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Cite this: Chem. Soc. Rev., 2014,
43, 5067
Quantum-chemical insights into mixed-valence
systems: within and beyond the Robin–Day scheme
M. Parthey and M. Kaupp*
In mixed-valence (MV) systems essentially identical, more or less electronically coupled, redox centres
are brought into formally different oxidation states by removal or addition of an electron. Depending on
the strength of electronic coupling, an electron or a hole is either concentrated on one of the redox
centres, or it is symmetrically delocalised onto several sites, or the situation is somewhere in between,
which leads to the classification system for MV systems introduced by Melvin Robin and Peter Day.
These different characteristics are of fundamental importance for the understanding of electron transfer
processes. Applications of quantum-chemical methods to aid the classification and to unravel the nature
of the electronic structure and spectroscopic data of both organic and transition-metal MV systems,
have gained tremendous importance over the last two decades. In this review, we emphasise the
prerequisites the quantum-chemical methods need to fulfill to successfully describe MV systems close
to the borderline between Robin–Day classes II and III. These are, in particular, a balanced treatment of
exchange, dynamical and non-dynamical correlation effects, as well as consideration of the crucial
influence of the (solvent or solid-state) environment on the partial localisation of charge. A large variety
of applications of quantum-chemical methods to both organic and inorganic MV systems are critically
appraised here in view of these prerequisites. Practical protocols based on a combination of suitable
density functional methods with continuum or non-continuum solvent models provided good
agreement with experimental data for the ground states and the electronic excitations of a large range
of MV systems close to the borderline. Recent applications of such methods have highlighted the crucial
importance of conformational effects on electronic coupling, all the way to systems where
conformational motion may cause a thermal mixing of class II and class III situations in one system.
1. Introduction
The importance of electron transfer (ET) in most areas of
chemistry, biology, and materials sciences can hardly be over-
estimated. In this vast field, mixed-valence (MV) compounds
have played a prominent role as central models for basic
understanding of ET, as well as for various applications, some
of which we will mention further below. The amount of
synthetic, electrochemical, and spectroscopic work that has
gone into the study of MV systems over the past 50 years is
dazzling and certainly outside the scope of this review. We will
focus here on the recently increased impact that quantum-
chemical (QC) methods have had on the unravelling of ET in
molecular, both organic and transition-metal, MV systems,
especially when considered together with the relevant spectro-
scopic studies. We will first motivate the need for QC studies
by discussing some basic models of ET together with the
difficulties in establishing the details by spectroscopy alone.
We will then point out the challenges for a quantitative
computational treatment of MV systems at or near the localised–
delocalised borderline, that have for a long time hampered more
accurate studies. This has led our group to design and validate
semi-empirically a scheme based on density functional theory
(DFT) and suitable modelling of solvent environment. Part of the
methodology and initial work on organic MV radical cations has
already been reviewed. Here we extend the discussion to a wider
range of organic MV systems and to improved solvent models. In
particular, however, we will focus on recent work on transition-
metal MV complexes, in the wider context of previous attempts of
QC modelling in the field. These recent studies have led, among
other things, to a new view of the importance of rotational
conformers, guiding us beyond the usual two-state or three-state
models of ET in MV systems.
1.1 Classifications of mixed-valence systems
A wide range of molecular and solid-state systems is embodied
in the MV concept, and various models have been devised to
classify the ET characteristics. Generally, one views a MV system
Technische Universita
¨t Berlin, Institut fu
¨r Chemie, Theoretische Chemie, Sekr. C7,
Straße des 17. Juni 135, 10623 Berlin, Germany.
E-mail: martin[email protected]
Received 30th December 2013
DOI: 10.1039/c3cs60481k
www.rsc.org/csr
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as a molecule or solid in which a given redox centre appears at
least twice, in two different oxidation states, connected by a
suitable bridging unit that typically provides for some electronic
coupling between the redox centres. The thermal or optically
induced ET between the (two or more) redox centres is at the
heart of attention, but ET from or to the bridging unit, which is
becoming increasingly more widely recognised, also plays a role
in the overall ET mechanisms and spectroscopic profiles. The
large variety of redox centres and bridge units and the different
coupling between them accounts for the multitude of MV
systems that can be envisioned or exist, either purposefully or
accidentally constructed by synthetic chemists or present in
nature. The ‘‘classical’’ MV systems are based on transition-
metal redox centres connected by bridging ligands. Examples are
Prussian Blue on the ‘‘serendipity’’ branch of the field or the
Creutz–Taube ion and its derivatives
1,2
that feature prominently
in the purposeful study of ET in transition-metal systems.
However, many other molecules have been known that were
classified as MV only much later. Examples are the radical
anions of aryl compounds with two or more nitro substituents.
These have been known and studied spectroscopically since the
early 1960s, but they were recognised as MV systems only much
later, where the nitro-substituted part of the aryl ring features as
redox centre, and the remaining aryl part plays the role of
bridging unit.
3–5
This served to introduce organic redox centres
to the field, which have received grown attention due to their
involvement in organic molecular electronics. Notable examples
are those based on triaryl-amine redox-active units (see below).
Organic MV systems are usually also based on organic bridges, but
‘‘inverted’’ MV systems with organic redox centres and transition-
metal-based bridges are also known as discussed below.
We may even widen the MV concept to such exotic species as
the H
2+
ion, which may be viewed as the smallest conceivable
MV system. We will generally restrict ourselves to MV systems
where the formal redox states differ by one unit, which for two
redox centres typically leads to open-shell compounds in the
ground and excited states.
6
Of course, MV systems with larger
electron-number differences are conceivable. Their closed-shell
variants have been classified as donor–acceptor systems.
6
Systems with non-identical redox centres further extend the
picture and may call for a looser MV definition.
7
In many cases
redox-non-innocence of the bridge or of terminal ligands of
transition metal complexes additionally complicates the redox-centre
assignment.
8–11
Here we will focus on organic and transition-metal MV
systems, which formally contain only two identical redox centres,
but are linked by bridges which may exhibit redox-non-innocent
behaviour. Such systems can be analysed within the famous
Robin–Day classification scheme (Fig. 1).
12
The three primary
Robin–Day classes correspond to (I) two completely decoupled
diabatic redox states and fully localised redox centres, (II)
moderate electronic coupling between the centres leading to a
double-well adiabatic ground-state potential-energy curve with
partly localised charges and a barrier for thermal ET (the electronic
coupling, 2H
ab
, is smaller than the Marcus reorganisation energy
l), and (III) strong coupling with 2H
ab
Zlleading to a single
ground-state minimum without ET barrier and the charge being
delocalised symmetrically over both redox centres. The reorganisa-
tion lis usually divided into the inner reorganisation energy l
i
arising from structural changes within the molecule during ET,
and the outer reorganisation energy l
o
arising from solvent
Fig. 1 Potential curves for the three primary Robin–Day classes: class I
(left), class II (middle) and class III (right).
M. Parthey and M. Kaupp
Matthias Parthey received his Diploma at Universita
¨tWu
¨rzburg in
2011 and PhD at Technische Universita
¨t Berlin in 2014 in the group
of Martin Kaupp, as member of the Berlin International Graduate
School of Natural Sciences and Engineering, associated with the
UniCat cluster of excellence. In 2012 he spent six months in the
group of Paul J. Low at Durham University, UK, for collaboration on
the optoelectronic properties of mixed-valence complexes. His
interests are the quantum-chemical treatment of mixed-valence
systems, computation of excited-state parameters, and
applications to catalysis and molecular electronics.
Martin Kaupp received his PhD in 1992 with P. v. R. Schleyer in
Erlangen and his Habilitation 1997 in Stuttgart, after a postdoc in
Montre
´al. He became professor at Universita
¨tWu
¨rzburg in 1999.
Since 2010 he is professor of Theoretical Chemistry at Technische
Universita
¨t Berlin. Martin Kaupp’s wide research interests include development and applications of quantum chemical methods to
calculate NMR and EPR parameters, density functional theory, relativistic effects, as well as computational bioinorganic, inorganic,
organic and organometallic chemistry. He has authored more than 220 publications. In 2001 he received the Dirac Medal of the World
Association of Theoretical and Computational Chemists (WATOC).
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rearrangements.
6,13
An intermediate class II/III introduced by
Meyer et al.
14
will be mentioned further below. Lear and
Chisholm introduced a new class IV, which represents a subclass
of class III taking into account vibronic progression.
15
This
review will focus largely on systems close to the borderline
between class II and class III, which is most challenging for
both experiment and theory.
1.2 Application examples
Here we give a brief glimpse of some application areas of MV
systems beyond their role as models to understand ET.
Biocatalysis. ET reactions are of tremendous importance in
heterogeneous, homogeneous, and enzymatic catalysis. In fact
it may be argued that ET is involved the majority of important
catalytic processes. Considering biocatalysis in particular,
metallo-proteins often contain multinuclear metal sites that
feature MV states during catalysis.
16
It is in particular the
conversion of small, stable molecules (e.g. H
2
O, CH
4
,N
2
,CO
2
)
that requires a sequence of elementary steps to circumvent
high activation barriers.
17
Multinuclear metal sites, sometimes in
combination with redox-active ligands are frequently involved here.
The oxygen-evolving complex of the photosystem II, which
catalyses the light-driven water oxidation, represents a typical
example for an active site with predominant MV active
states.
18,19
The Mn
4
Ca(m-O
n
) core features a tetranuclear transition
metal cluster, where different (MV) oxidation states are reached as
a consequence of light-induced electron removal (i.e. photo-
oxidation).
20
Broken-symmetry DFT calculations have clearly
demonstrated localised class II character for the relevant states,
and standard functionals like B3LYP have been found to provide a
reasonable picture of the electronic structure of these MV clusters,
which seem to be far from the borderline to class III.
Another type of multinuclear enzyme site is represented
by the dinuclear Cu
A
site in various copper enzymes, e.g.
cytochrome coxidase. Here the MV Cu
+I
Cu
+II
state appears to
be a delocalised class III system, but perturbations due to
mutation may alter the protein environment sufficiently to
cause partial localisation.
16,21–23
In contrast, partial delocalisation
characterises many of the important biological iron–sulfur clusters,
which renders these systems a challenge for both clear-cut experi-
mental and theoretical descriptions. Charge distribution in FeS
clusters depends crucially on coordination number of the metals
and the extent of magnetic coupling between them (ferromagnetic
coupling favours charge delocalisation, whereas antiferromagnetic
coupling appears to give rise to charge localisation).
