Introducing new reactivity descriptors: “Bond reactivity indices.” Comparison of the
new definitions and atomic reactivity indices
Jesús Sánchez-Márquez
Citation: The Journal of Chemical Physics 145, 194105 (2016); doi: 10.1063/1.4967293
View online: http://dx.doi.org/10.1063/1.4967293
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THE JOURNAL OF CHEMICAL PHYSICS 145, 194105 (2016)
Introducing new reactivity descriptors: “Bond reactivity indices.”
Comparison of the new definitions and atomic reactivity indices
Jesús Sánchez-Márqueza)
Departamento de Quı́mica-Fı́sica, Facultad de Ciencias, Campus Universitario Rı́o San Pedro,
Universidad de Cádiz, Puerto Real, Cádiz 11510, Spain
(Received 8 June 2016; accepted 26 October 2016; published online 17 November 2016)
A new methodology to obtain reactivity indices has been defined. This is based on reactivity functions
such as the Fukui function or the dual descriptor and makes it possible to project the information of
reactivity functions over molecular orbitals instead of the atoms of the molecule (atomic reactivity
indices). The methodology focuses on the molecule’s natural bond orbitals (bond reactivity indices)
because these orbitals (with physical meaning) have the advantage of being very localized, allowing
the reaction site of an electrophile or nucleophile to be determined within a very precise molecular
region. This methodology gives a reactivity index for every Natural Bond Orbital (NBO), and we
have verified that they have equivalent information to the reactivity functions. A representative set
of molecules has been used to test the new definitions. Also, the bond reactivity index has been
related with the atomic reactivity one, and complementary information has been obtained from the
comparison. Finally, a new atomic reactivity index has been defined and compared with previous
definitions. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4967293]
I. INTRODUCTION
Many publications justify the reactivity of a system
using canonical orbitals, which sometimes differ from frontier molecular orbitals (HOMO-1, HOMO-2, etc.).1–3 Also,
the definition of reactivity index may be ambiguous, as we can
see in Ref. 4. This led to a previous study5 to test new definitions for reactivity indices in which we defined Natural Bond
Orbital (NBO) reactivity indices and studied their advantages
and disadvantages. NBOs6,7 were chosen for two reasons: first,
they are very localized and allow the point of attack of an electrophile or nucleophile inside a concrete region of the atom
to be delimited and second, because of the ease of use of
Lewis structure diagrams. For example, consider the reaction,
Calculating the bond reactivity indices in such a case (NBO1
and NBO2 ) would be of great interest. In the current study,
a new methodology has been defined for obtaining reactivity
indices for the NBOs (bond reactivity indices) that is based
on an adequate least square fitting (Lagrange’s undetermined
multipliers8 ), which allows the projection of the information
of reactivity functions (Fukui function and the dual descriptor)
over molecular orbitals instead of the atoms of the molecule
(notice that the bond index concept is not a new idea, as can
be seen in the study by Bultinck et al.9 ). These indices have
the same information as reactivity functions but are easier
to use.
II. THEORETICAL BACKGROUND
where “Nu:” is a nucleophile with one or more occupied NBOs
(lone pairs, double-triple bonds, etc.) and “E” is an electrophile
with one or more unoccupied NBOs. Imagine that the reaction
can follow two paths:
a) Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0021-9606/2016/145(19)/194105/12/$30.00
To understand detailed reaction mechanisms such as
regio-selectivity requires, in addition to global descriptors,10–18 local reactivity parameters to differentiate the reactive behavior of the atoms forming a molecule. The Fukui
function19 (f (r)) and the softness20 (s(r)) are two of the most
commonly used reactivity descriptors,
!
∂ ρ (r)
f (r) =
,
∂N ν
!
!
!
(1)
∂ ρ (r)
∂ ρ (r)
∂N
s (r) =
=
·
= S · f (r) .
∂µ ν
∂N ν ∂ µ ν
The Fukui function is associated primarily with the
response of the density function of a system to a change in
the number of electrons (N) under the constraint of a constant
external potential [v(r)]. The Fukui function also represents
the response of the chemical potential of a system to a change
in external potential. As the chemical potential is a measure of
the intrinsic acidic or base strength, and softness incorporates
global reactivity, both parameters provide a pair of indices
145, 194105-1
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194105-2
Jesús Sánchez-Márquez
J. Chem. Phys. 145, 194105 (2016)
with which to demonstrate, for example, the specific sites of
interaction between two reagents.
Due to the discontinuity of the electron density with
respect to N, finite difference (FD) approximation leads to three
types of Fukui function for a system, namely, f + (r) Eq. (2),
f – (r) in Eq. (3) and f 0 (r) in Eq. (4) for nucleophilic, electrophilic, and radical attack, respectively. f + (r) is measured by
the electron density change following the addition of an electron, and f – (r) by the electron density change upon removal
of an electron. f 0 (r) is approximated as the average of both
previous terms, defined as follows:
f + (r) = ρN0 +1 (r) − ρN0 (r) , for nucleophilic attack,
(2)
f − (r) = ρN0 (r) − ρN0 −1 (r) , for electrophilic attack,
(3)
f 0 (r) =
1
ρN0 +1 (r) − ρN0 −1 (r) ,
2
for neutral (or radical) attack.
(4)
where Cν α are the molecular frontier orbital coefficients and
S χν are the atomic orbital overlap matrix elements. The subscripts “H” and “L” are referenced to the HOMO and LUMO
orbitals. This definition of the condensed Fukui function
Eqs. (9)–(12) has been pioneered by Pérez and Chamorro23,24
and has been used in a variety of studies yielding reliable
results.25–27
B. Reactivity indices of natural bond orbitals
In a previous study,5 we proposed the reactivity index
in the following equation for NBOi :
X
(13)
FFiNBO =
|Ci α | 2 fk(NBO)
i
that is based on the approximation,
X
fkα ≈
,
|Ciα | 2 fk(NBO)
i
(5)
where φα (r) is a particular frontier molecular orbital (FMO)
chosen depending upon the value of α = + or α = −. Eq. (5)
can be used to develop an approximated definition of the Fukui
function,
f − (r) = ρHOMO (r) , for electrophilic attack,
(6)
f + (r) = ρLUMO (r) , for nucleophilic attack,
(7)
where

