Molecular Dynamics Simulations of the Initial-State Predict Product Distributions of Dediazoniation of Aryldiazonium in Binary Solvents

Article abstract of DOI:10.1021/acs.joc.5b01289

The dediazoniation of aryldiazonium salts in mixed solvents proceeds by a borderline S_N1 and S_N2 pathway, and product distribution should be proportional to the composition of the solvation shell of the carbon attached to the -N₂ group (ipso carbon). The rates of dediazoniation of 2,4,6-trimethylbenzenediazonium in water, methanol, ethanol, propanol, and acetonitrile were similar, but measured product distributions were noticeably dependent on the nature of the water/cosolvent mixture. Here we demonstrated that solvent distribution in the first solvation shell of the ipso carbon, calculated from classical molecular dynamics simulations, is equal to the measured product distribution. Furthermore, we showed that regardless of the charge distribution of the initial state, i.e., whether the positive charge is smeared over the molecule or localized on phenyl moiety, the solvent distribution around the reaction center is nearly the same.

Full text of DOI:10.1021/acs.joc.5b01289

Article
pubs.acs.org/joc
Molecular Dynamics Simulations of the Initial-State Predict Product
Distributions of Dediazoniation of Aryldiazonium in Binary Solvents
Gustavo N. Cruz,† Filipe S. Lima,*,† Luís G. Dias,‡ Omar A. El Seoud,§ Dominik Horinek,∥
Hernan Chaimovich,† and Iolanda M. Cuccovia*,†
§
‡
†Departamento de Bioquímica, Instituto de Química, and Department of Química, Fac. Filosofia, Cien
̂
cias e Letras de Ribeirao
̃
Preto, Universidade de Sao Paulo, Sao Paulo, Brazil
∥Institute fur Physikalische und Theoretischen Chemie, University of Regensburg, 93053 Regensburg, Germany
̃
̃
̈
S
* Supporting Information
ABSTRACT: The dediazoniation of aryldiazonium salts in mixed solvents proceeds by a
borderline SN1 and SN2 pathway, and product distribution should be proportional to the
composition of the solvation shell of the carbon attached to the −N2 group (ipso carbon).
The rates of dediazoniation of 2,4,6-trimethylbenzenediazonium in water, methanol,
ethanol, propanol, and acetonitrile were similar, but measured product distributions were
noticeably dependent on the nature of the water/cosolvent mixture. Here we demonstrated
that solvent distribution in the first solvation shell of the ipso carbon, calculated from
classical molecular dynamics simulations, is equal to the measured product distribution.
Furthermore, we showed that regardless of the charge distribution of the initial state, i.e.,
whether the positive charge is smeared over the molecule or localized on phenyl moiety,
the solvent distribution around the reaction center is nearly the same.
methylbenzenediazonium ions in ethanol/water mixtures are
INTRODUCTION
■
ethyl methylphenyl ethers and cresols.9 All three compounds
produce equal amounts of products in a water molar fraction
(xwater = nwater/(nwater + nsolvent)) equal to 0.36. The authors
suggest that the observed product distribution reflects the
nucleophile distribution in the first solvation shells of the
arenediazonium ions. An underlying simplification is the
assumption that, in the case of an SN1 reaction mechanism,
the reaction is fast and the reorganization of the solvent when
the intermediate is formed is neglected. In butanol and water
mixtures, o-methyldiazonium decomposes producing o-methyl-
phenol and 1-butoxy-2-methylbenzene. In this reaction 50% of
each product is obtained at xwater = 0.34, suggesting that the
diazonium ion undergoes preferential solvation by water.