16
Molecular electronics. Class III MV systems are obvious
candidates for use as ‘‘molecular wires’’ or ‘‘nanojunctions’’ in
the field of molecular electronics.
24
Due to their electronically
delocalised character and vanishing thermal ET barriers, fast ET
over distances in the nm range is achievable.
25–30
Good energy
matching between the orbitals of the bridge and the redox
centres (‘‘end caps’’) is essential.
24,31
On the other hand, switching functions or data storage are
facilitated by a certain degree of localisation (‘‘trapping’’) of
charge carriers, pointing to class II situations with appreciable
barriers. Here the currently envisioned targets include the
controversially discussed ‘‘quantum-dot cellular automata’’
made up of MV complexes. These may be related to coding
information in quantum computers
32–34
and to molecular
transistors.
24
Driven by the technological and economic pursuit
of ‘‘Moore’s Law’’, the functional area of a transistor has been
halving every eighteen to twenty-four months over the last
decades, allowing the number of transistors per chip to double
in each technology generation, and thus giving rise to smaller
and more powerful electronic devices.
24,35
Although studies
suggest another two decades of potential progress in silicon
nanoelectronics,
36
a switch from the size-limited traditional
materials towards molecular components will be required to
continue this process of transistor scaling. Due to their versatile
adjustable electronic properties, MV systems represent promising
targets towards molecule-based electronics. Both organic and
inorganic MV compounds are used in photovoltaic devices, where
we may highlight dye-sensitised solar cells. Here MV systems may
act as dye, as well as redox shuttle, and remarkable efficiencies
have already been achieved.
37–39
2. Limits in the experimental
classification and the need for a
quantum-chemical description
Information on the extent of charge localisation–delocalisation
in an MV system can be obtained by various experimental
techniques. Most commonly the assignment to a Robin–Day
class is based on the analysis of the IVCT band, which typically
occurs in the NIR region of the electromagnetic spectrum. The
potential for multiple electronic transitions of similar energy
but different electronic origin, together with the asymmetric
IVCT band-shapes that characterise strongly coupled MV systems,
renders derivation of the ET characteristics and electronic structure
from NIR spectra alone very difficult in many MV transition-metal
complexes, despite the popularity of such analyses. In organic MV
systems the IVCT band usually appears as the lowest-energy band
in the spectrum, facilitating the assignment.
6
Other important spectroscopic techniques involve somewhat
different energy and time scales, e.g. vibrational spectroscopies
(IR, Raman), Stark spectroscopy, Mo
¨ssbauer spectroscopy, and
EPR spectroscopy. At the borderline between class II and III,
small activation barriers and fast ET processes may give rise to
contradictory findings with different spectroscopic techniques,
due to the different time scales of the spectroscopic methods,
which can be comparable to the rates of ET, inner-sphere
reorganisation processes, and solvent dynamics.
In some early studies the presence of symmetry-broken
crystal structures was used as classification criterion, but of
course the solid state may differ significantly from the situation
in solution. Despite the fact that crystals are normally grown at
low temperatures, at which the ET process is slowed down,
crystal structures may be distorted due to packing, counter ion, or
other crystal effects.
40
For example salen type complexes often
exhibit symmetry-broken structures in the crystal, but may feature
delocalised charge in solution.
41–44
Additionally, for complexes
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with free coordination sites, direct coordination of Lewis basic
solvents may occur.
IR and vibrational Raman spectroscopy represent very fast
and powerful methods to obtain information on the electronic
structure of the ground state of MV systems.
45
They are often used
in combination with UV-vis-NIR spectroscopyaspartofaspectro-
electrochemical approach.
46
Indications of charge localisation are
easily derived from the IR spectra of complexes, which are
symmetric in their non-mixed-valence forms. The splitting of
IR bands due to the energy difference in modes, which are
degenerate in the delocalised case (e.g. n(CRC) vibrations) or
the appearance of new IR bands, which do not involve changes
of the permanent dipole moment in the case of a symmetrically
distributed electron density, point to symmetry-broken struc-
tures.
47–49
Satellite groups, such as local auxiliary ligands, may
allow monitoring of modes, e.g. n(NO) or n(CO), which depend
strongly on the oxidation state of the metal centre and thus are
sensitive to the charge distribution.
50
Analogously, information
on electronic structure may be derived from Raman spectra.
45,51
Mo
¨ssbauer spectroscopy is sensitive to oxidation state and
charge distribution and corresponds to short time scales.
Limitations pertain to the Mo
¨ssbauer-active nuclei, where
57
Fe is the most prominent example.
52–54
The paramagnetic nature of most MV systems complicates
the use of NMR spectroscopy due to paramagnetic line
broadening.
Variable-temperature EPR spectroscopy is widely used to
determine ET barriers in organic MV systems.
5,55,56
Applications to
MV transition-metal complexes are more limited, due to difficulties
in extracting hyperfine coupling constants.
47
In principle the iso-
tropic hyperfine coupling constant of a MV system corresponds to
the value obtained for an analogous monometallic complex, if the
charge is localised. For a class III system it is close to half the value of
the monometallic analogue. In addition, line broadening effects in
the EPR spectrum may reveal ET processes that are slower than or
comparable to the EPR time scale (10
9
s).
57
The ET characteristics of MV compounds are most typically
analysed within the frameworks of the Marcus–Hush and
(generalised) Mulliken–Hush equations.
58–66
Assuming that
we are able to extract the necessary parameters of these models
(mainly) from the NIR spectra, a rather detailed view is provided. As
this article will focus on the direct computation of the adiabatic
potential energy surfaces rather than on their indirect construction
from experimentally estimated approximate diabatic potential
energy curves, we will not provide a detailed discussion of the
Marcus–Hush or Mulliken–Hush approaches. We refer the reader
to a recent, very detailed discussion.
6
However, problems with the application of these models
often arise from the lack of sufficient spectroscopic informa-
tion to support the requisite data analyses. A common way to
analyse NIR spectra is to fit the spectroscopic absorption band
profiles with Gaussian functions. In principle, the population
of both ground and excited state vibrational levels can, to some
extent, be taken into account assuming that these follow a
Boltzmann distribution. Unfortunately this assumption is
only valid for clear-cut class II or class III complexes in the
Robin–Day scheme, as pointed out by Brunschwig et al.
13
and
Lambert and Heckmann.
6
For systems close to the class II/class
III borderline, the NIR band gets more and more asymmetric,
as there is a cut-off at the smallest energy possible for an
electronic transition. Due to this asymmetry of the band
envelope, Gaussian functions are no longer a very suitable
approximation and caution with Gaussian deconvolutions is
advocated, when the investigated system is not a clear-cut class
II or class III complex.
In addition to the vibrational progression, an alternative
explanation for the band asymmetry of the Creutz–Taube ion
was suggested by Meyer et al.:
14
the asymmetry of the IVCT
band of the Creutz–Taube ion is explained by the coupling
of two of the proposed IVCT transitions at 6320 cm
1
and
7360 cm
1
. The fact that the band shape does not change in
low-temperature experiments was used as an argument
to rule out vibronic effects. High-level quantum-chemical
studies did not reproduce these transitions.
67
Asymmetric
band shapes are also observed for organic MV compounds
near the class II/class III borderline, ruling out d-orbital
contributionsasexplanationfortheasymmetryintheband
envelope.
In principle, the problems arising from asymmetric band
envelopes can be circumvented by fitting only the high-energy
part of the IVCT band with Gaussian functions. For this
analysis, the IVCT band must be both clearly assigned and
sufficiently well removed from other transitions to give an
unobstructed view of the band edge. While this is possible for
most organic systems with only two redox centres, for most MV
transition metal complexes multiple transitions close in energy
are observed. These transitions may lead to overlapping band-
envelopes, and render a direct assignment of transition energies
and bandwidth impossible.
Based on the semi-empirical model proposed by Meyer et al.,
experimental spectra of 3d and 4d complexes are normally
fitted with three Gaussian functions.
14
This localised model
assumes two interconfigurational (IC) bands, which arise from
excitations from orbitals localised at the hole-carrying (more
highly oxidised) metal centre to orbitals at the same centre, and
three IVCT bands originating from separate electronic excita-
tions from three dp-orbitals of the other metal centre. The IC
transitions are normally parity forbidden, and only gain intensity
if the system exhibits noticeable spin–orbit coupling. As the
latter tends to be significant only for 5d systems, 3d and 4d
metal complexes are typically expected to exhibit only the three
IVCT excitations.
Hence alternatives to analyses of UV-vis-NIR bands to under-
stand ET characteristics of MV systems are needed. This holds
also for other parameters inherent in the simplified models,
such as electron-transfer distances or the adiabatic dipole
moment of the excited states. These tend to be difficult to
extract from experiment. It is thus desirable to compute, e.g.,
excited-state parameters and compare them to the measured
spectra. This is only one of the many aspects that call for a full,
general, and quantitative quantum-chemical treatment of MV
systems.
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3. Challenges for a quantum-chemical
description of MV systems close to the
borderline
3.1 Wave-function-based methods
The proper treatment of Coulomb correlation is one of the
main challenges in the accurate description of MV systems.
UHF calculations provide a dramatic bias towards a localised
situation, usually accompanied by strong spin contamination
(in particular for the symmetrical transition state in a class II
situation). UMP2 calculations are reported to suffer from the
same unphysical exaggerated spin contamination.
68
We distinguish
dynamical and static (‘‘non-dynamical’’) correlation, keeping
in mind that the separation between these two contributions
is ill-defined.
69
From previous experience it is clear that a
quantitative description of localisation–delocalisation in MV
systems close to the borderline of classes II and III requires a
high level of treating both contributions. The static part may in
some cases (but probably not too frequently) lead to a break-
down of single-reference approaches. Including only the static
correlation contributions, e.g. within a complete-active-space
self-consistent-field (CASSCF) ansatz, does not cure the over-
localisation (sometimes, the symmetrical structure has been
enforced by constraints
70
).
Adequate treatment of the dynamical correlation part
requires good convergence with respect to the one-particle
basis set. Given the steep scaling of typical methods with
system size, their application to chemically relevant MV systems
is very limited. In fact, examination of the basis sets used in the
available ab initio post-HF studies suggests that the appreciable
fractions of dynamical correlation required to describe a class III
situation reasonably close to the borderline so far do not seem to
have been achieved, neither for organic
71–73
nor for transition-
metal MV
67,74,75
systems (see examples below in Section 4).
