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Laureates: Junior Prizes of the SCS Fall Meeting 2020 CHIMIA 2021, 75, No. 4 311
doi:10.2533/chimia.2021.311 Chimia 75 (2021) 311–318 © S. A. Grimmel, M. Reiher
On the Predictive Power of Chemical
Concepts
Stephanie A. Grimmel§ and Markus Reiher*
§
SCS-DSM award for best virtual poster in Computational Chemistry
Abstract: Many chemical concepts can be well defined in the context of quantum chemical theories. Examples
are the electronegativity scale of Mulliken and Jaffé and the hard and soft acids and bases concept of Pearson.
The sound theoretical basis allows for a systematic definition of such concepts. However, while they are often
used to describe and compare chemical processes in terms of reactivity, their predictive power remains unclear.
In this work, we elaborate on the predictive potential of chemical reactivity concepts, which can be crucial
for autonomous reaction exploration protocols to guide them by first-principles heuristics that exploit these
concepts.
Keywords: Automated mechanism exploration · Chemical concepts · Conceptual DFT · Reactivity prediction
Stephanie A. Grimmel completed her BSc stable intermediates, and products that are connected by transition
in Chemistry in Karlsruhe (Germany) in states. In the most basic formulation of transition state theory, it is
2016. Afterwards, she studied Physics necessary to identify the stationary points, i.e. local minima and
and Chemistry in Amsterdam (The first-order saddle points, on a potential energy surface to arrive
Netherlands), Rome (Italy) and Lyon at such a reaction network at full conformational resolution.
(France). For her master’s thesis she worked The mathematical definition of the potential energy surface
on the optimization of kinetic energy may be taken from the Born-Oppenheimer approximation which
functionals for subsystem DFT under the freezes the nuclear motion compared to that of the electrons and
supervision of Prof. Dr. Lucas Visscher at introduces the electronic energy. The electronic energy depends
Vrije Universiteit Amsterdam. Since 2018 on the positions of the nuclei frozen in space. It can serve as the
she has been a PhD student in the Theoretical Chemistry group of hypersurface for the description of reactive as well as diffusive
Prof. Dr. Markus Reiher at ETH Zurich. Her work revolves around processes. Quantum chemistry has provided routine tools to
the development of first-principles heuristics for automated calculate and explore such surfaces,[1–3] up to the point where
reaction space exploration. this is possible in an automated manner[4–6] in order to provide
the full detail of a chemical process in terms of thousands of
interconnected elementary steps. With subsequent inclusion
Markus Reiher received his PhD in of nuclear motion to arrive at free-energy differences for the
theoretical chemistry from the University of elementary reaction steps, we have all information, obtained in a
Bielefeld in 1998, working in the group of first-principles way with no experimental information required,
Juergen Hinze. After his habilitation in the to study the kinetics in the network.[7–15]
group of Bernd Artur Hess at the University While this procedure provides us with a quantitative
of Erlangen, he worked as a private docent description of chemical reactions in terms of relevant species
in Erlangen in 2003 and at the University involved, barrier heights, and concentration fluxes, it does
of Bonn in 2004. In 2005, he accepted not necessarily allow us to describe and understand what is
an offer for a professorship in physical actually going on. In particular, common reactivity patterns
chemistry at the University of Jena, where need to be identified from the quantitative data that is available
he worked until he moved to ETH Zurich in 2006. His research from experiment and from detailed calculations. In order to
covers many different areas in theoretical chemistry which range accomplish this, chemical concepts have been developed, many
from relativistic quantum chemistry to the development of new of them introduced in a rather ad hoc fashion to describe changes
electron-correlation theories and smart algorithms for the efficient in electronic structure upon a chemical reaction. Examples are
exploration of complex chemical reaction networks. the electronegativity concept as defined by Pauling[16,17] and the
hard and soft acids and bases (HSAB) concept by Pearson.[18–20]
In the past decades, it has become possible to root these concepts
1. Introduction in quantum chemical theories. While this is natural for electronic
Understanding chemical processes often starts with structure structure-related concepts such as partial charge and bond order,
elucidation of the relevant reactive species and their connection it is also possible for electronegativity and HSAB.
in terms of a reaction network. Quantum chemical calculations These concepts allow us to relate and discuss data obtained
allow us to detail the reaction mechanism by mapping it to a for different molecules and reactive systems. As such, they
network of elementary reaction steps characterized by reactants, have a categorizing value that can hardly be over-estimated.
*Correspondence: Prof. M. Reiher, E-mail: markus.reiher@phys.chem.ethz.ch
Laboratory of Physical Chemistry, ETH Zurich, Vladimir-Prelog-Weg 2, CH-8093 Zurich, Switzerland312
CHIMIA 2021, 75, No. 4 Laureates: Junior Prizes of the SCS Fall Meeting 2020
However, their predictive power remains obscure. Even though 2. Chemical Concepts
rigorous mathematical definitions for many concepts exist, most Electronegativity, hard and soft acids and bases, Fukui
concepts are not unique in a sense that they can be introduced functions, and the dual descriptor, all refer to the electronic
in different ways (e.g. there exist various ways to define atomic structure evaluated for a frozen molecular structure, typically
and hence partial charge as an atom is not rigorously defined in the equilibrium structure of a reactant. Hence, we limit
a molecule without ad hoc assumptions – even Bader’s elegant ourselves to the standard electronic-energy based discussion
atoms in molecules theory[21] is difficult to develop consistently for specific individual structures on the Born-Oppenheimer
in a fundamental relativistic framework without additional surface, typically those that mark local minima. Accordingly,
assumptions;[22,23] loosely speaking, due to the lack of second thermal and entropic effects are left aside so that one can expect
derivatives in the Dirac equation[24]). Moreover, the concepts are predictions for enthalpic but not for entropic contributions.
