Computational and Theoretical Chemistry 1010 (2013) 11–18
Contents lists available at SciVerse ScienceDirect
Computational and Theoretical Chemistry
journal homepage: www.elsevier.com/locate/comptc
Using conceptual density functional theory to rationalize regioselectivity:
A case study on the nucleophilic ring-opening of activated aziridines
Gilles Berger ⇑
Laboratoire de Chimie Pharmaceutique Organique, Faculté de Pharmacie, Université Libre de Bruxelles, Campus de la Plaine, CP 205-5, Bd du Triomphe, 1050 Brussels, Belgium
a r t i c l e
i n f o
Article history:
Received 30 November 2012
Received in revised form 27 December 2012
Accepted 27 December 2012
Available online 13 January 2013
Keywords:
Conceptual DFT
Aziridines
Fukui function
Fukui indices
Reactivity descriptors
Population analysis
a b s t r a c t
Density functional theory calculations have been performed to rationalize the regiochemistry of the
nucleophilic ring opening of activated aziridines. Atomic charges, lowest unoccupied molecular orbitals,
Fukui functions and Fukui indices were calculated at the B3LYP/6-311G++(2d,2p) level of theory. Frontier
molecular orbital theory, as well as the Fukui function were able to explain the experimentally observed
ratios of opening products and a surprising change in regioselectivity upon nitrobenzenesulfonyl activation on the nitrogen. In addition, robustness of atomic charges and Fukui indices to the basis set quality
was assessed.
Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction
Nitrogen activated aziridines are versatile synthetic intermediates easily produced from b-amino-alcohols [1]. Stereo and regiocontrolled nucleophilic opening of aziridines leads to highly
valuable, 1,2-bifunctionalized chiral compounds and among them
vicinal diamines are readily obtained [2]. Beside its interest in
medicine and pharmacy, the 1,2-diamine moiety has taken a large
place as chiral ligands in transition-metal catalyzed asymmetric
synthesis [3].
During the last decades, density functional theory (DFT) has
undergone fast development, especially in the field of organic
chemistry, as the number of accurate exchange–correlation functionals increased. Indeed, the apparition of gradient corrected
and hybrid functionals in the late 1980s greatly improved the
chemical accuracy of the Hohenberg–Kohn theorem [4] based
methods. The Kohn–Sham formalism [5] and its density-derived
orbitals paved the way to computational methods. In parallel, a
new field of application of DFT developed, the so-called conceptual
DFT [6]. Parr and Yang followed the idea that well-known chemical
properties as electronegativity, chemical potentials and affinities
could be sharply described and calculated manipulating the electronic density as the fundamental quantity [7,8]. Moreover, starting from the work of Fukui and its frontier molecular orbitals
(FMOs) theory [9], the same authors further generalized the con⇑ Tel.: +32 2 650 5248; fax: +32 2 650 5249.
E-mail address: [email protected]
2210-271X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.comptc.2012.12.029
cept and proposed the Fukui function f ð~
rÞ as a tool for describing
the local reactivity in molecules [10,11].
The present study makes the use of DFT-derived reactivity
descriptors to rationalize the regioselective ring opening of three
types of activated aziridines and makes a comprehensive assessment of the electronic effects arising upon the different methods
of activation. As aziridines consist in a three-membered ring with
two electrophilic positions, the relative reactivity of these two
electrophilic carbon atoms seems a perfect test for FMO theory
and the Fukui function.
