Computational and Theoretical Chemistry 1010 (2013) 11–18

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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 2012Received in revised form 27 December 2012Accepted 27 December 2012Available online 13 January 2013

Keywords:Conceptual DFTAziridinesFukui functionFukui indicesReactivity descriptorsPopulation analysis

2210-271X/$ - see front matter � 2013 Elsevier B.V.http://dx.doi.org/10.1016/j.comptc.2012.12.029

⇑ Tel.: +32 2 650 5248; fax: +32 2 650 5249.E-mail address: giberger@ulb.ac.be

a b s t r a c t

Density functional theory calculations have been performed to rationalize the regiochemistry of thenucleophilic 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. Frontiermolecular orbital theory, as well as the Fukui function were able to explain the experimentally observedratios of opening products and a surprising change in regioselectivity upon nitrobenzenesulfonyl activa-tion on the nitrogen. In addition, robustness of atomic charges and Fukui indices to the basis set qualitywas assessed.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

Nitrogen activated aziridines are versatile synthetic intermedi-ates easily produced from b-amino-alcohols [1]. Stereo and regio-controlled nucleophilic opening of aziridines leads to highlyvaluable, 1,2-bifunctionalized chiral compounds and among themvicinal diamines are readily obtained [2]. Beside its interest inmedicine and pharmacy, the 1,2-diamine moiety has taken a largeplace as chiral ligands in transition-metal catalyzed asymmetricsynthesis [3].

During the last decades, density functional theory (DFT) hasundergone fast development, especially in the field of organicchemistry, as the number of accurate exchange–correlation func-tionals increased. Indeed, the apparition of gradient correctedand hybrid functionals in the late 1980s greatly improved thechemical accuracy of the Hohenberg–Kohn theorem [4] basedmethods. The Kohn–Sham formalism [5] and its density-derivedorbitals paved the way to computational methods. In parallel, anew field of application of DFT developed, the so-called conceptualDFT [6]. Parr and Yang followed the idea that well-known chemicalproperties as electronegativity, chemical potentials and affinitiescould be sharply described and calculated manipulating the elec-tronic density as the fundamental quantity [7,8]. Moreover, start-ing from the work of Fukui and its frontier molecular orbitals(FMOs) theory [9], the same authors further generalized the con-

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cept and proposed the Fukui function f ð~rÞ as a tool for describingthe local reactivity in molecules [10,11].

The present study makes the use of DFT-derived reactivitydescriptors to rationalize the regioselective ring opening of threetypes of activated aziridines and makes a comprehensive assess-ment of the electronic effects arising upon the different methodsof activation. As aziridines consist in a three-membered ring withtwo electrophilic positions, the relative reactivity of these twoelectrophilic carbon atoms seems a perfect test for FMO theoryand the Fukui function.

2. Theoretical background

Chemical DFT started to develop in the late 1970s with theidentification of the chemical potential (l) as the first derivativeof the Kohn–Sham energy (E), with respect to the number of elec-trons (N) at constant external potential due to the nuclei (vð~rÞ)[12]:

l ¼ @E@N

� �vð~rÞ¼ �v ð1Þ

The chemical potential (l) is in fact the Lagrange multiplierfrom the normalization constraint of the DFT variational principle[13]. It should be noticed that the chemical potential is the nega-tive of a well-known empiric chemical quantity, the electronega-tivity (v). As a consequence of Eq. (1), electrons will have thetendency to flow from high l zones to low l zones until chemicalpotentials 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 potentialand applying the Maxwell relation, one can define the Fukui func-tion [10,14]:

f ð~rÞ �@qð~rÞ@N

� �vð~rÞ¼ dl

dvð~rÞ

� �N

ð2Þ

However, owing to the discontinuity of N, at least two types ofFukui functions should be defined, one being the right-hand sideand the other the left-hand side derivative at a given number ofelectrons (N = N0):

fþð~rÞ ¼@qð~rÞ@N

� �þvð~rÞ

ð3Þ

f�ð~rÞ ¼@qð~rÞ@N

� ��vð~rÞ

ð4Þ

A third type of Fukui function describing radical reactions canbe used, as a mean of the two others:

f 0ð~rÞ ¼ 12ðfþð~rÞ þ f�ð~rÞÞ ð5Þ

Exploring the chemical significance of these functions, it can beseen that a site where the nucleophilic Fukui function fþð~rÞ has alarge value is a site capable of accepting electronic density and thata large value of f�ð~rÞ indicates an electron donating site. In otherwords, one measures the site reactivity towards nucleophilic attackwhile the other measures the site reactivity towards electrophilic at-tack. It should be emphasized that in the frozen core orbital approx-imation where orbital relaxation is neglected, these functions can bereduced to the squares of the lowest unoccupied and highest occu-pied molecular orbitals (respectively, the LUMO and the HOMO). In-deed, expressing the electronic density in terms of the Kohn–Shamspin–orbitals (Eq. (6)) and taking the derivative with respect to N,gives a mathematical expression of the Fukui functions as the fron-tier molecular orbitals plus a correct term that includes the orbitalrelaxation effects as seen in Eq. (7) and (8) [15]:

qð~rÞ ¼XN

i¼1

j/ðiÞð~rÞj2 ð6Þ

fþð~rÞ ¼ j/Nþ1ð~rÞ j

2 þXN

i¼1

@j/ðiÞð~rÞj2

@N

!vð~rÞ

� j/LUMOð~rÞ j

2 ¼ qLUMOð~rÞ ð7Þ

f�ð~rÞ ¼ j/Nð~rÞj

2 þXN

i¼1

@j/ðiÞð~rÞj2

@N

!vð~rÞ

� j/HOMOð~rÞ j2 ¼ qHOMO

ð~rÞ ð8Þ

Practically, computing Fukui functions is not obvious and asolution is to take the finite difference approximation betweenthe N = N0 and N = (N0 ± 1) electronic total densities, as expressedin the following equations:

fþð~rÞ � qNþ1 � qN ð9Þ

f�ð~rÞ � qN � qN�1 ð10Þ

Nevertheless, other schemes can be used to compute reactivityindices avoiding this rough estimation of derivatives due to the useof an integer change on N. These methods introduce fractionaloccupation numbers from Janak‘s theorem [16] to produce infini-tesimal changes on the number of electrons [17–20]. The firstmethod using the finite difference approximation has been em-ployed in the present study.

A three dimensional representation of the functions is then ob-tained but a chemist’s dream would be to assign indices to atoms,

rather than dealing with spatially dependent functions. It thereforeneeds the integration of the Fukui function along the portion ofspace that could be attributed to a certain atom belonging to amolecule, leading to condensed to atoms Fukui functions (i.e. Fukuiindices) [21]. This procedure is analogous to the condensation ofthe electronic density to atoms, leading to the assignment of atom-ic 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):

Fþk ¼Z @qk

ð~rÞ

@N

!þvð~rÞ

d~r ¼ @pk

@N

� �þvð~rÞ¼ pk

Nþ1 � pkN ¼ qk

N � qkNþ1 ð11Þ

F�k ¼Z @qk

ð~rÞ

@N

!�vð~rÞ

d~r ¼ @pk

@N

� ��vð~rÞ¼ pk

N � pkN�1 ¼ qk

N�1 � qkN ð12Þ

Electrophilicity indices have recently been reviewed by Chatt-araj et al. [23]. Reactivity indices were comprehensively discussedby Chermette [24].

Carving the molecular electronic density into its atomic constit-uents is far from trivial [25]. Different methods can be employed,often leading to significantly different results and misleadingchemical interpretations. Moreover, it is not easy to choose themost appropriate method based on theoretical arguments. Threedifferent charges schemes were used in the present study to com-pute atoms-in-molecules (AIMs) properties: Mulliken populationanalysis (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 chargeswere not used as they are known to often produce unphysical re-sults, despite the good reproduction of the multipolar environmentof the molecule [31].