X 
X

(NBO) 2
( NBO)∗ (NBO)
Cν i +
=
Cχ i
Cν i
S χν  .

χ<ν
ν ∈ k 
(15)
The fk(NBO)
parameters were calculated with a modified
i
version of the UCA-FUKUI software28 and the C i coefficients of Eq. (16) were obtained by the least square method
by applying Eq. (17), which leads to the linear system of
Eq. (18),
X
φ(HOMO) ≈
Ci φi(NBO) ,
(16)
i
φ(HOMO) φ(NBO)
dτ
j
XX
(8)
Expanding the FMO in terms of the atomic basis functions,
the condensed Fukui function at the atom k is

X 
X

∗
α
2
 |Cν α | +
C χ α Cν α S χν  ,
(9)
fk =


χ<ν
ν ∈k

X 
X

2
∗
 |Cν H | +
=
C χH Cν H S χν 

χ<ν
ν ∈ k 
× (electrophilic attack),
(10)

X 
X


2
∗
+
C
C
S
|C
|
χL
νL
χν 
 ν L

χ<ν
ν ∈k
× (nucleophilic attack),
(11)
fk+ =
≈
X
Ck
φ(NBO)
φj(NBO) dτ,
k
(17)
Ck Bik Bl j Si l .
(18)
k
i
fk−
(14)
i
fk(NBO)
i
Under frozen orbital approximation (FOA) of Fukui, and
neglecting the orbital relaxation effects, the Fukui function can
be approximated as
1
ρLUMO (r) − ρHOMO (r) ,
2
for neutral (or radical) attack.
(12)
k
A. Frontier molecular orbital approximation
f 0 (r) =
1 +
fk + fk− (radical attack),
2
FFiNBO
The condensation of reactivity descriptors to atoms can be
carried out in several ways, such as the famous Yang-Mortier
scheme21 or other “Population strategies” as can be seen in
Refs. 4 and 22.
f α (r) ≈ φα (r)2 ,
fk0 =
l
Ai Bl j Si l ≈
XXX
k
i
l
The current study presents an improved methodology which
can use FMO and FD approximations (the definition of Eq. (13)
can only use the FMO approximation). Also, the new methodology, unlike the old one, can be based on other functions such
as the dual descriptor (the old methodology was based on the
f α (r) function).
III. INTRODUCTION TO THE METHODOLOGY
Below is a brief summary focused on the basis of
the new descriptors fi+ (NBO) and fi− (NBO) . The detailed discussion is presented in section S1 in the supplementary
material (ST1 in the supplementary material shows a summary of the nomenclature and definitions used in this
study).
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194105-3
Jesús Sánchez-Márquez
J. Chem. Phys. 145, 194105 (2016)
If we accept the approximation that NBOs do not change when the molecule loses an electron and only their occupancies
change, we get
all X
orbitals
2
*
f − (~r ) ≈
fi− (NBO) φ(NBO)
r .
(19)
i
i=1
The
(
fi− (NBO)
)
set can be estimated by the least square fitting of Eq. (20),
*f − (~r ) −
,
all X
orbitals
2
fi− (NBO)
i=1
φ(NBO) (~r )2 + d~r = MINIMUM,
i
-
and the linear system29 in Eq. (21) can be derived from Eq. (20),