2. Theoretical background
Chemical DFT started to develop in the late 1970s with the
identification of the chemical potential (l) as the first derivative
of the Kohn–Sham energy (E), with respect to the number of electrons (N) at constant external potential due to the nuclei (v ð~
rÞ)
[12]:
l¼
@E
¼ v
@N v ð~rÞ
ð1Þ
The chemical potential (l) is in fact the Lagrange multiplier
from the normalization constraint of the DFT variational principle
[13]. It should be noticed that the chemical potential is the negative of a well-known empiric chemical quantity, the electronegativity (v). As a consequence of Eq. (1), electrons will have the
tendency to flow from high l zones to low l zones until chemical
potentials are equalized, l being a global reactivity index of the
12
G. Berger / Computational and Theoretical Chemistry 1010 (2013) 11–18
molecule. Taking the derivative of l with respect to the potential
and applying the Maxwell relation, one can define the Fukui function [10,14]:
f ð~
rÞ @ qð~rÞ
@N
¼
v ð~rÞ
dl
dv ð~
rÞ
ð2Þ
N
However, owing to the discontinuity of N, at least two types of
Fukui functions should be defined, one being the right-hand side
and the other the left-hand side derivative at a given number of
electrons (N = N0):
f þ ð~
rÞ ¼
rÞ ¼
f ð~
F þk
@ qð~rÞ þ
@N
@N
ð4Þ
v ð~rÞ
1 þ
ðf ð~
rÞ þ f ð~
rÞÞ
2
ð5Þ
Exploring the chemical significance of these functions, it can be
seen that a site where the nucleophilic Fukui function f þ ð~
rÞ has a
large value is a site capable of accepting electronic density and that
a large value of f ð~
rÞ indicates an electron donating site. In other
words, one measures the site reactivity towards nucleophilic attack
while the other measures the site reactivity towards electrophilic attack. It should be emphasized that in the frozen core orbital approximation where orbital relaxation is neglected, these functions can be
reduced to the squares of the lowest unoccupied and highest occupied molecular orbitals (respectively, the LUMO and the HOMO). Indeed, expressing the electronic density in terms of the Kohn–Sham
spin–orbitals (Eq. (6)) and taking the derivative with respect to N,
gives a mathematical expression of the Fukui functions as the frontier molecular orbitals plus a correct term that includes the orbital
relaxation effects as seen in Eq. (7) and (8) [15]:
qð~rÞ ¼
j/ð~rÞ j2
ðiÞ
ð6Þ
i¼1
2
rÞ ¼ j/Nþ1
f ð~
ð~
rÞ j þ
þ
ðiÞ
N
X
@j/ð~rÞ j2
i¼1
rÞ ¼
f ð~
j/Nð~rÞ j2
@ qkð~rÞ
@N
v ð~rÞ
@ qð~rÞ N
X
¼
Z
ð3Þ
A third type of Fukui function describing radical reactions can
be used, as a mean of the two others:
f 0 ð~
rÞ ¼
rather than dealing with spatially dependent functions. It therefore
needs the integration of the Fukui function along the portion of
space that could be attributed to a certain atom belonging to a
molecule, leading to condensed to atoms Fukui functions (i.e. Fukui
indices) [21]. This procedure is analogous to the condensation of
the electronic density to atoms, leading to the assignment of atomic charges through population analysis or density partitioning [22].
We can thus describe, for the atom site k, Fukui indices (F+ and F)
in terms of atomic populations (p):
þ
@N
ðiÞ
N
X
@j/ð~rÞ j2
i¼1
!
@N
j/LUMO
j2 ¼ qLUMO
ð~
rÞ
ð~
rÞ
ð7Þ
j/HOMO
j2 ¼ qHOMO
ð~
rÞ
ð~
rÞ
ð8Þ
v ð~rÞ
!
v ð~rÞ
Practically, computing Fukui functions is not obvious and a
solution is to take the finite difference approximation between
the N = N0 and N = (N0 ± 1) electronic total densities, as expressed
in the following equations:
f þ ð~
rÞ qNþ1 qN
ð9Þ
f ð~
rÞ qN qN1
ð10Þ
Nevertheless, other schemes can be used to compute reactivity
indices avoiding this rough estimation of derivatives due to the use
of an integer change on N. These methods introduce fractional
occupation numbers from Janak‘s theorem [16] to produce infinitesimal changes on the number of electrons [17–20]. The first
method using the finite difference approximation has been employed in the present study.
A three dimensional representation of the functions is then obtained but a chemist’s dream would be to assign indices to atoms,
F k
¼
Z
@ qkð~rÞ
@N
!þ
d~
r¼
k þ
@p
¼ pkNþ1 pkN ¼ qkN qkNþ1
@N v ð~rÞ
ð11Þ
d~
r¼
k @p
¼ pkN pkN1 ¼ qkN1 qkN
@N v ð~rÞ
ð12Þ
v ð~rÞ
!
v ð~rÞ
Electrophilicity indices have recently been reviewed by Chattaraj et al. [23]. Reactivity indices were comprehensively discussed
by Chermette [24].