Most multi-determinantal approaches have thus been
applied within a semi-empirical framework, using the AM1
Hamiltonian for organic MV systems
68,76–81
or the INDO para-
meterisation for transition-metal complexes.
82,83
In fact, until
recently, AM1/CI calculations together with continuum solvent
models had been the only methods that allowed at least to
some extent a distinction between organic class II and class III
MV systems close to the borderline.
78
Limitations in the accuracy
of the approach necessarily arise from the semi-empirical para-
meterisation. Different CI schemes have been applied, including
either all valence single excitations
68
or a full CI in a very
small active space
81
(typically three electrons in two orbitals) of
the full-CI part of the calculation.
3.2 Density functional theory
The approximate inclusion of correlation via the exchange–
correlation functional at effectively single-determinant cost
makes Kohn–Sham DFT the most popular quantum-chemical
method of our days. With standard functionals it is assumed
that some of the static left–right correlation in bonds is
simulated by the (semi-)local exchange functional. This comes,
however, at the price of so-called self-interaction errors.
These are also termed, more suitably in the present context,
‘‘delocalisation errors’’.
84–91
Indeed, standard functionals of
the generalised-gradient-approximation (GGA) type or even
the most popular hybrid functional B3LYP,
92–94
where 20% of
the semi-local exchange is replaced by exact Hartree–Fock
exchange, generally arrive at a far too delocalised electronic
structure for MV systems close to the borderline and are thus
not suitable for analysis of these compounds.
95
That is, a
delocalised class III situation is erroneously produced on the
class II side when sufficiently close to the border, which is just
the opposite of the HF or CASSCF based approaches (see
above). We note, however, that excitation energies for class III
systems are typically reproduced well by B3LYP calculations.
We see thus that we deal with a rather sensitive balance
between opposing effects. We will come back to possible
solutions for this dilemma further below.
Van Voorhis advocated the use of constrained DFT (CDFT) in
such situations, where charge localisation is enforced by add-
ing system-dependent constraints to the Hamiltonian.
96–98
That is, the information about how many electrons should be
placed at a given redox centre is an input parameter of the
calculation. The predictive power of such an approach is
obviously very limited. It has been used mainly to extract
ET parameters needed for the Marcus–Hush and generalised
Mulliken–Hush treatments based on (enforced localised) diabatic
states.
33,96–102
Regarding the excitation energies, another, related problem
with standard functionals, e.g. of the GGA type, arises. It
is well known that for typical charge-transfer transitions,
such functionals underestimate the corresponding excitation
energies dramatically, and the reasons have been discussed
extensively.
103–106
But as nicely pointed out by Peach et al.,
107
some types of charge-transfer transitions are less critical in this
context. Even more importantly, the problem can be partly
remedied by including a suitable amount of exact exchange,
(E
exact
x
) into the functional, in a suitable variant of hybrid
functional (see Section 3.4).
As we move to transition-metal systems, we also have to
consider the problem of spin contamination. While an increase
of E
exact
x
admixture diminishes delocalisation errors, it also
tends to increase spin polarisation. It is well known that for
transition-metal complexes, too much exact exchange may
cause unphysical exaggerated valence-shell spin polarisation,
which in turn leads to spin contamination (i.e. hS
2
i4S(S+ 1),
where Sis the nominal total spin of the system).
108–112
This
effect is well known for hybrid DFT calculations on transition-
metal complexes, and it is most pronounced for systems
with significantly metal–ligand antibonding character of the
singly occupied molecular orbital(s).
29
Consequences of spin
contamination are discussed vigorously.
111–115
The S
2
expectation
value of the Kohn–Sham determinant strictly pertains to the
so-called ‘‘non-interacting reference system’’ and its physical
significance is thus being disputed.
113,114,116
Definitions more
in the spirit of Kohn–Sham DFT have been formulated.
117
Empirically, increased spin contamination has been found to
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notably deteriorate the quality of electronic structure and, e.g.,
EPR parameters.
109–112,114,115
Similar observations pertain to
TDDFT calculations of excitation spectra.
118
As spin polarisation
and spin contamination increase with the amount of E
exact
x
admix-
ture, caution with orbital assignments is advocated, and the S
2
expectation value should always be checked.
3.3 Environmental effects
Most of the relevant MV systems are charged, and experimentally
they are typically studied in a polar solvent environment. In fact,
experimental gas-phase data are essentially absent in this field. It is
thus obvious, that solvent and counter-ion effects are crucial ingre-
dients for a reliable quantum-chemical protocol. Polar solvents will
usually stabilise a localised, valence-trapped situation compared to a
delocalised charge distribution. Gas-phase calculations, which were
the standard approach until about 10 years ago, thus are clearly
biased towards a too delocalised description. In fact, there are clear
spectroscopic indications (see also further below for computational
results) that even a moderate increase of solvent polarity may change
a class III to a class II system, provided it is close to
the borderline.
119–121
While sometimes calculations included
continuum solvent models in single-point calculations of,
e.g., excitation spectra,
122
neglect of solvent effects in the
ground-state structure optimisation of a class II system often
provided (incorrect) symmetrical structures to start with,
thereby invalidating the subsequent spectroscopic calculations.
An explicit inclusion of the solvent into the quantum-
chemical model, or possibly a QM/MM ansatz, where the
solvent is treated classically together with a QM treatment of
the solute, can in principle provide a realistic description.
However, the dynamical nature of the solute–solvent inter-
actions requires such simulations to include the molecular
dynamics on a sufficient time scale. While this type of explicit
ab initio MD or QM/MM based MD studies has become an
important tool in other areas of research, they have not yet
entered the stage in the study of MV systems so far, at least not
to a notable extent. This is probably due to the complications
described in the previous two sections regarding the electronic-
structure treatment itself. Given the recent emergence of practical–
pragmatic solutions to the correlation problems (see next section),
we expect that over the next years dynamical studies will gain
substantial importance in the field of mixed-valency.
Thus, continuum solvent models so far dominate computa-
tions on MV systems, and we will provide examples throughout
this article. An extension beyond continuum models that allows
an adequate treatment also for protic solvents, has involved
Klamt’s D-COSMO-RS ansatz,
123
which will also be described
below. A very interesting approach based on force-field Monte-
Carlo simulations and an integral equation formalism is the
RISM-SCF procedure, which has also recently been applied to
MV systems.
124,125
There are no suitable continuum models to include counter-
ion effects. This is why these have so far been largely neglected.
It may be expected that counter ions will receive more scrutiny,
as MD studies will enter the field. The importance of counter-
ion effects depends of course on the system. Bulky MV systems
like the typical triarylamine radical cations are likely affected
less than smaller MV systems, e.g. some organic radical anions.
3.4 A reliable DFT-based protocol
The above discussion makes clear that, to reliably describe the
properties of MV systems close to the borderline between class
II and class III, a suitable quantum-chemical protocol has to
account in detail for dynamical and non-dynamical correlation
without suffering from extensive self-interaction errors, and it
has to cover at the same time the relevant environmental
effects. Notably, these features should already be included
accurately in the ground-state structure optimisation, in contrast
to many studies we find in the literature. Otherwise, computations of
excitation energies or other spectroscopic parameters may suffer
fromtheinaccuratestructures.Foranevenmoreaccuratedescrip-
tion of excitation spectra, vibronic coupling has to be considered. In
the absence of specific solvation effects, such as hydrogen bonding
or the dynamic exchange of solvent molecules within the coordina-
tion sphere of a metal centre, it is expected that the bulk solvent
effects may be covered adequately by dielectric continuum solvent
models like COSMO
126
or other ‘‘polarisable continuum models’’
(PCM).
127,128
As mentioned in Section 3.1, their combination with
the AM1/CI method provided the earliest semi-quantitative compu-
tational classification of organic MV systems close to the borderline.
Within a Kohn–Sham DFT setting, the judicial inclusion of
exact exchange into the functional should allow a balance to be
reached between the treatment of left–right correlation in
bonds (improved by semi-local ingredients of the exchange
functional) and a reduction of delocalisation errors due to
self-interaction (improved by more exact exchange). This leads
naturally into the realm of hybrid functionals, where we may
distinguish between constant exact-exchange admixture in
global hybrids, an admixture depending on the interelectronic
distance in so-called range-separated hybrids,
129–140
or a real-
space position-dependent admixture in so-called local
hybrids.
141–149
While detailed validation of the latter for MV
systems is still pending, the two former classes have been
evaluated in some detail (see below), initially for triarylamine
(TAA) radical cations, subsequently for organic MV radical
anions, and most recently for transition-metal complexes.
The initial validation of global hybrids for TAA radical
cations
120,121
(see below) in conjunction with continuum solvent
models led to an optimum exact-exchange admixture of 35% in a
customised one-parameter hybrid BLYP35, which was con-
structed analogous to the B1LYP model.
150
While this is an ad
hoc construction, later studies on smaller and thus computa-
tionally more convenient organic MV radical anions
151,152
allowed an even larger set of functionals to be screened and
showed, e.g., that the well-known BMK meta-GGA global hybrid
functional
153
also provides a very reasonable compromise. The
BMK functional incorporates 42% exact-exchange admixture,
somewhat ameliorated by contributions from local kinetic energy
density. Global hybrids with appreciably less exact exchange over-
estimate delocalisation. Functionals with larger exact-exchange
admixtures give too localised electronic structure.
120,121,151,152,154
Indeed, it turned out that the exact-exchange admixture is the by far
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most important aspect of a global hybrid that has to be selected to
achieve optimum performance for the treatment of MV systems.
Range-separated hybrids with 100% exact exchange in the
long range, such as oB97X,
155
and LC-BLYP,
136
so far tended to
provide a too localised description, whereas CAM-B3LYP
137
(which interpolates between 19% at short range and 65% at
long range
107,137
) appeared to be just slightly too localised for
organic MV radical anions. The ‘‘nonempirical tuning’’ of the
range-separation parameter has been suggested as a means to
adjust the delocalisation error, also recently for organic MV
systems.
156
A disadvantage is that such a tuning provides a
different range-separation parameter for different molecules or
even for different structures of one molecule, leading to size-
consistency problems.
157
The currently popular double hybrids exhibit too large exact-
exchange admixture to be accurate for MV systems close to the
border, and the MP2 contribution to the correlation functional
tends to be problematic at the transition state of thermal ET for
class II systems.
