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Density Functional Theory and Time-Dependent
Density Functional Theory Studies of Copper and
Silver Cation Complexes
Ricardo Oliveira Esplugas
Submitted for the degree of Doctor of Philosophy
University of Sussex
April 2009
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I hereby declare that this thesis has not been and will not be, submitted in whole or
in part to another University for the award of any other degree.
Signature:........................................................................
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Acknowledgment
Thanks to my supervisor, Dr. Hazel Cox, and also to other members of Sussex
University who supported me with helpful discussions during my PhD studies. These
are Prof. Tony Stace (now at the University of Nottingham), Prof. Malcolm Heggie,
Prof. John Murrell, Prof. Mike Lappert, Prof. John Venables, Prof. Tony McCaffery,
Dr. Peter Hitchcock, Dr. Liliana Puskar, Dr. Gianluca Savini, Dr. Jose Maria
Campanera, Dr. Lloyd Evans, Dr. Jingang Guan, Dr. Georgina Aitken, Jens Ryden,
David Wallis and Jeremy Maris.
I also want to mention scientists from outside the University of Sussex, namely Dr.
Erik van Lenthe from the ADF team (Scientific Computing and Modelling-
Amsterdam), Prof. Mike Robb (Imperial College London), Prof. Roberto Rivelino
(Universidade Federal da Bahia-Brazil) and Dr. Michael Seth (University of
Calgary-Canada).
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Abstract
The structure, stability and spectroscopy of the following complexes have been
investigated using density functional methods : [Ag pyridineN ]2+, [Ag acetoneN ]2+,
[Ag acetonitrileN ]2+, [Cu pyridineN]2+ with 1≤ N ≤ 6; [Cu ammoniaN ]2+, [Cu
ammoniaN]+ and [Cu waterN]+ with 1≤ N ≤ 8 ; [Cu waterN]2+ with 1≤ N ≤ 10 and
also copper (II) phthalocyanine.
A particular emphasis of this thesis has been to provide insight into the underlying
stability of these complexes and hence interpret experimental data, and to establish
the development of solvation shell structure and its effect on reactivity and excited
states. Energy decomposition analysis, fragment analysis and charge analysis has
been used throughout to provide deeper insight into the nature of the bonding in
these complexes. This has also been used successfully to explain observed
preferential stability and dissociative loss products.
Electronic excitation spectra have been obtained using Time-Dependent Density
Functional Theory (TDDFT), and this has included the evaluation of asymptotically
correct functionals in addition to standard functionals. Good agreement with UV/Vis
photodissociation spectra has been obtained in all cases (ranging from 0.1 to 0.3 eV).
The calculation of magnetic properties focused on the determination of the Landé g
factor for open-shell complexes with n=4 and 6. It was found that the g values are
sensitive to both the nature of the coordinating ligands (-O or –N) and the geometry
of the structure.
The effect of relativity on all these physical properties has been considered and it is
found that the effect is negligible for the Cu (I) and Cu (II) complexes but is quite
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significant for the Ag (II) complexes. For example, inclusion of relativistic effects in
TDDFT calculations on Ag (II) complexes can shift the dominant electronic
excitation energies by around 0.75 eV.
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INDEX
Chapter 1 Introduction
1.1 Computational chemistry
1.2 The chemistry of copper and silver
1.3 The generation and photodissociation of gas cationic complexes
1.4 Thesis overview
Chapter 2 Methods
2.1 The energetics of transition metal ions
2.2 Quantum chemistry methods
2.3 Relativity in chemistry
2.4 The adiabatic approximation
Chapter 3 The structures of gas phase copper and silver complexes
3.1 Background Theory
3.1.1 Geometry optimisation
3.1.2 Analysis of binding energies
3.1.3 Charges on atoms
3.1.4 The Jahn-Teller effect
3.2 Computational details
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3.3 Results: Calculated structures of the complexes studied in this thesis
3.3.1 Copper complexes
3.3.1.1 Cu (I) water and Cu (II) water
3.3.1.2 Cu (I) ammonia and Cu (II) ammonia
3.3.1.3 Cu (II) pyridine
3.3.1.4 Conclusions
3.3.2 Silver complexes- The relativistic effect
3.3.2.1 Ag (II) pyridine
3.3.2.2 Ag (II) acetone
3.3.2.3 Ag (II) acetonitrile
3.3.2.4. Conclusions
Chapter 4 Further studies of copper water and copper ammonia complexes:
ionization energies and fragmentation pathways
4.1 Ionization energies
4.1.1 Introduction
4.1.2 Background
4.1.3 Computational Details
4.1.4 Results
4.1.4.1 IE’s of copper water complexes
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4.1.4.2 IE’s of copper ammonia complexes
4.1.5 Conclusions
4.2 Fragmentation pathways of copper water and copper ammonia complexes
4.2.1 Introduction
4.2.2 Computational details
4.2.3 Results
4.2.3.1 Copper water complexes
4.2.3.2 Copper ammonia complexes
4.2.4 Conclusions
Chapter 5 The electronic spectra of gas phase copper and silver complexes
Introduction
5.1 Background theory
5.1.1 α and β electrons
5.1.2 HOMO and LUMO
5.1.3 Oscillator strength
5.1.4 TDDFT and asymptotically correct functionals
5.1.5 Review of previous work on TDDFT applied to transition metals
5.2 Results: the calculated electronic spectra
5.2.1 Copper complexes
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5.2.1.1 Copper (II) phthalocyanine
5.2.1.2 Copper (II) pyridine
5.2.1.3 Copper (I) water/copper (I) ammonia and
copper (II) water /copper (II) ammonia
5.2.1.3.1 Copper water complexes
5.2.1.3.2 Copper ammonia complexes
5.2.1.4 Conclusions
5.2.2 Silver complexes
5.2.2.1 Silver (II) pyridine complexes
5.2.2.2 Silver (II) acetone complexes
5.2.2.3 Silver (II) acetonitrile complexes
5.2.2.4 Conclusions
Chapter 6 Magnetic interactions of copper and silver complexes
Introduction
6.1. Background theory
6.2 Results: the calculated g values of copper and silver complexes
6.2.1 Copper complexes
6.2.2 Silver complexes
6.2.3 Conclusions
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Chapter 7 Summary
Appendix A Abbreviations
Appendix B Tables of calculated results concerning copper complexes
References
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Chapter 1 Introduction
This chapter contains a general introduction to the thesis, describing the main
motivations for this work and the perspectives in the area. Also, a brief description
of relevant aspects of the chemistry of the metals involved is provided and a
description of the unusual experiments that have produced the complexes that are
the main focus of this computational study.
1.1 Computational chemistry
Computational chemistry enables us to comprehend natural phenomena in more
depth. Also, it has the ability to simulate experiments that are too expensive, too
difficult or too environmentally harmful to be performed, so that it is a great tool
with the potential to be applied in all areas of chemical research.
The challenge of studying an interacting system consisting of dozens of electrons,
like a molecule or complex, has lasted since the early development of the quantum
theory at the beginning of the last century. Calculations involving transition metals
are particularly complex, especially if the metal atoms contain one or more unpaired
electrons.
Great developments in Density Functional Theory (DFT) during the past twenty
years enabled researchers to perform accurate calculations on systems containing up
to 100 atoms and even more. Parallel to this theoretical activity, large increases in
computing resources made available the multi-processor machines that are required
to perform very long and demanding calculations.
The study of transition metal containing complexes also became computationally
feasible, as DFT could deal with such systems in a relatively cheap manner;
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traditional high level ab initio methods are much more demanding and require a
hardware capability which is not available at present, if calculations on large systems
are to be made.
More recently, the development of time-dependent density-functional theory
(TDDFT) extended the power of DFT to the domain of excitation energies and
polarisabilities.
At the same time, experimentalists improved their techniques and equipment so that
they could perform more and more accurate measurements. This way the new
theories could be tested and developed further. Supersonic beam techniques are an
example of such experimental development.
Collaboration with Tony Stace and coworkers (1) was a unique opportunity to test
state-of-the-art theories like TDDFT and its implementations in quantum chemical
software. Furthermore, the theoretical results obtained in this thesis are very useful to
interpret such gas phase experimental results and to guide new research.
Stace and coworkers managed to produce various single and doubly charged metal
complexes in the gas phase (1-8). A wealth of data has been collected using
supersonic expansions and the pick-up technique, coupled with tunable lasers and a
range of mass spectrometers.
The supersonic expansion provides an excellent environment for the study of the
reactions of transition metal complexes, including photochemistry. The absence of
bulky solvent interactions enables accurate analyses of the metal-ligand interactions
and also electron transfer processes (ET), which are the simplest kind of process
where reactants become products (9). The fact that this sort of process doesn’t
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involve a re-arrangement of the nuclei (10) makes detailed theoretical calculations of
ET reactions possible. The gas phase environment also provides the opportunity of
producing symmetrical structures, which are rarely seen in the solid state because
deformations are always present (11).
The combination of these powerful theoretical and experimental methods can
provide important insight in areas like photochemistry of transition metals, which
deals with crucial issues like mutagenic effects of radiation, photosynthesis, solar
energy conversion and storage (12) and even the origin of life. In fact, ET chemistry
is ultimately responsible for life on Earth (13).
Furthermore, it is very important to develop an understanding of how ligand field
electronic transitions progress as a function of ligand number and type. Such
understanding would open the door to the preparation of transition metal complexes
with photophysical properties that are user-definable (14).
Finally, there is also the theoretical motivation which consists of testing new
functionals and software implementations to be employed in calculations involving
transition metals, and in particular complexes with the complications that arise as a
result of the presence of one or more unpaired electron.
This work will focus on the chemistry of singly and doubly charged copper and
silver. In order to study their complexes, a range of organic solvents were employed,
namely pyridine, acetone, acetonitrile, ammonia and water.
1.2 The Chemistry of copper and silver
The name copper, and also its symbol Cu, come from the Latin cuprum, possibly
named after the island of Cyprus, from where it has been mined for about 5,000
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years. The symbol of silver, Ag, comes from the Latin argentum. Silver is much less
abundant than copper, and that is why it is more expensive. The terrestrial
abundances of copper and silver are 68 ppm and 0.08 ppm, respectively. Cu and Ag
have two stable isotopes each (15).
Although copper and silver can be found in the native state, these elements are
normally found in ores. Copper is usually found as sulfides, oxides or carbonates,
whereas silver is mainly found as a sulfide. Malachite is an important copper ore,
which is shown in figures 1.1 and 1.2.
Figure 1.1: Crystal structure of malachite
(16), Cu2(OH)2(CO3) with copper represented in blue, oxygen in red, carbon in
black and hydrogen in white.
Figure 1.2: Photograph of malachite, which is copper ore and also gemstone.
Figures 1.1 and 1.2: Crystal structure of malachite and appearance (images produced
by the Author).
Copper and silver are located at the end of the transition metal series and, together
with gold, they form group 11 of the periodic table, also known as the coinage
metals because they have been used as money since very early times. These metals
are excellent conductors of heat and electricity, and these properties determine the
main uses of copper. Ag and also Au, the third member of the group, are too
expensive to be used in pipes and wires and they are mainly used in
jewellery. Copper is also very important because of its use in superconducting
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materials.
Cu and Ag can assume the oxidations states 1+, 2+ and 3+. The preferred oxidation
state of copper is 2+ and that of silver is 1+. That means that this group doesn’t
follow the usual trend, where the preferred oxidation state increases when moving
down a group (17).
The preferred oxidation state of gold is 3+, which is explained by relativistic effects
(18). The yellowish colour of gold is also explained by relativistic effects. The non-
relativistic band structures of silver and gold are very similar and ‘gold would look
silver’ (17) if the relativistic effect didn’t exist.
Property Cu Ag
Atomic number 29 47 Naturally occurring isotopes 2 2 Atomic weight 63.546 107.8682 Electronic configuration [Ar]3d104s1 [Kr]4d105s1 Electronegativity 1.9 1.9 metal radius / pm 128 144 ionic radius / pm I 77 115 II 73 94 III 54 75 Ionization energy 1st 7.72 7.57 (eV) 2nd 20.28 21.47
3rd 37.07 34.81
Table 1.1 Experimental values for selected physical properties of copper and silver (adapted
from (15))
Interestingly, the first ionization energy (IE) is higher for copper but the second is
higher for silver. In any case these IE’s are similar for both metals. They have two
isotopes each and also the electronegativities of these two elements are exactly the
same.
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The atomic radius of silver is larger than that of copper. Similarly, all the ionic radii
of silver are larger than the corresponding ones for copper. It is curious that the ionic
radius of copper is very similar for the Cu (I) and Cu (II) cations, considering that
the latter has one electron less. That is probably one of the reasons why these
cations are so useful to perform electron carrying tasks as it will be seen in the
biological section.
Oxidation state I (d10)
Both cations are diamagnetic in this oxidation state and usually colourless (15).
Cu (I) disproportionates in aqueous solution, unless complexed with ligands that
have π-acceptor properties (15). The Ag (I) ion, however, is stable in aqueous
solution.
Ag (I) has a rich chemistry and it forms complexes with oxygen, nitrogen, carbon
and sulphur donors. It also forms various halides, some of which have historical
importance as they were the basis of photography (17). Organometallic compounds
of silver only exist with Ag (I) and are usually air and moisture sensitive (17).
Despite its instability in solution, Cu (I) forms many compounds in the solid state. It
forms insoluble salts with each of the halogens. Perhaps the most well known Cu (I)
compound is the red cuprous oxide, which is produced in the Fehling’s test for
sugars (19).
If the Cu (I) ion is to form complexes with water, they are expected to be only 2-
coordinate, considering that the amine complexes of this ion are 2-coordinate (20).
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Clyde (21), on the other hand, states that the monocations of the coinage metals
group would be expected to have high coordination numbers “by any simple
electrostatic theory”.
However, this high coordination hasn’t yet been observed in the condensed phase, as
these cations, in addition to other d10 cations, seem to prefer a linear geometry,
which is only 2-coordinate. The following explanation has been proposed for the
linearity of these complexes: the 2 electrons that would occupy the 𝑑𝑑𝑧𝑧2 are placed
in a s- 𝑑𝑑𝑧𝑧2 hybrid orbital, so that charge is transferred into the xy plane and bonds
can form easily across the z axis (22). The limiting factor for this transformation to
occur is the energy gap between the (n-1)d and the n s orbitals. If this gap is very
large it will be difficult for an electron to be promoted. This energy is lowest for Au
(I) (1.9 eV) and it is followed by Cu (I) (2.7 eV).
Gas phase experiments
Although it is unstable in solution, Cu (I) forms a variety of compounds in the gas
phase.
A series of experiments have been performed with Cu (I) and small organic
molecules in the gas phase. Cu (I) binds to nitrogen and also oxygen as donor atoms
of structures which include urea, formamide and glycolic acid. These experiments
have been supported by DFT calculations and they envisage understanding the
mechanisms of biological processes involving Cu (I) and proteins (23-25).
Stace and coworkers have produce a range of Cu (I) (2)(26) compounds using the
pick-up technique (27), but didn’t manage to produce Ag (I) gas phase compounds.
These compounds were investigated for many years, before the breakthrough that
allowed the production of gas phase complexes containing doubly charged metals.
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Oxidation state II (d9)
Doubly charged copper and silver ions have a d9 electronic structure, and as a result
they are paramagnetic due to the unpaired electron. This feature makes difficult the
study of their compounds by nuclear magnetic resonance (NMR). The technique of
choice for the study of complexes containing these ions is electron spin resonance
(ESR), which will be described in detail in chapter six.
Cu (II) is the preferred oxidation state of copper, which has a very extensive
chemistry.These compounds are usually coloured (28) .
Cu (II) has the highest hydration energy of the divalent first row transition metals,
followed by Ni (II) (20). This is a consequence of the decrease in ionic radius in
moving to the right across the period.
Cu (II) complexes can be obtained, for instance, by dissolving solid copper in
sulphuric acid, nitric acid or ammonia. If ammonia is used as the solvent, formation
of [Cu(NH3)2]+ and subsequently [Cu(NH3)4]2+ takes place.
Cu (II) can form a large range of compounds like halides, carbonate, sulphate,
nitrate, hydroxide, oxides and sulphide. Among the complexes, the octahedral
[Cu(H2O)6]2+ is a very important one, which is formed when a Cu (II) salt is
dissolved in water. This solution has a typical blue colour. If excess ammonia is
added to the solution, the four planar water molecules are displaced to give the deep
blue [Cu(NH3)4(H2O)2]2+ (28), which illustrates the high affinity of copper for
ligands with nitrogen donors.
The d9 configuration is not very favourable for silver; however Ag (II) forms
condensed phase compounds with nitrogen and fluorine as donor atoms (17).
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A number of gas phase complexes of Cu (II) and Ag (II) have been produced by
Stace and coworkers (29) , with oxygen and nitrogen as donor atoms.
BIO-INORGANIC CHEMISTRY
Cu has an extensive bio-inorganic chemistry and is present in many proteins but Ag
has no role in sustaining life.
Metallic copper and silver have both anti-bacterial properties (15). Silver (I)
sulfadiazine is an agent to release Ag (I) slowly, and it is incorporated into a cream
to prevent infections in burns (17). Its anti-bacterial action is believed to be related to
the Ag-DNA interaction (30).
The biological importance of copper was first observed by Albert Szent-Gyorgyi’s in
1930, when he discovered in cabbage the enzyme ascorbic acid oxidase which was
later shown to contain copper as its prostetic group (31).
The crucial function of Cu in plants and animals is in the functioning of the principal
terminal oxidases, like cytochrome oxidase and ascorbic acid oxidase, where the
conversion between Cu (I) and Cu (II) plays the fundamental role in electron
transport. In fact, it is the conversion between these two states that configure the
most essential use of this element (32).
The cyanide ion (CN-) is an electron transport inhibitor because of its ability to
deactivate cytochrome oxidase, which is one of the enzymes that are responsible for
the process of respiration in the mitochondrial membrane (33). It is small enough to
penetrate all the way to the centre of the enzyme and attach itself to the Cu ion
moiety at its centre, so that the whole enzyme is inhibited. It illustrates the high
compatibility between copper and nitrogen. The main reason for the high toxicity of
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the cyanide ion, however, is its affinity to bond with iron (III) present in the oxygen
carrying protein haemoglobin (34).
Although the cyanide ion can attack the Cu ion and deactivate any known copper
enzyme, a number of other ligands have been employed in the study of copper in
these proteins (31). Most of them are nitrogen-based ligands like bipyridine,
phenantroline and cuproine. Each of these ligands inhibits the activity of copper
enzymes to different extents.
Lower forms of copper proteins (which don’t have enzymatic properties) can serve a
variety of functions, like carrying oxygen, as part of the blood of certain crustacea
and molluscs. They are called hemocyanins (31).
Most of the copper found in biological systems is in the form of proteins and very
little is found even in fluids like blood (31).
Some researchers have studied the ability of copper to bind to amino acids (35) and
also to DNA/RNA bases (36), which consist basically of nitrogen heterocycles, like
purine and pyrimidine, which have basic nitrogen atoms available to donate to a
metal cation. Cu is also known to bind to nitrogen heterocycles like pyridine.
Copper can also be employed in the treatment of biological damage caused by
ionising radiation (31).
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1.3 The generation and photodissociation of gas phase
cationic complexes
This section will describe the experimental setup used by Stace and coworkers, on
which the author of this thesis worked for a period just before the beginning of DPhil
studies.
The production of doubly charged metal complexes in the gas phase (37-39) is an
experimental development that allows for studying important ions/complexes
without the complications associated with the presence of a solvent. It allows for
studying various transition metals in their preferred oxidation state, which is usually
II or III. This way, it also makes possible the study of biologically important
transition metal containing molecules. Copper, in particular, is present in many
proteins and DFT studies of these have already appeared in the literature (40).
Experiments performed by Stace and coworkers combining the pick-up technique
and photodissociation (1,29,41,42), have been providing large amounts of
information about the chemistry of transition metal complexes. The apparatus to
perform such experiments will be described in this section.
Only in the case of the experiments in collision induced dissociation (CID) of
copper complexes, which is one of the subjects of chapter 4, has there been a
substantial difference in the experimental setup, as a collision gas is used to promote
fragmentation instead of a laser. In all cases the complexes are produced using the
pick-up technique. A brief description of this technique will be given next, as part of
the description of the apparatus used in the production and photodissociation of
silver complexes. This description will be based on figure 1.3:
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Figure 1.3 Apparatus used to detect photofragmentation of silver complexes (1).
The process starts at the cluster chamber (down on the left of figure 1.3) which
contains a mixture of gases containing argon and the chosen ligand. The mixture is
allowed to expand adiabatically into the vacuum through a hole smaller than the
mean free path of the gas molecules inside the chamber. As a result a supersonic
beam of ligand molecules, plus argon, is produced. This process promotes extremely
fast cooling of the beam so that clustering takes place. These newly formed clusters
of solvent and argon cross with a beam of silver atoms that are evaporated from an
electric oven kept at 1250 degrees, so that each cluster picks one metal atom. That is
the core of the pick-up technique.
Next, the neutral metal-ligand complexes move into the ion source, where they are
hit by an electron gun and then accelerated by a 5 keV potential into the first field
free region and then into a magnetic sector, where they are selected according to
their mass/charge ratio. The complexes selected enter the second field free region,
where they are hit by a tunable YAG (yttrium aluminium garnet) laser.
Photofragmentation can be promoted at particular frequencies of the laser and the
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fragments are recorded using an electrostatic analyser (ESA). This way, a spectrum
can be recorded.
1.4 Thesis overview
Chapter 2 describes the theoretical methods employed in this thesis. Background
information about Density Functional Theory (DFT) and also Time-dependent
Density Functional Theory (TDDFT) will be presented, along with a discussion on
exchange and correlation functionals, the adiabatic approximation in TDDFT, and
some of the shortcomings of TDDFT. Other associated concepts like relativity and
their implications in quantum chemistry will also be discussed.
Chapter 3 presents the calculated structures of all complexes studied in this thesis
and also the relevant background theories, like analysis of binding energies and their
decomposition into repulsive and attractive terms according to the Ziegler-Rauk-
Morokuma method. To provide further evidence for the existence of copper water
and copper ammonia structures, a study of fragmentation pathways, including proton
transfers, is carried out and compared to experimental results from Stace and
coworkers. The binding energy decomposition theory is used to explain some
unexpected geometries found.
The calculations performed in this thesis include the determination of the lowest
energy structures of all the copper and silver complexes studied, namely [Ag
pyridineN ]2+, [Ag acetoneN ]2+, [Ag acetonitrileN ]2+, [Cu pyridineN]2+ with 1≤ N ≤
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6; [Cu ammoniaN ]2+, [Cu ammoniaN]+ and [Cu waterN]+ with 1≤ N ≤ 8 ; [Cu
waterN]2+ with 1≤ N ≤ 10 and also copper (II) phthalocyanine.
Chapter 4 presents, firstly, the calculations of ionization energies of copper water
and copper ammonia complexes. These are of interest because they are related to
their redox properties and to the electron carrying ability of copper.
In the second part it presents calculations on fragmentation pathways of copper water
and copper ammonia complexes. The results are compared to experimental results
from Stace and coworkers who promoted electron capture dissociation of some
copper complexes (2) studied in chapter three.
Chapter 5 presents the calculated electronic spectra (UV/Vis) of all the complexes
presented in chapter 3. Spectra have been calculated using Time Dependent Density
Functional Theory (TDDFT). These results are used to interpret experimental gas
phase spectra, where available, and also to evaluate exchange-correlation
functionals, including the asymptotically correct functionals (ac) like SAOP and
LB94. Some background information concerning TDDFT and ac functionals is also
provided, along with a review of the relevant literature.
Chapter 6 is concerned with magnetic interactions of some of the doubly charged
complexes studied in this thesis. It present results of ESR calculations and in
particular the g-tensor is calculated using various different functionals and basis sets.
Background theory is presented along with a review of the relevant literature. The
potential applications of these results to support experiment are discussed.
Chapter 7 presents a summary and the main conclusions of the thesis.
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Chapter 2 Methods
This chapter will provided background information about various subjects that are
relevant for this thesis, namely the energetics of transition metals ions, quantum
chemistry methods and some associated concepts like relativity and the adiabatic
approximation in the calculation of excitation energies.
2.1 The Energetics of transition metal ions
The electronic energy of a free transition metal ion is determined by the interplay of
three factors (11):
1) Coulomb repulsion between d electrons
2) Exchange forces between d electrons
3) Spin-orbit coupling
If the ion is subjected to a “crystal field”, as a result of the presence of ligands, an
extra factor appears:
4) “ligand field splitting”.
The Coulomb interaction occurs between electrically charged particles (or objects)
and it has an inverse dependence on the distance between the particles (Coulomb
potential is proportional to 1 / r), so that the larger the distance, the weaker the
interaction. It can be attractive, when acting between opposite electrical charges, or
repulsive when acting between like charges, for instance negatively charged
electrons, where the Coulomb interaction raises the total energy of the system.
Exchange interaction only occurs between fermions (particles with half integer spin),
which in this context are the electrons. It has also an inverse-square dependence on
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the distance between the particles but, unlike the Coulomb interaction, it doesn’t
occur in the domain of classical physics. There will be an extended discussion about
exchange energy in the theoretical introduction of chapter three.
The spin-orbit energy is a magnetic interaction due to the coupling between the spin
and orbital angular momentum of an electron in an atom or molecule. These
magnetic fields arise as a result of the movement of the electron, and their directions
depend on the direction of rotation of the electron (43). There are two rotations: spin,
which is modeled as an intrinsic rotation of the electron, and orbital movement,
which is a rotation around the nucleus (44) (in practice, it is known that the electron
is neither spinning nor orbiting, but this semi-classical approach is successful in
explaining observed effects that arise as a result of spin-orbit coupling, like the
hyperfine splitting of the spectrum of hydrogen). As a result of spin-orbit coupling,
the electron energy will depend on the direction of rotation of its spin and also on the
direction of its orbital movement. There are two possibilities: the two rotations are in
the same direction or in opposite directions, so that the electron can assume two
different energies instead of only one when spin-orbit coupling is ignored. This issue
will be discussed further in the introduction of chapter 6.
The intensity of the spin-orbit coupling depends on the element (heavier ones
produce a stronger interaction) and also on its oxidation state. Cu (II) is, among first
row transition metals, the ion that presents the strongest spin-orbit coupling (11).
Ligand field splitting arises a result of the interaction between electrons in the metal
ion and electrons in the ligand(s). It is a directional effect, as the ligand can interact
more strongly with certain orbitals in the metal, at the centre of the complex,
depending on its angle of approach. The simple version of the theory, called crystal
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field, considers only Coulomb interactions between metal and ligand orbitals,
whereas ligand field theory is more complete and includes the formation of
molecular orbitals between metal and ligand(s) in addition to the electrostatic
(Coulombic) effects. Ligand field splitting depends on the type of ligand involved, as
it is well known that certain ligands cause a larger splitting than others. The
spectrochemical series show the ligands in order of increasing “splitting power”. A
more detailed discussion of these effects is presented in the next section.
The dynamics of d orbitals
The presence of d electrons makes the chemistry of transition metals very
interesting. Their complexes can assume different geometries, oxidation states,
colours, etc. Because of the crucial importance of d electrons and their orbitals in
determining geometries of transition metal complexes and also their prominent role
in the characterisation of electronic excitations of these complexes, as such processes
usually start or end on a d orbital, this section will be dedicated to them.
Crystal field theory
According to crystal field theory (CFT), which is based on electrostatic interactions,
and to ligand field theory (LFT), which improves on CFT by taking orbital
interactions into account, the behaviour and geometry of transition metal complexes
is determined by the number of d electrons present in the system.
In this section the basic consequences of CFT will be discussed briefly. In CFT the
metal and the ligands are treated as point charges. A free metal atom or ion has five
degenerate d orbitals, whose appearance is shown in figure 2.1. When other atoms or
molecules approach the metal centre, the degeneracy of its d orbitals is lifted. The
28
way in which their energies change will depend on these orbital shapes and on the
directions from which the ligands approach them.
d orbitals
Figure 2.1 is a representation of the five hydrogenoid d orbitals. The first three
orbitals shown on figure 2.1 have their lobes in between the cartesian axes, and the
last two have their lobes on the axes. The 𝑑𝑑𝑧𝑧2 orbital is aligned with the z-axis and
the dx2
- y2 has lobes along the x and y axes.
𝑑𝑑𝑥𝑥𝑥𝑥 dxz dyz
dz2 dx
2- y
2
Figure 2.1: Hydrogenoid d-orbitals.
Knowing the shapes of each d orbital, it is simple to figure out qualitatively in which
way their energies will shift in the presence of a crystal field of a given geometry if
only electrostatic interactions are to be taken into account. The Oh and Td geometries
29
will be studied first. Next, the D4h case will be analyzed as a distortion of the Oh
geometry. In each case, the degenerate unsplit d orbital energy is taken as zero.
The octahedral crystal field
In this geometry, the ligands will approach the metal along all three axes. It means
that the orbitals that lie along the axes (𝑑𝑑𝑧𝑧2 and dx2
- y2) will have their energy
increased due to electrostatic repulsion, whereas the other three orbitals will have
their energies lowered. Two ligands will approach along the z-axis and the other four
will approach along the x and y axes.
It is easy to understand why the two orbitals mentioned above have their energies
increased, but it is not as clear why the energy of the other three is lowered. This
lowering happens in order to preserve the “centre of gravity” of the levels (45). In
order to do so, the sum of all orbital energies must be zero and some orbitals acquire
negative energies to compensate for the positive energies acquired for the orbitals
that are repelled. This rule only works when the interactions are considered to be
purely electrostatic. Hence, the d orbitals in an octahedral complex will split in the
manner shown in figure 2.2:
Free metal Metal in an octahedral field
Figure 2.2: Ligand field splitting diagram for a transition metal in an octahedral geometry.
30
The energy difference between the new sets of levels is called crystal field energy
(CFE) and is commonly represented by the Greek letter ∆.
The tetrahedral crystal field
In the tetrahedral case, the ligands approach in directions that are at 45 degrees from
the main axes. It means that they come from the direction of the vertices of the boxes
drawn around the orbitals in figure 2.1. In this case, the orbitals that are going to be
repelled are the ones that are represented at the top of figure 2.1, i.e., the 𝑑𝑑𝑥𝑥𝑥𝑥 , 𝑑𝑑𝑥𝑥𝑧𝑧
and 𝑑𝑑𝑥𝑥𝑧𝑧 . The other two orbitals, 𝑑𝑑𝑧𝑧2 and dx
2- y
2 will be lowered in energy as shown
in figure 2.3.
It is important to note that the CFE associated with the tetrahedral geometry is
smaller than in the octahedral case.
Free metal Metal in a tetrahedral field
Figure 2.3: Crystal field splitting diagram for a transition metal in a tetrahedral geometry.
The square planar crystal field
In order to find out what the ligand field splitting in the square planar geometry is, it
is convenient to start from the octahedral splitting (figure 2.2). The square planar
geometry can then be considered as a deformation (elongation) of the octahedral
geometry, which will occur along the z-axis. The z-axis contains the two axial atoms
31
(top and bottom of the octahedron), so that four atoms stay in position and the
symmetry can be lowered to D4h. This will lift the degeneracy of most orbitals.
The dx2
- y2 orbital will still be high in energy, but the 𝑑𝑑𝑧𝑧2 orbital will be stabilised,
because there are no more ligands in this direction. Due to the breaking of the Oh
symmetry, the three orbitals at the bottom of the Oh diagram will no longer be
degenerate. One of them - the one along the xy plane - will acquire a higher energy
and the other two will be further stabilised. The result is shown on figure 2.4.
Figure 2.4: Crystal field splitting diagram for a metal in square planar and octahedral ligand
fields.
The position of the energy levels in the square planar geometry may vary depending
on the type of metal and ligand. In some cases, like PtCl42-, the energy of the 𝑑𝑑𝑧𝑧2
orbital can be at the bottom of the diagram, below the 𝑑𝑑𝑥𝑥𝑧𝑧 and 𝑑𝑑𝑥𝑥𝑧𝑧 .
This geometry will appear very often in this thesis. The symmetry labels that
correspond to each of the d orbitals in a square planar geometry (D4h point group)
are:
free metal
Octahedral Field
square planar field
32
dx2
- y2 b1g
𝑑𝑑𝑥𝑥𝑥𝑥 b2g 𝑑𝑑𝑧𝑧2 a1g
𝑑𝑑𝑥𝑥𝑧𝑧 𝑑𝑑𝑥𝑥𝑧𝑧 eg
The last pair is the only one that remains degenerate (eg) because they can be mixed
by the symmetry operation C4.
According to the diagrams above (figures 2.3 and 2.4), it can be seen that a d9 system
will prefer a square planar arrangement to a tetrahedral one, because the total energy
will be lower in that case, and the same applies to d7 and d8 metals. Metals with a
smaller number of electrons will prefer a tetrahedral geometry. Alternatively, if there
is no substantial CF energy involved, the ligands will prefer a tetrahedral geometry
because that is the situation where they can be the furthest apart from each other.
hole theory
According to hole theory (45), a d2 system will behave as a d8 system, but with the
energy levels inverted. For instance, a d2 system with octahedral symmetry will be
like a d8 system with tetrahedral symmetry. The idea is that the d8 system has 2 holes
(because d10 is the full shell) and those behave as electrons.
Likewise a d6 system can be treated as a d4 and so on. This approach facilitates the
analysis of spectra because a smaller number of electrons (or holes) can be treated,
like interpreting a d7 system based on a d3.
The spectra of d1 compounds, like those of Ti (III), are relatively simple. It would be
expected, from the theory just described briefly, that d9 compounds would have an
equally simple spectra (because they only have one hole), but unfortunately this is
not the case. The d9 ions Ag (II) and Cu (II) have very strong spin-orbit coupling,
33
which makes the spectra of their complexes complicated. Furthermore, these
complexes often exhibit Jahn-Teller distortion. It is observed that when the
degeneracy of orbitals is broken by spin-orbit coupling, the Jahn-Teller effect is not
needed, but in the case of these ions one of these two effects will certainly operate.
That is one of the reasons why a more sophisticated theory of electronic excitations
is needed to study these systems.
Ligand field theory
The next step in the study of the energy levels of a square planar d9 complex is to
introduce ligand field theory (LFT) which takes molecular orbital (MO) theory into
consideration, in addition to the electrostatic splitting of the d orbitals. The formation
of covalent bonding between the orbitals in the metal and in the ligand is considered
in this more accurate theory. Orbitals that overlap in space, have the correct
symmetry, and approximately the same energy, form MO’s that will contribute to the
bonding between metal and ligands.
LFT can produce more detailed energy level diagrams, which can be used to make
qualitative predictions of spectra, although these can only be accurate for the d1 ions
and complexes.
2.2 Quantum chemistry methods
In order to obtain a deep understanding of chemistry it is necessary to have a very
close look at matter, this implies going down to its basic constituents: protons and
electrons. Possessing a working knowledge of how these particles behave and
interact with each other in different molecular environments will be a major step
34
towards the ultimate goal of chemistry, which is to control the assembly of atoms in
order to produce materials with carefully chosen properties.
The application of the established physical laws to this problem is an ongoing
endeavour and one of its main difficulties is to deal with the multitude of different
interactions that are present in a molecular environment.
A major simplification of this problem, which is almost universally used as a first
approximation when tackling this question, is to assume that the nuclei don’t move
significantly during the timescale of the phenomena of interest, so that it can be
assumed that they stay at rest. This is a reasonable approximation because protons
are of order a thousand times heavier than electrons. This is known as the Born-
Oppenheimer approximation (46). By using this approximation, it is possible to
concentrate only on the movement of the electrons.
Wave-mechanics
The behaviour of electrons is controlled by the laws of quantum mechanics, and
therefore there is no concept of trajectory. Instead, the movement of electrons is
described by wavefunctions. Orbitals are represented by wavefunctions, so that
knowing the wavefunction it is possible to find out everything that is needed to know
about the system, e.g., all the observables. Observables include energy, position,
momentum, orbital angular momentum, spin angular momentum and projection of
the angular momentum along a particular axis.