typically evaluated for the unreactive, isolated reactants, i.e. in Naturally, these can only be sensible if the elementary step
their equilibrium structures. It is not at all clear how they could under consideration is governed by electronic effects. We note,
possibly point the way toward the minimum-energy reaction path however, that there exist generalizations of quantum theory as
and hence to the transition state and barrier height. Therefore, we well as chemical concepts towards thermodynamic ensembles
may anticipate that it will be difficult to define a predictive concept (see, e.g. refs. [36,37]).
that offers a rationale or better a quantitative relation between the
(molecular and/or electronic) structure of reactants and reactivity 2.1 Electronegativity
measures such as activation barriers (that eventually relate to rate
On the Predictive Power of
Electronegativity is among the oldest and most widely used
constants, and hence, also to equilibrium constants). chemical concepts. The term dates back to Berzelius’ work in the
Chemical Concepts
Reactivity predictions should allow one to answer questions early 19th century[38] and the concept even further.[39] It denotes
such as these: Will certain reactants react with one another under the tendency of atoms to attract electrons towards themselves.[16]
† ⋆
given reaction conditions? Which reaction paths and/or products Among Stephanie the many quantitative
A. Grimmel definitions
and Markus that Reiher
have been put
are to be expected, i.e. how do the reactants react with each other? forward, the first one by Pauling remains to be the most popular
[16]
From which direction is a given reactant attacked preferably? one,of which
Laboratory PhysicalisChemistry,
why we briefly consider
ETH Zurich, it here. Pauling2, proposed
Vladimir-Prelog-Weg 8093 Zurich,
We focus on the prediction of elementary steps as opposed to a thermochemical recipe Switzerland driven by the availability of such data at
more complex multistep reactions to which our discussion † can ofthe
Winner thetime.
Best His definition
Poster relates
Presentation Awarddifferences in the electronegativity
in Computational Chemistry at the SCS
be generalized in a straightforward manner. χ of two elements A Fall and Meeting
B to the 2020deviation of the dissociation
For planning chemical synthesis, it is important to know energy ⋆ of the heteroatomic diatomic molecule, Ed(AB), from
Corresponding author: markus.reiher@phys.chem.ethz.ch
whether and which of these concepts can be reliably employed. the average of the dissociation energies of the corresponding
Whereas one might argue that traditional chemical synthesis homoatomic molecules, Ed(AA) and Ed(BB):
planning has rather been based on chemical intuition that is
nourished by the vast amount of experimental knowledge (e.g.
Ed (AA) + Ed (BB)
· (eV−1/2 ) (1)
P P
by name reactions) – even when exploited in a data-driven χA − χB := Ed (AB) −
2
(1)
manner, [25–33] recent developments in automated reaction
exploration schemes[4–6] based on extensive quantum chemical
calculations would benefit from knowledge about the predictive With regard to the predictive capabilities of the electronegativity
value of reactivity concepts. it should be pointed out M that by I+ rearranging
A Eqn. (1) one obtains
χ := , (2)
We have put forward the idea of first-principles heuristics, an expression for the heteroatomic 2 dissociation energy in terms
i.e. the idea of exploiting features of the electronic wave function of Pauling electronegativities χ and homoatomic dissociation
P
to cut the deadwood from the reaction space,[34,35] which for energies. Considering the reliability of a heteroatomic dissociation
bimolecular reactions alone formally grows with the number energy calculated in this manner already sheds doubt on how
of pairs of atoms in both reactants. This can only be fruitful if reliable the electronegativities ∂Ecan
el potentially be. Hence, examples
χ := − = −µ (3)
reactivity predictions based on concepts are reliable (at least to a of poor predictions can be easily ∂N vfound, e.g. in the case of alkali
certain degree). Therefore, we elicit on what is known about the fluorides and chlorides.[40]
predictive power of chemical reactivity concepts in this work. Indeed, the definition of χP through Eqn. (1) has profound
For a concept to have predictive power, it should enable us to limitations, some of which are:
judge reliably on the reactivity of a given set of reactants without 1. If the evaluation χ− ≈ Eel (Nof −bond1) − E energies
el (N ) = I
relies on enthalpies (4)
any a priori information about possible reaction products and of formation + (as in Pauling’s work
χ ≈ Eel (N ) − Eel (N + 1) = A
[16]), it depends on
(5)
paths. Note, however, that it can be equally important – especially macroscopic environmental +
parameters such as temperature.
in automated mechanism exploration – to determine what atoms 2. The definition χ + χ I + A
χ ≈ marries electronic = with
χM nuclear dynamics (6)
−
≈
or functional groups in a molecule are unreactive. because experimental 2 data was2taken for the parametrization.
In the following section, we review mathematical definitions As a consequence, the data cannot be easily calculated with
of four chemical concepts. As examples we select the sufficient accuracy.
electronegativity and the chemical hardness, because they belong 3. It is not obvious whether only data on diatomics may be
ǫHOMO + ǫLUMO
to the core of chemical concepts taught to chemistry students used for the definition
χM ≈ − or whether bond . breaking processes (7)
2
from high school education onwards – however, typically, without in any molecule can be considered. That is, the definition
discussing their quantum chemical foundations. Furthermore, ignores the fact that atoms in different bonding situations
we consider the Fukui functions and the dual descriptor for the may not be comparable. 1In other words, one will be forced
identification of nucleophilic and electrophilic sites of reactants. to find a categorizing scheme that groups atoms in similar
We outline limitations of specific descriptors as well as overall binding situations, for example, by introducing different
constraints. Finally, we expand on how to gather further insights valence states (and we refrain here from a discussion of how
upon the predictive capabilities of chemical concepts. We that could possibly be accomplished in a rigorous way).