Carving the molecular electronic density into its atomic constituents is far from trivial [25]. Different methods can be employed,
often leading to significantly different results and misleading
chemical interpretations. Moreover, it is not easy to choose the
most appropriate method based on theoretical arguments. Three
different charges schemes were used in the present study to compute atoms-in-molecules (AIMs) properties: Mulliken population
analysis (MPA) [26] and natural population analysis (NPA) [27]
as density matrix based methods and Hirshfeld partitioning (HP)
[28–30] as a density method. Electrostatic potential fitted charges
were not used as they are known to often produce unphysical results, despite the good reproduction of the multipolar environment
of the molecule [31].
3. Computational details
All quantum mechanical calculations have been achieved using
the Gaussian09 software package [32]. Geometries of all the investigated systems were optimized at the density functional theory level using the B3LYP functional (combination of exchange from
Becke’s three parameter hybrid exchange functional (B3) with
the dynamical correlation functional of Lee, Yang and Parr (LYP))
[33,34]. The triple-zeta quality basis set with polarization and diffuse functions denoted 6-311G++(2p,2d) has been used. The bulk
solvent effects (acetonitrile, toluene and N,N-dimethylformamide
as in the experiments) have been included through the Integral
Equation Formalism version of the Polarizable Continuum Model
(IEF-PCM) [35]. All potential energy surface (PES) minima found
upon optimization were confirmed by frequency calculation.
Molecular orbitals and Fukui functions were rendered under
GaussView; cube densities were generated and manipulated using
the Cubegen and Cubman utilities from the Gaussian09 package.
Conformational search was achieved at the molecular mechanics
(MM) level of theory using the MM+ force field under Hyperchem
8.0 software, within a random generation scheme. Dihedral energy
plots were generated with the same program at the Parameterized
Model 3 (PM3) semi-empirical level of theory.
4. Investigated reactions
The herein studied reactions involve the nucleophilic ring opening of three activated aziridines, one neutral and two cationic species. Obviously, these reactions can occur on two positions (the
carbons C2 and C3, see Scheme 1) affording distinct regioisomers
with an inverse stereochemistry.
G. Berger / Computational and Theoretical Chemistry 1010 (2013) 11–18
13
aziridines A1 and A2, this regioselectivity is surprisingly lost when
aziridines are nosyl activated (A3). Indeed, 2-phenyl-3-methyl
substituted nosyl aziridines opened preferentially at the alkyl
substituted position. Notice that bulkier alkyl groups did not allowed the opening on that position and such aziridines no longer
open at room temperature, indicating a kinetically unfavorable attack on that position due to increased steric hindrance. Scheme 2
summarizes the experimental ratios of regioisomers [38].
5. Results and discussion
5.1. Atomic charges and electrostatic potential
Scheme 1. The nucleophilic ring-opening of nitrogen-activated aziridines; Y
represents the activating group on the nitrogen (Fig. 1a). Three types of aziridines
were compared (Fig. 1b).
Scheme 2. The regioisomeric products and their experimental ratios.
Although aziridine ring opening proceeds quite exclusively
through a SN2 (bimolecular nucleophilic substitution) mechanism,
the associated regiochemistry is not that simple. The latter is subject to changes depending on the N-activation method and the substituents present on the two ring carbons [36]. A comprehensive
review of nucleophilic aziridine ring opening reactions has been
published by Hu [37]. As our team was synthesizing chiral diamines as precursors of platinum(II) anticancer compounds, an
unexpected loss of regioselectivity was observed during the ring
opening of nosyl-activated aziridines [38]. Indeed, in the case of
2-phenyl-3-alkyl substituted aziridines, it is commonly accepted
that the preferred position for nucleophilic attack is the benzylic
position, here C2 [39–41]. However, if that was verified for
In order to study electrostatic arguments that would explain the
regiochemistry, atomic charges were determined by three different
methods: MPA, NPA and HP. Population analysis allows the attribution of net atomic charges in molecular systems. However, the
atomic charge is not a physical reality. In quantum mechanics neither an atom nor an atomic charge in a molecule is an observable.