120
These validation studies, which required the
adequate inclusion of solvent effects, will be discussed in some
more detail in the following sections, as will be the extension to
MV transition-metal complexes.
158–160
Overall, the BLYP35/
COSMO based protocol and its variations using either suitable
alternative functionals or alternative solvent models have
turned out to provide an unprecedentedly detailed bracketing
of localisation vs. delocalisation close to the borderline in a
wide variety of MV systems.
4. Validation for and applications to
organic MV systems near the
borderline
4.1 Bis-triarylamine radical cations
The initial validation studies of the protocol for purely organic
TAA systems have already been reviewed recently.
121
We therefore
only reiterate a few of the major conclusions of that work, which
resulted in the first version of the abovementioned computational
scheme. These systems had been studied earlier by a wide variety
of methods, and some of the problems discussed in Section 3
above were observed by a variety of authors.
121
TAA MV radical
cations are of interest due to a range of possible applications, and
they have been studied by a wide variety of spectroscopic and
computational methods.
6,79,120,121,154,161–172
Moreover, their
character can be tuned very well by the choice of bridge unit,
end cap, and even by the solvent, so that a wide variety of data for
cases close to the border between class II and class III situations
(from both sides) is available. Regarding the challenges for a
computational treatment, the TAA-based systems have advantages
and disadvantages: the relatively large size of the experimentally
studied systems puts appreciable demands on computational
efficiency, whereas the bulky aryl substituents at the amine redox
centres tend to shield the positive charge well, causing specific
solvation effects to be of minor importance in most cases. The
latter aspect supports the convenient use of continuum solvent
models.
The systematic variation of exact-exchange admixture in
B1LYP-style global hybrid functionals, and of the dielectric
constant of the continuum solvent model appropriate to the
condition of the experimental studies available for an appreci-
able variety of TAA-based MV radical cations led to the above-
mentioned preference for the BLYP35 functional combined
with continuum solvent models as the basis for the initial
version of the protocol discussed above (Section 3.4).
120,121
Notably, TDDFT calculations usingthesamefunctional,solvent,
and basis sets (in these initial studies SVP basis sets) provided
surprisingly good agreement with the experimental excitation
energies for the IVCT band. For class II systems, the excitation
energies were underestimated somewhat. This had to be expected,
as in this case the IVCT band has distinct charge-transfer
character, and DFT tends to underestimate such excitation
energies, sometimes dramatically so. However, the maximum
deviations were rather tolerable, about 1000–1500 cm
1
,suggesting
that 35% exact-exchange admixture turns out to be a reasonable
compromise. It may be assumed that the deviations are largest for
the most weakly coupled systems. Indeed, for strongly coupled
delocalised class III cases, where the character of the IVCT band is
more that of a delocalised single-chromophor p–p*excitation,the
excitation energies were overestimated by up to ca. 1000 cm
1
(blue
circles in Fig. 2). Most importantly, the very systematic performance
confirmed the accurate description of the underlying optimised
ground-state structures, as the excitation energies crucially depend
on whether a localised or delocalised structure is used.
These validation studies showed clearly that (a) standard
functionals like B3LYP exhibit too large delocalisation errors to
be reliable for class II TAA systems, and (b) the inclusion of
solvent effects at least by a continuum solvent model is crucial
for class II cases. We note, however, that B3LYP calculations,
even without solvent effects, provide good excitation energies
for clear-cut class III systems, where delocalisation errors and
neglect of solvent effects do not affect the results too much.
Somewhat lower exact-exchange admixtures compared to the
abovementioned BLYP35 functional tend to lower IVCT excita-
tion energies and thus actually improve agreement with experi-
ment for class III cases, as demonstrated by a recent study
173
at
PBE0
150
level (25% exact exchange; COSMO solvation), or by
work on multibranched TAA systems using B3LYP and a PCM
model.
174
In a TDDFT study of class III triarylamine-substituted
arylene bisimides,
122
the CAM-B3LYP range-separated hybrid
functional (at B3LYP/gas-phase optimised structures) yielded
reasonable excitation energies. Use of a PCM model in the
TDDFT single-points improved significantly the agreement
between computed and experimental ionisation potentials
and electron affinities in CH
2
Cl
2
.
A very recent DFT study on cross-conjugated TAA MV
systems dealt with conformational effects on ET,
175
which will
be discussed in Section 5.4 below. Notably, PBE0/6-31G(d)
calculations with a PCM acetonitrile solvent model correctly
reproduced localised ground-state minima for the relatively
weakly coupled class II systems involved. Not unexpectedly,
the IVCT transition energies were, however, underestimated by
a factor of about two, reflecting deficiencies of the TDDFT
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treatment with a functional containing only 25% E
exact
x
admixture.
Yet good agreement with experimentally determined electronic
coupling matrix elements was obtained, and these depended
significantly on conformation. Interestingly, an evaluation of the
range-separated hybrid functionals CAM-B3LYP and LC-oPBE
indicated a drastic overestimate of the experimental IVCT excita-
tion energies,
175
in particular for the latter functional (see also
Sections 4.3 and 4.4 below).
4.2 Metal-bridged bis-biarylamines and bis-triarylamines
Bis-biarylamines and bis-triarylamines incorporating a metal in
the bridge unit
176–180
are also covered in this section, in spite of
the metal involved. Their behaviour is very similar to purely
organic bis-triarylamine radical cations, as the bridge tends to be
involved only marginally/indirectly in the ET process.
176,179–181
Yet, the spectroscopy of these systems may become richer in case
of transition-metal systems, as in addition to the N -N
+
IVCT
transition also metal -N
+
charge-transfer transitions are
usually obtained in the UV-vis-NIR region. So far it appears that
mainly class II systems have been reported.
176,179–181
Com-
pounds in which thermal ET may proceed via the purely organic
2-20-bipyridine part of the bridge (coordinating to different
transition metals) may be a notable exception.
178
Here introduction
of iridium leads to a class II/class III borderline complex, whereas Ru
or Re bridges give class II behaviour. The BLYP35/C-PCM(CH
2
Cl
2
)
protocol was applied. It gave good account of the influence of
solvent effects, but the authors concluded that the distinction
between purely organic and transition-metal bridged systems
was not completely faithful.
178
For a weakly coupled class II Ga-bridged TAA MV system, it
was found that M06/def2-SV(P) calculations in an integral
equation formalism variant of the PCM model (IEFPCM, e.g.
reviewed in ref. 127) solvation model faithfully reproduced the
class II behaviour. However, the corresponding TDDFT calcula-
tions underestimated the IVCT energy significantly (3764 cm
1
compared to 6390 cm
1
).
181
This likely reflects the fact that
M06 exhibits only 27% exact-exchange admixture, i.e. less than
the BLYP35-based protocol delineated above.
Platinum-bridged TAA systems, had already been studied in
2004 by Jones et al.
176
Parthey et al. have recently applied the
BLYP35/COSMO protocol to such systems and found good
agreement with measured ground- and excited-state properties.
159
Introduction of the Pt bridge reduces electronic coupling compared
to purely organic TAA systems with comparable bridge length, and
thus moves the system even more into the class II range. Such
calculations aid decisively in the assignment of the optical
transitions. We note in passing the conceptual similarities of
these metal-bridged TAA systems to other MV compounds where
essentially organic radical ligands are coordinated to and thus
bridged by a diamagnetic metal centre, such as for example
certain salen-type complexes, where both class II and class III
examples may be found.
41–43,182,183
4.3 Dinitro-aryl radical anions
Dinitro aryl radical anions represent one of the earliest examples
of organic MV systems, studied widely since the 1960s (in fact
before the advent of the concept of mixed-valency), in particular
by EPR spectroscopy.
184–195
They cover both class II and class III,
as well as borderline cases (see below), in spite of being typically
much smaller than the bulky TAA systems discussed above. This
makes them also particularly suitable to evaluate methodological
aspects of electronic-structure methods. Their anionic nature and
the small substituents make them furthermore more sensitive to
specific solvent effects and possibly counter-ion effects, providing
appreciable challenges to computational studies.
We may consider the nitro groups as the redox centres,
albeit the spin densities extend considerably into the aryl
bridge moieties. The extent of electronic coupling depends
strongly on whether the system exhibits a Kekule
´or non-
Kekule
´substitution pattern. In general coupling is stronger
for an odd bond number (Kekule
´) between the redox centres
than for even numbers (non-Kekule
´). In most systems the
charged nitro groups are accessible for solvent coordination.
Indeed there are molecules such as 1,4-dinitrobenzene, 1
(cf. Fig. 2), which exhibit delocalised charge distributions and
vanishing thermal ET barriers (extracted from EPR experiments)
in aprotic solvents,
3,5,196
but class II behaviour and appreciable
Fig. 2 Dinitro radical anions investigated in ref. 3 and 151.
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barriers in alcohol solvents (e.g. 1exhibits free-energy ET barriers
DG*between222kJmol
1
in ethanol and 36 3kJmol
1
in
octan-1-ol
4
). Non-Kekule
´systems like 1,3-dinitrobenzene, 2
(Fig. 2) have sizeable barriers already in polar aprotic solvents,
but the barriers still increase significantly in alcohols, obviously
due to the effects of hydrogen bonding.
68
Due to their relatively small size, dinitro aryl radical anions
have been targeted by a number of quantum-chemical studies
at various computational levels, and with various degrees of
solvent modelling. Gas-phase unrestricted HF and MP2 calcu-
lations were found to suffer from exaggerated spin contamina-
tion. Nelsen, Clark et al. studied 2employing single-excitation
CI within the framework of the semi-empirical AM1 MO
method.
68
Gas-phase optimisations give a symmetrical struc-
ture, in contradiction with the experimentally observed class II
character in polar solvents like acetonitrile. Inclusion of solvent
effects by the COSMO model provided charge localisation for
appropriate dielectric constants.
An interesting study by Yoshida et al. used state-averaged
CASSCF calculations and included solvent effects (for MeCN
and MeOH) by the reference interaction site self-consistent-
field model (RISM-SCF).
124
This is an advanced solvent model
based on averaging over classical Monte-Carlo simulations for
solvent motion that is expected to account also for hydrogen-
bonding effects in alcohols. On the other hand, the neglect
of dynamical correlation effects at CASSCF level is expected
to bias the calculations towards a too localised picture (see
Section 3.1). Indeed, the computed ET barriers at this level
appear strongly overestimated (e.g., for 2in MeCN, a 298 K DG*
value of 60.63 kJ mol
1
was obtained, which is roughly a factor
3 too large). Other CASSCF calculations on 2gave gas-phase
activation energies on the order of 20 kJ mol
1
.