elaborate on why, besides the value of chemical concepts for 4. Moreover, Pauling’s definition only allows for the calculation
describing and understanding chemical processes, massively of relative electronegativity values, which requires an
automated reaction exploration algorithms will be important to arbitrary definition of an absolute reference. In his original
assess actual reactive behavior in full depth, shedding also light work, he chose 0.0 for hydrogen, which was later replaced
on the application range of reactivity concepts. by 2.1 and today is typically set to 2.2.[16,17,41,42] P
χA − χP
Chemical Concepts
B := Ed (AB) −
Ed (AA) + Ed (BB)
· (eV−1/2 ) (1)
2
Chemical Concepts χ − χP := Ed (AB) − d
P
†
E (AA) + E d (BB)
· (eV−1/2⋆) (1)
Laureates: Junior Prizes of the SCS Fall Meeting 2020 PA PB A. GrimmelE
Stephanie and Markus
2Ed (BB) Reiher
(1) 313
χA − χB := Ed (AB) − d (AA) + CHIMIA 2021, 75, No. 4
· (eV−1/2 )
2
nie A. Grimmel† and Markus Reiher⋆ Laboratory of Physical Chemistry, ETHMZurich, I +Vladimir-Prelog-Weg
A 2, 8093 Zurich,
χ :=
Switzerland2 , (2)
ical Chemistry, ETH Zurich, Vladimir-Prelog-Weg 2, 8093 Zurich, I + A
5. The parametrization results in an overdetermined
Switzerland †
Winner of thesystem when
Best Poster treating
Presentation χM := in Computational
isolated
Award ,
atoms or molecules.
ChemistryThisat issue
the SCS (2)
is discussed
I +2A
of equations.
Poster Presentation Award in Computational Chemistry at the SCS
in great detail
Fall M
elsewhere.
:= 2020, [47,48,50–54]
χMeeting (2)
2
6. Fall
Depending on which element pairs are employed the resulting
Meeting 2020 ⋆
CorrespondingInauthor:
practical applications, Eqn. (3) is most often evaluated
markus.reiher@phys.chem.ethz.ch
∂Eel
electronegativities may differ. within aχfinite
:= − difference=approach.
∂N v
−µ The electronegativity (3)is then
From
esponding author: this list, it becomes clear that Pauling’s scheme is assumedχto
markus.reiher@phys.chem.ethz.ch
:= −
be equal
∂Eelto the average of the left and right derivatives
+ [47]= −µ (3)
ambiguous for a couple of reasons. Most of these issues are not of Eqn. (3), χ− and ∂E∂Nelχ : v
present in the absolute electronegativity definition by Mulliken,
P [43]P χ := − = −µ (3)
χA − χB := Ed (AB) − E∂N d (AA) v
+ Ed (BB)
which may be rigorously traced back to quantum chemical · (eV−1/2 ) (1)
2
χP
Equantities
d (AA)
Ethat + be
can Ed (BB)
routinely calculated on modern computer χ− ≈ Eel (N − 1) − Eel (N ) = I (4) (4)
B := d (AB) − · (eV−1/2 ) (1)
2
hardware. The Mulliken electronegativity χM of some atom is the χ +
− ≈ Eel (N ) − Eel (N + 1) = A
χ ≈ E (N − 1) − E (N ) = I (5)
(4)
el el
average of its ionization energy I and electron affinity A, + χ−(N + χ) +− I(N+ (N
A ) ==M
χχ−χ ≈EE
≈≈ el (N−
elM I1)+ elE
E−A
≈ el + 1)
= χIA (5) (5)
(4)
(6)
+
χ − 2
:= + ,2 (2)
χ ≈ Eχel (N+)χ− E2el (N I ++A1) = A (5)
I +A
(2) χ≈ ≈ = χM (6)
χM := , (2) χ− +2χ+ I +2A
2 χ≈ ≈ =χ M
(6) (6)
2 2
M
ǫHOMO + ǫLUMO
which is a natural definition in the sense that it connects the χχ:= −≈ −∂Eel . (7)
2= −µ (3)
∂N
ǫ + ǫ
concept ∂Eto elthe two quantities that measure the energy required EvenχMthough
≈ − derived differently,
HOMO v LUMO this expression is equal
(7) to the
χ := − = −µ (3) .
or liberated
∂N whenv
an electron is removed or added to the Mulliken
χ ≈−
ǫHOMO +2ǫLUMO
M electronegativity χ M
.
given in Eqn. (2). Therefore,
(7)
finite
system. In our setting here, we may identify I and A as the differences clearly1 link 2 both approaches.
vertical electronic energy differences calculated between the Rather than calculating the electronic energy of the system
system’s ground state and the corresponding states at the same with 1
χ− ≈different
Eel (N −electron numbers
1) − Eel (N ) = I (i.e. N, N + 1, and N(4)– 1), one
− structure with one electron removed and added, respectively may
χ+ ≈replace
Eel (N ) ionization energy
= A and electron affinity(5)
by orbital
1
χ ≈ Eel (N − 1) − Eel (N ) = I (4) − Eel (N + 1)
(with proper sign conventions). With this definition, the concept energies−of the+ highest occupied molecular orbital (HOMO) and
+ χ +χ I +A
χ ≈ofEelelectronegativity
(N ) − Eel (N + 1) is =Ano longer limited to atoms (5) but can be theχ lowest
≈ unoccupied
≈ molecular
= χM orbital (LUMO),(6) ∈HOMO and
2 2
generalized
χ− + χ+ to I+entire
A molecules. ∈LUMO, respectively, of the system with N electrons by virtue of
χ≈ ≈ = χM (6)
An,
2 at first sight,
2 different approach to define electronegativity Koopmans’ theorem,[55] to obtain
is taken in the context of conceptual density functional theory
(cDFT).[44–46] For an extensive review of the foundations of cDFT ǫHOMO + ǫLUMO
see ref. [47] and for a recent perspective ref. [48]. cDFT provides χM ≈ − . (7) (7)
2
n the Predictive Power of
χM ≈−
ǫHOMO +definitions
mathematical ǫLUMO
. of numerous chemical (7) concepts (see
also below). Its basis is the standard procedure of physical
2
Chemical Concepts
modeling and that is to expand a quantity that depends on a set Whereas Koopmans’
1 theorem relies on error cancellation in the
of parameters in terms of a Taylor series. In this case, it is the case of Hartree-Fock calculations[56] (owing to the neglected orbital
electronic 1energy Eel[N,ν;{Rk}] that depends
phanie A. Grimmel† and Markus Reiher⋆ on the number of
electrons N and the external potential ν. The latter is typically given
relaxation upon oxidation vs. the missing electron correlation in
the independent particle model), it can be considered exact for
as the Coulomb
Physical Chemistry, ETH Zurich, potential of the underlying
Vladimir-Prelog-Weg nuclear framework
2, 8093 Zurich, ionization energies in exact Kohn-Sham DFT.[57–59] However, the
in Born-Oppenheimer
Switzerland theory, and hence, it also encodes the LUMO Kohn-Sham energy cannot be exploited that easily.[60,61] It
specific molecular structure given by the set of nuclear Cartesian refers to an excited Kohn-Sham electron rather than to an added
Best Poster Presentation Award in Computational Chemistry at the SCS
coordinates of all nuclei, {Rk}. Kohn-Sham electron[60,62] so that one might consider the HOMO
Fall Meeting 2020
Although this consideration is typically done in the framework energy of the N + 1 Kohn-Sham system and reverse its sign to
of conceptual
Corresponding DFT, we emphasize that such a Taylor expansion of
author: markus.reiher@phys.chem.ethz.ch obtain the electron affinity of the N Kohn-Sham system.[63]
the electronic energy may be written for any electronic structure Mulliken pointed out that the electronegativity is not only an
model. However, DFT may offer specific advantages over other atom’s ground state property, but instead depends on its valence
models
(e.g. as Koopmans’ theorem becomes exact; however, see state.[43] This notion was formalized by Hinze and Jaffé[64–67] by
our discussion
P Edbelow).