A situation where the wavefunction is known and as a result all the observables can
be determined is called a state of maximum knowledge about the system (47). It is
important to note that maximum knowledge in quantum mechanics is a limited
35
knowledge, as a consequence of Heisenberg's uncertainty relations and the statistical
(probabilistic) character of quantum mechanical laws.
The wavefunction ψ can be determined by solving the Schrödinger equation:
𝐻𝐻� 𝜓𝜓 = 𝐸𝐸 𝜓𝜓
Where 𝐻𝐻� is the Hamiltonian operator and E is the energy of the system. Similarly to
the classical Hamiltonian operator, in the quantum mechanical version it also
represents a sum of kinetic and potential energies, which in the one-dimensional
quantum mechanical picture looks like this:
𝐻𝐻� = −ħ2
2𝑚𝑚𝑑𝑑𝑑𝑑𝑥𝑥2 + 𝑉𝑉(𝑥𝑥)
Where the kinetic energy term (first term) is represented using the corresponding
quantum mechanical operator for momentum:
Px � = ħi
∂∂x
36
The potential 𝑉𝑉(𝑥𝑥) will depend on the characteristics of the system that is being dealt
with and determining the potential of a molecular system is not an easy task. That
means that it is already a challenge to determine the Hamiltonian operator and to
write the Schrödinger equation. To solve it is another problem.
The Schrödinger equation is an eigenvalue equation. In order to determine 𝜓𝜓 it is
usual to introduce a set of basis functions 𝜑𝜑𝑛𝑛 onto which 𝜓𝜓 can be projected, so that:
𝜓𝜓 = � 𝜑𝜑𝑛𝑛𝑛𝑛
This way, the problem is reduced to a set of simpler eigenvalue equations:
𝐻𝐻� 𝜑𝜑𝑛𝑛 = E 𝜑𝜑𝑛𝑛
After finding 𝜓𝜓 , the expectation values of observables (A) can be calculated in the
following way (for a normalised wavefunction):
< �̂�𝐴 > = ∫𝜓𝜓∗�̂�𝐴 𝜓𝜓 𝑑𝑑𝑥𝑥
The time-dependent Schrödinger equation, which is employed in time-dependent
problems like for instance an electronic excitation is shown below:
37
𝐻𝐻� 𝜓𝜓 = −𝑖𝑖ħ 𝜕𝜕𝜕𝜕𝜕𝜕
𝜓𝜓
In the case where the Hamiltonian doesn't depend explicitly on time, the time
evolution of the system can be described by multiplying the wavefunctions by an
exponential factor:
𝜑𝜑𝑛𝑛(𝜕𝜕) = 𝜑𝜑𝑛𝑛 𝑒𝑒−𝑖𝑖𝐸𝐸𝑛𝑛 𝜕𝜕
and the time dependent total wavefunction can be represented as linear combinations
of the time dependent eigenfunctions 𝜑𝜑𝑛𝑛(𝜕𝜕) , as in the time-independent situation.
In the case where the Hamiltonian depends explicitly on time the situation is more
complicated and it is necessary to introduce a new operator - the time evolution
operator (𝑈𝑈 � ):
𝑈𝑈 � (t) = 𝑒𝑒−𝑖𝑖 𝐻𝐻� 𝜕𝜕
ħ�
This operator represents the operation of "waiting a little bit" , so that:
𝜓𝜓 (t) = 𝑈𝑈 � (t) 𝜓𝜓 (0)
38
The equations above can describe the movement of particles when the wavefunction
has already been found.
The Hartree-Fock approximation
The most traditional wavefunction based method of solving molecular structure
problems is known as the Hartree-Fock (HF) approximation. The HF multielectron
wavefunction consists of a determinant formed of one-electron wavefunctions
(orbitals). Each of the electrons in the system is assumed to be moving in the field of
all the other electrons put together. This average field is dependent also on this
particular electron so that the equations must be solved in a self-consistent manner.
Many algorithms have been implemented to solve these equations using computers
and it has been a success for many years.
The HF multielectron wavefunction (𝛹𝛹𝐻𝐻𝐻𝐻) is conveniently represented by a Slater
determinant (50):
𝛹𝛹𝐻𝐻𝐻𝐻 =1√𝑁𝑁!
��
𝜓𝜓1(�⃗�𝑥1) 𝜓𝜓2(�⃗�𝑥1) … 𝜓𝜓N (�⃗�𝑥1)𝜓𝜓1(�⃗�𝑥2) 𝜓𝜓2(�⃗�𝑥2) … 𝜓𝜓N (�⃗�𝑥2)
⋮ ⋮ ⋮𝜓𝜓1(�⃗�𝑥N ) 𝜓𝜓2(�⃗�𝑥N ) … 𝜓𝜓N (�⃗�𝑥N )
��
Where 𝜓𝜓i(�⃗�𝑥N ) are spin orbitals, products of spatial and spin wavefunctions.
The determinant ensures that the resulting 𝛹𝛹𝐻𝐻𝐻𝐻 is an antisymmetric product of spin
orbitals and consequently it complies with Pauli principle.
39
According to the variational principle, the correct multielectron wavefunction (𝛹𝛹𝐻𝐻𝐻𝐻)
will be the one to which corresponds the lowest energy, and the iterative SCF
process will calculate the energy associated to a large number of determinants until
the minimum energy configuration is found.
The main shortcoming of the HF method is that it does not include electron
correlation. Improvements have been made to the theory, in order to include electron
correlation via Møller-Plesset perturbation theory (these methods are known as MP2,
MP3, MP4 and so on), configuration interaction, coupled cluster and others. They
offer good results but a high computational cost. They only work well in relatively
small systems.
The breakthrough, which opened the door for calculations involving larger and more
complex systems, like the ones containing transition metals, came with the
development of density matrix and density functional theories, which are described
next.
The density matrix
The problem of determining the electronic structure of an atom (with more than one
electron) or molecule is classified as a many-body problem. Curiously, this kind of
problem is easier to be solved quantum-mechanically than classically, and there are a
few techniques available to achieve this objective. The most used are Green’s
function and density matrix (DM) techniques and the latter has found most
applications in quantum chemistry (48). The density matrix has been used in various
different contexts, by different authors, and it is important to point out these
differences for an understanding of this important concept.
40
The DM formalism emerged in the context of theoretical physics, more specifically
statistical mechanics, and it subsequently found many applications in quantum
chemistry. According to Landau, “the density matrix is the most general way of
representing a quantum-mechanical system. The wavefunction description is a
particular case of the density matrix” (49).
The DM was introduced by von Neumann in 1927, to help to solve problems in
statistical mechanics. Dirac wrote his first papers using the DM in 1929 and 1930. At
this stage he wouldn’t call this entity a density matrix although he did it in his two
subsequent papers (1930, 1931) and all the others he came to write using this
formalism (48). The density matrix has been used in two contexts: statistical
and quantum mechanical systems.
It has found many applications in statistical mechanics because this branch of
physics deals with problems where there are large numbers of particles, so that it is
not possible to determine the momentum and position of each of them. That is the
case of a gas, for instance. Hence, statistical methods are necessary in this case
because of “our lack of knowledge” (48) about the system, knowledge that could, in
principle, be obtained by tracking the movement of each particle.
The nature of the statistics in quantum mechanics is essentially different. It is a
consequence of the impossibility of acquiring a complete knowledge about the
system (47). Even if the system under study is in a state of maximum knowledge,
which corresponds to a wavefunction, a statistical description would still be
necessary. Statistical methods must be used in a quantum mechanical system even if
the number of particles is small.
The density matrix was used exclusively in statistical mechanics until the sixties,
when many papers in quantum chemistry began to be published by Löwdin, Golden,
41
McWeeny and many others. Since then, the application of the density matrix
formalism to quantum mechanics, and in particular to the electronic structure of
atoms and molecules, has been developing continuously (48).
These two approaches to the density matrix, the statistical and the quantum
mechanical may cause confusion. For instance, Landau and Lifschitz use one
approach in their Statistical Physics (1958) and another in their Quantum Mechanics
(1958) (48)(49).
The quantum mechanical interpretation of the density matrix will be used from now
one, because that is the one that is relevant to the themes that will follow, namely
DFT and TDDFT.
There are two main reasons for using a density matrix. Each of them will be
described separately.
1- When dealing with interacting systems like mixtures, collisions or photon
absorption by an atom or molecule, where no wavefunction can be found.
2 - To avoid the complexity of the many-body wavefunction when dealing with an
isolated system.
1- The density matrix is very important because it describes systems which cannot be
described by a wavefunction, usually because they are part of a larger system and the
interaction between the systems cannot be appropriately described because of the
impossibility to determine such a Hamiltonian (50) . An important example of such
interacting systems is the interaction of an atom or molecule with a photon, which is
one of the important applications of TDDFT. This application of the density matrix
is treated in textbooks by Parr (50), Landau (49), d Espagnat (51), McWeeny (52),
42
ter Haar (48), Dreizler (53), Pilar (54) and others. This application of the density
matrix is exclusively quantum mechanical.
This approach can also be used to calculate the electronic structure of an isolated
atom or molecule (54). In that case, the electrons are considered to be interacting
with each other in pairs and there are no interactions involving three or more
electrons at the same time. This way the electron is considered as interacting with a
separate system which is the rest of the atom or molecules, whose details can be
averaged to save computing time. The rest of the atom or a molecule is treated as a
“heat bath” (48).
2- The density matrix is also commonly used in a more simple context: an isolated
atom or molecule, which can have its own wavefunction because it is not interacting
with anything. This wavefunction is often a Slater determinant (55). The density
matrix is used here to calculate expectation values of observables, in particular the
energy, without referring to the many-body wavefunction. The wavefunction is
expanded in basis functions in order to solve the Rootham-Hall equations in a SCF
calculation, and the density matrix consists of products between the expansion
coefficients. This approach is described in textbooks by Szabo (55) , Pople (56),
Frisch (57), Koch (58) and others.
In conclusion, the density matrix is essential to describe interacting systems and it is
also useful in calculating expectation values of observables in more simple systems
(isolated systems) which have a wavefunction which is too complex to be tackled by
ordinary methods. The first application is crucial for describing electronic
excitations, which is the interaction of a photon with the electronic shell of an atom
or molecule.
43
Consider a system with many degrees of freedom, all interacting with each other, of
which there is an interest in only a few of them. The interesting ones will be
represented by x and the rest will be represented by q. A typical example of such a
system is a heat bath (coordinates q), in which the interesting system (x) is inserted
(48) (50). The total wave function for such a system will be a function of both q and
x : ψ (q, x) .
The average value of an operator �̂�𝐴 in this case would be given by:
< �̂�𝐴 > = �𝜓𝜓∗ (𝑞𝑞, 𝑥𝑥′)�̂�𝐴 𝜓𝜓 (𝑞𝑞, 𝑥𝑥)𝑑𝑑𝑥𝑥 𝑑𝑑𝑞𝑞
Defining the density matrix ρ as (48) :
< 𝑥𝑥 | ρ | 𝑥𝑥′ > = ∫𝜓𝜓∗ (𝑞𝑞, 𝑥𝑥′) 𝜓𝜓 (𝑞𝑞, 𝑥𝑥)𝑑𝑑𝑞𝑞
The average value can then be rewritten as :
< �̂�𝐴 > = ∬ < 𝑥𝑥 | 𝜌𝜌 | 𝑥𝑥′ >𝑑𝑑𝑥𝑥 < 𝑥𝑥 � �̂�𝐴 � 𝑥𝑥′ 𝑑𝑑𝑥𝑥′ = Tr ρ �̂�𝐴
This way it is possible to calculate the expectation values of any observable, by
calculating the trace of the matrix obtained by multiplying the density operator by
the operator that represents the observable of interest. Hence, the density matrix
contains all the information about the system and there is no need for a
44
wavefunction.
The density matrix < 𝑥𝑥 | 𝜌𝜌 | 𝑥𝑥′ > is represented as γ (𝑥𝑥 , 𝑥𝑥′) by Parr (50) and as ρ
(𝑥𝑥 , 𝑥𝑥′) by Landau (49). In the case of an isolated system, there are no variables q,
and it can be described by a wavefunction which is a particular case of a density
matrix of the form
γ (𝑥𝑥 , 𝑥𝑥′) = 𝜓𝜓∗(𝑥𝑥′) 𝜓𝜓 (𝑥𝑥)
Time evolution of the density matrix
There is no time evolution operator that can be applied to the density matrix in order
to obtain its time dependence. The time evolution of the density matrix is obtained
by solving Eq :
�𝐻𝐻�(𝜕𝜕),𝜌𝜌(𝜕𝜕)� = 𝑖𝑖ħ 𝜕𝜕𝜕𝜕𝜕𝜕
𝜌𝜌(𝜕𝜕)
Which is similar to the Liouville equation of classical statistical mechanics (47).
This equation will be seen in the next section in the context of time-dependent
density-functional theory.
45
Density Functional theory (DFT) and Time-dependent density-functional
theory (TDDFT)
DFT started in the 1920’s and at that time it was called the Thomas Fermi Method
(48) (it later became the Thomas Fermi Dirac Method).
This method has undergone various refinements but the essence remains: electrons
are regarded as a collection of non-interacting particles (59) and the electronic
density plays the prominent role. That means that the energy of the N- electron
system can be determined without referring to the wavefunction and its 4N variables.
Instead, it uses the electron density which only depends on three variables (space
coordinates), regardless of the number of electrons in the system. The focus on the
electron density, along with density matrix techniques, allows for avoiding the
unnecessary complexity of the wavefunction. According to Gross (60) the
wavefunction is “overkill”. Hence, the DFT approach allows for faster calculations.
A major breakthrough came in 1964, when Hohenberg and Kohn set rigorous
foundations to DFT, by proving that there is a unique mapping between the electron
density and the external potential. In this landmark paper, they developed a theory
for a homogeneous electron distribution and also for a slowly varying electron
distribution (59).
The modern expression for the energy functional is shown below:
E[ρ] = T[ρ] + J[ρ] + Exc [ρ] + �𝜌𝜌 (𝑟𝑟) 𝑣𝑣(𝑟𝑟)𝑑𝑑𝑟𝑟
46
Where 𝑟𝑟 is the spatial coordinate, 𝜌𝜌 is the electron density and v (r) represents the
nuclear interactions. The three functionals on the right of the DFT equation are J[ρ] ,
representing Coulomb electron-electron repulsion, T[ρ] representing the kinetic
energy and Exc [ρ] that corresponds to the exchange-correlation energy.
The ground state energy of the many-electron system can be obtained by minimizing
the energy functional above (50) and the corresponding density (ρ) is the ground
state electron density. Similarly to other variational problems, it can be solved
employing the method of the Lagrange multipliers, using as a constraint the fact that
the electron density must integrate to the total number of electrons:
�𝜌𝜌 (𝑟𝑟) 𝑑𝑑𝑟𝑟 = 𝑁𝑁
Hohenberg and Kohn opened up the doors for the development of improved
functionals, like the LDAxc (Local Density Approximation) and others more
sophisticated, which also include gradients of the electron density like the GGA’s,
exact exchange (OEP), asymptotically correct (ac) and others.
The LDAxc was the first exchange-correlation functional to be conceived and it is
based on the homogeneous electron gas, which is an idealized situation where
electrons are almost uniformly distributed over a homogeneous background of
positive charge.
Thomas and Fermi (61) (62) had already used a similar model in the 1920’s, when
they proposed the first functional which could give the energy of an atomic or
molecular system based only on the electron density, so that no wavefunctions were
necessary. Based on the electron gas model, they found an expression for the kinetic
energy which depended only on the electron density, to which they added an
47
expression for the electron-electron repulsion and finally another for the nucleus-
electron attraction, which together constitute the Thomas-Fermi equation.
The Thomas-Fermi equation was a good start although it lacked any exchange and
correlation effects. Dirac worked on the development of the expression for the
exchange energy, and the resulting equation is known as Thomas-Fermi-Dirac (62).
Although this latest equation could include exchange effects, the correlation
wouldn’t be included until the formal development of DFT (59) and the advent of the
local density approximation (LDAxc).
Similarly to the HF approximation, the many-electron system in DFT is simplified
by introducing a set of one-electron equations, which are called the Kohn-Sham
equations and are normally solved iteratively by self consistent field calculations.
The result is a set of orbitals which are called Kohn-Sham orbitals.
Kohn-Sham (KS) orbitals are fundamentally different from the more traditional
orbitals that are obtained in wavefunction based methods, and that are normally
present in the chemistry literature. The physical meaning of KS orbitals has been a
subject of extensive debate, as they don’t represent “waves of probability” like the
traditional ones. Instead, they are entities introduced with the purpose of determining
the ground state electron density of a many-electron system. Although many refuse
to accept that KS orbitals possess any physical meaning, this issue has been settled
by a landmark paper from Baerends (63), who argues that KS orbitals not only have
physical meaning but they are actually more useful and interesting than traditional
orbitals. For instance, Baerends showed that the application of the variational
principle to solve the Schrodinger equation may lead to orbitals that are too diffuse,
because of the need to minimise the energy.
48
Despite the fact that for some researchers the KS orbitals have no physical meaning,
there is no argument about the stronger connection between DFT and chemistry.
Wavefunction based methods provide wavefunctions, which are very useful but at
the same time very abstract. They are waves of probabilities. DFT, on the other hand,
provides an exact electron density, which is something that can be directly detected
and measured in experiment.
Furthermore, DFT can provide some relevant chemistry parameters, like hardness
and electronegativity, in a direct manner. These parameters can be obtained by
differentiating the energy functional with respect to the number of electrons present,
as shown in the equations below, where µrepresents the electronegativity, in th
context of Pauling theory, and η represents the hardness, in the context of Pearson
theory (61).
𝛿𝛿𝐸𝐸𝛿𝛿𝑁𝑁
= μ
𝛿𝛿𝛿𝛿𝛿𝛿𝑁𝑁
= η
Perhaps the main shortcomings of DFT are twofold: (i) the fact that it is a theory that
only deals with the ground state of the system and (ii) the present absence of the
exact exchange correlation functional (Exc [ρ] ) . The first limitation has been dealt
with with the development of time-dependent DFT, which will be explained in the
next section. To address the second problem, a large number of functionals have
been developed in order to approximate the exact one. Until the exact exchange-
correlation functional is found, it is important to study the functionals that have been
49
created so far and be able to employ the one that is most appropriate for a particular
kind of calculation. A description of some of the most important exchange-
correlation functionals used today follows.
The Local Density Approximation functional (LDAxc) possesses a kinetic energy
term that corresponds to a system of non-interacting electrons. The difference
between the kinetic energies of the interacting and non-interacting electrons, together
with all exchange and correlation effects, are gathered in one term which is called
the exchange-correlation functional (58). The functional derivative of the LDAxc
exchange-correlation functional (LDAxc) provides the exchange-correlation energy,
which can be represented as:
LDAxcE = LDA
xE + LDAcE
The first term corresponds to the exchange energy, which is represented by(64) :
LDAxE = -9/4 αex [3/4π] 1/3 1
3/411 )]([ rdr ∑∫
γ
γρ ,
where the value of the αex factor is 2/3 and γ is the spin index, which represents α
and β spins.
The correlation term is represented by:
50
LDAcE = 1111111 )]()([)( rdrrr c
βα ρρερ∫ ,
where 𝜌𝜌1α and 𝜌𝜌1
β are the spin densities .
The factor )]()([ 1111 rrc βα ρρε is the correlation energy per electron and it cannot be
found analytically. However, a value for this term has been calculated by Ceperley
and Alder (65), who performed Monte Carlo calculations on the homogeneous
electron gas. Vosko, Wilk and Nusair (66) have fitted this parameter and obtained an
expression for LDAcE which is much used today.
Although the LDAxc is based on the homogeneous electron gas model, it applies
successfully to real systems where the electron densities are not homogeneous.
LDAxc is very useful to calculate a number of molecular properties, including
harmonic frequencies (58), metallic magnetism (67), surface electronic properties of
metals and semi-conductors (68), excitation energies (69) and also equilibrium
structures of molecules. On average, the LDAxc overestimates binding and hence
molecular bond lengths are too short and its accuracy is around 0.01 to 0.02 Å. For
bond angles the accuracy is 1 degree (58).
However, the LDAxc fails to calculate accurately many other important properties.
Molecular properties that cannot be successfully calculated using the LDAxc include
bond energies, ionization energies and electron affinities (62)(58). The LDAxc total
energy of a metal surface is too low and the LDAxc energy for atoms is too high.
The reasons for these failures of the LDAxc include the fact that exchange is
underestimated by about 10-15% and correlation is overestimated by up to 100-
51
200% (70). The LDAxc results can be improved by introducing self-interaction
correction and/or gradients of the electron density (58).
- The Generalized Gradient Approximation (GGA) :
This range of xc functionals was developed in order to overcome the limitations of
the LDAxc. Most of them are also based on the homogeneous electron gas model, but
they incorporate gradients of the electron density in order to account for the non-
homogeneous electron distributions that are observed in real systems. For this reason
they are called gradient corrected functionals (GGA’s) and can be written as (58):
GGAxcE )]()([ 1111 rr βα ρρ = ∫ (f 111111111 ))(),(),(),( rdrrrr βαβα ρρρρ ∇∇
GGA’s can also be split into an exchange and a correlation part, in the same fashion
it was shown for the LDAxc. There are many GGAxE and GGA
cE functionals available
nowadays, and these can be combined in various ways, by choosing the exchange
and correlation parts separately (71). A particularly important combination is found
in the functional B88X + P86C, which combines the exchange functional proposed by
Becke in 1988 (72) with the correlation functional proposed by Perdew in 1986 (73).
This functional has been used in this work because it is proven to provide accurate
results for a range of molecular properties, including total molecular energies, and it
is particularly good for transition metal complexes (58).
The B88X + P86C functional provides bond energies with an accuracy of 2 kcal/mol
(58). The method post SCF B88X + P86C has been employed extensively in this thesis
and it implies an optimisation with the LDAxc functional, where the B88X + P86C
functional is used only during the last SCF cycle. This way the calculated geometry
52
is not changed (it is still the LDAxc geometry) but there is an improvement in the
calculated energy. The B88X + P86C functional is also excellent for calculating IR
frequencies. Furthermore, it was the functional of choice of Ziegler et al. in their
study of excitation energies of d1 transition metal complexes (74).
There is a class of functionals called asymptotically correct, which is employed to
calculate excitation energies and polarisabilities. These functionals will be described
in chapter 5.
Although DFT is a mature subject, its time dependent version, called time dependent
DFT (TDDFT), is a theory that is not yet complete, although it is already very
developed. TDDFT can calculate the interaction of an atom or molecule with an
external field, usually the electromagnetic field of a photon. Hence, various dynamic
properties can be calculated, like excitation energies and polarisabilities. TDDFT has
been successfully applied to a variety of molecules (open and closed shell), radicals,
fullerenes, quantum dots and also solid state problems (60). Before the advent of
TDDFT, excitations were usually calculated using the configuration interaction (CI)
method (55), which is computationally very demanding, although there are variations
of the method which allow for faster calculations, like for instance CIS (CI –
Singles), which only considers determinants corresponding to single excitations,
instead of considering all possible excitations. Although CI has the potential to yield
exact results when an infinite basis set and an infinite number of determinants are
used (55), this sort of calculation is not viable to be applied in calculations involving
larger systems, like the complexes studied in this thesis, because it is too demanding
for present day computing power.
In the same way as rigorous DFT started with the Hohenberg-Kohn (HK) theorem
(59), rigorous TDDFT started with the Runge-Gross (RG) theorems (75) which
53
extend the Hohenberg-Kohn theorem to the time dependent situation.
The RG theorem has been proved in a reductio ad absurdum fashion, similarly to the
HK theorem. The RG invertibility theorems, which prove the uniqueness of the
representation of the time dependent density by a time dependent potential, was
proved by expanding the time dependent quantities in a Taylor series around the
initial time (76).
Some authors found counter examples where the invertibilty properties do not hold.
However the RG theorems have subsequently been proved by referring to other
mathematical theories like differential equations and density matrix (76). Chernyak
(76) presents a formulation of TDDFT which is based on the density matrix, where
the dynamics of the system is governed by the Liouville equation. It is an interesting
formulation because it provides a direct comparison to time dependent Hartree-Fock
theory (TDHF), which is a wavefunction based method. Furthermore, Casida (77)
has found an expression which converts wavefunctions into KS orbitals. A further
development of the application of the density matrix formalism to TDDFT was
proposed by Baerends in 2007 and it is now called time-dependent density-matrix-
functional theory (78).
Because a time dependent exchange correlation (xc) functional doesn’t exist, as DFT
is a ground state theory, researchers have to adopt the adiabatic approximation, in
which the time dependent xc functional is assumed to be the same as in the time
independent situation. This functional used in TDDFT calculations is called ALDA,
which stands for adiabatic local density approximation. That means that two
functionals are employed in a TDDFT calculation: one functional is used in the SCF
step, where the orbitals are found, and ALDA is used in the TDDFT step, when the
excitation energies are calculated. Although, at present, ALDA is the only functional
54
that can be used in the TDDFT step, there are many functionals that can be employed
in the SCF step. For instance, a TDDFT calculation may be performed using B88X +
P86C /ALDA, which means B88X + P86C in the SCF step and then ALDA, on the
TDDFT step.
2.3 Relativity applied to chemistry
Einstein’s theory of relativity offers many practical applications in the domains of
chemistry. Although it may look a bit surprising to some people, relativity is needed
to explain ordinary observations like the yellow colour of gold or unusual
characteristics of mercury, like its liquidity and its tendency to form amalgams with
different metals.
Electrons travel at relativistic speeds when they are close to a highly charged
nucleus. It is not totally clear what should be considered a heavy nucleus, as there
are different opinions about the need for relativistic corrections in copper systems
(46)(79) (80), for instance. However, there is little doubt that atoms like silver and
gold are to be considered heavy.
Relativistic effects have been used to perform geometry optimizations and frequency
calculations in ADF since the 1980' s, using initially the Pauli Hamiltonians (81) and
more recently the zeroth order regular approximation (ZORA) (82) (83) which is the
most used nowadays, and is the result of a zero order expansion of the Dirac
equation, which is the fully relativistic equation. The ZORA equation uses the same
exchange-correlation potential used in non-relativistic DFT but its kinetic energy
operator depends on the mass of the electron, so that it accounts for the variation of
the latter due to the relativistic effect.
The basis set used in ZORA calculations is different from the ones used in non-
55
relativistic calculations . Because the relativistic effects operate mainly in the
proximity of the nucleus, the ZORA s-type electrons do not move in Slater type s
orbitals. The relativistic calculation requires a different basis set, which accounts for
the wiggle of the s orbitals in the proximity of the nucleus, which is due to the very
strong potential (especially when it is a heavy nucleus). The basis set which is
implemented in ADF to be used in relativistic calculations is called DIRAC and it is
not a Slater-type orbital. DIRAC orbitals can have fractional exponents of the radial
dependence of the basis function (η) so that they account for the contraction of the s
and p orbital and to the diffuseness added to d orbitals as a result of relativistic
effects. Dirac basis functions have the following basic form: r η-1 e - ξ r .
The relativistic contraction of orbitals can be understood by referring to Heisenberg's
uncertainty principle. The mathematical expression of Heisenberg’s uncertainty
principle is shown below:
∆x ∆px ≥ ħ/2
It states that the product of the uncertainty in measuring position with the uncertainty
in measuring momentum must be equal or larger than ħ /2. That means that it is
impossible to determine accurately the position and momentum of an electron
simultaneously.
In a relativistic situation, the momentum of the electron is very high because, in
addition to the high speed, there is also an increase in mass, due to the relativistic
effect. The uncertainty in momentum becomes larger so that the uncertainty in
position is reduced and the electron can be located closer to the nucleus. The
contraction of orbitals causes a contraction of bond lengths in a molecule (81) .The
effect is more pronounced in small molecules, like metal hydrides (84), and it
56
increases down a group in the periodic table.
The use of relativistic effects is essential for the calculation of electron spin
resonance (ESR) parameters, like the g-tensors (85). These calculations will be
shown in chapter 6.
Relativistic effects are also very important in TDDFT calculations, and the way in
which they are implemented has been evolving continuously. The latest development
is due to Ziegler and Wang (86) who tested it in Pt (II) complexes. These authors
have also implemented spin flip TDDFT (SFTDDFT) which is a method that allows
for a change in spin state during the electronic excitation process (74). That means
that, for example, a doublet state can be converted into a quartet state upon
excitation. Ziegler has also developed an extension of TDDFT which allows for the
calculation of circular dichroism (87).
Although TDDFT has been developing constantly for twenty four years, since its
rigorous foundations were set by the RG theorems, there are still many limitations
which are criticized by some authors (60). Perhaps the main limitation of TDDFT is
the necessity of using the adiabatic approximation, which is the only way in which
the method has been implemented in quantum chemical codes.
2.4 The adiabatic approximation
Adiabatic phenomena are the ones in which there are no transfers of energy
among the constituent parts of the system. They are well known in many areas of
classical and quantum physics. Their applications include explaining the mechanical
equilibrium of the terrestrial atmosphere (43) and also the Carnot cycle, which is the
basis of internal combustion engines. In the domain of quantum mechanics, the
57
adiabatic approximation was already in use in the 50’s, when Gell-Mann & Low (88)
applied it to solve the time-dependent Schrodinger equation using perturbation
theory. They suggested that the perturbation could be switched on and off
adiabatically if it was done very slowly (89).
Levine and Bernstein, in their classical “Molecular reaction dynamics and chemical
reactivity” (90), have also made use of this concept, in the context of molecular
collisions. In their low-velocity, or adiabatic limit, energy transfer is very small. It
happens when the duration of the collision is much larger than the period of vibration
of the target molecule, so that it can “accommodate itself to the perturbation”
(90). It has the same meaning as Gell-Mann’s idea of switching the perturbation
slowly.
TDDFT is also based in perturbation theory and its adiabatic approximation has the
same meaning as the other quantum mechanical examples provided above, so that it
is valid when the perturbation is switched on slowly, with respect to the timescale of
the atomic and molecular dynamics. However, there are significant differences,
which make the TDDFT problem more complicated. The perturbation is now caused
by a photon (integer spin) instead of a colliding proton (half-integer spin) or small
molecule. Furthermore, the molecular dynamics is now considered in its full extent,
including electronic effects like Coulomb interaction and exchange and other spin
effects, so that it is more difficult to determine the timescale of events. For instance,
the period of precession of a spin-orbit interaction may be smaller or larger than the
time taken for an electronic excitation to occur (about 1 femtosecond). That will
depend on the amount of energy splitting caused by the spin-orbit coupling (47) .
This situation is more complicated than the case of the molecular collision, because
it involves electronic effects like Coulomb repulsion, exchange, spin and others.
58
Another problem that arises as a result of the adiabatic approximation is the loss of
memory (59) (60). "Much has yet to be understood” about the need for the memory
of the past history of the time dependent system. Wavefunction based methods don't
present this problem: the knowledge of the wavefunction at any instant of time will
be enough to calculate the time evolution of the system. However the time dependent
xc functional at time t depends on the electron density at all previous times (60), and
this information is lost when the ALDA functional is used. The memory problem is
the price paid for avoiding the complex wavefunction.
In conclusion, the adiabatic approximation assumes that the perturbation, which is
usually an external electromagnetic field, changes slowly in time. That is similar to
the LDAxc approximation, which assumes that the electron density varies very
slowly in space. It is seen that both of these approximations can work satisfactorily
even beyond their range of applicability.
59
Chapter 3 The structures of gas phase copper
and silver complexes
This chapter presents the lowest energy structural isomers of the complexes studied
in this thesis, calculated using DFT. The structures are discussed based on their
binding energies which are analysed using Incremental Binding Energies and also
the Ziegler-Rauk-Morokuma decomposition scheme. Charges on the atoms are
analysed using the Voronoi deformation density.
3.1 Background Theory
Cu (II) and Ag (II) have a high charge density so that they are not easily stabilized in
the gas phase. Pyridine, acetone, acetonitrile, water and ammonia are examples of a
limited number of ligands which are capable of stabilizing these ions (91). The
process of stabilization of such metal ions involves a delicate balance between the
ionization energies, polarisabilities and dipole moments of the ligands and the metal.
If this balance is not established, there will be an electron transfer from the ligand to
the doubly charged metal, and subsequent Coulomb explosion (3).
It is intriguing the fact that certain ligands provide such stability even when their
ionization energies are lower than the second ionization energy of the metal. Such
complexes are in fact metastable and this is explained by the presence of the
avoided crossing of the charge transfer curve with the bound state.
3.1.1 Geometry optimisation
The optimisation of geometry is the earliest and still the most common
computational chemistry calculation. In this thesis, they will be performed as the
60
starting point to other types of calculation, which include electronic excitation and
magnetic properties. This is a very important step because all the other calculations
will depend strongly on the quality of the geometries obtained. According to
Baerends, the quality of the ground state geometry is crucial for the outcome of a
TDDFT calculation (92), so that it is necessary to choose the right functional for the
geometry optimisation as well as for the calculations of excitation energies, magnetic
properties, ionization energies and fragmentation pathways.
3.1.2 Analysis of binding energies
The binding energies corresponding to the calculated structures are analysed
according to the Ziegler-Rauk-Morokuma scheme (93) (94), which will be briefly
reviewed here. In addition to that, incremental binding energies (IBE’s) have been
calculated, in order to provide further insight into the calculated structures of the
complexes.
IBE’s are calculated according to the formula: [MLN]2+→ [MLN-1 ]2+ + L . They
provide a measure of how favourable it is for a given complex to incorporate an
extra ligand in its solvation shells. These energies depend on how many ligands are
already on the metal. Usually the IBE’s go down steadily as more ligands are added
to the complex and the shape of the decaying curve may provide insight into
preferred coordination numbers for a given complex or molecule.
In the Ziegler-Rauk-Morokuma scheme (95), the binding energy is split into three
main components: preparation energy, orbital interactions and steric interaction. The
latter is composed of Pauli repulsion plus electrostatic interaction. The preparation
energy, which can also be called deformation energy, is the energy difference
between the isolated fragment, which can be a particular ligand or a metal, and the
61
fragment as part of a complex. The more the ligands get deformed as a result of the
interaction with the metal, the higher the preparation energy. This energy is usually
very small, of the order of a few hundredths of an eV, and it is often neglected.
However there are cases when it is important, like for instance in the incorporation
of the CH3 radical into an ethane molecule (95).
Pauli repulsion, which is the only positive (and consequently repulsive) component
of the binding energy, is a measure of the effects that operate as a result of the Pauli
Exclusion Principle, proposed by Wolfgang Pauli in 1925. The principle states that
particles that have half-integer spins, like the electrons for instance, cannot occupy
the same state. Basically it means that you cannot have two electrons with the same
spin in the same orbital. This is a consequence of the antisymmetry of the
wavefunction. Because there are a limited number of states inside a given region of
space surrounding an atom or molecule, electrons are forced to keep away from each
other and as a result they have to spread out in space. It is an extremely important
concept because it explains how matter can have strength and be hard to compress. It
is also a very important concept in chemistry, because it provides the only repulsive
electronic term needed to describe molecular interactions (there is only one more
repulsive term which is due to the nucleus-nucleus interaction). Most authors focus
on the attractive aspects of molecular interactions, but the repulsive components are
equally important. Pauli repulsion is a prominent factor in determining bond lengths
and molecular geometries. It is important to notice that the repulsion between bulky
ligands, for instance, is due to the Pauli principle and not to electrostatic repulsion
between electron clouds, as it is described in some major textbooks (e.g. (96)) ; in
fact interpenetrating electron clouds attract each other, and that gives rise to the
negative electrostatic energy at a typical molecular bond length . The electrostatic
62
energy only becomes positive at very short bond lengths, due to nucleus-nucleus
repulsion. Another common misconception is to assume that the repulsive wall in the
energy versus bond length curve is due to the nucleus-nucleus repulsion. That
repulsive wall is also a result of Pauli repulsion (95).