197
This would
be more in line with the solution ESR data. But as the neglected
effects of the polar solvent would increase the barrier
(Section 3.3), and dynamical correlation effects would lower it
(Section 3.1), this is obviously the right answer for the wrong
reason. Other gas-phase CASSCF and MRMP2 calculations on 2
gave very small ET barriers and somewhat non-intuitive energy
profiles.
99
While CASSCF calculations do not cover dynamical
correlation, the treatment of dynamical correlation at MRMP2
level has a strong basis set dependence (Section 3.1). The basis
sets used (6-31G*) appear too small in this context. Those
authors also applied the CDFT method to force localised
structures.
As one might expect, unconstrained standard B3LYP DFT
calculations on such systems are inadequate for class II situa-
tions, as they are biased towards delocalised structures. This
holds true even if solvent modelling is included.
198
One trick in
such a situation to still extract useful ET parameters may be the
use of Koopmans’ theorem for the neutral compound at the
structure of the radical ion.
3,5
The small size of the dinitro aryl radical anions has invited a
detailed study of DFT approaches combined with solvent
models.
151
The protocol described above, based on the BLYP35
functional and continuum solvent models, has been validated
for a test set of six dinitroaromatic radical anions, including
1and 2(Fig. 2). The results were then compared against those
obtained with other functionals, and the novel D-COSMO-RS
solvent model has been evaluated for protic solvent effects. The
range of functionals tested included meta-GGA global hybrid
functionals representing different exact-exchange admixtures
(the Minnesota M05,
199
M06,
200
M05-2X,
201
M06-2X,
200
and the
BMK
153
functional), range-separated hybrids (CAM-B3LYP,
137
oB97X,
155
and LC-BLYP
136
), and double-hybrids (B2PLYP
202
and B2PLYPD
203
). A wide range of ground-state properties were
considered, and TDDFT calculations were applied to the IVCT
excitation (cf. Fig. 3).
Performance of BLYP35 and continuum solvent models in
aprotic solvents was closely comparable to what had been
found previously for TAA-based radical cations (see above).
That is, experimental class II/class III distinctions were reproduced
very well for a given solvent. IVCT excitation energies for class II
systems were under-, those for class III systems overestimated
(cf. Fig. 3), typically by up to 1000 cm
1
(in case of 2, the agreement
between theory and experiment was closer than expected,
8140 cm
1
vs. 8320 cm
1
).
Among the other functionals tested, only the BMK meta-GGA
hybrid (42% exact-exchange admixture) and the CAM-B3LYP
range-separated hybrid (interpolating between 19% at short
Fig. 3 Computed excitation energies with different functionals for the
class III system 1(top) and for the class II system 2(bottom) in two solvents
(C-PCM) compared to experimental values in MeCN (11 000 cm
1
for 1
and 8320 cm
1
for 2).
151
Reprinted with permission from M. Renz, M. Kess,
M. Diedenhofen, A. Klamt and M. Kaupp, J. Chem. Theory Comput., 2012,
8, 4189. Copyright 2012 American Chemical Society.
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range and 65% at long range
107,137
) turned out to be more or less
competitive to BLYP35. Both functionals tended to give slightly
more pronounced localisation (e.g. larger ET barriers) and some-
what larger excitation energies for class II cases. This renders the
BMK functional rather attractive in this context, as it is a
thermochemically competitive functional, in contrast to BLYP35.
The other global hybrids either overdelocalised (M05, M06) or
overlocalised (M05-2X, M06-2X) significantly, and gave rather
unrealistic IVCT excitation energies and ET barriers for class II
systems, reflecting their rather different exact-exchange admixtures
(Fig. 3). The double hybrids also provided a too localised descrip-
tion, consistent with their rather large E
exact
x
admixture (53%).
Interestingly, B2PLYP nevertheless gave excessively low ET barriers,
apparently due to an overstabilisation of the symmetrical transition
states by the MP2-like correlation contribution, and it was the only
functional giving too low IVCT excitation energies for class III
systems (Fig. 3). The two range-separated hybrids oB97X and
LC-BLYP, which both feature 100% exact exchange at long
range, both gave a significantly too localised description,
accompanied by dramatic symmetry-breaking effects at several
of the ET transition states for class II systems. This suggested a
poor description of non-dynamical left–right correlation
effects, as similar but even more pronounced deficiencies were
found for HF wave functions.
151
Apart from the BMK functional,
the MPW1K hybrid functional
204
with similar E
exact
x
admixture
(42.8%) provided promising performance in preliminary tests.
This is interesting, as the functional has shown promise in the
study of transition-metal complexes.
205,206
A striking result was obtained regarding modelling of protic
solvent effects. While standard continuum solvent models are
clearly inadequate to capture the difference between aprotic
and protic solvents (e.g. the dielectric constants of MeCN and
MeOH are almost the same), the recent D-COSMO-RS
model
123,207
performed remarkably well.
151,152
For example, it
gave the experimentally observed switch from class III to class II
behaviour for 1upon going from MeCN to MeOH (see above), as
well as the experimentally observed increase of the ET barriers
for the more clear-cut class II cases 2and 2,7-dinitronaphthalene
radical anion 3by roughly a factor 2.
151
This is notable in view of
the fact that D-COSMO-RS does not explicitly account for the
individual solvent molecules, but brings in hydrogen bonding by
parameterisation within the relatively sophisticated underlying
statistical thermodynamics framework, for the cost of a standard
COSMO calculation. The self-consistent D-COSMO-RS model,
which so far had not been evaluated much elsewhere, thus
appears to provide a very convenient way to include protic
solvent effects into standard quantum-chemical calculations.
4.4 Diquinone radical anions
Diquinone radical anions are a related class of MV systems,
with similar features regarding size and solvent-accessibility, as
the dinitro aryl systems discussed above. Three quantum-
chemical studies with quite different methodologies have
recently focused on the tetrathiofulvalene bis-quinone radical
anion (Q-TTF-Q
)4(Fig. 4).
97,98,140,152
Interest in 4has been
fuelled by the possible importance of the TTF linker as a strong
p-donor bridge for organic molecular electronics.
208
Its NIR
spectrum in a 10 :1 mixture of ethyl acetate and tert-butanol is
dominated by a broad band at around 7700 cm
1
, and the
thermal ET barrier DH* was determined to be about 31 kJ mol
1
by ESR.
208
A class II situation was inferred also in other polar
solvents, but lack of solubility prevented a more detailed study
of the barriers.
Not unexpectedly, gas-phase optimisation at B3LYP level
gave a delocalised class III structure.
97
Wu and van Voorhis
used this system to evaluate the usefulness of their CDFT
approach to force the system to localise.
97,98
Indeed, a
double-well potential is then obtained.
97
Subsequent TDDFT
calculations at the same CDFT level gave a too low IVCT
excitation energy of 4575 cm
1
, which was attributed to the
lack of solvent reorganisation energy in the simulations.
In an independent DFT and CASSCF/CASPT2 study of 4in
various charge states, Ortı
´et al.
209
also operated under the
assumption that the mono-anionic MV form is a class II system
in the gas phase. While B3LYP and PBE0 provided a class III
situation in the gas phase and clearly too low ET barriers in
solution (dimethyl sulfoxide PCM solvent model), symmetry-
breaking already in the gas phase was obtained with BHHLYP
(50% E
exact
x
),
92
M06-2X (54%), and the range hybrids CAM-
B3LYP, oB97X, and oB97XD,
210
with appreciable barriers for
the last two functionals. Consequently, the barriers in solution
were drastically overestimated for oB97X and oB97XD, and still
appreciably too large for BHHLYP, M06-2X, and CAM-B3LYP
(somewhat surprisingly in view of the above discussion, the
CAM-B3LYP and BHHLYP barriers were even somewhat larger
than the M06-2X results).
209
As IVCT excitation energies were
all computed without consideration of solvent effects, they were
inevitably all far too low, except for the CAM-B3LYP data but
including the CASSCF(3,3)/CASPT2/6-31+G** results.
209
Fig. 4 Some diquinone radical anions investigated.
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5. Mixed-valence transition metal
complexes
In view of the importance of MV transition-metal complexes
for the general understanding of ET, based on pioneering
experimental/spectroscopic studies since the 1970s, it is not
surprising that such systems have also been in the focus of an
appreciable amount of computational studies. This section will
cover some of the historical development of such quantum-
chemical work, without claiming to be comprehensive, followed
by more recent, state-of-the-art studies. We will explicitly exclude
computational work on mixed-valency in extended solids.
5.1 The Creutz–Taube ion and its relatives
As the first synthetic transition-metal MV complex with pre-
dominantly delocalised character, and due to its importance in
the early understanding of ET processes between two
transition-metal centres, the famous Creutz–Taube ion
[{Ru(NH
3
)
5
}
2
(m-pz)]
5+
,6, has probably been studied more than
any other MV transition-metal complex, both experimentally
and computationally. In fact, a new Robin–Day class II/III has
been coined specifically for 6(see e.g. ref. 14 and references
therein). We do not intend to cover the extensive literature on
the various different aspects of ET in the Creutz–Taube ion
(vibronic effects, solvent effects, class II/III), which have been
reviewed many times, for example in ref. 14 and 211. In
particular, we will not attempt to evaluate the importance of
vibronic effects, which have been studied in impressive detail
as described in the references cited above. Our focus will be on
the electronic-structure methods and on the effects of environ-
ment. Also, we will not attempt a comprehensive coverage of
quantum-chemical studies, all the way back to the first
extended-Hu
¨ckel MO study.
69
The wide range of methods
applied and their advantages and disadvantages can probably
be illustrated as follows:
The earliest studies at X
a
and related levels did not attempt
to optimise ground states, but focussed generally on electronic
structure and electronic transitions. Given the local DFT and
gas-phase character of these calculations, a clear bias towards
class III behaviour is obvious.
212–214
Indeed, the first DFT-based
structure optimisations used ‘‘pure’’ (LDA and/or GGA-type)
functionals and thus inevitably provided equal Ru–N(pyr) bond
lengths and overall symmetric structures.
215,216
Depending
somewhat on the basis sets and functionals, the optimised
Ru–N(pyr) distances were clearly larger than the experimental
values, but the computed DSCF calculations of excitation
energies gave reasonable results.