Here, we examined the rate and products of the
dediazoniation of 2,4,6-trimethylbenzenediazonium ion in
mixtures of water/acetonitrile and water/alcohols: water/
methanol, water/ethanol, and water/propanol (Scheme 1, for
sake of brevity, SN1 mechanism is presented). In addition, we
have carried out classical molecular dynamics (MD)
simulations (i.e., no bond breaking/forming in the simulations)
The capacity of some probes to distinguish different media has
been used to investigate micelles, membranes, cell compart-
ments, enzyme active sites, etc.1 Diazonium salts have been
used to determinate the concentration of anions at the surface
of micelles, liposomes, and reverse micelles.2 The low
sensitivity of the diazonium salts reaction rates to solvent
polarity and the reaction of the intermediate carbocation with
different nucleophiles, following first-order kinetics,3 has been
used as a good argument for using these probes as a tool for the
determination of anions concentration in several condensed
media.2
On the basis of extensive experimental evidence, a SN1
mechanism, with the intermediate formation of a very reactive
phenyl cation, was thought to be the only reaction pathway.3
Alternative mechanisms, assuming specific water and nucleo-
phile interactions in the initial state, have been presented, and
the nucleophilic displacement process for aryldiazonium ions in
water is proposed to be at the boundary between SN2 and
SN1.4−6 Recently, excellent agreement was found between in-
water theoretical activation free energies calculated according to
the Marcus theory with the corresponding experimental data
when assuming the SN2 rather than SN1 mechanisms.7
For solvent mixtures, as in the solvolysis of 4-methylbenze-
nediazonium ions in 2,2,2-trifluoroethanol−water, the kinetic
and thermodynamic results suggest that the rate-determining
step for dediazoniation is the formation of a highly reactive aryl
cation that traps any nucleophile available in its solvation shell.8
The major products of the dediazoniation of 2-, 3-, and 4-
+
of solvent distribution around 1-ArN2 for water/solvent
mixtures showing that the preferential solvation around ipso
carbon reflects the experimental product yields of dediazonia-
tion. Such theoretical evidence should be valid when
nucleophilic reactivities are similar and reactions occur at the
Received: June 8, 2015
© XXXX American Chemical Society
A
DOI: 10.1021/acs.joc.5b01289
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The Journal of Organic Chemistry
Article
Scheme 1
SN1 (or SN1/SN2 borderline) mechanism.6,10 Such a simple
protocol is a shortcut for the more extensive reactive
trajectories calculations at the femtosecond time scale.6,11−13
Interestingly, preferential solvation was only marginally
dependent on ipso carbon charge distribution.
RESULTS
■
+
The Decomposition of 1- ArN2 in Solvent Mixtures.
Product Yields in Methanol. The rate constant for the
+
solvolysis of 1-ArN2 in water and methanol (30 °C) were
8.2 × 10−3 and 9.9 × 10−3 min−1 ( 5%), respectively.The main
reaction products were 2,4,6-trimethylphenol (1-ArOH) and
2,4,6-trimethylanisole (1-ArOMe). A small percentage (<0.5%)
of 2,4,6-trimethylbenzene (1-ArH), the reduced product of 1-
+
ArN2 , and traces of 2,4,6-trimethyl chloride (1-ArCl), due the
presence of ca. 10−3 mol dm−3 HCl, were formed, consistent
with other studies reported with similar diazonium salts.14
The increase in water mole fraction [(water moles)/(water
moles + MeOH moles)] in water/methanol mixtures produced
a linear increase of the 1-ArOH molar concentration and an
equivalent decrease in the molar concentration of 1-ArOMe
(Figure 1). In Figure 1, the amount of products formed is given
+
Figure 1. Effect of water mole fraction in the % Products of 1-ArN2
+
□
○
and 16-ArN2 reactions in Methanol/Water: 1-ArOH ( , ); 1-
ArOMe ( , ) (Two sets of experiments); 16-ArOH ( ); 16-ArOMe
■
●
△
▲
(
). MD simulations, (red solid circle) M1 model, and (green solid
circle) M2 model. Black lines are linear fits to % 1-ArX.
+
in percentage (%) of the total diazonium salt [1-ArN2 ]T. As
both solvent mixtures were not linear. The formation of 50% of
each product occurred at 0.36 and 0.32 water mole ratio for
water/ethanol and water/n-propanol, respectively (Figure 2).
As the chain of the alcohol increased from ethanol to propanol,
50% of each product occurred at even lower water mole ratios.
The effect was observed previously with water/butanol
mixtures by Bravo and co-workers in monosubstituted
methyl-benzenediazonium salts.15
only traces of 1-ArH and 1-ArCl were formed in all water/
methanol mixtures, their concentrations were subtracted from
+
the total concentration of 1-ArN2 . Thus, the sum of [1-ArOH]
+
and [1-ArOMe] was equal to [1-ArN2 ]T:
[1‐ArN+2 ]T = [1‐ArOH] + [1‐ArOMe]
(1)
The %1-ArOH linearly increased and that of %1-ArOMe
linearly decreased with the water mole fraction. With a water
mole fraction of 0.5, the percentages of 1-ArOH and 1-ArOMe
were identical (i.e, 50%) (see Figure 1).
