembly
Self-assembly methodology provides an important tool to synthetic chemists. Based on the
spontaneous association of two or more molecules or ions, self-assembly allows to create a
larger aggregate species through the formation of reversible non-covalent interactions
previously described. Starting from relatively simple molecules with complementary
functionalities, sophisticated supramolecular complexes can be generated, as represented in
Figure 16.
Self-assembly methodology has the ability to correct mistakes during the synthesis and
gradually get closer to the most thermodynamically stable products. When multiple molecules
are able to join together, usually there exist more than one combination. And it is the
thermodynamically most stable species which will be obtained as the most predominant one.
This means, the self-assembly process is considered as thermodynamically selective.2,82
32
GENERAL INTRODUCTION
Figure 16. Schematic representation of a traditional covalent synthesis [(1)(2),(3)], followed by a spontaneous
self-assembly process to obtain the final complementary supramolecular aggregate [(3)(4)].2
Many natural systems are based on the self-assembly methodology. One of the most relevant
examples is the DNA double helix formation. As it is known, the formation of the DNA
structure is a spontaneous and reversible process, capable of self-correct any mistake during
the DNA synthesis. The process is always directed to the thermodynamically most stable
product, and then the kinetic products are not obtained.85
Host-Guest chemistry
Host-guest chemistry is a kind of supramolecular recognition in which, for instance, enzymes
are based on. The latter is a perfect example of a biological host-guest complex. In the enzimes
the host component is defined as an organic or inorganic molecule whose binding sites
converge in a complex, and the guest component is defined as any molecule whose binding
sites diverge in the complex (see Figure 17).
Figure 17. Scheme of a large molecule (2) covalent synthesis from smaller ones (1), which will act as a host for a
smaller guest (3) molecule, to obtain the final host-guest complex (4).2
As is evident, many factors must be taken into account to tune the host-guest interactions,
being the two most important ones, the size of both the host and guest, and complementary
33
MASTER THESIS
binding sites and interactions. By modifying and tuning these parameters, one can obtain
different degrees of selectivity towards different species involved. This selectivity can arise
from different factors, such as the complementarity of the both binding sites and electronic
structures. Thus, the formation of the final host-guest complex does not require a significant
conformation change of the host molecule to accommodate the guest species.81,82
The size of the host molecule can be modified by using different molecules in the first covalent
synthesis (referring to the scheme represented in Figure 17), taking into account that
electronic complementarity also must be satisfied. The electronic complementarity can be also
tuned, by changing the different substituents presents in both host or guest molecules, or by
changing the metal ions if it is the case.86
We have to bear in mind that usually the host-guest interactions take place in solution. That
mean, host and guest molecules are not isolated from other possible influences. In real
systems, guests are competing with surrounding solvent molecules, and to get the final
binding, many interactions between receptor and solvent molecules must be broken producing
both enthalpic and entropic important consequences. Then, solvent plays a key role in the
recognition process.84
Molecular recognition based on supramolecular
chemistry
Self-assembly methodology allows us to synthesize large and complex structures inspired by
nature, which can be used for interact specifically with a family of molecules or a unique one
by host-guest interactions. The key role of this final recognition process, as we have seen in the
previous sections, is the complementarity between the host and guest species.
The final objective of the host-guest recognition can be only the interest in a particular
molecule. Nevertheless, the host-guest recognition can also be used, for instance to perform a
reaction over only one specific molecule.
In the present study, we will focus on three dimensional metallo-supramolecules, which posses
empty inner cavities and have great potential to perform host-guest chemistry. Our major
interest is the study of a molecular recognition based on host-guest chemistry using metallosupramolecules as nanovessels (hosts) which are able to selective recognize compatible
substrates.
In the next section, we will describe some examples of molecular recognition based on hostguest chemistry where the guest molecules are fullerene compounds. We will see the first
hosts proposed early after fullerene discovery, and we will review how the type and shape of
hosts have evolved until nowadays.
34
GENERAL INTRODUCTION
Purification and isolation of fullerenes based on host-guest interactions
The raw product obtained by the evaporation of graphite is soot and slag. Next to soluble
fullerenes the soot and slag contain other kinds of closer carbon structures, as giant fullerenes
or nanotubes, and amorphous carbon. Fullerenes can be isolated from the soot either by
sublimation or by extraction.46 But the main problem appears when one wants to separate
different fullerenes.
To separate fullerenes predominantly chromatographic methods are used. However, as
fullerenes are not very soluble in some solvents used, enormous amounts of solid phase and
solvent are needed, rendering this method inefficient. That is the reason why different
alternative methods are proposed. But until now no one has found the method that combines
the effectiveness and efficiency desired.
Some methodologies are based on specific functionalization of one fullerene type, in order to
get different polar species, or in the same way, to reduce or oxidate one unique fullerene type
of the mixture. But this is a hard way because of the similar reactivity that different fullerene
species present.
Other proposals are based on the different size and shape that fullerene have. And more
specifically, they are based on molecular recognition. In this sense, there exist several studies
which present host-guest specific interactions between supramolecular structures developed
and different fullerenes. One can tune the size of the host system to get a more specific and
selective interaction with the guest molecule. The complementarity of the structure and
electronic configuration of the macromolecule and the fullerene are the two most important
features to separate, efficiently, one fullerene type from a mixture.
A large collection of receptors which can selectively trap fullerenes has been reported in the
literature. Initially, the first specific molecular receptors for fullerenes were functionalized
macrocycles, as for example, the azacrown-ether receptors (Figure 18) described by Ringsdorf,
Diederich and co-workers.87,88
Figure 18. Schematic representation of the structures of the receptors for fullerenes reported by Diederich,
Ringsdorf and co-workers. 87
35
MASTER THESIS
But later, there has been reported several and different structures based on molecules with
large π systems that offer large surfaces to interact with fullerene structures which maximize
the van der Waals interactions and π-π stacking.89 For example, one type of these receptors
are based on corannulene structures90 (see Figure 19), but maybe the most important are
those based on metalloporphyrines.89
Figure 19. Structure of the crystal structure of the host-guest complex obtained with a corannulene-based
receptor.90
Based on this π-π stacking complementarity described, Nazario Martín and co-workers have
reported a specific receptor for C60 fullerene which is based on a p-extended analogue of
tetrahiafulvalene (TTF), 2-[9-(1,3-dithiol-2-ylidene)anthracen-10(9H)-ylidene]-1,3-dithiole
(exTTF).91 This receptor is composed of two units of exTTF connected through an isophtalic
diester spacer. The large and concave aromatic surface of the exTTF units serves as a
recognizing motif for the convex surface of C60. By using these TTF analogues, they have also
reported a supramolecular-based linear polymer which includes C60 molecules and has
interesting optoelectric properties.92
Aida reported for the first time the formation of very stable inclusion complexes between
fullerenes and a dimeric construct in which two metalloporphirins are linked by flexible alkyl
spacers (see Figure 20).93 Posterior structural variations on this design have lead to what is
surely the richest collection of receptors for fullerenes.89,94
The nature of the metal ion affects the binding energy between the host and the guest. In the
case of C60, it has been reported that for this dimer metalloporphyrinic structures of Aida’s lab
the most stable structure, i.e. that having the biggest binding constant, structure is the one
which has Rhodium(III) cations, and where a methyl group completes the coordination sphere
of the trivalent ion.95 This is an important feature to take into account when new
metalloporphyrinic-based host to trap fullerenes are proposed, together with the molecular
spacers used to link the two metalloporphyrins.
36
GENERAL INTRODUCTION
Figure 20. Schematic representation of the first stable host-guest structure with metalloporphyrinic host reported
by Aida and co-workers.93
One of the latest metalloporphyrin-based hosts to encapsulate fullerenes reported by Meng,
Nitschke and co-workers, are a series of porphyrin-faced M8L6 hollow cubic architectures (see
Figure 21,(a)), where metal ion changes.96 The inner void space of these cubes and their
electronic π structure (like previous described cases) make them a good option to selective
encapsulate C60 and C70 as guests (see Figure 21,(b)).
Figure 21. Schematic representation of the M8L6 structure reported by Meng, Nitschke and co-workers and the
host-guest interaction with C60 and C70 fullerenes. 96
37
MASTER THESIS
38
COMPUTATIONAL METHODOLOGIES
CHAPTER II.
COMPUTATIONAL METHODOLOGIES
In the last years, the fast evolution of computers has increased enormously the computational
power available to scientists. This, together with the new developments and advances in
quantum chemistry theory, have allowed chemists to obtain very accurate and useful
information about geometrical structures or physical and chemical molecular properties.
There exist different methodologies in quantum and computational chemistry. First, we found
the ab initio methods, which are also called the wavefunction based methods. They are based
on solving the Schrödinger equation. See the diagram where different ab initio based on HF
methods are ordered in terms of energy (see Figure 22). Second, there are the semi-empirical
methods which are based on the Hartree-Fock (HF) formalism. Some approximations are
introduced here, and also, some empirical parameters. And finally, we have the molecular
mechanics methods, based on the classical physics of forces and electrostatic interactions.
Figure 22. Schematic representation of ab initio methods based on Hartree-Fock approximation ordered by
chemical accuracy.
Below, the ab initio Hartree-Fock method will be explained, which is the starting point for the
called post Hartree-Fock methods. But we can not use these methods because the
computation effort required to treat large systems we have proposed to study, the fullerenes,
involves a computational cost that is not affordable with our resources. So Density Functional
Theory (DFT) will be also described, as it allows us to treat our large and complex chemical
system with a lower computational cost but with a similar accuracy.
39
MASTER THESIS
40
COMPUTATIONAL METHODOLOGIES
1. The Hartree-Fock approximation
In this section the Hartree-Fock approximation as well as some fundamental concepts such as
exchange, self-interaction and electron correlation will be explained. We will base our
description on the book Modern Quantum Chemistry by Szabo and Ostlund.97
Solving the Schrödinger equation is the main objective of quantum mechanics, and HF
formalism gives a tool to find approximate solutions to the electronic structure of molecules.
But it gives also an important conceptual understanding of the physics behind this
approximation, which will be of great help when discussing the Density Functional Theory.
Hartree-Fock is an extension of the molecular orbital theory in which correlated electronelectron repulsion is not specifically taken into account, only its average effect. But it also has
an important role as a starting point for more accurate approximations that include electron
correlation. It assumes that the motion of each electron can be described by a simple function
called orbital which is not explicitly dependent of the motion of other electrons.
There are three major approximations in the HF method:
i.
The Born-Oppenheimer approximation, which assumes that the electronic motion
can be decoupled from nuclear motion. The movement of the electrons is far
faster than the nucleus movement. Then, although the electron cloud depends on
the nucleus positions, it can be assumed that it does not depend on the nucleus
movement. So, the total wavefunction can be written as a product of its electronic
and nuclear (vibrational, rotational and translational) components.
Ψ = ψnuc × ψel
ii.
(13)
Each molecular spinorbital (𝜒𝑎 (𝑥⃗1 )) is assumed to be a linear combination of a
finite number of basis functions (𝜙𝜈 (𝑥⃗1 )). The one electron functions 𝜒𝑎 (𝑥⃗1 ) are
chosen to be orthonormal. This approximation was proposed by Roothaan and
Hall, who introduced the atomic orbitals as basis functions. That is, the Linear
Combination of Atomic Orbitals (LCAO).
𝑘
𝜒𝑎 (𝑥⃗1 ) = ∑ 𝐶𝜈𝑎 𝜙𝜈 (𝑥⃗1 )
(14)
𝜈=1
⟨𝜒𝑎 (𝑥⃗1 )|𝜒𝑏 (𝑥⃗1 )⟩ = 𝛿𝑎𝑏
(15)
In (14) k is the number of atomic orbitals. If a complete set of 𝜙𝜈 is used, equation
(14) is an exact expansion.
41
MASTER THESIS
iii.
Each eigenstate is described by a single Slater determinant formed by a set of
spinorbitals. Then, the HF wavefunction is given by an antisymmetrized product of
one-electron wavefunctions, which fulfill the Pauli principle:
Ψ = ΦSD
𝜒1 (𝑥⃗1 ) 𝜒2 (𝑥⃗1 ) ⋯ 𝜒𝑁 (𝑥⃗1 )
𝜒 (𝑥⃗ ) 𝜒2 (𝑥⃗2 ) ⋯ 𝜒𝑁 (𝑥⃗2 )
=
| 1 2
|
⋮
⋮
⋱
⋮
√N!
𝜒1 (𝑥⃗𝑁 ) 𝜒2 (𝑥⃗𝑁 ) ⋯ 𝜒𝑁 (𝑥⃗𝑁 )
ΦSD =
1
1
√N!
det{𝜒1 (𝑥⃗1 )𝜒2 (𝑥⃗2 ) … 𝜒𝑁 (𝑥⃗𝑁 )}
(16)
(17)
where N is the number of electrons.
The variational principle specifies that the best set of spinorbitals are those which minimize
the electronic HF energy 𝐸0 :
1
̂ |Ψ0 ⟩ = ∑⟨𝑎|ℎ|𝑎⟩ + ∑⟨𝑎𝑏‖𝑎𝑏⟩
𝐸0 = ⟨Ψ0 |𝐻
2
𝑎
(18)
𝑎𝑏
The spinorbitals 𝜒𝑎 are systematically varied until the energy reaches a minimum value. The
equation for the best spinorbitals (which minimizes the energy) is the Hartree-Fock equation,
𝑓𝑖 𝜒𝑖 = 𝜀𝑖 𝜒𝑖
(19)
̂𝑏 (𝑘))
𝑓̂𝑎 (𝑘) = ℎ̂(𝑘) + ∑ (𝐽̂𝑏 (𝑘) − 𝐾
(20)
where 𝑓𝑖 is the Fock operator
𝑏
The first component of the Fock operator ℎ̂(𝑘) includes the kinetic energy and the attractive
̂ the Exchange term. The Fock
electron-nucleus potential, 𝐽̂ is the Coulomb operator and 𝐾
operator is a monoelectronic operator which by definition is build from the spinorbitals that
we are optimizing.
The Coulomb operator in an exact theory includes the two-electron potential operator 𝑟𝑖𝑗−1 .
But in the HF approximation the latter operator is replaced by a one-electron potential,
−1
obtained by averaging the interaction 𝑟12
of electron 1 and electron 2 over all space and spin
coordinate 𝑥⃗2 of electron 2, weighted by the probability that electron 2 occupies the volume
element 𝑑𝑥2 at 𝑥2 (𝑑𝑥2 |𝜒𝑏 (2)|2 ). By summing over all 𝑏 ≠ 𝑎, we obtain the total averaged
42
COMPUTATIONAL METHODOLOGIES
potential acting on the electron in 𝜒𝑎 , arising from the 𝑁 − 1 electrons in the other
spinorbitals. The Coulomb operator associated with this approximation is represented by
equation (21).
𝐽̂𝑏 (1) = ∫ 𝑑𝑥2 𝜒𝑏∗ (2)
1
𝜒 (2)
𝑟12 𝑏
(21)
In the other side, the exchange term does not have a classical physical interpretation, and is
due to the antisymmetric nature of the Slater determinant (see equation (22)).
̂𝑏 (1)𝜒𝑎 (1) = [∫ 𝑑𝑥2
𝐾
𝜒𝑏∗ (2)𝜒𝑎 (2)
] 𝜒𝑏 (1)
𝑟12
(22)
Now, taking into account the Roothaan and Hall introduction of LCAO approximation (equation
(14)), the calculation of the HF orbitals is reduced to the calculation of the expansion
coefficients 𝐶𝜈𝑖 . So, the following expression is described:
𝑘
𝑘
𝑓̂(1) ∑ 𝐶𝜈𝑖 𝜙𝜈 (1) = 𝜀𝑖 ∑ 𝐶𝜈𝑖 𝜙𝜈 (1)
𝜈
(23)
𝜈
If we multiply the equation (23) by 𝜙𝜇∗ (1) and integrate, a matrix equation can be obtained,
𝑘
𝑘
∑ 𝐶𝜈𝑖 ∫ 𝑑𝑟𝜙𝜇∗ (1)𝑓̂(1)𝜙𝜈 (1)
𝜈
= 𝜀𝑖 ∑ 𝐶𝜈𝑖 ∫ 𝑑𝑟𝜙𝜇∗ (1)𝜙𝜈 (1)
(24)
𝜈
where the overlap matrix S is defined as
𝑆𝜇𝜈 = ∫ 𝑑𝑟𝜙𝜇∗ (1)𝜙𝜈 (1)
(25)
This overlap matrix is Hermitian. Furthemore, because of the normalization of the atomic basis
functions 𝜙𝜈 , the value of its elements are between zero and one, 0 ≤ |𝑆𝜇𝜈 | ≤ 1. Then, the
Fock Matrix F can be written as:
43
MASTER THESIS
𝐹𝜇𝜈 = ∫ 𝑑𝑟𝜙𝜇∗ (1)𝑓̂(1)𝜙𝜈 (1)
(26)
which is also Hermitian, and represents the Fock operator applied to 𝜙𝜈 (1) in a matrix form.
Taking this into account, the equation (23) can be written as:
𝑘
𝑘
∑ 𝐹𝜇𝜈 𝐶𝜈𝑖 = 𝜀𝑖 ∑ 𝑆𝜇𝜈 𝐶𝜈𝑖
𝜈
𝑖 = 1, 2 … , 𝑘
(27)
𝜈
The previous equation is called the Roothaan-Hall equation. An equivalent matrix equation is:
𝑭𝑪 = 𝑺𝑪𝜀
(28)
where the matrix C and the diagonal matrix 𝜀 contain the coefficients 𝐶𝜈𝑖 and the orbital
energies 𝜀𝑖 , respectively. So an iterative methodology is needed to solve the Roothaan
equations, as they are not linear.
The Hartree-Fock method is also called the Self-Consistent method (SCF) because this iterative
procedure is used to solve the Roothaan equations (eq. (27)). An initial set of spinorbitals is
needed to start the process of solving the Fock equations, which step by step generate a new
set of orbitals until the optimum coefficients that lead to the minimum of the energy are
found.
The SCF method does not give an accurate description of the most of the chemical systems.
This is because the average potential used does not describe the correlation of the motion of
the electrons. An important consequence of the approximate treatment of the electronelectron repulsion is that the true wavefunction of a many electron system is never a single
Slater determinant.
This problem is usually solved by introducing the post Hartree-Fock methods. The correlation
energy is described as the difference between the exact non-relativistic Born-Oppeheimer
energy for a determined basis set (i.e. Full-CI) and the HF limit energy (equation (29)).
𝐸𝑐𝑜𝑟𝑟 = 𝐸𝑒𝑥𝑎𝑐𝑡 − 𝐸𝐻𝐹
44
(29)
COMPUTATIONAL METHODOLOGIES
2. The Density Functional Theory
In this section, we will describe the Density Functional Theory (DFT) based on the books
Theoretical and Computational Chemistry98 and Introduction to Computational Chemistry.99
The DFT method is a quantum mechanical theory that gives accurate results. It is the best
choice for the study of large molecules with a moderate computational cost. DFT gives a good
prediction for the molecular properties, as long as we choose the appropriate functional for
our purposes.
Traditional methods in electronic structure theory, in particular the Hartree-Fock theory and
its descendants, are based on many-electron wavefunctions, as we have discussed before. The
main objective of DFT is to replace these wavefunctions with the electronic density of the
systems as the basic expression to get information about these systems. Whereas the
wavefunction is dependent on 3N variables, where N is the number of electrons, the density is
only a function of three variables, dramatically reducing the complexity of the problem.