(AA) + E The identification
d (BB) of reactivity concepts introduction of orbital electronegativities. The electronegativity
A − χB := Ed (AB) − · (eV−1/2 ) (1)
as responses of Eel towards2 changes in N and ν is in line with the χi of orbital i with Ni being the number of electrons in that orbital
understanding that they vary upon interactions between reactants is given by
during chemical reactions.
In 1961 – and hence even before cDFT evolved as a field –
∂Eel
I +A χi := − . (8) (8)
IczkowskiχMand:= Margrave, identified the electronegativity
(2) with the ∂Ni
negative response2 of the energy towards changes in the number of
v
electrons at constant external potential:[49]
Dramatically different electronegativity values were found,
∂Eel
for example, for χ2 different valence states of carbon.[68]
χ := − = −µ (3) (3) ω := (9)
∂N Although 2η there is a decent correlation between Mulliken
v
and Pauling electronegativities,[64,69,70] the theoretical basis of
Mulliken-Jaffé electronegativities χM has the salient feature that it
Later on, this expression was recognized as the negative of links directly to quantities of electronic structure theory, in which
the chemical potential µ by Parr.[44] The minus sign is introducedAh Bany s+A (fixed)−−
s Bh ↽ molecular
⇀
−− Ah Bh + A scaffold
s Bs represents a point on the Born-
≈ Eela(Nhigh
χto− have − 1)(positive)
− Eel (N ) electronegativity
=I (4) tendency Oppenheimer surface given by the electronic energy, as opposed
flag a strong
χto+ ≈
attract
Eel (Nelectrons,
) − Eel (N +and
1) =hence,
A it should be associated
(5) with a to the profound problems associated with Pauling’s definition.
decrease
χ− + inχenergy
+
I+upon
A an M increase in N (note that for sufficiently Electronegativities of atoms are widely used to identify
χ ≈ molecular
large ≈ systems,= the
χ electronic energy can be (6)considered reactive
2 2 1 ∂ 2 Ecenters 1within
∂µ molecules. For example, the fact that
el
to decrease upon reduction of the system by one electron). Despite η̃ := tetrahedral
2 ∂N 2 v
carbon
= has
2 ∂N v
a lower electronegativity
(10)than chlorine
appearing mathematically concise, Eqn. (3) suffers from the fact (8.07 vs. 10.05 eV according to the Mulliken-Jaffé values from
that the number of electrons N is restricted to integer numbers ref. [68]) allows us to predict that a nucleophilic attack on
ǫHOMO + ǫLUMO
χM ≈ − . (7)
2
∂ 2 Eel
∂µ
η := = (11)
∂N 2 ∂N2η
−−
Ah Bs + As Bh ↽ −⇀
− Ah Bh + As B χs2
ω := (9)
314
CHIMIA 2021, 75, No. 4 Laureates
Ah Bs + A −−
s Bh ↽−− :AJhunior
⇀ 2ηA
Bh + Psrizes
Bs of the SCS Fall Meeting 2020
Ah Bs + As Bh ↽ −− −⇀− Ah Bh + As Bs
2
1 ∂ Eel 1 ∂µ
η̃ := = (10)
chloromethane should preferably occur at the carbon center. 2This ∂N1 means
A
∂ 2vEel this
2h Bs + A hprefactor
2 s B∂N −− ⇀ Ah Bemerging
1↽−− v∂µ h + As Bs in front of any second-
The question whether electronegativity can be used to predict η̃ derivative
:=
2 ∂Nterm 2 of= a Taylor series expansion is to (10) be understood
1 ∂2E 1 v ∂µ2 ∂N v
reaction barriers in a quantitative manner was tackled, e.g.η̃ :=
in as included
el
= in the definition, implying a scaling
(10) of the hardness.
the context of hydrogen abstraction reactions: Nguyen et al. 2 In ∂N 2 following, 2 ∂N
the
2
v we
continue
v to apply the scaled hardness.
established a correlation between the reaction barriers of theη := ∂InEelanalogy =
∂µ
1 ∂to
2 electronegativity, the chemical hardness is
E el
1 ∂µ
(11)
∂N 2η̃ := ∂N = (10)
abstraction of a given hydrogen atom from propene and methane often calculated ∂ vEel2 ∂Nwithin
2 a2 finite
∂N vdifference approach, which
2 v∂µ
η := 2 = v (11)
and the electronegativity of attacking radicals. A larger share eventually
[71]
∂ 2 Eel ∂N
reads as
∂µ ∂N v
v
of chemical space was investigated in a study of the relation η := 2
= (11)
∂N ∂N v
between different cDFT properties and 216 electrophilicity v
values from the Mayr database[72–75] by Lee et al.[76] They η ≈ I − A. ∂ 2 Eel
∂µ
(12) (12)
η := = (11)
observed only a very poor correlation (R = 0.539) between
2
η ≈ I −∂N A. 2
v ∂N v (12)
Mayr’s electrophilicities and electronegativities obtained with
η ≈ I − A. (12)
Eqn. (6).
