A conceptual DFT approach for the evaluation and interpretation of redox potentials.

Abstract

Conceptual DFT aims at describing the properties of molecules in interactions by using chemical reactivity descriptors. Herein, the redox behaviour of a given species, as quantified by the redox potential, is linked to DFT-based descriptors. We made use of a hierarchical decomposition of the corresponding half-reactions into one-electron reduction, protonation, dissociation and water-forming or dissociation reactions. Most of these reactions can be readily described through reactivity descriptors, such as the electrophilicity, nucleofugality and electrofugality, as defined in conceptual DFT. The final expression linking the corresponding free energy changes to the redox potential seems to give correct predictions for the redox potentials of bromo, chloro and nitro oxo acids in the gas phase, as in a polarised continuum model.

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DOI: 10.1002/chem.200601896
A Conceptual DFT Approach for the Evaluation and
Interpretation of Redox Potentials
Jan Moens, Paul Geerlings,* and Goedele Roos[a]
Introduction
Computational chemistry has evolved to the point that it is
sometimes competitive to experiment to obtain precise
values for certain molecular properties. Density functional
theory (DFT)[1]has played a predominant role in this evolu-
tion in the last decade.[2,3]Moreover, by giving precision to
widespread but often rather vaguely defined concepts, such
as electronegativity, hardness and softness, it affords a non-
empirical, numerical evaluation of these concepts. This con-
ceptual DFT branch, as termed by its protagonist Parr,[4]
provoked an avalanche of papers[5,6]in which these concepts
were used both qualitatively and quantitatively for a variety
of molecular properties. Applications to “chemical reactivi-
ty” have played a key role in this field since the 1980s and
early 1990s, and great interest was paid (also by our group)
to a variety of classical organic reactions (electrophilic sub-
stitutions,[7]nucleophilic additions,[8]concerted reactions,[9]
radical reactions,[10]and acid–base reactions[11]; for a review,
see references [5,12]). The study of redox reactions, which
often involve complex reaction mechanisms, fell until re-
cently out of the scope of conceptual DFT. Nonetheless, re-
actions in which electrons are consumed are clean examples
of the change in energy of an atom or molecule with the
number of electrons. This is one of the key relations on
which reactivity indices are based in the framework of con-
ceptual DFT.[13]It is, therefore, surprising that the study of
these reactions is still an underdeveloped field. The central
problem in the description of electrochemical reactions is
that most of these processes take place on the surface of the
electrodes. Modelling these types of reactions through quan-
tum chemical means should include diffusion and adsorption
processes at the electrode surface, although this model can
be simplified if we limit the study to reactions in solution.
As a consequence, the electrodes accomplish the role of cat-
alytic surfaces. Their effect will be merely of a kinetic in-
stead of thermodynamic nature. Hence, the calculation of
the redox potential will not depend on the nature of the
electrodes, which can be replaced by a fictitious electron re-
servoir. In this study, we approximated this reservoir
through an ideal sea of electrons with zero chemical poten-
tial. As a result, the reservoir behaves as a perfect electron
donor. In recent years it has been shown that accurate calcu-
lations of redox potentials are possible by using ab initio
methods.[14–20]The aim of the present article is not so much
another accurate calculation of redox potentials, but the in-
troduction of a methodology based on descriptors from con-
ceptual DFT, which affords a chemical understanding of the
energetics of the redox process. We propose a strategy
which makes it possible to link conceptual DFT and compu-
tational electrochemistry with the aid of a theoretical de-
Abstract: Conceptual DFT aims at de-
scribing the properties of molecules in
interactions by using chemical reactivi-
ty descriptors. Herein, the redox be-
haviour of a given species, as quanti-
fied by the redox potential, is linked to
DFT-based descriptors. We made use
of a hierarchical decomposition of the
corresponding half-reactions into one-
electron reduction, protonation, disso-
ciation and water-forming or dissocia-
tion reactions. Most of these reactions
can be readily described through reac-
tivity descriptors, such as the electro-
philicity, nucleofugality and electrofu-
gality, as defined in conceptual DFT.