The DFT gives another approach for solving the Schrödinger equation, and it is based on the
Hohenberg and Kohn (HK) theorems. The first HK theorem states that the electronic density
can fully determine the energy of a non-degenerate electronic ground state.100 In fact, there
exists a one-to-one relationship between the electronic density and the Hamiltonian. This
means not only the energy but also other observable properties of the system can be
determined by the electronic density.
Further, the second HK theorem establish the DFT variational principle and it proves that the
energy of the system 𝐸[𝜌] is a minimum (𝐸0 ) when the exact electronic density of the system
𝜌0 is considered.
𝐸0 [𝜌0 ] ≤ 𝐸[𝜌]
(30)
𝛿𝐸[𝜌]
−𝜇 =0
𝛿𝜌(𝑟⃗)
(31)
Thus the variational equation is obtained:
where 𝜇 is the electronic chemical potential. In fact, μ is the Lagrange multiplier that ensures
the normalization of the electronic density.
∫ 𝑑𝑟 𝜌(𝑟⃗) = 𝑁
45
(32)
MASTER THESIS
The Kohn-Sham formulation
Kohn and Sham introduced the orbital concept within the DFT framework101 and proposed a
self-consistent method similar to the SCF for the HF theory. That is, a set of orthogonal
spinorbitals which minimizes the energy obtained. The KS orbitals may be expanded in a set of
basis functions, analogous to the HF method.
𝑀
𝜒𝑖 = ∑ 𝑐𝜇𝑖 𝜙𝜇
𝑖 = 1, 2 … , 𝑀
(33)
𝜇
To compute the kinetic energy, they proposed to divide the kinetic energy in an exact part and
a correction term. A system of non-interacting electrons is considered as it can be exactly
solved, and the remainder is merged with the non-classical contributions to the electronelectron repulsion, which are unknown. The Hamiltonian is assumed to have the following
expression,
𝐻𝜆 = 𝑇 + 𝑉𝑒𝑥𝑡 (𝜆) + 𝜆𝑉𝑒𝑒
0≤𝜆≤1
(34)
where 𝑇 is the kinetic energy, 𝑉𝑒𝑒 is the Coulomb repulsion, 𝜆 is a coupling parameter that
varies from 0 (non-interacting system) to 1 (interacting system), and 𝑉𝑒𝑥𝑡 (𝜆) is the external
potential. The last described term is equal to the electron-nuclear attraction 𝑉𝑛𝑒 when 𝜆 = 1,
that is an electron interacting system.
|𝜒 𝜆 ⟩ is the interacting ground state wavefunction that leads to the electron density 𝜌 of the
real system. But when 𝜆 = 0, |𝜒 𝜆 ⟩ is the single determinant wavefunction built with the KohnSham orbitals 𝜒𝑖 , and 𝑉𝑒𝑥𝑡 (𝜆) is the so-called Kohn-Sham effective potential (𝜐(𝑟)). The kinetic
energy functional of the reference system of non-interacting N particles follows the equation
(35).
𝑁
1
𝑇𝑠 = ∑ ⟨𝜒𝑖 |− 2 ∇2i |𝜒𝑖 ⟩
(35)
𝑖=1
The functional of the energy has to include the kinetic energy (𝑇[𝜌]), the nucleus-electron
attraction potential (𝐸𝑛𝑒 [𝜌]) and finally the electron-electron repulsion potential (𝐸𝑒𝑒 [𝜌]).
Because of the Born-Oppenheimer approximation, the repulsion potential between nuclei is
not taken into account as is independent of the charge density. As it happened in the HF
46
COMPUTATIONAL METHODOLOGIES
approximation, the electronic correlation can be divided into a Coulomb and an Exchange part,
as equation (36) shows.
𝐸[𝜌] = 𝑇[𝜌] + 𝐸𝑛𝑒 [𝜌] + 𝐽[𝜌] + 𝐾[𝜌]
(36)
This equation does not provide the total kinetic energy. However, the difference between the
exact kinetic energy and that calculated by assuming non-interacting particles is small. The
remaining kinetic energy is included into an exchange-correlation term, and a general DFT
energy expression can be written as:
1
𝜌(𝑟⃗)𝜌(𝑟⃗′)
𝐸[𝜌] = 𝑇𝑠 [𝜌] + ∫ 𝜌(𝑟⃗)𝜐(𝑟⃗)𝑑𝑥 + ∫ 𝑑(𝑟⃗)𝑑(𝑟⃗′)
+ 𝐸𝑥𝑐 [𝜌]
|𝑟⃗ − 𝑟⃗′|
2
(37)
The first term 𝑇𝑠 [𝜌] is the kinetic energy functional for the non-interacting electron system,
and the second term gives the electron-nucleus interaction. The classical 𝐽[𝜌] Coulomb
repulsion of the electron cloud, plus its self-interaction energy error is represented by the third
term. And finally, the last term which is the exchange-interaction energy functional 𝐸𝑥𝑐 [𝜌] and
includes non-classical effects of the electron correlation, as well as the difference 𝑇[𝜌] −
𝑇𝑠 [𝜌].
If equation (37) is rearranged and we now apply the variational principle, the effective
potential Kohn-Sham 𝜐𝑒𝑓𝑓 is obtained.
𝛿𝑇𝑠 [𝜌]
=𝜇
𝛿𝜌(𝑟⃗)
𝛿𝐽[𝜌] 𝛿𝐸𝑥𝑐 [𝜌]
𝜐𝑒𝑓𝑓 (𝑟⃗) = 𝜐(𝑟) +
+
𝛿𝜌(𝑟⃗)
𝛿𝜌(𝑟⃗)
𝜐𝑒𝑓𝑓 (𝑟⃗) +
(38)
where the effective potential Kohn-Sham 𝜐𝑒𝑓𝑓 and the chemical potential 𝜇 have been
introduced. Then, the effective KS potential together with the kinetic energy operator form the
Hamiltonian for the non-interacting system:
𝑁
1
̂𝑠 = ∑ [− ∇2i + 𝜐𝑒𝑓𝑓 (𝑟⃗)]
𝐻
2
𝑖=1
1
[− ∇2i + 𝜐𝑒𝑓𝑓 (𝑟⃗)] 𝜒𝑖 = 𝜖𝑖 𝜒𝑖
2
47
(39)
MASTER THESIS
Thus, once we know the various contributions in this equation, we have a grid on the potential
𝜐𝑒𝑓𝑓 which we need to insert into the one-particle equations which in turn determinate the
orbitals and hence, the ground state density and the ground state energy by employing the
̂𝑠 forms a set of orbitals 𝜒𝑖 whose associated
energy expression (𝐸[𝜌]). The solution of 𝐻
electron density is equal to the exact one.
It is clear that there are many similarities between the Kohn-Sham methodology to the SCF for
the Hartree-Fock theory. In both cases, one has to make a guess for the orbitals 𝜒𝑖 , to build the
electron effective potential, and to solve the iterative equations (19) or (39) until selfconsistency to finally obtain the HF or DFT energy from equations (18) or (37), respectively.
The use of electronic density has a very important advantage in relation to the wavefunction
because whereas the former only depends on three coordinates, the latter depends on the 3N
coordinates, where N is the number of electrons. If the form of the exchange-correlation
functional 𝐸𝑥𝑐 [𝜌] was known, DFT methodology would provide us the exact total energy. But,
of course, this is not known yet, although great efforts have been and are being done to find
more accurate expressions. Since exact solutions are generally not available in either
approach, the important fact is the computational cost for generating a solution of, in general,
a good accuracy. In this respect, DFT methods have very favorable characteristics.
The expression for the exchange-correlation functional
In the early of DFT theory, before the HK theorems and the KS formulation were introduced,
the first attempt was to consider a non-interacting uniform electron gas. The classical
formulation of 𝐸𝑛𝑒 [𝜌] and 𝐽[𝜌] were used, and the following kinetic 𝑇[𝜌] and exchange 𝐾[𝜌]
expression were proposed.
5
𝑇𝑇𝐹 [𝜌] = 𝐶𝐹 ∫ 𝑑(𝑟⃗)𝜌3 (𝑟⃗)
4
𝐾𝐷 [𝜌] = −𝐶𝑥 ∫ 𝑑(𝑟⃗)𝜌3 (𝑟⃗)
2
3
(3𝜋 2 )3
10
1
3 3 3
𝐶𝑥 = ( )
4 𝜋
𝐶𝐹 =
(40)
Taking into account the above mentioned, the Thomas-Fermi-Dirac (TFD) model is obtained.102
But this model does not give a correct description of the chemical bond. Then, the functionals
𝑇[𝜌] and 𝐾[𝜌] had to be improved. One solution is to include in the expressions of the
functionals not only the electronic density, but also its derivatives.
In the Kohn-Sham formulation DFT, the exchange-correlation energy is given by:
48
COMPUTATIONAL METHODOLOGIES
2
⃗⃗𝜌(𝑟⃗)| )
𝐸𝑥𝑐 [𝜌] = ∫ 𝑑(𝑟⃗) 𝜌(𝑟⃗)𝜖𝑥𝑐 [𝜌(𝑟⃗)] + Ο (|∇
(41)
where 𝜖𝑥𝑐 [𝜌(𝑟)] is the exchange-correlation energy functional of a non-interacting electron
system, but applied on the electronic density of the system 𝜌(𝑟) instead of uniform electronic
density.
In the Local Density Approximation (LDA) only the first term of the equation (41) is considered.
LDA assumes that the density locally can be treated as a uniform electron gas, that is, the
density is a slowly varying function. The first LDA method, the 𝑋𝛼 method proposed by
Slater,103 divided the exchange-correlation term into two parts. The first is exchange part given
as:
1
1
9
3 3
𝜖𝑥𝐿𝐷𝐴 [𝜌(𝑟⃗)] = − 𝛼 ( ) 𝜌(𝑟⃗)3
4 4𝜋
(42)
The second is the correlation part, which is neglected. On the other hand, for spin polarized
systems the alpha and beta electrons are considered independently, and the local spin density
approximation (LSD) is obtained by:
1
1
1
9
3 3
𝜖𝑥𝐿𝑆𝐷 [𝜌(𝑟⃗)] = − 𝛼 ( ) [𝜌𝛼 (𝑟⃗)3 + 𝜌𝛽 (𝑟⃗)3 ]
4 4𝜋
(43)
Later, Vosko, Wilk and Nusair (VWN) proposed a expression104 for the correlation part, and the
final equation of the exchange-correlation energy functional is represented by:
𝐿𝑆𝐷 [𝜌(𝑟
𝐸𝑥𝑐
⃗)] = ∫ 𝑑(𝑟⃗)𝜌(𝑟⃗)[𝜖𝑥𝐿𝑆𝐷 [𝜌(𝑟⃗)] + 𝜖𝑐𝐿𝑆𝐷 [𝜌(𝑟⃗)]]
(44)
Although the LDA approach works moderately well for all kinds of systems, it is not accurate
enough for some type of calculations. Some improvements can be introduced considering a
non-uniform electron gas. If one considers the LDA method as the zero-order term in Taylor
expansion of the electron density, higher-order terms might be included. So the gradient
expansion approximation (GEA) is introduced.
49
MASTER THESIS
𝐸𝑥𝐺𝐸𝐴 [𝜌(𝑟⃗)] = 𝐸𝑥𝐿𝑆𝐷 [𝜌(𝑟⃗)] + 𝐶𝑥 ∑ ∫
⃗⃗𝜌𝜎 (𝑟⃗)|
|∇
𝜎
3
𝜌𝜎4 (𝑟⃗)
𝑑(𝑟⃗)
(45)
This low order gradient expansion was proposed by Becke,105 where 𝐶𝑥 is a constant. But
Becke corrections do not improves the LSD results. That is the reason why Perdew and Wang106
considered not only the local density, but also their local gradients.
𝐺𝐺𝐴 [𝜌]
⃗⃗𝜌𝜎 ) ;
𝐸𝑥𝑐
= ∫ 𝑑(𝑟⃗) 𝑓 𝐺𝐺𝐴 (𝜌𝜎 , ∇
𝜎 = 𝛼, 𝛽
(46)
where the function 𝑓 𝐺𝐺𝐴 has to be defined. There have been two different ways to design
suitable approximations to this function, as the exact form is not known. First, a widely used
semi-empirical exchange functional (B or B88) was proposed by Becke.107 In B88 exchange
functional a parameter is included, which is chosen on the basis of a least-squares fit to the
exact HF exchange energy of the noble gases. This exchange part is usually used with the
correlation functional proposed by Lee, Yang, Parr108 (to get the BLYP functional), and with the
gradient correction proposed by Perdew109 in 1986, the P86 to achieve the BP86 functional.
On the other hand, Perdew and Wang suggested a non-empirical approach for define the
exchange density functional, giving rise to the PW86106 and the extended PW91.110,111 This nonempirical approach was later on simplified in the PBE functional, which contains only physical
constants as parameters.
More accurate than GGA functionals are the meta-GGA and hybrid functional. The former
functionals include a further term in the expansion, depending on the density, the gradient of
the density and the second derivative (Laplacian) of the density (equation (47)).
𝑚𝐺𝐺𝐴 [𝜌]
𝐸𝑥𝑐
= ∫ 𝑑(𝑟⃗) 𝑓 𝐺𝐺𝐴 (𝜌𝜎 , ⃗∇⃗𝜌𝜎 , ⃗∇⃗2 𝜌𝜎 , 𝜏𝜎 ) ;
𝜎 = 𝛼, 𝛽
(47)
where 𝜏𝜎 is the KS orbital kinetic energy density for an electron of spin 𝜎.
𝑀
2
⃗⃗ψi (𝑟⃗)| ;
𝜏𝜎 (𝑟⃗) = ∑|∇
𝜎 = 𝛼, 𝛽
(48)
𝑖=1
DFT hybrid functionals are usually a linear combination of the HF exchange (𝐸𝑥𝐻𝐹 ) and some
other exchange and correlation functionals. The parameters relating amount of each
functional can be arbitrarily assigned, and are usually fitted to correctly reproduce some set of
50
COMPUTATIONAL METHODOLOGIES
observables (band gaps, structural parameters, etc.). The most extended and used DFT
functional is B3LYP. It is an adaptation of the hybrid approach of Becke,
𝐵3𝐿𝑌𝑃
𝐸𝑥𝑐
= (1 − 𝑎)𝐸𝑥𝐿𝑆𝐷 + 𝑎𝐸𝑥𝐻𝐹 + 𝑏∆𝐸𝑥𝐵88 + 𝑐𝐸𝑐𝐿𝑌𝑃 + (1 − 𝑐)𝐸𝑐𝐿𝑆𝐷
(49)
where = 0.20; 𝑏 = 0.72; 𝑎𝑛𝑑 𝑐 = 0.81 , which are the three empirical parameters.
Figure 23. The Jacob’s ladder of density functional approximations to the exchange-correlation energy adds local
ingredients successively, leading up in five steps from the Hartree world (𝑬𝒙𝒄 = 𝟎) of weak or no chemical
bonding to the heaven of chemical accuracy (with errors in energy differences of order 1 kcal/mol).112,113
Figure 23 represents Jacob’s ladder, as proposed by J. Perdew112,113 with the five rungs
representing the hierarchy of density approximation: the local spin approximation or LDA (first
rung), the generalized gradient approximation (GGA) (second rung), the meta generalized
gradient approximation mGGA (third rung), the hybrid generalized gradient approximation
hGGA (fourth rung) and the hybrid meta generalized approximation or mhGGA (fifth rung) and
finally the fully non-local description (last rung).
DFT methodology represented an important improvement in the computational chemistry
field because of the surprisingly accurate results that can be obtained with a moderate
computational cost. As we have told, in our case to treat the large fullerene compounds DFT
will be used.
51
MASTER THESIS
Basis Functions
As we have just seen in the latest sections, the computed orbitals are usually expanded in a set
of known functions. The one electron atomic orbital expression is given by:
∞
𝜙𝑖 = ∑ 𝐶𝑖𝑗 𝜑𝑗
(50)
𝑗=1
The expansion of the atomic orbitals can be done in terms of Slater type orbitals STOs or in
Gaussian type orbitals GTOs. STOs expression is:
𝑆𝑇𝑂
𝜑𝑛𝑙𝑚
= 𝑁𝜉 𝑟 𝑛−1 𝑒 −𝜉𝑟 𝑌𝑙𝑚𝑙
𝑙
(51)
where 𝑁𝜉 is the normalization constant, 𝑌𝑙𝑚𝑙 is the spherical harmonics and 𝜉 is a variational
parameter which is related to the radial function and indicates the orbital compression. They
reproduce accurately electronic behavior round the nucleus. But the Slater functions present
the disadvantage that the two-electron integrals cannot be solved analytically, so a numerical
procedure is required. For the GTOs:
2
𝐺𝑇𝑂
𝜑𝑛𝑙𝑚
= 𝑁𝜉 𝑟 𝑛−1 𝑒 −𝜉𝑟 𝑌𝑙𝑚𝑙
𝑙
(52)
The only difference of the GTOs with respect to STOs is a quadratic dependence on r in the
exponential part. That means the analytical form of the two-electron integrals can be
analytically solved. GTOs doesn’t have a cusp at the nucleus and decay to zero too rapidly.
Thus, a linear combination of GTOs are needed to represent accurately the electronic behavior
close to the nucleus and in the tail.
Usually, the charge distribution is not uniform as more electronegative atoms might be more
negatively charged, and the respective occurs with the electropositive case and positive
charge. This fact has an effect on the shape of the atomic orbitals which can be described
more correctly by introducing polarization functions. That is adding d-type functions to the
first row atoms Li-F and p-type functions to H and He. The introduction of polarization
functions leads to hybridized orbitals.
Finally, diffuse functions can also be added to increase the flexibility of the basis set which is
specially needed in the case of anionic species. That is, to include an extra set of s- and p- or dtype functions with low 𝜉 value for each atom.
52
COMPUTATIONAL METHODOLOGIES
Core electrons treatment
For systems involving elements from the third row or higher in the periodic table, there is a
large number of core electrons which in general are chemically unimportant. But it’s necessary
to use a large number of basis functions to expand the corresponding orbitals, otherwise the
valence orbitals will not be properly described due to a poor electron-electron repulsion
description. There exists one strategy included in the computational package ADF called frozen
core approximation to describe the core electrons at low computational cost.
In the frozen core approximation, deep-core atomic orbitals are kept frozen in the molecular
calculation as they practically do not change upon molecule formation. They are obtained from
very accurate single-atom calculations using large STOs basis sets.114 The deep-core orbitals are
explicitly orthogonalized against the valence orbitals and then, the frozen core density can be
included in the calculation.
Relativistic effects: ZORA approximation
The zero-order regular approximation (ZORA) implemented ion ADF is a quite good option to
take into account the relativistic effects which are important in the more internal orbitals of
heavy metals. The formulation and deduction of ZORA methodology is described in the book
Introduction to Computational Chemistry by Jensen.99 The ZORA method99,114 is obtained by
rewriting the energy expression and expanding in the parameter 𝐸 ⁄(2𝑚𝑐 2 − 𝑉), which
remains small even close to the nucleus. Retaining only the zero order term we get:
𝐸=
𝑝2 𝑐 2
−𝑉
2𝑚𝑐 2 − 𝑉
(53)
where c is the light velocity, p is the linear momentum and m is the mass of the electron. V is
the electric potential from the nuclei.