An improved, but still weak correlation (R2 = 0.737)
∂Eel
The application of Koopmans’ theorem then yields
χwas found with. the cDFT electrophilicity (8) ω, which is a η ≈ ǫLUMO − ǫHOMO .
i := − [77,78]
(13)
∂Ni v
joint property calculated from the electronegativity χ and the η ≈ I − A. (12)
chemical hardness η: η ≈ ǫLUMO − ǫHOMO . (13) (13)
η ≈ ǫLUMO − ǫHOMO . (13)
χ2 The global
1
softness, S, is identified as the inverse of the
ω := (9) (9) hardness,
S := . (14)
2η η η ≈ 1ǫLUMO − ǫHOMO . (13)
S := . (14)
1 η
The concept of chemical hardness will be considered in the S := .
η
(14) (14)
following
Ah Bs + A s Bh −
↽ −
−− section.
⇀ Ah Bh + The
As Bpoor correlation coefficients may be taken
s LiI(g) + CsF(g) ↽−−−⇀
− LiF(g) + CsI(g) 1
as a clear indication that kinetic predictions based on Eqns. (6) S := . (14)
and (9) are unreliable. However, it should be pointed out that withLiI(g) +ACsF(g)
prototypical
−−
↽ −⇀ example
− LiF(g) of Pearson’s HSAB principle is the
η+ CsI(g)
Mayr’s electrophilicities being obtained from experimental data,+ CsF(g)
LiI(g) reaction
−
↽−⇀
−−between
LiF(g) +LiICsI(g)
and CsF:[81]
deviations
may not be exclusively attributed to shortcomings 2
1 ∂ 2 Eel
1 ∂µ
η̃ := of the 2
concepts’
= definitions: Measuring(10)inaccuracies and 2
2 the∂Nprotocol
v 2 for
∂N the
v calculation of electrophilicities from LiI(g) + CsF(g) − ↽ −
−⇀− LiF(g) + CsI(g)
2
experimental rate constants (see ref. [73]) as well as limitations
of the applied electronic structure method can also contribute to
the
discrepancies.
∂ 2 Eel
∂µ
This issue is considered in some more detail With I– being softer
2 than F– and Cs+ softer than Li+, the HSAB
η := in Section
2
=4. (11) principle correctly predicts that the conversion should be exothermic.
∂N ∂N
This is especially notable considering that a prediction based on
v v∂Eel
χi := − . (8)
2.2 Hard and Soft∂N Acids
i v and Bases Pauling electronegativities and a rearranged form of Eqn. (1) fails
Based on experimental observations Pearson formulated the for this example.[81]
hard and soft acids and bases (HSAB) principle as follows: “Hard Nevertheless, the HSAB principle should by no means be applied
η ≈ I − A. (12)
acids prefer to bind to2hard bases and soft acids prefer to bind to blindly. Most importantly, as Pearson himself pointed out,[82] it only
soft bases.” χ
ω This
:= means, that with A denoting an acid,
(9) B a constitutes one amongst many effects determining the interaction
[18–20]
2η h and s hard and soft, respectively, the
base, and the subscripts between two reactants. Considering the terms in the Taylor series
exchange reaction expansion of the electronic energy, this means that an exact
η ≈ ǫLUMO − ǫHOMO . (13) correspondence between chemical reactivity and predictions based
on the HSAB principle can only be expected if all other derivatives
Ah Bs + As Bh ↽ −− ⇀
− − Ah Bh + As Bs are negligible – a condition that is hardly ever met in chemical
reactions. For example, it is known that the HSAB principle tends
∂Eel
χi := − . (8)
1 ∂E∂N
el i v
to be dominated by the preference of strong acids to recombine with
S := χ .
should beηiexergonic.
:= − In Pearson’s
. (14)
early work, soft acids and bases
(8) strong bases.[83] Applications of the HSAB principle to ambident
2 ∂N i v
were defined 1 ∂to Ebe
el those 1 that∂µare easily polarizable, while hard reactants were heavily criticized due to notable failures occurring
:= = [18] (10)
ones η̃are 2hard∂N to2 polarize.
v2 2 ∂NIt vis rather surprising that such a even for prototypical systems.[74,84]
phenomenological ω := concept that was proposed as an attempt
χ
(9) to Although its now established roots in quantum chemistry
categorize
(g) + CsF(g) − −⇀
↽−− ω :=a wealth
LiF(g) + χ of
22η
CsI(g) experimental observations can actually be allow for explicit calculations, the HSAB principle should not
based on rigorous (9)
2 mathematical
2η definitions in a quantum chemical be mistaken as a universal tool for reactivity predictions. It may
framework. cDFT
∂ Eel provided ∂µsuch a framework, in which hardness be regarded as one measure for reactivity in the context of all
η := = (11)
η appears as ∂N the2resistance
v ∂Nof vthe chemical potential µ towards such concepts evaluated for a reactant.