+
Product Yields in Acetonitrile. The rate constant of 1-ArN2
with acetonitrile was 10.1 × 10−3 min−1 ( 5%). In water/
acetonitrile mixtures, 1-ArN2+ reacts with acetonitrile producing
the acetamides (1-ArNHAc).17 1-ArH was not formed in a
significant amount in those mixtures. The %1-ArOH and %1-
ArNHAc were nonlinearly related to the water mole fraction
(Figure 3), and 50% of each product was formed when water/
acetonitrile molar fraction was about 0.7.
Molecular Dynamics Simulations. The θZ is the theoretical
fraction (in percentage) of a given nucleophile relative to the
total nucleophiles in the first solvation shell of the ipso carbon.
Simulations were carried out with a charge model (M1)
Product Yields in Ethanol and Propanol. The solvolysis of
+
1-ArN2 was also studied in mixtures of water/ethanol and
water/n-propanol where 1-ArOH and the ethers 1-ArOEt and
1-ArOPr were formed, respectively.The rate constants of 1-
+
ArN2 dediazoniation with ethanol and propanol were 8.2 ×
10−3 and 7.1 x10−3 min−1 ( 5%), respectively.
Figure 2 shows the % products formed in those mixtures, and
different from the results obtained with water/methanol
mixtures, the plots of % products vs water mole fraction in
B
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The Journal of Organic Chemistry
Article
+
Figure 3. Effect of water mole fraction in the % products of 1-ArN2
+
and 16-ArN2 in Acetonitrile/Water. Different symbols for the same
product are related to independent sets of experiments: 1-ArOH
△ ▽
▲ ▼
(
,
); 1-ArNHCN (
; ) (Two sets of different experiments), 16-
□
■
ArOH ( ); 16-ArNHAc ( ); (red solid circle) M1 model and (green
solid circle) M2 model. Solid black lines are third-order polynomials
adjust to % 1-ArX.
observed, the x1/2 values decreased in the order acetonitrile >
methanol > ethanol > propanol.
DISCUSSION
■
The low sensitivity of dediazoniation reactions can be related to
the nucleophilic scales. Ritchie used substituted aryldiazonium
cations in his N+(≡log(k/kH O)) scale.16,17 Using our kinetic
2
+
data for pure solvents and water, N+ (in this case, cation
dependent) resulted in the values 0.091, 0.082, 0, and −0.063
for acetonitrile, methanol, ethanol, and propanol, respectively.
Such N+ estimated values suggest very similar nucleophilicity
among solvents.
Do solvent mixtures exhibit the same behavior? Minegishi et
al. evaluated first-order kinetic constants of different benzhy-
drylium ions in binaries mixtures of water/alcohol and water/
acetonitrile and also concluded that relative nucleophilicity of
these solvents and mixtures are similar and fairly independent
of carbocation eletrophilicities.10 Our kinetic data of
aryldiazonium solvolysis in the solvent mixtures (not shown)
also support similar nucleophilicities. Bravo-Diaz et al.9,18
analyzed the decomposition of substituted aryldiazonium
cations mixture of solvents, observed low selectivity, and
argued that product ratios must reflect the local concentration
around the aryldiazonium cation.
Is there a relationship between preferential solvation and
product ratios? We calculated local concentration of the
solvents around diazonium ion from MD simulations. The
local solvent composition depended linearly on xwater for water/
methanol binary mixtures, but varied nonlinearly for the other
solvent mixtures. The O atom (or N) solvated the ipso carbon
or surrounded the − N2 group, and there was no preference to
interact with any other groups of the diazonium ion (Figure 4).
Our classical simulations showed, assuming an equal
probability of reaction, that the calculated local number of
solvent molecules around the aryldiazonium cation can predict
the product ratios, when preferential solvation shell on ipso
carbon site is considered (Figures 1−3). In addition, our
simulations have also considered two different atomic charge
Figure 2. Effect of water mole fraction in the % products of 1-ArN2
+
and 16-ArN2 reactions in solvent mixtures. MD simulations, (red
solid circle) M1 model and (green solid circle) M2 model. (A)
Ethanol/water: 1-ArOH ( , ); 1-ArOEt ( , ). Solid black lines are
third-order polynomials adjust to % 1-ArX. (B) Propanol/water:
Different symbols for the same product are related to independent sets
□
■
●
○
□
◇
■ ◆
● ▲
○
△
of experiments: 1-ArOH ( , ,
(
, ); 1-ArOPr ( , , , );16-ArOH
▽
▼
),16-ArOPr ( ).
modeling the rupture of ipso carbon and nitrogen bond.