The electrostatic interaction, which is the negative (i.e. favours bonding) component
of the steric interaction, consists of the nucleus-nucleus, nucleus-electron and
electron-electron interaction. Although it has considerable positive components, like
the nucleus-nucleus interaction, the electrostatic interaction will normally be
negative at typical molecular distances.
Finally, the last component of the binding energy is the orbital interaction, which can
also be called charge transfer. It consists of the transfer of electrons between the
frontier orbitals of the species involved. In the case of the complexes studied here,
charge flows from the water to the positively charged copper atom. More
specifically, there is an interaction between the low lying LUMO of the metal and
the HOMO of the ligand (water). Orbital interactions comprise the most negative
contribution to the binding energy and, together with the electrostatic interaction; it
must overcome Pauli repulsion so that bonding can take place.
3.1.3 Charges on atoms
The calculation of atomic charges is a controversial issue and there are many
schemes available to perform this task. Difficulties arise because atomic charges are
not observables in the sense that they cannot be directly calculated by using an
operator and wavefunction, as it can be done for energy, momentum and others.
Furthermore, it invokes the contentious issue of the existence of atoms inside
molecules (AIM) (61). All schemes devised to calculate atomic charges possess
63
some arbitrary parameters and their differences are essentially in the way they define
where an atom begins and where an atom ends, so that the total charges in a
molecule can be partitioned appropriately.
The Mulliken charge analysis is the oldest and most traditional scheme. It is
implemented in most quantum chemistry codes and it is based on Hartree-Fock-like
wavefunctions. More recently, more accurate schemes have been devised in order to
overcome some of the deficiencies of the Mulliken scheme. The main problem with
the Mulliken analysis is that it uses non-orthogonal orbitals, and that causes it to give
different results for each basis set used. If diffuse functions are used the results may
become totally meaningless.
The charge analyses performed in this thesis are based on the Voronoi deformation
density (VDD), which is a scheme based on the electron density rather than on
wavefunctions. This scheme produces consistent and meaningful results when
compared to other traditional schemes (97). In particular, it is basis set consistent and
also it gives results that correspond to chemical intuition. For instance, it doesn’t
yield a high polarisation between atoms that are joined by a covalent bond (97).
The Voronoi charges scheme divides the intermolecular space in cells, which are
called Voronoi cells. They are analogous to the Wigner-Seitz cells that are used in
solid state physics. The Voronoi calculation starts with the “protomolecule”, which
is comprised of non-interacting atoms brought together. The VDD is obtained by
measuring how much electron density crosses the boundaries of the Voronoi cells
when the interaction is switched on. In an ADF calculation the interaction between
the atoms is switched on in two steps: firstly the atomic orbitals are orthogonalised
and the Pauli principle is applied. Subsequently, the system relaxes through the SCF
64
cycles. Voronoi charges are calculated for each of these steps, but in this work only
the charges obtained after the final SCF relaxation (VDD) will be considered.
3.1.4 The Jahn-Teller effect
According to a theorem by E. Teller and H.A. Jahn, dated back to 1937, a molecule
in a degenerate ground state will become distorted in order to lift this degeneracy.
This effect is very important in transition metal complexes, in particular when the
coordination number is five or six. The effect is well documented for octahedral
transition metal complexes (15).
Jahn-Teller distortion occurs in certain octahedral transition metal compounds in
order to lift the orbital degeneracy that would be observed if the structure was
perfectly octahedral. It happens as if “nature dislikes orbitally degenerate ground
states” (98). More specifically, in a d9 octahedral complex the 𝑑𝑑𝑧𝑧2 and dx2
- y2 orbitals
have the same energy and there are 3 electrons to be shared between them. That can
be done in two different ways and both configurations will have the same energy
(98). There are two ways of avoiding this degeneracy: the equatorial bonds can be
shortened and the axial bonds elongated or vice-versa. Both situations are possible,
but Cu (II) and Ag (II) normally prefer the first (elongated axial bonds)(98).
3.2 Computational details
Calculations were performed using the Amsterdam Density Functional (ADF) (71)
program, which perform self consistent field DFT calculations employing Slater type
basis sets and a density fitting scheme.
Preliminary calculations were performed, employing various basis sets and two
different methods in order to find out what conditions provide the best agreement to
experimental data available for the singly charged copper water complexes.
65
Geometries have been optimised using the LDAxc and also the B88X + P86C
functional (more information about those functionals is provided in chapter 2).
For each case, a relativistic optimisation was performed, employing the ZORA
equation, in addition to the non-relativistic calculation, because it is not completely
clear if relativistic corrections are needed in calculations involving copper. This is a
controversial issue. Although some authors (46) state that relativistic effects are
“normally negligible for the first three rows in the periodic table”, which means up
to Kr (with Z=36), other authors have found that Cu is the only element in the first
transition series that is substantially affected by relativity (79)(80).
In order to obtain further insight into the application of the Ziegler-Rauk-Morokuma
energy decomposition in this chemistry, fragment calculations have been performed
for complexes that are considered of high interest, like the four, six and eight-
coordinate structures. Fragment calculations can be performed using the ADF code
and, in the way they are applied more often in the present work, they employ whole
ligands as a basic unit in the calculation, so that the characteristics of the interactions
of these with the metal centre can be abstracted, avoiding unnecessary complications
introduced by internal processes peculiar to each ligand. This kind of calculation can
normally include restricted fragments only and that would rule out calculations on
the doubly charged complexes which have an unpaired spin. To overcome this
limitation, some advanced settings have to be used in the ADF program, like the
“fragoccupations” key.
66
Basis sets
Various basis sets were used and in all cases a frozen core was used. The core was
frozen at the 2p level in the case of copper and at the 1s level in the case of nitrogen
and oxygen.
The basis sets used were the following: TZ2P, which is a valence triple ξ basis set
with two polarisation functions, TZ2P+, which is a TZ2P basis set with extra 3d
functions on the metal, QZ4P, which is a valence quadruple ξ with four polarization
functions. Also, two types of even tempered basis sets (ET) have been employed.
The Even tempered basis sets currently implemented in ADF have been developed
by De Chong and van Lenthe (99) in order to offer an alternative to the large QZ4P
basis set. The problem with the QZ4P basis set is that it is very costly
computationally, although it is very useful for benchmark calculations on small
molecules. Other problem frequently associated with calculations involving very
large basis sets, like QZ4P, is that some functions may become linearly dependent.
To overcome this problem ADF offers the “Dependency” key. This problem has
also been addressed during the development of the latest ET basis sets. Using design
principles based on even-tempering and completeness profiles, the ET suite has been
developed to perform, at a lower cost, almost as well as a very large (almost
complete) basis set.
The two ET basis sets employed in these calculations were the ET-pVQZ, which is a
valence quadruple ξ basis set, and the ET-QZ3P diff, which is a valence quadruple ξ
basis set with 3 polarization functions and one set of diffuse s, p, d and f Slater type
orbitals (STO’s).
67
The ET-pVQZ basis set is the latest and the best ET STO basis set of quadruple ξ
quality, and it is the one recommended (by ADF developers (71)) to follow TZ2P+
in the hierarchy of basis set used to perform standard calculations like geometries
and energies.
The Dirac basis sets TZ2P, TZ2P+ and QZ4P were used for the relativistic
calculations. These basis sets have special characteristics which account for the
relativistic effects, like for example the wiggle of s orbitals in regions very close to
the nucleus, as described in chapter two.
3.3 Results: Calculated structures of the complexes
studied in this thesis
3.3.1 Copper complexes
3.3.1.1 Cu (I) water and Cu (II) water
Previous studies of the solvation of copper by water and ammonia include the work
of Ziegler et al. (100), with doubly charged ion Cu (II) and the work of Feller et
al.(101) with the singly charged ion Cu (I).
Ziegler et al. have performed static and dynamical DFT calculations on [Cu(L)N]2+ ,
for N = 3 to 8 and L= H2O and NH3. Static DFT calculations were carried out
relativistically using the ADF program and employed the TZP basis set, with a core
frozen at 2p for copper, and the B88X + P86C functional. Dynamical calculations used
the Car-Parrinello method, which allowed determining the lowest energy structures
out of several starting structures.
68
These calculations were motivated by the intriguing result obtained by
experimentalists led by Stace (4) (8), who discovered the first evidence of aqueous
Cu (II) in the gas phase. Furthermore, Stace et al. have found that the most stable
geometry for solvated Cu (II) in the gas phase is the eight-coordinate structure. This
result contradicts the common belief that the preferred geometry of solvated copper
(II) is the six-coordinate Jahn-Teller distorted octahedral structure, which yields the
famous blue colour of solvated copper (II) in the liquid phase.
Ziegler could successfully explain this result, based on DFT and Car-Parrinello
calculations. His group found that the most stable structure for the eight coordinate
complex has four water molecules directly coordinate to the metal plus another four
molecules that are hydrogen bonded, providing a flat shape to the complex. Other
interesting conclusions were also achieved, based on the calculations of complexes
with other coordination numbers. It was found that the solvation energies depend on
the number of primary ligands (i.e., ligands in the first solvation shell), the number
of axially bonded ligands and finally on the number of hydrogen bonds.
The overestimation of hydrogen bond energies by DFT methods is not an issue in
this context because the preference for a hydrogen bonded position, instead of an
axial one, is very strong in this kind of system (100). The maximum ion intensity
found in the mass spectrum obtained by Stace and coworkers is further explained by
noting the instability of the smaller clusters with respect to electron transfer from the
water to the copper ion.
Calculations by Feller and coworkers have been performed on [M(H2ON)]+ , for N= 1
to 5 and M= Cu, Ag and Au. This group employed a variety of software packages
(Gaussian 94, Gamess, Molpro) and wave function based methods (RHF, MP2,
69
CCSD (T)), employing various basis sets, including the large correlation consistent
basis set. The solvation enthalpies of these complexes were calculated and it was
found that they are in good agreement to experimental results obtained in the gas
phase (101).
In agreement with the conclusions of Ziegler and his group for the divalent
complexes, it was found that for the copper complexes the water ligands prefer
hydrogen bonded positions to positions in the primary solvation shell of the metal.
However, in the case of singly charged copper this preference is manifest with only
three water ligands. It was also found that the bond lengths of the hydrogen bonds
were shortened with respect to the length found in pure water, due to the polarization
caused by the metal ion (this finding also agrees with the trends found by Ziegler and
coworkers on the divalent system). Feller has also concluded that MP2 was the
wavefunction based method that gave the best results, when compared to
experiment, for calculations in these systems.
By looking at the calculated structures, which are displayed in figure 3.1, two
solvation patterns can be identified in complexes with coordination numbers 3 and 4.
In one pattern, all the available ligand molecules are directly coordinated to the
metal, constituting the first solvation shell. In other cases, some of the ligand
molecules are hydrogen bonded to the molecules located in the first solvation shell
instead of coordinating directly to the metal. These ligand molecules constitute the
second solvation shell.
Hence, the number of possible structures for these gas phase complexes increases
substantially. Furthermore, the complexes cannot be identified by the coordination
number N only. To differentiate between different isomers of complexes of the same
70
coordination number, the following notation will be used : x + y , where x is the
number of ligand molecules in the first solvation shell and y is the number of solvent
molecules in the second solvation shell. This way we can refer to complex 2+1, for
instance, which has 2 ligands directly coordinate to the metal and a third ligand in a
second solvation shell coordinated via a hydrogen bond, and also to complex 3+0,
which has all 3 ligands directly coordinate to the metal centre.
The same notation will be used for copper ammonia complexes, in the next section.
Preliminary calculations
Due to existence of experimental results for the incremental binding energies of
singly charged copper water complexes, it was possible to carry out a preliminary
study of methods and basis sets in order to determine the conditions under which
these structures can be calculated best.
The results of all preliminary calculations are displayed in table 3.1, which displays
the total energy of each structure under different conditions.
Table 3.1 shows that for all levels of theory considered, the calculated IBE’s for
Cu+(H2O)2 range between 42-48 kcalmol-1 and therefore they are in good agreement
with the experimental values of 40.0 ± 3. This is particularly so for the non-
relativistic calculations which range from 42 – 45 kcal mol-1. In the case of
Cu+(H2O) the IBE’s calculated non-relativistically, which range between 42-50
kcalmol-1 are in better agreement to experimental values of 38.4 ± 1.4 kcalmol-1
than the IBE’s calculated relativistically which predicted increased bonding.
71
Table 3.1: Comparison between calculated and experimental IBE’s for the singly charged
copper water complexes, optimised using both LDAxc, with post SCF B88X + P86C
corrections to the energy, and B88X + P86C with a variety of basis sets. Relativistic corrections
were also employed in some optimisations, as indicated. Data refers to structures with N=1
to 5, namely structures 1, 2, 2+1, 2+2 and 3+2.
It is unexpected that the calculations involving the largest basis set, QZ4P, along
with the most sophisticated method here which is B88X + P86C with relativistic
Calculated and experimental IBE’s in kcal / mol Exp.
non-relativistic relativistic
post SCF B88X + P86C B88X + P86C post SCF B88X + P86C B88X + P86C
TZ2P TZ2P+ ET-
pVQZ ET-QZ3P diff TZ2P
ET-pVQZ TZ2P TZ2P+ QZ4P TZ2P TZ2P+ QZ4P
1 43.4 42.1 45.9 46.1 44.5 50.6 46.9 45.2 46.6 47.7 46.1 61.5 36.0 ± 3.0
(102)
38.4 ± 1.4
(103)
2 43.4 42.2 44 44.5 43.6 44.8 46.6 45.6 47 47.9 46.5 47.4 40.0 ± 3.0
(102)
40.7 ±1.6
(103)
2+1 18.4 18.5 17.7 17.8 18.O 18.2 18.7 18.6 18 19.5 18.9 18.5 16.4 ± 0.2
(104)
17.6 ± 2.0
(102)
13.7 ± 1.8
(103)
2+2 16.2 16.2 16 16.7 16.1 16.3 16.6 16.4 16 17.2 16.6 16.5 16.7 ± 0.2
(104)
16.0 ± 2.0
(103)
12.8 ± 1.0
(104)
3+2 6.2 6.1 6 7.1 6.5 7.3 5.1 5.6 5 7.2 6.5 6.5 14.0 ± 0.1
(104)
72
corrections, provided results of bad quality for the 1st IBE (61.5 kcal / mol
compared to an experimental result of 36.0 ± 3.0 kcal / mol). The result regarding the
2nd IBE is also not very satisfactory but in the case of the other IBE’s the results
achieve the expected quality.
All calculated values for the third and fourth IBE’s (corresponding to Cu+(H2O)3 and
Cu+(H2O)4 ) are in even better agreement with experiment than the first two IBE’s,
regardless of the level of theory used. For Cu+(H2O)3 the calculated values range
between 17-19 kcalmol-1 therefore in good agreement with experimental values of
17.6 ± 2.0 kcalmol-1 . One of the best values obtained for the third IBE is the one
calculated non-relativistically using the TZ2P basis set and the LDAxc functional
with post SCF B88X + P86C corrections (shown in table 3.1 first column). This
calculated value is within 0.8 kcal/mol of the experimental result obtained by
Magnera (102) in 1989.
Similarly, calculations of the fourth IBE’s are in good agreement to experiment.
Calculated values range between 16-17 kcalmol-1 and they are in agreement with
experimental values of 16.0 ± 2.0.
Calculations of the fifth IBE’s, however, were the ones that had the worst agreement
with experiment although the trend with respect to the lower coordinate structures is
correct. Considering all conditions tested, there has been an energy difference of at
least 7.8 kcal/mol between theory and experiment.
In conclusion, the best overall agreement between theory and experiment has been
found in the non-relativistic calculations employing the LDAxc functional with post
SCF B88X + P86C corrections to the energy. Under these circumstances, more than one
73
basis set performed well, but the TZ2P and TZ2P+ were the best. Their results were
very similar.
In this case the TZ2P has been taken as the basis set of choice for all calculations in
this thesis involving copper complexes because of its lower computational cost when
compared to TZ2P+, given that both provided similar results in this context.
Although the following sections will only show the lowest energy isomer for a given
coordination number, optimisations were initiated employing different starting
geometries in each case, so that various possible outcomes could be studied. These
different geometries included mainly variations of bond lengths and bond angles. In
the case of structures containing a second solvation shell, different arrangements of
ligands were considered and in general the structures containing double hydrogen
bonds were found to be more stable than the corresponding structures containing
single hydrogen bonds (it is curious that even ammonia, which has only one lone
pair of electrons, could also engage in double hydrogen bonds) . This strategy is
needed because the geometry optimisation algorithms employed in this work locate a
local energy minimum and it is often difficult to be sure if this local minimum is
also the global energy minimum for a particular structure.
The calculated structures of the monovalent copper water complexes have been
calculated and are shown in figure 3.1.
74
N=1 N=2
N=3(a) -38.86 eV N=4(a) -53.72 eV
N=3(b) -38.58 eV N=4(b) -53.54 eV
N=4(c) -53.11 eV N=5
75
N=6 N=7
N=8
Figure 3.1: Structures of Cu+(H2O)N complexes, 1 ≤ N≤ 8 .Total energies are shown for
competing structures only.
The two-coordinate structure is found to be linear, as in the case of [CuCl2]- , which
is made by dissolving CuC1 in hydrochloric acid (15). The three-coordinate
structure is asymmetric but is nearly a trigonal planar, as in K[Cu(CN)2], which in
the solid contains Cu(CN)3 units linked in a polymeric chain (15).
The four-coordinate structure is tetrahedral like the condensed phase [Cu(CN)4] 3- ,
[Cu(py)4] +, and [Cu(L-L)2] + (e.g. L-L = bipy, phen) (15), although this is not the
lowest energy N=4 complex.
76
Complex 2+1 has an energy which is lower than the energy from the 3+0 complex
by 0.3 eV. Surprisingly, the structures with a second solvation shell are preferred.
That seems to contradict the chemical intuition based upon solution chemistry, where
it would be expected otherwise.
An even larger difference in energy is observed between the four-coordinate
complexes, which are presented here as 4+0, 3+1 and 2+2. The 4+0 complex is less
stable than the 2+2 structure by 0.61 eV, whereas the 3+1 complex remained in
between the two, with an energy only 0.18 eV above the 2+2 structure.
This trend persists for higher coordination numbers. The structures with hydrogen
bonded ligand molecules are preferred and in fact, double hydrogen bonds are
preferred over the single hydrogen bonds optimised for lower N. This preference
becomes so strong that it becomes very difficult to optimise a structure with all
ligands in the first solvation shell. That is why the structures 5+0, 6+0 and so on are
not presented. In the doubly charged case, however, these structures will be
presented.
Calculations from Feller and coworkers, which employed the method MP2, have
also found that the formation of a second shell is favoured even before the saturation
of the first solvation shell (101).
Metal-ligand bond lengths increase steadily as more ligands are added. They start at
1.90 Å, in the case of the structure with only one ligand, and reach a maximum of
2.42 Å in the six-coordinate 4+2 structure. Interestingly, the metal-ligand bond
lengths of all the structures containing four ligands in the first solvation shell are
very similar, irrespective of the number of ligands in the second solvation shell. This
issue will be discussed further later in this chapter. It will also be considered in the
77
context of electronic excitations, in chapter five. It is also curious the fact that the
metal-ligand bond length of the one-coordinate complex is slightly larger than the
corresponding bond length of the two-coordinate complex.
The lengths of the hydrogen bonds also vary according to number of ligands present
in the structure. Such bond lengths range from 1.45 Å in the three-coordinate 2+1
structure to 1.80 Å in the eight-coordinate 4+4 case. All these hydrogen bonds have
lengths that are smaller than the ones observed in liquid water, without the presence
of metals or any other polarising entity. The hydrogen bonds in these calculated
structures are stronger than ordinary hydrogen bonds because of the polarising
influence of the metal. They have been named charge-enhanced hydrogen bonds (2).
They will be discussed further in a section later in this chapter.
Using the same computational details, the structures of the doubly charged copper
water complexes have been calculated and are displayed in figure 3.2.
N=1 N=2
N=3 N=5(a) -56.19 eV
78
N=4 N=5(b) -56.38 eV
N=6(a) -71.75 eV N=7
N=8 N=5(c) -56.16 eV
79
N=9 N=6(b) -71.12 eV
N=10
Figure 3.2: Structures of [Cu (H2O)N]2+ complexes, 1 ≤ N ≤ 10 . Total energies are shown
for some competing structures.
Although these doubly charged structures have been studied extensively using DFT
(100) further analyses have been made, in order to compare to the singly charged
structures, and in chapter 5 their electronic spectra will be discussed.
The doubly charged one-coordinate structure has a bond length that is similar to the
bond length in the corresponding singly charged structure (the bond length in the
singly charged complex is 0.05 Å larger). Although the ionic radii of these singly
and doubly charged ions are very similar, 0.77 and 0.73 Å respectively (according to
table 1.1), the charge changes dramatically. The effect of the charge is reflected in
80
the average binding energy of the complexes, i.e. the energies involved in Cu+ +
H2O Cu+ (H2O) and Cu2+ + H2O Cu2+ (H2O), which are 1.88 and 5.80 eV for
the singly and doubly charged respectively, according to tables B1 and B2 in
appendix B.
The main difference between the geometries of these two complexes lies in the
dihedral angle, which is much larger in the singly charged complex. An attempt to
optimise these structures with C2v symmetry, in which the dihedral angles would be
180 degrees, resulted in a less stable structure. A small enlargement of the O-H
bond is also noticed in the doubly charged complex, as a result of the migration of
charges from the hydrogens towards the oxygen, resulting in an activation of the O-
H bond. Fragment calculations have been performed on both the singly and the
doubly charged complexes and it has been found that the Pauli repulsion term is
higher, by 2.91 eV, in the singly charged complex. This can be explained by the
presence of the extra electron in the bonding region. What is more curious is that the
electrostatic interaction terms are almost equal; they only differ by 0.2 eV. The
relatively small electrostatic interaction energy on the doubly charged complex is
possibly due to its relatively long bond length. It is possible that the optimum
distance could not be reached because of Pauli repulsion, as the lone pair of
electrons, in the 2p orbital of the oxygen atom, is interacting with 3d orbitals in the
copper. It has already been described by Baerends (105) that lone pairs donated by
ligands interact with upper core orbitals in the metal (3s, 3p), which causes strong
Pauli repulsion. This effect is particularly strong in copper because its 3d orbitals are
at the same distance from the nucleus as the 3s and 3p orbitals. This distance for the
copper atom is 0.32 Å. Baerends (95) has also shown examples of compounds
where the electrostatic interaction would be optimum at bond lengths smaller than
81
the actual ones, like in the case of N2C2. Furthermore, an analysis of the charge
distribution around the complexes, using the Voronoi scheme, reveals that the
polarisation between copper and oxygen is larger in the doubly charged complex but
only by 40 % (the difference in metal charge is 100%). That is a result of the
movement of charge from the hydrogen to the oxygen, in the doubly charged
complex, and consequent activation of the O-H bond.
The two-coordinate structure is linear like in the condensed phase
Diacquadithiocyanatocopper (II), which is shown in figure 3.3.
By looking at all the structures it can be observed that again two solvation patterns
are available to certain coordination complexes, namely the ones that have at least
three ligands in the first solvation shell and one or more in the second solvation
shell. Similarly to the singly charged complex, it has been found that structures with
a second solvation shell are more stable than structures of the same coordination
number which have only the first solvation shell, i.e., the 5th water ligand finds it
more favourable to be hydrogen bonded to the first solvation shell than be attached
directly to the metal centre. For instance, it can be seen in figure 3.2 that the complex
4+2 is more stable than the six-coordinate 6+0 complex by 0.63 eV and that the five-
coordinate 4+1 structure is more stable than the 5+0 structure by 0.22 eV.
Structures like the 2+1 structure obtained in the singly charged case are not stable in
the doubly charged case. The higher charge on the metal centre promotes proton
transfer and the complex suffers a Coulomb explosion, i.e. the complex dissociates.
Such fragmentation pathways will be discussed later in chapter 4.
However, a larger number of structures have been determined in the doubly charged
case. The higher charge on the metal centre permits a higher coordination number on
82
the first solvation shell (up to six) and this gives rise to structures like 5+0, 6+0 and
even the ten-coordinate 6+4 structure. None of these structures could be obtained in
the singly charged case.
Metal-ligand bond lengths are generally shorter than in the singly charged structures,
as expected. Likewise, the lengths of the hydrogen bonds have also decreased, when
compared to the singly charged complexes, as can be seen in figures 3.1 and 3.2.
That is due to the stronger polarisation of the ligands in the first solvation shell in the
presence of a doubly charged metal atom. This fact is further evidence of their status
as charge-enhanced hydrogen bonds. It is curious that metal-ligand bond lengths
remain more or less constant (at about 1.8 or 1.9 Å) despite the addition of more
ligands, except in the cases where the complex is strongly distorted, due to the Jahn-
Teller effect. In this case the metal-ligand bond length can achieve 2.57 Å e.g. as
calculated in the 6+4 structure.
With respect to the three-coordinate 3+0 structures, the irregular structure found in
the singly charged case becomes more regular and acquires an approximate T-shape
when the metal is doubly charged. This T-shape is also found for calculated
structures of doubly charged three-coordinate copper argon complexes (7).There is
little change in the bond lengths, however, as only one of them is larger in the singly
charged case (2.18 Å compared with 1.87 Å). The two other bond lengths remain the
same whatever the charge on the metal (1+ or 2+).
The four-coordinate 4+0 structures also show major structural differences when the
charge on the metal changes from 1+ to 2+. The singly charged structure displays a
tetrahedral geometry whereas in the doubly charged case a pseudo square planar
structure is more favourable. All four bond lengths are larger in the tetrahedral case.
83
As more water ligands are added to Cu (I), the tetrahedral structure tends to be
flattened, but the corresponding doubly charged structures are always more flat. This
is due to the unpaired electron at the metal centre. More on these differences will be
discussed in chapter four, under ionization energies.
Finally, the basic structures obtained in the Cu (II) that couldn’t be obtained in the
singly charged case are the five-coordinate 5+0 and the six-coordinate 6+0, which
are pseudo square based pyramid and pseudo octahedral respectively. The nine and
ten-coordinate structures are derived from these two: ten-coordinate is an octahedral
structure with four waters in the second solvation shell.
Hexaacquacopper (II) dinitrate (106) Diacquadithiocyanatocopper (II)
(107)
Thalium hexaacquacopper (II) sulphate
(108) Copper (II) hydroxide (109)
Figure 3.3: Experimental structures of some solid state copper (II) water complexes. The
colour coding employed is the following: copper atoms are represented in dark blue colour,
84
nitrogen in light blue, oxygen in red, carbon in black, hydrogen in grey and sulphur in
yellow.
For the sake of comparison, crystal structures of substances containing copper water
bonds are shown in figure 3.3. The metal-water bond lengths in these solid structures
are on average larger than the ones that have been calculated in this thesis and that
correspond to the gas phase (the calculated metal-water bond lengths that correspond
to the Jahn-Teller distortions have been left aside as none of the solid structures
presented here have such distortion, not even the octahedral one which is
hexacquacopper (II) dinitrate). The crystallised doubly charged copper structures,
presented in figure 3.3 present the following metal-water bond lengths:
Hexaacquacopper(II) dinitrate - [Cu (H2O)6)(NO3)2 ] 2+ - presents 2.07, 2.08 and
2.14 Å (106) , Diacquadithiocyanatocopper(II) - Cu (N C S)2 (H2O)2 - presents
2.00 Å (107) , Thalium hexacquacopper(II) sulphate - Tl2 Cu (H2O)6 (SO4)2 -
presents 1.95 and 2.01 Å (108). Interestingly, the hydroxide - Cu (O H)2 - presents a
similar bond length which is 1.97 Å (109), so that the extra hydrogen present in
water doesn’t seem to make a lot of different in the solid state. It will be seen in
chapter four that the removal of a proton causes significant changes in the structure
of the calculated gas phase complex.
Perhaps the most interesting aspect of these calculations on the doubly charged
copper complexes is their preference for structures with only four ligands in the first
solvation shell; if additional ligands are to be present, they are preferably hydrogen
bonded to the first solvation shell. This trend culminates with the 4+4 structure being
particularly stable and preferred over an octahedral-type 6+2 arrangement.
85
In order to obtain further insight into this chemistry, some energy decompositions
have been analysed. Firstly, it can be seen in table 3.2 that, as the coordination
number of the complexes increases, the total energy also increases steadily, as
expected. Also, the three components of the binding energy, namely Pauli repulsion,
electrostatic interaction and orbital interaction, also increase steadily (in the case of
the electrostatic interaction, it becomes more negative).
It is interesting to observe that in all cases there is a very strong Pauli repulsion.
Pauli repulsion is typically very high in complexes containing water as a ligand
because of the strong repulsion between the water lone pair of electrons and the
upper core shells, 3s in metals like copper (first transition series)(95). This repulsion
cannot be outweighed by the electrostatic interaction, which is the attractive part of
the steric interaction. The bonding in these complexes is only possible because of
strong orbital interactions.
structure 4+4 6+0 4+0
Total Pauli Repulsion: 323.69 230.53 158.83
Electrostatic
-71.90 -50.65 -37.67
Total Steric Interaction
251.79 179.88 121.16
Orbital Interactions -353.87 -251.05 -162.02
Total Bonding Energy: -102.11 -71.12 -40.86
Table 3.2: Energy decompositions extracted from optimisations of doubly charged copper
water complexes (total energies in eV are shown).
Table 3.2 shows the energy decomposition of selected doubly charged copper
complexes. It is seen that all components of the binding energy of a complex
increase steadily, as more ligands are added. This happens because every time you
86
add a ligand, the total energy of that ligand is added. That means that each of the
components of the internal binding energy of the ligand are added and that explains
the trends seem in table 3.2.
To have a better insight of what is happening during the formation of the various
complexes it is interesting to check only what happens to the energy decomposition
as a result of the new interactions created as a result of the formation of the complex.
That means that internal interactions of the ligands may be left aside, as they are
predominantly unchanged in complex formation. The ligands normally suffer small
deformations but the energies associated to these preparation energies are very small
in the systems studied here.
In order to obtain this information, fragment calculations have been performed,
where each water molecule behaves as a single unit during the calculation. Fragment
calculation on an open shell transition metal complex is an advanced feature of the
ADF software package, as explained in the introduction to this chapter.
The energy decompositions obtained from some of these calculations are listed on
table 3.3.
4+4 4+0 2
2+ 1+ 2+ 1+ 2+ 1+
Total Pauli Repulsion: 8.53 5.83 9.72 7.17 6.29 6.84
Electrostatic Interaction: -16.20 -9.52 -13.64 -8.90 -8.16 -6.91
Total Steric Interaction: -7.67 -3.69 -3.92 -1.73 -1.87 -0.07
Orbital Interactions -12.49 -3.39 -9.93 -2.95 -7.81 -3.76
Total Bonding Energy: -20.16 -7.08 -13.85 -4.68 -9.68 -3.83
87
Table 3.3 (previous page): Energy decompositions (eV) extracted from fragment
calculations involving singly (1+) and doubly (2+) charged copper water complexes, where
each water molecule is treated as a fragment. Total Steric interaction is the sum of the Pauli
repulsion and the Electrostatic energy.
First thing to notice is that the energies are much smaller than in table 3.2. That is
because the figures in table 3.2 add up to the total energy of the molecule (sometimes
called the absolute energy), including the total energies of each ligand and metal,
whereas in table 3.3 they represent only the binding energy, i.e. the energies
involved in the process of bringing the fragments together. Hence, another
interesting feature of fragment calculations is that they provide directly the binding
energies of the complexes involved.
Binding energies are often calculated by subtracting the total energies of each ligand
plus the metal from the total energy of the complex. This alternative approach
neglects effects like preparation energy and in future it will be referred to as the
standard way of calculating binding energies of complexes. All the energies shown
in this thesis, except the ones in the tables of energy decompositions, have been
calculated in this standard manner.
By observing the total binding energies at the bottom of table 3.3, it is seen that these
energies increase steadily, as expected. However, this steady growth masks some
subtleties within their components, more precisely the Pauli repulsion. Although this
component of the bonding energy increases when going from the two to the four-
coordinate complex, it actually decreases when going to the eight-coordinate
complexes. And that happens to both the singly and doubly charged complexes.
Focusing on the doubly charged case, where the increase in the binding energy when
going from the 4+0 to the 4+4 structure is 6.3 eV, it is noticed that the decrease in
88
Pauli repulsion (1.19 eV) is responsible for a substantial amount of the increase in
binding energy along with the increase in both the electrostatic interaction and
orbital interactions, which are exactly the same: 2.56 eV each. Therefore the
reduction in Pauli repulsion contributes almost 20% of the total increase in binding
energy in going from 4+0 to 4+4 structures. The answer to why the presence of four
hydrogen-bonded water molecules reduces the Pauli repulsion of the complex lies in
their polarising power. The oxygen atoms in water have the power to draw charge
away from the centre of the molecule, diminishing the Pauli repulsion and increasing
the strength of the metal-ligand chemical bonds.
3.3.1.2 Cu (I) ammonia and Cu (II) ammonia
The structures of the singly charged copper ammonia complexes have been
calculated and are displayed in figure 3.4:
N=1 N=2
N=3 N=4(a) -75.33 eV
89
N=4(b) -75.73 eV N=5
N=6 N=7
N=8
Figure 3.4: Structures of [Cu (NH3)N]+ complexes, 1 ≤ N ≤ 8.
The bond length of the one-coordinate copper ammonia complex is very similar to
the corresponding water complex (it is 0.02 Å shorter). For N=3, the copper
ammonia structure has three bond lengths that are approximately the same
90
(averaging 1.97 Å), unlike the corresponding water structure that has very different
bond angles and lengths. The four-coordinate 4+0 structure is tetrahedral, like in the
copper water case, but in the ammonia case the bond lengths are longer (up to 0.18
Å).
In the case of singly charged copper ammonia complexes the calculated structures
also present two solvation shells, but the preference for having ligands in a second
shell, rather than in the first, only starts with N= 5. In the case of N=4, the 3+1
structure is less stable than the 4+0 by 0.4 eV. That contrasts with singly charged
copper water complexes which prefer having a second solvation shell for N = 3 or
higher. However, from N=5 the presence of a secondary solvation shell is preferred,
as it was not possible to obtain stable structures for complexes of the form 5+0 or
6+0. Such structures would always break down into 4+1 and 4+2 structures
respectively.
This strong preference for the 4+0 tetrahedral structure, in complexes containing
copper and nitrogen, is also observed in biology. A good example is the protein
plastocyanin, which carries electrons from cytochrome bf complex to photosystem I
(110). It has a copper atom bound to hystidine residues, through nitrogen atoms, and
to cysteine and methionine residues through sulphur atoms. As more water ligands
are added to the second solvation shell the complex tends to a flattened shape, as it
was seen in the case of the calculated copper water structures.
91
Figure 3.5: Structure of Plastocyanin (110).
In the case of the five-coordinate 4+1 complex, the metal-ligand bond lengths are
similar to the ones in the corresponding copper water complex, however the
hydrogen-bond lengths are significantly larger for copper ammonia complexes
(around 0.3Å larger). This trend continues for the larger structures shown in figure
3.6. The structures of doubly charged copper ammonia complexes have been
calculated and are displayed in figure 3.6:
N=1 N=2
N=3 N=4
92
N=5(a) -85.00 eV N=6(a) -105.26 eV
N=7 N=8
N=5(b) -84.90 eV N=6(b) -104.74 eV
Figure 3.6: Structures of [Cu (NH3)N]2+ complexes, 1 ≤ N ≤ 8.
The three-coordinate complex is approximately T-shaped, as observed in the case of
copper water. The four-coordinate complex is now square-planar and its bond
lengths are 2.04 Å. This is almost the same value found for the length of the four Cu-
N bonds of the condensed phase Cu(NO3)2 (15) .