And finally, the ZORA Hamiltonian is represented by
𝑐2
−𝑉
2𝑚𝑐 2 − 𝑉
𝑐2
𝑐2
= ∑ 𝑝𝑖
𝑝
+
𝜎(∇𝑉 × 𝒑) + 𝑉
𝑖
(2𝑚𝑐 2 − 𝑉)2
2𝑚𝑐 2 − 𝑉
̂𝑍𝑂𝑅𝐴 = 𝜎 𝑝2
𝐻
𝑖
53
(54)
MASTER THESIS
where 𝜎 is a parameter defined by the Dirac’s relativistic equation. In the zero order
approximation, the spin coupling is already included.
Classical corrections: Dispersion Energy
London dispersion forces are intermolecular weak forces due to the interaction between
temporary induced multipoles in molecules which don’t have permanent dipole moment.
These interactions can take place too between large distance separate zones of the same
molecule.
Dispersion energy is usually not good described by DFT functionals as they underestimate its
value. Then, the DFT interaction energies between two molecules are quite smaller than the
really ones. For example, in our studied systems, a reactants adduct between cis-butadiene
and fullerene molecule is formed. But it can only be well described if we take into account
London interactions. This fact is quite important because has a direct effect over the energy
barriers and reaction energies computed for our studied reactions.
The method of dispersion correction as an add-on to standard Kohn-Sham density functional
theory (DFT-D) proposed by Grimme115-117 (D2 and D3 methods) describes the total energy as,
𝐸𝐷𝐹𝑇−𝐷 = 𝐸𝐾𝑆−𝐷𝐹𝑇 + 𝐸𝑑𝑖𝑠𝑝
(55)
where 𝐸𝐾𝑆−𝐷𝐹𝑇 is the usually self consistent KS energy as obtained from the chosen DF, and
𝐸𝑑𝑖𝑠𝑝 is the dispersion correction as a sum of two- and three-body energies:
𝐸𝑑𝑖𝑠𝑝 = 𝐸 (2) + 𝐸 (3)
(56)
with the most important two-body term given by:
𝐸 (2) = ∑
∑
𝐴𝐵 𝑛=6,8,10,…
𝑠𝑛
𝐶𝑛𝐴𝐵
𝑛 𝑓𝑑,𝑛 (𝑟𝐴𝐵 )
𝑟𝐴𝐵
(57)
Here, the first sum is over all atom pairs in the system, 𝐶𝑛𝐴𝐵 denotes the averaged (isotropic)
𝑛
nth-order dispersion coefficient (orders n=6, 8, 10,…) for atom pair AB, and 𝑟𝐴𝐵
is their
𝐴𝐵
internuclear distance. 𝐶𝑛 and 𝑠𝑛 coefficients are necessary to adapt the potential specifically
54
COMPUTATIONAL METHODOLOGIES
to the chosen functional. The damping function 𝑓𝑑,𝑛 (𝑟𝐴𝐵 ), which determine the range of the
dispersion correction, is given by
𝑓𝑑,𝑛 (𝑟𝐴𝐵 ) =
1
−𝛼𝑛
1 + 6(𝑟𝐴𝐵 ⁄(𝑠𝑟,𝑛 𝑅0𝐴𝐵 ))
(58)
where 𝑠𝑟,𝑛 is the order-dependent scaling factor of the cuttof radii 𝑅0𝐴𝐵 .
In the DFT-D3 case, instead of using an empirically derived interpolation formula, the
dispersion coefficients have been computed by time-dependent (TD) DFT calculations. Then
we can conclude that D3 method is less empirical than D2 case.
Grimme’s method to calculate the dispersion energy is considered a classical correction
because of the formulas that it is based on. These formulas are referring to the typical classic
method to evaluate long distance forces as London, van der Waals, etc.
From previous studies presented by Osuna, Swart and Solà,118,119 we know the important paper
which plays the dispersion energy correction in the calculation of the energy barriers for
reactions involving fullerene compounds. They showed that to improve the theoretical results
and to get results close to the experimental values, it is essential to include Grimme’s
dispersion corrections in the energy profile of the reaction.
Energy Decomposition Analysis
In the Kohn-Sham approximation, the total bonding energy between the different fragments
that forms the molecule can be decomposed in different contributions. One possible method
to do so is the approach proposed by Morokuma,114,120,121 which is implemented in the ADF
package.
First of all, the total bonding energy is defined as:114
∆𝐸𝐵𝑜𝑛𝑑𝑖𝑛𝑔 = 𝐸𝑇 − 𝐸𝑟𝑒𝑓
(59)
where 𝐸𝑇 is the absolute energy of the final system, and the reference energy (𝐸𝑟𝑒𝑓 ) is
calculated as the sum of the energies of all the defined fragments:
55
MASTER THESIS
𝑁
𝑖
𝐸𝑟𝑒𝑓 = ∑ 𝐸𝑓𝑟𝑎𝑔
(60)
𝑖=1
where N is the number of fragments.
The total bonding energy is divided into two major components: preparation energy and
interaction energy (equation (61)). The preparation energy is the necessary energy to deform
the initial fragments from their equilibrium geometries to the final geometry they have into
the molecule. And the interaction energy is the energy due to the interaction between the
different fragments.
∆𝐸𝐵𝑜𝑛𝑑𝑖𝑛𝑔 = ∆𝐸𝑝𝑟𝑒𝑝 + ∆𝐸𝑖𝑛𝑡
(61)
In the next step, interaction energy, ∆𝐸𝑖𝑛𝑡 , is decomposed into three new terms:
∆𝐸𝑖𝑛𝑡 = ∆𝑉𝑒𝑙𝑠𝑡𝑎𝑡 + ∆𝐸𝑃𝑎𝑢𝑙𝑖 + ∆𝐸𝑜𝑖
(62)
Each term is referred to different physical concepts. The ∆𝑉𝑒𝑙𝑠𝑡𝑎𝑡 term corresponds to the
classic electrostatic interaction between the charge distribution of the fragments when they
are already prepared, that is, deformed. This term is usually attractive for neutral fragments.
The second term, ∆𝐸𝑃𝑎𝑢𝑙𝑖 , is due to the antisimetrization and renormalization of a hypothetical
intermediate wavefunction taken as the product of the different wavefunctions of all
fragments {Ψ1 , Ψ2 , … , ΨN }. It is the necessary energy to construct the wavefunction:
Ψ 0 = N𝐴̂[Ψ1 Ψ2 … ΨN ]
(63)
where N is the normalization constant, and 𝐴̂ is the antisymmetrization operator. It comprises
the destabilizing interactions between occupied orbitals, and it is responsible for any steric
repulsion.
Finally, the wavefunction is allowed to relax from Ψ 0 to the fully converged Ψ. The associated
orbital interaction energy, ∆𝐸𝑜𝑖 , accounts for electron pair bonding, charge transfer (including
the HOMO-LUMO interaction) and polarization (empty/occupied orbital mixing on one
fragment due to the presence of another fragment). This can be further decomposed into the
contributions from the distinct irreducible representations Γ of the interacting system,122,123
which is very useful in systems with a clear 𝜎/𝜋 separation.
∆𝐸𝑜𝑖 = ∑ ∆𝐸𝑜𝑖,Γ
Γ
56
(64)
COMPUTATIONAL METHODOLOGIES
3. Fullerene compounds using the
Computational Chemistry tools
For the computational study of reaction mechanisms one has to locate the different stationary
points involved in the corresponding reaction path. The reaction path is the minimum energy
path which connects the reactants and products through the transition state. Each stationary
point has an optimized geometry obtained after applying a series of iterations which modify
the geometry until the energy of the system has reached a point on the potential energy
surface (PES) with gradient zero. Reactants and products are minima on the PES, whereas
transition states are first-order saddle-points (Figure 24).
Figure 24. Representation of a Potential Energy Surface (PES) with minima, first-order saddle points (transition
state), and second-order saddle points.
To find the reactants and products geometries, we have to find the minima in the reaction
coordinate path. To know the path we should follow, we have to calculate the gradient. The
gradient is the first derivative of the energy respect to the Cartesian coordinates of all the
nuclei. Following the gradient slope downhill, a point where the slope will be equal to zero will
be found. That point corresponds to a minimum of the PES.
The procedure that is behind the geometry optimization is the following: one starts with an
initial geometry, computes the potential energy and the gradient, and afterwards the
geometry of the molecule is changed by moving downhill. Once a new geometry is obtained,
the cycle starts again until the change on the geometry as well as on the energy of the system
is below a defined threshold value.
The procedure for the localization of the transition states is slightly different, as they
correspond to a maximum along the reaction coordinate but minima in the other directions.
Although the gradient is used to locate transition state geometries, the second derivatives of
57
MASTER THESIS
the energy (the Hessian Matrix) are also required. The computational cost of the construction
of the Hessian is too high, but there exists different strategies to solve this problem. One
solution consists in computing the Hessian only at the first point of the geometry optimization,
and then updating it following a certain scheme such as the Bofill updating method.124 Other
strategy is constructing a model Hessian with the correct number of eigenvalues
corresponding to the approach of reacting molecules. The latter is the approach used in our
calculations and which is implemented in QUILD (ADF) package.125
For the final characterization of the different stationary points, an evaluation of the vibrational
normal modes is required. The structures corresponding to a minimum in the PES must have
all the frequency values real (positive) i.e. all the diagonal Hessian Matrix elements positive.
For the transition state geometries, as they are first-order saddle-points, we must have one
(and only one) imaginary frequency (negative) which means having one negative Hessian
Matrix eigenvalue.
Reaction, Activation, Deformation and Interaction
Energies in fullerene reactivity
From the energies of the optimized structures belonging to the reaction path of our interest,
we can extract and calculate different energy parameters. That is the reason why sometimes
single point energy calculations (SP) are performed using higher basis sets or different
methods. As we know, by the variational principle the obtained energies will be more
accurate. This strategy is usually performed for large systems, such as fullerenes or
metalloporphyrines, where the use of highly accurate methods or large basis sets in the
optimization procedure is computationally expensive. If we assume that geometry is much less
dependent on the size of the basis set than the energy, the latter hypothesis is valid.
The energies obtained from reactants, products and transition states can be very useful to
extract information about the thermodynamics and kinetics of a certain reaction as can be
seen in equations (65), (66) and Figure 25.
∆𝐸𝑟𝑥 = 𝐸𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑠 − 𝐸𝑟𝑒𝑎𝑐𝑡𝑎𝑛𝑡𝑠
(65)
∆𝐸 ⱡ = 𝐸𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛 𝑠𝑡𝑎𝑡𝑒 − 𝐸𝑟𝑒𝑎𝑐𝑡𝑎𝑛𝑡𝑠
(66)
where ∆𝐸𝑟𝑥 is the reaction energy and ∆𝐸 ⱡ is the reaction barrier.
58
COMPUTATIONAL METHODOLOGIES
Figure 25. Schematic reaction profile where reactants, transition state and products are represented as well as
the activation energy (∆𝑬ⱡ ) and reaction energy (∆𝑬𝒓𝒙 ).
Activation and reaction energies are better represented in terms of enthalpies or Gibbs free
Energies. They are obtained using the equivalent expressions to equations (65) and (66).
Gibbs free energies have significant interest when the number of reacting molecules changes
along the reaction coordinate, which implies a modification of the total entropy of the system.
To obtain the enthalpies or Gibbs free Energies using the thermostatistic expressions we have
to perform a frequency calculation, i.e. computing the second derivates of the energy respect
Cartesian coordinates.
The deformation energy is the energy required to deform the molecule from the minimum
geometry to the geometry of the transition state. Deformation energy is an effective tool to
understand the chemistry of fullerene compounds. Usually, the most reactive bonds of the
(endo)fullerene cage are those which involve lower deformation energies to reach the TS. In
addition to that, the insertion of a large compound inside the fullerene cavity implies also a
deformation energy. An extremely high deformation energy found theoretically for the
encapsulation of the moiety inside the fullerene, means that the endohedral fullerene will not
be formed.
In this sense, we have introduced a new energy parameter, the non-cluster deformation
energy, which allows us to understand the reactivity differences between the equivalent bonds
of free cage and endohedral fullerenes. The electronic effects due to the encapsulate moiety
can be computed as the difference between a SP energy calculation of the transition state
optimized structure without the cluster inside, and the SP energy of the reactant structure also
without the cluster inside. Bonds which have highest values for this parameter are those with a
different reactivity between the free and endohedral fullerene cages.
59
MASTER THESIS
Predicting the fullerene chemistry
Bond distances
In order to describe and predict the regioselectivity of a cycloaddition reaction on a fullerene
cage, one can look to the carbon-carbon bond distances. The shortest bond distances indicate
a higher 𝜋 electronic density and 𝜋 bonding orbital interactions on the highest occupied
molecular orbital (HOMO). The [6,6] bonds are usually the shortest bonds in the cage. On the
other hand, [5,6] bonds are usually the larger as a consequence of the antibonding 𝜋
contribution of the orbitals present in the HOMO. Shorter C-C distance is associated with more
reactivity than the longer distance. That is because C-C bond has more double bond character
that facilitates the interaction with dienes/dipoles in cycloaddition reactions. This is not an
unchangeable law, and indeed there are some cases described where the most favorable
addition site corresponds to the largest C-C bond, but it is useful to understand the fullerene
compounds reactivity.
Pyramidalization angles
The relationship between chemical reactivity and the carbon pyramidalization angle was
proposed by Haddon.76,77 The carbon pyramidalization angle is a simple a measure of the local
curvature of carbon containing systems, which is represented by:
𝜃𝑝 = 𝜃𝜎𝜋 − 90∘
(67)
For each carbon atom, one can define a vector with its origin on the atom and with direction
chosen to equalize the three angles between it and the three C-C bonds. 𝜃𝜎𝜋 are these
equalized angles.
Figure 26. Representation of the pyramidalization angle. A vector (black colored in the figure) that equalizes the
α, β and δ angles is defined in order to compute the final pyramidalization angle value (θp=α-90°). It is 0° and
19.47° for sp2 and sp3 centers respectively.
60
COMPUTATIONAL METHODOLOGIES
The pyramidalization angle for sp2 centers is 0° and 19.47° for sp3 systems. The more
pyramidalized the carbon atoms are, the closer to the final sp3 situation and the lower the
deformation of the system during the transit from reactants to products. In principle, those
bonds with higher pyramidalization angles exhibit a higher reactivity. But as in the bond
distance case, there are some cases where pyramidalization angles don’t describe correctly the
reactivities found.
All pyramidalization angle measures presented in this work, were performed using the πorbital axis vector approach (POAV1)126 as implemented in the POAV3 program.127
Frontier orbitals
Cycloaddition reactions are usually described using the FMO theory. Hence, the interaction
that takes place in the Diels-Alder reaction is between the HOMO of fullerene and the LUMO
of diene or between the LUMO of fullerene and the HOMO of the diene. In most cases, the
most prominent contribution consists of the HOMO of the diene and the LUMO of the
fullerene compound, due to the stabilized LUMO orbitals of the fullerenes.
A theoretical analysis of the shape of the LUMO orbitals of fullerene compounds might give a
prediction of the most reactive C-C bonds. Only the bonds which present a suitable shape (an
antibonding 𝜋 contribution) to interact with the HOMO of the diene will be reactive. We have
to take into account that fullerene orbitals are highly delocalized, so a large number of C-C
bonds with suitable orbitals for interacting with diene are usually found.
The difference in energy between the HOMO and LUMO (HOMO-LUMO gap) gives an idea of
how reactive the species is. Small gaps are usually found for highly reactive compounds.
Introducing moieties inside the fullerene cages to obtain the endohedral fullerene compounds,
usually implies an increase of the HOMO-LUMO gap, which usually leads to a reduction of the
exohedral reactivity.
61
MASTER THESIS
62
OBJECTIVES
CHAPTER III.
OBJECTIVES
In the present work, there are included two different projects. The different objectives
proposed for each project are described below.
Exohedral reactivity of Ti2C2@C78 endofullerene: Diels-Alder addition on all non-equivalent
bonds:
The main objective of this project is to extend the previous studies of Diels-Alder reactivity of
endofullerenes carried out by our group78,79 to the Ti2C2@C78 system. We want to evaluate the
different regioselectivity of the Diels Alder reaction between 1,3-cis-butadiene and Ti2C2@C78.
We aim not only compare the obtained results with the previous ones, but also to understand
why different regioselectivities are found when the endohedral moiety changes. In order to
reach this goal, we will to employ the same tools used in the previous cases (for example,
fullerene deformation energies). Nevertheless, we also plan to define new tools to
complement the analysis of endofullerene Diels-Alder regioselectivity.
A recent study has shown the important role of the inclusion of the dispersion energy
corrections for the evaluation of the energy barriers for this type systems.119 We propose to
compute these corrections for the present system in order to see how the energy barriers are
modified.
Fine-tunable metalloporphyrinic nanocages as hosts for fullerene encapsulation:
This project is part of a collaboration between the experimental group of X. Ribas and M.
Costas (UdG) and our computational chemistry group. The main goal of the global project is to
design metalloporphyrin-based nanocages for selective encapsulation of fullerene compounds
by host-guest chemistry. This Master Thesis has three concrete objectives related with the
main goal of this collaborative project. The first one is to reproduce, using QM-DFT
methodology, the experimental structure of the nanocage systems synthesized for which
crystal structures are available.2 Our second goal will be to determine, by using DFT
methodology, the structures of the other nanovessels synthesized by our experimental
colleagues, but which crystal structures are not available.
Due to the large computational cost that implies the treatment of a system which includes
more than 660 atoms and some metallic atoms with QM methods it is very difficult to study
63
MASTER THESIS
the host-guest interactions between the nanovessels and the fullerenes. Then, our third goal is
to design a model system which correctly describes the interactions between the nanocages
and the fullerenes. In order to check our model, we will compare the obtained results with the
ones obtained using the entire real systems.
Once we have a good model defined, our interest will be to study the effect on the final
interaction energy of the orientation of the fullerene molecule inside the cavity of the cage,
and the effect of the porphyrinic metal ions. In that sense, we want to use the Energy
Decomposition Analysis to extract useful information of the different interaction energies. We
expect that this information will be very useful for the synthesis of new nanocages.
64
RESULTS
CHAPTER IV.
RESULTS
In the present chapter, and ordered in two different sections, we will present the results
obtained for the two proposed studies described in the previous chapter.
65
MASTER THESIS
66
RESULTS
1. Exohedral reactivity of Ti2C2@C78 endofullerene: DielsAlder cycloaddition on all non-equivalent bonds
Introduction
The main goal of this work is to discuss the change in reactivity induced by the change in the
encapsulated metallic cluster. In order to do so, we first locate all the stationary points
(minima and saddle points) on the potential energy surface of the Diels-Alder cycloaddition
reaction between 1,3-butadiene and Ti2C2@D3h-C78 for all nonequivalent bonds of the EMF.
The results are compared to those found for the fullerenes D3h-C78, Sc3N@D3h-C78 and
Y3N@D3h-C78.78,79 First of all, the correct characterization of Ti2C2@D3h-C78 is explored, as
bibliography is not clear about this aspect.
In the Ti2C2@D3h-C78 case, there are 7 [6,6] and 6 [5,6] nonequivalent bonds to be studied in
order to take into account all possible final reaction products (see Figure 27).
Figure 27. Different nonequivalent bonds of Ti2C2@D3h-C78 are represented. Numbers denote the [6,6] bonds and
lowercase letters the [5,6] bonds. Different colors are used to mark the different bond types: orange, type A;
green, type B; red, type C; blue, type D. The Schlegel diagram, which converts the 3D structure into 2D
representation, is also shown. The position of the titanium atoms are marked with red arrows.