Simulations were also carried using a charge model based on a
stable 1-ArN2 cation (the M2 model). Such charge models
+
gave different atomic charges at ipso carbon (M1: +0.61|e|; M2:
−0.46|e|). However, the composition of the solvation shell
around the ipso carbon was only slightly dependent on the M1
and M2 charge models. An additional rule must be provided to
link the solvation shell composition with products percentage
(% product). The assumption was that nucleophiles have equal
probabilities to react on ipso carbon site (in low selectivity
case). Therefore, comparisons between θZ and % products (in
measurements described above) were straightforward and
presented in Figures 1, 2, and 3. For the four binary mixtures
studied here, the θZ values of the organic solvents decreased as
the xwater increased. In methanol/water mixture, both θZ and %
products showed a nearly linear behavior, and the water
fraction at which θwater = 50 (x1/2) was equal to 0.5. However, a
nonlinear dependence of θZ with xwater was found for the other
solvent mixtures, as can be seen in Figures 1 and 2. The MD
results showed a good agreement with experimental values,
regardless of the nature of the organic solvent and the charge
distribution model used for the probe. As experimentally
C
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Article
Even though a product prediction ratio is possible, a
molecular understanding of the complete process is still
lacking. Probably, calculations of reactive trajectories (as in
ref 13) will be necessary if we want to deeply understand the
assumption of equal probabilities.
EXPERIMENTAL SECTION
Materials. Methanol, ethanol, n-propanol, 2-propanol, and
acetonitrile were HPLC grade, and 2,4,6-trimethylphenol (1-ArOH)
and trimethylbenzene (1-ArH) were used as received. 2,4,6-
■
Figure 4. Spatial distribution of the oxygen atom of water (red
surface) or methanol (green surface) in MD simulations with xwater
equal to (A) 0.25; (B) 0.50; and (C) 0.75.
+
trimethylbenzenediazonium (1-ArN2 ) tetrafluoroborate and 2,6-
+
dimethyl-4-hexadecylbenzenediazonium tetrafluorborate (16-ArN2 )
were synthesized as described.1,2 The compounds 2,6-dimethy-4-
distributions, i.e., an atomic charge distribution considering
electrons delocalized over all molecule (M2 model) and atomic
charge distribution considering electrons delocalized over aryl
cation and a neutral N2 moiety (M1 model). Such models
changed the atomic charge on ipso carbon of −0.46|e| to +0.61|
e| going from M2 to M1 model, respectively. As presented
above (Figures 1−3), our product predictions were only slightly
dependent on the ipso carbon atomic value.
The charge sign inversion on the ipso carbon is linked to the
total dipole moment vector behavior: an inversion of its
direction changes the atomic charge on the carbon site (Figure
5). Total electrostatic potential was also calculated at the
+
hexadecyl-aniline (used in the synthesis of 16-ArN2 ), 2,6-dimethyl-4-
hexadecyl-phenol, 16-ArOH, and 2,6-dimethyl-4-hexadecyl-benzene
(15-ArH) were kindly furnished by Dr. L. Romsted. 2,4,6-
benzeneacetamide (1-ArNHAc) and 2,6-dimethy-4-hexadecyl- benze-
neacetamide, 16-ArNHAc, were synthesized as described.18,19
Synthesis of 2,4,6-trimethylanisole (1-ArOMe) and 2,6-dimethyl-4-
+
+
hexadecylanisole (16-ArOMe): 500 mg of 1-ArN2 or 16-ArN2 , were
added to 100 mL of methanol containing 50 μL of an aqueous solution
of HCl 1.0 mol dm−3 and maintained at 30 °C for 24 h. After solvent
evaporation, water was added, and the product extracted with
chloroform. Na2SO4 was added to the chloroform to remove traces
of water and filtered. Chloroform was evaporated, and the product was
purified in a silica gel column G60 with CHCl3/hexane 75:25 (v/v).
The standard curves for the determination of the concentration of
the compounds 2,4,6-trimethyl-1-ethoxy-benzene (1-ArOEt), 2,6-
dimethyl-4-hexadecyl-1-etoxybenzene (16-ArEt), 2,4,6 trimethyl-1-
propoxy-benzene (1-ArOPr), and 2,6-dimethyl-4-hexadecyl-1-propox-
ybenzene (16-ArPr) were prepared from a solution of weighed 1-
ArN2+ which was reacted in ethanol or propanol in the presence of 1.0
× 10−3 mol dm−3 HCl. The percentage of 1-ArOH and 1-ArCl were
<5% of the initial concentration of the diazonium salt.