3. Computational details

All quantum mechanical calculations have been achieved usingthe Gaussian09 software package [32]. Geometries of all the inves-tigated systems were optimized at the density functional theory le-vel using the B3LYP functional (combination of exchange fromBecke’s three parameter hybrid exchange functional (B3) withthe dynamical correlation functional of Lee, Yang and Parr (LYP))[33,34]. The triple-zeta quality basis set with polarization and dif-fuse functions denoted 6-311G++(2p,2d) has been used. The bulksolvent effects (acetonitrile, toluene and N,N-dimethylformamideas in the experiments) have been included through the IntegralEquation Formalism version of the Polarizable Continuum Model(IEF-PCM) [35]. All potential energy surface (PES) minima foundupon optimization were confirmed by frequency calculation.Molecular orbitals and Fukui functions were rendered underGaussView; cube densities were generated and manipulated usingthe 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 Hyperchem8.0 software, within a random generation scheme. Dihedral energyplots were generated with the same program at the ParameterizedModel 3 (PM3) semi-empirical level of theory.

4. Investigated reactions

The herein studied reactions involve the nucleophilic ring open-ing of three activated aziridines, one neutral and two cationic spe-cies. Obviously, these reactions can occur on two positions (thecarbons C2 and C3, see Scheme 1) affording distinct regioisomerswith an inverse stereochemistry.

Scheme 1. The nucleophilic ring-opening of nitrogen-activated aziridines; Yrepresents the activating group on the nitrogen (Fig. 1a). Three types of aziridineswere compared (Fig. 1b).

Scheme 2. The regioisomeric products and their experimental ratios.

G. Berger / Computational and Theoretical Chemistry 1010 (2013) 11–18 13

Although aziridine ring opening proceeds quite exclusivelythrough a SN2 (bimolecular nucleophilic substitution) mechanism,the associated regiochemistry is not that simple. The latter is sub-ject to changes depending on the N-activation method and the sub-stituents present on the two ring carbons [36]. A comprehensivereview of nucleophilic aziridine ring opening reactions has beenpublished by Hu [37]. As our team was synthesizing chiral dia-mines as precursors of platinum(II) anticancer compounds, anunexpected loss of regioselectivity was observed during the ringopening of nosyl-activated aziridines [38]. Indeed, in the case of2-phenyl-3-alkyl substituted aziridines, it is commonly acceptedthat the preferred position for nucleophilic attack is the benzylicposition, here C2 [39–41]. However, if that was verified for

Fig. 1. Computed ESP and atomic charges on C2 and C3; MPA (red), NPA (blue) and HP (breferred to the web version of this article.)

aziridines A1 and A2, this regioselectivity is surprisingly lost whenaziridines are nosyl activated (A3). Indeed, 2-phenyl-3-methylsubstituted nosyl aziridines opened preferentially at the alkylsubstituted position. Notice that bulkier alkyl groups did not al-lowed the opening on that position and such aziridines no longeropen at room temperature, indicating a kinetically unfavorable at-tack on that position due to increased steric hindrance. Scheme 2summarizes the experimental ratios of regioisomers [38].

5. Results and discussion

5.1. Atomic charges and electrostatic potential

In order to study electrostatic arguments that would explain theregiochemistry, atomic charges were determined by three differentmethods: MPA, NPA and HP. Population analysis allows the attri-bution of net atomic charges in molecular systems. However, theatomic charge is not a physical reality. In quantum mechanics nei-ther an atom nor an atomic charge in a molecule is an observable.Consequently, different mathematical treatments can be applied toassign a fraction of the electronic density to a particular atom. Re-sults obtained from these methods should therefore be carefullyanalyzed, keeping in mind strengths and weaknesses of each meth-od. Mulliken population analysis is known to be quite inaccurate,sensitive to conformational equilibriums and strongly dependenton the basis set choice, so more reliable and robust values shouldbe obtained from NPA and HP [42–44]. If peculiar charges assign-ments are often obtained from Mulliken’s scheme, Hirshfeldcharges are usually too small and too close to the isolated atom[45,46]. A solution to these underestimated results is to computea self-consistent variant of the Hirshfeld scheme, the iterativeHirshfeld method. This method is known to produce good results,exhibiting high robustness against basis set choice and reproduc-ing efficiently the ESP for organic molecules [47]. Unfortunately,this method is not yet implemented in quantum mechanical soft-ware packages and needs an in-house code to be computed.