all orbitals
X
2
φ(NBO) (~r )2 · φ(NBO) (~r )2 d~r ; j = 1, 2, 3, ..., all orbitals.
(NBO)
− (NBO)
−
f (~r ) · φj
(~r ) d~r =
fi
i
j
(20)
(21)
i=1
Finally, for the f + (~r ) function, we obtain the system of Eq. (22),
all orbitals
X
2
φ(NBO) (~r )2 · φ(NBO) (~r )2 d~r ; j = 1, 2, 3, ..., all.
(NBO)
+ (NBO)
+
f (~r ) · φj
(~r ) d~r =
fi
i
j
(22)
i=1
A. A new descriptor based on a dual descriptor
In previous studies,22,30,31 the dual descriptor function (∆f (r)) has been defined as Eq. (23), where ρN (r), ρN+1 (r), and
are the densities of the neutral molecule, the anion, and the cation. This function was used in a previous study32 to
partition the real space into non-overlapping reactive domains that feature a constant ∆f (r) sign that makes it possible to identify
the nucleophilic and electrophilic regions inside a molecule,
2
2 *
*
(23)
∆f (r) = f + (r) − f − (r) = ρN+1 (r) − 2ρN (r) + ρN−1 (r) ≈ φ(LUMO) r − φ(HOMO) r .
By using the approximation Eqs. (S7) and (S13) (see supplementary material), we can write the (f + (r) − f – (r)) function as
all X
orbitals 2 .
∆f (r) = f + (r) − f − (r) =
2αi − αi− − αi+ · φ(NBO)
(r)
(24)
i
ρN−1 (r)
i=1
That can be rewritten as
∆f (r) =
all X
orbitals
i=1
2 ,
∆fi(NBO) · φ(NBO)
(r)
i
(25)
where the parameters ∆fi(NBO) have been obtained by means of the condition,
all X
orbitals 2
2
(NBO) (NBO)
*∆f (r) −
∆fi
· φi
(~r ) + d~r = MINIMUM.
,
i=1
This is very similar to Eq. (20). Finally, Eq. (27) can be derived from Eq. (26),
!
all orbitals
X
2
2 (NBO)
2
(NBO)
(NBO)
(NBO)
φi
∆f (r) · φj
(~r ) d~r =
∆fi
·
(~r ) · φj
(~r ) d~r , j = 1, 2, 3, ..., all.
(26)
(27)
i=1
B. Calculation details
Previous indices were dimensionless (and this is important) but it is also important that they were normalized:
all orbitals
all orbitals
P
P
fi− (NBO) = 1,
fi+ (NBO) = 1, and
∆fi(NBO) = 0. The method of Lagrange multipliers8 was used to normalize
all orbitals
P
i=1
i=1
i=1
the reactivity indices of Eqs. (20, 22 and 26). Finally, we obtained
−
*f (~r ) −
,
all X
orbitals
i=1
fi− (NBO)
2
all X

orbitals
φ(NBO) (~r )2 + d~r + λ 
− (NBO)
 = MINIMUM,
f
−
1

i
i

 i=1
(28)
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194105-4
Jesús Sánchez-Márquez
*f + (~r ) −
,
J. Chem. Phys. 145, 194105 (2016)
all X
orbitals
fi+ (NBO)
i=1
2
all X

orbitals
φ(NBO) (~r )2 + d~r + λ 
+ (NBO)
 = MINIMUM,
f
−
1

i
i

 i=1
(29)
2

all X
all X
orbitals
orbitals
2 + d~r + λ 
(NBO) 
*∆f (~r ) −
∆fi (NBO) φ(NBO)
(~
r
)
∆f
(30)
 = MINIMUM.