216
Hardesty et al.
70
obtained
a similar overestimate of the Ru–N(pyr) distance at B3LYP
hybrid level but much closer agreement with experiment at
MP2 level (these calculations enforced C
2v
symmetry and thus
necessarily equal bond lengths). Subsequent DFT calculations
that included solvent effects at the PCM level gave much
shorter Ru–N(pyr) bonds and thus closer agreement with
experiment.
217,218
Yet, these calculations used symmetry, and
thus a possible symmetry breaking could not be evaluated. It
appears likely that the much shorter bond lengths obtained at
MP2 level were a compensation between sizeable basis-set
superposition errors due to the too small basis sets used at
that time
70
and the neglected environmental effects.
Symmetry breaking at the HF level and a tendency towards a
more delocalised description upon including electron correla-
tion for the Creutz–Taube ion had been mentioned earlier in a
combined semi-empirical CNDO and ab initio CASSCF study.
74
Yet, at that time full optimisations at adequate levels could not
yet be done. Nevertheless, these studies were the first that
incorporated continuum solvent models in the quantum-
chemical treatment. The first full optimisations probably per-
tain to an INDO-based study, used to demonstrate a new set of
INDO parameters for Ru.
82
A somewhat asymmetric structure
was obtained, consistent with the HF-like nature of the single-
determinant INDO wavefunction. Consequently, subsequent
INDO-CISD single-point calculations gave a more delocalised
wave function.
82
The computed IVCT excitation energy was
nevertheless significantly too low.
The probably most sophisticated ab initio computations on 6
were Bolvin’s single-point CASSCF and CASPT2 calculations
(done at a slightly idealised structure derived from crystallographic
data).
67
In accordance with the findings of Broo and Larsson,
74
CASSCF yields rather poor agreement with experimental UV-vis-NIR
data due to the missing dynamical correlation in the ground-state
wavefunction. Subsequent MS-CASPT2 calculations (with PCM
solvent) gave excellent agreement with measured excitation bands
(e.g. 6400 cm
1
compared to the experimental IVCT band at
6370 cm
1
). Interestingly, even transitions accessible only by
magnetic circular dichroism spectroscopy were nicely reproduced,
and EPR g-tensors in good agreement with experiment were
obtained.
30
The aforementioned first DFT calculations including solvent
effects were part of a multi-step procedure aimed at evaluating
the solvent contributions to the reorganisation energy and thus
to the IVCT energy.
217
While this involved conceptually diabatic
localised states, the structure was taken to be delocalised.
This relatively complicated procedure, which involved both
TDDFT and CASSCF steps (with solvent effects included only
in the latter), afforded overall also a good IVCT excitation
energy (6170 cm
1
).
217
Even more sophisticated solvent treatments used classical
molecular dynamics (MD) for explicit water molecules, within a
model approach.
220
MD for the solvent motion was also used at
the PBE0 DFT level for the related but more localised cyano-
bridged ruthenium complex [(H
3
N)
5
Ru-m-NC-Ru(CN)
5
]
,
221
in
the context of simulating the L
3
-edge X-ray absorption data of
this class II system. The comparison of experimental UV-vis-
NIR data and TDDFT calculations at PBE0 and B3LYP level with
COSMO(H
2
O) solvent model for this class II system had been
performed earlier.
222
As part of a study on the importance of rotamers for ET in
MV complexes (Section 5.4), the BLYP35/COSMO(MeCN) proto-
col has recently been applied also to the Creutz–Taube ion.
219
Notable spin contamination (hS
2
i= 0.99) and extensive negative
spin density on the pyrazine ligand were found at this level with
35% E
exact
x
admixture. This is a known feature of such hybrid
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functionals for open-shell transition-metal complexes in cases
when the singly occupied molecular orbital(s) exhibit(s) significantly
metal–ligand antibonding character.
109–111
This is clearly the case for
6(cf. Fig. 5, top right). The overall results for ground-state structure
and IVCT band of 6nevertheless matched experimental observations
rather well. In particular, the computations confirmed a description
as a system exceedingly close to the borderline between class III
and class II. That is, two slightly different Ru–N(pyr) bond
lengths (2.027 Å vs. 2.017 Å, rather close to experimental values
in two different solid-state structures
40
) indicated the onset of
localisation, but the computed spin density remained largely
symmetrical and delocalised (Fig. 5, bottom left). The TDDFT
results for the IVCT excitation energy at this optimised structure
depended somewhat on the treatment of non-equilibrium
solvation, with the CPCM model in Gaussian09 providing the
lower value (E
IVCT
=6250cm
1
) compared to the COSMO
implementation in TURBOMOLE 6.4 (E
IVCT
=7046cm
1
), and
thus somewhat better agreement with experiment. Most notably,
a conformational relaxed scan of the relative orientation of redox
centres and pyrazine bridge at the same BLYP35/COSMO(MeCN)
level gave partial localisation of the spin density onto one of the
Ru centres when the ligand was rotated away from the minimum
structure by 451,i.e. when it eclipses the equatorial ammonia
ligands.
219
This confirms the borderline Robin–Day character of
6and opens questions about the role of conformational motion,
which is addressed further below.
5.2 Polyynediyl-bridged organometallic complexes
Organometallic MV complexes have recently received increased
interest due to their often appreciable thermal and chemical
stability, which may make them suitable as building blocks in
molecular electronics applications. Here we will focus on the
subclass of polyyne-bridged complexes (Fig. 6, left; the
diethynylphenyl-bridged complexes on the right will be dis-
cussed below in Section 5.3), which have been in the focus of
significant computational work and give rise to interesting
spectroscopic features and questions. Moreover, these com-
plexes are considered in the context of long-range ET materials
(molecular wires),
28,50,223,224
and as mimics of linear carbon
allotropes.
225
A variety of reviews and book sections address these
all-carbon bridged MV systems.
28,223,224,226–229
We note in passing
the importance of bridged oligoferrocene complexes.
52,230–233
Comparable to the purely organic MV systems covered in
Section 4, the entire range from weakly coupled class I or II
to strongly coupled class III systems is accessible for these
organometallic examples by chemical modification and by changing
the solvent environment. For example, change of either the metal
centres,inthecaseofthediynediyl-bridged
28,30,47–50,57,224,234–241
and
diethynyl-benzene-bridged complexes,
28,242,243
and/or modification
of the diethynylaromatic bridging ligand (e.g. in iron
33,100,224,244–251
and ruthenium complexes
206,243
),arewaystotunethebehaviour.
While remarkable chain sizes have been achieved synthetically, e.g. a
28 carbon-atom polyynediyl bridge between two platinum centres,
252
we will focus here on the diynediyl-bridged complexes. In these
compounds the choice of the terminal metal redox centre is a
critical determinant of whether a system exhibits localised
(e.g. [{Mo(Z-C
7
H
7
)(dppe)}
2
(m-CRCCRC)]
+
,9)
57
or delocalised
charge distribution (e.g. [{FeCp*(dppe)}
2
(m-CRCCRC)]
+
,7).
28,224,240
Two metal atoms coupled by a ‘‘carbon wire’’ made such
systems attractive targets for computational studies. The majority of
this work dealt with strongly coupled class III systems, where
obviously the standard gas-phase DFT treatments with functionals
like BP86 or B3LYP correctly reproduced the delocalized, symme-
trical ground-state character of the mixed-valence species. Examples
are the class III MV iron complexes [{FeCp(CO)
2
}
2
(m-(C)
n
)]
+
(n=4–8),
models for the related Fe–C
4
–Fe and Re–C
4
–Re monocationic
complexes 7and 10 and their mixed Fe–Re complex,
48
a series
of related diiron and dirhenium molecular wires,
253
diynediyl-
bridged manganese MV systems,
30,241,254
the ruthenium
complex 8,
235
as well as iron/ruthenium and iron/rhenium
analogues of this system.
237
Interestingly, a relatively early HF calculation on the dirhe-
nium MV complex 10 indicated SCF convergence problems.
47
While the DFT calculations were generally consistent with the
delocalised class III situations derived experimentally for the
homodinuclear complexes, some localisation onto the iron
Fig. 5 Isosurface plots of spin density (bottom left, 0.002 a.u.) and b-
SOMO (top right)/b-HOMO (bottom right) (0.03 a.u.) of the Creutz–
Taube ion, 6, calculated at the BLYP35/def2-SVP/COSMO(MeCN) level.
219
Adapted with permission from M. Parthey, J. B. G. Gluyas, M. A. Fox, P.
J. Low and M. Kaupp, Chem. – Eur. J., DOI: 10.1002/chem.201304947.
Copyright @ 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. Fig. 6 Diynediyl-bridged complexes 7–11, diethynylphenyl-bridged
complexes 12–16, and the truncated models 13-Me and 14-Me, which
were used to computationally investigate the influence of different con-
formers on the UV-vis-NIR and IR spectra of 13 and 14.
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centre was computed (B3LYP, gas phase) on the mixed Fe–Re
complex.
48
Yet, a too delocalised spin-density distribution
compared to the experimentally established class II character
49
may be discerned.
Often, the extent of spin density on the carbon bridge vs. the
metal centres has been of central interest. For the dirhenium
complex 10 and analogues with longer bridges up to 20 carbon
atoms, Reiher et al. investigated spin-state energies (also for the
neutral and dicationic complexes) and spin densities at BP86
and partly B3LYP* level (15% E
exact
x
admixture in the latter
case).
255
In addition to the doublet-quartet splitting of the MV
cationic form, the singlet–triplet splitting of the dication was
investigated for the full systems. Not unexpectedly, spin-state
splittings were found to decrease with increasing chain length.
While the doublet ground state of the MV system had some
tendency to localise the spin at the Re centres, the quartet
exhibited more spin on the bridge, in particular for the longer
bridges. The gas-phase nature of the calculations and the
functionals used may exaggerate the delocalization onto the
bridge somewhat.
Computationally studied counterexamples to the predominantly
class III molecular wires discussed above are provided by the weakly
coupled class II dimolybdenum complex 9,
57
by the diruthenium
complex 16,
206
and by some carborane-bridged molybdenum
256
and
ruthenium
205
complexes. Neither B3LYP calculations on 9nor
application of the MPW1Khybridfunctional(E
exact
x
admixture
42.8%) in the gas-phase calculations apparently reproduced the
class II character of this MV cation.
57
For the particularly weakly
coupled carborane-bridged dimolybdenum and diruthenium
systems, MPW1K calculations gave a localised structure and
spin-density distribution, even though solvent effects were not
considered.