Consequently, different mathematical treatments can be applied to
assign a fraction of the electronic density to a particular atom. Results obtained from these methods should therefore be carefully
analyzed, keeping in mind strengths and weaknesses of each method. Mulliken population analysis is known to be quite inaccurate,
sensitive to conformational equilibriums and strongly dependent
on the basis set choice, so more reliable and robust values should
be obtained from NPA and HP [42–44]. If peculiar charges assignments are often obtained from Mulliken’s scheme, Hirshfeld
charges are usually too small and too close to the isolated atom
[45,46]. A solution to these underestimated results is to compute
a self-consistent variant of the Hirshfeld scheme, the iterative
Hirshfeld method. This method is known to produce good results,
exhibiting high robustness against basis set choice and reproducing efficiently the ESP for organic molecules [47]. Unfortunately,
this method is not yet implemented in quantum mechanical software packages and needs an in-house code to be computed.
Calculated atomic charges for the three aziridines are given in
Fig. 1. In addition, electrostatic potential has been plotted on the
0.0004 isodensity surface. This value is known to give a good representation of reactivity indicators, as it models the van der Waals surface [48,49]. It can be seen from Fig. 1 that only MPA delivers
significantly different charge values for the two ring carbons of the
aziridines. Charges obtained from NPA and HP calculations are not
discriminative between the two electrophilic positions. However,
the difference between C2 and C3 atomic charges from MPA scheme
is clearly in contradiction with the experiments for A1, as we would
expect to obtain more positive values at the reaction site leading to
the majority isomer. Indeed, for aziridine A1 MPA leads to a more
negative value at the experimentally highest reactive site and this
Fig. 1. Computed ESP and atomic charges on C2 and C3; MPA (red), NPA (blue) and HP (black). (For interpretation of the references to color in this figure legend, the reader is
referred to the web version of this article.)
14
G. Berger / Computational and Theoretical Chemistry 1010 (2013) 11–18
cannot account for a more favorable electrostatic interaction with
the electron rich nucleophile.
It seems then clearly impossible to get a prediction of site reactivity through a simple electrostatic analysis for two main reasons:
firstly because atomic charges deliver a far too simple picture of
the interaction between the nucleophile and the electrophile and
secondly, variability over population analysis schemes gives uncertain results.
5.2. Frontier molecular orbitals and Fukui functions
If electrostatic arguments failed to explain the observed regiochemistry, FMO theory and conceptual DFT should be more successful. DFT derived reactivity descriptors have been recently and
intensively used in the field of organic chemistry, either as global
reactivity indices or as a tool for describing the local reactivity in
molecules [50–58]. These descriptors have been successfully applied to aziridines and aziridinium species [36,59,60].
Within the FMO context, one expect the LUMO to be localized in
the neighborhood of the most reactive carbon atom toward nucleophilic attack, as the interaction with the nucleophile’s HOMO
would be maximized in that situation [9]. FMO theory estimates
the slopes to the possible transition states at the early stage of a
reaction and the preferred reaction path will thus have the best
FMO overlap when the contact intermediates are formed. The
interaction energy at this stage of a reaction can be estimated
according to the following equation [61–63]:
DE ¼
qrðnucÞ qsðelecÞ ðcrðnucÞ csðelecÞ bÞ2
HOMO
Rrs
ELUMO
ðnucÞ EðelecÞ
ð13Þ
With the above results obtained from atomic charges calculation, one would expect the coulombian term of Eq. (13) to be very
similar for the two ring carbons (at least from NPA and HPA
charges) and moreover close to zero, consequently leading to an
interaction energy governed by the LUMO localization (i.e. the second part of Eq. (13)). In a similar way, the most reactive site for aziridine nucleophilic opening should exhibit a higher value of the
Fukui function f þ ð~
rÞ in its local environment [6]. As stated in paragraph 2, the Fukui function goes beyond FMO theory by adding
a correct term including orbital relaxation. These electronic properties being clearly geometry dependent, quantum mechanical calculations were performed for all chemically relevant conformers.