i
i
,
i=1

 i=1
Also, instead of calculating the integrals of Eqs. (28)–(30), we have used a grid of equally spaced points to make the fitting.
In all cases, we used 182 points/Å3 and verified that by increasing the amount of points, the results do not change significantly
(for example, for the A1 reagent we have used 180 000 points). This approximation converts Eq. (21) into Eq. (31), Eq. (22) into
Eq. (32), and Eq. (27) into Eq. (33),
X
X
X
2
φ(NBO) (r )2 · φ(NBO) (r )2 , j = 1, 2, 3, ...,
(rm ) =
f − (rm ) · φ(NBO)
fi− (NBO)
(31)
m m j
i
j
m
X
m
2
(rm ) =
f + (rm ) · φ(NBO)
j
X
2
(rk ) =
∆f (rk ) · φ(NBO)
j
X
X
k
m
i
fi+ (NBO)
X
m
i
∆fi(NBO)
X
i
k
IV. COMPUTATIONAL METHODS
All the structures included in this study were optimized at
the B3LYP/6-31G(d)33,34 level of theory using the Gaussian09
package.35 For the FMO approximation, the electrophilic
Fukui function was evaluated from a single point calculation
TABLE I. Sample 1: dienes bearing an electron withdrawing group (EWG)
at C1 or/and C2 .
Reagent
R (C1 )
R’ (C2 )
A1
B1
C1
D1
E1
F1
G1
Cl
H
Cl
H
NO2
CN
H
H
Cl
Cl
CHO
H
H
CN
TABLE II. Sample 2: dienes bearing an electron donating group (EDG) at
C1 or/and C2 .
Reagent
R (C1 )
R’ (C2 )
H1
I1
J1
K1
L1
CH3
H
CH3
−OCH3
H
H
CH3
CH3
H
−OCH3
TABLE III. Sample 3: dienophiles bearing an EWG.
Reagent
R
A2
B2
C2
D2
Cl
CHO
NO2
CN
φ(NBO) (r )2 · φ(NBO) (r )2 , j = 1, 2, 3, ...,
m m i
j
(32)
φ(NBO) (r )2 · φ(NBO) (r )2 , j = 1, 2, 3, ....
k k i
j
(33)
in terms of molecular orbital coefficients and overlap integrals.
For the FD approximation, the densities were calculated for the
neutral molecule, cation and anion from single point calculations (the wave functions were calculated with Gaussian09).
The condensed Fukui functions, fk(NBO) Eq. (15), fk− Eq. (9)
and fk− , calculated under the FD approximation and atomic
populations,36 were obtained using UCA-FUKUI software.28
The fi− (NBO) in Eq. (21), fi+ (NBO) in Eq. (22) and ∆fi (NBO) in
Eq. (27) parameters were obtained with a modified version of
UCA-FUKUI.
V. RESULTS AND DISCUSSION
A. Testing the new descriptors fi −(NBO) and f i +(NBO)
in a sample of representative molecules:
Rationalization of Diels-Alder reactions
The interaction between unsymmetrical dienes and
dienophiles can give two isomeric adducts, depending upon
the relative position of the substituent. The selectivity for
the formation of one adduct over the other is called regioselectivity, and this kind of isomer is called a regioisomer.
In this class of cycloaddition, the degree of regioselectivity
is often high;37–42 yet, it is well established that the more
powerful the electron-donating and electron-withdrawing substituents on the diene/dienophile pair, the more regioselective
the reaction.3 Regioselectivity has been described in terms of
a local hard and soft acid and base (HSAB) principle, and
some empirical rules have been proposed to rationalize the
TABLE IV. Sample 4: dienophiles bearing an EDG.
Reagent
R
E2
F2
CH3
−OCH3
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194105-5
Jesús Sánchez-Márquez
J. Chem. Phys. 145, 194105 (2016)
TABLE V. fi− (NBO) and fi+ (NBO) parameters in Eqs. (21) and (22) calculated under finite difference (FD) and frontier molecular orbital (FMO) approximations
(HOMO and LUMO) for some important NBOs of Fig. 1 (sorted by energy, the highest values are printed in boldface). Partial orbital occupancies have been
added in the last column.
Schematic representations of NBOs
Level
NBO
Type
fi+ (NBO)
(FD)
fi+ (NBO)
(FMO: LUMO)
Occupancies
25
24
82
87
BD C1−C4
BD C6−C8
0.116
0.156
0.274
0.315
0.0586
0.1258
fi− (NBO)
(FMO: HOMO)
Occupancies
0.351
0.442
0.175
-0.003
0.023
1.9424
1.9433
1.9204
1.9755
1.9946
Level
NBO
Type
fi− (NBO)
(FD)
23
22
21
20
10
4
9
23
22
21
BD C1−C4
BD C6−C8
LP Cl10
LP Cl10
LP Cl10
0.350
0.399
0.153
0.035
0.074
TABLE VI. fi− (NBO) and fi+ (NBO) in Eqs. (21) and (22) under FD approximation and FMO approximation for some important NBOs of Fig. 2 (sorted by energy).
Partial orbital occupancies have been added in the last column.
Schematic representations of NBOs
Level
NBO
Type
17
51
BD C1−C2
fi+ (NBO)
(FD)
fi+ (NBO)
(FMO: LUMO)
Occupancies
0.0736
0.406
0.730
fi− (NBO)
fi− (NBO)
Level
NBO
Type
(FD)
(FMO: HOMO)