93
As in the singly charged case, the doubly charged 4+1 structure is preferred over the
5+0. Unlike the case of doubly charged water complexes, however, there is not a
strong preference for this geometry, as the difference in energy between the two, 4+1
and 5+0, is only 0.10 eV (0.22 eV in the case of water). Metal-ligand bond lengths
are slightly longer than in the case of the corresponding water complexes (around 0.1
Å) and hydrogen-bond lengths are significantly longer (around 0.3 Å), like in the
case of singly charged complexes.
As more ligands are added, the preference for structures with a second solvation
shell becomes stronger. For the six-coordinate complexes, the 4+2 structure is more
stable than the 6+0 by 0.48 eV (0.63 eV in the case of water). The preference for a
second solvation shell follows on to the larger complexes, as no other structures
could be obtained for those.
The octahedral complex is strongly Jahn-Teller distorted, as its axial bonds lengths
(2.62 and 2.41 Å) are much larger than the equatorial ones (around 2.03 Å).
Figure 3.7 shows the incremental binding energies, defined as the energies involved
in the step CuLn2+ CuLn-1
2+ + L, for doubly charged copper water and copper
ammonia complexes. These profiles indicate that ammonia has a stronger tendency
to form complexes with lower coordination number (1 or 2). For coordination
numbers 3, 4 and 5 this tendency is equal for both types of complex, and finally the
water complexes become more favourable for coordination numbers 6 and above.
This graph also illustrates why the larger ammonia complexes (N=9, 10) couldn’t be
obtained in the calculations. At N=8 it is a weakly bounded structure and the larger
structures tends to disintegrate in the course of the geometry optimisation. It also
94
shows preferential stability at N = 5 for NH3 whereas no such preference is seen for
H2O.
Figure 3.7: Incremental binding energies comparison between doubly charged copper water
and copper ammonia complexes (only minimum energy structures considered).
For the sake of comparison to these calculated geometries, figure 3.8 shows some
examples of copper ammonia complexes observed in the solid state. The doubly
charged structure has a larger metal-nitrogen bond length than the single charged
one. This is the contrary to what was observed in the gas phase calculations and it is
probably due to the effect of the other atoms present in the structure or a different
coordination number. The one on the left, amminecopper (I) chloride - Cu (NH3) Cl
– has a metal-nitrogen bond length of 1.89 Å (111) and the one in the right,
amminecopper(II) nitrate – (Cu (NH3)) (NO3)2 - has a metal-nitrogen bond length
of 1.95 Å (112).
These solid state bond lengths are slightly smaller than the calculated ones.
Calculated metal-ligand bond lengths range from 1.86 to 2.08 Å for singly charged
95
ammonia complexes and from 1.89 to 2.05 Å for the doubly charged (omitting Jahn-
Teller distorted bonds).
Figure 3.8: Experimental structures of solid state copper (I) ammonia (left) and copper (II)
ammonia (right) complexes. LEFT: amminecopper (I) chloride (111); RIGHT:
amminecopper (II) nitrate (112). The colour coding employed is the following: copper atoms
are represented in dark blue colour, nitrogen in light blue, oxygen in red, hydrogen in grey
and chlorine in green.
Decomposition of charge-enhanced-hydrogen-bond energies
As described previously, the hydrogen bonds presented in this chapter are charge-
enhanced. In order to obtain further insight into the nature of these bonds, a more
detailed study is presented in this section.
Hydrogen bonds are a very important subject because they define the internal
structures of biological macromolecules, where metals are often present, and also the
interaction between molecules (113).
The 4+1 complex was chosen as the prototype system for this study. Fragment
calculations have been performed on the singly and doubly charged 4+1 copper
96
water and also copper ammonia complexes. The binding energy decompositions
obtained from these calculations are presented in table 3.4.
These calculations are a bit different from other fragment calculations performed in
this thesis, where each ligand has been considered as one fragment. In this case, each
calculation performed has only two fragments: one is the ligand that is hydrogen
bonded and the other is the rest of the complex, i.e., the four-coordinate structure.
This way it is possible to obtain specific information about the hydrogen bonding.
Cuz+(H2O)4(H2O) Cuz+(NH3)4(NH3) 2+ 1+ 2+ 1+
Total Pauli Repulsion: 1.89 1.22 1.23 0.63
Electrostatic Interaction: -2.06 -1.3 -1.42 -0.74
Total Steric Interaction: -0.17 -0.08 -0.20 -0.11
Orbital Interactions -1.89 -0.7 -1.22 -0.35
Total bonding energy* -2.06 -0.78 -1.41 -0.46 Hydrogen bond length** 1.60 1.73 1.90 2.17
* Energy of the double hydrogen bond.
**Average of two bond lengths, as they are double hydrogen bonds (angstroms).
Table 3.4: Hydrogen bond energies (eV) calculated using two fragments: the hydrogen
bonded water as one fragment and the rest of the complex as the other. Total Steric
interaction is the sum of the Pauli repulsion and Electrostatic energy.
The strongest hydrogen bonds were found in doubly charged copper water
complexes. The energy of the double hydrogen bond in this case is -2.06 eV, so that
each individual bond energy in this case is -1.03 eV or -96.2 kJmol-1. That is much
97
larger than the energies of hydrogen bonds in pure water, in the condensed phase,
which are of the order of 10-40 kJmol-1 (113). The charge enhanced hydrogen bonds
studied here have energies of the order of covalent bonds. The hydrogen bond
lengths in water in the condensed phase are about 3.1 Å (113). This length is
considerably larger than the ones found in the case of the charge-enhanced hydrogen
bonds studied here.
These hydrogen bond energies, like all molecular energies in this thesis, have been
calculated using the LDAxc functional plus post SCF GGA corrections (B88X + P86C).
GGA corrections to the exchange energy in these systems are much larger than the
corresponding corrections to the correlation energy. GGA corrections to the
exchange energy are represented by an added positive term, which means that they
destabilise the complex, whereas GGA corrections to the correlation energy
correspond to an added negative term, so that they stabilise the complex. GGA
corrections only affect the Pauli repulsion and the orbital interaction terms of the
binding energy. The electrostatic energy is unaffected. It is observed that the Pauli
repulsion term is affected to a much greater extent than the orbital interactions term.
It is also observed that GGA corrections to the exchange (positive) are much larger
than GGA corrections to the correlation energy. That is expected as Pauli repulsion
and exchange interactions are intrinsically connected.
As a result, the increase in the total Pauli repulsion as a result of GGA corrections to
the exchange energy leads to a situation where the total steric interaction becomes
close to zero, as shown in table 3.4. That means that the electrostatic force alone is
not capable of binding a water molecule tightly enough in these cases. It is
commonly assumed that hydrogen bonding is a purely electrostatic phenomenon but
that is not the case in these complexes. An extra force is needed in order to bind the
98
water molecule, and that comes through orbital interactions. The importance of
orbital interactions for the strength of hydrogen bonds has also been discussed by
Baerends in his landmark paper about hydrogen bond interactions in DNA (114).
However, the covalent character of hydrogen bonds is still a controversial issue
(113).
It is readily seen in table 3.4 that hydrogen bond energies in copper water complexes
are greater than in the corresponding copper ammonia complexes. There is a
correlation between bond energies and bond lengths: the greater the energy the
smaller the bond length.
It is also seen that in both ammonia and water complexes, there is a significant
increase in Pauli repulsion when going from a singly charged to doubly charged
complex. Pauli repulsion increases from 1.22 eV to 1.89 eV in the case of water and
in the case of ammonia it almost doubles, going from 0.63 eV to 1.23 eV. This is a
consequence of the shortening in bond length that provokes an increase in electron
density in the region between the atoms. However, there is a large increase in the
electrostatic interaction term, as a result of the larger charge (2+ instead of 1+).
The total steric interaction is slightly more negative (attractive) for the doubly
charged complexes (-0.17 eV for water and -0.20 eV for ammonia) than for the
singly charged complexes (-0.08 eV for water and -0.11 eV for ammonia).
The amount of steric interaction is about the same for both copper water and copper
ammonia complexes, and the increased hydrogen bond energy found in the water
case comes as a result of stronger orbital interactions, as shown on table 3.4.
99
3.2.1.3 Cu (II) pyridine
A preliminary optimisation of the four-coordinate copper (II) pyridine complex
employing relativistic corrections, performed as part of this work, have shown that
there is no substantial difference in the geometry obtained using these corrections.
The difference in geometry was less than 0.01 Å in the Ag-N bond. This evidence
adds to the ones presented in section 3.2.1.1 regarding the IBE’s of singly charged
copper water, so that the relativistic corrections are not significant in the context of
these calculations involving copper.
The structures of copper pyridine complexes are shown in figure 3.9, along with the
corresponding symmetries.
N=2 C2v N=3 C2v
100
N=4 D4h N=4 D2h
N=4 D2d N=5 5A (C2v)
N=5 5B(C2v) N=5 5C(C2v)
101
N=6 6A (D2h) N=6 6B(D2h)
N=6 6C(D2h) N=6 6D(C2v)
Figure 3.9: Calculated structures of [Cu(pyridine)N]2+ complexes, with 1≤ N ≤ 6.
The structure of the one-coordinate complex could not be produced because the optimisation
process failed to achieve an Aufbau distribution of electrons.
The two-coordinate structure presents almost the same bond length calculated for the
corresponding doubly charged copper ammonia, which is only 0.04 Å larger.
However, the main difference between the geometry of these two complexes is the
nitrogen-metal-nitrogen angle, which is much smaller in tha case of the pyridine
complex (142o instead of 179o), giving this complex a bent shape.
102
The three-coordinate complex is T-shaped. That is the same geometry observed for
the analogous doubly charged complexes of copper and water. The bond lengths are
slightly smaller than the ones calculated for the corresponding ammonia complex
(1.93 and 1.87 Å compared with 2.01 and 1.95 Å).
For the coordination numbers four, five and six, more than one possible structure has
been calculated in each case. For N=4, four possible structures have been calculated,
namely D4h, D2h and D2d. For N=5, there are also three possibilities, all having C2v
symmetry. Finally, for N=6 there are four possibilities: three structures of D2h
symmetry (A, B and C) and one of C2v symmetry (D).
The relative energies of these structures are shown in table 3.5.
[Cu (pyridine)N]2+ N 4 5 6 0.0 (D4h) 0.0 (5A-C2v) 0.0 (6A- D2h) 32.9 (D2h) 20.1 (5B-C2v) 59.6 (6B- D2h) 34.2 (D2d) 27.4 (5C-C2v) 13.6 (6C- D2h) - - 42.9 (6D-C2v)
Table 3.5: Energies of copper (II) complexes (kJmol-1) relative to most stable structural
isomer.
According to table 3.5 , the most stable four-coordinate structure is the square planar
(D4h). Its bond length (figure 3.9) is slightly smaller that observed in copper
ammonia (1.97 instead of 2.03 Å). The difference in energy between the competing
four-coordinate structures is only of the order of 30 kJmol-1, which is relatively
small.
The energies of these three four-coordinate complexes have been calculated
previously by Cox and coworkers(3) and the same trend in stability was found,
although the magnitude of the differences was different because a smaller basis set
103
was employed. In their unrestricted calculation, they have found that the D2h
structure is less stable than the D4h by 36 kJmol-1 and the D2d , the least stable of the
three, being 38.3 kJmol-1 above D4h.
The most stable five-coordinate structure is a square based pyramid, structure 5A,
where the axial bond length is 3.23 Å and the equatorial bond lengths are 1.96 and
1.97 Å . The competing structures, 5B and 5C, are very close in energy. All present
C2v symmetry and the difference between them is of the order of 20-30 kJmol-1 ,
which make them closer in energy than the four-coordinate structures.
In the case of the coordination number six there are four competing structures. The
favoured structure (6A- D2h) is pseudo-octahedral and presents a strong Jahn-Teller
distortion, as seen on figure 3.9. The axial bond lengths are 3.88 Å long and the
equatorials are 2.01 - 2.03 Å. The 6C structure, also of D2h symmetry, is slightly
higher in energy (13.6 kJmol-1 above 6A). The other two geometries considered,
however, are considerably higher: the 6B – D2h structure is 59.6 kJmol-1 above 6A
and the 6D, which is a C2v structure, is 42.9 kJmol-1 above 6A.
Jahn-Teller (JT) distortion in copper (II) complexes
Tables 3.6 and 3.7 present the bond lengths of JT distorted Cu(II) complexes.
Copper pyridine complexes suffer the largest distortions. The distortion is maximum
in the six-coordinate copper pyridine complex, which presents a 1.85 Å difference
between the longest axial bond length and the shortest equatorial bond length. This
difference was also considerable in the five-coordinate case: 1.27 Å.
104
Five-coordinate
ligand Bond lengths (Å)
equatorial axial
water 1.92, 1.97 2.10
ammonia 2.09, 2.04, 2.02 2.13
pyridine 1.96 3.23
Table 3.6 Bond lengths of five-coordinate copper (II) complexes.
Six-coordinate
ligand Bond lengths (Å)
Equatorial axial
water 1.95,1.96 2.30,2.31
ammonia 2.02,2.03,2.04 2.41,2.62
pyridine 2.01,2.03 3.88
Table 3.7 Bond lengths of six-coordinate copper (II) complexes.
The five-coordinate copper water and copper ammonia complexes exhibit very small
JT distortions, which are 0.13 – 0.18 and 0.04 – 0.11 Å respectively. In the six-
coordinate case, however, the distortions are larger and the ammonia presents a
larger distortion, 0.58 Å compared with that of water, 0.35 Å.
105
Incremental binding energies
Figure 3.10: Incremental binding energies of copper (II) complexes.
The IBE’s of copper complexes, presented in figure 3.10, are calculated using the
4+0, 5+0 and 6+0 structures for copper water and copper ammonia complexes. In
the case of copper pyridine complexes the curve has a steep descent from N=4 to
N=5 and 6, suggesting that N=4 is a preferred situation. The very low energies
associated with N=5 and 6 are a consequence of the strong Jahn Teller distortion
presented by these complexes, which makes their axial bonds very long.
The next step in this binding energy analyses is the energy decomposition, according
to the Ziegler-Rauk-Morokuma scheme presented in section 3.1.2.
106
Binding Energies decomposition
Cu (pyridine)N Cu (ammonia)N Cu (water)N
N 6 4 6 4 6 4
Total Pauli Repulsion: 15.41 14.46 12.79 10.44 7.91 9.72
Electrostatic Interaction: -22.56 -19.68 -20.64 -16.56 -14.64 -13.64
Total Steric Interaction: -7.15 -5.22 -7.84 -6.12 -6.73 -3.92
Orbital Interactions -12.48 -13.86 -10.31 -10.69 -9.01 -9.93
Total Binding Energy: -19.63 -19.08 -18.15 -16.81 -15.74 -13.85
Table 3.8: Energy decomposition for doubly charged copper complexes from fragment
calculations, where each ligand and also the metal are considered as different fragments.
Only the four-coordinate 4+0 and the six-coordinate 6+0 structures are considered. The
binding energy decomposition according to the Ziegler-Rauk-Morokuma scheme,
described in section 3.1.2, is presented in table 3.8 for the four and six-coordinate
copper complexes. For N=6, only the lowest energy copper pyridine complex has
been considered. For the sake of consistency, the pseudo-octahedral copper ammonia
and copper water complexes have been included in this comparison, although they
are less favourable than the corresponding 4+2 complexes. For N=4, all complexes
considered in the energy decomposition have a square planar structure. That is not
the preferred structure for the four-coordinate copper water complex but it has been
used in this particular study for the sake of consistency.
The total binding energies obtained from a fragment calculation differ slightly from
the energies obtained using the standard approach, described earlier in this section,
and displayed in Appendix B in table B2. For instance, the binding energy of the
pseudo-octahedral copper water complex is 15.74 eV, when calculated using a
107
fragment calculation. When using the standard approach, this energy is 15.85 eV.
The difference between the two, which is 0.11 eV, is mainly due to the fact that the
standard approach neglects the preparation energy. For the square planar copper
water complex (N=4), the fragment calculation gives a binding energy of 13.85 eV
whereas the standard approach gives a binding energy of 13.72 eV. This time the
difference is 0.13 eV, which is very slightly more than the difference seen in the case
where N=6. Curiously, the increased binding energy of the complex with N=6, when
compared to the complex with N=4, is mainly due to a decrease in Pauli repulsion.
This is very unusual because this term normally increases as more ligands are added
to the complex and the increased binding energy is normally due to the increase in
the electrostatic interaction. The latter only increases by 1 eV by increasing N from 4
to 6. That is probably a result of the strong JT distortion observed in the pseudo-
octahedral structure.
Copper pyridine and copper ammonia complexes behave in a more ordinary manner.
When N increases from 4 to 6 the Pauli repulsion terms also increase. In the case of
ammonia it increases by 2.35 eV but in the case of pyridine it only increases by 0.95
eV. That is because the octahedral copper pyridine structure is strongly JT distorted
so that the axial ligands are kept far from the metal, avoiding the build up of
electrons in the central region of the complex, and consequently causing only a small
increase in Pauli repulsion. The bonding contribution from orbital interactions is also
reduced in the process of going from N=4 to 6. In the case of pyridine it reduces by
1.38 eV and in the case of ammonia it is reduced by 0.38 eV. The main contribution
in the bonding of these complexes is the electrostatic interaction, which increases by
2.88 eV in the case of pyridine and 4.08 eV in the case of ammonia.
108
When going from N=4 to N=6, the total binding energy of all three types of complex
also increase, as expected. However, in the case of pyridine the increase is very
small, and that is probably due to the strong Jahn-Teller distortion suffered in the
case of the octahedral-type structure, which caused the electrostatic interaction term
to increase little. The increase in total binding energy, when going from N=4 to N=6
is only 0.58 eV in the case of the pyridine complexes. In the cases of ammonia and
water these increases are 1.34 eV and 1.89 eV respectively.
Ligand IE (eV) α (Å3) µ (D)
acetone 9.71 6.39 2.88
acetonitrile 12.19 4.40 3.92
ammonia 10.7 2.26 1.47
pyridine 9.25 9.18 2.21
water 12.6 1.48 1.85
Table 3.9: Ionization energies (IE), polarisabilities (α) and dipole moments (µ) of ligands
(from (115).
Table 3.9 displays some physical properties of the ligands considered here. By
analysing this table it is possible to draw a correlation between the polarisabilities of
the ligands and some components of the binding energy decomposition. Water is the
ligand with lowest polarisability (1.48 Å), followed by ammonia (2.26 Å) and finally
pyridine (9.18 Å). The electrostatic contribution to the bonding and the orbital
interaction term, also follow this trend: they are the lowest in the case of water and
the highest in the case of pyridine.
109
Finally, table 3.8 shows that the copper water complexes are the ones with the lowest
binding energies. That is evidence of the stronger affinity of copper for ligands that
possess nitrogen donors. The poisonous character of cyanide is a notorious evidence
of this strong affinity, as described in section 1.2. Actually, the pseudo-octahedral
copper water complex could only exist because of this unusual reduction of Pauli
repulsion, as the main contributor to the bonding in these complex, the electrostatic
interaction, increases very little by adding axial ligands to its square planar structure:
only 1 eV (to be compared to 4.08 eV in the case of ammonia and 2.88 eV in the
case of pyridine).
3.3.1.4 Conclusions
Relativistic corrections are not important in the study of singly and doubly charged
copper complexes with water or pyridine, despite the fact that some authors argue
that copper is the only first row transition metal to require such corrections. This
conclusion has been extended to the complexes with ammonia, although evidence
has been presented for copper water and copper pyridine complexes only.
Furthermore, a comparison between theory and experiment has shown that the
LDAxc functional with post SCF Becke and Perdew gradient corrections, and TZ2P
basis set, with a frozen core on copper at the 2p level can satisfactory describe the
bonding of copper water complexes.
Regarding copper water and copper ammonia complexes, the calculated structures
showed patterns of solvation that are markedly different from the ones found in the
condensed phase. It has been found that singly charged copper water complexes have
a preference for a structure containing two solvation shells, when the number of
110
ligands is equal or greater than three. In the doubly charged case the threshold for
such preference is five. That suggests that the high charge density on the doubly
charged metal will promote proton transfer to water and subsequent Coulomb
explosion if water is hydrogen bonded to a complex containing less than four ligands
in the first solvation shell. In the case of ammonia the threshold is five whatever the
charge on the metal. The calculated structures have bond lengths slightly larger than
the ones found in similar structures in the solid state. The high stability of the doubly
charged eight-coordinate copper water complex has been attributed to the reduction
in Pauli repulsion, with respect to the square planar structure, probably due to the
charge withdrawing action of the hydrogen bonded waters.
Copper water and ammonia complexes, singly or doubly charged, having five or
more ligands exhibit charge-enhanced hydrogen bonds, which are stronger than the
ones found in liquid water. It has been found that the orbital interaction term is
fundamental for this bonding to exist, so that it is not mainly electrostatic as
commonly pictured in textbooks. For instance, Stryer (19) claims that hydrogen
bonds are “fundamentally electrostatic interactions”.
Copper (II) pyridine complexes have copper nitrogen bond lengths that are usually
larger than in the case of ammonia. They also present the largest JT distortions,
particularly in the six-coordinate case. That fact contributes to the high stability of
the four-coordinate complex.
The electrostatic interaction is the largest component of the binding energy of copper
complexes, and it is correlated with the polarisability of the ligands, so that copper
pyridine complexes have the largest binding energies, followed by copper ammonia
complexes and finally copper water complexes. These results suggest that ligands
111
with nitrogen donor atoms like ammonia and pyridine are more effective at
stabilising Cu (II) than ligands with oxygen donor atoms like water.
3.3.2 Silver complexes- The relativistic effect
Preliminary calculations involving doubly charged silver pyridine complexes,
performed as part of this work, indicated that relativistic effects can have a
considerable effect on the outcome of geometry optimisations. For instance, in the
case of the square planar [Ag(py)4]2+ complex the reduction in the silver-nitrogen
bond lengths due to relativistic corrections is 0.05 Å. As a result, the relativistic
optimised geometry of this complex presents Ag-N bond lengths of 2.16 Å. The
corresponding bond length in the non-relativistic geometry would be 2.21 Å.
Although this may not look like a large difference, it can have a significant impact
on further calculations that will be performed on these structures, in particular on the
calculation of excitation energies. Nemykin (116) has previously verified, in his
study of molybdenum complexes, that very small differences in geometry (of the
order of a few hundredths of an Ångstrom) can have a large effect (around 0.5 eV or
more) on the calculated excitation energies. The calculated excitation energies of
these silver complexes will be presented in chapter five, which will include a
comparison between the relativistic and non-relativistic spectra.
Computational details
Geometry optimisations on silver complexes were performed using the TZP DIRAC
basis set and the LDAxc functional with post SCF Becke and Perdew corrections to
the energy. Relativistic corrections were taken into account by means of the ZORA
equation (82) (83).
112
The calculated structures of doubly charged silver pyridine complexes are presented
in figure 3.11.
3.3.2.1 Ag (II) pyridine
The bond length for N=1 is 2.10 Å. It is reduced for the bent structure calculated for
N=2, where each bond length is 2.06 Å. Curiously, for N=3 the featured bond angle
increases by 20 degrees while the corresponding bond lengths are kept the same. The
new formed bond has bond length 2.15 Å. Moving from N=3 to 6, the bond lengths
increase as N increase.
N=1 N=2
N=3 N=4
113
N=5 N=6
Figure 3.11: Structures of [Ag (py)N]+ complexes, 1 ≤ N ≤ 6, optimised using LDAxc
functional with post SCF Becke and Perdew corrections to the energy . Relativistic
corrections were taken into account by means of the ZORA equation.
For N=4 the optimum structure has a D4h symmetry, similarly to the four-coordinate
copper pyridine complex. The bond lengths are larger than in the case of the
corresponding copper complexes ( 2.16 Å instead of 1.97 Å) and that is a
consequence of the different ionic radii of silver and copper, which are 0.94 Å and
0.73 Å respectively for the doubly charged case ( table 1.1). The difference in the
bond length of the complex is 0.19 Å whereas the difference in ionic radiuses is 0.21
Å. Surprisingly the calculated Ag-N bond length of the four-coordinate silver
pyridine complex is identical to the Ag-N bond length in the condensed phase
complex Ag[meso-Me6[14]ane](NO3)2 (17).
For N=5, the lowest energy structure has a C2v symmetry and corresponds to the
copper pyridine structure named 5A (figure 3.9), which is also the lowest energy
structure when copper is the metal. Likewise, this structure is strongly Jahn-Teller
distorted.
114
For N=6 the optimum structure has a D2h symmetry and corresponds to the 6C
structure in figure 3.9. Unlike previous cases, for N= 2 to 5, this structure doesn’t
coincide with the lowest energy structure for the six-coordinate copper pyridine,
which is the 6A structure. This structure also has a strong Jahn-Teller distortion.
3.3.2.2 Ag (II) acetone
Optimised structures of silver acetone complexes are presented in figure 3.12. With
the exception of the complex with N=1, which has a relatively large bond length
(2.21 Å) all the other Ag (II) acetone complexes have bond lengths similar to those
found in silver pyridine complexes, except for the JT distorted bonds in the N=5 and
N=6 complexes, where it is found that JT distortions are smaller than in the case of
pyridine.
N=1 N=2
N=3 N=4
115
N=5 N=6
Figure 3.12: Structures of [Ag (acetone)N]2+ complexes, 1 ≤ N ≤ 6, optimised using LDAxc
functional with post SCF Becke and Perdew corrections to the energy . Relativistic
corrections were taken into account by means of the ZORA equation.
Optimised structures for N=3 to 6 present a bent configuration with respect to the O-
Ag-O angle, and no local minima could be found for a corresponding linear
configuration. For N=2, however, optimised geometries could be found for both the
linear and bent configurations. The linear configuration, with an O-Ag-O bond angle
of 179 degrees, was found to be more stable than the bent configuration by just 11.4
kJmol-1.
It is expected that the bent Ag-O=C bond in these oxygen coordinating acetone
complexes will influence the bonding and spectrum when compared to the
complexes with nitrogen donor atoms, like pyridine and acetonitrile (shown below in
figure 3.13). The Ag-O= C bond angle drops from 180 degrees for N=1 and 2 to
~126 degrees for the equatorial bonds in the cases where N=5 and 6.
116
3.2.2.3 Ag (II) acetonitrile
The structures of the silver acetonitrile complexes have been calculated and are
displayed in figure 3.13:
N=1 N=2
N=3 N=4
N=5 N=6
117
Figure 3.13 (previous page): Structures of [Ag (acetonitrile)N]2+ complexes, 1 ≤ N ≤ 6,
optimised using LDAxc functional with post SCF Becke and Perdew corrections to the
energy . Relativistic corrections were taken into account by means of the ZORA equation.
Ag (II) acetonitrile complexes have roughly the same structures as the previous cases
(acetone and pyridine). However, the bond lengths of its complexes are consistently
shorter.
For N=2 the complex is linear, like the acetone complex and unlike the case of
pyridine, where the complex is bent. For N=3 the complex is T-shaped, like the other
two previous cases.
For N=4, the structure is also square planar. It is square based pyramid for N=5 and
pseudo-octahedral for N=6. The JT distorted bonds are also shorter than in the case
of acetone or pyridine complexes.
Charges
The charge in the metal atom at the centre of a complex is expected to decrease
steadily as more and more ligands are added. The incremental binding energy
profiles, discussed later and presented in figure 3.15, are good evidence that the more
ligands added, the lower becomes the incremental binding energy. That is a direct
consequence of the lowering of the charge in the metal through electron donation
from the ligands, which is essential for the formation of a strong bond with the
ligands.
118
Two charge distribution profiles have been calculated for the silver complexes,
namely Mulliken and Voronoi, and they are shown in figure 3.14. It is easily seen
that the two profiles are very different. Based on the evidences just presented it is
possible to choose the Voronoi profile as the more appropriate for this context (as the
Mulliken populations are somewhat erratic). It produces the expected result which is
a curve that decreases steadily. It also provides other interesting information, like the
fact that the curves for acetone and pyridine converge to the 0.4 value but the
acetonitrile curve converges to 0.5. The curve for the silver acetonitrile complexes is
quite different from the other two, and this fact will be used to explain experimental
observations in chapter 5.
Voronoi charge on the metal Mulliken charge on the metal
Figure 3.14 Calculated charges on the metal, according to the Voronoi and Mulliken
schemes, plotted against the number of ligands N. Data are shown for acetonitrile,
acetone and pyridine Ag(II) complexes.
119
Jahn-Teller distortion in silver complexes
Tables 3.10 and 3.11 display bond lengths of the silver complexes and demonstrate
the effect of JT distortion.
Five-coordinate
ligand bond lengths (Å)
equatorial axial
acetone 2.16, 2.18 2.40
acetonitrile 2.10 2.34
pyridine 2.16, 2.20 3.20
Table: 3.10 Bond lengths of five-coordinate silver (II) complexes.
Six-coordinate
Ligand bond lengths (Å)
equatorial axial
acetone 2.16, 2.18, 2.20, 2.23 2.59, 2.46
acetonitrile 2.11 2.52
pyridine 2.26 2.56
Table 3.11 Bond lengths of six-coordinate silver (II) complexes.
The pseudo-octahedral complexes usually display the largest distortions, however
the complex with the largest distortion is the five-coordinate silver pyridine, which
presents a difference of 1.04 Å between the axial and the shortest equatorial bond.
120
Overall, pyridine complexes are the ones that exhibit the largest distortions, followed
by acetone complexes. Complexes that have acetonitrile as a ligand exhibit the
lowest degree of distortion. Acetone complexes tend to have different equatorial
bond lengths but acetonitrile complexes have all these bond lengths equal.
Incremental binding energies
Incremental binding energies have been calculated for all three silver complexes and
the result is plotted in fig. 3.15. The curves show that for N=1 the pyridine complex
is the most preferred and acetonitrile complexes the least, but this situation is
reversed for N=2 onwards.
It can be seen that the curves have a plateau at N=4, which represents the preferential
stability of this coordination number. Towards N=5 and 6 the curve goes down
sharply, as a result of the little amount of energy gained by adding more ligands to
the four-coordinate structures. This is a consequence of JT distortion. The
preferential stability of the four-coordinate compounds has been confirmed by
experimental data, like for instance the recorded intensity of mass spectra (6).
Experimentally (1) the acetone complexes are the only ones that don’t have a
preferential stability at N=4 ; they are more stable at N=5.
121
Figure 3.15: Incremental binding energies of silver complexes.
Binding energies decomposition
The results of the binding energy decomposition, according to the Ziegler-Rauk-
Morokuma scheme are presented in table 3.12.
Ag (acetone)N Ag (acetonitrile)N Ag (pyridine)N
N 6 4 6 4 6 4
Total Pauli Repulsion: 11.02 8.52 12.98 12.43 13.91 13.28
Electrostatic Interaction: -16.84 -12.84 -19.83 -16.44 -20.08 -17.41
Total Steric Interaction: -5.81 -4.32 -6.85 -4.01 -6.17 -4.13
Orbital Interactions -11.47 -12.06 -10.52 -12.02 -12.49 -13.77
Total Bonding Energy: -17.29 -16.38 -17.37 -16.03 -18.67 -17.91
Table 3.12: Binding energies decomposition (eV) of silver (II) complexes.
0.0
200.0
400.0
600.0
800.0
1000.0
1200.0
1 2 3 4 5 6
IBE
(kJ/
mol
)
IBE's of Ag(II) complexes
Ag(II) acetone
Ag(II) acetonitrile
Ag(II) pyridine
122
It can be seen that pyridine complexes have the largest binding energies. It is higher
than in other complexes by at least 1.30 eV.
The acetone complex has a higher binding energy than the corresponding acetonitrile
complex for N=4. In the case of N=6, however, it is the acetonitrile complex that has
the largest binding energy. The relative reduction of the binding energy of the silver
acetone complex when N changes from 4 to 6 is attributed to the dramatic increase in
its Pauli repulsion term. It increases by 2.50 eV while the corresponding contribution
for pyridine and acetonitrile only increases by 0.63 and 0.55 eV respectively. This
difference cannot be due to the reduced bond lengths of the six-coordinate acetone
complex, when compared to the other octahedral complexes, because those are very
similar to the ones found in the acetonitrile case. The reason must be the presence of
the two lone pairs in the oxygen atom of each of the two extra acetone ligands, that
are added to move from N=4 to N=6, that experience repulsion from the metal in the
bonding region.
Similarly to what was observed in the copper complexes, there is a correlation
between ligand polarisability and the intensity of the orbital interactions. These are
highest for the silver pyridine complexes, as pyridine is the ligand with the higher
polarisability and lowest for the silver acetonitrile complexes, as acetonitrile has the
lowest polarisability among the ligands studied in this section. The results for the
acetone complexes are in between. Table 3.13 shows physical data concerning the
ligands studied in this section.
123
Ligand IE (eV) α (Å3) µ (D)
Acetone 9.71 6.39 2.88
Acetonitrile 12.19 4.40 3.92
Pyridine 9.25 9.18 2.21
Water 12.6 1.48 1.85
Table 3.13: Physical data of some ligands used to form complexes (115).
Pyridine complexes also present the highest electrostatic interaction among the
complexes studied in this section, but the correlation between ligand polarisability
and electrostatic interactions in the corresponding complex breaks down in the case
of acetone and acetonitrile, as the data for silver acetonitrile complexes for N=4 and
6 shows that it has higher electrostatic interactions than the corresponding acetone
complexes. These low absolute values of the electrostatic interaction for acetone
complexes is evidence of the difficulty of stabilising Ag (II) with oxygen based
ligands. That is probably why there are no solid state examples of silver (II) with
oxygen based ligands (1). It is interesting to note that, although silver acetone
complexes present the lowest absolute values for the electrostatic interaction, they
are the ones that undergo the largest change in electrostatic energy in going from
N=4 to N=6, which corresponds to 4 eV. The corresponding energy in the case of the
pyridine complexes is 2.67 eV and 3.39 eV in the case of acetonitrile complexes.
124
3.3.2.4 Conclusions
Relativistic corrections are needed to perform accurate calculations in these
complexes. It has been found that in the four-coordinate pyridine complex the
presence of relativistic corrections shortens the Ag-N bonds by ~0.05 Å.
All silver complexes present a T-shaped structure in the 3-coordinate case, square
planar in the four-coordinate case, square-based pyramid in the five-coordinate case
and finally pseudo-octahedral in the six-coordinate case. Bond lengths of acetone
and pyridine complexes are similar whereas the bond lengths in acetonitrile
complexes are slightly shorter. Acetonitrile complexes also exhibit the lowest JT
distortions, whereas pyridine complexes show the highest.
The study of the charge on the silver atoms, for all complexes, has shown that the
Voronoi deformation density (VDD) is the appropriate scheme to employ in studies
of silver complexes. The traditional Mulliken approach has provided a picture that
goes against chemical intuition, as the charge on the metal hardly changes as more
ligands are added; in fact, the charge on the acetonitrile complex increases. VDD
shows that the charge in the metal is reduced as more ligands are added, and that
happens in all cases. Acetone and pyridine complexes show a similar behaviour but
acetonitrile complexes have charges on the metal that are higher by about 0.1.
Considering complexes with N=4, the pyridine complex has the highest binding
energy, followed by the acetone complex. For N=6, the pyridine complex still has
the largest binding energy but this time it is followed by the acetonitrile complex,
because the acetone complex suffers a large increase in Pauli repulsion when it goes
form N=4 to N=6, possibly because of the presence of two lone pairs of electrons in
125
the oxygen donor atom. Furthermore, the acetone binds to the metal with a bent Ag-
O-C bond angle contrary to expectation. Perhaps in order to reduce this repulsion
and facilitate better overlap. Eitherway, this suggests that ligands with nitrogen
donors are more efficient at stabilising Ag (II), as observed in condensed phase
chemistry (117).
126
Chapter 4 Further studies of copper water and
copper ammonia complexes: ionization energies
and fragmentation pathways
In this chapter DFT is used to calculate ionization energies of copper compounds,
and how they vary as a function of coordination number. These values are related to
redox properties and are related to the electron transportation capability of copper.