The easiest case comes with aziridinium A1 as it seems obvious
that only one bond rotation is relevant. Indeed, if the two aziridine
ring substituents could be rotated, the methyl is from the one hand
highly isotropic and from the other hand very slightly implicated in
the position of the LUMO and f þ ð~
rÞ. As indicated in Fig. 2, one dihedral was varied to rotate the phenyl group and the PM3 energy was
plotted every 10°. Obviously, due to the C2 symmetry of the mono-
Fig. 2. Dihedral plot, LUMO and Fukui function for aziridinium A1. LUMO isovalue = 0.04; f þ ð~
rÞ isovalue = 0.0015; contour plots in the plane formed by the three aziridine ring
atoms (N, C2 and C3). LUMO contour isovalues: 0.01(1), 0.02 (2), 0.04 (3), 0.08 (4). f þ ð~
rÞ contour isovalues: 0.005 (1), 0.01 (2), 0.02 (3), 0.04 (4).
G. Berger / Computational and Theoretical Chemistry 1010 (2013) 11–18
substituted benzene ring, half of the plot would have been sufficient. Two conformers, indicated by the arrows, were run through
quantum calculations, both single point and optimization schemes.
Geometry optimization at the B3LYP level of the above mentioned
conformers led to a single rotamer for which are represented local
reactivity descriptors. For the two conformers, both the LUMO and
the Fukui function are located around the benzylic position, maximizing the interaction with the approaching nucleophile’s HOMO
on that position. This is in total accordance with the observed excess (80:20) of product 1 resulting from the nucleophilic attack on
the C2 position (Scheme 2). Contour plots are represented in the
plane defined by the three ring-atoms, considered as the most relevant plane as the nucleophile approach backward to the leaving
group (here the nitrogen) in a SN2 reaction mechanism.
The conformational landscape of aziridinium A2 is more complicated and shows several local minima. A conformational search
was therefore performed with a maximum energy gap of 3 kcal/
mol between the highest and the lowest energy conformers [64].
Eight conformations were found below the 3 kcal/mol limit and
therefore considered for quantum calculations. The dihedrals that
were varied are represented in Fig. 3 and all conformers can be
found in the supplementary material. All of them show a clear
preference for nucleophilic attack on C2 as observed experimentally and the lowest energy structure and its associated LUMO
and f þ ð~
rÞ densities are represented in Fig. 3.
15
Thus, for both aziridiniums considered (A1 and A2), the LUMO
and the Fukui function are positioned in the neighborhood of the
aryl substituted carbon and moreover on the phenyl ring itself. If
the phenyl is clearly not a site for nucleophilic attack (high electronic density of the aromatic system, no ring tension), it drives
the opening on C2 by positioning the LUMO on the benzylic carbon.
This explains the well-known activation effect of an aromatic ring
on aziridines.
The third aziridine that was considered (A3) shows a very different picture in terms of LUMO and f þ ð~
rÞ localization (Fig. 4); both
being now completely shifted to the nosyl group, since the nitrobenzene ring has become the part of the molecule that will better
stabilize an increase of the electronic density. Having this in mind,
the relevant conformational equilibrium is now the free rotation of
the nosyl group around the nitrogen–sulfur r bond, as it will
change the position of the LUMO and f þ ð~
rÞ on the aziridine ring
carbons. The dihedral map for this rotation shows a relatively flat
potential for values with the bulky nosyl group away from the aziridine substituents and two local minima were found upon DFT
optimization, corresponding to the two lowest energy conformations of the nosyl group; one is orienting the nucleophilic attack
on C2, when the other orients it on C3. The energy gap between
the two conformers is of 0.31 kJ/mol (A3a being located at the absolute minimum), leading to a Maxwell–Boltzmann two-state distribution of 89:11 at 298 K, the reaction temperature. The orbital
Fig. 3. LUMO surface and Fukui function for aziridinium A2. LUMO isovalue = 0.04; f þ ð~
rÞ isovalue = 0.0015; contour plots in the plane formed by the three aziridine ring atoms
(N, C2 and C3). LUMO contour isovalues: 0.01(1), 0.02 (2), 0.04 (3), 0.08 (4). f þ ð~
rÞ contour isovalues: 0.005 (1), 0.01 (2), 0.02 (3), 0.04 (4).
16
G. Berger / Computational and Theoretical Chemistry 1010 (2013) 11–18
configuration is then much more balanced that what we have seen
for the aziridinium species A1 and A2 and the observed loss of regiochemistry is rationalized by this shift of the LUMO to the nosyl
group and by the rotational freedom of this group around the r
S–N bond; the same conclusion being made for the Fukui function.