Computational Details
The computational procedures used in the present work are the same that were used for the
previous related studies75,76 in order to be able to directly compare the obtained results. All
density functional theory (DFT) calculations were performed with the Amsterdam Density
Functional (ADF) program.114,128 The molecular orbitals (MOs) were expanded in an
67
MASTER THESIS
uncontracted set of Slater-type orbitals (STOs) of double-𝜉 (DZP) and triple-𝜉 (TZP) quality
containing diffuse functions and one set of polarization functions. Core electrons (1s for
second period and 1s2s2p for third and fourth period) were not treated explicitly during the
geometry optimizations (frozen core approximation),114 because it was shown to have a
negligible effect on the obtained geometries.129 Scalar relativistic corrections were included
self-consistently using the zeroth-order regular approximation (ZORA).
An auxiliary set of s, p, d, f and g, STOs was used to fit the molecular density and to represent
the Coulomb and exchange potentials accurately for each self-consistent field cycle. Energies
and gradients were calculated using the local density approximation (Slater exchange and
Vosko-Wilk-Nusair correlation)104 with nonlocal corrections for exchange (Becke88)107 ad
correlation (Perdew86)106 included self-consistently (i.e., through the use of the BP86
functional). All of the energies reported in this work, are obtained from single-point energy
calculations with the TZP basis at geometries that were obtained with the DZP basis (i.e., at the
BP86/TZP//BP86/DZP level).
Although it is well-documented that BP86 functional underestimate energy barriers130 (in the
case of parent Diels-Alder BP86/TZP predicts a barrier of 18.6 kcal mol-1, which implies an
underestimation of the experimental value by about 6 kcal mol-1), this underestimation should
be similar for all Diels-Alder transition states (TSs) reported in this work.
All geometric optimizations and TS searches were performed with the QUILD125 (Quantumregions Interconnected by Local Descriptions) program, which functions as a wrapper around
the ADF program. The QUILD program constructs all input files for ADF, runs ADF, and collects
all needed data. ADF is used only for the generation of the energy and gradients. Furthermore,
the QUILD program uses improved geometry-optimization techniques, such as adapted
delocalized coordinates and specially constructed model Hessians with the correct number of
eigenvalues.125 The latter is of particular use for TS searches. All TSs have been characterized
by computing the analytical131 vibrational frequencies, and checking that they have one (and
only one) imaginary frequency corresponding to the approach of the two reacting carbons.
Pyramidalization angles, described in the introductory chapters, were calculated using the
POAV3 program.127
Energy dispersion corrections are introduced using Grimme’s methodology115,117 (D3)
implemented in ADF 2010.01 version.128 Only initial adducts are fully optimized using these
corrections in each optimization step. Dispersion corrections for the rest of structures are
introduced by single point calculations on the stationary point structure found in a usual
optimization procedure. We have not reoptimized all structures as differences in structural
parameters are minimal.119
Structural characterization of Ti2C2@D3h-C78 endofullerene
As we have described in the introductory chapters, it is reported in bibliography that
Ti2C2@D3h-C78 endofullerene is a metallic carbide EMF instead of the dimetallofullerene
68
RESULTS
Ti2@C80. But it was no clear the position that C22- adopts inside de cage. So the first step of this
study was to correctly characterize the Ti2C2@D3h-C78 reactant.
Figure 28. D3h-C78 structure representation. There are also represented the C3 axis (red), and all symmetry planes
(grey).
The two Titanium atoms are placed on the C3 axis and fixed in the center of the hexagons
perpendiculars to the C3 axis. But in the consulted bibliography,42,44 we have found three
different possible orientations of the C22- moiety between the two Ti atoms inside the D3h-C78
cage. It is not clear which of these orientations is the most energetically favorable. So we have
optimized at BP86/DZP level all three structures (see a schematic representation in Figure 29)
and comparing the obtained energies we have chosen the best option.
Figure 29. Schematic representation of the different possible orientations of the C 22- moiety between two
Titanium atoms inside D3h-C78 fullerene cage reported in bibliography.42,44
The obtained results are presented in Table 1, where we can see that most stable isomer is the
third, which has linear Ti2C2 (they are numbered as schematic representation in Figure 29).
Isomer
E (kcal mol-1)
(a)
-16200.75
(b)
-16222.37
(c)
-16222.76
Table 1. Relative energies obtained for the optimized geometries at BP86/DZP level.
As we can see in Table 1, energy differences between having linear orientation of Ti2C2 cluster
(isomer c, the most stable) and perpendicular orientation (isomer a) is about 22 kcal mol-1, not
a small gap. And taking into account the energy value for the intermediate structure (isomer
b), very close to the linear one (difference is smaller than 1 kcal mol-1), we can conclude that
C22- moiety can slightly oscillate between the two titanium atoms, from conformation c to
69
MASTER THESIS
conformation b, at room temperature but not freely rotate. For the end on conformation, we
have also checked the possible rotation of the Ti2C2 cluster as a whole along two axes
perpendicular to the molecular C3 axis and passing through the center of mass of the Ti2C2
cluster. To this end, we have performed single point in energy calculations rotating 15º in each
step, around these two perpendicular axes, while keeping frozen the geometry of the cluster
and the fullerenic cage. We have seen that such rotations are hindered by high energy barriers.
Indeed a small rotation of 15º already requires about 80 kcal mol-1. This proves that the Ti2C2
cluster in the Ti2C2@C78 EMF has an almost fixed end on conformation. Consequently, we will
perform our study considering the end on conformation which is the lowest energy structure,
without taking into account possible rotations of this cluster.
Figure 30. Ti2C2@D3h-C78 most favorable isomer structure at BP86/DZP level.
Based on the ionic model defined by Poblet and co-workers,38 we have calculated the number
of electrons transferred from the metallic cluster to the fullerene cage. Taking into account the
energies of the HOMO and HOMO-n electrons of the isolated Ti2C2 metallic cluster, and the
LUMO and LUMO+n of the D3h-C78 cage presented in Table 2, we can see that there will be an
electronic transfer from metallic cluster to the fullerene cage. Looking the energy values, we
can conclude that electrons from HOMO (2 electrons) and HOMO-1 (4 electrons, degenerated)
of Ti2C2 moiety are transferred to the LUMO and LUMO+1 (degenerated) molecular orbitals of
C78 cage when the metallic carbide is encapsulated. This occurs because the LUMOs energies of
the fullerene cage are lower in energy than the occupied HOMOs orbitals of the cluster (see
representation in Figure 31).
D3h-C78
MO
Symm. label
nº occ. (e)
E (eV)
HOMO-5
HOMO-3
HOMO-2
HOMO
LUMO
LUMO+1
LUMO+3
LUMO+4
LUMO+5
EEE1
EE1
AAA1
EEE1
AA2
EE1
EEE1
AAA2
AA1
4
4
2
4
0
0
0
0
0
-6.84
-6.83
-5.87
-5.75
-5.13
-4.86
-3.68
-3.38
-3.31
HOMO-3
EE1
4
-6.01
HOMO-1
EEE1
4
-3.52
Ti2C2
HOMO
AA1
2
-3.50
LUMO
AAA2
0
-3.46
LUMO+1
EE1
0
-3.15
Table 2. Relative energies (eV) of the molecular orbitals of interest of Ti2C2 moiety and D3h-C78 cage. All values are
obtained from a BP86/TZP//BP86/DZP calculation.
70
RESULTS
So, according to these observations and in the ionic model scheme, the treated Ti2C2@D3h-C78
system can be seen as a (Ti2C2)6+@(D3h-C78)6- ionic complex.
Figure 31. Schematic representation of the frontier molecular orbitals of Ti2C2 metallic cluster and D3h-C78 free
cage.
In the next sections we will present all values obtained from the computation of the DielsAlder reaction on all nonequivalent bonds. We will refer to each different bond according to
the nomenclature used in Figure 27, where numbers and lowercase letters refer to [6,6] and
[5,6] bonds respectively. For example, the label 7 represents the pyracylenic (type A) [6,6]
bond situated in the position as indicated in Figure 27, and f refers to the [5,6] corannulene or
type D bond, the situation of which is also marked in the same figure.
Reaction energies for the Diels-Alder reaction on Ti2C2@D3h-C78
The reaction energies obtained at BP86/TZP//BP86/DZP level are listed in Table 3. As can be
seen, in general the Diels-Alder reaction is more favored over the [5,6] bonds than [6,6] bonds.
Although [6,6] bonds of empty fullerenes are generally more dienophilic, Diels-Alder reactions
of EMFs take place preferently on [5,6] bonds.132 Reactions energies are always exothermic for
[5,6] type D bonds except for the b case, and are endothermic for [6,6] bonds except for the 1
and 3 bonds. The most favorable addition is produced over the [5,6] bond c, for which reaction
energy is -18.1 kcal·mol-1, next comes [5,6] bond f, with a reaction energy equal to -14.6
kcal·mol-1, and [6,6] bond 3, with -14.1 kcal·mol-1.
Short bond distances together with high pyramidalization angles have been used as reactivity
descriptors to predict the exohedral reactivities of fullerene derivates. Shorter bonds, exhibit
more double-bond character that facilitates the interaction with the diene. And high
pyramidalization angles means that C atom is closer to the final sp3 geometry, and hence, the
system has to be deformed less during its transit from reactants to TS and products. As we
have seen in the introductory chapters, bonds with rather long C-C bond distances have always
been considered to be non- or much less reactive. The same occurs for lowest pyramidalization
71
MASTER THESIS
angles, where fullerene has to be too much deformed to adopt the final structure, which
implies a decrease of the exohedral reactivity. In Table 4 are presented all bond lengths and
pyramidalization angles for the present studied species, Ti2C2@D3h-C78, the previous studied
species, Sc3N@D3h-C7878 and Y3N@D3h-C78,79 and D3h-C78 free cage. We can see how the
encapsulated moiety induces structural changes into the fullerene cage, obtaining different
bond length and pyramidalization angle values for the different considered fullerenes.
Ti2C2@D3h-C78
product
bond type
ΔER
Rfull
RCC
1
A [6,6]
-2.6
1.562
1.575
1.575
2
C [6,6]
4.2
1.654
1.575
1.575
3
B [6,6]
-14.1
1.625
1.574
1.567
4
B [6,6]
8.5
1.629
1.571
1.579
5
B [6,6]
1.6
1.598
1.576
1.580
6
B [6,6]
11.1
1.604
1.571
1.591
7
A [6,6]
5.1
1.667
1.564
1.564
a
D [5,6]
-7.1
1.728
1.553
1.553
b
D [5,6]
3.4
1.623
1.567
1.564
c
D [5,6]
-18.1
1.628
1.562
1.562
d
D [5,6]
-6.8
1.616
1.563
1.563
e
D [5,6]
-5.5
1.640
1.564
1.566
f
D [5,6]
-14.6
1.630
1.564
1.548
Table 3. Reaction energies (ΔER, kcal mol-1) and bond lengths in the final products of the C-C bonds over which the
reaction took place (Rfull, Å) and of the two new C-C bonds formed (RCC, Å). Bold values indicate the bonds that
are the most reactive under thermodynamic control.
D3h-C78a
Sc3N@D3h-C78 a
Y3N@D3h-C78 up a
Y3N@D3h-C78 down a
Ti2C2@D3h-C78
product
Rfull
θp
Rfull
θp
Rfull
θp
Rfull
θp
Rfull
θp
1
1.369
10.46
1.440
13.80
1.463
14.03
1.463
14.03
1.388
11.02
2
1.465
8.58
1.466
8.33
1.475
8.46
1.474
8.55
1.456
8.92
3
1.432
9.62
1.450
9.26
1.460
9.83
1.448
9.64
1.413
9.71
4
1.415
9.60
1.426
9.44
1.432
9.22
1.435
9.34
1.446
8.66
5
1.418
9.53
1.432
8.97
1.443
8.56
1.443
8.67
1.416
9.75
6
1.420
9.44
1.400
9.99
1.396
9.66
1.398
9.78
1.423
8.68
7
1.388
11.64
1.400
11.21
1.398
11.17
1.401
11.06
1.457
13.53
a
1.438
11.64
1.437
11.21
1.439
11.17
1.440
11.06
1.477
13.53
b
1.410
10.49
1.446
9.73
1.455
10.10
1.457
9.89
1.447
10.66
c
1.465
10.32
1.423
9.27
1.423
9.02
1.423
9.02
1.422
10.81
d
1.446
10.56
1.452
12.00
1.468
12.62
1.464
12.38
1.431
10.76
e
1.438
10.38
1.449
10.92
1.454
10.60
1.453
10.70
1.435
8.64
f
1.442
11.13
1.432
10.88
1.431
10.58
1.431
10.59
1.452
10.96
Table 4. Bond lenghts (Rfull, Å) and pyramidalization angles (θp, deg) for the different bonds in free and
endohedral fullerenes. Reported pyramidalization angles represent the average over the two atoms that
constitute the bond under consideration. a Values from ref. 79.
72
RESULTS
Based on bond length and pyramidalization angles presented in Table 4, we could consider
bond 1 as the most reactive one, because it has the shortest bond length and a high
pyramidalization angle. However, looking into Table 3 we can see that bond 1 is far to be the
most reactive option on Ti2C2@D3h-C78 case. Indeed for the most reactive bonds found, c, f and
3, their geometric parameters do not indicate that they would be the most preferred bonds to
the cycloaddition reaction. So, in the present case these descriptors cannot help us to give a
reasonabley explanation about the thermodynamic controlled reactivity of the Diels-Alder
reaction on Ti2C2@D3h-C78 system. In next sections we will try to explain what is happening.
Figure 32. Comparison between Sc3N@D3h-C78, Y3N@D3h-C78 and Ti2C2@D3h-C78 structures, where can be seen the
different orientation of metallic clusters and deformation effects over fullerene cage they have.
The inclusion of the Ti2C2 moiety inside the fullerene cage induces different structural changes
than the Sc3N and Y3N cases. The origin of these differences lies into the position that the
cluster adopts inside the cage. For the TNT moieties, fullerene cage is deformed along the
horizontal symmetry plane, however when titanium metallic carbide is introduced, fullerene
cage is deformed along C3 symmetry axis (see Figure 32). The deformation energy of the C78
cage for encapsulating Sc3N is 32.2 kcal·mol-1, for Y3N is significantly higher (67.5 kcal·mol-1),
and for Ti2C2 moiety is 39.58 kcal·mol-1 at BP86/TZP//BP86/DZP level of theory. Direct
comparison between TNT endofullerenes and the present metallic carbide endofullerene is
quite difficult because of the different emplacement and orientation of clusters inside the
cage, which leads different bonds to be more modified in each case. However, we can see that
more important changes are produced because of the inclusion of the Y3N cluster inside the
D3h-C78 fullerene than because of the inclusion of Ti2C2 moiety.
The principal changes in the fullerene structure induced by the encapsulation of Ti2C2 metallic
cluster are found in the most proper bonds to the titanium ions. Bond 7 (type A, [6,6]) and
bond a (type D, [5,6]), which are the closer bonds to the titanium atoms, increase their bond
lengths significantly with respect to the free cage and other endofullerenes. Furthermore, their
pyramidalization angles also increase (see Table 4). The bond length enlargement plays against
their reactivity as an increase of the bond length decrease the reactivity for the cycloaddition
reaction. Bond 4 (type B, [6,6]), which is placed quite near from Ti ions, also increases the bond
length but not so much.
The most reactive bonds, c (type D [5,6]), f (type D, [5,6]), and 3 (type B, [6,6]), are slightly
favored by the inclusion of the titanium cluster inside the C78.For bonds c and 3 the cause is
73
MASTER THESIS
their bond lengths decrease. In the bond c case, in addition, pyramidalization angle is also
modified being much higher, i.e. favoring its reactivity. Bond f remains more or less the same
with a small increase in the bond length. However, it clearly increases its reactivity, being one
of the most reactive bonds for Ti2C2@D3h-C78 endofullerene, because of electronic (not
structural) effects due to the presence of the metallic cluster inside the fullerene cage, as we
will see in next sections. All remaining bonds are not so modified by the inclusion of the
metallic carbide, having similar structural parameters respect C78 free cage.
Therefore, the Diels-Alder cycloaddition over Ti2C2@D3h-C78 is usually favored for type D [5,6].
The reaction energies for the more reactive [5,6] c and f bonds are -18.1 kcal mol-1 and -14.6
kcal mol-1 respectively. All the rest of [5,6] bonds that present intermediate bond distances,
present exothermic reaction energies (from -5.5 to -7.1 kcal mol-1), except for bond b, which
reaction energy is slightly endothermic (3.4 kcal mol-1).
[6,6] bonds in free fullerene systems are usually favored, but not EMFs. For the studied
system, all reaction energies found for [6,6] bonds type are endothermic (from 1.6 to 11.1 kcal
mol-1) with the exception of type A bond 1 (-2.6 kcal mol-1), which a priori seems to be the
most reactive bond because presents the shortest bond length (1.38 Å) and high
pyramidalization angle (11.2º). The other exception is bond 3, which is the third most reactive
bond. Bond 3 (type B) presents an exothermic reaction energy of -14.1 kcal mol-1, although its
bond length and pyramidalization angle are quite similar to the other [6,6] bonds which
present endothermic reaction energies.
But if we locate the three most reactive bonds (c, f, and 3) on the endofullerene structure, we
can see that these three bonds are parallel to the inside cluster orientation. Taking into
account that Ti2C2 moiety causes a big strain on fullerene carbon structure, we can think that
fullerene energy strain plays an important role in the final reactivity of Ti2C2@D3h-C78
compound.
Energy barriers for the Diels-Alder reaction on Ti2C2@D3h-C78
We have also determined the TS structures for the cycloaddition reaction for each of the
nonequivalent bonds. In all cases, the TS search has started from a symmetric structure in
which the two C-C bonds to be formed have the same bond length, leading in most cases to an
asynchronic but still concerted TS. All results are presented in Table 5.
The attack on the bond c is also the kinetically most favorable among all of the bonds
considered (with an energy barrier of 17.4 kcal mol-1). The other two most exothermic
products (f and 3) also have the lowest barriers, but in different order (18.2 kcal mol-1 for bond
3 and 19.3 kcal mol-1 for bond f) as can be seen in Figure 33.
74
RESULTS
Ti2C2@D3h-C78
product
bond type
ΔEⱡ
1
A [6,6]
23.6
2.229
2.131
9.2
2
C [6,6]
26.8
1.696
2.622
19.6
3
B [6,6]
18.2
2.509
2.083
8.8
4
B [6,6]
29.6
1.688
2.397
20.3
5
B [6,6]
20.6
1.843
3.123
10.6
6
B [6,6]
27.0
1.669
2.428
23.7
7
A [6,6]
27.5
1.800
2.490
15.6
a
D [5,6]
21.4
2.073
2.219
9.0
b
D [5,6]
26.5
2.502
1.678
17.8
c
D [5,6]
17.4
2.067
2.562
7.2
d
D [5,6]
22.6
1.850
2.739
11.8
e
D [5,6]
20.5
1.889
3.078
12.7
D [5,6]
19.3
1.944
2.970
12.2
f
(ΔEⱡ,
ΔEⱡ def. ful.
RCC
mol-1)
Table 5. Reaction barriers
kcal
and bond lengths (RCC, Å) for the bonds being formed at the TS together
with the deformation energies of the endofullerene (ΔEⱡdef. ful., kcal mol-1). Bold values indicate the bonds that are
the most reactive under kinetic control.
∆E
ΔE#.ⱡ
∆EΔE
reac.
R
30
E (kcal mol-1)
20
10
0
-10
-20
1
2
3
4
5
6
7
a
b
c
d
e
f
Figure 33. Reaction energies (red) and reaction barriers (blue) found for all nonequivalent bonds of the
Ti2C2@D3h-C78 endohedral metallofullerene.