Occupancies
16
15
14
8
2
16
15
14
BD C1−C2
LP Cl6
LP Cl6
LP Cl6
0.407
0.212
-0.040
0.031
0.664
0.365
0.003
0.012
1.9970
1.9246
1.9754
1.9946
experimental regioselectivity pattern observed in some DielsAlder (D-A) reactions.43,44 There is not a unique criterion;
however, it explains most of the experimental evidence that
has accumulated in cycloaddition processes involving fourcenter interactions. An excellent discussion of regioselectivity
in concerted pericyclic reactions can be found in Ref. 44.
Eqs. (21) and (22) were applied to a sample of molecules
that are related to the D-A reaction type (to obtain the fi− (NBO)
and fi+ (NBO) parameters). This set of molecules was divided
into four samples (Tables I–IV) depending on the substituents
(electron withdrawing group (EWG) or electron donating
group (EDG)).45
As an example, SF. 1 (supplementary material) shows
some NBOs of the reagents A1 and A2 and the schematic
representations of these orbitals (this kind of representations
will be used in Tables V and VI and Figs. 5–10).
FIG. 1. (a) fi− (NBO) and (b) fi+ (NBO) parameters in Eqs. (21) and (22) calculated under frontier molecular orbital approximation for reagent A1 . The NBOs
are represented on the “x” axis (sorted by energy).
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Jesús Sánchez-Márquez
J. Chem. Phys. 145, 194105 (2016)
FIG. 2. (a) fi− (NBO) and (b) fi+ (NBO) parameters in Eqs. (21) and (22) calculated under frontier molecular orbital approximation for reagent A2 . The NBOs are
represented on the “x” axis (sorted by energy).
FIG. 3. ∆fi (NBO) indices (based on the dual descriptor function) for the reagent A1 . Calculations were made with Eq. (33) and (a) frontier molecular orbital
approximation or (b) finite difference approximation. The NBOs are represented on the “x” axis (sorted by energy).
Figs. 1 and 2 show parameters fi− (NBO) and fi+ (NBO) of
Eqs. (21) and (22) calculated for the reagents A1 and A2 (samples 1 and 3). Tables V and VI show these parameters for some
important NBOs of Figs. 1 and 2 (higher values) under finite
difference (FD) and frontier molecular orbital (FMO: HOMO
and LUMO) approximations. In the supplementary material,
FIG. 4. Bond reactivity indices ∆fi (NBO) based on the dual descriptor function for reagent A2 . (a) Frontier molecular orbital approximation, (b) finite difference
approximation. The NBOs are represented on the “x” axis (sorted by energy).
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TABLE VII. ∆fi (NBO) indices (based on the dual descriptor function) for some important NBOs of Fig. 3 (reagent A1 ). Calculations were made with Eq. (33)
and frontier molecular orbital approximation.
NBO
4
9
23
22
21
Type
∆fi (NBO)
(FMO)
∆fi (NBO)
(FD)
Occupancies
NBO
BD C1−C4
BD C6−C8
LP Cl10
LP Cl10
LP Cl10
-0.3972
-0.4784
-0.1479
-0.0101
-0.0150
-0.4575
-0.5310
-0.1756
-0.0093
-0.0187
1.9424
1.9433
1.9204
1.9755
1.9946
89
82
87
Type
∆fi (NBO)
(FMO)
∆fi (NBO)
(FD)
Occupancies
BD C8−Cl10
BD C1−C4
BD C6−C8
-0.0237
0.2945
0.3190
0.0107
0.2549
0.2763
0.0210
0.0586
0.1258
TABLE VIII. Bond reactivity indices ∆fi (NBO) for some important NBOs of Fig. 4 (reagent A2 ). Calculations made with Eq. (S14) (supplementary material).
NBO
2
16
15
14
Type
∆fi (NBO)
(FMO)
∆fi (NBO)
(FD)
Occupancies
NBO
BD C1−C2
LP Cl6
LP Cl6
LP Cl6
-0.7451
-0.3236
-0.0122
-0.0272
-0.7451
-0.3763
-0.0366
-0.0011
1.9970
1.9246
1.9754
1.9946
55
51
the same parameters calculated for all the reagents can be seen
in ST 2-20. A high fi− (NBO) or fi+ (NBO) index indicates that the
NBO is very reactive and that the orbital is very electrophilic
(high fi+ (NBO) ) or very nucleophilic (high fi− (NBO) ). For example, NBOs 9, 4, and 23 have the highest values for the fi− (NBO)
in Fig. 1(a) and NBOs 2 and 16 in Fig. 2(a). This indicates
that they are the most nucleophilic NBOs. On the other hand,
NBOs 87 and 82 in Fig. 1(b) and NBO 51 in Fig. 2(b) have the
highest fi+ (NBO) values and so they are the most electrophilic.
These results are reasonable because the NBOs that show
the highest nucleophilic behavior are those that have higher
energies and high partial occupancies.
On the other hand, the most electrophilic NBOs are
those that have less energies and minor partial occupancies.