The final expression linking the corre-
sponding free energy changes to the
redox potential seems to give correct
predictions for the redox potentials of
bromo, chloro and nitro oxo acids in
the gas phase, as in a polarised continu-
um model.
Keywords: density functional calcu-
lations · electrophilicity · nucleofu-
gality · oxo acids · redox chemistry
[a] J. Moens, Prof. Dr. P. Geerlings, G. Roos
Eenheid Algemene Chemie, Faculteit Wetenschappen
Vrije Universiteit Brussel (VUB)
Pleinlaan 2, Brussels (Belgium)
Fax: (+ +32)2-629-3317
E-mail: pgeerlin@vub.ac.be
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composition scheme constructed from recurrent steps, such
as one-electron reduction reactions, protonation reactions
and dissociation reactions. Central in this approach is the
idea of writing a thermodynamic property, such as the redox
potential, as a sum of reactivity descriptors from conceptual
DFT. The value of each descriptor will then measure the
propensity of a certain chemical process to take place. We
should, however, emphasise that this reaction scheme is not
based on any kind of kinetic analysis. In our approach, we
focused on the properties of single molecules involved in re-
duction half-reactions, and tried to define a theoretical de-
scriptor for describing electron uptake by the recipient. Re-
actions in which charge transfer takes place will be de-
scribed here through the use of a complete set of reactivity
descriptors, namely the electrophilicity,[21,22]nucleofugality[23]
and electrofugality,[23]as recently introduced in the context
of conceptual DFT.[4]Each reactivity index is capable of
measuring the susceptibility of a molecule or molecular frag-
ment to a certain chemical process. Central in this study is
the use of electrophilicity to quantify the energy change by
electron uptake. This descriptor will act as an approximation
of the adiabatic electron affinity, which gives an exact de-
scription of one-electron reduction reactions. The nucleofu-
gality and electrofugality will figure here as measures of the
leaving-group capacity of the nucleofuges and the electro-
fuges. Although the term nucleofugality has been known for
several decades,[24]it is only recently that Ayers et al. have
proposed a rigorous theoretical foundation and an elegant
working equation the use of which has also been advocated
by the present authors.[25]In this context, the term electrofu-
gality arose in a natural way from the definitions of electro-
philicity and nucleofugality.
Redox reactions include a vast class of reaction types,
varying from simple cases in which the composition of reac-
tant and product remains the same to complex reactions in
which the composition changes drastically. Here, we concen-
trate on reduction reactions of the oxo acids, such as chloro,
nitro and bromo oxo acids. These species form a coherent
group with molecules that have a certain analogy in the
form of their redox reactions and for which electrochemical
properties are well described on an experimental basis.[26]
The nitro oxo acids, however, do not behave identically to
the chloro and bromo oxo acids, and can therefore be used
to test the transferability of the applied method.
Electrochemical reactions are highly dependent on the en-
vironment. It is therefore extremely important to take the
solvent adequately into account. Several different ap-
proaches have already been studied to deal with solvation
effects, such as molecular simulations,[27]Langevin dipole
moments,[28]integral equation techniques[29]and dielectric
continuum methods.[30–32]The dielectric continuum method
is, in regard to the other methods, broadly applicable and
capable of calculating the solvation energy with high accura-
cy.[31]Herein, we therefore used such an approach with the
polarised continuum model (PCM) as a dielectric continuum
method.[33]
Theoretical Background
Conceptual DFT provides sharp definitions of a number of
reactivity descriptors for atomic and molecular systems, thus
helping us to gain insight into the chemical properties of
these systems. Here, the electrophilicity, nucleofugality and
electrofugality will be used as reactivity indices for the de-
scription of the electrochemical process.