In the second part, it is presented an interpretation of the latest experimental results
from Stace and coworkers on copper complexes (2). These experiments aim to study
fragmentation pathways of copper complexes.
4.1 Ionization energies
4.1.1 Introduction
In this section, calculated values of the ionization energy (IE) of each of the copper
(I) water and copper (I) ammonia complexes that have been studied in this thesis are
presented. The IE’s of copper (I) water complexes are calculated in different ways
and a comparison between various methods, adiabatic, vertical and Koopmans’, is
carried out. The IE’s of copper (I) ammonia complexes have only been calculated in
one way, using the most accurate technique, which is the adiabatic. Furthermore, it
has also been studied how the IE’s vary depending on which method and basis set is
used and the number of coordinating ligands, N.
4.1.2 Background
The conversion of Cu (I) to Cu (II) plays a very important role in chemistry, and in
particular in the chemistry of life. Many electron transport metabolic pathways rely
127
on the conversion between these two cations, including respiration (31). Various
proteins (including enzymes) provide a range of environments, by varying ligands
and geometries, in order to tune the redox properties of these cations. The ionization
energy (IE) is one of the most important features that control the functioning of the
Cu (I) / Cu (II) couple. In this section, a study of the variation of the IE of copper (I)
with the ligand environment (water and ammonia ligands) is presented.
4.1.3 Computational Details
Preliminary calculations of energies of the copper atom and ions (Cu (I) and Cu (II))
have been performed with and without relativistic corrections, using the ZORA
equation. A variety of methods and basis sets have been employed. The functionals
used were the LDAxc with post SCF B88X + P86C corrections, the B88X + P86C
functional and also LB94, an asymptotically correct functional. The following basis
sets have been used: DZ, TZP, TZ2P, TZ2P+ and QZ4P. Calculations with these
basis sets have been performed in an all electron basis and also with a frozen core at
2p on copper and 1s on oxygen and nitrogen. Some even–tempered (ET) basis sets
have also been used. These are the ET-pVQZ (valence quadruple ξ) and the ET-
QZ3P (quadruple ξ with polarisation) with one, two and three sets of diffuse
functions. The use of the very large ET-QZ3P-3DIFFUSE basis set requires the use
of the key “Dependence” which avoids the problem of linear dependence of the basis
functions.
Calculations have also been performed for the determination of the ionization
energies of complexes. Single point calculations were performed on the singly
charged complexes using the same method used in chapter 3 (LDAxc and post SCF
B88X + P86C gradient corrections) and also using the LB94 functional. The basis set
128
used in calculations involving complexes was always the TZ2P, as in chapter 3.
None of the calculations involving complexes included relativistic corrections.
4.1.4 Results
Calculations have initially been performed to estimate the IE’s of copper, and in
particular the IE of Cu (I), i.e. Cu (I) Cu (II) + e- . It has been found that this
process consist of the removal of an electron located in the dx2
- y2 orbital.
The energies of the copper atom and its ions (Cu(I) and Cu(II)) , along with their
IE’s, have been calculated and are presented in appendix B, tables B5 to B10. The
first IE of copper has been calculated by subtracting from the calculated energy of
the singly charged ion the energy of the atom. The second IE is obtained in an
analogous manner, i.e., by subtracting from the energy of the doubly charged ion the
energy of the singly charged. In summary:
1st IE = ECu(I) – ECu and 2nd IE = ECu(II) – ECu(I)
These atomic and ionic energies have been calculated under various different
conditions, as described in section 4.1.3, and the resulting IE’s have been compared
to experimental results (presented in table 1.1).
The best agreement with experiment was obtained in calculations involving the
LB94 functional. Regarding the first IE, the best result was obtained in a non-
relativistic calculation employing the LB94 functional and the TZ2P+ basis set, with
a frozen core at 2p. The error with TZ2P+ was 3.89% (0.31 eV) and the best result
for the second IE is also obtained employing the LB94 functional with TZ2P+, but
using relativistic corrections. In this case the error was 3.30% (0.67 eV). In fact, the
relativistic calculations consistently provide the best results for the second IE only
129
and the LB94/TZ2P+ non-relativistic calculation provided the best overall results
(with errors of 3.89% and 3.99% for the 1st and 2nd IE’s of copper, respectively).
Nevertheless, for the sake of consistency with calculations in chapter three, the
calculations in this section, in addition to LB94, employed the LDAxc functional
with post SCF B88X + P86C gradient corrections and the TZ2P basis set with a frozen
core on copper at 2p, and without relativistic corrections. Under these conditions the
error in the calculated IE’s was slightly larger than using LB94 and TZ2P+. The
error is 9.33% on the first IE and 5.08 % on the second.
Interestingly, the use of the large ET, all electron basis sets, although providing
consistent 1st and 2nd IE errors ranging from 5 – 9% did not dramatically improve the
calculated energies. This would imply that the errors arise from the DFT method
rather than the basis set employed.
4.1.4.1 IE’s of copper water complexes
The IE’s of singly charged copper water complexes have been calculated in a variety
of manners, namely vertical, adiabatic and using Koopmans’ theorem.
Vertical IE’s correspond to the difference between the absolute energy of the
monocation minus the absolute energy of the dication, assuming that there is no
change in geometry when going from monocation to dication. That means that a
single point energy calculation is performed on the structure of the monocation with
the charged increased from one to two (+1 to +2). The shortcoming of this approach
is that the structure associated with the dication is not optimised, so that it is a crude
approximation to the adiabatic IE. However, this is perfectly suitable to the optical
transitions where the timescale of events is very small and there is not enough time
for a rearrangement of the geometry of the complex to take place. This
130
approximation is eliminated in the case of the adiabatic method for the calculation
of IE’s. Adiabatic IE’s correspond to the difference between the absolute energy of a
monocation and the corresponding dication. In this case, absolute energies are
obtained by performing full optimisations of both cation and dication and therefore
allowing the geometry of the dication to relax completely. This is expected to be the
most accurate manner of calculating an IE.
Koopmans’ IE’s correspond to minus the energy of the HOMO of the cation
complex under study, i.e. the energy to remove an electron from the HOMO is –
EHOMO. Koopmans theorem has been developed in the context of the Hartree-Fock
approximation and interestingly it has been found to work in the context of DFT as
well, as long as an asymptotically correct functional is used (118).
Calculated IE’s, in eV, are presented in table 4.1 (using complexes discussed in
chapter 3):
Koopmans - LDAxc post SCF B88X + P86C
Koopmans - LB94 Vertical Adiabatic
[Cu(H2O)]+ 12.40 19.05 17.44 17.40 [Cu(H2O)2]+ 11.11 17.28 15.58 15.53 [Cu(H2O)3]+ 3+0 9.44 15.70 14.52 13.62
2+1 10.36 16.35 14.39 * [Cu(H2O)4]+ 4+0 8.09 14.60 12.91 12.25
3+1 8.56 14.85 13.02 * 2+2 9.40 15.51 13.46 *
[Cu(H2O)5]+ 4+1 8.08 14.15 11.77 11.38
3+2 8.24 14.26 Not
converged 11.97 [Cu(H2O)6]+ 4+2 7.70 13.83 11.74 10.83 [Cu(H2O)7]+ 4+3 6.86 12.96 10.74 10.16 [Cu(H2O)8]+ 4+4 5.47 11.74 9.60 9.53
*Cases where there is no adiabatic IE because the corresponding doubly charged complex was not
stable with respect to proton transfer.
131
Table 4.1(previous page): IE’s of copper complexes calculated using various techniques and
also different functionals. The vertical and adiabatic energies were calculated using LDA
with post-SCF B88X+P86C and the LB94 energies were obtained by performing a single point
energy calculation on the LDA with post-SCF B88X+P86C optimised structure.
Adiabatic IE’s are the ones whose calculation is more laborious and that are
expected to be the most accurate. Hence, they will be the reference for these
analyses.
All techniques reproduce the same trend, which is a reduction of the IE of the
complex as more ligands are added. Koopmans’ IE’s can have substantial variations,
depending on what functional is used. The IE’s obtained using the LDAxc
functional are considerably smaller (underestimated) than the ones obtained using
the LB94 functional. This difference is of the order of 6 to 7 eV on average.
Koopman’s IE’s obtained using the LB94 functional are the most accurate, when
compared to the ones obtained using LDAxc. That is because the LB94 functional,
like other asymptotically correct functionals, bring the orbital energies to a lower
level, which is closer to the real energies. As a result, it can be successfully used in
the context of Koopmans’ theorem. Asymptotically correct functionals, derived for
TDDFT studies, provide realistic orbital energies because they are used to calculate
excitation energies; if the orbital energies are very high they will be largely in the
virtual region and the accuracy of the calculation will deteriorate.
Koopmans’ IE’s, obtained using the LB94 functional, are about 2 eV above the
adiabatic IE’s. This pattern is fairly consistent throughout the range of complexes
studied. On the other hand, Koopmans’ IE’s obtained using the LDAxc functional
are about 4 eV below adiabatic excitation energies. The pattern here is also
132
consistent throughout the range. Interestingly, the LDAxc functional seems to be
more sensitive to the geometry of the complex, for a given coordination number. For
instance, the difference between the Koopmans’ IE’s of the singly charged 4+0 and
3+1 complexes is 0.47 eV using LDAxc and 0.26 eV if the LB94 functional is used.
It is surprising the fact that vertical IE’s are very close to adiabatic IE’s, considering
that they correspond to very different structures in some cases, e.g. [Cu(H2O)4]+ is
approximately tetrahedral (Td) whereas [Cu(H2O)4]2+ is approximately square planar
(D4h).
For the one-coordinate complex the difference is only 0.04 eV. Although the
structures of these one-coordinate singly and doubly charged complexes are very
similar, there is a difference of 0.05 Å in the Cu-O bond length and also differences
in the Cu-O-H angles. So, a more significant energy difference might be expected.
In the case of the three-coordinate geometries a meaningful difference in IE’s is
observed despite the fact that two of the three bond lengths remained equal in going
from singly charged to doubly charged. The difference lies in one bond length that is
increased by 0.30 Å, in the singly charged complex and in the difference in the O-
Cu-O bond angles. The singly charged complex presents an asymmetry and the three
O-Cu-O angles differ from each other. In the doubly charged case the molecule is
symmetric, so that these three angles are equal. These differences in angles and one
of the bond lengths cause the adiabatic and vertical IE’s of the three-coordinate 3+0
complex to differ by 0.9 eV.
In the case of the 4+0 complex, a more dramatic structural change takes place: in
going from singly to doubly charged, a tetrahedral complex becomes square planar.
However, the differences in bond length are not large. While some of the bond
133
lengths only change by 0.04 Å, the largest difference observed was 0.17 Å. The
difference between adiabatic and vertical energies in this case was only 0.66 eV.
The 4+1 complex presents very little change in bond lengths in going from singly
charged to doubly charged. All bond lengths, including the hydrogen bond length,
only change by about 0.10 Å. The main difference that takes place is the increase in
the O-Cu-O bond angles from 156 and 164 degrees to 177 degrees. As a result, the
difference between vertical and adiabatic IE’s is 0.39 eV.
Going from 4+2 to 4+3 and 4+4, a progressive reduction of the difference between
adiabatic and vertical IE’s is observed; that difference starts at 0.91 eV, for the 4+2
complex, then diminishes to 0.58 eV, in the case of the 4+3 complex, and finally
goes down to 0.07 eV in the case of the 4+4 complex. By observing the structures of
copper water complexes in figures 3.1 and 3.2, it can be seen that the Cu-O bond
lengths of the complexes change little in going from 4+3 to 4+4. In the singly
charged case, they are about 2.04 Å in the 4+3 complex and 2.07 Å in the 4+4
complex. In the doubly charged systems these bond lengths average 1.89 Å for both
4+3 and 4+4 complexes. That means that the difference between the Cu-O bond
lengths in singly and doubly charged is constant for both 4+3 and 4+4 complexes.
Hence, the reduction in the difference between adiabatic and vertical IE’s (0.51 eV)
in going from 4+3 to 4+4 must be attributed to the differences in bond angles. Singly
charged complexes of the form 4 + Y, for Y=0 to 4, are more bent whereas the
doubly charged complexes have an almost planar structure, as a result of their d9
configuration. However, as more water ligands are attached to the second solvation
shell, the more difficult it is for the singly charged structure to keep the bent
configuration. The result is that the structure of singly charged becomes steadily
flattened as more water ligands are added to the second solvation shell.
134
It is interesting to observe that the IE’s of singly charged copper water complexes
only become lower than the IE of water (12.6 eV) when the number of water ligands
added is equal or larger than 4.
4.1.4.2 IE’s of copper ammonia complexes
For the sake of comparison, the adiabatic IE’s (eV) of the copper (I) ammonia
complexes are shown in table 4.2:
IE (eV)
[Cu(NH3)]+ 16.64
[Cu(NH3)2]+ 14.83
[Cu(NH3)3]+ 3+0 11.91
[Cu(NH3)4]+ 4+0 11.57
[Cu(NH3)5]+ 4+1 10.3
[Cu(NH3)6]+ 4+2 9.77
[Cu(NH3)7]+ 4+3 9.31
[Cu(NH3)8]+ 4+4 8.71
Table 4.2: Adiabatic IE’s (eV) of the copper (I) ammonia complexes
The first thing to notice is that these adiabatic IE’s energies are lower than the
corresponding energies in the case of copper (I) water complexes. The first IE is 0.96
eV lower than its water analogue. In the case of the last calculated IE (complex 4+4)
the difference is 0.82 eV. Overall, it can be seen from table 4.2 that the average
difference between IE’s of singly charged copper water and copper ammonia
complexes is about 0.9 eV. This difference is small, considering that the IE of
ammonia is 10.7 eV and, therefore, 1.9 eV lower than the IE of water.
The IE of the complex becomes lower than the IE of ammonia after five or more
ligands are added.
135
4.1.5 Conclusions
The calculated first and second IE’s for copper are in good agreement with
experiment, and the LB94 provided the best results. Relativistic corrections proved
to be helpful in the case of the second IE only.
IE’s of copper complexes calculated using DFT and the Koopmans’ approximation
are only appropriate if an asymptotically correct functional is employed.
Vertical IE’s have been in good agreement with IE’s obtained using more accurate
methods (adiabatic). That was surprising considering that it is only a crude
approximation, considering the fact that they do not consider an optimised geometry
regarding the dication. Nevertheless, the results can deteriorate if there is a
substantial difference in geometry between the singly charged and the corresponding
doubly charged structure, and some trends could be found in such situations. There
seems to be a correlation between the error in the vertical IE and the difference in
bond length between the singly and doubly charged complexes. For instance, the
difference between adiabatic and vertical IE’s is 0.9 eV in the case of the 3+0
complex, where the only difference between the singly and doubly charged
structures lies in a bond length that changes by 0.30 Å. In the case of the 4+0
complexes the largest change in bond length has been 0.17 Å and there has been a
0.66 eV difference in the corresponding IE’s, even though there are major structural
changes in going from the tetrahedral to the square planar configurations. Finally, in
the case of the 4+1 complexes, a 0.39 eV IE difference is associated with a 0.10 Å
difference in bond length. Hence, the difference in IE’s between singly and doubly
charged structures seem to decrease steadily as the difference in bond length between
the corresponding structures gets smaller.
136
Trends in the differences between vertical and adiabatic IE’s in copper water
complexes are also detected in cases where the differences between the Cu-O bond
lengths of singly and doubly charged complexes are kept constant when adding more
ligands to the second solvation shell, and there are changes in bond angles only.
That has been observed in the case of the 4+3 and 4+4 complexes, where both the
doubly and singly charged complexes become flattened as more ligands are added to
the second solvation shell and the geometries of both types of complex become
increasingly similar, eliminating the large difference seen in the 4+0 case
(tetrahedral and square planar). As the structures get flattened, the differences
between vertical and adiabatic IE’s of the complexes is also reduced steadily.
All the adiabatically calculated IE’s for copper water complexes are in the range
between 17.40 and 9.53 eV and in all cases they are below the second IE of copper
which is 20.28 eV (experimental) or 21.30 eV(calculated at this same level of
theory). In most cases the calculated IE’s have values in between the second IE of
copper and the first IE of water, which is 12.6 eV (experimental). From N=4 the IE
of the complex becomes lower than the IE of water alone. In the case of ammonia,
five ligands are needed to bring the IE of the complex below the IE of ammonia
alone, which is 10.7 eV (experimental).
4.2 Fragmentation pathways of copper water and
copper ammonia complexes
4.2.1 Introduction
The results presented so far, concerning copper water and copper ammonia
complexes, have been applied to investigate some of the latest results produced by
137
Stace and coworkers concerning the behaviour of these gas phase complexes (2).
This research group has produced a detailed study of fragmentation pathways of
copper water and also copper ammonia complexes. Experimental results include
various pathways, like unimolecular decay, proton transfer, collision induced
dissociation (CID) and electron capture dissociation (ECD). The latter can be
induced using a variety of gases, and the choice of these may determine the
occurrence of ECD or CID. These studies employ xenon as the collision gas; this gas
has the property of inducing both CID and ECD, depending on the conditions.
Among other motivations, the computational study presented here aims to provide
further evidence of the preferred [Cu(X)8]2+ unit as a very stable configuration, and
to add to this investigation of Stace and coworkers.
The theoretical determination of how favourable are the reactions studied in this
chapter would involve calculations of transition states and also a full thermodynamic
treatment which would include the calculation of Gibb’s free energy. This complete
analysis is beyond the scope of this work. Instead, a more simple approach will be
adopted, similar to the approach adopted by Kebarle and coworkers in their study of
ion-molecule clusters with doubly charged metal ions (119), which proved to yield
useful results. Such an approach deals with enthalpy considerations only. It means
that the total energies of products and reactants are calculated and their difference is
computed as the enthalpy of reaction. In this simplified approach it will be assumed
that exothermic reactions are more likely to occur than the corresponding
endothermic pathways.
138
4.2.2 Computational details
The computational details of the calculations of dissociation products were exactly
the same as described in the previous section, so that results are consistent. In
summary, that is a TZ2P basis set and the LDAxc functional with post SCF Becke
and Perdew corrections to the energy. Preliminary calculations involved different
possibilities for the outcome of the dissociation process by varying parameters like
the spin states and the exact location of the hydroxyl group.
4.2.3 Results
4.2.3.1 Copper water complexes
New calculated structures: the hydroxyl group
Although the experimental results comprise [Cu(X)N]2+ complexes, X = H2O or
NH3, with N=1 to 20, the theoretical calculations are limited to 1 ≤ N ≤ 10 for water
complexes and 1 ≤ N ≤ 8 for ammonia complexes. It hasn’t been possible to
calculate structures larger than those, not even using DFT methods, which are the
most suitable to treat accurately large systems. In addition to the structures presented
in chapter three, new structures containing the hydroxyl group (Cu+(H2O)LOH) have
been calculated specifically for this study of fragmentation and are displayed in
figure 4.1. It can be seen that the removal of the proton causes substantial
deformation of the structures.
Cu+OH Cu+(H2O)OH
139
Cu+(H2O)2OH Cu+(H2O)3OH
Cu+(H2O)4OH L=4, proton removed from second
solvation shell
Cu+(H2O)4OH L=4, proton removed from first
solvation shell
Cu+(H2O)5OH Cu+(H2O)6OH
L=6, proton removed from first solvation shell
Cu+(H2O)6OH
L=6, proton removed from second solvation shell
Figure 4.1: Structures of copper (I) water complexes containing a hydroxide group.
140
In complexes with N ≥ 5, a second solvation shell starts to appear. In those cases,
there are basically two possibilities for the location of the hydroxyl group: the first or
the second shell. It has been found that in all structures calculated the more
favourable structures, i.e. the ones with the lowest total energy, are the ones which
contain the hydroxyl group in the first solvation shell. Figure 4.2 shows the two
competing structures for N=6 and the respective energies. The difference in energy
between the two structures is 0.41 eV.
-92.94 eV -92.53 eV
Figure 4.2: Comparison of two [Cu(H2O)6OH] structures and their respective energies.
Dissociation pathways of copper water complexes
The first two pathways to be studied, unimolecular decay and proton transfer, occur
without the need of a collision gas, a laser or any exciting device.
Unimolecular decay:
This process consists of expelling a whole water molecule from the complex. In this
case, the remaining fragment will still be doubly charged, as the molecule expelled is
neutral. The general equation for this process is shown below:
[Cu(H2O)N]2+ [Cu(H2O)N-1]2+ + H2O
It is believed (2) that these reactions are driven by the energy transferred during the
process of electron impact ionization.
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In fact, the calculations performed have shown that about 1 eV must be supplied to a
seven-coordinate complex, so that unimolecular decay can occur. Also according to
these calculations, smaller complexes need a far larger energy input if unimolecular
decay is to be observed. For N=10 the energy needed goes down to about 0.5 eV.
These results can be visualised in figure 2 (blue line). Negative values in the graph
correspond to the energy released in the reaction (exothermicity), so that reactions
corresponding to points in negative regions of the graph would be expected to occur
spontaneously.
The curve seems to converge to the value of 1eV when N is around 7 and this fact
agrees with experimental observations, as neutral molecule loss has been observed
for N > 7 (2). According to these calculations, the energy transferred to the
complexes during the electron impact ionization process are estimated to be in the
order of 1 eV.
Figure 4.3: Plot of the calculated enthalpies of reaction of two competing dissociation
pathways: unimolecular decay and proton transfer. The enthalpies (eV) are plotted against
the number of ligand molecules (N).
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Proton transfer followed by Coulomb explosion
In this section, the energies involved in the proton transfer pathway have been
calculated. These are the energies involved in the following process:
[Cu(H2O)N]2+ CuOH+(H2O)N-1 + H3O+
These results are represented by the red line in figure 2. It can be readily seen that
proton transfer is highly favourable when N is small, and this tendency steadily
diminishes until N=7 (exothermicity close to zero). For higher N values the reaction
is not expected to occur spontaneously.
This finding agrees with the experimental results of Stace and coworkers, who
concluded that N=8 is the minimum number of solvent molecules “required to
transform [Cu(X)N]2+complexes from being in a metastable state to a situation where
the dication is in a stable solvent environment” (X=H2O).
This is an important finding because it provides further theoretical evidence of the
favoured [Cu(X)8]2+ structure for the solvation of doubly charged copper in water.
Electron capture dissociation (ECD)
To further probe the structure and reactivity of these doubly charged copper
complexes, Stace and coworkers have also performed experiments using a collision
gas in order to promote fragmentation of the complexes. In some cases, the collision
gas promoted electron transfer (ECD) also.
The collision gas used in these particular experiments was xenon, which is a noble
gas and therefore has high ionization energy. This characteristic makes it very useful
143
for these purposes, because it can promote ECD and also CID reactions. The latter
are possible because there may not be enough energy to ionise the xenon, in which
case the outcome would be a mechanical collision only, without the transfer of an
electron.
In this section, ECD processes involving the 6 and 8-coordinate doubly charged
copper water complexes, as well as the 8-coordinate doubly charged copper
ammonia complexes, will be studied.
Stace and coworkers have also raised the question of “why electron capture yields
both CuOH+ (H2O)N and Cu+ (H2O)N fragments”. To address this issue, a calculation
of the energies of both sets of complexes has been performed, and the results
obtained are listed in table 4.3.
N CuOH+(H2O)N + H Cu+(H2O)N+1 diff
1 -19.18 -23.9 4.72
2 -34.45 -38.86 4.41
3 -49.32 -53.72 4.40
4 -64.16 -68.15 3.99
5 -78.98 -82.58 3.60
6 -93.89 -97.12 3.23
7 -108.4 -111.64 3.24
Table 4.3: Calculated energies of singly charged copper water complexes and the
corresponding hydroxide containing copper water complex; the difference between each pair
is shown in the column on the left.
It can be readily seen that the hydroxides are less stable. This result agrees with
Vukomanovic and Stone who have concluded that the Cu+-OH bond is weaker than
the Cu+ - H2O bond when in presence of one or two water molecules (120). The
work presented here extends this estimation to up to 7 water molecules, and in all
cases the hydroxide is less stable. The total energy of the CuOH+(H2O) complex
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(plus the expelled hydrogen) is higher than the energy of the Cu+(H2O)2 complex by
4.72 eV. As the number of water molecules in the complex increases, the difference
between the two diminishes steadily, as the hydroxide forming pathway becomes
gradually more favourable. This calculated trend agrees with experimental results
from Stace and coworkers who have found that “in smaller fragments Cu+(H2O)N is
the dominant electon capture product” and when N>8 the hydroxide is the only
product.
ECD of [Cu(H2O)6]2+
The Mass Analysed Ion Kinetic Energy Spectrum (MIKE) obtained by Stace and
coworkers following ECD of [Cu(H2O)6]2+ and using xenon as the collision gas, is
reproduced in figure 4.4. They refer to the following processes:
[Cu(H2O)6]2+ + Ar CuOH+(H2O)6-K-1 + KH2O + H + Ar+ and
[Cu(H2O)6]2+ + Ar Cu(H2O)6-K + KH2O + Ar+
Figure 4.4: MIKE spectrum of ECD of [Cu(H2O)6]2+ (taken from (2)).
145
The peaks corresponding to the formation of the hydroxide are smaller than the
corresponding peaks produced by the Cu+(H2O)N for K=4 and 3. That is not the case
for K=2.
The calculated results, presented in figure 4.5, show that all reactions would be
unfavourable on the basis of the total energies of reactants and products only. The
fact that all energies plotted are positive means that an energy input would be
necessary to perform these reactions. It also shows that such energy input is larger
for the hydroxides, so that calculations predict that the formation of the hydroxides
would be disfavoured in all three cases. The prediction only fails for K=2, which is
the only case where the peak corresponding to the hydroxide is the highest.
The calculation also provides an approximate value of the magnitude of the energy
gap between the products of the two competing pathways, which is of about 5 eV.
The energy input needed to perform this reaction is not small, and that is a
consequence of the high ionization energy of xenon. The energy input could come
from different sources which include the interaction of the complexes with the
collision gas, the process of insertion of the metal atom into the ligand cluster which
is produced during supersonic expansion and also from the process of electron
impact ionization, in which the complexes are bombarded by electrons that have
energies of the order of various keV.
146
Figure 4.5: Enthalpies of reaction (eV) for competing outcomes of ECD of
[Cu(H2O)6]2+ for different values of K using Xe as the collision gas.
ECD of [Cu(H2O)8]2+
The MIKE spectrum obtained by Stace and coworkers following ECD of
[Cu(H2O)8]2+ , using xenon as the collision gas, is reproduced in figure 4.6.
Figure 4.6: MIKE spectrum of ECD of [Cu(H2O)8]2+ (taken from (2)).
147
Similarly to the [Cu(H2O)6]2+ case , the peaks corresponding to hydroxides are
higher than the peaks for Cu+(H2O)N for the higher values of K ( 5 and 4) but not for
the lower ones (3 and 2). In fact, for K=2 there is only the hydroxide peak.
Calculated results for these reactions, which are written below, are presented in
figure 4.7.
[Cu(H2O)8]2+ + Ar CuOH+(H2O)8-K-1 + KH2O + H + Ar+
and
[Cu(H2O)8]2+ + Ar Cu(H2O)8-K + KH2O + Ar+
Figure 4.7 : Enthalpies of reaction (eV) for competing outcomes of ECD of [Cu(H2O)8]2+ for
different values of K using Xe as the collision gas.
148
The calculations, however, predicted lower peaks for the hydroxides in all cases.
They also predicted that a substantial energy input is necessary to perform these
reactions, similar to the case of [Cu(H2O)8]2+ .
4.2.3.2 Copper ammonia complexes
New copper ammonia structures
Cu+(NH3)NH2 Cu+(NH3)2NH2
Cu+(NH3)3NH2 Cu+(NH3)4NH2
Figure 4.8: Structures of CuNH2+ (NH3)N complexes, 1 ≤ N≤ 4.
ECD of [Cu(NH3)8]2+
The MIKE spectrum obtained by Stace and coworkers following ECD of
[Cu(NH3)8]2+ , using xenon as the collision gas, is reproduced in figure 7. It can be
seen that the peaks follow the same pattern seen for water, as there is also
competition between two products of ECD from the doubly charged copper complex,
in this case CuNH2+(NH3)N and Cu+(NH3)N.
149
However, there is a difference: in the case of ammonia all the peaks corresponding to
complexes of the form Cu+(NH3)N are larger than the peaks corresponding to
CuNH2+(NH3)N .
Figure 4.9: MIKE spectrum of ECD of [Cu(NH3)8]2+ (taken from (2)).
Calculations have also been performed for this system, and the results are displayed
in figure 4.10. The reactions in this case are the following:
[Cu(NH3)8]2+ + Ar CuNH2+(NH3)8-K-1 + KNH3 + H + Ar+
and
[Cu(NH3)8]2+ + Ar Cu(NH3)8-K + KNH3 + Ar+
By analysing the calculatd results, it can be seen that the trends are similar to the
ones obtained for the copper water complexes, where the CuNH2+(NH3)N are less
likely to be formed than the corresponding Cu+(NH3)N complexes because their
reaction enthalpies are more positive .
150
Figure 4.10: Enthalpies of reaction (eV) for competing outcomes of ECD of [Cu(NH3)8]2+
for 4 values of K, using Xe as the collision gas.
In this case the agreement between theory and experiment is better than in the case
of water, because theory predicts that the pathway that produces Cu+(NH3)N
complexes is more likely to prevail, and that is what has been observed, as the peaks
corresponding to this reaction are higher than the ones from the competing reaction
in all cases (K=3,4,5). For K=6, no peak is observed for the CuNH2+(NH3) species.
This may be related to the higher gradient of the corresponding graph, on figure 8
(blue line), in going from K=5 to K=6.
Like in the case of copper water complexes, the difference in energy between the
products of the competing pathways is around 5 eV.
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4.2.4 CONCLUSIONS
The simplified approach adopted in this section, which analysed the outcome of gas
phase reactions based on the enthalpies of reactions associated with different
dissociation pathways, has provided various predictions that agree with experimental
results, although in some cases the agreement was poor.
Firstly, the calculations about unimolecular decay have led to the conclusion that this
pathway is likely to occur when the number of ligands is seven or more. This finding
agrees with experimental observations. Furthermore, the background energy needed
to promote these reactions has been estimated to be of the order of 1 eV.
The calculations on proton transfer provided further evidence for the experimental
observation that eight is the minimum number of ligand water molecules needed to
produce a stable doubly charged complex. It was observed that N=8 was the first
configuration to be above the energy axis on the proton transfer graph of figure 2,
indicating a non-favourable reactivity.
The calculations involving ECD processes have shown that a large input of energy is
needed to drive these reactions, mainly because of the high ionization energy of
xenon, which is used as the collision gas. They have also predicted that the
pathways producing OH+ or NH2+ species are less likely to occur; the blue lines that
correspond to these products are always high on the graphs indicating a very positive
enthalpy associated to the reaction. This prediction proved to be correct in the case
of the ammonia complexes. In the case of water complexes it has been only partially
correct. These calculations also predicted that the difference in energy between the
competing pathways studied is about 5 eV. In addition, the energy needed to drive
the reactions is around 5 eV and 10 eV, for reactions starting from eight -coordinate
152
complexes, and around 3 eV and 7 eV for reactions starting from the six-coordinate
copper water complex.
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Chapter 5 The electronic spectra of gas phase
copper and silver complexes
This chapter presents the results of TDDFT calculations involving copper and silver
complexes. Calculations are performed on all structures presented in chapter 3.
Analyses are carried out based on molecular orbitals involved in the electronic
excitations and different functionals are evaluated. Background information on
electronic excitation of open shell transition metal complexes is provided at the
beginning of the chapter. Calculated excitation energies of copper and silver
complexes will be presented and compared to experimental results obtained by Stace
and coworkers using the type of apparatus described in chapter 2. It is assumed that
the absorption of a photon will cause dissociation of the complex so that
photofragmentation channels can be associated with electronic excitation energies.
Introduction
There are various motivations to perform these TDDFT calculations. Firstly, they
provide invaluable information for interpreting the experimental results (1).
Secondly, they make possible the evaluation of asymptotically correct (ac) exchange
correlation functionals to be used in future calculations of excitation energies
involving open shell gas phase transition metal complexes. Transition metal systems
are a hard test for theory for reasons that include the high electron correlation.
The gas phase environment alleviates some difficulties associated with the
condensed phase, like the interaction with solvent, counter ions etc. Some of the
complexes studied in this thesis have their experimental spectra recorded, but this
154
hasn’t been accomplished for all of them yet. Thirdly, this study will analyse the
need for relativistic corrections in TDDFT calculations involving copper, which has
not been established as yet although for ground state calculations it is a controversial
issue (79)(46). Finally, it is important to find out how the excitation energies change
as a result of an increasing number of ligands, i.e. the effect of solvation on
electronic excitations, and also to characterize the nature of the transitions, i.e. ligand
to metal charge transfer (LMCT), metal to ligand charge transfer (MLCT) or ligand
field transitions (dd).
5.1 Background theory
5.1.1 α and β electrons
Unrestricted calculations present two sets of electrons that are classified as β
electrons and α electrons. The unpaired electron will be an α electron, as well as all
other electrons that have the spin pointing in this same direction. Exchange
interaction only takes place between electrons that have the same spin; because α
electrons are more numerous- as it is always the case in open shell systems - their
exchange energy is higher and their total energy is lower (exchange provides a
negative contribution to the total energy of an electronic system). Exchange energy
is a consequence of the indistinguishability of electrons and it is a purely quantum
phenomenon, without any classical analogue.
This way, each full orbital will have one β and one α electron. In the case where the
orbital is singly occupied, it will have an α electron only.
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5.1.2 HOMO and LUMO
In the context of unrestricted calculations, the concepts of highest occupied
molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO)
acquire some extra complexity. That is because α and β electrons are often dealt with
separately, and it becomes convenient to introduce a new terminology that allows for
that. Therefore, the concepts of α LUMO, α HOMO, β LUMO and β HOMO must be
introduced. For instance, the α LUMO is the lowest unoccupied molecular orbital
with α spin. It may not be the LUMO overall, i.e. there may be another unoccupied
molecular orbital with a lower energy, but with a β character. The β LUMO will be
particularly important in the study of electronic excitations in d9 complexes because
dominant transitions will usually end on this orbital. It is important to note that the
traditional concept of HOMO-LUMO gap, which normally involves a relatively
large gap in orbital energies, will not apply to a β HOMO-LUMO gap.
5.1.3 Oscillator strength
The molar extinction coefficient, which is the number experimentalists use to
quantify the intensity of an electronic transition, is not easily handled theoretically
(45). Because of this issue, the oscillator strength (f) has been introduced. The
oscillator strength is proportional to the area under an absorption peak in a plot of
extinction coefficient versus frequency. It can adopt values between 0 and 1; the
higher its value, the stronger the corresponding electronic transition.
This chapter will adopt the following terminology: a large peak has an oscillator
strength of 0.09 or above; a medium peak has an oscillator strength between 0.08 and
0.02 and a small peak has an oscillator strength between 0.02 and 0.01.
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Tables of excitation energies
There are a lot of excitation energy data that must be shown in this chapter, as this is
the main focus of this thesis. For each structure, the lowest twenty electronic
excitations have been calculated. Not all of those are important, i.e. have a
significant intensity. To make easier to focus on the dominant transitions, the tables
display peaks that are proportional to the oscillator strength of the corresponding
transition. These peaks are normalized in such a way that when they are at their
maximum (when the peak fills the whole cell) they correspond to an oscillator
strength f = 0.25. Very few complexes, however, will display such high oscillator
strength.
5.1.4 TDDFT and asymptotically correct (ac) functionals
The DFT calculation of excitation energies and also polarizabilities required the
development of a new class of functionals, which have been named ac functionals.
Asymptotically correct functionals
These new functionals have the right behaviour in the asymptotic regions, e.g., far
from the centre of an atom or molecule. For this reason they are called ac
functionals. They are supposed to improve on the results provided by the LDAxc and
GGA’s in the calculation of certain molecular properties like excitation energies and
polarisabilities (58).