These orbitals often change during the chemical reaction (for
example, the rupture and formation of bonds in ST 21 in
the supplementary material). Also, Tables V and VI (and
ST 2-20 in the supplementary material) show that the FMO
Type
∆fi (NBO)
(FMO)
∆fi (NBO)
(FD)
Occupancies
BD C2−Cl6
BD C1−C2
-0.0563
0.7894
-0.0489
0.6978
0.0277
0.0736
approximation leads to the same results (qualitatively) as the
FDapproximation. They did not display exactly the same values but showed the same tendencies. Often, the fi− (NBO) and
fi+ (NBO) values for the FD approximation are lower than the
FMO ones (see ST2-20), the Fukui function under FD approximation has negative zones, and under FMO approximation not.
This affects the coefficients but qualitatively it does not change
the conclusions obtained with both approximations.
B. Testing the bond reactivity index ∆ fi (NBO)
based on dual descriptor
Figs. 3 and 4 show the ∆fi (NBO) indices obtained with
Eq. (27) for the A1 and A2 reagents, where the dual descriptor
function (∆f (r)) is used instead of the Fukui function. Tables
VII and VIII show the ∆fi (NBO) indices for some important
NBOs of Figs. 3 and 4 (the ST 41-57 in the supplementary
material show the same parameters for the B1 -F2 reagents).
Regarding the meaning of the index, a very small value
TABLE IX. fk(NBO)
parameter (Eq. (15)) for the most important NBOs of Fig. 1 (reagent A1 ).
i
Atom number
1
2
3
4
5
6
7
8
9
10
NBO 9
NBO 4
NBO 87
NBO 82
Z
BD C6−C8
BD C1−C4
BD C6−C8
BD C1−C4
6
1
1
6
1
6
1
6
1
17
0.000
0.000
0.000
0.010
0.000
0.471
0.000
0.513
0.000
0.005
0.487
0.000
0.000
0.501
0.000
0.011
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.007
0.000
0.516
0.000
0.473
0.000
0.003
0.481
0.000
0.000
0.511
0.000
0.008
0.000
0.000
0.000
0.000
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Jesús Sánchez-Márquez
J. Chem. Phys. 145, 194105 (2016)
TABLE X. fk(NBO)
parameters (Eq. (15)) for the most important NBOs of
i
Fig. 2 (reagent A2 ).
NBO 2
NBO 51
Atom number
Z
BD C1−C2
BD C1−C2
1
2
3
4
5
6
6
6
1
1
1
17
0.479
0.517
0.000
0.000
0.000
0.004
0.501
0.495
0.000
0.000
0.000
0.003
(negative) indicates a nucleophilic behavior and a very large
index shows an electrophilic behavior. The results obtained
for
bond
reactivity index based on the dual descriptor
the
∆fi (NBO) are consistent with the bond reactivity indices based
on Fukui function ones (fi− (NBO) and fi+ (NBO) in Tables V and
VI). Both indices determine the most reactive orbitals and both
methodologies predict the expected reactivity for the D-A
reaction of ST 21 (supplementary material). An excellent
source for the comparison of previous results is Refs. 32, 48,
and 49.
As an example, ST 21 in the supplementary material
shows some points of the intrinsic reaction coordinate (IRC)
for the D-A reaction: A1 + A2 . The NBOs included in ST 21
(NBOs 4 and 9 from A1 and NBO 2 from A2 ) obtained the
highest values for the fi− (NBO) , fi+ (NBO) , and ∆fi (NBO) indices
(Tables V–VIII). The orbitals related to the new bonds are also
included. It is seen that NBOs 4 and 9 from A1 and NBO 2
from A2 disappear in step 39/81 (transition state-1), and then
new NBOs appear (to the right in ST 21). Of course, these new
NBOs are related to the formation of the new bonds.
C. Theoretical regioselectivity
The indices fi− (NBO) and fi+ (NBO) calculated in Sec. V A
(Figs. 1 and 2) showed that the higher indices (for the A1
reagent) belong to NBOs 4 and 9 (see fi− (NBO) ) and NBOs 82
and 89 (see fi+ (NBO) ), and we will accept that these are the most
FIG. 5. Schematic representations of the main NBOs studied. In this case, the arrows represent theoretical polarization of the orbitals due to EWGs or EDGs
(conclusions based on the parameters fk(NBO)
).
i
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194105-9
Jesús Sánchez-Márquez
reactive orbitals (more nucleophilic and electrophilic).
Regarding A2 , the higher fi− (NBO) corresponds to NBO 2 and
the higher fi+ (NBO) to NBO 51. The parameter fk(NBO)
Eq. (15)
i
will help to study the regioselectivity of the D-A reaction.
This estimates how a NBO is “polarized.” Tables IX and X
for the most reactive NBOs of A1
show calculated fk(NBO)
i
and A2 (main NBOs in Figs. 1 and 2). Tables ST 22-40, in