The other TSs are found at slightly higher energies (20.5-23.6 kcal mol-1) or considerably higher
energies (26.5-29.6 kcal mol-1). The highest barrier is found for bond 4, which has the second
most endothermic reaction energy value. Bond 6, which provided the highest endothermic
reaction energy, has also a large reaction barrier (27.0 kcal mol-1) (see also Figure 33).
75
MASTER THESIS
Therefore, the cycloaddition reaction over [5,6] bonds are usually favored respect [6,6], not
only from the thermodynamic control, but also from the kinetic one. However, there are
several exceptions, such as bond b (type D, [5,6]), which has a endothermic reaction energy
and high reaction barrier, or bond 3, as we said, one of the three most reactive bonds.
In next section we will do an analysis of the molecular orbitals (MOs) which will help us to
understand the regioselectivity found for this EMF. Also we will discuss the deformation
energies of the fullerene and we will introduce a new energy parameter, the non-cluster
energy barrier, which will provide us with useful information to understand the reactivity of
this endohedral metallofullerene.
Effects on the reaction barrier due to the presence of Ti2C2 inside the D3h-C78
fullerene cage. Deformation energies and Molecular Orbitals.
We have seen that the three most reactive bonds under thermodynamic control and kinetic
control are the same. But, we have also seen that their bond lengths and pyramidalization
angles are not too short or too high, respectively, to justify their high reactivity. So, where the
high reactivity of these bonds come from? We will try to answer this question in the present
section.
An analysis of the molecular orbitals (MOs) as a descriptor for the observed reactivity is done.
In the case of fullerene compounds, the main interaction occurs between the HOMO of
butadiene (at -5.77 eV) and the LUMOs of the fullerene. In the Ti2C2@D3h-C78 case, LUMOs
energies are found between -4.49 and -4.00 eV, and for the free C78 cage values go from -5.13
and -3.69 eV. Note that because of the formal charge transfer upon metal-cluster
encapsulation, the three lowest LUMOs of the free fullerene are occupied and only higherlying unoccupied orbitals are available for interacting with the HOMO of the diene. As a
consequence, the HOMO(diene)-LUMO(fullerene) gap increases from 0.64 eV for free fullerene
to 1.28 eV for the endohedral Ti2C2. This larger gap is an indication of a more stable and less
reactive species. As we will see in the comparative section, that is what we observe.
Comparing now the low-lying unoccupied orbitals of either the free fullerene cage or the Ti2C2
derivative, we can find one possible explanation to the large reactivity found for some bonds.
The larger contributions of the C atoms for a certain C─C bond in the LUMOs, means that the
bond of interest will be more reactive. But the lobes of the LUMOs within any one of these
bonds should be of opposite sign in order to have a good interaction with the HOMO of the
diene on the [4+2] cycloaddition.
In Figure 34 the shape of the lowest unoccupied molecular orbitals of C78 free cage and
Ti2C2@C78 derivative are presented. There, it can be seen that, for the Ti2C2 compound, most
favorable bonds under themodynamic and kinetic conditions (bonds c, 3, and f) present
favorable LUMO (degenerate) orbitals to interact with the HOMO of diene. The LUMO orbitals
also present an adequate shape for the Diels-Alder reaction on bond a. Focusing now on the
76
RESULTS
LUMO+2 orbital, we can see that bonds e, 5 and 4 have the correct lobe disposition to react.
Bonds e and 5 have only slightly large reaction barriers than c, 3, and f bonds. In contrast, bond
4 have the largest energy barrier of all nonequivalent bonds. Finally, in LUMO+3 orbitals bonds
1, 2, 3, f and 7 presents favorable orbitals to react. Bonds 1, 2 and 7 have higher reaction
barriers than bonds having correctly oriented lobes in LUMO+2 and LUMO MOs. Bonds 3 and f
have smaller barriers, but they were present in the LUMO MO case.
Therefore, we have found that the principal reactive bonds present favorable orbitals to
interact during the cycloaddition reaction and they are ordered by decreasing reactivity over
the 5 first LUMO orbitals of endohedral metallofullerene. That means, molecular orbital
analysis is a useful tool to understand the final reactivity of nonequivalent bonds, but not
always as a descriptor for predicting the reactivity. From this MO analysis, one could predict
that bond 4 must have an energy barrier similar to bonds 5 or e, instead of having the highest
one. Bond a should also be quite reactive. Another important observation to take into account
is that bond d, which is quite reactive, is not present in any of the LUMO orbitals analyzed.
Thus, as a predictive tool, molecular orbital analysis is qualitative correct but quite imprecise
as we have seen.
Figure 34. Representation of the C78 LUMO, LUMO+1, and LUMO+3;78 and Ti2C2@C78 LUMO, LUMO+2, and
LUMO+3 molecular orbitals (isosurface value 0.02 au) where those bonds with favorable orbitals to interact with
the HOMO of the diene are marked with ellipses.
77
MASTER THESIS
As molecular orbital analysis cannot explain by itself the reactivity of all nonequivalent bonds
of Ti2C2@C78 fullerene compound, we will use additional tools to try to understand the results
found.
Ti2C2@D3h-C78
product
bond type
ΔEⱡ def. ful.
ΔEⱡ no Ti2C2
ΔEⱡ - ΔEⱡ no Ti2C2
ΔEⱡ
1
A [6,6]
9.2
6.8
16.7
23.6
2
C [6,6]
19.6
25.6
1.2
26.8
3
B [6,6]
8.8
17.4
0.9
18.2
4
B [6,6]
20.3
6.8
22.9
29.6
5
B [6,6]
10.6
8.1
12.5
20.6
6
B [6,6]
23.7
21.5
5.6
27.0
7
A [6,6]
15.6
3.5
24.0
27.5
a
D [5,6]
9.0
11.0
10.4
21.4
b
D [5,6]
17.8
-1.2
27.7
26.5
c
D [5,6]
7.2
14.1
3.4
17.4
d
D [5,6]
11.8
11.7
10.9
22.6
e
D [5,6]
12.7
7.7
12.9
20.5
f
D [5,6]
12.2
25.2
-5.8
19.3
(ΔEⱡdef. ful.,
mol-1),
Table 6. Fullerene deformation energy barriers
kcal
fullerene non-cluster energy barriers
(ΔEⱡno_Ti2C2, kcal mol-1), and difference between actual energy barriers (Table 5) and fullerene non-cluster energy
barriers (ΔEⱡ- ΔEⱡno_Ti2C2, kcal mol-1) for all nonequivalent bonds of Ti2C2@D3h-C78 Diels-Alder reaction.
35
ⱡ
∆E#
Fullerene
ΔEdef.
def. Fullerene
∆EΔE
#.ⱡ
ⱡ Ti2C2
∆E#
ΔENo
no_Ti2C2
30
E (kcal mol-1)
25
20
15
10
5
0
-5
1
2
3
4
5
6
7
a
b
c
d
e
f
Figure 35. Comparison of the different fullerene deformation energy barriers (red), fullerene non-cluster energy
barriers (green), and energy barriers (purple) for all nonequivalent bonds of Ti2C2@D3h-C78 Diels-Alder reaction.
In Table 6 are presented the different deformation energies obtained for all nonequivalent
bonds of Ti2C2@C78 system for the Diels-Alder reaction. First, we will focus on fullerene
78
RESULTS
deformation energies (ΔEⱡdef. ful.). As can also be seen in Figure 35, bonds which present lower
fullerene deformation energies (from 7.2 to 12.7 kcal mol-1) are 1, 3, 5, a, c, d, e, and f. The rest
of nonequivalent bonds (2, 4, 6, 7, and b) present fullerene deformation energies comprised
between 15.6 and 23.7 kcal mol-1. Here, it is important to remark that the two less reactive
bonds (thermodynamic and kinetically) 4 and 6, are those which have the highest fullerene
deformation energies. This means that their poor reactivity is determined by the energy
required to distort the fullerene cage to the geometry it has in the TS when reaction takes
place over one of these two bonds.
Observing now the obtained values for the non-cluster energy barriers, we will be able to
finally understand the reactivity of our fullerene compound. The non-cluster energy barrier is
the energy difference between the TS and reactants when the Ti2C2 cluster is removed keeping
the geometry of the reactants and TS frozen. The non-cluster energy barrier gives an idea of
the geometric effect of the cluster in the energy barrier. The difference between the actual
energy barrier and the non-cluster energy barrier gives an idea of the pure electronic effect of
the Ti2C2 cluster in the energy barrier. If this difference is negative it means that the electron
transfer from the Ti2C2 cluster favors this addition and vice versa. Bonds which have low
energy differences are those in which the charge transfer from the Ti2C2 cluster does not have
a great effect on their final reactivity. These bonds are 2, 3, 6, c, and f, with difference energy
values between -5.8 and 5.6 kcal mol-1. For the rest of nonequivalent bonds (1, 4, 5, 7, a, b, d
and e), the presence of the metallic cluster implies an increase of the reaction barrier with
values going from 10.4 to 27.7 kcal mol-1 and, therefore, a reduction of reactivity that has to be
attributed to charge transfer from the metallic cluster. Indeed for all bonds, except f, there is a
reduction of reactivity due to charge transfer from the Ti2C2 unit.
Let us now to draw our attention to bond f, which is the only one that the absence of the
metallic cluster implies an increase of the reaction barrier. That is the reason why the
difference calculated for this bond is negative (-5.8 kcal mol-1), and this means that reactivity of
bond f is enhanced because of charge transfer from the Ti2C2 moiety. If we look at the MOs of
the free C78 cage in Figure 34, LUMOs have not the right shape for a good interaction with the f
bond. On the other hand, the LUMOs of Ti2C2@C78 have contributions to the f bond with lobes
having the required opposite signs on each C atom. Thus charge transfer changes the LUMO
orbitals and make them more suitable for the Diels-Alder reaction with the f bond.
A second and not less important observation is that there are only three bonds (c, f, and 3)
which have low fullerene deformation energies and low differences between reaction barriers
and non-cluster energy barriers (see Figure 36). As we have explained previously, these three
bonds are the ones with lowest energy barriers and highest exothermic reaction energies.
They are bonds c, f and 3, the three most reactive bonds found for this system (see Figure 36).
On the other side, there are only three bonds (4, 7, and b) that are disfavored by these two
parameters. They are also those which have the highest energy barriers and endothermic
reaction values, together with bond 6. The latter, bond 6, has the highest (23.7 kcal mol-1)
fullerene deformation energy, and that is the reason of its lower reactivity. Something similar
occurs for the bond 2 (fullerene deformation energy equal to 19.6 kcal mol-1).
79
MASTER THESIS
Figure 36. Representation of the distribution of favorable (green) and not favorable (red) fullerene deformation
energies and fullerene non-cluster energy barriers (differences with respect reaction barriers for the latter). Only
favored bonds in both cases are marked.
The similar reactivity found for bonds a, d and e can be explained in terms of deformation
energies. They have not large fullerene deformation energies (values range between 9.0 and
12.7 kcal mol-1), and their differences between reaction barriers and non-cluster deformation
energies are very close as can be seen in Table 6.
As we have show, only one criterion is not sufficient to predict and explain the reactivity that
takes place in the endohedral fullerene compound. But the combination of all, structural
parameters (bond lengths and pyramidalization angles), molecular orbital analysis and
deformation energies, gives a complete vision of what happens in our studied system. The
results obtained in our work can be successfully explained only when all these criteria are
taken into account together.
Effect of the cluster nature and fullerene strain on the exohedral
reactivity of D3h-C78 endofullerene compounds. The reactivity of
Ti2C2@D3h-C78 and XN3@D3h-C78 (X = Sc, Y) metallofullerenes.
Previously in our group, Diels-Alder cycloaddition over all nonequivalent bonds of the free
fullerene cage D3h-C78, and its endohedral derivatives Sc3N@D3h-C78 and Y3N@D3h-C78 have
been studied.78,79 As the same isomer is used (D3h) in all different cases, any differences in
reactivity can be directly attributed to the nature of the cluster atom encapsulated inside the
cage, and the strain energy it causes.
80
RESULTS
The Diels-Alder reaction energies as well as the activation barriers for all species are reported
in Table 7 and Figure 37, Table 8 and Figure 38 respectively. As it can be seen there, in all cases
except for the Sc3N compound, most favorable cycloadditions took place over [5,6] bonds, in
contrast to C60 fullerene, where most reactive bonds are type A [6,6] bonds. As we have seen,
when metallic clusters are encapsulated inside the cage, in general the reaction becomes less
exothermic and reaction barriers increases. This is something expected because of the charge
transfer that takes place from the metallic cluster to the fullerene cage, which implies an
increase of the HOMO-LUMO gap (as we have seen in previous sections). Thus, there is a clear
general reduction in exohedral reactivity going from C78 free cage to encapsulated derivatives.
Between Ti2C2@C78 and previous studied systems, there exist many differences in reactivity,
mainly caused because of the metallic cluster shape and its orientation inside fullerene cage. In
most cases, it is found that the reactivity decreases from in the order free C78 > Sc3N@C78 >
Y3N@C78 (see for instance bonds 5, 7, b, c, and e). The effect of the Ti2C2 cluster it is in general
to decrease the reactivity as compared to the free C78. However, it does so in a different way.
For instance, as compared to Sc3N@C78 and Y3N@C78 EMFs, the Ti2C2 cluster decreases the
reactivity of the 4, 6, 7, and b bonds while for bonds 3, c, and f it makes them more reactive.
Starting from bond c, it is the worth noting that the presence of the Ti2C2 moiety reduces its
bond length and causes a very large increase of c bond reactivity, becoming one of the most
reactive bonds in this derivative while for the other EMFs cases was one of the less reactive
bonds. Interestingly, the gain of reactivity has a geometric origin (bond length reduction), and
electronic effects do not play a big role there, as confirmed by the low difference between
actual energy barrier and fullerene non-cluster energy barrier.
ΔER (kcal mol-1)
product
bond type
D3h-C78 a
Sc3N@D3h-C78 a
Y3N@D3h-C78 a,b
Ti2C2@D3h-C78
1
A [6,6]
-16.0
4.0
-5.4
-2.6
2
C [6,6]
12.5
6.1
6.6
4.2
3
B [6,6]
1.8
9.2
-0.9
-14.1
4
B [6,6]
-9.2
-9.7
-7.8
8.5
5
B [6,6]
0.9
5.9
8.7
1.6
6
B [6,6]
4.0
-12.7
-11.0
11.1
7
A [6,6]
-18.8
-7.6
-6.1
5.1
a
D [5,6]
-8.8
-6.0
-4.1
-7.1
b
D [5,6]
-23.9
-4.3
-1.9
3.4
c
D [5,6]
-13.3
-10.4
-6.1
-18.1
d
D [5,6]
-5.2
-7.6
-15.0
-6.8
e
D [5,6]
-13.8
-4.9
-4.1
-5.5
f
D [5,6]
-5.9
-6.5
-6.6
-14.6
Table 7. Reaction energies (ΔER, kcal mol-1) for the free cage and the endohedral derivatives. Bold values indicate
the bonds that are the most reactive. a Values from ref. 79. b Only results for the down region are reported here.
81
MASTER THESIS
D3h-C78
D3h-C78
15
Sc3N@D3h-C78
Sc3N@D3h-C78
Y3N@D3h-C78
Y3N@D3h-C78
Ti2C2@D3h-C78
Ti2C2@D3h-C78
10
E (kcal mol-1)
5
0
-5
-10
-15
-20
-25
A [6,6] C [6,6] B [6,6] B [6,6] B [6,6] B [6,6] A [6,6] D [5,6] D [5,6] D [5,6] D [5,6] D [5,6] D [5,6]
1
2
3
4
5
6
7
a
b
c
d
e
f
Figure 37. Comparison of the reaction energies (in kcal mol-1) found for the Diels-Alder reaction over the
nonequivalent bonds of the free D3h-C78 cage (dark blue),78 Sc3N@D3h-C78 (blue),78 Y3N@D3h-C78 (light blue),79 and
Ti2C2@D3h-C78 (grey-blue). Only results for the down region of Y3N@D3h-C78 are reported here.
ΔEⱡ (kcal mol-1)
product
bond type
D3h-C78a
Sc3N@D3h-C78 a
Y3N@D3h-C78 a,b
Ti2C2@D3h-C78
1
A [6,6]
12.2
23.8
20.1
23.6
2
C [6,6]
30.2
27.1
27.0
26.8
3
B [6,6]
21.7
28.9
27.1
18.2
4
B [6,6]
14.8
20.0
21.1
29.6
5
B [6,6]
14.4
27.6
30.2
20.6
6
B [6,6]
17.2
18.5
18.3
27.0
7
A [6,6]
13.5
20.1
20.6
27.5
a
D [5,6]
17.2
21.5
23.0
21.4
b
D [5,6]
12.5
20.7
23.1
26.5
c
D [5,6]
16.7
20.1
22.5
17.4
d
D [5,6]
22.1
19.7
17.1
22.6
e
D [5,6]
15.3
22.3
17.1
20.5
f
D [5,6]
18.0
21.5
21.9
19.3
Table 8. Reaction energies (ΔEⱡ, kcal mol-1) for the free cage and the endohedral derivatives. Bold values indicate
the bonds that are the most reactive. a Values from ref. 79. b Only results for the down region are reported here.
82
RESULTS
D3h-C78
D3h-C78
35
Sc3N@D3h-C78
Sc3N@D3h-C78
Y3N@D3h-C78
Y3N@D3h-C78
Ti2C2@D3h-C78
Ti2C2@D3h-C78
30
E (kcal mol-1)
25
20
15
10
5
0
A [6,6] C [6,6] B [6,6] B [6,6] B [6,6] B [6,6] A [6,6] D [5,6] D [5,6] D [5,6] D [5,6] D [5,6] D [5,6]
1
2
3
4
5
6
7
a
b
c
d
e
f
Figure 38. Comparison of the reaction barriers (in kcal mol -1) found for the Diels-Alder reaction over the
nonequivalent bonds of the free D3h-C78 cage (dark green),78 Sc3N@D3h-C78 (bright-green),78 Y3N@D3h-C78 (green),79
and Ti2C2@D3h-C78 (light-green). Only results for the down region of Y3N@D3h-C78 are reported here.
Bonds 4, 6 and 7 in the Ti2C2 derivative case increases their reaction barriers and have large
endothermic reaction energies in contrast to fullerene free cage and other studied endohedral
derivatives. We have to keep in mind that bond 6 is the one having the largest fullerene
deformation energy in Ti2C2@C78 compound, and bonds 4, b, and 7 are those which where
disadvantaged from the point of view of the deformation energy and difference in energy
barrier parameters. In contrast bonds c and f (and bond 3) where those most favored by these
two energy descriptors.
Therefore, more than because structural predisposition (small bond length and high
pyramidalization angles) of nonequivalent bonds the final reactivity of the different fullerene
derivatives is totally determined by the nature of the metallic cluster. Fullerene deformation
energies depend directly on the cluster shape, and its position and orientation inside the cage,
which determines the fullerene structure strain. And the presence of metallic ions close to one
determined zone inside of the fullerene cage, induces changes in fullerene electronic structure
(i.e. changes in LUMO molecular orbitals) which define the bond disposition to interact with
diene during the cycloaddition reaction. As we have seen, most reactive bonds are those which
are most favored by the metallic cluster presence, which plays a key role in the final
endofullerene chemistry.
83
MASTER THESIS
Inclusion of dispersion corrections: changes in the energy profile of
Diels-Alder reaction
Dispersion corrections change dramatically the energy profile of the Diels-Alder reactions
involving fullerene compounds, as our group have recently reported.119 Taking into account
dispersion corrections, we can reproduce better the reaction path that Diels-Alder reaction
follows (see Figure 39).