Calculated atomic charges for the three aziridines are given inFig. 1. In addition, electrostatic potential has been plotted on the0.0004 isodensity surface. This value is known to give a good repre-sentation of reactivity indicators, as it models the van der Waals sur-face [48,49]. It can be seen from Fig. 1 that only MPA deliverssignificantly different charge values for the two ring carbons of theaziridines. Charges obtained from NPA and HP calculations are notdiscriminative between the two electrophilic positions. However,the difference between C2 and C3 atomic charges from MPA schemeis clearly in contradiction with the experiments for A1, as we wouldexpect to obtain more positive values at the reaction site leading tothe majority isomer. Indeed, for aziridine A1 MPA leads to a morenegative value at the experimentally highest reactive site and this

lack). (For interpretation of the references to color in this figure legend, the reader is

14 G. Berger / Computational and Theoretical Chemistry 1010 (2013) 11–18

cannot account for a more favorable electrostatic interaction withthe electron rich nucleophile.

It seems then clearly impossible to get a prediction of site reac-tivity through a simple electrostatic analysis for two main reasons:firstly because atomic charges deliver a far too simple picture ofthe interaction between the nucleophile and the electrophile andsecondly, variability over population analysis schemes gives uncer-tain results.

5.2. Frontier molecular orbitals and Fukui functions

If electrostatic arguments failed to explain the observed regio-chemistry, FMO theory and conceptual DFT should be more suc-cessful. DFT derived reactivity descriptors have been recently andintensively used in the field of organic chemistry, either as globalreactivity indices or as a tool for describing the local reactivity inmolecules [50–58]. These descriptors have been successfully ap-plied to aziridines and aziridinium species [36,59,60].

Within the FMO context, one expect the LUMO to be localized inthe neighborhood of the most reactive carbon atom toward nucle-ophilic attack, as the interaction with the nucleophile’s HOMOwould be maximized in that situation [9]. FMO theory estimatesthe slopes to the possible transition states at the early stage of areaction and the preferred reaction path will thus have the bestFMO overlap when the contact intermediates are formed. The

Fig. 2. Dihedral plot, LUMO and Fukui function for aziridinium A1. LUMO isovalue = 0.04;atoms (N, C2 and C3). LUMO contour isovalues: 0.01(1), 0.02 (2), 0.04 (3), 0.08 (4). fþð~rÞ

interaction energy at this stage of a reaction can be estimatedaccording to the following equation [61–63]:

DE ¼qrðnucÞ � qsðelecÞ

�Rrs� ðcrðnucÞ � csðelecÞ � bÞ2

ELUMOðnucÞ � EHOMO

ðelecÞð13Þ

With the above results obtained from atomic charges calcula-tion, one would expect the coulombian term of Eq. (13) to be verysimilar for the two ring carbons (at least from NPA and HPAcharges) and moreover close to zero, consequently leading to aninteraction energy governed by the LUMO localization (i.e. the sec-ond part of Eq. (13)). In a similar way, the most reactive site for azi-ridine nucleophilic opening should exhibit a higher value of theFukui function fþð~rÞ in its local environment [6]. As stated in par-agraph 2, the Fukui function goes beyond FMO theory by addinga correct term including orbital relaxation. These electronic prop-erties being clearly geometry dependent, quantum mechanical cal-culations were performed for all chemically relevant conformers.

The easiest case comes with aziridinium A1 as it seems obviousthat only one bond rotation is relevant. Indeed, if the two aziridinering substituents could be rotated, the methyl is from the one handhighly isotropic and from the other hand very slightly implicated inthe position of the LUMO and fþð~rÞ. As indicated in Fig. 2, one dihe-dral was varied to rotate the phenyl group and the PM3 energy wasplotted every 10�. Obviously, due to the C2 symmetry of the mono-

fþð~rÞ isovalue = 0.0015; contour plots in the plane formed by the three aziridine ringcontour isovalues: 0.005 (1), 0.01 (2), 0.02 (3), 0.04 (4).