205,256
Together with the clear-cut class III examples
discussed above, this shows that, as we move away from the
borderline between class II and class III, some of the short-
comings of the computational treatment become less serious.
5.3 Complexes containing diethynylaromatic bridges
Increasing the length of the all-carbon chains was found to
deteriorate the chemical stability of the polyynediyl-bridged MV
complexes. Introduction of aromatic spacers has therefore
become a welcome means to increase the metal–metal distances
while maintaining good stability and still appreciable, albeit possibly
somewhat weaker, electronic coupling between the metal
centres.
245,257
Substantial efforts have thus been invested into the
study of organometallic complexes containing diethynylaromatic
bridges (cf. Fig. 6, right), including computational work.
Interestingly, some of these dinuclear iron and polyyne-
bridged ruthenium complexes (and some trinuclear iron species)
proved to be sufficiently stable to be studied by scanning
tunnelling microscopy (STM) on a gold surface, including their
mixed-valence states.
33,100,258,259
This allowed their single-
molecule characterisation in such an adsorbed environment,
complementary to the usual spectroscopic studies in solution or
in the solid state. These investigations were accompanied by DFT
calculations to simulate the STM images. Fig. 7 shows examples
of measured and simulated images of two diiron complexes,
namely 12 with a 1,4-diethynylbenzene bridge and 15 with a
1,3-diethynylbenzene bridge (cf. Fig. 6). The stronger electronic
coupling by the para-substituted linker in 12 makes this a
class III – class II borderline case in polar aprotic solution,
246,260
and class III on the surface, as indicated by a relatively symmetrical
image in the latter case (Fig. 7).
33,100
In contrast, the meta-substitution in 15 leads to a class II
situation, again both in polar solution
249
as well as on the gold
surface
33,100
(Fig. 7). Standard BP86 or B3LYP gas-phase calculations
gave delocalised situations for both complexes
33,100,244,249
and thus
failed to reproduce symmetry breaking for 15 or for a corresponding
trimetallic 1,3,5-triethynylbenzene-bridged iron complex.
249
Therefore constrained DFT simulations were subsequently applied.
Ofcoursethepredictivevalueislimited,asthesameconstraints
applied to 12 will also give an asymmetric image.
33,100
Simplified models of 15 and of closely related diethynyl-
pyridine-bridged complexes were also computed at B3LYP level
by Costuas et al.
261
Some structural symmetry breaking could
be observed in some of the optimisations. However, negligible
energy differences (around 1 kJ mol
1
) between localised and
delocalised structures indicated problems in the descriptions,
which is not surprising at the given gas-phase DFT level.
Subsequent CASSCF and MR-CI calculations provided localised
electronic structures, in spite of missing environmental effects.
This is unsurprising given that the CASSCF calculations do not
include dynamical electron correlation, and the MR-CI calcula-
tions recovered very little of it, due to the very small basis sets
used (STO-3G).
261
Similarly to its iron analogue, the 1,3-diethynylbenzene-
bridged ruthenium complex 16 (Fig. 6) exhibits class II beha-
viour in solution. Fox et al. modelled this system and a related
complex with Ru(dppe)
2
Cl end caps, at the MPW1K gas-phase
level.
206
In spite of the neglected solvent effects, partially localised
structures were obtained with this enhanced E
exact
x
admixture, yet
substantially more delocalisation of spin density onto the bridge
was computed compared to analogous iron complexes.
5.4 Rotamers: the importance of conformational motion
The importance of conformations for ET transfer rates has been
acknowledged for a long time in various contexts, way beyond
the scope of the present article, but of course including the field
of MV systems.
117,262–271
One means to restrict conformational
Fig. 7 Comparison of simulated and experimental STM images under
opposite biases for 12 (a, left) and 15 (b, right). CDFT (B3LYP) was employed
to obtain charge localisation.
100
Reprinted with permission from R. C.
Quardokus, Y. Lu, N. A. Wasio, C. S. Lent, F. Justaud, C. Lapinte and S. A.
Kandel, J. Am. Chem. Soc., 2012, 134, 1710. Copyright 2012 American
Chemical Society.
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freedom is the introduction of steric constraints,
270,271
e.g. by
tethering.
272–274
Linkers have been constructed specifically, e.g.
to force certain dihedral angles within the bridge. In not too
restricted situations, the results of spectroscopic experiments
almost always correspond to an average over a sampling of
conformational degrees of freedom.
Here quantum-chemical calculations may aid in the evalua-
tion of the effects of such conformational motion, and a variety
of studies along these lines has been performed.
239,261,275–277
With respect to certain optoelectronic properties, it is in
fact important to describe correctly not only the ground-state
conformational energy surface but also that in crucial excited
states. Shortcomings of TDDFT in case of charge-transfer-type
excited states have been noted,
278,279
and more sophisticated
methods
280
may have to be applied. In this section, we will
focus on conformational effects in the types of organometallic
systems discussed in the two preceding sections but note that
conformational effects are increasingly also in the focus of
quantum-chemical studies on other MV systems. This includes
recent work on organic TAA-based systems (cf. Section 4.1
above) with partly saturated bridges, where the predominant
pathways for ET depend crucially on conformation.
175
Experimental research on conformational effects in organo-
metallic carbon-bridged species of the type discussed above
has been reviewed by Low.
257
We will focus our attention on
the corresponding computational work. As a prerequisite for
studies on binuclear MV systems, a number of computational
studies focused on spin-density delocalisation as a function of
conformation in mononuclear Fe and Ru complexes bearing a
diethynylaromatic ligand that becomes the bridge in corres-
ponding dinuclear species.
54,243,277,281–283
This provides insight
into the coupling between metal d-orbitals and the ligand
p-system as a function of rotation angle.
A number of computational studies have evaluated the
relative orientation of the two end caps in all-carbon-bridged
MV complexes. As for the abovementioned mononuclear com-
plexes, most of these studies used standard functionals and
gas-phase conditions (but see below) and thus are expected to
provide an overall too delocalised description. BP86/gas-phase
calculations on a truncated model of the C
2
-linked complex
[{Ru(dppe)Cp}
2
(m-CRC)]
+239
provided a lowest-energy minimum
structure with a Cp(midpoint)–Ru–Ru–Cp(midpoint) torsion angle
of 551, likely due to direct steric interactions between the end
caps. A transoid minimum (torsion angle 1801)atsomewhat
higher energy was also found. In addition to a main excitation
near 16200 cm
1
, TDDFT calculations for this latter minimum
provide low-intensity peaks at lower energies. However, due to the
relatively large barriers the NIR spectra are likely dominated by
the lowest-energy minimum in this case. Rotamers were also
found in similar BP86/gas-phase computations on trimetallic
ruthenium complexes.
276
The BLYP35/COSMO computational protocol was applied to
the class III complex [{Ru(PPh
3
)
2
Cp}
2
(m-CRCCRC)]
+
,8
(Fig. 6),
158
to evaluate the origin of an additional high-energy
shoulder found in its NIR spectrum (Fig. 8). The latter could not be
explained in the original analysis.
235
Full BLYP35/COSMO(CH
2
Cl
2
)
structure optimisation gave an almost C
i
-symmetric trans con-
former trans-8. TDDFT calculations at the same level provided a
p–p* transition corresponding excellently to the principal com-
ponent in the NIR spectrum (Fig. 8).
A subsequent relaxed scan of the P–Ru–Ru–P dihedral O
from 01to 1801(1801corresponds to trans-8), and structure
optimisation of resulting local minima led to two further
class III rotamers: a cisoid form cis-8 is almost isoenergetic
with trans-8 and provides almost identical transitions. In
contrast, the slightly higher-energy conformer perp-8, in which
the Cp moieties are nearly perpendicular to each other, exhibits
an additional, less intense MLCT excitation at 13 982 cm
1
(Fig. 8). This provides an explanation for the experimentally
Fig. 8 Isosurface plots (0.03 a.u.) of the orbitals involved in the NIR
transitions of 8calculated at the BLYP35/(def2-)SVP/COSMO(CH
2
Cl
2
) level
and experimental spectrum collected during spectroelectrochemical oxi-
dation in CH
2
Cl
2
/0.1 M NBu
4
PF
6
.
158
The excitation around 14 000 cm
1
only gains intensity for rotameric forms, which exhibit approximately
orthogonal disposition of the Cp rings around the RuC4Ru axis. Adapted
with permission from M. Parthey, J. B. G. Gluyas, P. A. Schauer, D. S. Yufit,
J. A. K. Howard, M. Kaupp and P. J. Low, Chem. – Eur. J., 2013, 19, 9780.
Copyright @ 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
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observed shoulder in the NIR spectrum and indicates the
importance of conformation of the redox centres even for such
a simple ‘‘all-carbon wire’’.
In contrast to other examples discussed below, however, all
conformations retain a fully delocalised class III character,
explaining the absence of new vibrational bands and the
relative solvent-independence of the IVCT band. Notably, the
assignment of the NIR shoulder to an MLCT transition of
the perpendicular conformation in 8was experimentally confirmed
by the synthesis and spectroscopic analysis of the tethered
system [{RuCp}
2
(m-CRCCRC)(m-Ph
2
P(CH
2
)
5
PPh
2
)
2
]
+
.This
pseudo-macrocyclic complex is conformationally restricted to a
limited rotamer subspace around the cisoid form. While it
exhibits largely similar spectroscopic properties as cis-8,the
shoulder assigned to perpendicular rotamers is absent in the
NIR spectrum of the tethered complex.
In case of diethynylaromatic bridges, both the relative
orientation of the redox centres and of the bridge may be
important. DFT studies of the 1,4-diethynylbenzene-linked
diiron complex 12 and its heterobimetallic analogue containing
one Fe(dppe)Cp* and one Mo(Z-C
7
H
7
)(dppe) end cap indicated
an appreciable dependence of the spin-density distribution of
the MV cationic form on the relative conformation of the two
redox centres.
57
The closely related 1,4-diethynylbenzene-bridged MV
diruthenium complexes 13 and 14
284,285
(cf. Fig. 6) were experi-
mentally identified as class III systems, but several features in
their spectra raised puzzling questions: (a) similar to the
situation for 8above, the NIR bands exhibited high-energy
shoulders that lacked a convincing explanation; (b) the IR
spectrum exhibited not only the features expected for class III
systems but extra CRC stretching modes and aryl breathing
modes that should be absent for a centrosymmetric complex.