3
15
10
1
2
E (kcal/mol)
5
0
4
-5
-10
-15
-20
Reaction Coordinate
Figure 39. Energy profile of the Diels-Alder cycloaddition between Ti2C2@D3h-C78 and 1,3-cis-butadiene where are
represented: (1) reactants; (2) reactant complex structure; (3) transition state; and (4) product.
To reproduce the energy profile (Figure 39) we have to obtain the intermediate reactants
complex that is formed. First of all, we tried to obtain these complex structures (numbered as
2 in Figure 39) from a simple optimization procedure, as the same we have done for
characterization of reactants and products. The fact was that not all initial complexes could be
found using this procedure. Only when dispersion corrections are taken into account during
the optimization procedure all intermediates can be characterized correctly.
All reaction energies (from 1 to 4 in Figure 39 nomenclature) and reaction barriers (from 1 to
3) including dispersion corrections are presented in Table 9. There can also be found reaction
energies (from 2 to 4) and reaction barriers (from 2 to 3) when the initial complex formed is
taken into account.
Results in Table 9 show that inclusion of dispersion corrections lead to, in general, products
with an extra stabilization of about 11.2-13.5 kcal mol-1 with respect to reactants. Something
similar happens with reaction barriers, where the inclusion of dispersion corrections implies a
84
RESULTS
decrease of the barrier of about 11-13.5 kcal mol-1. This means TSs are much stabilized by
dispersion corrections than infinite separate reactants.
If we now take into account the existing interactions between the two reactants, that is the
initial intermediate complex formed, and compare reaction energies and barriers obtained
from this adduct, we can see that differences between the reaction energies and energy
barriers obtained with respect to the intermediate complex or separated reactants are not
very large. Differences in the reaction energies and barriers are comprised between 1.25 and
3.6 kcal mol-1, which is the stabilization energy that implies the formation of the initial
intermediate.
If we analyze the new order of different reactivities found for all nonequivalent bonds, we can
see that although the numbers change, the trend is always the same, obtaining the same
ordering in all three different reaction energies/reaction barriers compared.
Therefore, the inclusion of dispersion energy corrections is essential to correctly reproduce the
Diels-Alder reaction. If we want to compare theoretical predictions with experimental
observations, we must take them into account, as it is reported in the bibliography.119 But we
have also shown that although reaction barriers and reaction energies change, the reactivity
trend remains. This fact allows us to use the computed energies without dispersion corrections
to get an accurate prediction of fullerene reactivity. Finally, we have found that having into
account the initial complex formed by London dispersion interactions only causes a
stabilization of reactants of about 1.25-3.6 kcal mol-1, which are in the borderline of the
precision of the methodology.
Ti2C2@D3h-C78
product
bond type
ΔER
ΔER - D3
ΔER int. - D3
ΔEⱡ
ΔEⱡ - D3
ΔEⱡ int. - D3
1
A [6,6]
-2.6
-13.8
-12.6
23.6
10.9
12.1
2
C [6,6]
4.2
-8.6
-6.4
26.8
14.3
16.4
3
B [6,6]
-14.1
-26.8
-23.6
18.3
5.4
8.6
4
B [6,6]
8.5
-4.2
-0.8
29.7
16.4
19.8
5
B [6,6]
1.6
-10.8
-8.2
20.6
9.4
12.0
6
B [6,6]
11.1
-1.5
2.2
27.0
13.5
17.2
7
A [6,6]
5.1
-7.8
-5.6
27.5
15.7
17.9
a
D [5,6]
-7.1
-19.6
-16.9
21.4
9.1
11.8
b
D [5,6]
3.4
-8.9
-6.4
26.5
14.2
16.7
c
D [5,6]
-18.1
-29.3
-27.2
17.4
5.1
7.2
d
D [5,6]
-6.8
-19.0
-16.2
22.6
10.6
13.4
e
D [5,6]
-5.5
-19.0
-16.0
20.5
8.2
11.2
f
D [5,6]
-14.6
-28.0
-24.8
19.3
7.3
10.5
Table 9. Reaction energies (ΔER, kcal mol-1) and reaction barriers (ΔEⱡ, kcal mol-1) without and with energy
dispersion corrections (D3), and those ones corresponding to the reaction starting from first reactants complex
intermediate formed (int.).
85
MASTER THESIS
Final remarks and conclusions
The present study of the Diels-Alder reaction on Ti2C2@D3h-C78 has shown that most stable
regioisomers are obtained when the attack is produced over a type D [5,6] bonds c, f, and type
B [6,6] bond 3. These results could have not been predicted by the initial bond lengths and
pyramidalization angles of these bonds.
First of all, we correctly characterized the Ti2C2@D3h-C78 endofullerene, as it was not clear
which position C22- moiety adopts between the two titanium ions inside the fullerene cage.
After that, we tested over all nonequivalent bonds the regioselectivity of the Diels-Alder
cycloaddition with 1,3-cis-butadiene. Our results showed that the regioselectivity of this
exohedral addition is extremely modified by changing the nature of the cluster encapsulated
inside, as we have seen by comparing with previous studies. The preference for reacting with a
given bond is due to two different factors: first, fullerene deformation energy caused because
of the presence of metallic cluster inside, and reduction of the strain energy of the fullerene
cage by Diels-Alder attack on selected bond; second, changes induced by metallic cluster in
MOs (LUMOs) of fullerene cage that determines the final predisposition of different bonds to
interact with the HOMO of diene.
We find that, as compared to free C78 for which [6,6] 1 and 6 bonds and the [5,6] b bond are
the preferred addition sites, the Ti2C2 cluster favors the addition to the [6,6] 3 bond and the
[5,6] c and f bonds. The effect is similar to that found for the Sc3N cluster. In both clusters we
have similar encapsulation energies (not especially high indicating relative low fullerene cage
strain) and similar changes in reactivity, the most favored bonds being [5,6] bonds close to the
metals and [6,6] bonds far from the cluster. Indeed, the Sc3N cluster favors the reactivity of the
6 and d bonds, while the 3, c, and f additions sites are the preferred ones in Ti2C2@D3h-C78. The
change in reactivity is due to the different MOs of the C78 fullerenic cage that receive the
charge transfer from the metal cluster.
As a whole, compared to free C78, for Ti2C2@D3h-C78 we find:
1. Bond 1. This bond is quite reactive in free C78 and its reactivity is clearly reduced in the
Ti2C2@D3h-C78 EMF. The reason is not structural (bond length and pyramidalization
angles are similar) but electronic (change in the LUMOs shape and energy).
2. Bond 2. It is quite unreactive for all systems because of large C–C bond length and low
pyramidalization angles.
3. Bond 3. In free C78 this bond is not especially reactive. With the inclusion of the Ti2C2
cluster it becomes much more reactive. The charge transfer from the cluster reduces
its C–C bond length, while the strain slightly increases the pyramidalization angles.
Electronic effects are close to zero for this bond.
4. Bond 4. It is not particularly reactive in free C78 and it is clearly deactivated in
Ti2C2@D3h-C78. This is due in part to structural effects (bond length increases and
pyramidalization angles are reduced) but particularly to electronic effects.
5. Bond 5. Its reactivity is reduced in Ti2C2@D3h-C78 due to electronic effects.
6. Bond 6. It is reactive neither in free C78 nor in Ti2C2@D3h-C78. No LUMOs available with
the correct lobes signs for the Diels-Alder addition.
86
RESULTS
7. Bond a. It is quite unreactive for all systems because of the large C–C bond length.
8. Bond b. For free C78 this is a very reactive [5,6] bond. It is the [5,6] bond having the
shortest C–C bond length. Encapsulation of Ti2C2 reduces dramatically its reactivity due
to both structural (longer C–C bond length) and electronic effects.
9. Bond c. In free C78 is quite reactive. In Ti2C2@D3h-C78 this bond is shortened and has a
similar reactivity to that of free C78. It is the [5,6] bond in this EMF having the shortest
C–C bond length. In addition, the electronic effect is one of the smallest for all the
series.
10. Bond d. This bond with a relatively larger C–C bond length is unreactive in both free C78
and Ti2C2@D3h-C78. It becomes reactive in Y3N@D3h-C78 because of strain.
11. Bond e. This bond is not very reactive in free C78 and it is clearly deactivated in
Ti2C2@D3h-C78. This is not due to structural effects (bond length decrease and
pyramidalization angle increases) but to electronic effects.
12. Bond f. It is not reactive in free C78 and becomes one of the most reactive in Ti2C2@D3hC78 due to electronic effects. It is the only bond favored (energy barriers are reduced)
by the charge transfer from the metallic cluster.
Finally, we have studied the inclusion of dispersion corrections to the present cycloaddition
reaction. Only when dispersion corrections are taken into account, an initial reactant complex
is found, which implies a stabilization of reactants about 1-3 kcal mol-1, and respective changes
in barriers and reaction energies. We have also shown that dispersion corrections make
reaction barriers and reaction energies change, but the reactivity trend remains the same. This
fact allows us to use the computed energies without dispersion corrections to get an accurate
prediction of fullerene reactivity. However, if the objective is to compare theoretical
predictions with experimental observation values, we must include these corrections in order
to get the correct energy and barrier values.
87
MASTER THESIS
88
RESULTS
2. Fine-tunable metalloporphyrinic nanocages as hosts
for fullerene encapsulation
Introduction
The search of hosts for fullerenes is an active area of research due to their potential use for
purification of fullerenes, as we have seen in introductory chapters where some recent
examples are presented.93,95,96 M. Costas and X. Ribas’s group recently reported 3D trigonal CuII
coordination cages based on tricarboxylate linkers.133 Following this line, in this work we
present a DFT study of 3D cubic nanostructures based on the self-assembly of 5,10,15,20tetrakis-(4-carboxyphenyl)-porphyrins and macrocyclic di-copper(II) complexes. The size of the
cavity and the supramolecular affinity of fullerenes guest molecules towards porphyrin-based
hosts convert these superstructures as candidates for fullerene purification. We have been
tuning the dimensions, and electronic structure of the nano-cube by changing: i) the length of
the organic macrocycle (Me2m, Me2p, Me2pp which will be described later); ii) the transition
metal bound into the macrocycle (CuII, PdII); iii) and the transition metal coordinated to the
porphyrin moieties (none, PdII, MnIII, CaII, CoII, RhIII, MgII).
Whereas Me2p and largest Me2m nanovessels bear the proper inner void space for C20,
Me2pp nanocapsules allow the hosting of C60. Besides the role of the porphyrin transition
metal on the binding affinity between the nanovessels and the fullerenes, we have also
analysed the influence of the type of bond or ring of the fullerene that lies closer to the
metalloporphyrin ([6,6] bond and [5,6] bond taking into account two possible orientations, and
6 member ring). Finally, we have also investigated in detail the type of the host-guest
interaction using energy decomposition analysis (EDA) in order to understand which terms
determine the success of the host-guest process. The main goal of this computational study is
to obtain the necessary and useful information to be able to synthesize in the lab 3D
nanostructures which present high affinity for C60 substrate.
Figure 40. Representation of the Host-guest interaction between Pd_Pd_Me2pp nanocapsule and C60 fullerene.
89
MASTER THESIS
Computational Details
All density functional theory (DFT) calculations were performed with the Amsterdam Density
Functional (ADF) program.114,128 The molecular orbitals (MOs) were expanded in an
uncontracted set of Slater-type orbitals (STOs) of double-𝜉 (DZP) quality containing diffuse
functions and one set of polarization functions. Core electrons (1s for second period and
1s2s2p for third, fourth and fifth period) were not treated explicitly during the geometry
optimizations (frozen core approximation),114 because it was shown to have a negligible effect
on the obtained geometries.129 Scalar relativistic corrections were included self-consistently
using the zeroth-order regular approximation (ZORA).
An auxiliary set of s, p, d, f and g, STOs was used to fit the molecular density and to represent
the Coulomb and exchange potentials accurately for each self-consistent field cycle. Energies
and gradients were calculated using the local density approximation (Slater exchange and
Vosko-Wilk-Nusair correlation)104 with nonlocal corrections for exchange (Becke88)107 and
correlation (Perdew86)106 included self-consistently (i.e., through the use of the BP86
functional). But also another nonlocal correction for correlation (LYP)108 was used by the BLYP
GGA functional.
All of the energy analysis are obtained from single-point energy calculations with the TZP basis
at geometries that were obtained with the DZP basis (i.e., at the BLYP-D/TZP//BLYP-D/DZP
level).
All geometric optimizations were performed with the QUILD125 (Quantum-regions
Interconnected by Local Descriptions) program, which functions as a wrapper around the ADF
program. The QUILD program constructs all input files for ADF, runs ADF, and collects all
needed data; ADF is used only for the generation of the energy and gradients. Furthermore,
the QUILD program uses improved geometry-optimization techniques, such as adapted
delocalized coordinates and specially constructed model Hessians with the correct number of
eigenvalues.125 The latter is of particular use for TS searches.
Energy dispersion corrections are introduced using Grimme’s methodology115,117 (D2 and D3)
implemented in ADF 2010.01 version.128 All structures fully optimized using BLYP functional,
have introduced these corrections in each optimization step. Dispersion corrections for the
rest of structures (BP86 optimizations) are introduced by a single point calculation over the
stationary point structure found in a usual optimization procedure. We did not reoptimize all
structures as differences in structural parameters are minimal.119
Experimental crystallographic structures and DFT structures
As a master’s thesis project of the PhD. student Cristina García Simón, several 3D nanocages
structures have been recently synthesized in Ribas and Costas’s lab (UdG) by using
supramolecular chemistry self-assembly methods.2 These structures are based on 5,10,15,20-
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RESULTS
tetrakis-(4-carboxyphenyl)-porphyrins and different palladium- and copper-based macrocycle
molecular clips: Pd_Me2m, Cu_Me2m, Pd_Me2p, Cu_Me2p, Pd_Me2pp and Cu_Me2pp (see
Figure 41). Each nanocage consists in two parallel porphyrins linked by four macrocyclic
dinuclear metallic complexes. Each porphyrin’s carboxilate residue is linked by means of η1-O
monodentate coordination to one metallic center of the linker. Altogether, the overall
structure is defined as a tetragonal prismatic cage, where carboxyphenyl moieties act as
vertices of the polyhedron (see Figure 42).
Figure 41. Schematic representation of: (a) Cu_Me2m ligand; (b) Pd_Me2p ligand; (c) Pd_Me2pp ligand; and (d)
MnCl 5,10,15,20-tetrakis-(4-carboxyphenyl)-porphyrin.
Experimentally, they have synthesized five different nanocages: Pd_Cu_Me2p, Pd_Pd_Me2p,
Pd_Cu_Me2pp, Pd_Pd_Me2pp and MnCl_Cu_Me2pp (where first metal is referred to
porphyrins moieties, the second metal refers to the ligand used). The characterization by X-ray
diffraction has been possible only for the Pd_Pd_Me2pp nanostructure.
Figure 42. X-ray structures for (a) Pd_Pd_Me2p and (b) Pd_Cu_Me2pp synthesized nanocapsules.
91
MASTER THESIS
We have optimized at BP86/DZP level all of these structures, and some more which are about
to be synthesized (see Figure 43). In Table 10 there is a comparison between crystallographic
and DFT optimized structural parameters. We have chosen the metal-metal porphyrin distance
to be compared as it is a parameter which correctly describes the inner void cavity.
Dist. M-M (Å)
X-ray structure
DFT (BP86/DZP)
Pd_Cu_Me2m
-
7.05
Pd_Pd_Me2m
-
5.18
MnCl_Cu_Me2m
-
8.26
MnCl_Pd_Me2m
-
6.10
Pd_Cu_Me2p
7.02
8.11
Pd_Pd_Me2p
7.49
8.85
MnCl_Cu_Me2p
7.91
8.98
MnCl_Pd_Me2p
-
8.93
Pd_Cu_Me2pp
11.21
13.07
Pd_Pd_Me2pp
-
13.11
MnCl_Cu_Me2pp
-
13.00
MnCl_Pd_Me2pp
-
13.24
Table 10. Porphyrinic metal-metal distances for X-ray structures and DFT optimized at BP86/DZP level.
Figure 43. Schematic representation of the BP86/DZP optimized structures of: (a) Pd_Cu_Me2m; (b)
Pd_Cu_Me2p; and (c) Pd_Cu_Me2pp
92
RESULTS
As it can be seen, differences between solid and DFT structures are around a constant value of
1 Å in each case. For Pd_Cu_Me2pp case, the cage has almost twice the length of Pd_Cu_Me2p
cage, so the difference is also twice (about 2 Å). Even so, ordering of the different inner void
spaces remains equal for both structural types. The difference between the crystallographic
and DFT structures can be due to the more compact packing that solid crystal structures have
in comparison with gas-phase DFT structures. Moreover, as we can see in Figure 42, X-ray
structures are a bit distorted, but for the computed ones, we have imposed D4h symmetry
which prevents to suffer any of these distortions. Therefore, the optimized BP86/DZP
structures can be considered as a correct reproduction of the experimental synthesized
systems.
The metals used in the linkers have a small effect in the molecular structure, and it is even
smaller when the size of the cage increases. Metal-Metal distance for Pd_Cu_Me2p cage is
8.11 Å, and when the metals of the linker are changed from Cu to Pd, the distance increases to
8.85 Å, about 0.7 Å of difference (see Table 10). The major differences are found on Me2m
ligands, where the distances increases about 2 Å when the metals of the linker are changed
from Pd to Cu, showing that Me2m-based cages are more ligand dependent and flexible due to
the shape of the ligand. But when we focus on Pd_Cu_Me2pp and Pd_Pd_Me2pp
nanocapsules, the difference is reduced to 0.04 Å. So, the metal ligand effect over final
structural size and shape is minor. A different issue is the porphyrin metal effects on the
selective encapsulation of C60 fullerene, which will be center of the present study.
The main objective of this project is to design new nanocapsules which may be able to
encapsulate C60 fullerene, which diameter is about ~7 Å. This measure automatically discards
all Me2m and Me2p ligand-based nanocages, as they are not sufficient large to host this
molecule. Only C20 fullerene has the correct shape and size to be encapsulated by the Me2p
and larger Me2m nanocages. Therefore, we will be centered on Me2pp ligand-based
nanocages, which inner void space seems to be optimal to host C60 fullerene molecules.
Host-Guest studies: Effects of the fullerene orientation and porphyrin
metal ions
Once we have correctly characterized the synthesized nanocapsules structures, we are able to
study the host-guest interaction between M_M_Me2pp nanocages (where M refers to the
metallic ions) and C60 molecule.
We have studied 5 different orientations of C60 fullerene with respect porphyrininc cages, as
represented in Figure 44. Besides investigating which orientation is most favorable, we also
want to study the effect of the metal of the porphyrin in the host-guest interactions. We
expect that metal coordination with the fullerene will be an extra stabilization of the hostguest formed complex, in addition to the van der Waals and electrostatic interactions (see the
“4. Supramolecular Chemistry” section in the introductory chapters). Furthermore, we have
93
MASTER THESIS
also investigated the effect of different metallic porphyrin types. By changing the metal ions
present in the porphyrins we want to study which metals exhibit stronger interactions with the
fullerene compound, and that is, which will be the best option to encapsulate the C60
molecule. The different porphyrins chosen for the present work are: porph-Pd, porph-MnCl,
porph-Co, porph-RhCH3, porph-H, porph-Ca, porph-Mg, and [porph]2-.