Electrophilicity: A thermodynamic interpretation of the
electrophilicity has been given by Parr et al. as a validation
of the qualitative suggestion made by Maynard et al.[34]The
electrophilicity is a measure of the ability of a molecule to
accept electrons from a perfect electron donor (that is, a sea
of electrons of zero chemical potential and zero hardness at
zero temperature). On the basis of a second-order model for
the variation of the energy versus the change in the number
of electrons DN with constant external potential n(r) (that
is, the potential due to the nuclei) the energy change is as
follows:
DE ¼ mDN þ1=2hðDNÞ2
ð1Þ
with m, the chemical potential, and h, the chemical hardness,
defined by m=(@E/@N)nand h=(@E2/@N2)n. The system will
become saturated with electrons when DE/DN equals zero.
Equation (1) reduces to DE=?m2
ing gain in energy has been identified as the electrophilicity,
w, of the system:
2h, in which the correspond-
w ¼m2
2h
ð2Þ
In a finite difference approximation, yielding m to ?
and h to I?A,[3]one obtains:
?
IþA
2
?
w ¼ðI þ AÞ2
8ðI?AÞ
ð3Þ
wherein I and A are the ionisation potential and the elec-
tron affinity, respectively.
The electrophilicity encloses on the one hand, the tenden-
cy of electrons to escape the equilibrium system with a
factor m2, while the chemical hardness can be seen as the re-
sistance to electron transfer. As shown in Equation (3), the
electrophilicity and electron affinity are correlated but not
equal. While the electron affinity quantifies the energy
change due to the uptake of a single electron, w is associat-
ed with the energy lowering for maximal electron flow.
Nucleofugality and electrofugality: The quality of a leaving
group can be quantified through the use of the nucleofugali-
ty and electrofugality concepts. A group will act as a good
nucleofuge when it readily accepts an entire electron from a
system. As compared to the electrophilicity, the nucleofuge
is forced to accept an entire electron upon dissociation,
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while an electrophile just has to accept a “piece” of an elec-
tron from the reservoir.[35]Therefore, the nucleofugality can
be seen as a sort of activation energy that has to be over-
come upon dissociation. Here, we used the definitions of
electrofugality and nucleofugality as proposed by Ayers
et al.[23]They considered these indices as fundamental prop-
erties of the leaving group, thereby ignoring the complica-
tions associated with intermolecular and intramolecular in-
teractions.[36,37]The nucleofugality is defined as a measure of
the relative stability of an electron acceptor of charge q?1
in comparison with the acceptor fragment with charge q+ +
Dqidealin the presence of a perfect electron donor. The ex-
pression for this energy change is:
DEnucleofuge¼ Eðq?1Þ?Eðq þ DqidealÞ¼ ?A þ wðqÞ¼ðI?3AÞ2
8ðI?AÞ
ð4Þ
Similarly, the electrofugality is a measure of the relative sta-
bility of an electron donor with charge q+ +1 in comparison
with the donor fragment of charge q+ +Dqidealin the presence
of a perfect electron donor:
DEelectrofuge¼ Eðq þ 1Þ?Eðq þ DqidealÞ ¼ 1 þ wðqÞ ¼ð3I?AÞ2
8ðI?AÞ
ð5Þ
expressed in terms of ionisation potential and electron affin-
ity of the electrofugal or nucleofugal fragment.
Redox potentials: The most appropriate way of calculating
the redox potential is by using a thermodynamic cycle link-
ing the process in the gas phase with that in solvent.[14–20]
The calculation of the Gibbs free energy is summarised in
Equations (6) and (7) which show the thermodynamic cycles
for the redox potential of oxo acids in the case of an acid
[Eq. (6)] and basic [Eq. (7)] environment, respectively.