2
changes
Ah Bs +inAthe s Bhnumber
−
↽−−⇀− Ah of Bh electrons
+ As Bs N. Accordingly, a derivative
term
Ah Bsin+ the
As Baforementioned
−−
h ↽ −⇀− Ah Bh + ATaylor s Bs
expansion of the electronic 2.3 Fukui Indices and the Dual Descriptor
energy is interpreted as the hardness:[79] To obtain information about where a reactant is attacked,
η ≈ I − A. (12) local reactivity descriptors are required. One prominent example
1 ∂ 2 Eel
1 ∂µ
is the Fukui function f(r) with the Cartesian coordinate r denoting
η̃ := 2
= (10) (10) some position in space. It was introduced by Parr and Yang as a
1 2 ∂ 2∂N
Eel v 1 2 ∂µ∂N v
η̃ := = (10) density-functional generalization of Fukui’s Frontier Molecular
2 ∂N 2 v 2 ∂N v
η ≈ ǫLUMO − ǫHOMO . (13)
Orbital (FMO) theory.[85]
However, the prefactor 1/2 is frequently omitted:[47,80] The Fukui function is the mixed second derivative of the
electronic energy with respect to ν and N at a given position r. It can
2
∂ Eel ∂µ
η := = (11)
∂ 2∂N
Eel
2
v ∂N v
∂µ be represented either as the response of the chemical potential to a
η := = (11) (11) change in the external potential at position r or as the response of the
∂N 2 v 1 ∂N v
S := . (14) electron density ρ(r) towards a change in the number of electrons, i.e.
η
η ≈ I − A. (12)f − (r) ≈ ρHOMO (r) (20)
f + (r) ≈ ρLUMO (r), (21)
Laureates: Junior Prizes of the SCS Fall Meeting 2020 CHIMIA 2021, 75, No. 4 315
f − (r) ≈ ρHOMO (r) (20)
+
f (r) ≈ ρLUMO (r), (21)
δ ∂Eel δµ ∂ δEel ∂ρ(r) (2) δη ∂f (r)
f (r) :=
δv(r) ∂N
=
δv(r)
=
∂N δv(r)
=
∂N
. (15) (15) f (r) :=
δv(r) N
=
∂N v(r)
(22)
(22)
δη ∂f (r)
Regions where f(r) is large are supposed to be those
from which f (2) (r)Within =difference approximation, ƒ (r)(22)
the finite can be written
(2)
δ ∂Eel δµ ∂ δEel [85] ∂ρ(r) :=
f (r) :=reactive
δ ∂E attacks =are δµthe most
=− favorable.
∂ δEel = Due to the
∂ρ(r) . derivative
(15) δv(r)
as the difference
(2) Nbetween
+ ∂N the− electrophilic
v(r) and nucleophilic Fukui
el f (r) ≈ f (r) − f (r). (23)
f (r) := δv(r) ∂N − = δv(r) ∂ρ(r) = ∂N δv(r) = ∂N . (15)
discontinuities
δv(r) ∂Nf (r)with respect
:= δv(r) to ∂N
≈ ρleft
N, and
δv(r)
N (r) − ρright derivatives are not(16) functions,
∂N
N −1 (r)
equal, and hence, two∂N different
v(r) quantities are typically reported,
ƒ–(r) and ƒ+(r), which can be approximated by finite differences:
+
∂ρ(r)
f + (r) := ≈ ρN +1 (r) − ρN (r) (17) f (2) (r) ≈ f + (r) − f − (r). (23) (23)
∂N v(r)
−
∂ρ(r) −
f − (r) := ∂ρ(r)
f − (r) :=
≈ ρN (r) − ρN −1 (r) (16)
(16)
∂N v(r) ≈ ρN (r) − ρN −1 (r) (16)
∂N +v(r) ƒ(2)(r) takes values between –1 and 1 with negative and positive
+
δ ∂Eel f + (r) :=
∂ρ(r) +
δµ ∂ρ(r) ∂− δE
≈ elρN +1 (r) ∂ρ(r)
− ρN (r) (17) values indicating susceptibility towards electrophilic and
f=(r) :=
δv(r) ∂N δv(r) ∂N ∂N
= fv(r)
∂N k ≈
ρN−1
≈qk,N +1=(r) − ρN
− qk,N (r). (15) (17)
(17) (18) nucleophilic attacks, 3 respectively.[92] The dual descriptor was
v(r)δv(r) ∂N
(19) reported to be more robust than Fukui functions, especially in the
δ ∂Eel δµ ∂+ δEel ∂ρ(r)
= = fk ≈ qk,N −=qk,N +1 . (15)
δv(r) ∂N δv(r) ∂N δv(r) ∂N frozen-orbital approximation.[94]
Hence, ƒ–(r) describes the response of the system towards a As a demonstration
3 for the identification of reactive sites
decrease of N, i.e. towards electrophilic attacks and ƒ+(r) the through the inspection of ƒ(2)(r), we consider propanone in the finite-
response
∂ρ(r)towardsfk− ≈ anqk,N
increase
−1 − qk,Nof N, i.e. the reactivity with respect
(18) difference approximation (see Fig. 1). We obtained the underlying
− −
f − (r) := f≈ ≈Nq(r)
k+ ρ k,N− −1ρ− q (r)
N −1k,N (16) (18)
to nucleophilic
∂N − attacks.
fk+ ≈ qk,N To allow
− − qk,N +1 for an easier interpretation, the electronic densities from a DFT/PBE[95,96]/def2-SVP[97,98]
(19)
v(r) f (r)
− q≈k,NρHOMO (r) (20)
f − (r) := ∂ρ(r)
Fukui + f≈
functions ≈ qk,N
kare
ρNoften
(r)+− not
ρN −1analyzed
+1
(r) in full spatial(16) (19)
resolution, but calculation with the program Orca.[99] The condensed-to-atom
∂ρ(r)
∂N f (r) ≈ ρ (r), (21)
+
f (r) := instead in a condensed-to-atom
v(r) ≈ ρN +1 (r) − ρN (r) representation. Staying
LUMO
(17) in the finite values given in Table 1 were determined from Hirshfeld charges.[87]
∂N +
∂ρ(r) approximation, these Fukui indices ƒk can be obtained The fact that they turned out to be negative at the oxygen atom
difference v(r) ±
+
f (r) := atom k ≈ ρN (r) − ρN (r) (17)
δ ∂Efor
any from the
∂ atomic ∂ρ(r)qk and are given in atomic and positive at the adjacent carbon atom agrees with the notion that
charges
∂N +1
el δµv(r) δEel
units=as
δv(r) ∂N δv(r)
= = . (15) these are the primary attack sites for electrophilic and nucleophilic
− ∂N δv(r) ∂N
f− (r) ≈∂ ρHOMO (r) ∂ρ(r) (20) attacks, respectively.