METHODS
Figure 5. Dipole moments (blue arrows) of the aryl-diazonium cation
in M1 (left) and M2 (right) charge models.
■
Product Analysis by HPLC. Products of dediazoniation reaction
were analyzed in a HPLC equipped with a C-18 reverse phase column,
using a loop of 30 μL. Products were detected by absorption at 220
nm.
A solvent mixture of methanol/water 82:18 (v/v) was used to
separate the products of dediazoniation of 1-ArN2 in the HPLC
(retention times: 1-ArOH, 4.6 min; 1-ArOMe, 6.7 min; 1-ArOEt, 7.7
min; 1-ArOPr, 9.8 min; 1-ArH, 13.0 min) using a flux of 0.8 mL min−1.
For experiments with acetonitrile/water mixtures, the solvent mixture
used was methanol/water 75:25 (v/v), 0.8 mL min−1 (retention times:
1-ArNHAc, 4.6 min; 1-ArOH, 6.8 min). The products 16-ArOH, 16-
ArOMe, 16-ArOPr, and 16-ArNHAc were eluted with methanol/2-
propanol 82:18 (v/v), with a flux of 1 mL min−1 (retention times: 16-
ArNHAc 5.1 min, 16-ArOH, 6.0 min, 16-ArOMe, 8.7 min, 16-ArOPr
9.2 min, and 16-ArH, 9.5 min).
B3LYP/TZVPP level for both aryldiazonium cation and a
nonrelaxed aryl cation systems (Figure 6). Although the total
electrostatic potentials are different on the center of ipso
carbon nucleus, comparison between electrostatic potentials
does not show significant deviation above 0.2 nm.
+
The concentration of each product eluted in the HPLC was
determined through a standard curve of area versus concentration
obtained in the same HPLC conditions used for the samples. The
percentage of products was calculated by dividing the concentration of
each product by the sum of all products multiplied by 100.
+
+
Dediazoniation reaction of 1-ArN2 and 16-ArN2 . All the
solvents mixtures used in the dediazoniation reactions contained HCl
0.001or 0.002 mol dm−3, added from HCl 12.5 mol dm−3 (ca. 0.010
mL per 100 mL of solvent). Pure solvents contained 0.55 mol dm−3 of
water due to the addition of HCl. All solvents mixtures used in the
reactions were prepared by weight in 2 mL volumetric tubes.
Reactions were started by adding a small aliquot (0.025 or 0.030
mL) of a concentrated 1-ArN2+ solution (between 8 and 4 × 10−3 mol
dm−3), diluted in the organic solvent, to the appropriate solvent
mixture with a final volume of 1.0 mL in 2 mL tubes equipped with
Teflon stoppers. The stock solutions of the diazonium salts were
maintained in an ice-bath to avoid significant decomposition prior to
Figure 6. Total electrostatic potential at B3LYP/TZVPP level
calculated on the ipso carbon site and above it. Black and red lines
represent the aryldiazonium cation and the phenyl cation, respectively.
D
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Article
the addition to the solvent mixtures. A layer of 50 μL of hexane was
added over the reactions tubes with 1-ArN2 to avoid products
ASSOCIATED CONTENT
* Supporting Information
The Supporting Information is available free of charge on the
ACS Publications website at DOI: 10.1021/acs.joc.5b01289.
■
+
S
evaporation when the water content in the mixtures was higher than
80%. The addition of cyclohexane does not affect product distribution
or solvent composition due to its low solubility in water.14 After 24 h
at 30 °C, the samples were diluted to 2 mL with n-propanol, to
homogenize the solutions which were maintained at 8 °C, until HPLC
analysis.