In the asymptotic region, a DFT potential is expected to decay as 1/r, which is the
expression that corresponds to the Coulomb potential due to a point charge.
However, functionals like LDAxc and GGA’s don’t present this asymptotical
behaviour. LDAxc potential decays exponentially and B88X+P86C decays as –c/r2
(121). That’s why this new generation of functionals had to be developed.
157
The first ac functional was proposed by Leween and Baerends in 1994 (122) and it is
known as LB94. It has been tested on a range of organic molecules and it yields
results that for high lying excitations are better than those calculated using non-ac
functionals. However, the results from LB94 for states close to the centre of the
molecule were not as good. In fact, standard functionals may perform better in this
respect. Therefore, new functionals were needed to integrate the advantages of both
types of functionals (ac and non-ac). These new functionals, also introduced by
Baerends and coworkers, are called SAOP- statistical averaging of (model) orbital
potentials (123) and GRAC - gradient correct asymptotically correct functional
(124) .
Casida and Salahub (125) also tried to improve on the LB94 functional. They
introduced the AC-LDA (asymptotically correct local density approximation) in
2000. This new functional was tested on small organic molecules and it provided
better excitation energies than the LB94 functional.
Tozer and Handy (126)(127) have also introduced ac functionals in 1998. Their
functional is called HCTC-AC and it has been successfully tested on a range of small
organic molecules and also on some neutral transition metal complexes. These
researchers are also engaged in developing functional that offer an increased
performance regarding charge transfer excitations, particularly in the cases where
there are large distances between ligands and metal centres.
Ahlrichs and coworkers (128) have proposed a different approach to the calculation
of high lying excitations. Instead of using an ac functional, they use an auxiliary
basis set expansion for the treatment of Coulomb type matrix elements, which has
also been successfully tested on a range of small organic molecules.
158
Frisch and coworkers (129) have also calculated accurate excitation energies for a
range of organic molecules, including C70. Their calculations were performed using
the Gaussian (130) code, which hasn't got any ac functionals implemented. Their
functional of choice was B3LYP.
Most of the new functionals have been tested on organic molecules only. The quest
for functionals to yield accurate excitation energies for open shell transition metal
compounds, in the gas phase in particular, is ongoing.
Although much progress has been made with the use of ac functionals, not everyone
agrees that this is the way forward. Burke and coworkers (131) have found that the
most simple of the functionals, the LDAxc, can also yield accurate Rydberg
excitation energies. Chelikowsky (132) has also demonstrated that TDLDA can be
used to calculate accurate excitation energies on a variety of systems. Both research
groups have investigated the conditions under which the TDLDA approach can be
effectively used.
It has been found that the conditions for successfully applying the TDLDA are:
1) the excitation energy is lower than minus the energy of the HOMO (highest
occupied molecular orbital) (132).
2) the transitions occur between bound orbitals (131). (In some cases the TDLDA
yields accurate excitation energies even for bound free transitions, as in the π → π*
excitation in benzene).
It is convenient to use the TDLDA when possible because it offers advantages, with
respect to other more complicated functionals. It requires no empirical parameters,
although it is fitted to a Monte Carlo simulation, like for instance the B3LYP
functional (uses three empirical parameters). Also it has a more simple form, without
derivatives of the potential, so that it saves computing time.
159
5.1.5 Review of previous work in TDDFT applied to transition metals
Since the first TDDFT calculations involving transition metal compounds were
performed by the Theoretical Chemistry Group of Vrije University (Amsterdam) in
1999, more than 60 papers have been published in this area. The applications of the
method include the assignment of experimental spectra (1) (133) (134), the
development of materials of technological importance (135) (136) and also the
understanding of biological processes (137).
The first TDDFT calculations involving transition metal complexes were performed
on MnO4-, Ni(CO)4 and Mn2(CO)10, because the assignments of the spectra of these
complexes were controversial (92) . The calculation on MnO4- showed that the older
DFT method used to calculate excitation energies, ∆-SCF, may give results that
differ substantially from TDDFT, although in some cases the two methods give
similar results. TDDFT has stronger theoretical foundations and is capable of
providing oscillator strengths, which are not readily available in a ∆-SCF calculation.
The results obtained for MnO4- and Ni(CO)4 where of a quality comparable to
results from highly correlated methods like CI, CASPT2 and CASSCF. The results
for Mn2(CO)10 are the highest-level theoretical results ever obtained for that
compound (92). Such large system cannot be treated by CASSCF because the
number of active orbitals necessary to perform the calculation would be too large.
Equally, it would be very computationally expensive, if not prohibitive, to treat this
system with other highly correlated methods. In the same year, the Vrije University
Theoretical Chemistry Group published another article reporting a relativistic
TDDFT calculation, performed on transition metal hexacarbonyls (138). The metals
chosen were Cr, Mo and W, so that the extent of the relativistic effect can be
evaluated as one moves down the group. All the calculations reported in these two
160
papers were performed using the ADF program, and the functionals used were B88X
+ P86C /ALDA and LB94/ALDA. Although it was made clear that the choice of
functional affects significantly the results, there was no general conclusion on which
functional is the best to be used in a TDDFT calculation involving transition metals.
Other important developments in the implementation of TDDFT in ADF came from
the group at the University of Calgary in 2004 and 2005. Firstly, the first TDDFT
calculation performed on an open-shell transition metal complex, including metals
like Mo, W, Tc, Re and Cr (139). Using B88X + P86C /ALDA and LDA/ALDA , an
accuracy of 0.3-0.5 eV on excitation energies was achieved. This level of accuracy is
the same found in closed shell transition metal systems. Secondly, the development
of the spin-flip TDDFT (SFTDDFT), allowed for more types of transitions to be
studied, for instance singlet to triplet transitions (74). Finally, in 2005, the group
proposed an implementation of TDDFT which accounts for spin-orbit coupling (86).
This new approach was tested on square-planar Pt (II) complexes and results proved
that spin-orbit coupling must to be taken into account in TDDFT calculations (140).
(Although currently, this is only implemented for closed-shell complexes, thus it is
not possible to do spin-orbit coupled TDDFT calculations on the Cu (II) and Ag (II)
complexes presented here.) The functionals used in these calculations were LDAxc
/ALDA and SAOP/ALDA. In many cases results were similar but SAOP proved to
be better to deal with electronic excitations which depart from a d orbital.
Although the implementation of TDDFT in ADF has been developed to a more
advanced stage, the vast majority of the TDDFT research published so far is carried
out using Gaussian, which is another quantum chemistry software package which has
a TDDFT implementation. The main disadvantage of Gaussian, in the context of
TDDFT calculations, is the absence of any asymptotically correct functionals. Most
161
TDDFT calculations involving transition metals performed on Gaussian have
employed the LANL2DZ (141) (133) (142) (143) (136) (144) basis set on the metal
whereas ADF calculations use TZP or TZ2P. Virtually all Gaussian TDDFT
calculations use the B3LYP functional for geometry optimisation and also for the
TDDFT calculation itself.
TDDFT has been applied to a large variety of systems which include solids, gases
and solutions. The calculated frequencies are usually in the UV/Vis range but there
have been calculations of x-ray spectra (145) as well.
The system to which most attention has been devoted by the TDDFT researchers is
the porphyrin ring (146) (147) (148) (141) (149) (150) and its derivatives. It is a
large system, as it is comprised of four pyrrole subunits, so that it couldn’t be treated
by methods like CI which are computationally very expensive. The advent of
TDDFT allowed for the elucidation of the properties of this important system which
has crucial importance for life and also various technological applications, the most
prominent being light harvesting (151) (152) and photodynamic therapy (PDT) (153)
(154).
This cavity at the centre of the porphyrin ring can accommodate a metal ion, and
TDDFT calculations have been performed on this system using a variety of metals ,
including all members of Group 4A (148), Zn (150) and Ni (147). The spectra of
bacteriochlorin (141), which is a porphyrin system where two of the pyrrole rings
have been reduced, has been studied using all the different transition metals in its
cavity, in order to find the optimum complex for a given application, which in this
case is PDT. Also, there have been calculations envisaging to determine if the
distortions of the porphyrin ring shift the excitation energies significantly (147). The
162
problem of how changes in geometry affect the excitation energies of a given
complex has also been addressed by Nemykin, who performed TDDFT calculations
on Mo carbonyl complexes (116).
In addition to porphyrin, many other systems of biological importance have been
studied using TDDFT. For instance, the binding of transition metals to DNA (142),
which is experimentally investigated using spectrophotometric titration. An
understanding of these binding mechanisms helps in the development of new cancer
drugs and other complexes of biochemical activity. Works that contribute to the area
of metalloproteins include the electron paramagnetic study of Cu complexation in a
hemicarcerand (137). Other examples include a study of the binding of NO, which is
an important neurotransmitter, to diimine-iron complexes (155) and a study of
clusters involving iron and sulfide (145), which are important mediators in one-
electron redox processes in respiration and photosynthesis.
TDDFT calculations in transition metals have also included solvent effects (149)
(143) (156), by using the polarizable continuum model (PCM) which is implemented
in Gaussian. One author uses the model to calculate shifts on the spectrum of
porphyrins (149), due to the presence of water. There are also calculations of
solvatochromic shifts, the shift in excitation energies due to solvent interactions, in
systems like (Me(2)Pipdt)Mo(CO)(4) and (Me(2)Pipdt = N,N '-dimethylpiperazine-
2,3-dithione) which was performed by Nemykin et al. (156) and also tetracarbonyl
tungsten complex of 2-(2 '-pyridyl)quinoxaline (143), where a linear correlation
between the solvent dipole moment and the shift in the excitation energies was
found.
163
Many other calculations focused on complexes with technological applications. For
instance, polypyridine complexes of Ru, Fe and Os (157) and also
[Cu(pqx)(PPh3)2]+ (158) are used in solar energy conversion. Another area in which
TDDFT is applied often is electroluminescence. Examples of calculations in this area
include the divalent Os and Ru complexes and a series of Cu singly charged
complexes (137) . Another area of application of TDDFT is the development of
synthons to be used in synthetic laboratories, like the tetracarbonyl tungsten complex
of 2-(2 '-pyridyl) quinoxaline (143). Finally, TDDFT is applied to the development
of complexes to be used in catalysis. Examples of this application include the study
of transition metal monoxides (136), Re cyanide complexes (159) rhodium
tetracarboxylate complexes (160) used in hydrogenation, hydroformylation,
oxidation and carbene reactions- and also the half-sandwich arene Ru complexes
(135), used in hydrogenation and in ring opening metathesis polymerisation.
Finally, there are TDDFT works envisaging to help in the interpretation of
experimental spectra and also to assess the performance of different functionals.
Examples of those are the study of RuO4 and OsO4, which are important systems
because of their similarity to MnO4 - , and also the study of vanadium and
molybdenum oxide (133) , which are interesting subjects because they have plenty of
experimental data available. The spectra of silver and zinc complexes have also been
successfully explained using TDDFT (1) (134).
164
5.2 Results: the calculated electronic spectra
5.2.1 Copper (I) complexes / copper (II) complexes
Computational Details
Electronic excitation energies of copper complexes have been calculated using the
same basis set that was used in the geometry optimisations of the structures. That is a
TZ2P basis set with a frozen core at 2p for copper and 1s for oxygen and nitrogen. A
trial calculation performed on the four-coordinate copper (II) pyridine complex of
symmetry D2d employed relativistic corrections using the ZORA equation.
In the TDDFT calculations two functionals were employed: LDAxc and LB94.
5.2.1.1 Copper (II) phthalocyanine
Introduction
Phthalocyanine is a nitrogen based multidentate which is also called tetra aza tetra
benzoporphine. The complex is shown in figure 5.1.
Figure 5.1: Calculated structure of copper (II) phthalocyanine.
165
Calculations have been performed on this complex in order to provide an indication
of the adequacy of the methods chosen in this thesis, as this is a system which
possesses a well known spectrum in the condensed phase. Copper phthalocyanine
blue is perhaps the most important blue pigment in industry; it is the choice of ink
makers, paint makers and plastic colour formulators throughout the world (161). It is
also very flexible as its chlorinated derivative, copper phthalocyanine green, is used
when a mixture of blue and yellow is not convenient. The brominated version
provides a yellowish green colour.
Copper (II) phthalocyanine is a large transition metal complex, but its spectra can be
calculated using TDDFT. The complex was successfully optimised using the same
conditions (method and basis set) that were used for all the copper complexes in this
thesis.
Results
This complex has D4h symmetry. After the optimisation, TDDFT calculations were
performed. Although the TDDFT calculation using LB94/ALDA failed to achieve an
end for such a large system, the LDAxc /ALDA calculation was successfully
completed, and its results are displayed in table 5.1.This calculation also illustrates
the power of TDDFT to deal with large systems involving transition metals.
166
Table 5.1: Excitation energies (E / eV) and oscillator strengths (f) calculated for the Cu (II)
phthalocyanine complex using the LDAxc functional. Symmetry labels correspond to
transition symmetries in D4h.
The calculated spectrum of copper phthalocyanine has a very large peak (f=0.295) at
1.72 eV, which is located in the low energy limit of the visible spectrum, and it
corresponds to the colour red. A strong absorption of wavelengths in the red causes
the complex to exhibit its complementary colour, which is blue, as observed.
In fact, the prominent feature of the experimental spectrum of copper (II)
phthalocyanine vapour is a strong and sharp absorption at 658 nm (162), which
167
corresponds to an energy of 1.82 eV. Hence, there has been a very good agreement
between theory and experiment.
Dominant transitions in this complex are of the LMCT type. This finding agrees with
previous calculations of electronic transitions (163) that employed the ZINDO
method.
These results suggest that the methodology employed to study the electronic
excitations of copper complexes is appropriate.
5.2.1.2 Copper (II) pyridine
Introduction
Calculations on these complexes have already been performed by Cox and
coworkers (3), who used the older DFT - ΔSCF method to calculate the excitation
energies of the four and six-coordinate complexes as open-shell TDDFT was not
available at that time. The purpose of the present work is to use TDDFT, which is the
best DFT based method at the moment, in order to calculate these energies.
Furthermore, the calculations will also be performed on the five-coordinate
structures, presented in chapter three.
The experimental spectrum
The electronic excitation spectra of copper pyridine complexes, for N=4 to 6, have
been obtained by Stace and coworkers and are presented in figure 5.2
168
Figure 5.2: Photofragmentation spectra of doubly charged copper pyridine complexes
obtained by Stace and coworkers (3). The energies are shown in wavenumbers and also in
eV (at the bottom of the graph).
These spectra comprise broad bands centred in the range from 1.9 eV (15323 cm-1)
for N=4 to 1.55 eV (12500 cm-1) for N=5. The peak for N=6 seems to be in between
those for N=4 and N=5, and it is centred around 1.86 eV (15000 cm-1). These spectra
will be compared to the calculations performed, which are shown in the next
sections.
169
Results
As discussed in chapter 3, copper pyridine structures have been successfully
calculated for N= 2 to 6.
In the cases of N= 4 to 6, a number of structures have been found for each N. The
differences in energy between all these different structures are relatively small (these
energies are displayed in table 3.5) and they lie within the background energy of this
kind of experiment. That means that all these structures can be accessible during the
experiment.
Hence, excitation energies will be calculated for all the structures just mentioned, so
that a comparison can be made to experimental results in order to determine which of
the structures are most likely responsible for the observed photofragmentation
spectra. The calculated spectra of the four-coordinate complexes will be presented
first.
The four-coordinate complexes
According to chapter 3, there are three possible geometries for the four-coordinate
doubly charged copper pyridine complex, namely D4h , D2h and D2d . The square
planar configuration (D4h), whose image is displayed in chapter 3, is the most stable.
The D2d geometry is the least stable, but it lies only 34.2 kJ/mol above the D4h
structure. The D2h structure is obtained by rotating by 90 degrees two of the pyridine
rings on the D4h structure.
To attempt this assignment, the calculated spectra of these complexes will be
compared to the experimental results.
Calculated excitation energies are shown in table 5.2.
170
Table 5.2: Excitation energies and oscillator strengths calculated for the [Cu(pyridine)4 ]2+
complexes using the LDAxc and LB94 functionals.
Based on the data in table 5.2 the following analyses can be made: firstly, it is
readily seen that the excitation energies obtained using the LB94 functional are
always lower than the corresponding LDAxc energies. Comparison to the
experimental spectrum (figure 5.2) suggests that LB94 energies show better
agreement. Furthermore, the LB94 functional yields higher oscillator strengths in the
area that coincides with the experimental data (1.24-2.48 eV). As a result, further
analyses of the spectra of four-coordinate complexes will be carried out based on
LB94 results.
171
Secondly, it can be also seen that although the D4h and D2h structures are very
similar, their spectra are quite different. The D2h structure presents more bands that
match the experimental region as it presents a large peak at 2.16 eV . The D4h
configuration has its first significant peak at 2.35 eV, and that is a medium peak.
Finally, it is seen that the configuration whose spectrum has the best agreement with
experiment is the D2d. It presents a large peak at 2.13 eV (slightly closer to
experiment than the peak from the D2h structure which is located at 2.16 eV) and also
two medium peaks at 1.52 eV, which are therefore located in the area under the
experimental curve. This conclusion agrees with that found by Cox and coworkers
using the ΔSCF method (3).
Further evidence for the assignment of this experimental spectrum to the D2d
structure comes from the fact that photofragmentation of this complex doesn’t yield
charged fragments and therefore it is associated with a d d transition, which
doesn’t involve charge transfer. In fact, among the three structures considered here
for the four-coordinate doubly charged copper pyridine complex, the D2d structure is
the only one capable of producing a dipole-allowed d d transition. That is because
this type of transition is forbidden in the other two geometries due to symmetry
considerations (point group having inversion symmetry).
The need for relativistic corrections in calculations of the electronic spectra of
copper complexes
In chapter 3 it was concluded that relativistic effects were not important for
geometries of copper complexes but in order to eliminate this doubt in the context of
the calculations performed here in this chapter, a preliminary calculation of
excitation energies of the four-coordinate doubly charged copper pyridine complex
172
(D2d geometry) has been performed relativistically and non-relativistically using, in
each case, the LDAxc and also the LB94 functionals. These energies are presented
in table 5.3.
Table 5.3: Relativistic and non-relativistic energies of the four-coordinate D2d copper (II)
pyridine complex.
Table 5.3 shows that the relativistic and non-relativistic energies are very similar. In
the LDAxc case, the difference in energy observed in the dominant B2 transition (at
2.79 eV on the relativistic case) was 0.11 eV. This difference decreased to 0.07 eV
in the case of the E dominant transition. Non-relativistic energies were lower than
their relativistic counterparts.
173
In the LB94 case, the difference in the dominant B2 excitation (located at 2.21 eV in
the relativistic calculation) was only 0.08 eV. The difference in the dominant E
excitation energy was extremely small: 0.01 eV.
Interestingly, smaller differences between relativistic and non-relativistic results are
observed when the LB94 functional is used instead of the LDAxc functional.
Furthermore, the small differences in energy observed as a result of relativistic
corrections being applied to the calculation did not lead to improvement with respect
to the comparison to experimental results. Therefore, relativistic corrections don’t
seem to be relevant in this context and therefore they have not been applied to
further calculations of excitation energies on these copper systems.
The spectra of the five-coordinate complexes will be presented next.
The five-coordinate complexes
According to chapter three, there are three possible structures for the five-coordinate
complexes. The most stable structure is 5A. Structures 5B and 5C lie 20.1 and 27.4
kJmol-1 above 5 A respectively. These energies are very low if compared to the
differences in energy among competing structures in the four-coordinate and six-
coordinate cases. As a result, all three structures are likely to be found in an
experiment.
The electronic excitation spectra of all three possible five-coordinate structures have
been calculated and they are presented in table 5.4.
174
Table 5.4: Excitation energies and oscillator strengths calculated for the [Cu(pyridine)5 ]2+
complexes using the LDAxc and LB94 functionals.
The LB94 spectra present lower excitation energies and a better agreement to
experimental results. These facts were also observed in the case of four-coordinate
complexes. Again, LB94 energies are closer to the peaks of the experimental spectra
and oscillator strengths in this range are higher than the corresponding ones
calculated using the LDAxc functional.
The experimental curve for this complex consists of a broad band extending from
1.24 eV (10000 cm -1) to 2.48 eV (20000 cm -1). The highest region of the curve,
which corresponds to the strongest absorption, is centred around 1.55 eV
(12500 cm -1 ).
175
The spectrum of the 5C structure presents a very good agreement to experiment, as it
presents two medium peaks in the experimental region just described, one at 1.58 eV
and the other at 1.99 eV. Both peaks have A1 symmetry.
The six-coordinate complexes
It has been explained in chapter three that there are four possible structures for the
six-coordinate complexes: 3 are pseudo-octahedral (6A, 6B and 6C) and have D2h
symmetry. The other (6D) has C2v symmetry. The structure that is lowest in energy
is 6A. Structure 6C is very close in energy; it is only 13.6 kJmol-1 above 6A. The
other two structures, however, are considerably higher in energy (59.6 and 42.9
kJmol-1 ). Hence, structures 6A and 6C are very likely to be found in the experiment,
whereas 6B and 6D will have a much reduced probability of being formed.
176
Table 5.4: Excitation energies and oscillator strengths calculated for the
[Cu(pyridine)6]2+ complexes using the LDAxc and LB94 functionals.
The experimental curve for the six-coordinate complex consists of a broad band
extending from 1.3 eV (10485 cm-1) to 2.48 eV (20000 cm-1). The region with the
highest absorption goes from around 1.55 eV (12500 cm-1) to 2.10 eV (16936 cm-1).
Considering the two most likely structures for this complex, the 6C structure is the
one whose calculated spectrum presents the best agreement to experiment. The
calculated excitation spectrum of the 6A structure, the lowest in energy, has its
lowest dominant transition located at 2.87 eV, which is well outside the experimental
band.
177
The 6C structure presents a large peak, of B2u symmetry, located at 2.14 eV. This is
the closest to the experimental curve. The agreement between theory and experiment
in this case is not as good as in the previous cases (four and five-coordinate
structures), but it is reasonable.
The 6B structure presents a large peak, also of B2u symmetry, at 2.22 eV, which is
further from the experimental region than the peak from 6C. Furthermore, the 6B
structure is unfavourable as it is almost 60 kJ/mol above the 6A structure.
5.2.1.3 Copper water / copper ammonia
5.2.1.3.1 Copper water
Cu(I) water
The calculated spectra of Cu (I) complexes are shown in table 5.5, which displays
the results obtained for each complex. The most evident feature of these spectra as a
whole is the absence of any strong peaks. The calculated oscillator strengths are very
low, except in very few cases. For instance the one-coordinate complex presents a
sequence of five strong peaks, using the LDAxc functional, with oscillator strengths
between 0.020 and 0.138, starting at 7.12 eV. These were the strongest peaks
calculated for singly charged copper water complexes.
Other strong peaks include the one calculated for the 2+1 complex, located at 8.46
eV (f=0.05190) and also another one calculated for the 3+0 complex , which is
located at 5.15 eV (f=0.0478) using the LDAxc functional.
Unfortunately the strong peaks calculated for gas phase singly charged copper water
complexes are located at energies that are very high to be detected experimentally.
178
The available tunable lasers work at much lower energies. Stace and coworkers, for
instance, record photofragmentation spectra using equipment that scans the UV-VIS
part of the spectrum, comprising frequencies / wavelengths between 1.73 and 2.85
eV (14000 to 23000 cm-1).
The lowest energy at which a strong peak has been calculated for these complexes is
3.29 eV. It corresponds to the 4+1 complex and it has oscillator strength of 0.0337.
Hence, there are no meaningful peaks that lie in the visible range.
179
180
Table 5.5: Excitation energies calculated for Cu (I) water complexes using the LDAxc and
LB94 functionals.
Character of transitions
Although most of the transitions studied for this type of complex have very low
oscillator strengths, typically around 0.002, the highest peaks corresponding to each
complex have been analysed and it has been found that they correspond to metal to
ligand charge transfers (MLCT).
This fact is illustrated in figure 5.3, which shows the molecular orbitals involved in
the transitions.
181
Figure 5.3: Molecular orbitals involved in electronic excitations involving the 4+1 (on top)
and 4+4 (below) singly charged copper water complexes.
On the left hand side of figure 5.3 (top), complex 4+1 and its molecular orbitals are
shown in detail. The pictured transition, from orbital 29 (HOMO -1) to orbital 31,
which is the LUMO, is the one located at 2.55 eV with oscillator strength 0.00784.
It clearly illustrates the MLCT character of the transition. The molecular orbital
where the transition ends (31) has virtually zero amplitude at the metal centre.
On figure 5.3 (bottom), an electronic transition in the 4+4 complex is pictured. It is
located at 1.55 eV (f=0.0025). The transition goes from molecular orbital 41
(HOMO -1) to molecular orbital 43 (LUMO). It is clearly seen that orbital 43
(destination) has no amplitude at the metal centre so that is also a MLCT.
Electronic transitions in figure 5.3 show the general trend: dominant electronic
transitions in singly charged copper complexes are of the MLCT type. These two
cases have been chosen to illustrate this general trend, among the dominant
excitations of this type of complex.
182
This trend has also been observed in other contexts. For instance, the spectra of the
condensed phase hexanuclear copper (I) complex, with trithiocyanuric acid as a
ligand, has been reported to display electronic transitions of the MLCT type (164).
Cu(II) water
The spectra of the doubly charged copper water complexes is much more interesting
than those of the singly charged complexes, because it presents a number of
dominant transitions, many of them lying in the visible range. This fact enables the
experimental study of these complexes in the gas phase, using techniques like the
one described in chapter one (although they are yet to be performed).
All the spectra calculated for the doubly charged copper complexes are presented in
table 5.6:
183
184
185
Table 5.6 (previous page): Excitation energies calculated for Cu (II) water complexes using
the LDAxc and LB94 functionals.
The lowest energy transitions, which are the first ones shown in each column in table
5.6, are usually of dd character and they are considerably weaker than the
dominants (LMCT) transitions, because of Laporte selection rule.
It can be seen that the absorption maxima change considerably when more ligands
are added to the doubly charged copper. There are many strong absorptions in the
visible range and it appears that the predominant colour of the complexes moves
from red to blue as more ligands are added. The colour that corresponds to the low
energy part of the spectrum, around 1.7 eV (700 nm) is the red and the colour that
corresponds to the high energy end, located at around 3.1 eV (400nm) is the blue.
Hence, complexes that absorb in the low energy end of the visible spectrum (reddish
colours) will appear blue, because it is the complementary colour. Likewise,
complexes that absorb in the high energy end will appear red.
Although copper (II) compounds are often blue, like many of its salts, minerals and
also proteins, it is a misconception to assume that they are always blue. They can
assume various colours and this calculation illustrates how the colour of copper (II)
complexes varies as a function of the number of water ligands present. It is
interesting to remember that chromium (III) compounds, for instance, can assume a
variety of colours, as seen on the gem stones emerald, ruby and alexandrite (green,
dark red and red respectively) and also on its carbonate, chloride and oxide (grey,
purple and green respectively)(161) .
186
The complex [Cu(water)]2+ has its largest peak outside the visible range, located at
4.6 eV (5.23 using LDAxc). There is also a medium peak located at 2.76 eV (2.96
eV using LDAxc) which is at the high energy end of the visible spectrum and
associates the colour red to this complex.
The complex [Cu(water)2]2+ has a very strong absorption (f= 0.12) at 2.34 eV (2.61
eV using LDAxc) and that corresponds to colours located around the middle of the
visible spectrum. The highest peaks in the electronic excitation spectrum of
[Cu(water)3]2+ are outside the visible range (4.86 and 5.50 eV) but there are also two
medium peaks at 2.47 and 2.66 eV, which confer a reddish colour to the complex.
The addition of two hydrogen bonded water ligands to the three-coordinate copper
water structure, to form the 3+2 structure, changes the appearance of the spectrum
considerably. In this case there are two medium peaks in the visible range, at both
ends of the spectrum; the first is located at 1.6 eV (1.88 eV using LDAxc) and the
second is located at 2.71 eV (3.06 eV using LDAxc).
The electronic excitation spectrum of the four-coordinate copper water complex
displays two small peaks at the lower energy end of the visible spectrum (at 1.69 and
1.88 eV) and that suggests that the colour of this complex is blue. The highest peaks
in this spectrum are again beyond the visible range, at 5.04 and 5.06 eV. That agrees
with the fact that copper proteins of Type I (where copper is four-coordinate) are
often called copper blue proteins.
When more ligands are added to the [Cu(water)4]2+ complex in such a way as to
form new complexes with a second solvation shell, namely complexes 4+1, 4+2, 4+3
and 4+4, the peaks just described move up in energy, progressively. This trend can
be observed in table 5.6.
187
The small peak located at 1.88 eV for the 4+0 complex moves to 2.67 eV, in the case
of the 4+1 complex, 2.73 eV in the case of the 4+2 complex, 2.93 eV in the case of
the 4+3 complex and finally 3.12 eV in the case of the 4+4 complex. The peak also
gains intensity, as the hydrogen bonded water ligands are added. The oscillator
strength starts at 0.011, for 4+0 and then progressively increases to 0.020, 0.025 and
0.05 for the 4+3 complex. In the case of the 4+4 complex the oscillator strength of
the peak is 0.034.
Likewise, the peaks located at 5.04 and 5.06 eV in the 4+0 complex move to 5.20
and 5.25 eV, 5.33 and 5.50 eV, 5.47 and 5.59 eV and finally 5.64 and 5.67 eV
respectively when water ligands are added to form the 4+1, 4+2, 4+3 and 4+4
complexes.
This systematic blue shift that takes place as a consequence of the addition of water
ligands to the second solvation shell of the complex can be interpreted by analysing
the orbitals involved in the transitions. It has been found that all the dominant
transitions, in all of the copper (II) complexes studied in this thesis end on the β
LUMO of the corresponding complex. Because there is no spin flip, all the
transitions also start in a β orbital.
What is interesting is that the β LUMO of the 4+0 complex is very similar to the β
LUMO’s of the 4+1, 4+2, 4+3 and 4+4 complexes, i.e. it is the half-filled dx2
- y2 metal
based orbital, and that accounts for the similarities in their dominant transitions. This
fact is illustrated in figure 5.4 for the 4+0, 4+1, 4+2 and 4+4 complexes. Figure 5.4
shows the orbitals involved in the strongest electronic transitions of the complexes.
By analysing the excitation energies data in table 5.6 it is found that there are two
prominent bands in each case, and these are the transitions depicted in figure 5.4. On
188
the left side of the arrow is shown the orbital where the transition originates and on
the right side is the orbital where the transition ends.
Although the orbital where the transitions of 4 + X (X=0 to 4) complexes end are
very similar in all cases, the orbital where these transitions originate is not. The latter
spreads over all the ligands and has virtually zero amplitude at the metal centre, so
that it is different in each configuration. That is probably the factor that causes the
small blue shift observed when water ligand molecules are added to the second
solvation shell of these complexes.
4+0 4+1
4+2 4+4
Figure 5.4: Molecular orbitals involved in dominant electronic excitation of the doubly
charged 4+0 (top left), 4+1 (top right), 4+2 (bottom left) and 4+4 (bottom right) copper
water complexes.
189
Furthermore, the dominant excitations of the 6+0 complex are very close to the ones
from the 4+0 complex. The strongest peaks on the calculated electronic excitation
spectrum of the 6+0 complex are located at 4.96 eV (both), so that there is only a
difference of 0.1 eV from the corresponding transitions in the 4+0 complex. Again
the reason for that is the appearance of the β LUMO in the 6+0 complex, which is
the orbital where the relevant transitions end. It is very is similar to the β LUMO of
the 4+0 complex. In this case, the similarity is due to the fact that the β LUMO in the
6+0 complex doesn’t include the axial ligands. The orbitals involved in the dominant
transitions of the 6+0 complex are illustrated in figure 5.5. Furthermore, the orbitals
where the transitions originate don’t include axial ligands either. That makes this
transition very similar to the ones in the 4+0 complex, and explains why the energies
are so close.
In fact, the dominant transitions of the 6+0 complexes are closer in energy to the
corresponding ones from the 4+0 complex than the corresponding ones from the
4+X (X=1 to 4) complexes. That is because the dominant transitions in the
complexes with second solvation shell start in orbitals that are delocalised over all
the ligands whereas the corresponding transitions in the 6+0 complex start in orbitals
based only on the four equatorial ligands. That makes it more similar to the
transitions in the 4+0 complex. Furthermore, the spectrum of the 6+0 complex is
more similar to the spectrum of the 4+0 than to the spectrum of the 4+2 complex,
which has the same number of water ligands. The 5+0 complex doesn’t follow this
trend because its geometry is very different from these ones that are based on the
square planar structure.
190
Figure 5.5: Molecular orbitals involved in dominant excitation of the 6+0 complex.
Another feature of copper (II) electronic excitations that can be learnt from figures
5.4 and 5.5 is the character of the transitions. It is clear, in all cases depicted there,
that these transitions are of the LMCT type. In fact, all the dominant electronic
excitations calculated for the copper (II) water complexes are of the LMCT type.
This fact agrees with other studies of this metal dication. For instance, the electronic
transitions in the copper proteins azurin and plastocyanin are associated to LMCT,
which originates in a nitrogen or sulphur atom (165). It is also observed that all
dominant transitions in these systems end on the β LUMO. That means that
dominant transitions don’t have to cross the HOMO-LUMO gap in the molecule, as
the β LUMO is already singly occupied and the electron that comes in a transition
simply pairs up with the electron that is already there. In general the LB94 functional
provides larger transition energies for singly charged complexes and lower energies
for doubly charged complexes when compared with the data obtained using LDAxc.
191
5.2.1.3.2 Copper ammonia
Cu(I) ammonia
192
Table 5.7: Excitation energies calculated for Cu (I) ammonia complexes using the LDAxc
and LB94 functionals.
The calculated excitation energies of the singly charged copper ammonia complexes
are shown in table 5.7.
Similarly to the case of copper (I) water complexes, the spectra of copper (I)
ammonia complexes don’t present any strong absorptions in the visible range.
Considering all excitation energies in the range that has been calculated, a few small
peaks can be found in the spectra of these complexes. Strong peaks have only been
found in the spectra of the one, two and three-coordinate complexes. These are: 2
peaks around 9.54 eV in the one-coordinate complex, one peak at 9.91 eV in the
spectrum of the two-coordinate complex. This complex has also presented two
193
medium peaks at 4.08 and 6.68 eV. Also similarly to the copper (I) water complexes,
the electronic transitions in copper (I) ammonia complexes are of the MLCT type.
Cu (II) ammonia
194
Table 5.8: Excitation energies calculated for Cu (II) ammonia complexes using the LDAxc
and LB94 functionals.
195
The calculated excitation energies of the doubly charged copper ammonia complexes
are shown in table 5.8.
Like in the case of copper water complexes, the electronic spectra of the doubly
charged doublet complex is much more interesting than the closed-shell singly
charged complex, as it presents a series of large and medium absorptions, many of
which lie in the visible range.
The one and the two-coordinate doubly charged copper ammonia complexes don’t
present any absorption in the visible range, although the one-coordinate complex has
two medium peaks at 4.08 and 6.68 eV and the two-coordinate complex has a
medium peak at 0.99 eV, which is just below the range of energies of visible light.
The three-coordinate complex, however, presents two significant absorptions in the
visible range: one at 1.79 eV (red), with oscillator strength f= 0.0183, and another at
3.06 eV (blue-violet), with oscillator strength f=0.061. As a result, this complex is
expected to have a reddish coloration.
Unlike the 4-coordinate copper (II) water complex, which is expected to have a blue
colour as many other 4-coordinate copper (II) complexes, the 4-coordinate copper
(II) ammonia complex is expected to exhibit a reddish colour as its absorption
maximum, in the visible range, is a medium peak located at 2.75 eV, which is in the
upper half of the visible light energy spectrum. This important structure also presents
one large degenerate peak located at 3.86 eV.