243,285
This motivated a detailed computational study of both the bridge
and redox centre rotation at the BLYP35/COSMO level for 13 and
14,aswellasfortheCreutz–Taubeion6.
219
Two-dimensional
relaxed scans on the truncated model complexes 13-Me and 14-Me,
(dppe replaced dmpe), full optimisations of ground-state minima
for truncated and untruncated complexes, as well as subsequent
TDDFT calculations were carried out.
The 2D relaxed scan for 13-Me (Fig. 9) indicated that the
bridge rotation (angle Y
eff
) has a greater influence on the spin-
density distribution than the redox centre rotation (angle O,
Fig. 10), but that it is also associated with a larger energy
penalty. Yet the entire conformational space covers only an
energy range of less than 30 kJ mol
1
(Fig. 9). Minima are
obtained for the transoid (O= 1801,Y
eff
=01) and cisoid (O=01,
Y
eff
=01) conformations, which exhibit the strongest electronic
coupling due to the optimal overlap between the bridging
ligand p-system and the metal d-orbitals of similar symmetry.
The largest energies are obtained when the bridge is nearly
perpendicular to the optimal orientation and thus the spin
density is localised on one redox centre and the neighbouring
ethynyl unit (Fig. 10). When only the relative conformation of
the two redox centres is varied, while the bridge is kept in its
optimal position, the diethynylphenyl part of the molecule
bears significant amounts of spin density for all rotamers. In
contrast the relative contribution of the two redox centres gets
more and more shifted towards one metal moiety due to
the decreased electronic coupling. The charge-localised local
maximum in conformer space is less than 15 kJ mol
1
above
the minima (Fig. 9).
In contrast to 8,13 exhibits a solvatochromic NIR shoulder,
which gains intensity in the more polar acetone/CH
2
Cl
2
(6:1)
mixture. This can be explained by comparing the BLYP35/
COSMO(CH
2
Cl
2
) TDDFT results for the entire relaxed scan of
13-Me. A Boltzmann-weighted averaging of computed excita-
tion energies and intensities in combination with Gaussian
broadening gives good agreement with the measured band
envelope. The shoulder arises from structures with partially
localised class II character, while the main band gains intensity
mainly from class III type conformers. Hence the shoulder
exhibits the typical solvent dependence of class II systems,
while the main feature stays nearly unchanged in different
solvents.
Full optimisation of 13 provided three minima, trans-13,cis-
13, and perp-13. Both trans-13 and cis-13 give rise to only one
intense p–p* excitation corresponding to the main peak in the
NIR spectrum (TDDFT-BLYP35/COSMO(CH
2
Cl
2
) level), whereas
the partially localised (class II !) perp-13 conformer provides
excitations that may be associated with the high-energy
shoulder.
219
This has been further confirmed by harmonic
vibrational frequency analyses of the three conformers. While
trans-13 and cis-13 account only for two closely spaced n(CRC)
stretching bands, due to its non-symmetrical localised struc-
ture perp-13 explains the occurrence of more strongly split
n(CRC) bands, as well as of an aryl breathing mode.
243
A similar outcome was obtained for the related complex 14.
An even shallower conformational potential energy landscape,
and somewhat less pronounced charge localisation has been
computed. Again the low-energy shoulder in the NIR spectrum
arises from partially localised portions of the conformational
space, and vibrational analysis of fully optimised structures
Fig. 9 Computed potential energy surface of truncated 13-Me (BLYP35/
COSMO(CH
2
Cl
2
) level).
219
Orepresents the dihedral angle for the redox
centres and Y
eff
for the bridge rotation. Adapted with permission from
M. Parthey, J. B. G. Gluyas, M. A. Fox, P. J. Low and M. Kaupp, Chem. –
Eur. J., DOI: 10.1002/chem.201304947. Copyright @ 2014 WILEY-VCH
Verlag GmbH & Co. KGaA, Weinheim.
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accounts for the simultaneous observation of features in the IR
spectra indicating both class II and class III behaviour.
219
A similar relaxed scan for the classical Creutz–Taube ion 8
also revealed structures exhibiting both delocalised and localised
spin densities.
219
However, in this case only one conformational
minimum has been found. Moreover, contributions from the
partially localised regions of conformational space to the NIR
band were found to be too small to affect a notable modification
of the overall band shape.
6. Summary and outlook
Close to the borderline between class II and class III mixed-valence
systems, the challenges for both experimental and computational
classification are obviously much larger than for more clear-cut
delocalised or localised situations. Quantum-chemical methods that
allow us to move very close to the borderline and allow essentially
quantitative predictions have emerged only very recently, and they
clearly will require further refinement. Nevertheless, the examples
discussed in this article demonstrate the remarkable potential of
quantitative quantum-chemical treatments of MV systems.
Due to their favourable scaling with system size, DFT
approaches have become the most widely used methods also
in this field. Yet, most standard functionals are biased towards
a too delocalised description, whereas some other highly
advertised functionals overshoot into the overlocalised direc-
tion. Judicious inclusion of exact exchange into the functional
is mandatory if DFT is to be applied successfully in this field.
Global hybrid functionals with about 30–45% exact-exchange
admixture so far turned out to provide a reasonable compro-
mise, when augmented by an appropriate treatment of (solvent
or solid-state) environmental effects, as the latter often favour a
more localised charge and spin distribution (see below). Then
both DFT treatment of ground-state structures and properties
and subsequent TDDFT calculations of excitation spectra pro-
vide valid connections to experimental observation and even
predictive quality. Further improved functionals, either of the
range-separated hybrid variety, local hybrid functionals with
position-dependent exact-exchange admixture, or even more
sophisticated constructions, may well provide further enhanced
accuracy. Post-Hartree–Fock approaches are an obvious further
direction to pursue in this field. However, many of the applica-
tions of such methods reviewed here suffered in particular from
far too small single-particle basis sets. In view of the steep
basis-set dependence of the post-HF methods, these lead to a
dramatic underestimation of dynamical correlation effects. As
the latter tend to favour more delocalised electronic structure,
too small basis sets tend to bias the calculations towards a too
localised description. Sometimes compensation with errors
arising from neglect of environmental effects (which tends to
favour too delocalised charge and spin) might then even
provide the qualitatively right answer for the wrong reason. In
many MV systems examined so far, the consideration of a
multi-configurational character of the ground-state wave func-
tion was less important than expected and less important than
the influence of the other factors discussed, in particular those
due to the environment. It will thus be interesting to evaluate in
more detail than done so far the usefulness and performance of
predominantly single-reference approaches like coupled-
cluster theory for MV systems. A caveat might apply to the
symmetrical transition states for thermal electron transfer in
class II systems, where symmetry breaking of the underlying
UHF wave functions might present a more serious difficulty.
Related problems have been identified for double-hybrid func-
tionals or for certain density functionals with too excessive
exact-exchange admixture.
The importance of environmental effects for charge localisa-
tion can hardly be overemphasised. As most spectroscopic
experiments on MV systems tend to be performed in polar
solvents, solvent models become central to a reasonable com-
putational description, already during ground-state optimisation.
Continuum solvent models appear to provide a straightforward
first approximation for most aprotic solvents, provided that
specific solvent effects like hydrogen bonding or the exchange
of solvent molecules within the coordination sphere of a
transition-metal centre are not of overriding importance. In many
computational studies, the importance of non-equilibrium solva-
tion effects during TDDFT computations of vertical excitations
has been highlighted. We have also emphasised that a change of
solvent polarity may change the character of an MV system
fundamentally. This is known from spectroscopic experiments
with different solvents, and it is supported strongly by the most
recent computational studies. In fact, many class II MV systems
with reasonably strong electronic coupling are probably localised
only due to the solvent influence and might represent class III
cases in the gas phase or in a non-polar environment. This should
be kept in mind when assessing the suitability of a given
electronic-structure method in correctly describing MV systems.
In fact, hydrogen-bonding from protic solvents may localise even
systems that are still delocalised in polar aprotic solvent environ-
ments. Continuum solvent models fail to cover hydrogen-bonding
effects adequately and thus are not suitable to distinguish suffi-
ciently between such situations. One may then have to augment
the electronic-structure calculations by explicit solvent modelling,
Fig. 10 Spin density isosurface plots (0.002 a.u.) of truncated 13-Me for
different points on the potential energy surface (BLYP35/COSMO(CH
2
Cl
2
)
level; cf. Fig. 9 for the potential energy surface).
219
Adapted with permis-
sion from M. Parthey, J. B. G. Gluyas, M. A. Fox, P. J. Low and M. Kaupp,
Chem. – Eur. J., DOI: 10.1002/chem.201304947. Copyright @ 2014
WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
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which usually involves time-consuming molecular dynamics or
Monte-Carlo simulations, e.g. within a QM/MM framework. Very
few successful applications of such approaches to MV systems
have been reported so far, e.g. within the RISM-SCF framework.
Interestingly, the much cheaper Direct-COSMO-RS approach also
appears to provide a reasonably faithful description of the influ-
ence of protic solvents in this context. An area requiring substan-
tially more efforts in the future is how to appropriately include
counter-ion effects into such quantum-chemical treatments of
electron transfer in MV systems.
Finally, this article has emphasised the importance of con-
formational motion for the coupling between the redox centres.
While conformational effects on electron transfer have been
appreciated for a long time, a quantitative computational
procedure to model them had been lacking. Recent computa-
tional studies with appropriate methodology showed how dif-
ferent thermally accessible conformations may alter the
spectral characteristics of organic or inorganic MV systems.
In case of a series of organometallic diruthenium complexes,
the computations demonstrated that conformational motion
may even average to some extent localised and delocalised
electronic and molecular structures, leading us beyond the
traditionally more one-dimensional understanding of the
Robin–Day classification scheme.
Acknowledgements
Special thanks are due to Manuel Renz, mixed-valence pioneer
in the Kaupp group. Paul J. Low and Christoph Lambert are
acknowledged for the excellent collaborations and discussions.
The authors’ own work in this field has been supported by DFG
project KA1187/13-1, by the Berlin DFG cluster of excellence on
‘‘Unifying Concepts in Catalysis’’ (UniCat), and initially by DFG
graduate research school GRK 1221 in Wu
¨rzburg. MP is grateful
to UniCat, the Berlin International Graduate School of Natural
Sciences and Engineering (BIG-NSE), and the German Aca-
demic Exchange Service (DAAD) for funding.
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