G. Berger / Computational and Theoretical Chemistry 1010 (2013) 11–18 15

substituted benzene ring, half of the plot would have been suffi-cient. Two conformers, indicated by the arrows, were run throughquantum calculations, both single point and optimization schemes.Geometry optimization at the B3LYP level of the above mentionedconformers led to a single rotamer for which are represented localreactivity descriptors. For the two conformers, both the LUMO andthe Fukui function are located around the benzylic position, maxi-mizing the interaction with the approaching nucleophile’s HOMOon that position. This is in total accordance with the observed ex-cess (80:20) of product 1 resulting from the nucleophilic attack onthe C2 position (Scheme 2). Contour plots are represented in theplane defined by the three ring-atoms, considered as the most rel-evant plane as the nucleophile approach backward to the leavinggroup (here the nitrogen) in a SN2 reaction mechanism.

The conformational landscape of aziridinium A2 is more compli-cated and shows several local minima. A conformational searchwas 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 andtherefore considered for quantum calculations. The dihedrals thatwere varied are represented in Fig. 3 and all conformers can befound in the supplementary material. All of them show a clearpreference for nucleophilic attack on C2 as observed experimen-tally and the lowest energy structure and its associated LUMOand fþð~rÞ densities are represented in Fig. 3.

Fig. 3. LUMO surface and Fukui function for aziridinium A2. LUMO isovalue = 0.04; fþð~rÞ is(N, C2 and C3). LUMO contour isovalues: 0.01(1), 0.02 (2), 0.04 (3), 0.08 (4). fþð~rÞ contou

Thus, for both aziridiniums considered (A1 and A2), the LUMOand the Fukui function are positioned in the neighborhood of thearyl substituted carbon and moreover on the phenyl ring itself. Ifthe phenyl is clearly not a site for nucleophilic attack (high elec-tronic density of the aromatic system, no ring tension), it drivesthe opening on C2 by positioning the LUMO on the benzylic carbon.This explains the well-known activation effect of an aromatic ringon aziridines.

The third aziridine that was considered (A3) shows a very differ-ent picture in terms of LUMO and fþð~rÞ localization (Fig. 4); bothbeing now completely shifted to the nosyl group, since the nitro-benzene ring has become the part of the molecule that will betterstabilize an increase of the electronic density. Having this in mind,the relevant conformational equilibrium is now the free rotation ofthe nosyl group around the nitrogen–sulfur r bond, as it willchange the position of the LUMO and fþð~rÞ on the aziridine ringcarbons. The dihedral map for this rotation shows a relatively flatpotential for values with the bulky nosyl group away from the azi-ridine substituents and two local minima were found upon DFToptimization, corresponding to the two lowest energy conforma-tions of the nosyl group; one is orienting the nucleophilic attackon C2, when the other orients it on C3. The energy gap betweenthe two conformers is of 0.31 kJ/mol (A3a being located at the abso-lute minimum), leading to a Maxwell–Boltzmann two-state distri-bution of 89:11 at 298 K, the reaction temperature. The orbital

ovalue = 0.0015; contour plots in the plane formed by the three aziridine ring atomsr 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 seenfor the aziridinium species A1 and A2 and the observed loss of reg-iochemistry is rationalized by this shift of the LUMO to the nosylgroup and by the rotational freedom of this group around the rS–N bond; the same conclusion being made for the Fukui function.However, the lowest energy state (A3a conformer) has a higherlocalization of the LUMO and the Fukui function around C3 andthus the relative proportion of A3a and A3b could maybe explain

Fig. 4. LUMO surface for aziridinium A3a and A3b (a). Nucleophilic Fukui function densityfþð~rÞ isovalue = 0.004; contour plots in the plane formed by the three aziridine ring atomcontour isovalues: 0.005 (1), 0.01 (2), 0.02 (3), 0.04 (4).