However, the lowest energy state (A3a conformer) has a higher
localization of the LUMO and the Fukui function around C3 and
thus the relative proportion of A3a and A3b could maybe explain
the slight excess (65:35) of the product resulting from the attack
of the nucleophile on C3.
5.3. Fukui indices: Condensed to atoms Fukui functions
Fukui indices resulting from the condensation of the Fukui function to the atoms C2 and C3 were calculated as given by Eq. (11)
using the three atomic charges schemes (Table 1).
Fig. 4. LUMO surface for aziridinium A3a and A3b (a). Nucleophilic Fukui function density surface for aziridinium A3a and A3b, positive values only (b). LUMO isovalue = 0.05;
f þ ð~
rÞ isovalue = 0.004; contour plots in the plane formed by the three aziridine ring atoms (N, C2 and C3). LUMO contour isovalues: 0.005 (1), 0.01 (2), 0.02 (3), 0.04 (4). f þ ð~
rÞ
contour isovalues: 0.005 (1), 0.01 (2), 0.02 (3), 0.04 (4).
Table 1
Fukui indices on carbons C2 and C3.
Aziridine
A1
A2
A3a
A3b
Fþ
C2
Fþ
C2
þ
Fþ
C2 F C2
MPA
NPA
HP
MPA
NPA
HP
MPA
NPA
HP
0.385
0.021
0.067
0.014
0.034
0.026
0.020
0.009
0.056
0.027
0.003
0.007
0.248
0.010
0.013
0.120
0.026
0.013
0.007
0.002
0.048
0.012
0.004
0.004
0.137
0.031
0.054
0.026
0.006
0.013
0.027
0.007
0.008
0.015
0.001
0.003
0.031
0.017
0.032
0.030
0.045
0.020
0.039
0.061
0.137
0.381
0.030
0.020
0.053
0.081
0.025
0.058
0.051
0.018
0.355
0.477
0.001
0.370
0.017
0.024
0.054
0.053
0.002
0.036
0.039
0.020
0.021
0.024
0.045
0.020
0.017
0.086
0.090
0.002
0.058
0.026
0.005
0.015
0.019
0.027
NPA
MPA
0.034
0.038
0.071
0.068
0.001
0.053
0.065
0.035
0.036
0.038
0.054
0.006
0.003
0.157
0.152
0.004
0.096
0.034
0.008
0.006
0.009
0.028
0.019
0.019
1.085
1.185
0.017
1.043
0.385
0.022
0.150
0.035
0.409
HP
NPA
MPA
0.065
0.065
0.029
0.030
0.082
0.044
0.032
0.061
0.059
0.057
0.035
HP
NPA
0.028
0.055
0.601
0.881
0.060
0.609
0.348
0.019
0.021
0.063
0.134
0.046
0.041
0.004
0.006
0.073
0.018
0.015
0.037
0.035
0.033
0.017
0.087
0.033
0.214
0.156
0.049
0.100
0.033
0.004
0.008
0.034
0.047
MPA
HP
NPA
0.099
0.041
0.134
0.074
0.043
0.043
0.004
0.009
0.009
0.036
0.013
Fþ
C2
Fþ
C2
2
qCNþ1
0.019
0.001
0.595
0.791
0.014
0.667
0.248
0.016
0.019
0.058
0.102
HP
G. Berger / Computational and Theoretical Chemistry 1010 (2013) 11–18
17
Due to the large discrepancies between the atomic charges obtained from the three calculation methods (MPA, NPA and HP), Fukui indices are highly depending on the population analysis
method and the results should therefore be analyzed with caution.
A positive value of the Fukui index emphasizes that an increase of
atomic population arises on that position due to the additional
electron on the molecule and the more positive the value the more
reactive the carbon toward nucleophilic attack. It is then expected
to observe a higher value on C2 (and thus a positive value of DF+)
for A1 and A2 where A3 should give a more balanced pattern. This
is actually true for NPA and HPA calculation but MPA failed to give
a correct condensed value of the Fukui function for A1 (i.e. a positive DF+ value) despite the clear location of f þ ð~
rÞ around C2 as seen
in Fig. 2. Indices calculated for A3 indicates a favorable attack on C3
for the A3a conformer whereas that tendency shifts to C2 for the A3b
conformer in agreement with the experimental loss of regioselectivity and the f þ ð~
rÞ localization.