Figure 44. Schematic representation of the different porphyrin-C60 orientations tested: two different orientations
on [6,6] and [5,6] bonds, and over 6 member ring.
First of all, we started to study the experimentally synthesized nanocages, and Pd_Pd_Me2pp
in particular. When the C60 molecule is included, the total number of atoms grows to more
than 660. That is a too large number of atoms, including different metal ions, to be treated
only with pure DFT quantum methods. However, thanks to the BSC computational resources
we expected to be able to study the total host-guest complex. But even using the BSC
computational resources, the increase of the computational cost when the total host-guest
complex is considered forced us to change the DFT level of computation. Then, we decided to
study the inclusion of C60 inside the capsules with the BLYP-D functional is used. BLYP
functional when the dispersion corrections are included has been described by Grimme and
co-workers117 as a good option to correctly describe fullerene complex with large π-stacking
interactions. Furthermore, for our huge host-guest systems, the computational cost of BLYP-D
calculations is far lower than BP86 functional calculations.
We have successfully optimized, the [6,6]_2 and [5,6]_2 orientations of the
C60@Pd_Pd_Me2pp complex at BLYP-D/DZP level, which will be described in the next
subsection.
94
RESULTS
Figure 45. Representation of the C60@Pd_Pd_Me2pp-C2h [5,6]_2 structure optimized at BLYP-D/DZP level.
The large computational requirements involved in these calculations and the related technical
difficulties appeared have forced us to model the total system. In the next section we will
present the model system which will be used in our study. We will show that this model offers
an accurate description of the total host-guest system.
Modeling the system: porphyrinic dimer
As we have explained, the big size of the present systems increases exponentially the difficulty
for treating them. In that sense, we have modeled the total system in order to decrease the
number of atoms to be included in the QM calculations reducing substantially their complexity.
We have optimized the total C60@Pd_Pd_Me2pp complex at BLYP-D/DZP level with [6,6]_2
and [5,6]_2 orientations of fullerene. These results are our starting point to compare with
those obtained for the proposed model system.
To define our proposed model system, we have eliminated all four linkers which form the
nanocage. Thus, we can exclude from the computation 100 atoms per linker, i.e. 400 atoms
including 8 metallic ions. Moreover, from each porphyrinic ring, we have also excluded from
the model system all the substituted aromatic rings (in total 4-carboxyphenyls per porphyrin).
In Figure 46 we present a schematic representation of the model system in comparison with
the total one. One could think that the model system is not able to correctly describe the
complete system behavior, but surprisingly, the results show the opposite.
In Table 11 there are presented the metal-fullerene distances (measured as represented in
Figure 45) for both complete and model systems. As it can be seen, differences between
complete and model optimized structures are minimal, less than 0.06 Å, indicating a very good
correspondence between the behaviors of both systems. Moreover, the computational effort
95
MASTER THESIS
to treat the model system in comparison with the total one is dramatically reduced. Thus, the
model gives us a good description of the complete system with a simplest model which can be
easily treated from the computational point of view.
Figure 46. Schematic representation of the total C60@Pd_Pd_Me2pp system treated (left) and our model
Pd_Me2pp system (right).
C60 - M (Å)
[6,6]_2
[55,6]_2
2.739
2.721
Pd_Pd_Me2pp
(2.799)
(2.763)
Table 11. Metal - C60 distances for Pd_Pd_Me2pp nanocapsule optimized at BLYP-D/DZP level. Model results are
in parenthesis.
Of course, that the host-guest interaction energies obtained with the proposed model are not
the total ones. Our model does not include the effect of the macrocycles and aryl rings of the
porphyrins. But obtain the total interaction energies is not our objective in this project. Our
aim is to study the effect of the metal of the porphyrin moieties on the host-guest interaction,
and which C60 orientation is more favorable. So, we can use the model interaction energies to
study these two effects. The results obtained for the compared geometries (Table 11) show
that our model correctly describes the interactions between porphyrinic dimer and the
fullerene molecule that occurs in the complete host-guest complex.
Therefore, the proposed model system is a good option to study the two different effects
proposed: fullerene orientation and porphyrinic metal change. So, in next section we will use
this model system to study both parameters.
96
RESULTS
Study of the fullerene orientation and porphyrin metal effects
In this section, we will study the effect of the porphyrin metal change and fullerene orientation
on the formation of the final host-guest complex. As we said before, we will use our proposed
model system in order to reduce the high computation requirements.
C60 - M (Å)
[6,6]_1
[6,6]_2
[5,6]_1
[55,6]_2
Ca_Ca (M)
Ca_Ca (Porph)
H_H
Mg_Mg
Pd_Pd
Pd_RhCH3 (Pd)
Pd_RhCH3 (Rh)
[Porph]2-_ [Porph]2RhCH3_RhCH3
Co_Co
Co_Pd (Co)
Co_Pd (Pd)
MnCl_MnCl
3.502
2.594
2.778
2.599
2.834
3.316
2.474
2.567
2.476
2.315
2.312
2.828
2.978
3.420
2.509
2.653
2.717
2.799
2.776
2.739
2.440
2.379
2.200
2.196
2.795
2.919
3.582
2.680
2.740
2.848
2.815
2.806
2.626
2.691
2.614
2.498
2.518
2.813
3.019
3.454
2.541
2.643
2.739
2.763
2.524
2.749
2.488
2.519
2.374
2.378
2.754
2.935
[ring]
3.042
3.055
Table 12. Metal-C60 distances (Å) for all model systems studied taking into account fullerene orientation inside
the host-guest complex. For mixed systems (i.e. different porphyrins used) metallic ion taken as a reference is
indicated in parenthesis. Structural optimization performed at BLYP-D/DZP level.
Table 12 lists the metal – C60 distances found for all the different systems studied. There are
the systems containing the eight different porphyrins previously described, and two additional
ones which are constructed by combining two different porphyrin moieties, such as Co_Pd and
Pd_RhCH3 systems (where each metal is referred to one porphyrin respectively). Only two ring
orientations have been obtained (see Figure 44), Pd_Pd and MnCl_MnCl. For the rest of
different porphyrin type cases, during the optimization process the fullerene molecule rotates
loosing ring orientation in favor of another one.
Figure 47. Schematic representation of the C60@Ca_Ca structure optimized at BLYP-D/DZP level, where calcium
ions goes out of porphyrin planes.
When the porphyrin moieties includes inside them Ca+2 ions, the final conformations obtained
are those with the metal ions placed outside of the porphyrin rings planes giving the
Ca2+[Porph]2- adduct (see Figure 47). So, in this case the measurement of the fullerene-metal
distance loses sense as a descriptor of the inner void space of the cage. Then, in this particular
97
MASTER THESIS
case we took as a reference the center of the four nitrogen atoms of the porphyrin ring instead
of the metallic ion.
If we compare the distance between the two manganese atoms in the porphyrin dimer
present in the optimized [6,6]_2 complex C60@MnCl_MnCl, which is 12.84 Å, and for the
isolated MnCl_Cu_Me2pp nanocage (Table 10, 13.00 Å), we can see that differences are
minimal (about ~0.15 Å) confirming the optimal structural conditions of the cage to host the
C60 molecule inside.
From C60-M distances reported in Table 12, there can be seen that largest distances are found
for the over ring orientation of Pd_Pd and MnCl_MnCl systems. But is the latter, MnCl_MnCl
system, which has the largest distances independent from C60 orientation. On the opposite
case we found C60@Co_Co systems, which are those that have the shortest C60-M distances.
One special case is the rhodium system, which clearly exhibit shorter distances for the [6,6]
orientations than for [5,6] ones. In general, [6,6]_2 and [5,6]_2 C60 orientations exhibit slightly
lower C60-M distances than [6,6]_1 and [5,6]_1 orientations, respectively.
Table 13 results show that the most favorable host-guest complexes are obtained with
manganese, cobalt and rhodium porphyrins. The latter metal, rhodium, was expected to have
good association with the fullerene molecule as it is reported by Aida and co workers in similar
porphyrin-based systems.95 The M-C60 distance, then, it is not here always a good criterion to
determine the strength of the interaction, since for the Mn metalloporphyrin presents a quite
large distance with C60. The preferred orientation of the fullerene inside these three complexes
are the [6,6]_2. In general, most energetically favorable orientations coincide with those which
have the shortest metal-fullerene distance. But the comparison between Table 12 and Table
13 results show that is not a rule that is always fulfilled. From the total collection studied,
C60@MnCl_MnCl complex is the energetically most stable, showing that the best host-guest
interactions take place between fullerene and Mn ions.
The worst option that we have found as C60 host is the one involving two charged porphyrins.
Electrostatic repulsions between these two charged molecules seem to be large and they could
destabilize the final formation of the host-guest complex. If we compare the C60@[Porph]2_[Porph]2- interaction energies with those found for the C60@Ca_Ca system, we see that the
inclusion of the Ca2+ ions as contra-ions is determinant: from ~+5 kcal mol-1 to -70 kcal mol-1
(see Table 12).
ΔEint. (kcal mol-1)
[6,6]_1
[6,6]_2
[5,6]_1
[5,6]_2
[ring]
Ca_Ca
-70.1
-71.6
-70.7
-71.6
H_H
-63.6
-62.1
-62.0
-62.0
Mg_Mg
-60.3
-64.1
-60.6
-63.0
Pd_Pd
-53.8
-55.1
-57.4
-59.3
-59.2
Pd_Rh
-65.4
-66.0
-61.8
-64.8
[Porph]2-_ [Porph]29.7
7.8
5.7
5.1
Rh_Rh
-71.3
-75.6
-66.3
-70.1
Co_Co
-72.7
-79.1
-66.4
-73.4
Co_Pd
-63.9
-63.4
-61.7
-66.0
MnCl_MnCl
-82.2
-84.2
-80.4
-82.6
-78.3
Table 13. Interaction energies (kcal mol-1) for all model systems studied taking into account fullerene orientation
inside the host-guest complex. Structural optimization performed at BLYP-D/DZP level.
98
RESULTS
From preliminary experimental results,2 we expected to find a good interaction between Pd
porphyrins and C60 molecule. But the results obtained, although they are not bad, are not as
good as we expected. Interaction energies are quite poor compared with C60@MnCl_MnCl
case, about 20-25 kcal mol-1 lower (see H-H results in Table 13). In the next section, by using
the Energy Decomposition Analysis tools we will try to understand the different interaction
which takes place when Pd atoms are present in the system.
The studied mixed systems, Co_Pd and Pd_RhCH3, also show this decrease of the interaction
energies that implies having palladium porphyrin in the system. Both cases exhibit lower
interaction energies that their equivalent single-metal systems. This fact is directly derived
from the presence of palladium phorphyrin in the system, as the metal-fullerene distance also
shows. Focusing on the later parameter, M-C60 distances, we can see that each metal
maintains his optimal distance which is found for the single-metal systems. There is only one
exception, the [6,6]_1 Pd_RhCH3, where Pd-C60 distance is quite large because of the
displacement of the Pd porphyrin respect the fullerene molecule. In this case, placing the
palladium atom over the six-membered ring of the fullerene instead of remaining on the [6,6]
bond is preferred; and we could not optimize the structure with the two porphyrins eclipsed
and completely parallel. So, the present structure has not the shape to correctly describe the
total real system. Fortunately, this is the only case where we had this problem.
The last two systems investigated were those which contain Mg porphyrins and simple nonmetallated porphyrins. These two systems don’t exhibit large interaction energies, and their
M-C60 lengths oscillate in the same range.
As we have seen, differences between fullerene orientation isomers exist but in terms of
energy and metal-fullerene distance, are small. In general, for one particular system these
interaction energies are in a 5 kcal mol-1 range. So that means that at room temperature
fullerene can freely rotate inside the nanocage, instead of being in a fixed position. That is in
correspondence with the different M-C60 distances found for one system depending of the
fullerene orientation. In general they are around 0.2 Å of differences, a very small value.
In solution nanocapsules are vibrating and suffering constant conformational changes, which
modify the metal distance respect the encapsulated C60. Therefore, the supposition of having a
dynamic host-guest interaction, which consists in an interconversion between the different
orientational isomers, becomes strong.
In the next section, we will analyze these interaction energies in the sense of EDA tool. We will
try to explain the differences found for different metals and to understand why one metal is
better than another. This information will be very useful in order to achieve our final goal: be
able to design a specific nanocage to selectively host a C60 molecule.
99
MASTER THESIS
Energy Decomposition Analysis applied to the study of host-guest interactions
The interaction between the porphyrin dimer with C60 fullerene has been studied by using
Energy Decomposition Analysis (EDA) tools. All the modeled systems have been treated with
this methodology, but until now, we have not been able to compute the EA of structures
containing manganese ions because of problems with non-paired electrons and SCF
convergences.
We have to take into account that all fragments used to define the final interacting complex
must be defined as restricted closed-shell. This may be a problem because most of the studied
systems have metallic ions, which present unpaired electrons in the calculations. Because of
that, we have to prepare the fragment inputs carefully, giving the correct electron occupations
to the orbitals, although they are defined as restricted fragments.
All EDA values computed are presented in Table 14. There we can find the total interaction
energies (ΔEint), all different terms in which are decomposed this interaction energy: Pauli,
Electrostatic, Orbital Interaction and Dispersion terms (EPauli , EElstat, EO.int, ED2 and ED3),
respectively. We have also computed the different geometric deformation energies for both
porphyrin and C60 molecules.
The interaction between the two porphyrins when they are in the final optimized position, i.e.
giving the porphyrin model dimer (see schematic representation in Figure 48) has also been
analyzed. And finally, we have added the C60 molecule inside the porphyrin dimer to get the
final model. Therefore, we have analyzed two different interaction energies: first, dimer
formation; and second, final host-guest interactions between our model and C60 fullerene.
Figure 48. Schematic representation of the fragments used in EDA analysis: (1) each isolated porphyrin is
computed; (2) interaction between the two porphyrins in the position they have in the final system are
calculated; (3) C60 molecule is prepared as a new fragment; finally, (4) interaction energy between porphirin
dimer and C60 molecule is computed.
100
RESULTS
System
ΔEint
-71.6
-71.6
Ca_Ca
-70.7
-70.1
-64.1
-63.0
Mg_Mg
-60.6
-60.3
-62.1
-62.0
H_H
-62.0
-63.6
7.8
[Porph]2_[Porpf]2-
5.1
5.7
9.7
-55.1
-59.3
Pd_Pd
-57.4
-59.2
-53.8
-66.0
Pd
_RhCH3
-64.8
-61.8
-65.4
-75.6
RhCH3
_RhCH3
-70.1
-66.3
-71.3
-79.1
-73.4
Co_Co
-66.4
-72.7
EPauli
0.0
117.5
0.0
119.1
0.0
114.7
0.0
111.3
0.0
110.9
0.0
108.7
0.0
103.0
0.0
100.0
0.0
91.5
0.0
94.9
0.0
99.6
0.0
107.9
0.0
113.9
0.0
104.7
0.0
113.3
0.0
102.5
0.0
110.5
0.0
122.0
0.0
111.1
0.0
93.3
0.0
107.7
0.0
178.9
0.0
153.2
0.0
131.1
0.0
146.7
0.0
232.5
0.0
182.4
0.0
152.7
0.0
197.5
0.0
251.0
0.0
224.1
0.0
179.4
0.0
207.6
EElstat
0.1
-48.8
0.1
-49.2
0.1
-47.6
0.1
-46.3
-0.1
-48.0
-0.1
-45.7
-0.1
-42.7
-0.1
-43.0
0.0
-37.2
0.0
-38.1
-0.1
-41.0
-0.1
-43.6
104.9
-51.1
104.0
-47.2
103.8
-50.5
103.7
-45.1
-0.1
-53.2
-0.1
-58.2
-0.1
-52.7
-0.1
-41.1
-0.1
-51.4
-0.1
-91.3
-0.1
-74.4
-0.1
-62.8
-0.1
-72.1
-0.1
-122.9
-0.1
-89.6
-0.1
-73.7
-0.1
-101.7
-0.1
-135.9
-0.1
-112.9
-0.1
-88.4
-0.1
-109.3
EDA values
EPauli+Elstat
EO.int
0.2
0.1
68.7
-25.3
0.1
0.1
70.0
-26.0
0.1
0.0
67.2
-24.6
0.1
0.0
64.9
-21.8
-0.1
0.0
62.9
-26.2
-0.1
0.0
63.1
-26.5
-0.1
0.1
60.3
-23.7
-0.1
0.1
57.0
-21.8
0.0
0.0
54.3
-17.9
0.0
0.0
56.8
-20.9
0.0
0.0
58.6
-22.7
-0.1
0.0
64.3
-23.4
104.9
-0.6
62.8
-53.8
104.0
-0.6
57.5
-54.8
103.8
-0.6
62.8
-55.2
103.7
-0.6
57.4
-47.0
-0.1
0.1
57.3
-22.1
-0.1
0.1
63.9
-27.7
-0.1
0.1
58.5
-23.6
-0.1
0.1
52.1
-24.7
-0.1
0.1
56.2
-20.5
-0.1
0.1
87.6
-49.4
-0.1
0.1
78.9
-40.8
-0.1
0.1
68.4
-32.3
-0.1
0.1
74.6
-38.6
-0.1
0.1
109.6
-74.0
-0.1
0.1
92.8
-53.7
-0.1
0.1
79.0
-41.7
-0.1
0.1
95.7
-60.0
-0.1
-4.1
115.1
-89.3
-0.1
-9.4
111.2
-83.2
-0.1
-12.0
91.1
-67.0
-0.1
-16.0
98.3
-70.1
101
ED2
-0.1
-84.3
-0.1
-85.0
-0.1
-83.5
-0.1
-83.5
-0.1
-78.7
-0.1
-78.1
-0.1
-76.5
-0.1
-75.3
-0.1
-71.8
-0.1
-71.2
-0.1
-74.5
-0.1
-77.2
-0.1
-74.7
-0.1
-70.6
-0.1
-75.5
-0.1
-74.8
-0.1
-71.2
-0.1
-74.9
-0.1
-73.1
-0.1
-75.8
-0.1
-72.0
-0.1
-83.1
-0.1
-81.6
-0.1
-77.8
-0.1
-81.6
-0.1
-89.6
-0.1
-87.5
-0.1
-82.7
-0.1
-86.3
-0.1
-88.7
-0.1
-85.5
-0.1
-79.7
-0.1
-84.2
ED3
-0.1
-73.2
-0.1
-73.9
-0.1
-74.1
-0.1
-73.5
-0.1
-68.4
-0.1
-67.4
-0.1
-66.7
-0.1
-66.3
-0.1
-66.3
-0.1
-65.5
-0.1
-68.1
-0.1
-70.2
-0.1
-67.3
-0.1
-64.2
-0.1
-68.5
-0.1
-68.1
-0.1
-64.7
-0.1
-66.4
-0.1
-65.4
-0.1
-65.1
-0.1
-65.1
-0.1
-70.8
-0.1
-69.4
-0.1
-67.6
-0.1
-68.3
-0.1
-72.9
-0.1
-71.8
-0.1
-68.5
-0.1
-71.7
-0.1
-72.1
-0.1
-70.7
-0.1
-68.8
-0.1
-70.6
ETotal-D2
0.1
-41.0
0.1
-41.0
0.0
-40.9
0.0
-40.4
-0.2
-41.9
-0.2
-41.6
-0.2
-39.9
-0.2
-40.1
-0.1
-35.5
-0.1
-35.4
-0.1
-38.6
-0.1
-36.3
104.2
-65.7
103.3
-67.9
103.1
-67.9
103.0
-64.4
-0.1
-36.0
-0.1
-38.6
-0.1
-38.2
-0.1
-36.3
-0.1
-36.3
-0.1
-44.9
-0.1
-43.5
-0.1
-41.7
-0.2
-45.5
-0.2
-54.0
-0.2
-48.4
-0.2
-45.3
-0.2
-50.6
-4.3
-62.8
-9.6
-57.6
-12.1
-55.6
-16.2
-56.1
ETotal-D3
0.1
-29.9
0.1
-29.9
0.1
-31.5
0.1
-30.4
-0.1
-31.6
-0.1
-30.8
-0.1
-30.1
-0.1
-31.0
-0.1
-30.0
-0.1
-29.6
-0.1
-32.2
-0.1
-29.3
104.2
-58.4
103.4
-61.5
103.1
-60.9
103.0
-57.7
-0.1
-29.4
-0.1
-30.2
-0.1
-30.5
-0.1
-29.5
-0.1
-29.5
-0.1
-32.6
-0.1
-31.3
-0.1
-31.5
-0.1
-32.2
-0.1
-37.2
-0.1
-32.7
-0.1
-31.1
-0.1
-35.9
-4.3
-46.2
-9.6
-42.7
-12.1
-44.7
-16.2
-42.4
Edef.