DG?
gasis the Gibbs free energy in the gas phase, DG?
the Gibbs free energy in the aqueous phase and DG?
aqis
solvis the
solvation Gibbs free energy. The standard Gibbs free energy
of each state in the gas phase is obtained by using Equa-
tion (8):
DG?
gas¼ E0Kþ ZPE þ DDG0!298K
ð8Þ
The energy at 0 K (E0) is calculated by using DFT at the op-
timum geometry. Zero-point energies (ZPEs; unscaled) and
thermal contributions (DDG0!298K) together with entropies
have been used to convert internal energies to Gibbs free
energies at 298.15 K.[38]Hereby, we assume that reacting
species behave as ideal gases within the rigid rotator–har-
monic oscillator approximation. In Equation (8), an extra
term should be introduced to convert the DG?
1 atm to 1m:
gasstate from
DG?
gasð1mÞ ¼ DG?
gasð1 atmÞ þ RT lnð24:46Þ
ð9Þ
The connection between the gas and aqueous phases is
made through the calculation of the solvation Gibbs free
energy of the specific species. In this study, we used a polar-
ised continuum approach (PCM-UAHF; UAHF: united
atom Hartree–Fock) to describe the solvent and the interac-
tions with the solute. The DG?
Equation (10):
solvvalues were computed from
DG?
solv¼ G?
aq?G?
gas
ð10Þ
in which G?
solution and G?
take into account small changes in geometry when going
from gas to solvent, we reoptimised the geometry of the
molecule in PCM.
Through the calculation of the Gibbs free-energy change
of the complete reaction, the standard redox potential can
be determined through the Nernst equation [Eq. (11)]:
aqis the total Gibbs free energy of the system in
gasis the equivalent quantity in a vacuum. To
E?¼ ?DG?
redox,aq
nF
ð11Þ
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in which n is the number of exchanged electrons and F the
Faraday constant. The calculated value of E8 8 is thereby rela-
tive to the reduction potential of a reference electrode.
Here we will use the normal hydrogen electrode (NHE):
HþðaqÞ þ e?!1=2H2ðgÞ
ð12Þ
with an associated free energy change of ?4.44 eV.[39]
Methodology
Studied redox reactions: A survey of the redox reactions
used in this study can be found in Table 1.[26,40]Central to
the idea of this article is the possibility of dividing a global
reduction reaction in a couple of recurrent steps. These
steps will be described energetically and for some of them
we will use reactivity descriptors from conceptual DFT. It
seems at first sight that this reaction path is chosen arbitrari-
ly. Nevertheless, we will show that this strategy could be
used in each reaction and is also chemically justified. In the
following paragraphs, our approach will be elucidated and
the necessary reaction steps will be introduced.
Decomposition scheme: The global reaction will be divided
into one-electron reduction reactions, protonation reactions,
dissociation reactions and the formation or dissociation of
water. Concerning the one-electron reduction reactions, the
chemical composition of reagent and product remains the
same during this reduction. As mentioned in the Introduc-
tion, the electrophilicity will be used as an approximation to
the adiabatic electron affinity. In comparison with the elec-
tron affinity, the electrophilicity has major advantages for
the description of electron uptake. First of all, the possibility
of making this descriptor local (through the Fukui func-
tion[41–43]) offers the opportunity to look at the position in
the molecule, which is reduced. Secondly, this descriptor
contains extra information about the system?s polarisability,
which is intimately related to its chemical softness. Thirdly,
it conceals inherently the system?s electron affinity through
the presence of the chemical potential in its definition. The
contribution to the reaction Gibbs free energy could then be
written as:
DG?/ ?w
ð13Þ
An electron uptake will be energetically more favourable
when the system possesses a higher electrophilicity. Neglect-
ing the contribution of the entropy change during reaction,
Equation (13) could be rewritten as an equality:
DG?¼ ?aw
ð14Þ
by introducing a proportionally constant a, which will be
considered as a parameter in this work.