δ ∂Eel δµ (2) δE
δη
f (r) ≈ ρHOMO (r)el ∂f (r) (20)
= f +=(r) := == . (15) (22)
δv(r) ∂N δv(r) f + (r) ∂N δv(r) (r),
≈ ρLUMO
δv(r) N ∂N
∂N v(r) (21)
f −(r)
fk− ≈ qk,N −1 ≈ ρLUMO
qk,N (r), (18) (21)
(18)
+
f− ≈ qk,N − q−k,N +1 (19)
(18)
fkk ≈ q k,N −
−1 qk,N
∂ρ(r)
+
f (r) := k
− f ≈ q −
k,N ≈ k,Nq ρf (r)
+1 − ρN −1 (r)
(2) + (19)
(16) (19) (23)
N (r)
≈ f (r) − f (r).
−
∂N − v(r) δη
∂ρ(r)
(2)
∂f (r)
f (r) := ∂ρ(r)
− f (r)+ :=≈ ρNδη(r) − ρN = (r) ∂f (r) (16) (22)
f (2) (r)v(r)
∂N := δv(r) N =−1 ∂N v(r) (22)
f + (r) := N +1 (r)N− ρN (r)
≈ ρδv(r) ∂N v(r) (17)
∂N +
However, v(r) atomic charges are yet another concept without
∂ρ(r)
(r) ≈
f + (r) := f ρHOMO≈ ρN(r) (20)
−
a unique
∂N definition. Numerous
+1 (r) − ρN (r) mathematical definitions
(17) exist
+ v(r)
because (r)of≈
ff − (r) ≈theρ difficulty
LUMO (r), to define spatial boundaries
HOMO (r) +
ρ(2) (21) of an atom
(20)
in a fquantum (r) ≈ fsuch
f (2)system (r) − f a (r). (23)
f + (r) −asf − molecule (see above). Therefore,
−
+
(r) ≈fρLUMO (r) ≈(r), (r). (21) (23)
the choice among these can significantly affect the values of the
resulting
fk− ≈ Fukui
qk,N −1 indices.
− qk,N [86] Hirshfeld
3 charges[87] are
(18)a commonly Fig. 1. ƒ(2)(r) = ±0.01 a0–3 isosurface for propane. Blue encodes negative
recommended
fk−+ ≈ qk,N −choice.
+1[88]
qk,N (19) values, orange positive ones.
f ≈δη qk,N −1 − qk,N
∂f (r) (18)
Ifk+
f (2) (r) := frozen = are assumed, ƒ–(r) and ƒ+(r) can
orbitals (22)be expressed
δv(r) ∂N
fk ≈ of
(2) in terms δη
qk,N
theN qk,N +1
− removed incoming orbital upon(19)
∂f (r)or v(r) change of the Table 1. Condensed dual descriptor indices for propanone.
f (r) := = (22)
numberδv(r)of electrons,
N ∂N v(r)
Atom ƒk(2)
f (2)f(r) 3
− ≈ f + (r) − f − (r). (23)
(r) ≈ ρHOMO (r) 3 (20) (20) OCO –0.1
+
(2)
f− (r)
≈≈
f f(r)(r) f +ρ − f(r), (21)
ρ(r) (r). (23) CCO 0.1
−
LUMO
≈ HOMO (r) (20)
+
f (r) ≈ ρLUMO (r), (21) (21) Cα 0.0
H 0.0
δη ∂f (r)
in terms
i.e.:=
f (2) (r) of HOMO
= and LUMO densities, ρHOMO(r) and ρLUMO(r),
(22)
δv(r) calculated
respectively, N ∂Nas the v(r) absolute square of the respective Notably, the dual descriptor allowed for a back-on-the-
δη ∂f (r)
f (2) (r) := which3 manifests
orbital, = (22) Naturally,
δv(r) N ∂Nthe v(r)
connection to FMO theory. envelope discussion of the Woodward-Hoffmann rules[100,101]
this interpretation is prone to fail for quasi-degenerate frontier refuting the common belief that one requires an orbital model to
3
orbitals.
(2)
For +example, it cannot reliably predict the regioselectivity predict pericyclic reactions.
f (r) ≈ f
of electrophilic (r) − f − (r). substitutions.[89,90] In orbital-weighted
aromatic (23) Unfortunately, the applicability of Fukui indices and the dual
Fukui functions
f (2) (r) ≈ f + (r) this
− f −behavior
(r). is cured by the inclusion
(23) of further descriptor for predicting nucleophilicity and electrophilicity is
orbitals – however, at the cost of introducing an empirically limited to orbital-controlled soft–soft interactions.[102] The most
motivated weighting factor.[91] reactive sites for hard–hard interactions can be better predicted
Instead of considering ƒ–(r) and ƒ+(r) separately to identify using atomic charges or with descriptors extracted from the
nucleophilic and electrophilic sites, Morell et al. suggested the molecular electrostatic potential.[102,103] For example, Fukui
use of the dual descriptor ƒ(2)(r) defined as the derivative of functions are known to poorly predict protonation sites.[102]
the chemical hardness with respect to the external potential or, Another level of complexity arises due to the fact that for many
equivalently,3the response of the Fukui function towards changes chemical reactions neither charge control nor orbital effects are
in the number of electrons:[92,93] highly dominating, but both have to be considered.[103–105]
3316
CHIMIA 2021, 75, No. 4 Laureates: Junior Prizes of the SCS Fall Meeting 2020
3. Limitations of Reactivity Prediction by Local out and analyze vast numbers of quantum chemical calculations,
Concepts inaccessible to manual work with quantum chemistry programs, and
As we outlined for the examples above, the predictive power can therefore deliver the data required.
