EIGHT RULES OF AROMATICITY

EIGHT RULES OF AROMATICITY
Miquel Solà
Institute of Computational Chemistry and Catalysis
Universitat de Girona
EIGHT RULES OF AROMATICITY5/2/2015

3
1825
1900
Benzene synthesis
Polycyclic Aromatic
Hydrocarbons
Heterocycles
1938 Aromatic Transition States
1959 Homoaromaticity
1964Möbius Aromaticity
1972
Triplet Aromaticity
4n Baird rule
1978 Three-dimensional
Aromaticity
1979-Aromaticity
1872
Kekulé
Benzene Structure
1855Chemical
Aromaticity
1931
1959
4n+2 Hückel rule
p-sextets
150 YEARS OF AROMATICITY
5/2/2015 EIGHT RULES OF AROMATICITY

4
Aromaticity is and old concept that is still very useful!
1985
1991
2001
Fullerenes
2007
All-metal Aromaticity
2005d-orbital Aromaticity
Nanotubes
2008
-Aromaticity
f-Aromaticity
1982
Metallabenzenes
2000 2(N+1)2 Hirsch’s rule
…and more recently

What is aromaticity?
“Aromaticity is a manifestation of electron delocalization in closed
circuits, either in two or three dimensions, which results in
energy lowering, often quite substantial, and a variety of unusual
chemical and physical properties. These include a tendency
toward bond length equalization, unusual reactivity, and
characteristic spectroscopic features as well as distinctive
magnetic properties related to strong induced ring currents”
Chen and Schleyer et al. Chem. Rev. 2005, 105, 3842.
In 2014, in every 42 minutes appeared a paper in which the
word “aromatic*” was in the title, keywords or the abstract!

R. Hoffmann, V.I. Minkin, B.K. Carpenter, Hyle, Int. J. Philos. Chem. 1997, 3, 3
“Widely applied for the characterization of specific features of
conjugated cyclic molecular systems, the notion of
aromaticity lacks a secure physical basis. Not that this has
stopped aromaticity from being a wonderful source of
creative activity in chemistry. We can think of no other
concept that has led to so much exciting chemistry! Yet,
although numerous indices of aromaticity have been
designed, based on energetic, geometrical and magnetic
criteria, no single property exists whose measurement could
be taken as a direct, unequivocal measure of aromaticity.”

Aromaticity is not an observable. Consequently, it is not well
defined and there is not a unique and generally accepted
measure of aromaticity.
5/2/2015
Many criteria have been used to develop indices of aromaticity:
– Energetic (ASEs)
– Structural or Geometrical (HOMA,…)
– Magnetic (NICS, ring currents, 1H NMR…)
– Electronic (hardness, electron delocalization…)

5/2/2015
Indices of aromaticity
 

bondsn
1i
2
iopt RR
n
α
1HOMA
Ropt = 1,388 Å
a = 257,7
 
3
1
, 3
3
n
n n
PDI





Symmetry Delocalization
1.389
0.074
0.100
Magnetic shielding tensor
DI or 2c-ESI
QTAIM partition
E. Matito, M. Duran and MS, J. Chem. Phys. 2005, 122, 014109
J. Poater, X. Fradera, M. Duran and MS, Chem. Eur. J. 2003, 9, 400

5/2/2015
Indices of aromaticity
Multicenter delocalization indices
 For monodeterminantal WFs:
M. Giambiagi, M. S. de Giambiagi, C. D. dos Santos Silva and A. P. de Figuereido, Phys. Chem.
Chem. Phys. 2000, 2, 3381
P. Bultinck, R. Ponec and S. van Damme, J. Phys. Org. Chem. 2005, 18, 706
A = {A1, A2, …, AN}
     Nii
iii
iiiiiiring ASASASnnI N
N
N 1
21
32211
,,,
21)( A
)(
2
1
)(
)(
AA
A

P
ringI
N
MCI

Rules of aromaticity. Where do they come from?
Aromatic compounds are usually symmetric and they have
degenerate orbitals.
5/2/2015
In this case only some special numbers of electrons leads to a
closed-shell or half-filled same spin open-shell structures.
Closed-shell or half-filled same spin open-shell structures
provide an extra stabilization of the system (aromatic
stabilization) and justifies the existence of a series of aromaticity
rules.

1. 4n+2 Hückel’s rule
5/2/2015
It strictly holds for monocyclic systems like benzene and
cyclooctatetraene (annulenes with Dnh symmetry)
Dnh
Stability comes from the closed-shell electronic configuration
(equivalent to the octet rule)
even n

5/2/2015
Cyclohexene
1.828
1.008
0.976
0.978
1.389
Benzene
 Symmetry and significant delocalization
(C,C’)m=0.074
(C,C’)p=0.100
1.008
0.976
1.389
1.3891.389
1.389
1.389
 meta vs para-delocalization
Electronic delocalization in aromatic species
(C,C’)m=0.043
(C,C’)p=0.009
(C,C’)m=0.059
(C,C’)p=0.014