0.2
0.8
0.6
0.0
1.1
0.6
0.8
0.7
2.1
0.7
2.5
0.2
2.7
0.4
2.5
0.7
0.8
0.4
1.1
0.0
1.7
0.3
1.4
0.7
0.7
0.2
0.8
0.5
2.1
0.3
1.9
0.3
1.6
1.0
1.9
0.4
2.4
0.7
2.4
1.3
2.2
0.2
2.1
0.6
1.8
0.6
2.3
0.3
2.5
0.3
2.1
0.8
1.9
0.4
2.1
0.1
2.6
0.6
3.9
1.1
2.1
0.0
2.1
0.2
3.5
0.9
[6,6]2
[5,6]2
[5,6]1
[6,6] 1
[6,6]2
[5,6]2
[5,6] 1
[6,6] 1
[6,6]2
[5,6]2
[5,6] 1
[6,6] 1
[6,6]2
[5,6]2
[5,6] 1
[6,6] 1
[6,6]2
[5,6]2
[5,6] 1
[ring]
[6,6] 1
[6,6]2
[5,6]2
[5,6] 1
[6,6] 1
[6,6]2
[5,6]2
[5,6] 1
[6,6] 1
[6,6]2
[5,6]2
[5,6] 1
[6,6] 1
MASTER THESIS
-63.4
-66.0
Co_Pd
-61.7
-63.9
System
ΔEint
0.0
122.8
0.0
173.6
0.0
129.8
0.0
159.1
EPauli
-0.1
-57.9
-0.1
-85.9
-0.1
-62.6
-0.1
-81.3
EElstat
-0.1
64.9
-0.1
87.8
-0.1
67.2
-0.1
77.8
EPauli+Elstat
2.2
-30.8
-2.6
-55.2
2.2
-33.6
-8.0
-45.7
EO.int
-0.1
-78.0
-0.1
-80.6
-0.1
-76.2
-0.1
-78.4
ED2
-0.1
-66.7
-0.1
-68.9
-0.1
-67.0
-0.1
-68.1
ED3
2.0
-43.8
-2.7
-48.1
2.0
-42.7
-8.2
-46.3
ETotal-D2
2.0
-32.5
-2.7
-36.3
2.1
-33.5
-8.1
-36.0
ETotal-D3
2.5
0.0
2.1
0.6
2.4
0.3
3.5
0.9
Edef.
[6,6]2
[5,6]2
[5,6] 1
[6,6] 1
Table 14. Energy Decomposition Analysis (kcal mol-1) for all model systems studied. Total interaction energies for
all isomers (ΔEint, kcal mol-1, computed at BLYP-D/DZP level), and all interaction terms (Pauli, Electrostatic, Orbital
Interactions, and Grimme’s dispersion D2 and D3, in kcal mol-1 and computed at BLYP-D/TZP level) are given. In
white-lines, the obtained values for porphyrin dimmers are listed. In grey-lines, are showed values for the total
systems (interactions between porphyrins and C60). Deformation energies (Edef., kcal mol-1) presented for each
isomer refers to the deformation cost for each porphyrin from equilibrium geometry to the final in the present
system (first value), and for the C60 fullerene (second value).
First of all, we will analyze the results obtained for the two most favorable systems (as MnCl
results are not yet available): C60@RhCH3_Rh_CH3 and C60@Co_Co. As it can be seen in Table
14, these systems present the highest values for the orbital interaction terms. They also have
large destabilizing values for the Pauli terms, which can be understand as a steric repulsion,
and are mainly caused by the presence of occupied metal d-orbitals which points to occupied
MOs of the fullerene cage.
The opposite case is found in palladium porphyrin-based systems. They have very low orbital
interaction term values, which agree with the total interaction energies found and discussed in
the latter section. Their orbital interaction values are about 40-60 kcal mol-1 lower than those
found for cobalt and rhodium systems. That is a very large difference, and shows the noncoordination capacity of palladium in these host-guest systems. The sum of their electrostatic
and Pauli terms is close to the ones found for rhodium and cobalt systems. Therefore, as it was
expected, it is found that orbital interaction term due to the coordination of fullerene by the
metal is the most important to define the host-guest interactions in the studied systems.
The low capacity of palladium porphyrins to coordinate the C60 molecule is also showed by the
mixed study cases. Both Pd_RhCH3 and Pd_Co systems present much low orbital interaction
values than single-metal Co and Rh systems. Comparing the orbital interaction terms, we can
see how they have decreased an average of 20 kcal mol-1 due to the substitution of one Co or
RhCH3 metallic porphyrin, respectively, for a palladium one. A schematic representation of the
Co metal coordination and Pd non-coordination can be seen in Figure 49.
Focusing now on charged porphyrins system, it is found that electrostatic terms are not as
favorable as they were in the cobalt, rhodium, or even in palladium case. The [porph]2- dimer is
the only one which is found to have very high electrostatic repulsion energies. This repulsion is
the main cause of the high unfavorable interaction energy found for this dimer formation.
Orbital interaction energies are quite large, which means that lone pair electrons disposed
over nitrogen atoms in the porphyrin charged moieties can effectively interact with the C60
LUMOs. This point is corroborated by the inclusion of calcium(II) contraions, in Ca_Ca systems,
where orbital interaction terms are 50% smaller. This is because of the presence of the positive
charge outside the porphyrins (as we saw in the previous section) which implies a new
102
RESULTS
electronic organization of the porphyrin moiety. Similar results are found for Mg_Mg studied
models.
H_H model systems have shown the lowest Pauli repulsion energy values. We have to keep in
mind that Pauli energy terms, usually are interpreted as a descriptor of the steric repulsions
present in the system. So, in this study, the results found are according to the chemical
intuition which tell us that hydrogen atoms are quite smaller and don’t have big electron cloud
around it as metallic ions have, causing less steric repulsion. Indeed, the low steric repulsion of
the H_H structure is the key fact to get a better interaction energy with the fullerene than
Pd_Pd case.
In general, for most favorable cases studied (i.e. Rh and Co cases), [6,6] orientations are more
favored and have highest orbital interaction values than [5,6] ones. Moreover, [6,6]_2 and
[5,6]_2 orbital interaction values are higher than [6,6]_1 and [5,6]_1 respectively. These
observations give us an idea about how is the coordination between the metal and fullerene
molecule.
Figure 49. Representation of an example of molecular orbital (MOs, isosurface value 0.02 au) interaction by metal
coordination of the C60 molecule. The present system is the [5,6]_1 C60@Co_Pd, where coordination interactions
are only found for the partially occupied MO of Porph-Co moiety and occupied MO of C60.
Deformation energies described in Table 14, show which is the energetic cost to adapt
porphyrin and fullerene geometries to the final ones. As it can be seen, deformation energy
values are quite small, for C60 always less than 1.1 kcal mol-1 and 3.9 kcal mol-1 for porphyrins.
103
MASTER THESIS
These small values are in agreement with the hypothesis of the geometric complementarity
between metallic porphyrins and C60 molecule.
Besides, we have computed two different dispersion energies: Grimme’s D2 and D3.115,117 D3
dispersion corrections are about ~10 kcal mol-1 lower than D2 corrections. D2 energy
dispersion corrections take values between 71 and 89 kcal mol-1, and D3 between 64 and 74
kcal mol-1.Therefore, the total energies obtained by using these two different corrections
reproduce the same trend in studied systems series, although the final number values
changes.
Final remarks and conclusions
A study of the host-guest interactions due to the encapsulation of C60 fullerene inside a
metalloporphyrin-based 3D-nanostructure has been carried out. Our results show that
manganese, rhodium and cobalt metallic porphyrins are those which interact most favorably
with C60 molecule.
First of all, we correctly reproduced at DFT level the X-ray structures obtained from
synthesized nanocages in the lab. Small differences were found, but we attribute them to the
higher symmetry we have in the calculations and the high packing that crystallographic
structures have. Taking into account the inner void space measures found, we discarded all
Me2p and Me2m ligand-based nanocages as possible hosts for C60 molecule. On the other
hand, we have seen that all Me2pp ligand-based nanostructures were good candidates to
encapsulate C60 fullerene. After that, we started the host-guest interaction studies between all
different metalloporhyrin nanocages proposed and C60. We found several difficulties to
compute at DFT level the whole C60@M_M_Me2pp systems, related with the total number of
atoms which are present in the system (more than 660 atoms, some of them metallic ions). So
we proposed a model system in order to study the differences in the nanovessel-C60
interaction energy when the metal of the metalloporphyrin is changed and C60 molecule has
different orientations. This model consisted in a porphyrin dimer which has the fullerene inside
as the equivalent in nanocages. Our results showed that the model system correctly describes
coordination distances between the metal and C60 for the C60@Pd_Pd_Me2pp nanocapsule.
Based on this proposed model, we have done a complete study of the effects on the hostguest interactions due to the fullerene orientation inside the nanocage and metal ion of the
porphyrin moieties. Energy Decomposition Analysis (EDA) was used to complete the analysis.
Results found, showed that most favorable interactions take place in systems which metals
have large capacity to coordinate the C60 fullerene, such are Mn, Rh and Co. A priori we
expected to find good interactions between palladium-based systems and C60, but our
calculations showed that Pd_Pd_Me2pp is not the best option to host C60 molecule as its
interaction orbital energy term is quite poor.
104
RESULTS
Moreover, energy differences found for fullerene orientational isomers in host-guest
complexes (C60@M_M) were relatively small (about 1-9 kcal mol-1), which may indicate that at
room temperature fullerene can rotate inside the nanocapsule instead of being in a fixed
position in the host-guest complex.
From the information obtained in this work, new synthetic and host-guest experiments are
proposed. The new nanovessels suggested by our theoretical calculations will be synthesized in
the aim of accomplish the major and final goal of the project: selective encapsulation of C60 by
host-guest interactions by metalloporphyrin-based 3D nanostructures.
105
MASTER THESIS
106
CONCLUSIONS
CHAPTER V.
CONCLUSIONS
The most important conclusions obtained from the two projects involving fullerene
compounds and presented in this work, have already been described in previous chapters.
However, they will be briefly summarized here.
Exohedral reactivity of Ti2C2@C78 endofullerene: Diels-Alder addition on all non-equivalent
bonds:
We have studied the Diels-Alder cycloaddition of 1,3-cis-butadiene over all nonequivalent
bonds on Ti2C2@D3h-C78 endohedral metallofullerene. Results showed that most stable
regioisomers are obtained when the attack is produced over a type D [5,6] bonds c and f, and
type B [6,6] bond 3. However, it was not expected to find this high reactivity for these bonds as
they bonds length and pyramidalization angles are not the usual ones (short lengths, large
pyramidalization angles) for the most reactive sites.
We have described the geometric structure of the inner Ti2C2 cluster, and we have found that
regioselectivity of this cycloaddition reaction is extremely modified by the nature of the
encapsulated cluster. By comparing with previous studies and from the analysis performed, it
is found that the preference for reacting over one bond is determined by two major factors:
first, fullerene deformation energy caused by the presence of a metallic cluster inside, and
attack over selected bonds reduces the strain energy of the cage; second, changes in MOs
(LUMOs) of fullerene cage induced by the metallic cluster determines the final predisposition
of different bonds to interact with the HOMO of diene.
Finally, we have studied the effect of the inclusion of dispersion corrections in the present
reaction. These dispersion corrections make reaction barriers and reaction energies values
change, but the reactivity trend remains constant. This fact allows us to use the previously
computed energies without dispersion corrections to get an accurate prediction of fullerene
reactivity. However, if the objective is to compare theoretical predictions with experimental
observation values, we must include these corrections in order to characterize the initial
reactant complex (which implies a stabilization of reactants about 1-3 kcal mol-1) to get the
correct energy and barrier values.
107
MASTER THESIS
Fine-tunable metalloporphyrinic nanocages as hosts for fullerene encapsulation:
We have correctly reproduced at DFT level the X-ray structures obtained from synthesized
nanocages in the lab, and moreover, we have predicted the geometry for some nanocages
which crystal structure could not be obtained. From the inner void space measures, we have
discarded all Me2p and Me2m ligand-based nanocages as possible hosts for C60 molecule.
However, we have seen that all Me2pp ligand-based nanostructures are good candidates to
encapsulate C60 fullerene.
Then, we performed a host-guest interaction studies between all different metalloporhyrin
nanocages proposed and C60. Despite of use BSC computational resources, we found several
difficulties to compute at DFT level the total C60@M_M_Me2pp systems, related with the total
number of atoms that are present in the system (more than 660 atoms, some of them metallic
ions), which induced us to propose a model system. This model consisted in a porphyrin dimer
which has the fullerene inside as the equivalent in nanocages. Our results showed that the
model system correctly describes the coordination distances between the metal and C60
molecule. We have done a complete study of the effects on the host-guest interactions due to
the fullerene orientation inside the nanocage and metal ion of the porphyrin moieties by using
the model system.
Energy Decomposition Analysis (EDA) was used to analyse the obtained results. The EDA
analysis showed that most favorable interactions take place in systems which metals have
large capacity to coordinate the C60 fullerene, such are Mn, Rh and Co. Our calculations
showed that Pd_Pd_Me2pp is indeed a bad option to host C60 molecule as its interaction
orbital energy terms are quite poor, contrary to what we expected initially. Moreover, energy
differences found for fullerene orientational isomers in host-guest complexes (C60@M_M)
were relatively small (about 1-9 kcal mol-1), which may indicate that at room temperature
fullerene can rotate inside the nanocapsule instead of being in a fixed position of the hostguest complex.
108
ACKNOWLEDGEMENTS
CHAPTER VI.
ACKNOWLEDGEMENTS
We acknowledge the computer resources, technical expertise, and assistance provided by the
Barcelona Supercomputing Center (Centro Nacional de Supercomputación). Excellent service
by the Centre de Serveis Científics i Acadèmics de Catalunya (CESCA) is also gratefully
acknowledged.
Agraïments
Aquest treball posa punt i final al Màster Interuniversitari en Química Teòrica i Computacional
que durant el darrer any he cursat. Ha estat un any llarg, i perquè no reconèixer-ho, molt dur.
Primer de tot m’agradaria donar les gràcies per tot a en Josep Maria i a en Miquel. M’han ofert
la possibilitat de tenir un finançament durant tots aquests mesos, així com la possibilitat de fer
una petita estada al grup de’n Manthos Papadopoulos al National Helenic Research Fundation,
on tot i només estar-hi un mes vaig poder aprendre moltíssim amb en Heribert!! Moltes
gràcies per tot! Estic segur que us podré tornar tota la confiança que m’heu i m’esteu
demostrant cada dia en forma de treball i esforç durant els anys que duri la meva tesi.
A en Marcel i a la Sílvia. Mira que puc arribar a ser pesat i mai m’heu posat mala cara per cap
mail ni per cap pregunta, per més repetitiu que pogués ser. Durant aquest primer any he pogut
aprendre moltíssim dels dos, gràcies per la vostra paciència..!!
A la Carme i a en Dani, per “robar-me” galetes ;) !! I perquè sense ells, no sé com estaria
l’IQC..! Gràcies per tots aquests moment.. i els que vindran! A tots els companys del 177 (ara ja
molt pocs!) i a tots els del Parc, amb els que durant aquest últim mes hem compartit les
“vacances d’estiu”. Perquè treballar al vostre costat, dóna gust!! Els partits de bàsquet, els de
futbol, els sopars, els congressos, ....! A tots els formeu part de l’IQC perquè entre tots, m’heu
acollit a la gran vostra gran família com a un més i m’heu fet sentir com a casa.. que bé que ho
passarem aquests anys que vénen de doctorat!!!
109
MASTER THESIS
A tots els amics i amigues que sempre heu estat al meu costat, donant-me suport i suportantme en els moments més difícils, que no han estat pocs... Sobren les paraules quan ja tots
sabeu lo importants que sou per mi. Estic orgullós de tenir amics com vosaltres, i treballo cada
dia per què pugueu sentir-vos igual. Gràcies a tots i a totes..!!!!
Als meus pares. Heu patiu com ningú.. i per això, tota aquesta feina és per vosaltres. I tota la
que vindrà també.. Tot el que sóc us ho dec a vosaltres. Heu treballat sempre per donar-nos, a
l’Ivan i a mi, tot el que hem necessitat inculcant-nos el valor de l’esforç i el sacrifici. I poc a poc,
entre tots dos, us demostrarem que no us heu equivocat. Perquè us ho mereixeu com ningú.
A l’Ivan, el ñiño! Sempre m’has suportat, amb més o menys crits, i sé que sempre ho faràs (Por
la cuenta que te trae.. ;) ). Hem crescut junts, vivint un munt d’experiències plegats. Poc a poc
els nostres camins es van separant, però nosaltres seguim igual d’aprop! I que no canviï.. Ets
un crack..! I.. ara que ja saps on és la biblioteca, et va tocant a tu agafar el relleu en això
d’estudiar durant els caps de setmana eh? Jeje! Però no et pots queixar, estàs molt ben
acompanyat per la Mireia. Una abraçada molt forta, i gràcies a tots dos per tot!!
A l’avi Joan, a l’avi Lluís, a la iaia Antònia i a la iaia Paquita. Perquè cada vegada que us explico
alguna cosa i veig com se us encenen els ull, me’n adono que tot l’esforç val la pena. Us estimo
molt!
Al tio David, la tia Marisa, a la tia Montse, i al tio Francesc. Als meus cosins, i en especial a la
meva fillola, la Nina, la meva debilitat!! Sempre esteu al meu costat, i això és molt important
per mi.
Gràcies a tots!!!
I ara, sento que comença una nova aventura.. el doctorat!
M’hi acompanyeu?
Marc
110
ACKNOWLEDGEMENTS
111
MASTER THESIS
CHAPTER VII.
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CHAPTER VIII.
SUPPORTING INFORMATION
As the results presented for both projects included in this work are not yet published, we are
not able to share all geometries and optimized structures.
However, all the tables and figures present in this work are contained in the CD enclosed.
Exohedral reactivity of Ti2C2@C78 endofullerene: Diels-Alder addition on all non-equivalent
bonds:
SI_Ti2C2C78_table.xlsx : different tables used, and graphic comparisons.
SI_Ti2C2C78_images.pptx : different images and schematic representations.
Fine-tunable metalloporphyrinic nanocages as hosts for fullerene encapsulation:
SI_Nanocages_table.xlsx : different tables used, and graphic comparisons.
SI_Nanocages_images.pptx : different images and schematic representations.
120
SUPPORTING INFORMATION
121