Followingtheone-electron
protonation reaction will be interpreted as the inverse of an
acid dissociation reaction. The energy change of this reac-
tion can be described through the pKaof the conjugated
acid:
reductionreactions,the
DG?¼ ?RT ln
?1
Ka
?
ð15aÞ
DG?¼ RT lnðKaÞ and DG?¼ ?2:303RT pKa
ð15bÞ
The pKacalculation is a troublesome and highly demanding
computational burden and, therefore, we chose to calculate
the proton affinity and calibrate it to known experimental
pKa values. This method produces Gibbs free energy
changes that are of the correct order in size and gives the
correct trends in pKavalues of the oxo acids.
For the dissociation step, the following reaction can be
taken as an example:
HClO2
?! ClO þ OH?
ð16Þ
Table 1. Survey of the studied reduction reactions of chloro, nitro and bromo oxo acids with their corresponding experimental redox potentials.
Chloro oxo acidsExptl E8 8 [V]Bromo oxo acidsExptl E8 8 [V]
acidic environment
?+ +3H+ ++ +2e?!HClO2+ +H2O
2) HClO2+ +2H+ ++ +2e?!HClO+ +H2O
3) HClO+ +H+ ++ +2e?!Cl?+ +H2O
Chloro oxo acids
basic environment
?+ +3H2O+ +6e?!Br?+ +6OH?
?+ +2H2O+ +4e?!BrO?+ +4OH?
12) BrO?+ +H2O+ +2e?!Br?+ +2OH?
Nitro oxo acids
1) ClO3
1.214
1.645
1.482
10) BrO3
11) BrO3
0.61
0.492
0.761
basic environment
?+ +H2O+ +2e?!ClO?+ +2OH?
5) ClO?+ +H2O+ +2e?!Cl?+ +2OH?
6) ClO3
acidic environment
?+ +4H+ ++ +3e?!NO+ +2H2O
14) HNO2+ +H+ ++ +e?!NO+ +H2O
15) NO3
4) ClO2
0.66
0.81
0.33
13) NO3
0.957
0.983
0.934
?+ +H2O+ +2e?!ClO2
Bromo oxo acids
?+ +2OH?
?+ +3H+ ++ +2e?!HNO2+ +H2O
Nitro oxo acids
acidic environment basic environment
?+ +H2O+ +2e?!NO2
17) NO2
7) HBrO+ +H+ ++ +2e?!Br?+ +H2O
8) BrO3
9) BrO3
1.331
1.423
1.447
16) NO3
?+ +2OH?
0.01
?0.46
?+ +6H+ ++ +6e?!Br?+ +3H2O
?+ +5H+ ++ +4e?!HBrO+ +2H2O
?+ +H2O+ +e?!NO+ +2OH?
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The energy change during this reaction is described through
the electrofugality and nucleofugality. Here, the nucleofugal-
ity of the leaving group, OH?, and the electrofugality of
ClO have to be calculated. These descriptors both relate to
the thermodynamic stability of the nucleofuge/electrofuge,
and thus the contribution to the Gibbs free energy is de-
scribed as follows:
DG?¼ bðDEelectrofugeþ DEnucleofugeÞ
ð17Þ
This equation introduces a proportionally constant b that
will be considered as a parameter in this work.[44]Through-
out, OH?will systematically be used as a leaving group in
the dissociation reaction. A point of criticism would be that
OH?is not an appropriate leaving group and therefore
other groups, such as H2O, should be considered. Let us un-
derline that the nucleofugality of OH?is merely used as a
constant-energy contribution throughout the different reac-
tions. If H2O was used as the leaving group instead of OH?,
the electrofuge would remain the same and as a conse-
quence also its electrofugality. As a result, Equation (17)
would only change with a constant term.
The last reaction step that has to be introduced into the
reaction scheme is the formation or dissociation of water. In
an acid environment, dissociation reactions producing OH?
ions would result in a net production of OH?ions. To pre-
vent this net production, reactions wherein water is formed
needed to be included in the global decomposition scheme.