Table 1Fukui indices on carbons C2 and C3.

Aziridine FþC2FþC2

MPA NPA HP MPA

A1 �0.385 0.034 0.056 �0.248A2 0.021 0.026 0.027 �0.010A3a �0.067 �0.020 0.003 �0.013A3b 0.014 0.009 0.007 �0.120

the slight excess (65:35) of the product resulting from the attackof the nucleophile on C3.

5.3. Fukui indices: Condensed to atoms Fukui functions

Fukui indices resulting from the condensation of the Fukui func-tion to the atoms C2 and C3 were calculated as given by Eq. (11)using the three atomic charges schemes (Table 1).

surface for aziridinium A3a and A3b, positive values only (b). LUMO isovalue = 0.05;s (N, C2 and C3). LUMO contour isovalues: 0.005 (1), 0.01 (2), 0.02 (3), 0.04 (4). fþð~rÞ

FþC2� FþC2

NPA HP MPA NPA HP

0.026 0.048 �0.137 0.006 0.0080.013 0.012 0.031 0.013 0.0150.007 0.004 �0.054 �0.027 �0.0010.002 0.004 0.026 0.007 0.003

Tabl

e2

Ato

mic

char

ges

and

Fuku

iin

dice

sfo

raz

irid

ine

A1

calc

ulat

edw

ith

diff

eren

tba

sis

sets

.

Bas

isse

tqC

2N

qC2

NqC

2Nþ

1qC

2Nþ

1Fþ C

2Fþ C

2

MPA

NPA

HP

MPA

NPA

HP

MPA

NPA

HP

MPA

NPA

HP

MPA

NPA

HP

MPA

NPA

HP

STO

3G0.

035

0.09

30.

080

0.04

70.

119

0.08

20.

016

0.08

70.

046

0.02

80.

099

0.06

50.

019

0.00

60.

034

0.01

90.

020

0.01

73-

21G

�0.

096

�0.

036

0.07

9�

0.05

6�

0.02

40.

089

�0.

115

�0.

033

0.04

1�

0.05

5�

0.04

10.

065

0.01

9�

0.00

30.

038

�0.

001

0.01

70.

024

6-31

G+

0.02

5�

0.05

70.

075

�0.

006

�0.

048

0.08

31.

335

�0.

214

0.00

4�

0.60

1�

0.13

40.

029

�1.

085

0.15

70.

071

0.59

50.

086

0.05

46-

311G

+�

0.27

1�

0.00

40.

074

0.09

00.

016

0.08

30.

914

�0.

156

0.00

6�

0.88

1�

0.07

40.

030

�1.

185

0.15

20.

068

�0.

791

0.09

00.

053

6-31

G+(

d,p)

�0.

009

�0.

053

0.07

20.

074

�0.

045

0.08

00.

008

�0.

049

0.07

30.

060

�0.

043

0.08

2�

0.01

7�

0.00

4�

0.00

10.

014

�0.

002

�0.

002

6-31

1G+(

d,p)

�0.

417

�0.

004

0.07

10.

058

0.01

50.

080

0.62

6�

0.10

00.

018

�0.

609

�0.

043

0.04

4�

1.04

30.

096

0.05

30.

667

0.05

80.

036

6-31

1G++

(2d,

2p)

�0.

294

0.00

10.

080

0.10

00.

022

0.07

10.

091

�0.

033

0.01

50.

348

�0.

004

0.03

2�

0.38

50.

034

0.06

5�

0.24

80.

026

0.03

9cc

-pV

DZ

0.03

3�

0.00

40.

072

�0.

003

0.01

40.

081

0.01

10.

004

0.03

7�

0.01

90.

009

0.06

10.

022

�0.

008

0.03

50.

016

0.00

50.

020

cc-p

VTZ

�0.

059

�0.

014

0.07

1�

0.00

20.

024

0.08

0�

0.09

1�

0.00

80.

035

0.02

10.

009

0.05

9�

0.15

0�

0.00

60.

036

�0.

019

0.01

50.

021

cc-p

VQ

Z0.

020

�0.

043

0.07

10.