the supplementary material, show calculated parameters for
all the reagents of this study. The fk(NBO)
parameters depend
i
on the electron withdrawing and donating groups. For example, in Table IX (A1 reagent), the fk(NBO)
parameters for C6
i
and C8 indicate that
NBO
9
(BD
C
−C
6
8 ) is displaced to
(NBO)
C8 f6(NBO)
<
f
and
NBO
87
(BD
C
6 −C8 ) is displaced
89
9
(NBO)
(NBO)
to C6 f8 87 < f6 87 .
The parameters for C1 and C4 indicate that NBO 4 (BD
C1 −C4 ) and NBO 82 are displaced to C4 (f1(NBO)
< f4(NBO)
and
4
4
(NBO)
(NBO)
f1 82 < f4 82 ). Regarding A2 , Table X shows that NBO 2
(BD C1 −C2 ) is displaced to C2 and NBO 51 (BD C1 −C2 ) is
displaced to C1 . The general conclusions for the four samples
(Tables IX and X; ST 22-40 in the supplementary material) are
summarized in Fig. 5.
Taking into account Fig. 5, we can imagine the charge
transfer between the NBOs of A1 and A2, and as a consequence, the theoretical regioselectivity of Fig. 6 can be
obtained. The general information obtained from all the samples (ST 2-40 in the supplementary material) is shown in
Fig. 6 (and SF 2-5 in the supplementary material), where the
arrows represent the charge transfer between a very nucleophilic orbital and a very electrophilic one. The theoretical
regioselectivity showed in Fig. 6 and SF 2-5 (donation and
acceptance of electron pairs) is as expected for this wellknown reaction and allows us to predict most of the products
of the reactions.46,47 Notice that the systems E2 and F2 have
a LUMO with positive orbital energies. A new electron added
to the molecule might be located in most diffuse orbital of the
J. Chem. Phys. 145, 194105 (2016)
system giving in these two cases meaningless results for the
prediction of a nucleophilic attack.
D. Comparison of atomic and bond reactivity indices
Under the FMO approximation, Eq. (19) can be written
as
φ(HOMO) (r)2 ≈
X
i
2
fi− (NBO) · φ(NBO)
(r) .
i
(34)
2
2
Condensing the φHOMO (r) and φi(NBO) (r) functions by
means of Eqs. (10) and (15), we obtain Eq. (35),
X
)
fi− (NBO) · fk(NBO
.
(35)
fk− ≈
i
i
Eq. (35) allows us to relate the atomic reactivity indices fk−
(Table XI) and the bond reactivity indices fi− (NBO) . This leads
us to the useful conclusion that the atomic reactivity index is
the sum of several contributions: the bond reactivity indices of
each orbital of the molecule. We think that the fk(NBO)
coeffii
− (NBO)
cients (Table XII) could determine what portion of the fi
index belongs to each atom. Also, Eq. (35) is a direct connection between atomic reactivity and bond reactivity indices
and can be useful for comparing both. Table XI shows that the
atomic reactivity index of the chlorine atom is similar to the
carbons (C1 , C4 , C6 , and C8 ). This seems incoherent because
chlorine does not react (at least not like the carbons). However,
the bond reactivity index of NBO 4 (or NBO 9) is in the order
of 1:2 regarding the chlorine NBO 23. Eq. (35) allows us to
analyze why different conclusions were obtained from the two
types of indices: the double-bonds (NBO 9: BD C6 ==C8 and
NBO 4: BD C1 ==C4 in Table V) have a bond reactivity index
bigger than the lone-pair of the chlorine (NBO 23). The double bond index is distributed between the two carbons (equally
shown in Table XII). On the
divided, as the parameter fk(NBO)
i
other hand, NBO 23 (LP) “only belongs” to the chlorine and
FIG. 6. Theoretical donation and acceptance of electron pairs at the transition state (based on the parameters fi− (NBO) , fi+ (NBO) , and fk(NBO)
). Reagents: diene
i
bearing an EWG at C1 and dienophile bearing an EWG.
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Jesús Sánchez-Márquez
J. Chem. Phys. 145, 194105 (2016)
FIG. 7. Relationship between atomic reactivity indices (fk− ) for some dienes (A1 , B1 , K1 , and H1 ) in some particular reactions.
only contributes to the chlorine index, and this justifies why the
atomic reactivity index of the chlorine is bigger than expected.
Table XI shows that the fk− indices obtained with Eq. (9)
are very different to the ones obtained with Eq. (35), and they
even have a different tendency. The atoms with the highest
reactivity indices must form the new bonds. For example, in