As water ligands are added to the second solvation shell to form the 4 + X (X=1 to
4) complex, an interesting trend appears: both the medium peaks, which lie in the
196
visible range, and the large peaks that are higher in energy, are progressively blue
shifted as the number X increases. That is exactly what was observed in the case of
the doubly charged copper ammonia complex having the same coordination patterns.
The medium peaks remain inside the visible range until X=4, when they reach 3.21
and 3.27 eV. The large peaks start at 3.86 eV, for X=1 and end at 4.28 eV for X=4.
Interestingly, the spectra of the 5+0 and 6+0 copper (II) ammonia complexes are
very similar to the spectrum of the 4+0 complex. The dominant excitations of the
5+0 complex are only 0.07 and 0.11 eV higher than the dominant excitations of the
4+0 complex. Furthermore, the dominant excitations of the spectrum of the 6+0
complex are 0.01 eV and 0.02 eV lower than the corresponding ones in the spectrum
of the 5+0 complex.
5.2.1.4 Conclusion
The TDDFT calculation reproduced the trends of experimental spectra of the
condensed phase complex copper phthalocyanine, which is a blue dye.
The calculated excitation energies of complexes of the form [Cu(py)N]2+ have been
compared to gas phase experimental values for N=4, 5 and 6. For each N, a number
of isomers have been considered. It was found that structural isomers can have very
different spectra. Overall, the spectra obtained employing the LB94 functional show
a better agreement with experiment. The four-coordinate complex of symmetry D2d
showed a good agreement with experimental values, so it was also used for a
comparison between relativistic and non-relativistic excitation energies. It was found
that relativistic corrections are not needed for calculation of excitation energies in
doubly charged copper complexes as relativistic corrections did not alter
substantially the non-relativistic electronic spectra, and this result was applied to all
197
subsequent calculations involving copper. The calculated spectra of the five-
coordinate 5C complex also showed good agreement with experimental values. Only
in the case of the six-coordinate complexes, a good agreement with experiment
couldn’t be found; the complex corresponding to the 6C structure showed a
reasonable agreement.
With respect to copper water and copper ammonia complexes, it has been observed
that in the singly charged complexes none of the lowest excitations are intense.
Strong transitions only appear at energies well beyond the UV/Vis experimental
range explored by Stace and coworkers. Dominant transitions in singly charged
complexes are of the MLCT type.
The calculated spectra of doubly charged copper ammonia and copper water
complex are very interesting and show various absorptions in the UV visible range
among the lowest excitations, particularly in the area covered by experiments like
the ones performed by Stace and coworkers (described in chapter one). That
suggests that it would be very interesting to experiment with these complexes in
future. The lowest excited states are predicted to shift from red to blue as the number
of ligands increase from one to eight. According to Ramamurthy, “it is highly
interesting that the lowest excited electronic states can be shifted over the large
energy range from the UV to the IR by chemical variation of the ligands” (14).
It has also been observed, for doubly charged copper ammonia and copper water
complexes, that dominant excitations of complexes based on the 4+0 square planar
structure, that is 4+1, 4+2, 4+3 and 4+4 and 6+0 have similar excitation energies.
The corresponding excitation energy is slightly blue shifted as more ligands are
added. It has been explained, based on images of orbitals involved in the transition,
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that it is due to similarity of the β LUMO in all these structures, as in all cases this
orbital involved the metal dx2
- y2 and only the ligands in the first solvation shell, and
the dominant excitations in all these complexes end on the β LUMO. The dominant
excitations in doubly charged complexes are always of the LMCT type.
Finally it has been observed that, in the case of doubly charged complexes,
excitation energies calculated using the LB94 functional are always lower than the
ones calculated using the LDAxc functional. In the case of singly charged complexes
the situation is reversed: LB94 provides larger excitation energies than LDAxc.
5.2.2 Silver complexes
Computational details
TDDFT calculations of silver complexes involved the same computational
conditions used in the geometry optimisations of these complexes, presented in
chapter three. They employed an all electron TZP DIRAC basis set and relativistic
corrections using ZORA. TDDFT calculations used three different functionals,
namely LDAxc, SAOP and LB94, for the coordination numbers four, five and six.
For coordination numbers one, two and three, for which there are no experimental
results, only the LDAxc and LB94 functionals were used.
Preliminary calculations were carried out non-relativistically on the four, five and
six-coordinate silver pyridine complexes to evaluate the need for relativistic
corrections. This preliminary work involved non-relativistic geometry optimisations
of each complex plus TDDFT calculations with all three functionals. Only the lowest
energy structures, for each N, have been involved in the TDDFT calculations.
199
5.2.2.1 Silver pyridine complexes
This section presents TDDFT calculations on complex of the form [Ag(pyridine)N]2+
with 1≤ N ≤ 6.
The experimental spectrum
The electronic excitation spectra of silver pyridine complexes, for N=4 to 6, have
been obtained by Stace and coworkers and are presented in figure 5.6:
Figure 5.6: Photofragmentation spectra of doubly charged silver pyridine complexes
obtained by Stace and coworkers (3) . The energies are shown in wavenumbers and also in
eV (below).
200
Interestingly, all the peaks are around the same location; however there is a small red
shift as N increases in value.
The need for relativistic corrections in the TDDFT calculation
Although the application of relativity to copper complexes is still controversial, and
it has been shown earlier in this chapter that it is not relevant for the calculations
performed in this thesis, there is little doubt that this concept must be applied in
calculations involving heavy atoms like silver. This section evaluates the need for
relativistic corrections by calculating differences between relativistic and non-
relativistic calculations of silver pyridine complexes with N from 4 to 6.
Table 5.9 presents a comparison of calculated excitation energies obtained with and
without relativistic corrections for the silver pyridine complexes with coordination
numbers 4 to 6. These are the only values of N for which there are experimental
results available for comparison. Three functionals were used for this purpose,
namely LDAxc, SAOP and LB94. The twenty lowest excitation energies are
displayed in all cases.
Four –coordinate complexes
Firstly, in the case of the four-coordinate complex, it can be seen in table 5.9 that
dipole-allowed Eu transitions are dominant.
Relativistic
In the relativistic case, the first and third set of doubly degenerate Eu excitation
energies, which are the ones with the highest oscillator strengths, consist mainly of
transitions between the highest occupied eu molecular orbitals, namely the β 15 eu
(HOMO -6) and the β 16 eu (HOMO -3), and the β LUMO (β 13 b1g ).
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β 15 eu 13 b1g and β 16 eu 13 b1g
Dominant transitions, for N=4, in the relativistic (and non-relativistic) approaches
The second set of Eu excitation energies, which have a much lower oscillator
strength, consists mainly of transitions between the α 13 b1g (α HOMO) and the α
17 eu orbital (α LUMO +1).
Non-relativistic
In the non-relativistic case, the first and second set of doubly degenerate Eu
excitation energies, which are the ones with the highest oscillator strengths, consist
mainly of transitions between the highest occupied eu molecular orbitals, β 15 eu
(HOMO -7) and β 16 eu (HOMO -4), and the β LUMO (β 13 b1g ).
Therefore, the orbitals participating in the transitions are the same as in the
relativistic case. The only difference is the position of these orbitals; for instance, the
β 15 eu orbital is the HOMO -6 in the relativistic calculation, whereas in the non-
relativistic calculations it is the HOMO -7.
The third set of Eu excitation energies, which have a much lower oscillator strength
than the other two, consists mainly of transitions between the α 13 b1g (α HOMO)
and the α 17 eu orbital (α LUMO +1). This is totally similar to the relativistic case.
202
Comparison
Having assigned the transitions, a comparison between relativistic and non-
relativistic excitations can now be carried out, initially with reference to the SAOP
data. Firstly, regarding the transition β 16 eu 13 b1g , it is found that the relativistic
result is 0.76 eV above the non-relativistic. The relativistic calculation obtained an
oscillator strength for this transitions that is almost twice the value found in the non-
relativistic calculation.
Secondly, regarding the transition β 15 eu 13 b1g , which is the strongest
transition calculated for the four-coordinate silver pyridine complex, it is found that
the relativistic calculation again yields the largest energy. The difference is almost
the same as before: 0.74 eV. However, this time the non-relativistic calculation gives
an oscillator strength that is slightly larger than the one obtained relativistically (0.30
instead of 0.29).
Finally, with respect to the weakest symmetry allowed transition, which is α 13 b1g
α 17 eu , the difference in excitation energy obtained as a result of the
introduction of relativistic corrections is 0.58 eV. The oscillator strength is about ten
times larger in the non-relativistic case.
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scale Relativistic
non-relativistic
Figure 5.7: Contour lines representation of the β LUMO ( β 13 b1g ) calculated with and
without relativistic corrections. The colour coding is also shown on the left; amplitudes are
the lowest on the red regions and higher on the blue/green region. The arrows point at areas
where the two pictures differ most.
In both the relativistic and non-relativistic situations, the two strongest transitions
end on a partially occupied orbital (13 b1g ) while the weakest symmetry allowed
transition starts on this orbital and ends on a totally unoccupied orbital (17 eu ). The
13 b1g orbital is predominantly dx2
- y2 in character (metal orbital), so that it is highly
influenced by relativity (it is well known that d orbitals get more diffuse as a result
of relativity). This explains the higher shift regarding excitation energies associated
to transitions that end on this orbital, when compared to the one that ends
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on the LUMO +1, which is predominantly of 𝑝𝑝𝑥𝑥 character on the pyridine. The 13
b1g is represented in figure 5.7 where the differences that arise as a result of the
relativistic effect can be appreciated. With respect to calculations performed
employing the other two functionals, LDAxc and LB94, the comparison will be
drawn based on the dominant transition only. The LDAxc relativistic excitation
energy (3.51 eV) is 0.76 eV higher than the non-relativistic. As described above, the
SAOP relativistic value (3.28 eV) is 0.74 eV above the non-relativistic. In the LB94
case the relativistic value (3.00 eV) is higher by 0.68 eV.
Five coordinate complexes
For the five and six-coordinate complexes the comparison between relativistic and
non-relativistic results will be limited to the strongest transition.
In the case of five-coordinate complexes the dominant transitions have B1 and B2
symmetry. The strongest one has B2 symmetry. The LDAxc functional gives a non-
relativistic excitation energy of 2.68 eV for this transition. The corresponding
relativistic energy (not shown in the table because it is the 21st excitation) is 3.31 eV.
The difference between the two calculations is 0.63 eV.
The SAOP functional gives non-relativistic and relativistic excitation energies of
2.46 and 3.07 eV respectively. The difference between the two calculations is 0.61
eV.
In the case of the LB94 functional these energies are 2.05 and 2.57 eV respectively,
so that the difference between the two calculations is 0.52 eV.
Six-coordinate complexes
In the case of six-coordinate complexes the dominant transitions have B2u and B1u
symmetry.
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The LDAxc functional gives relativistic excitation energies of 3.06 and 3.17 eV, for
the dominant transitions of symmetry B2u and B1u respectively (these transitions are
not shown on table 5.9 because they are the 27th and 31st excitations respectively, so
that they are outside the range of lowest twenty excitations displayed). The
corresponding non-relativistic energies are 2.40 and 2.44 eV. In both cases there is a
difference of 0.66 eV between relativistic and non-relativistic calculations.
In the case of the SAOP functional, the relativistic energies are 2.88 and 2.93 eV
while the non-relativistic energies are 2.17 and 2.22. In either case there is a
difference of 0.71 eV between relativistic and non-relativistic calculations.
Finally, in the case of the LB94 functional, these energies are 2.53 and 2.63 in the
relativistic case and 1.95 and 1.98 in the non-relativistic. The difference between
relativistic and non-relativistic calculations is 0.58 eV (B2u) and 0.65 eV (B1u).
Comparison to experimental result – [Ag(pyridine)N]2+
Table 5.9 shows that the LB94 functional yields lower excitation energies than
LDAxc in all cases. This fact has also been observed for doubly charged copper
compounds, earlier in this chapter. The SAOP functional, which was used in this
section for the first time because of the fact that these calculations demand an all
electron basis set, almost always yields excitation energies that are in between the
ones obtained using LDAxc and LB94. That was not a surprise because this
functional was developed with the purpose of improving the behaviour of LB94 in
the valence regions by mixing it with LDAxc and other potentials (more details
about ac functionals are given in section 5.1.5).
The previous section has already pointed out that relativistic corrections are
necessary for calculations involving silver complexes, so that all the analyses carried
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out from now on, involving pyridine and other ligands, will also have these
corrections.
By analysing all the calculated excitation energies on table 5.9 and comparing those
to the experimental spectra it can be concluded that the SAOP functional gives the
best agreement between theory and experiment, and that will be shown in detail in
this section. Hence, all subsequent comparisons between theoretical and
experimental results will be based on the results obtained using the SAOP functional.
The calculated spectra of silver pyridine complexes with N=4 to 6 present two
dominant excitations in all cases. These bands are shifted to the red as the number of
ligands increase from four to six, and that is in agreement with experimental results.
However, the size of the shift on the calculated spectra is larger than in the
experimental result.
The four-coordinate complex presents dominating bands located at 2.75 eV and 3.28
eV. The first of those is in good agreement with the experimental result, which
consists of a broad band located at 2.65 eV. The second dominant band, however,
falls outside the experimental range.
The same happens with the pairs of dominant bands of the five and six-coordinate
complexes which are located at 2.98 eV / 3.07 eV and 2.88 eV and 2.93 eV
respectively. A good agreement with experiment is also observed here.
The splitting between the dominant peaks is reduced from 0.09 eV, when N=5, to
0.05 eV when N=6. Table 5.9 is shown on the next pages.
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N=4
208
N=5
209
N=6
210
Table 5.9 (previous pages): TDDFT excitation energies (E / eV) of [Ag(pyridine)N]2+ ,N=4
,5 and 6 using the functionals LDAxc, SAOP and LB94. Calculations were performed with
and without relativistic corrections.
Table 5.10 presents TDDFT calculations on complexes of the form [Ag(pyridine)N]2+
with 1≤ N ≤ 3 with the LDAxc and LB94 functionals. No experimental data exists
for the spectra of these complexes but it can be seen that they could also have
photofragmentation spectra recorded as they present strong absorptions in the
experimental range. Based on the results obtained employing the LB94 functional, it
is observed that the excitation spectrum for for N=1 presents two medium peaks at
2.63 eV and 2.81 eV. For N=2 there is a medium peak at 2.48 eV and for N=3 there
is a large peak at 2.02 eV .
Table 5.10: Excitation energies (E / eV) calculated for the one, two and three-coordinate
silver pyridine complexes using the functionals LDAxc and LB94.
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5.2.2.2 Silver acetone complexes
The experimental spectrum
The electronic excitation spectra of silver acetone complexes, for N=4 to 7, have
been obtained by Stace and coworkers and are presented in figure 5.8. They consist
of a series of broad bands, all centred at around 2.5 eV. A blue shift is observed
when moving from N=4 to N=7. The calculated spectra are presented in table 5.11.
Figure 5.8: Photofragmentation spectra of doubly charged silver acetone complexes obtained
by Stace and coworkers (1). The energies are shown in wavenumbers and also in eV
(below).
212
Calculated results
Table 5.11: Excitation energies (E / eV) calculated for the four, five and six-coordinate silver
acetone complexes using the functionals LDAxc, SAOP and LB94.
213
For N=4 to 6 the presence of two dominant bands is a main feature of all the spectra,
although for N=4 there is an extra strong absorption at a lower energy and with a
lower intensity than the dominant two. The two strongest bands of the calculated
spectrum for N=4 are located at 2.06 and 2.10 eV. They are in close agreement with
the experimental spectrum, which has its highest peak at around 2.30 eV. The
estimated error is of 0.2 eV or about 11%. Such agreement in a TDDFT calculation
has been considered highly accurate according to Ziegler (86).
For the complex with N=5 the dominant bands are located at 2.13 and 2.18 eV. For
N=6, these bands are located at 1.93 and 2.15 eV. Hence, all calculated bands are in
good agreement with experiment, which start recording absorptions at around 1.9
eV. Those grow steadily until reaching a maximum at around 2.5 eV on average.
The calculated blue shift of the spectra, when going from N=4 to N=6, is also in
agreement with experiment. However, it is slightly larger than in the experiment.
The size of the splitting between the dominant bands increases when going from
N=4 to 6. The splitting is 0.04 eV for N=4, 0.05 eV for N=5 and finally 0.22 eV for
N=6, in which case it is of the order of the energy that binds the sixth acetone ligand
(0.27 eV), according to the incremental binding energies shown in chapter 3.
Table 5.12 presents TDDFT calculations on complexes of the form [Ag (acetone)N]2+
with 1≤ N ≤ 3 with the LDAxc and LB94 functionals. Although there are no
experimental spectra recorded for these complexes, the calculated spectra indicates
that complexes with N=1 and N=3 are suitable to be studied experimentally.
214
Table 5.12: Excitation energies (E / eV) calculated for the one, two and three-coordinate
silver acetone complexes using the functionals LDAxc and LB94.
According with results obtained using the LB94 functional, the complex with N=1
presents a medium peak at 2.40 eV and the complex with N=3 present a medium
peak at 2.34 eV and small peaks at 2.28 eV and 2.39 eV.
5.2.2.3 Silver acetonitrile complexes
Absence of experimental spectrum
No electronic excitation spectra could be recorded using acetonitrile as a ligand
despite the existence of such complexes. The calculated results presented here
provide an explanation for this lack of observation. It is found that there are no
strong dipole-allowed absorptions of photons in the experimental energy range for N
= 4 to 6. The TDDFT data for [Ag(acetonitrile)N]2+, N= 1 – 6 are displayed in table
5.13.
215
Table 5.13: Excitation energies (E / eV) calculated for all silver acetonitrile complexes using
the functionals LDAxc and LB94.
216
It is seen from table 5.13 that for the complexes studied experimentally, namely
those with N=4 to 6, there are no significant absorptions below 4 eV (= 32,262 cm-1).
The four-coordinate complexes exhibit the lowest absorption, which is a medium
peak, located at 4.62 eV and the five-coordinate complex has its lowest absorption,
which is a small peak, located at 4.04 eV, both of which are outside the experimental
range. The six coordinate complex doesn’t present any significant absorptions
whatsoever in the range of the calculations.
Complexes with N=1 to 3 do present absorptions in the range of the experiment, but
these structures have not been probed yet experimentally. According to the results
obtained using the LB9 functional, the complex with N=1 presents a medium peak at
2.19 eV, the complex with N=2 presents a medium peak at 2.72 eV and finally the
complex with N=3 presents a medium peak at 2.88 eV.
Excitation energies and JT distortion
Splitting
It has been observed that in the case of silver pyridine complexes the size of the
splitting between dominant peaks in the spectra decreases, when going form N=4 to
6. In the case of silver acetone complexes, however, the pattern is reversed, i.e., the
splitting becomes larger as N increases.
A possible explanation would be to associate the splitting between dominant peaks
of the spectra to the degree of Jahn-Teller distortion in the corresponding structures.
Tables 3.10 and 3.11 show that in the case of the acetone complexes the degree of JT
distortion increases in going from N=5 to N=6 whereas in the case of the silver
pyridine complexes the trend is the opposite: JT distortion decreases in going from
N=5 to N=6.
217
However, an analysis of the molecular orbitals involved in these transitions
dismisses this explanation. This analysis of the MO’s involved in dominant
transitions of silver pyridine complexes with N=4 to 6, shows that those ligands
whose bond lengths to the metal cation suffer JT distortions don’t participate in the
relevant MO’s. Figure 5.9 shows the MO’s:
[Ag (pyridine)4]2+
[Ag (pyridine)5]2+
[Ag (pyridine)6]2+
Figure 5.9: Molecular orbitals involved in the dominant transitions of the four-coordinate
(top), five-coordinate (middle) and six-coordinate doubly charged silver pyridine complexes
(bottom).
The images show that the MO’s where the transitions start and end don’t involve
those ligands that suffer JT distortions. Hence, it doesn’t particularly matter the exact
218
position of those ligands in order to determine the excitation energies. All the
orbitals where such transitions end are similar to those of the four-coordinate
complex, where axial ligands are absent. This also explains why the experimental
absorption profile in the photodissociation spectra for N = 4 - 6 (figure 5.6) are so
similar.
The situation is analogous for the silver acetone complexes.
The images in figure 5.9 also show that dominant transitions in silver pyridine
complexes are of the LMCT type. It can be clearly seen in all images that the MO’s
where the transitions start have no orbital amplitude at the metal centre and that the
orbitals where the transitions end have a high amplitude at this position.
In fact, dominant transitions in all silver complexes studied here are all of the
LMCT type.
The knowledge of the orbitals involved in the lowest excited states of a complex is
very important because they “determine the photo physical and photochemical
properties and thus the specific use of the compound” (14).
A qualitative relation exists between the fact that the dominant transitions are LMCT
and the ionization energy of the ligands involved. Acetonitrile has the highest IE
(12.19 eV) whereas the IE of pyridine and acetone are 9.25 and 9.71 eV,
respectively. Thus, the ionization energies of the ligands can be used to interpret the
presence or absence of a measured spectrum. This fact also suggests/confirms that
these transitions are LMCT, because the experiment couldn’t record spectra for those
ligands with higher IE’s. The laser could only scan a limited range of frequencies, so
219
that in some cases the photons used didn’t have enough energy to remove an electron
from the ligand in order to promote a LMCT.
This can be used to rationalise the lack of spectra for the case Ag (II) complexes with
ligands with IE’s of 11.0 eV or above, namely 1,1,1,3-fluoroacetone, acetonitrile,
and CO2.
Although there is a trend between IE’s of ligands and the ability of recording a
photodissociation spectrum for the corresponding silver complexes, this trend seems
to breakdown when the relative excitation energies of the coordinating ligands which
do exhibit a spectrum are compared. Among the ligands which exhibit a spectrum in
this experiment, acetone and methyl-vinyl-ketone have the highest IE’s.
Figure 5.10: Experimental excitation spectra recorded by Stace and coworkers (1) of 4-
coordinate Ag(II) complexes.
220
According to the trend just described, it would be expected that complexes having
these ligands would present the highest excitation energies among the complexes
studied, however that is not the case. Figure 5.10 shows the experimental spectra of
doubly charged silver in different ligand environments. It can be seen that acetone
and methyl-vinyl-ketone are on the left side of the graph, which means that their
complexes have dominant excitation energies that are red-shifted to those of
complexes with the other ligands, which have lower IE’s.
Ligand IE (eV)a α (Å)b µ (D)c Pyridine 9.25 9.18 2.21
2-Pentanone 9.38 9.93 2.74 4-Picoline 9.46 2.7
Methyl vinyl ketone 9.65 Acetone 9.71 6.39 2.88 1,1,1,3-
Fluoroacetone 11 Acetonitrile 12.19 4.4 3.92
Carbon dioxide 13.77 2.91 0
Table 5.14: Physical properties of ligands studied experimentally (115).
There are two main reasons that explain why the correlation between IE’s of a ligand
and the detection of its spectrum in this experiment doesn’t apply to the relative
excitation energies of complexes with different ligands. Firstly, the process of
electronic excitation is more complex than ionization. TDDFT calculations have
shown that a single excitation may involve the participation of various MO’s that can
be based on more than one ligand, as it is often the case (see for example the MOs
pictured in fig 5.9, it can clearly be seen that electron density resides on 2 or more
pyridine ligands). Hence, the traditional image of an electronic excitation as the
movement of a single electron from one point to another is too simplistic. The
second reason is based on the concepts of α and β HOMO’s and LUMO’s which
221
have been introduced at the beginning of this chapter, in section 5.1.2. All these
complexes are open shell and the dominant electronic excitations end up at the β
LUMO, which is the orbital that pairs up with the α HOMO. This is the dx2
- y2 orbital
on silver. That means that this electron doesn’t have to move across the α HOMO- α
LUMO gap, that corresponds to the HOMO-LUMO gap in closed shell molecules.
Hence, there is no reason why the extent of the α HOMO - α LUMO gap should
influence the magnitude of these excitations.
The sizes of all these gaps are displayed in table 5.15:
ΔE HOMO-LUMO (eV)
Ligand (L) SPIN [AgL4]2+ [AgL5]2+ [AgL6]2+ Acetone α 3.68 3.57 3.41
β 1.5 0.81 0.34 Pyridine α 3.02 3.13 3.13
β 2.11 1.64 0.66 Acetonitrile α 5.08 5.46 5.49 β 2.49 1.73 1.24
Table 5.15: α and β HOMO-LUMO gaps of silver complexes.
Table 5.15 shows that the α HOMO-LUMO gaps of the complexes are well
correlated with the ionizations energies of the ligands, where acetonitrile has the
highest IE and also the largest α HOMO-LUMO gap, followed by acetone and then
pyridine.
However, most interestingly, the trend on the magnitude of the excitation energies is
well correlated with the extent of β-HOMO-LUMO gaps.
222
The excitation energies of silver pyridine complexes are red shifted with respect to
the corresponding acetone complexes in all cases, which agrees with the higher β-
HOMO-LUMO gaps of the latter. This fact is also observed in experiment.
As explained previously, all the dominant transitions are LMCT’s so that they
involve the movement of an electron from a ligand based molecular orbital to the
half-filled antibonding dx2
- y2 metal-based molecular orbital. These transitions are
detailed in table 5.16:
ACETONE ACETONITRILE PYRIDINE energy energy energy
N transition (cm-1) f transition (cm-1) f transition (cm-1) f 4 h−1l 16 629 0.18 h−17l 43 462 0.23 2Eu·h-6l 26 468 0.29 h−2l 16 963 0.11 h−18l 43 468 0.23 2Eu·h−3l 22 177 0.04 5 h−2l 17 622 0.17 h−25l 44 704 0.15 2B2·h−12l 24 724 0.19 h−1l 16 201 0.13 h−26l 44 537 0.14 2B1·h−11l 24 043 0.16 6 h−4l 17 367 0.14 h−23l 42 852 0.25 2B1u·h−12l 23 620 0.2 h−2l 15 543 0.12 h−22l 42 823 0.25 2B2u·h−10l 23 248 0.16
Table 5.16: Excitation energies and oscillator strengths (f) of all silver complexes with N=4
to 6. The orbitals responsible for the transitions are also detailed (they are all β orbitals): h
stands for HOMO and l for LUMO.
In the case of the four-coordinate silver acetone complex the dominant transitions
start at the h−1 and h−2, which is the MO below (and second below) the HOMO,
and end on the LUMO (all orbitals in the table are β so that this is the β LUMO). In
the five-coordinate case the pattern is the same and in the six-coordinate case the
transitions start at the h−4 and h−2.
Dominant transitions in four-coordinate silver pyridine complexes start at h-6 and h-
3. In five-coordinate complexes the transitions start in orbitals even lower down,
223
h−12 and h−11. In six-coordinate complex the transitions start at the h−12 and
h−10.
Hence, this is another explanation for the higher energies associated with the silver
pyridine complexes in comparison to silver acetone complexes: they are higher
because they start at orbitals that are lower down so that it involves a larger energy
gap. It also explains why the acetonitrile complexes have such high excitation
energies: their transitions start at orbitals that are very low, like for instance the
h−25.
5.2.2.4 Conclusions
A comparative study has shown that relativistic corrections have a large influence on
the outcome of TDDFT calculations on silver complexes. For instance, in the case of
four-coordinate silver (II) pyridine complexes the difference due to the relativistic
effect on the two dominant transitions was 0.75 eV on average, using LDAxc
/ALDA. In the case of the LB94/ALDA calculation the difference is slightly smaller
(0.68 eV) which suggests that the relativistic effect is taking place mainly in the
valence regions. In both the relativistic and the non-relativistic case the transitions
take place between the same orbitals. Furthermore, the shape of these orbitals
changes little as a result of the relativistic correction, as the contour lines in figure
5.6 shows. That means that the relativistic effect is acting mainly in the TDDFT part
of the calculation, rather than in the SCF step where the orbital energies are
calculated. Also, due to the fact that the dominant transition that ends on the α
LUMO+1 suffers less influence from relativity, it can be concluded that transitions
that end on silver 𝑑𝑑 orbitals, are more influenced by relativity than the ones that end
on ligand 𝑝𝑝 orbitals.
224
In the case of the five and six-coordinate silver (II) pyridine complexes the effect of
relativity was similar to the four-coordinate case. The differences in the energy of the
dominant excitations, due to relativistic corrections, was 0.63 and 0.66 eV
respectively, using the LDAxc functional. With LB94 these differences were slightly
smaller, as in the case of the four-coordinate complexes.
The SAOP functional provided the best agreement with experiment, which was
observed in all cases, for N=4 to 6. As a result, all the subsequent analyses of
excitation energies of silver complexes were based on this functional.
Calculated excitation energies for [Ag (acetone)N]2+, with N=4 to 6, are also in good
agreement with experimental results.
In the case of silver acetonitrile complexes, none of the lowest excitation energies
have substantial intensity, in particular the ones in the experimental range. Dominant
transitions in these complexes have a much higher energy. This fact agrees with the
experimental finding that no photodissociation could be recorded for acetonitrile
complexes.
Silver pyridine and silver acetone complexes, with N=5 and 6, suffer JT distortions
but their spectra has not been affected by them, as an analysis of the MO’s involved
has shown. That is because all dominant transitions in these complexes end on the β
LUMO’s. The β LUMO’s of the five and six-coordinate complexes do not include
the axial ligands, which are JT distorted, so that these excitations have very similar
energies. In the case of doubly charged copper complexes a similar trend was also
detected.
225
It has also been pointed out that the magnitude of dominant excitation energies are
related to β HOMO-LUMO gaps, rather than to α HOMO-LUMO gaps as is the case
in closed shell molecules.
226
Chapter 6 Magnetic interactions of copper and
silver complexes
This chapter presents calculation of the g tensor for copper and silver complexes,
using a variety of methods and basis sets. Calculated results are compared to
experimental values from the condensed phase.
Introduction
Electron spin resonance spectroscopy (ESR) is a very important tool in the study of
transition metals, and that is why this last chapter will introduce DFT calculations of
ESR parameters. According to Carrington (11), “the theory of electronic structure of
transition metals is both satisfying and successful. In large measure the successes are
due to the comprehensive and precise results of thousands of ESR studies”.
Furthermore, ESR is the main physical method for the study of open shell transition
metal complexes, because the nuclear magnetic resonance technique (NMR) cannot
be satisfactorily applied to these paramagnetic species. Many of the complexes
studied in this thesis, namely the doubly charged silver and copper complexes,
present an unpaired electron which makes them paramagnetic regardless of the
coordination geometry (166).
ESR techniques are very important also in the study of radical ions and organic
molecules which have a triplet ground state. ESR can be used in the study of
structure and reactions of metalloproteins. Furthermore, it has been found (167) that
the ESR spectra of type 2 copper proteins is similar to the spectra of simple copper
complexes, which will be studied in this section.
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6.1. Background theory
The magnetic moment of the electron has two components: orbital magnetic moment
and spin magnetic moment.
A calculation employing classical mechanics, and without any relativistic
corrections, can provide a good approximation for the orbital magnetic moment (43).
It works as follows: the electric current along the electron orbit is given by the
electron charge times its velocity and divided by the circumference of the orbit:
𝐼𝐼 = 𝑞𝑞 𝑣𝑣
2𝜋𝜋𝑟𝑟
The magnetic moment μ will be given by the current times the area of the orbit:
�⃗�𝛿 =𝑞𝑞 v�⃗ x r⃗
2
Where v�⃗ x r⃗ is a vector product.
Introducing the classical expression for the angular momentum 𝐽𝐽 :
𝐽𝐽 = m v�⃗ x r⃗
Rearranging the expressions for 𝐽𝐽 and �⃗�𝛿 :
�⃗�𝛿 = 𝑞𝑞
2𝑚𝑚 𝐽𝐽
Because the charge of the electron is negative the result is that the magnetic moment
(𝛿𝛿) and the angular moment (𝐽𝐽) point in opposite directions:
�⃗�𝛿 = − 𝑞𝑞
2𝑚𝑚 𝐽𝐽
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Interestingly, this result that has been obtained using classical arguments only, and it
is also valid in quantum mechanics (43). Unfortunately this classical analogy doesn’t
go too far and in order to calculate the spin magnetic moment it is necessary to use
quantum mechanics.
Quantum mechanical calculations have found that, in the case of electron spin, the
ratio between magnetic and angular momentum is given by:
�⃗�𝛿 = −𝑞𝑞𝑚𝑚
𝐽𝐽
That means that it is twice as large as in the case of orbital movement.
The total magnetic moment is found by adding together spin and orbital magnetic
moments. However, the way these two moments add up depends on other factors,
which include the intensity of the spin-orbit coupling. As a result, the total magnetic
moment is given by the following formula:
�⃗�𝛿 = − 𝑔𝑔 𝑞𝑞
2𝑚𝑚 𝐽𝐽
Where g is the Landé g-factor.
If the electron is free there is no orbital angular momentum so that the value of g is
2. At present, the most accurate value of g for a free electron, including corrections
from quantum electrodynamics, is 2.0023.
The spin orbit constant is a measure of the intensity of spin orbit coupling. Some
values are listed in table 6.1:
229
d electrons ion Spin orbit constant
(cm-1) total spin 3d2 V3+ 209 1 3d3 V2+ 167 3/2 3d4 Cr2+ 230 2 3d4 Mn3+ 352 2 3d5 Mn2+ 347 5/2 3d6 Fe2+ 410 2 3d8 Ni2+ 649 1 3d9 Cu2+ 829 ½
Table 6.1: Spin orbit constants for selected first row transition metals. Adapted from
Carrington (11).
Table 6.1 shows that the intensity of the spin orbit coupling changes considerably
from one transition metal ion to another. It also shows that Cu (II) has the highest
spin orbit coupling among the first row transition metals, and that is one of the
reasons why this ion is so important in the context of ESR. It is often used as a
probe in the study of other substances which cannot produce an ESR spectrum alone.
The ESR experiment can determine the Landé g -factor, which will be referred to as
g from now on, and also the hyperfine couplings, which characterizes the interaction
of the unpaired spin with the nuclei around the molecule it is immersed into. Only
the g factor will be studied in this thesis.
Atoms that present strong spin-orbit coupling have g values that differ substantially
from the free electron value. These cases are very interesting to be studied by ESR
techniques.
The landmark experiment performed by Stern and Gerlach, in the 1920’s, proved the
spatial quantization of the electron spin, i.e., proved that the magnetic moment of the
electron, when inserted in a magnetic field, can only assume two orientations:
230
aligned with the field or against the field. This behaviour contradicts the classical
expectation, in which the magnetic moment of the electron would be able to assume
any orientation inside the field.
In order to understand the ESR experiment, is important to consider the energy of a
magnetic moment �⃗�𝛿 inserted into a magnetic field 𝐵𝐵�⃗ .This energy (U) is given by:
𝑈𝑈 = �⃗�𝛿 . 𝐵𝐵�⃗
Assuming that the magnetic field is applied in the direction z, the energy expression
simplifies to:
𝑈𝑈 = µ𝑧𝑧 𝐵𝐵
Where µ𝑧𝑧 is the projection of the magnetic moment 𝛿𝛿 n the direction z.
As a result, there are two possible energies for the electron in the magnetic field,
depending on whether component µ𝑧𝑧 is with or against the field.
This idea is the basis of the ESR experiment, in which the electron inside the
magnetic field is hit by electromagnetic waves. When the energy of the
electromagnetic field matches the difference in energy between the two positions
allowed for the electron inside the magnetic field the electron will flip and there will
be absorption of energy, and that is the resonance point. The magnetic field is varied
until the resonance is found, and then the value of g can be determined. In fact, the
magnetic moment of a molecule depends on the direction from which it is measured
and as a result this parameter is best described by a tensor, which is called the g -
tensor.
231
The measurement of g -tensors in the gas phase, in the kind of experiments discussed
in section 1.3 hasn’t yet been performed. However, this possibility of making
measurements of magnetic properties of compounds produced using supersonic
expansions have already been considered by Stace (168), based on the innovative
experiments from Becker and de Heer who managed to incorporate a Stern and
Gerlach type magnet into a supersonic beam experiment, so that they could measure
magnetic properties of clusters containing as little as a hundred atoms (169).