005

�0.

017

0.08

10.

055

�0.

034

0.03

30.

063

�0.

036

0.05

7�

0.03

5�

0.00

90.

038

�0.

058

0.01

90.

024

aug-

cc-p

VTZ

�0.

164

�0.

019

0.07

10.

032

0.01

40.

080

�0.

245

�0.

047

0.01

70.

134

�0.

013

0.03

5�

0.40

90.

028

0.05

4�

0.10

20.

027

0.04

5

Mea

n�

0.10

9�

0.01

30.

074

0.03

10.

008

0.08

10.

237

�0.

053

0.03

0�

0.13

7�

0.02

50.

051

�0.

355

0.03

90.

045

0.00

10.

032

0.03

1Sd

0.15

70.

041

0.00

40.

048

0.04

50.

004

0.49

90.

081

0.02

00.

381

0.05

80.

018

0.47

70.

061

0.02

00.

370

0.03

00.

017

G. Berger / Computational and Theoretical Chemistry 1010 (2013) 11–18 17

Due to the large discrepancies between the atomic charges ob-tained from the three calculation methods (MPA, NPA and HP), Fu-kui indices are highly depending on the population analysismethod and the results should therefore be analyzed with caution.A positive value of the Fukui index emphasizes that an increase ofatomic population arises on that position due to the additionalelectron on the molecule and the more positive the value the morereactive the carbon toward nucleophilic attack. It is then expectedto 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. Thisis actually true for NPA and HPA calculation but MPA failed to givea correct condensed value of the Fukui function for A1 (i.e. a posi-tive DF+ value) despite the clear location of fþð~rÞ around C2 as seenin 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 regioselec-tivity and the fþð~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 ba-sis set choice, and bad results are often obtained when large basissets are used [65]. Atomic charges were calculated for aziridiniumA1 using different basis set to make a robustness assessment of theatomic charges and Fukui indices. The results are presented in Ta-ble 2 (means and standard deviations over the basis sets are bolded).Uncertain charges are indeed obtained from Mulliken populationanalysis scheme with very high standard deviation over the basissets. Moreover, for some bases unphysical trends are produced. Nat-ural bond order derived scheme (NPA) which like MPA is based onthe density matrix, is much more robust than MPA to the changesof basis set. Finally, Hirshfeld charges are practically insensible tobasis set quality. Fukui indices thus obviously suffer from the samedrawbacks than atomic charges as they rely on qk

N and qkNþ1 values.

6. Conclusion

In the present article, both FMO theory and the Fukui functionwere able to rationalize the experimentally observed ratios of iso-mers resulting from the opening of the three activated aziridines.For the systems above studied, both LUMO and fþð~rÞ are indeedbetter localized near the most reactive carbon. It should be high-lighted that the frozen core orbital picture (i.e. FMO theory) doesnot suffer from the neglect of relaxation effects for the systemsstudied here. There was no clear advantage of the use of the Fukuifunction over FMO theory and the localization of the LUMO.

Condensed to atoms Fukui function led to somewhat good pre-diction of reactivity, however, intrinsic limitation and low robust-ness of calculated atomic charges does not provide highly reliableresults. As previously mentioned, Mulliken population analysis didnot afford reliable charge values. Indeed, it is the only scheme thatfailed to predict the highest reactive carbon for aziridine A1

through the condensation of its Fukui function. The lack of robust-ness of this method is also demonstrated by its instability upon ba-sis set changes. As expected, NPA gives more robust Fukui indicesand its variability upon basis set changes is far reduced when com-pared with MPA. Hirshfeld partitioning delivers charge values thatseem to be somewhat independent from the atoms local environ-ment and too close to the isolated atom, computed Fukui indicesfrom this method are therefore very low and make the comparisonbetween two reactive sites tricky.

Finally, this work emphasizes again the growing importance ofmodern DFT reactivity descriptors in organic chemistry as it showsits efficacy in solving and rationalizing concrete and practical is-sues 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, inthe online version, at http://dx.doi.org/10.1016/j.comptc.2012.12.029.

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