the case of the reagent A1, the double bonds (NBOs) C1==C4
and C6==C8 are the most nucleophile. C4 and C8 are the most
nucleophiles atoms (fk− (C1 ) < fk− (C4 ) and fk− (C6 ) < fk− (C8 )
according to Eq. (35)). This could show that atom C8 attacks
the dienophile and the C4 the atom C6. Then we obtain the
bonds: C8-dienophile (simple) and C4==C6 (double). To close
the ring, the most nucleophile atom of the dienophile (that
is bonded to the EWG) attacks to the C1 and the bond C1dienophile (simple) is assembled. This interpretation of the
regioselectivity based on the indices fk− (Eq. (35)) (and we
have tried to summarize in Fig. 7) is coherent with the experimental results. The B1 (ST 58), H1 (ST 63), K1 (ST 67), and
many other cases (ST 58-69 of the supplementary material)
show similar data behaviour. Broadly speaking, the new index
calculated with Eq. (35) shows better results than the other
studied indices (Eq. (9) and populations of the NBO analysis).
Probably, electron withdrawing and donating groups
polarize electronic density and this changes the orbitals
(canonical and NBOs), but does not change all the occupied orbitals in the same way. The “standard methodologies”
(Eqs. (6)–(8)) use the information of only one frontier orbital
TABLE XI. Condensed Fukui function fk− calculated with Eq. (9) (frontier
molecular orbital approximation), finite difference (FD) approximation, and
atomic population from a NBO analysis, and Eq. (35), for the reagent A1 .
Atom
1
2
3
4
5
6
7
8
9
10
fk−
fk−
fk−
Z
FMO Eq. (10)
FD (NBO analysis)
FMO Eq. (35)
6
1
1
6
1
6
1
6
1
17
0.236
0.001
0.001
0.105
0.005
0.177
0.002
0.236
0.001
0.234
0.234
0.025
0.044
0.055
0.055
0.116
0.050
0.130
0.036
0.255
0.185
0.000
0.000
0.194
0.000
0.213
0.000
0.231
0.000
0.186
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194105-11
Jesús Sánchez-Márquez
J. Chem. Phys. 145, 194105 (2016)
TABLE XII. fk(NBO)
parameter in Eq. (15) for the most nucleophile NBOs of
i
reagent A1 .
NBO 21 NBO 22 NBO 23
Atom number
1
2
3
4
5
6
7
8
9
10
Z
6
1
1
6
1
6
1
6
1
17
NBO 9
NBO 4
LP Cl10 LP Cl10 LP Cl10 BD C6−C8 BD C1−C4
0.000
0.000
0.000
0.000
0.000
0.003
0.000
0.040
0.001
0.955
0.000
0.000
0.000
0.000
0.000
0.002
0.000
0.023
0.001
0.974
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.014
0.000
0.985
0.000
0.000
0.000
0.010
0.000
0.471
0.000
0.513
0.000
0.005
0.487
0.000
0.000
0.501
0.000
0.011
0.000
0.000
0.000
0.000
(HOMO or LUMO) and there is no guarantee that this orbital
was polarized in the same way as the other orbitals (especially,
if it has a different symmetry). The bond reactivity methodology (with NBOs) uses the information of all the orbitals
and they are also localized orbitals and, consequently, more
independent (not like the canonical ones). A perturbation in a
localized area of the molecule only affects some bond orbitals
or only one (canonical orbital behaviour is different).
VI. CONCLUSIONS
A new descriptor has been defined: “bond reactivity
index.” Considering that a chemical reaction is determined
by the breaking and creation of bonds, the new index has the
advantage that it is related with bonds instead of atoms. These
bond reactivity indices based on the Fukui function and the
dual descriptor function have been calculated and the results
obtained from the two kinds of indices have been equated.
The new indices have been tested with a set of representative molecules and the Diels-Alder reaction type, and we have
found a good description of the reactivity. Also, the FD and
FMO approximations have been compared and both showed
the same tendencies.
A new atomic reactivity index has been defined and it
provided the best results for the sample of molecules studied.
Also, it provided a relationship that we used to compare the
atomic reactivity and bond reactivity indices, obtaining useful information about the relationship between the two kinds
of indices.
SUPPLEMENTARY MATERIAL
See supplementary material for the complete reactivity
index data of the studied reagents.
ACKNOWLEDGMENTS
Calculations were performed through CICA (Centro
Informático Cientı́fico de Andalucı́a). We thank Professor M.
Fernández Núñez, whose suggestions were greatly appreciated.
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