In a basic environment, a net consumption of H+ +will take
place; therefore, an extra dissociation of water has to be in-
cluded. The contribution to the Gibbs free energy is then
DG8 8=?RTlnACHTUNGTRENNUNG(KW/55.56) when water dissociates and DG8 8=
?RTlnACHTUNGTRENNUNG(55.56/KW) when water is produced, with KWequal to
1.00?10?14at 298.15 K.
The total energy change on the basis of these different re-
action steps becomes:
DG?¼ ?a
X
diss:
red:
i
wi?2:303RT
X
prot:
i
pKai
þb
X
i
DEleaving,i?jðRT lnðK0ÞÞ
ð18Þ
in which the summation runs over the number of reduction,
protonation and dissociation reactions, respectively. The
factor j equals the number of water dissociation or forma-
tion reactions in the reaction scheme. In this equation a and
b are two parameters and K’ is equal to KW/55.56 when
water is dissociated or 55.56/KWwhen water is formed. The
standard redox potential then becomes:
E?¼
1
nF
?
a
X
i
wiþ 2:303RT
X
i
pKai?bDEleavingþ RT lnðK0Þ
?
ð19Þ
On the basis of the foregoing discussion of the various reac-
tion steps, it must be clear that this scheme leaves many
ways open for writing a reaction scheme of a reduction reac-
tion. However, a number of considerations based on chemi-
cal intuition significantly reduce the possibilities and leave
just one possibility open to use as reaction scheme. In order
to propose a design for a reaction scheme that can be used
in each redox reaction, the identification of the types of
starting products is important. As can be seen from Table 1,
anions and neutral molecules are starting products from
which further reaction takes place. Defining rules for these
species will consequently lead to rules for cations.
Anions, cations and neutral molecules: The basic reactions
for anions, cations and neutral molecules in the reaction
scheme considered are proposed and analysed. An anion is
considered to dissociate with the formation of an OH?
group and a corresponding electrofuge if it possesses a
proton. Otherwise, the formation of OH?as leaving group is
impossible. We consider protonation reactions of an anion
when it does not already possess a proton. Although this
principle seems weak at first sight, it will result in a signifi-
cant simplification of the problem. Protonation reactions
could find place on different positions in the molecule (such
as on an OH group or an oxygen atom). Taking these possi-
bilities into account will only result in a more complex prob-
lem. Reduction reactions for anions are not considered be-
cause it will force a negative system to accept another elec-
tron.
For neutral molecules neither protonation nor dissociation
reactions are considered. A dissociation reaction will result
in a positive electrofuge, which may cause problems when
forming reagents such as Cl+ +. These reactions form products
which are chemically improbable. A protonation reaction
will result in a positively charged molecule. As a conse-
quence, subsequent reactions must be a reduction or dissoci-
ation reaction. However, in the case of H2ClO+ +this would
lead to dissociation reactions for which the only possible
leaving group will be water. We wanted to preserve OH?as
a transferable leaving group over the different reaction
schemes, and therefore the protonation of neutral molecules
is not considered. Reduction reactions are the only reactions
examined for neutral molecules.
As a consequence of the previous discussion about anions
and neutral molecules, it is impossible to generate positively
charged molecules. Therefore, no rules have to be set for
cations. These basic principles are summarised in Table 2.
Taking these principles into account, one possible reaction
scheme remains which can be used in each reaction of the
studied oxo acids. The resulting scheme for the case of the
reduction of chlorite in acid and basic environments is given
in Table 3.
It is, however, possible to construct new pathways if other
choices are made in Table 2. As the Gibbs free energy is a
state function, one is free to select the reaction path. Con-
sider, for example, a protonation step for neutral molecules;
this will lead to the formation of species such as R?H2O+ +.
These structures turned out to give computational problems
during optimisation and therefore were not used to build up
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