ESR parameters can be calculated using DFT but it is often difficult to obtain an
accurate g value for transition metal atoms (170) and van Lenthe has observed that
“further systematic studies are therefore needed in order to judge the ability of the
available DFT approaches to describe the ESR parameters for heavy metal systems”
(171).
In this thesis the main values of the tensor will be calculated, which are usually
referred to as g ║ and g ┴ , which are the parallel and the perpendicular component
respectively.
The implementation of the ESR program in ADF was carried out by van Lenthe and
coworkers (172). It employs Gauge Including Atomic Orbitals and the g -tensor is
obtained in a relativistic calculation (ZORA) with spin-orbit effects included.
The functionals used in this study are the LDAxc, B88X + P86C, LB94 and SAOP. The
B88X + P86C functional is often used for ESR calculations on transition metal
complexes and it was the functional of choice in the calculations of the g-tensor in
iron systems by van Lenthe (173) and also in the calculations on copper proteins by
Swart (167).
232
Because gas phase measurement of the g values are not yet possible, these calculated
results will be compared to experimental results obtained in the condensed phase.
6.2 Results: the calculated g values of copper and silver
complexes
6.2.1 Copper complexes
Computational details
The functionals employed in this section are the LDAxc, B88X + P86C and LB94. The
basis sets used are the TZ2P and the large QZ4P, and relativistic effects were
included via ZORA (the spin-orbit option was employed). In all cases the core has
been frozen in copper at 2p and in oxygen and nitrogen at 1s. Preliminary
calculations using an all electron basis set didn’t make any difference for the g
values.
Preliminary calculations also employed higher values for the “integration” key,
which controls the size of the integration grid in ADF. It showed that increasing its
value to 5.0 and then 6.0 didn’t make any difference to the results obtained.
Results
The main g values of the octahedral [Cu (H2O)6]2+ complex have been previously
calculated by Tachikawa (174) using a very demanding wavefunction based method
called multi reference single and double excitation configuration interaction
MRSDCI, which is a variation of the well known configuration interaction method
(CI). He obtained two sets of values for g, one for an elongated structure and the
other for a compressed structure. These values are : g ║ = 2.300 and g ┴ = 2.080
and g ║ = 2.210 and g ┴ = 2.000 respectively. They agree reasonably well with the
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experimental values (175), obtained in solid glass matrices, which are g ║ = 2.400
and g ┴ = 2.095.
The g values calculated in this work, and presented in table 6.2, don’t agree well
with experiment, but they are close to the values obtained using MRDSCI which is
a method that is much more “expensive”, that means, it consumes much more
computing time than DFT . The comparison is drawn between the values calculated
in this work and the values of the elongated structure calculated by Tachikawa. That
is because the gas phase structure calculated in this work, and shown in chapter
three, has axial bond lengths of 2.30 and 2.31 Å, which are in close agreement with
the experimental value of 2.34 Å. The equatorial bond lengths are also in good
agreement; they are 1.95 and 1.96 Å in this work and the experimenta1 is 1.99 Å.
Table 6.2 shows that agreement between g values calculated in this work and
experiment increases, for both components of g, when a large basis set is used. The
LDAxc functional provided the best results, followed by B88X + P86C and LB94. The
LDAxc functional combined with the QZ4P basis set provided the closest agreement
to experiment and to MRSDI calculations. The low value provided by LB94
indicated that the asymptotic regions are not important for a calculation of g.
[Cu(ligand)N]2+
method and
basis set water Ammonia pyridine
4 6 4 6 D4h D2d
g║ g┴ g║ g┴ g║ g┴ g║ g┴ g║ g┴ g║ g┴
LDAxc TZ2P 2.178 2.048 2.233 2.071 2.117 2.03 2.17 2.05 2.099 2.027 2.164 2.065
B88X + P86C TZ2P 2.159 2.045 2.197 2.064 2.112 2.03 * * 2.098 2.027 2.156 2.061
LB94 TZ2P 2.148 2.038 2.182 2.056 2.083 2.02 * * * * 2.098 2.031
LDAxc QZ4P 2.2 2.053 2.264 2.08 2.132 2.034 * * * * * *
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Table 6.2 (previous page): Calculated main values of the g tensor of copper (II) complexes
with water, ammonia and pyridine, using different functionals and basis sets.
* Calculations that failed to converge. Six-coordinate ammonia complex failed
because the SCF cycles would not converge within 300 cycles. In the case of the
LDAxc functional the calculation converged moderately within 500 cycles, but
that number of cycles would be not viable to perform with the more complicated
functionals. The calculations on the pyridine complexes failed, in the case of the
QZ4P basis set, because these systems are too big to employ such a large basis
set in an ESR calculation.
Table 6.2 shows that the calculated g values of copper complexes are higher when
the LDAxc functional is employed. The B88X + P86C functional provides values that
are slightly lower and the values from the LB94 functional are even lower, in all
cases. The use of the large QZ4P basis set has also influenced substantially the
calculated values.
The g values of the complexes with nitrogen donor ligands, square planar complexes
copper pyridine (D4h) and copper ammonia, are similar. They are 2.099 / 2.027 and
2.117/ 2.030 respectively. The water complexes have higher values of g than the
complexes with nitrogen donor atoms in all cases.
The results for D2h copper complexes have not been listed because they are very
close to values for the D4h complex. However, it is interesting to note how much the
g values from the D2d complex differ from the values of the D4h complex.
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6.2.2 Silver complexes
Computational details
The functionals employed in these ESR calculations are the LDAxc, B88X + P86C,
LB94 and SAOP and the basis sets are the TZP, TZ2P and QZ4P.
As in the case of copper, the use of an increased integration grid, with integration
values rising to 5.0 and also to 6.0, didn’t make any difference to the results. Also,
the use of the TZ2P basis set provided exactly the same g -values as those obtained
using TZP.
ESR calculations of silver complexes involving the SAOP functional proved to be
unviable because they require over 400 cycles to achieve convergence, so that they
consume a great number of computer hours. Results were only obtained for some
small complexes so that a comparison wouldn’t be possible and so these results were
not considered for further analyses.
Results
Calculated g values for silver complexes are shown in table 6.3.
The ESR calculations on six-coordinate complexes could only be completed in the
case of acetonitrile. In the case of acetone and pyridine the calculations stopped
because of memory problems. The same happened in the case of calculations
involving the large QZ4P basis set, where it was successful only in the case of the
square planar silver acetonitrile complex.
It’s interesting to note that there is a similarity between the g values of four-
coordinate silver complexes having nitrogen donor ligands, i.e., silver acetonitrile
and silver pyridine. Surprisingly, the silver pyridine D2d structure is the one that has
236
g values that are close to the square planar four-coordinate acetonitrile complex.
Silver acetone complexes also show similar g values.
[Ag(ligand)N]2+ method and
basis set acetone Acetonitrile pyridine 4 4 6 D4h D2d g║ g┴ g║ g┴ g║ g┴ g║ g┴ g║ g┴
LDAxc TZP 2.141 2.032 2.156 2.036 2.191 2.051 2.108 2.029 2.163 2.031 B88X + P86C
TZP 2.142 2.033 2.162 2.038 2.196 2.053 2.111 2.03 2.165 2.033 LB94 TZP 2.11 2.025 2.14 2.029 2.174 2.044 2.078 2.022 2.108 2.016
LDAxc QZ4P * 2.16 2.036 * * * * *
Table 6.3: Calculated main values of the g tensor of silver (II) complexes with acetone,
acetonitrile and pyridine, using different functionals and basis sets.
* Calculations that failed to finalize for shortage of computer memory.
The B88X + P86C functional provided results higher than the ones obtained using the
LDAxc, which is the opposite of what happened in the case of copper complexes.
The LB94 provided the lowest results, as in the case of copper. The use of the large
QZ4P basis set, which was only possible in the case of the square planar acetonitrile
complex, proved to make a substantial difference.
A comparison between calculated g values and experimental values obtained in the
condensed phase has been carried out, based on the data shown on table 6.4.
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Compound Conditions g║ g┴ References
[Ag(bipy)2](NO3)2 Solid 2.168 2.047 (176) [Ag(picolinate)2] Solid 2.244 2.072 (176)
[Ag(pyridine)4]S2O8 doped in [Cd(pyridine)4]S2O8 2.18 2.04 (177) Ag(SO3F)2 80K solid 2.407 2.086 (178)
AgPt(SO3F)6 80K solid 2.486 2.134 (178) AgSnF6 80K solid 2.61 2.135 (179)
[Ag(bipy)2](SO3F)2 80K solid 2.17 2.051 (178) [Ag(bipy)2](SO3F)2 80K MeCN 2.166 2.054 (178)
[Ag(bipy)2](SO3CF3)2 80K MeCN 2.16 2.057 (178)
Table 6.4: Experimental g values for silver complexes.
The complexes with nitrogen donor ligands only are [Ag(bipy)2](NO3)2 and
[Ag(pyridine)4]S2O8 . Their g values are 2.168/ 2.047 and 2.180/ 2.040 respectively,
so that they are in the same range as the calculated values for this type of complex.
Experimental values of complexes that have, in addition to nitrogen based ligands,
ligands containing sulphur, fluorine and oxygen have values of g that are
considerably higher.
The experimental value for the condensed phase four-coordinate silver pyridine
complex is in reasonable agreement with the calculated values. The experimental
value is 2.180/ 2.040 and the closest calculated values, which are the ones obtained
using the B88X + P86C functional, are 2.111/ 2.030 in the case of D4h symmetry and
2.165/ 2.033 in the case of D2d symmetry. It is curious that the calculated values for
the D2d geometry are in better agreement than the square planar, which is the
favoured structure in the case of d9 silver complexes. The experimental values
deviate from the g value for the free electron by 0.1777 / 0.0377 whereas the
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calculated values differ by 0.1087/ 0.0277 in the D4h case and 0.1627/ 0.307 in the
D2d case. In the case of the D4h geometry this represents an error of 38.8 % regarding
the parallel component of g and an error of 26.5% regarding the perpendicular
component. In the case of the D2d geometry these errors are 8.4 % and 18.5 %
respectively. The authors of the experiment (177) could not determine the structure
of their silver pyridine complex but they stated that a square planar arrangement is a
“good working hypothesis”. The calculations performed here contradict their
hypothesis.
6.2.3 Conclusions
It has been found that the increase in the grid of integration from the default value of
4.0 to 5.0 and also 6.0 doesn’t affect the calculated g values. The use of a large basis
set, like the QZ4P, does affect substantially the results. However, in the case of the
calculations involving silver complexes, the use of the TZ2P basis set doesn’t bring
any difference with respect to the results obtained using the TZP basis set.
The LDAxc and B88X + P86C functionals provide results that are often close to each
other. In the case of copper complexes the B88X + P86C functional gives lower results
and in the case of silver complexes it gives higher results. The LB94 functional, in
all cases, provides the lowest results, which don’t lead to improvement in the
comparison with experimental results. This means that the asymptotic region of the
complex is not relevant in the calculation of g values of copper and silver complexes.
A similar conclusion has been reached by Swart (167) in the context of copper
proteins.
Comparison to experimental results showed that the LDAxc functional provided the
best agreement in the case of the octahedral copper water complex. The use of a
239
large basis set, namely QZ4P made a substantial difference in the g values and
increased the agreement with experiment. The agreement with experimental values
was reasonable.
In the case of the silver complexes, the comparison to experiment considered the
four-coordinate pyridine complex, in which case the B88X + P86C functional provided
the closest agreement to experiment. The error with respect to experimental
deviations from the free electron g value was a minimum in the case of the
calculations involving the D2d geometry, and it was 8.4% with respect to the parallel
component of g and 18.5% with respect to the perpendicular component of g. The
fact that the B88X + P86C functional provides the best g values agrees with work from
van Lenthe and Swart, as described in the introduction.
It is interesting that the calculated g values of D4h and D2d structures differ
substantially, because it means that the ESR technique can be applied to determine
geometries of complexes in the cases where they cannot be determined by standard
techniques like x-ray diffraction. That is the case for experiments in the gas phase.
Furthermore, Tachikawa (174) has shown that calculated values for a compressed
structure of the octahedral copper water complex differ substantially from the values
calculated for the elongated structure, which suffers JT distortion. That means that
the ESR technique has the potential to assess the degree of JT distortion in transition
metal complexes, which is very interesting considering that JT distortion cannot be
assessed using UV/Vis spectroscopy, in the case of various copper and silver
complexes, as described in chapter five.
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Chapter 7 Summary
Computational chemistry is a relatively new branch of chemical research and it has
been promoting the development of chemistry knowledge in a very clean and
environmentally friendly manner. Traditional chemistry research often makes use of
substances that are harmful to the environment and the need to reduce those is
increasing by the day, as the limited capacity of the planet to absorb this kind of
waste is becoming evident. Computational chemistry is becoming more and more
viable because of the development of more powerful computer hardware and
software. Furthermore, theoretical developments in quantum chemistry, like the
advent of Density Functional Theory, have enabled the study of more and more
complex systems.
Quantum chemistry has existed since the beginnings of the 20th century but for many
decades it could only deal with atomic systems and also small molecules, usually
organic. Thanks to the recent developments in quantum theory and in computational
resources, described above, it is now possible to deal with much larger systems, even
when transition metals are present. Calculations involving transition metals are
known to be more complex than calculations on organic molecules because of the
difficulty of evaluating correlation energy of d orbitals. In this thesis, calculations
have been successfully performed in large systems like [Cu water10]2+ , [Ag
pyridine6]2+ and also [Cu phthalocyanine]2+.
This thesis focused on the study of copper and silver complexes, including the
doubly charged complexes which are open shell and therefore pose extra
complications as the electrons are divided in two sets (α and β) which have different
energies.
241
Collaboration with experimentalists led by Tony Stace at the University of
Nottingham has been crucial to evaluate the quality of the results obtained and the
adequacy of the levels of theory used, including exchange-correlation functionals
and basis sets. This group performs experiments in the gas phase so that bulk solvent
interactions are excluded. That makes their results very convenient to be studied
theoretically. Many of the results presented here have been compared to
experimental values.
The calculations performed in this thesis start with the determination of the lowest
energy structures of all the copper and silver complexes studied, namely [Ag
pyridineN ]2+, [Ag acetoneN ]2+, [Ag acetonitrileN ]2+, [Cu pyridineN]2+ with 1≤ N ≤
6; [Cu ammoniaN ]2+, [Cu ammoniaN]+ and [Cu waterN]+ with 1≤ N ≤ 8 ; [Cu
waterN]2+ with 1≤ N ≤ 10 and also copper (II) phthalocyanine.
The calculated structures of copper water and copper ammonia complexes show
patterns of solvation that are markedly different from the ones found in the
condensed phase. This is in agreement with, and extends, previous theoretical studies
and provides a quantitative explanation for gas phase experimental mass
spectroscopy data. Singly charged copper water complexes have a preference for a
structure containing a secondary solvation shell, when N ≥ 3. In the doubly charged
case the threshold for such preference is five. In the case of ammonia the threshold is
five whatever the charge on the metal. The higher threshold for doubly charged
cation complexes is attributed to ease of proton transfer for lower N (and higher
proton affinity of ammonia) and this is explored further is chapter 4.
Energy decomposition analysis of the hydrogen bonds in cation and dication copper
water and ammonia complexes attribute the charge-enhanced hydrogen bonds to
242
increased orbital interaction, as the enhanced electrostatic interaction is moderated
by the increased Pauli repulsion.
Incremental binding energies revealed that [Cu pyridine4]2+ complexes are
preferentially stable, which is not the case when the solvating ligand is water or
ammonia. A contributing factor is that larger Cu (II) pyridine complexes present
larger Jahn-Teller distortions when compared to the 6+0 water or ammonia
complexes, but of course, in these latter cases the complexes are able to take the
distortion to the extreme and preferentially form a 4+2 complex.
A comparison of the contributing energy terms of the 4+0 and 6+0 [CuLN]2+, L
=H2O, NH3, and pyridine, suggests that ligands with nitrogen donor atoms like
ammonia and pyridine are more effective at stabilising Cu (II) than ligands with
oxygen donor atoms such as water. Favourable electrostatic contributions to the
bonding and orbital interactions correlate well with the ligands’ polarisability.
In the case of doubly charged silver complexes, the lowest energy structural isomers
of [AgLN]2+ , L = pyridine, acetone and acetonitrile, and N = 1 – 6, have been
determined. In all cases, the axial bonds of the N=5 and 6 complexes is severely
Jahn-Teller distorted. Calculated incremental binding energies show that, for all
three ligands, the N=4 Ag (II) complexes are preferentially stable. This agrees with
experimental mass spectroscopy evidence, except in the case of acetone where the
preferred coordination number is 5. Binding energy analysis shows that acetone
Ag(II) complexes containing oxygen-donating ligands (acetone) are less stable
(lower electrostatic interaction and incremental binding energy) than those where the
coordinating atom is nitrogen (pyridine and acetonitrile). This is attributed, in part, to
the presence of bent C=O-Ag bonds for acetone complexes with N > 2. This
243
observation may account for the lack of condensed phase complexes where Ag (II) is
coordinated to oxygen.
Ionization energies have been calculated for singly charged copper water and copper
ammonia complexes. The asymptotically correct functional, LB94, calculated the
ionization energies of copper and its complexes more accurately than the standard
LDA or BP86 functional. Furthermore, a Koopmans’ like approximation is
appropriate when using LB94 due to its correct asymptotic behaviour that stabilizes
the frontier orbitals.
The vertical and adiabatic ionization energies of Cu (I) water complexes are in fairly
good agreement. This was surprising given the very different Cu (I) and Cu (II)
geometries in many cases. In the gas phase experiments, the neutral or singly
charged complexes are ionized to form the doubly charged complexes and so in this
context the vertical ionization energies are probably more significant. This work
shows that even when significant re-arrangements of geometries on ionization occurs
it does not dramatically influence the ionization process and calculated vertical
ionization energies (or Koopmans’ if an asymptotically correct functional is used)
are adequate to provide trends. Increased solvation reduces the ionization energy of
the Cu (I) complexes. Formation of a second solvation shell further reduces the
energy required to remove an electron from the metal (to form the Cu(II) d9
complex). The IE of the complex becomes lower than the IE of the ligand, water (or
ammonia), only after four (or five) or more ligands are added. Experimentally it is
difficult to form the gas phase dication complex for N < 8. This result may provide a
possible explanation: ionization of the ligand (rather than the cation complex to form
the dication) is a competing process.
244
The study of fragmentation pathways has shown that only for N > 8 are the proton
transfer dissociative products energetically less favourable than formation of the
dication complex. Thus, the reason for the instability of Cu (II) clusters with N < 7 is
due to the ease (thermodynamic stability) of proton transfer. This provides further
evidence of the experimentally preferred [Cu(L)8]2+ unit, L = H2O as a very stable
configuration.
The energy transferred to the complexes during the electron impact ionisation
process is estimated to be in the order of 1 eV from calculations of the enthalpy of
reaction of unimolecular decay. The reaction energy converges to a value of 1 eV
when N =7 and this fact agrees with the experimental observation of neutral ligand
loss for N > 7.
The loss products of electron capture dissociation were modeled and compared with
MIKE spectra of Cu (II) hydrates. The electron capture was modeled using a Xe
atom as the electron source, to model the Xe used experimentally as the collision
gas. It was found that several eV are required for the formation of all the Cu (I)
products considered, due to the high IE of Xe. Experimentally both the hydroxide
and hydrate are formed but it is the hydroxide that is favoured in the case of larger
solvation shells. The calculations performed also reflect this trend.
It has also been found that the hydroxide containing Cu (I) water complexes are less
stable than the Cu (I) hydrates. The work presented here agrees with Vukomanovic
and Stone who have concluded that the Cu+-OH bond is weaker than the Cu+ - H2O
bond when in presence of one or two water molecules (120) and extends this
conclusion to up to 7 water molecules.
245
Calculations on Cu (II) ammonia complexes have shown that the pattern is similar to
that of Cu (II) water complexes, i.e., the CuNH2+(NH3)N are less likely to be formed
than the corresponding Cu+(NH3)N complexes because their reaction enthalpies are
more positive. This is in agreement with experimental observation (MIKE spectra).
The calculated TDDFT spectrum of copper (II) phthalocyanine has a very large peak
(f=0.295) at 1.72 eV. This corresponds to a wavelength in the red, which causes the
complex to exhibit its complementary colour, which is blue, as observed, and is in
excellent agreement (0.1 eV) with the prominent feature of the experimental
spectrum of copper (II) phthalocyanine vapour at 1.82 eV. This helped validate the
methodology used here.
The electronically excited states of the lowest 3- 4 structural isomers of
[Cu(pyridine)N]2+, N=4-6 have been calculated and compared with the
photodissociation spectra of Stace et al.. It was found that the structural isomers for a
particular N have very different spectra. The N=4 spectra was assigned to the D2d
structure and the excitations were d d transitions which agreed well with the
experimentally observed neutral loss products. The N=5 spectra was assigned to the
C2v (5C) structure. The agreement between theory and experiment for the N=6
complex is not quite as good but the spectra was assigned to the D2h (6C) structure as
this is the structure which exhibited the lowest lying strong dipole-allowed transition
(at 2.14 eV) which still falls within the broad experimental band. In all cases the
LB94 functional performed best.
TDDFT calculations of Cu (I) water complexes and Cu (I) ammonia complexes
showed that there are no dominant transitions in the experimental range. The lowest
energy at which a strong peak has been calculated for these complexes is 3.29 eV. It
246
corresponds to the 4+1 Cu (I) hydrate and it has oscillator strength of 0.034. For this
closed shell complex, dominant transitions were MLCT.
The calculated spectra of the doubly charged copper water and ammonia complexes
present a number of dominant transitions, many of them lying in the visible range,
and thus potentially observable. It was found that increased solvation results in a
blue-shift of the dominant excitations. For these open shell complexes, all dominant
transitions were LMCT.
For N = 4 to 6, [AgLN]2+, L = acetone, pyridine and acetonitrile, all functionals
considered (LDAxc, LB94, SAOP) result in excitation spectra that reflect the
experimental observations in that: (i) The acetone spectra are red-shifted relative to
the pyridine spectra; (ii) All acetone complexes have strong bands between 15000 –
18000 cm-1 and absorption maxima blue-shifts as N increases; (iii) All pyridine
complexes have strong bands between 22000 – 28000 cm-1 and absorption maxima
red-shifts as N increases; (iv) All acetonitrile complexes do not have a strong
absorption in the experimental range (first dominant peak at ~ 42000 cm-1).
Although the calculated structures exhibit considerable Jahn-Teller distortion, the
recorded spectra show no evidence of this having an influence on photofragment
yields as a function of excitation wavelength because the ligands that are subject to
distortion do not contribute electron density to the molecular orbitals responsible for
the charge transfer transitions. This pattern of behaviour is common to both nitrogen-
and oxygen-donating ligands.
All dipole-allowed excitations are LMCT from an electron localised on the ligands to
the half-filled anti-bonding dx2-y2 orbital based on the metal cation which agrees with
the observed pyridine cation fragment.
247
The High IE of acetonitrile explains lack of spectra in UV/Vis . However, given the
IE’s of pyridine and acetone (9.25 eV and 9.71 eV respectively), the electronic
spectra of pyridine complexes would be expected to be red-shifted relative to the
electronic spectra of acetone complexes, but they are not. This is due to the facts
that: (i) transitions involve the movement of a β-electron from a ligand based orbital
that involves density on several ligands, (ii) the bonding in the Ag - nitrogen
containing - ligand complexes (pyridine and acetonitrile) is stronger than in the Ag -
oxygen containing - acetone complexes, and also (iii) the transitions on the pyridine
complexes arise from deeper lying orbitals and as a result larger energy gaps are
involved.
A comparison with condensed phase silver complexes gives support to the
theoretical findings presented in this thesis: (i) the complex Ag[MoF6].4CH3CN is
reported as being a white solid (189). This observation gives qualitative support to
the evidence presented here that [Ag(CH3CN)4]2+ does not absorb at visible
wavelengths. (ii) The [Ag(pyridine)4]S2O8 complex is reported to form reddish
crystals (117), which would match the observation that [Ag(pyridine)4]2+ absorbs in
the blue/near UV region of the spectrum and Wasson (190) reports a λmax value of
400 nm (25,000 cm-1).
The study of magnetic properties of doubly charged copper and silver complexes
has shown that reasonable calculations of the g tensors require the use of large
relativistic basis sets such as QZ4P but this was prohibitively large for the complexes
with larger ligands such as pyridine.
The calculated g values of the octahedral [Cu (H2O)6]2+ complex of g ║ = 2.26 and
g ┴ = 2.08 agree reasonably well with high-level MRSDCI calculations on an
248
elongated structure (g ║ = 2.300 and g ┴ = 2.080) but are a little lower than the
experimental values of g ║ = 2.400 and g ┴ = 2.095 obtained in solid glass matrices
(175).
The calculated values for the four-coordinate Ag(II) pyridine complex are in
reasonable agreement with the condensed phase experimental values. The
experimental value is 2.180/ 2.040 and the calculated values are 2.111/ 2.030 in the
D4h symmetry and 2.165/ 2.033 in D2d symmetry. It is not clear why the D2d values
are in better agreement with experiment than the lower energy D4h structure. That
indicates that there is a possibility that the experimental structure, which could not be
determined by the authors of the experiment (177), is not exactly square planar as it
is normally expected in the case of d9 metals.
It is clear that the g values are sensitive to both the nature of the coordinating ligands
(-O or –N) and the geometry of the structure. This means that the ESR technique has
the potential to assess the degree of JT distortion in transition metal complexes,
which is very interesting considering that JT distortion did not influence the UV-VIS
spectra, in the case of various copper and silver complexes, as described in chapter
five.
249
Appendix A Abbreviations
ac Asymptotically Correct (functional)
B88X + P86C Functional with exchange from Becke and correlation from Perdew
CFT Crystal Field Theory
CIS Configuration Interaction using Singles excitations only
DM Density Matrix
DFT Density Functional Theory
EPR Electron Paramagnetic Resonance
ESA Electrostatic Analyser
ESR Electron Spin Resonance
ET Electron Transfer
ET Even Tempered (basis set)
ET-pVQZ ET Valence Quadruple ξ Basis Set
ET-QZ3P ET Valence Quadruple ξ basis set with 3 polarization functions
and one set of diffuse s, p, d and f orbitals
FFR Field Free Region
HF Hartree-Fock
HOMO Highest Occupied Molecular Orbital
IBE Incremental Binding Energy
250
IE Ionization Energy
LB94 Functional (ac) by van Leween and Baerends
LDAxc Local Density Approximation Exchange-Correlation
LMCT Ligand to Metal Charge Transfer
LUMO Lowest Unoccupied Molecular Orbital
MLCT Metal to Ligand Charge Transfer
MO Molecular Orbital
MP2 Møller-Plesset Perturbation Theory- Second order
QZ4P Valence Quadruple ξ with four polarization functions
SAOP Statistical Averaging of Orbital Potentials
STO Slater Type Orbitals
TDDFT Time Dependent Density Functional Theory
TZP Valence triple ξ basis set with one polarisation function
TZ2P Valence triple ξ basis set with two polarisation functions
TZ2P+ TZ2P basis set with extra 3d functions
UV/Vis Ultraviolet/visible
ZINDO Zerner’s intermediate neglect of differential overlap
ZORA Zeroth Order Regular Approximation
xc Exchange-Correlation (functional)
251
Appendix B Tables of calculated results
concerning copper complexes
Copper water binding energies (calculated using the “standard” approach
described in chapter 3)
Complex Config. abs. (eV) b e (eV)
b e (kJ/mol)
[Cu(H2O)]+ -7.86 1.88 181.42 [Cu(H2O)2]+ -23.9 3.76 362.84 [Cu(H2O)3]+ 3+0 -38.58 4.28 413.02
2+1 -38.86 4.56 440.04 [Cu(H2O)4]+ 4+0 -53.11 4.65 448.72
3+1 -53.54 5.08 490.22 2+2 -53.72 5.26 507.59
[Cu(H2O)5]+ 4+1 -67.76 5.14 496.01 3+2 -68.15 5.53 533.64
[Cu(H2O)6]+ 4+2 -82.58 5.8 559.70 [Cu(H2O)7]+ 4+3 -97.12 6.18 596.37 [Cu(H2O)8]+ 4+4 -111.64 6.54 631.11
Table B1: Binding energies (be) of [Cu(H2O)N]+ with N= 1 to 8, calculated using post SCF
B88X + P86C and a TZ2P basis set with the core frozen at 2p on Cu and 1s on O.
252
Complex Config. abs. (eV) b e (eV)
b e (kJ/mol)
[Cu(H2O)]2+ 9.54 5.8 559.70 [Cu(H2O)2]2+ -8.37 9.55 921.57 [Cu(H2O)3]2+ 3+0 -24.96 11.98 1156.07 [Cu(H2O)4]2+ 4+0 -40.86 13.72 1323.98 [Cu(H2O)5]2+ 5+0 -56.05 14.75 1423.37 4+1 -56.38 15.08 1455.22 [Cu(H2O)6]2+ 6+0 -71.31 15.85 1529.52
4+2 -71.75 16.29 1571.98 [Cu(H2O)7]2+ 4+3 -86.96 17.34 1673.31 [Cu(H2O)8]2+ 4+4 -102.11 18.33 1768.84
Table B2: Binding energies (be) of [Cu(H2O)N]2+ with N= 1 to 8, calculated using post SCF
B88X + P86C and a TZ2P basis set with the core frozen at 2p on Cu and 1s on O.
Copper ammonia binding energies
Complex Config. Abs. (eV) b e (eV) b e (kJ/mol)
[Cu(NH3)]+ -13.90 2.72 254.32 [Cu(NH3)2]+ -35.96 5.42 506.77 [Cu(NH3)3]+ 3+0 -54.89 4.99 466.56 [Cu(NH3)4]+ 4+0 -75.73 6.47 604.94
3+1 -75.33 6.07 567.54 [Cu(NH3)5]+ 4+1 -95.30 6.68 624.58 [Cu(NH3)6]+ 4+2 -115.03 7.05 659.17 [Cu(NH3)7]+ 4+3 -134.63 7.29 681.61 [Cu(NH3)8]+ 4+4 -154.04 7.34 686.29
Table B3: Binding energies (be) of [Cu(NH3)N]+ with N= 1 to 8 , calculated using post SCF
B88X + P86C and a TZ2P basis set with the core frozen at 2p on Cu and 1s on N.
253
Complex Config. abs. (eV) b e (eV)
b e (kJ/mol)
[Cu(NH3)]2+ 2.73 7.41 692.65 [Cu(NH3)2]2+ -21.13 11.91 1113.82 [Cu(NH3)3]2+ 3+0 -42.98 14.4 1346.40 [Cu(NH3)4]2+ 4+0 -64.16 16.22 1516.57 [Cu(NH3)5]2+ 5+0 -84.90 17.6 1645.60 4+1 -85.00 17.7 1654.95 [Cu(NH3)6]2+ 6+0 -104.74 18.08 1690.48
4+2 -105.26 18.6 1739.10 [Cu(NH3)7]2+ 4+3 -125.32 19.3 1804.55 [Cu(NH3)8]2+ 4+4 -145.32 19.94 1864.39
Table B4: Binding energies (be) of [Cu(NH3)N]2+with N= 1 to 8, calculated using post
SCF B88X + P86C and a TZ2P basis set with the core frozen at 2p on Cu and 1s on N.
Copper pyridine binding energies
N abs. (eV) b e (eV)
b e (kJ/mol)
1 -50.86 10.40 972.12 2 -125.71 15.29 1429.31 3 -198.75 18.37 1717.77 4 -270.56 20.22 1890.73 5 -341.18 20.88 1951.87 6 -411.63 21.37 1998.23
Table B5: Binding energies (be) of [Cu(py)N]2+with N= 1 to 6, calculated using post SCF
B88X + P86C and a TZ2P basis set with the core frozen at 2p on Cu and 1s on N and C.
254
Ionization energy of Cu atoms / ions
Absolute energies (eV) IE's (eV) ERROR (%)
Cu Cu(I) Cu (II) 1st 2nd 1st 2nd basis set DZ -0.27 8.4 29.62 8.67 21.22 12.31 4.64 TZP -0.26 8.44 29.63 8.71 21.19 12.82 4.49 TZ2P -0.26 8.44 29.63 8.71 21.19 12.82 4.49 TZ2P + -0.26 8.31 29.47 8.58 21.16 11.14 4.34
Table B5: Relativistic energies calculated using LDAxc post SCF B88X + P86C. The core was
frozen at 2p in all basis sets used. The error is calculated with respect to experimental values
shown in table 1.1 (chapter 1).
Absolute energies (eV) IE's (eV) ERROR (%)
Cu Cu(I) Cu(II) 1st 2nd 1st 2nd basis set
DZ -0.27 8.14 29.53 8.4 21.39 8.81 5.47
TZP -0.26 8.18 29.5 8.44 21.31 9.33 5.08
TZ2P -0.26 8.18 29.5 8.44 21.31 9.33 5.08
TZ2P + -0.26 8.07 29.36 8.33 21.29 7.90 4.98
Table B6: Non-relativistic energies calculated using LDAxc post SCF B88X + P86C.
The core was frozen at 2p in all basis sets. The error is calculated with respect to
experimental values shown in table 1.1 (chapter 1).
255
Absolute energies (eV) IE's (eV) ERROR (%)
Cu Cu(I) Cu(II) 1st 2nd 1st 2nd basis set
DZ -0.26 8.17 29.51 8.44 21.34 9.33 5.23
TZP -0.26 8.18 29.5 8.44 21.31 9.33 5.08
TZ2P -0.26 8.19 29.5 8.45 21.31 9.46 5.08
TZ2P + -0.26 8.07 29.36 8.33 21.29 7.90 4.98
Table B7: Non-relativistic energies calculated using LDAxc post SCF B88X + P86C. All
electron basis sets were used. The error is calculated with respect to experimental values
shown in table 1.1 (chapter 1).
Absolute energies (eV) IE's (eV) ERROR (%) Cu Cu(I) Cu(II) 1st 2nd 1st 2nd
basis set
ET-pVQZ -0.26 8.11 29.41 8.38 21.29 8.55 4.98
ET-QZ3P-1DIFFUSE -0.26 8.19 29.64 8.45 21.45 9.46 5.77
ET-QZ3P-2DIFFUSE -0.26 8.04 29.34 8.3 21.3 7.51 5.03
ET-QZ3P-3DIFFUSE -0.26 8.09 29.44 8.35 21.35 8.16 5.28
Table B8: Non-relativistic energies calculated using LDAxc post SCF B88X + P86C. The
even-tempered (ET) all electron basis sets were used. The error is calculated with respect to
experimental values shown in table 1.1 (chapter 1)
256
Absolute energies (eV) IE's (eV) ERROR (%) Cu Cu(I) Cu(II) 1st 2nd 1st 2nd
basis set
DZ -0.23 8.66 29.73 8.43 21.07 9.20 3.90
TZP -0.30 8.76 29.79 8.45 21.03 9.46 3.70
TZ2P -0.30 8.76 29.79 8.45 21.03 9.46 3.70 TZ2P + -0.56 8.83 29.78 8.27 20.95 7.12 3.30
Table B9: Relativistic energies calculated using the model potential LB94. The cores were
frozen at 2p. The error is calculated with respect to experimental values shown on table 1.1
(chapter 1).
Absolute energies (eV) IE's (eV) ERROR (%) Cu Cu(I) Cu(II) 1st 2nd 1st 2nd
basis set
DZ -0.23 8.38 29.62 8.15 21.25 5.57 4.78 TZP -0.30 8.49 29.66 8.19 21.17 6.09 4.39 TZ2P -0.30 8.49 29.66 8.19 21.17 6.09 4.39 TZ2P + -0.58 8.59 29.68 8.02 21.09 3.89 3.99
Table B10: Non-relativistic energies calculated using the model potential LB94. The cores
were frozen at 2p. The error is calculated with respect to experimental values shown in table
1.1 (chapter 1).
257
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