of a single chemical concept is very limited. Following the Taylor The compilation of such a test set is non-trivial: Typically
series expansion of the electronic energy, which explicitly states researchers hand-pick reference reactions based on their prior
that various partial derivatives, now understood as different knowledge or the literature or make use of databases such as the
chemical concepts, are to be added up, it is not sufficient to limit aforementioned Mayr database. However, this approach bears the
oneself to one chemical concept when trying to understand the risk of assembling biased test sets: Chemical concepts are integral
change in energy upon reaction, but several must be considered parts of chemistry education. That is why it appears reasonable to
simultaneously. They all need to be incorporated into a reactivity assume that reactions conforming to these are overrepresented in
prediction protocol. The fact that machine-learned models based the literature and, hence, in the test sets. This should not affect the
on several indicators clearly outperform schemes based on a comparison of different evaluation schemes for a given concept.
single selected descriptor underlines this necessity.[106,107] However, there might be a tendency to overrate the predictive
The key difficulty of all reactivity descriptors defined in terms power of chemical concepts as such in a circular-argument
of a Taylor series expansion of the electronic energy around a situation. Automated reaction network exploration algorithms allow
reference point is that this is always local information in the sense one to enumerate large numbers of different reaction coordinates
that it is valid for the unreactive and non-activated equilibrium – including those that appear to be unpromising from a chemist’s
structure of a reactant, i.e. for the reference point: The local slope perspective. Hence, there is hope that the bias in favor of known
and curvature of the energy functional is supposed to correlate chemical concepts is less pronounced.
with the barrier height of eventual transition states. As formulated Furthermore, reaction databases typically only contain
in Klopman’s non-crossing rule different reaction curves shall not information on successful reactions, but no explicit information
cross between reactants and transition states.[108] In practice, this about unproductive reaction coordinates. The possibility to sort out
means that the behavior at the onset of reactions has to be decisive clearly unreactive reaction guesses is, however, one of the key tasks a
for the overall course of the reaction. This is why Geerlings et useful reactivity predictor must fulfill, which is why this information
al. state that cDFT descriptors are expected to be unreliable for should be included in a test set. A brute-force enumeration of
late transition states.[48] Making direct use of this statement for reaction paths by automated exploration algorithms will certainly
reactivity predictions, however, is impossible because transition also include unreactive scans. Finally, the fact that the target quantity
states are obviously unknown prior to the analysis. of the reactivity prediction (e.g. reaction barriers) is calculated with
This issue may be considered more severe for all but the same quantum mechanical method as the reactivity descriptor
intramolecular reactions due to the fact that to project towards itself, opposed to experimental data, offers the possibility to study
reaction paths and energy barriers the interactions between all the reliability of the reactivity scheme separated from possible
reactants have to be considered, which will disturb the information shortcomings of the underlying method compared to experiment.
gathered at the reference point. cDFT descriptors can be evaluated So far, such extensive data has not been produced yet, but it can
for associated reactive complexes and even along entire reaction be anticipated that this will change.[4–6,110]
paths,[109] but the relevant complex geometry and especially the path Once available, the machine learning evaluation of reactivity
are, obviously, not known a priori in prediction tasks. For a list of descriptors faced with explicit knowledge of vast reaction networks
further approaches which incorporate the effect of a second reactant, will provide us with means to assess the extent to which we may
see ref. [48]. In principle, it would be possible to evaluate derivatives prune quantum chemical reactivity explorations by first-principles
of Eel[N,ν;{Rk}] up to arbitrary order. In practice, however, there are heuristics[34,35] in order to dramatically accelerate the automated, but
mathematical and technical difficulties associated with taking such computationally costly exploration process.
derivatives.[48] Moreover, even if these derivatives were evaluated,
there is no reason to believe that the radius of convergence of 5. Conclusions
the series expansion is such that it encodes information up to the In this work, we discussed why chemical concepts are not only
transition state structure. In other words, the series expansion is not valuable to explain, describe, and categorize reactions a posteriori,
expected to converge for structures farther away from the reference but also bear hope for reactivity prediction. Mathematical definitions
point on the Born-Oppenheimer surface. allow for the quantification of well-known chemical concepts by
Due to these arguments, a perfect matching between reactivity means of quantum chemistry. In particular, partial derivatives of the
descriptors, which are evaluated at equilibrium structures, and electronic energy with respect to the decisive parameters of a system
activation barriers (and even less so for reaction energies) cannot (such as the number of its electrons) allow us to clearly define
be expected. They might, however, still be a valuable tool in order reactivity concepts relying on electronic structure information.
to (de-)prioritize more or less promising reaction guesses for further However, ambiguities may arise due to competing definitions of
analyses. I.e. if one can tell (for instance, in an automated mechanism the same concept as well as different approximation and localization
exploration algorithm), which sites are very likely to be unreactive, schemes. An exact correspondence between chemical concepts and
then this knowledge accelerates the exploration as unreactive paths reactivity (measured, e.g. in terms of the activation barrier associated
can be omitted by the exploration protocol without the necessity to with that process) cannot be expected because of the fact that the
consider them explicitly. concepts are evaluated for a specific molecular structure, whereas
barriers require the comparison of energies of two sets of structures,
4. Reactivity Descriptors and Automated Reaction i.e. reactants and transition state structures (for an illustration of this
Space Exploration problem, see Fig. 2). The usefulness of chemical concepts critically
To verify the usefulness of a reactivity prediction scheme and depends on whether, despite these expected uncertainties, they still
gather profound knowledge about the associated uncertainties allow for discriminating between reactive and unreactive pathways
detailed statistical analyses have to be carried out. Automated with a high degree of confidence.
reaction space exploration schemes cannot only benefit from the Especially automated quantum chemical reactivity exploration
exploitation of chemical concepts in the context of first-principles can effectively exploit this situation. If, due to the aforementioned
heuristics,[34,35] they also offer the potential to help validating these. uncertainties, we do not have clearly separable clusters of
For statistically rigorous analyses large and chemically diverse test reactive and unreactive pairs of atoms, this will be a sign that our
sets have to be investigated. Automated algorithms allow to carry methodology cannot cut deadwood in reaction space with sufficientLaureates: Junior Prizes of the SCS Fall Meeting 2020 CHIMIA 2021, 75, No. 4 317
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