Arenediazonium cation structure and atomic charges;
radial distribution function between ipso carbon and the
oxygen atom of methanol and water in the methanol/
water 50/50 simulation; description of the procedure to
obtain the spatial distribution figures; mass spectra of the
products (PDF)
Input files containing the force field parameters used in
all simulations are available as well as the spatial
configuration files for simulations with xwater = 50
(ZIP)
Molecular Dynamics Simulations. All simulations were
performed with GROMACS 4.5.5 simulation package in double
precision in a NPT ensemble.20 The 4 × 4 × 4 nm edge initial box was
built with Packmol package21 and consisted of one 1-ArN2 and
+
different numbers of solvent molecules (water, methanol, ethanol,
propanol, and acetonitrile), depending on the water molar fraction. A
background negative charge is neutralizing the system. The number of
solvent molecules was calculated from the molar fraction and the
density of the systems.22−24 The water model used was SPC/E, and
the force field of the alcohols was OPLS-AA.25 The force field for
acetonitrile was taken from Wick et al.l,26 since it reproduces the
amount of hydrogen-bonded acetonitrile molecules.27 The equilibrium
AUTHOR INFORMATION
■
Corresponding Authors
*E-mail: filipe.lima@usp.br.
*E-mail: imcuccov@iq.usp.br.
+
geometry of 1-ArN2 was calculated with Gaussian 03 at B3LYP/6-
31+G* level.28−33 Then, CHELPG34 charges were calculated for the
molecule with and without the −N2 group (nonrelaxed geometry),
models M2 and M1, respectively, using ORCA v3.0.1 at B3LYP/
TZVPP level using RIJCOSX technique in vacuum.35−38 Note that in
MD simulations with M1 model the nitrogen atoms have no partial
charge (see Scheme SI1 and Table SI1). For modeling the
intermolecular interactions, it is common practice to define and
compute an atomic charge distribution representing the electrostatic
potential around the molecule or their multipole moments. For such
task, different protocols and ideas have been proposed, e.g., the
RESP,39 MK,40,41 CHELP,42 CHELPG,34 CHELMO,43 DDEC/cn,44
CMx,45,46 and multipole-derived charges.47 In special case of an OPLS-
consistent atomic charge model, Jorgensen et al. have suggested the
CMx charge model and Canongia−Lopez et al. have used the
CHELPG. Commonly, atomic charge models are based on RESP39 or
MK40,41 protocol at HF/6-31G* level in vacuum, but these were
developed for aqueous systems, not mixtures. Alternatively, Jorgensen
et al. have suggested the CMx charge model as an OPLS-consistent
atomic charge model.48 We are following Canongia−Lopez et al.49
who have suggested the CHELPG protocol as a convenient way to
generate an OPLS-consistent charge model. The charge distribution
for M1 and M2 and other atomic parameters of 1-ArN2+ can be found
in Supporting Information. All other 1-ArN2+ parameters were OPLS-
AA.
The geometric combination rule was used for Lennard-Jones
interactions. Coulomb forces were calculated using smooth particle-
mesh Ewald summation with periodic boundary conditions. The cutoff
radius of the Lennard-Jones interactions was equal to 1.2 nm, with a
switching function starting at 1.1 nm. All simulations were performed
at 303 K and 1 bar of pressure, maintained by isotropic pressure
coupling using Parrinello−Rahman algorithm. Simulations were
carried out for 20 ns, and the first 2 ns was not used for calculations.
The number of solvent molecules in the first solvation shell of the
carbon in the aromatic ring (ipso) bound to the N atom was calculated
as in eq 2:
Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENTS
■
This work was partially financed by FAPESP Grant 2013/
08166-5, a Post Doctoral Fellowship to F.S.L. (FAPESP, 2014/
06073-2), CNPq for scholarships (to I.M.C. and H.C.), the
Instituto Nacional de Cien
Complexos (INCT-FCx, CNPq-FAPESP) and the Nucleo de
̂
cia e Tecnologia de Fluidos
́
̀
Apoio a Pesquisa de Fluidos Complexos (NAP-FCx, USP). We
acknowledge the Dr. Paolo Di Mascio for the use of the Mass
Spectrometer and Dr. Fernanda Manso Prado, Luciana
Coutinho de Oliveira, and Rafael Y. Vavrick for helpful
́
discussions. We acknowledge also the Central Analitica do
Instituto de Quimica da USP and MSc Janaina Vilcachagua and
MSc Cristiane P. M. Xavier. We thank the anonymous referees
for their careful revision and exceedingly helpful criticism of a
previous version of this manuscript.
́
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■
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NZ = 4πρZ ∫ r g(r)r dr
2
l
(2)
0
(9) Pazo-Llorente, R.; Bravo-Diaz, C.; Gonzalez-Romero, E. Eur. J.
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distribution function between the atom ipso carbon and the oxygen
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minimum of g(r) or the first inflection point in the g(r)*r2 function.
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