5.4. Sensitivity of atomic charges and Fukui indices to the basis set
Mulliken charges are known to be particularly sensitive to the basis set choice, and bad results are often obtained when large basis
sets are used [65]. Atomic charges were calculated for aziridinium
A1 using different basis set to make a robustness assessment of the
atomic charges and Fukui indices. The results are presented in Table 2 (means and standard deviations over the basis sets are bolded).
Uncertain charges are indeed obtained from Mulliken population
analysis scheme with very high standard deviation over the basis
sets. Moreover, for some bases unphysical trends are produced. Natural bond order derived scheme (NPA) which like MPA is based on
the density matrix, is much more robust than MPA to the changes
of basis set. Finally, Hirshfeld charges are practically insensible to
basis set quality. Fukui indices thus obviously suffer from the same
drawbacks than atomic charges as they rely on qkN and qkNþ1 values.
0.237
0.499
0.081
0.004
0.008
0.045
0.031
0.048
0.074
0.004
0.013
0.041
0.109
0.157
Mean
Sd
MPA
0.016
0.115
1.335
0.914
0.008
0.626
0.091
0.011
0.091
0.055
0.245
0.082
0.089
0.083
0.083
0.080
0.080
0.071
0.081
0.080
0.081
0.080
HP
NPA
0.119
0.024
0.048
0.016
0.045
0.015
0.022
0.014
0.024
0.017
0.014
0.047
0.056
0.006
0.090
0.074
0.058
0.100
0.003
0.002
0.005
0.032
MPA
0.080
0.079
0.075
0.074
0.072
0.071
0.080
0.072
0.071
0.071
0.071
HP
NPA
0.093
0.036
0.057
0.004
0.053
0.004
0.001
0.004
0.014
0.043
0.019
0.035
0.096
0.025
0.271
0.009
0.417
0.294
0.033
0.059
0.020
0.164
MPA
STO3G
3-21G
6-31G+
6-311G+
6-31G+(d,p)
6-311G+(d,p)
6-311G++(2d,2p)
cc-pVDZ
cc-pVTZ
cc-pVQZ
aug-cc-pVTZ
qCN2
qCN2
Basis set
Table 2
Atomic charges and Fukui indices for aziridine A1 calculated with different basis sets.
2
qCNþ1
6. Conclusion
In the present article, both FMO theory and the Fukui function
were able to rationalize the experimentally observed ratios of isomers resulting from the opening of the three activated aziridines.
For the systems above studied, both LUMO and f þ ð~
rÞ are indeed
better localized near the most reactive carbon. It should be highlighted that the frozen core orbital picture (i.e. FMO theory) does
not suffer from the neglect of relaxation effects for the systems
studied here. There was no clear advantage of the use of the Fukui
function over FMO theory and the localization of the LUMO.
Condensed to atoms Fukui function led to somewhat good prediction of reactivity, however, intrinsic limitation and low robustness of calculated atomic charges does not provide highly reliable
results. As previously mentioned, Mulliken population analysis did
not afford reliable charge values. Indeed, it is the only scheme that
failed to predict the highest reactive carbon for aziridine A1
through the condensation of its Fukui function. The lack of robustness of this method is also demonstrated by its instability upon basis set changes. As expected, NPA gives more robust Fukui indices
and its variability upon basis set changes is far reduced when compared with MPA. Hirshfeld partitioning delivers charge values that
seem to be somewhat independent from the atoms local environment and too close to the isolated atom, computed Fukui indices
from this method are therefore very low and make the comparison
between two reactive sites tricky.
Finally, this work emphasizes again the growing importance of
modern DFT reactivity descriptors in organic chemistry as it shows
its efficacy in solving and rationalizing concrete and practical issues often encountered by synthetic chemists.
18
G. Berger / Computational and Theoretical Chemistry 1010 (2013) 11–18
Appendix A. Supplementary material
Supplementary data associated with this article can be found, in
the online version, at http://dx.doi.org/10.1016/j.comptc.2012.12.
029.
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