5/2/2015
1-2
1-3
1-4
1-2
1-3
1
2
3
4
5
6
Sp(A) = 0
1-2
1-3
1-41-5

5/2/2015
N-2 N N+2 D1 D2 diff
(1-2) 1.270 1.390 1.353 0.120 -0.037 -0.157
p
(1-2) 0.300 0.425 0.387 0.125 -0.038 -0.163
(1-3) 0.148 0.073 0.121 -0.075 0.048 0.123
p
(1-3) 0.107 0.036 0.083 -0.071 0.047 0.118
(1-4) 0.050 0.103 0.062 0.054 -0.042 -0.095
p
(1-4) 0.039 0.094 0.052 0.054 -0.041 -0.096
AROMATIC
4N+2
ANTIAROMATIC
4N
AROMATIC
4N+2
ANTIAROMATIC
4N
(1-2)  (1-3)  (1-4)  (1-2)  (1-3)  (1-4) 
1-2
1-31-4
C6H6
+2e-
C6H6
+2
4p-e antiaromatic
C6H6
6p-e aromatic
C6H6
-2
8p-e antiaromatic
+2e-
D1 D2
D1=[N] - [N-2], D2=[N+2] – [N] diff=D2 - D1

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ANTIAROMATIC
(N)
C8H8 N-2 N N+2 D1 D2 diff
(1-2) 1.332 1.401 1.361 0.069 -0.040 -0.109
p
(1-2) 0.344 0.430 0.409 0.087 -0.022 -0.108
(1-3) 0.113 0.064 0.094 -0.048 0.030 0.078
p
(1-3) 0.074 0.029 0.060 -0.046 0.031 0.077
(1-4) 0.020 0.043 0.026 0.023 -0.017 -0.040
p
(1-4) 0.016 0.040 0.023 0.023 -0.016 -0.040
(1-5) 0.056 0.008 0.042 -0.048 0.034 0.082
p
(1-5) 0.054 0.007 0.041 -0.047 0.034 0.082
AROMATIC
(N+2)
ANTIAROMATIC
(N)
AROMATIC
(N-2)
(1-2)  (1-3)  (1-4)  (1-2)  (1-3)  (1-4)  (1-5) (1-5) 
1-2
1-3
1-41-5
C8H8
+2e-+2e-
8p-e antiaromatic
C8H8
10p-e aromatic
C8H8
2-
6p-e aromatic
C8H8
2+
D1 D2
1
5
4
dC-C(1-5) > dC-C(1-4)
(1-5) > (1-4)
planar

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0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
-0.03
-0.01
0.01
0.03
0.05
0.07
0.09
0.11
0.13
0.15
0.17
1-2 1-3 1-4 1-2 1-3 1-4 1-5
4,5-MR 6,7-MR 8,9-MR
1-2 Decrease Increase Decrease
1-3 Increase Decrease Increase
1-4 Increase Decrease
1-5 Increase
AROMATIC
4N2
ANTIAROMATIC
4N
C8H8
2-
C8H8
2+
C8H8
C6H6
2-
C6H6
2+
C6H6
4,5-MR 6,7-MR 8,9-MR
1-2 Increase Decrease Increase
1-3 Decrease Increase Decrease
1-4 Decrease Increase
1-5 Decrease
AROMATIC
4N2
ANTIAROMATIC
4N
F. Feixas, E. Matito, M. Solà and J. Poater Phys. Chem. Chem. Phys. 2010, 12, 7126

a2ub2ga1g
HOMO ( pocc)HOMO-1 (occ)HOMO-2 (occ) LUMO+
eu
LUMO (  unocc)
N-2(2p e-) N-2(0p e-) N-2(2p e-) N(2p e-)
Al4 (a1g)a Al4 (a2u)a Al4 (b2g)a Al4
2-
MCI 0.197 0.182 0.325 0.356
MCI 0.010 0.182 0.138 0.169
MCIp 0.187 0.000 0.187 0.187
Aromaticity p  p p
a Orbital from which two electrons have been removed
  p
radial tangential
F. Feixas, E. Matito, J. Poater and M. Solà WIREs Comput. Mol. Sci. 2013, 3, 105

2. Clar’s p-sextet rule
5/2/2015
The 4n+2 Hückel rule strictly holds for monocyclic
systems like benzene and cyclooctatetraene
Coronene
24 p electrons / 4n
Pyrene
16 p electrons / 4n
The Clar’s p-sextet rule is a generalization of the 4n+2
Hückel rule to polycyclic aromatic hydrocarbons
(PAH)

“Aromatic p-sextets are defined as six p-electrons localized to
a single benzene ring separated from adjacent benzene rings
by formal CC single bonds”
“Clar’s valence structure of a benzenoid hydrocarbon is a
valence structure having the maximal number of disjoint p-
aromatic sextets”
phenanthrene

chrysene
Migrating p-sextets
Reactions in
anthracene occur over
9,10-positions
Picene
Anthracene

Acenes more reactive than phenacenes
coronene pyrene
4n+2 rule fails
For n larger than ca. 7
acenes become biradicals

The fewer the Clar sextets the deeper the color
More stable, larger
HOMO-LUMO gaps
A. T. Balaban and D. J. Klein J. Phys. Chem. C 2009, 113, 19123

One isomer reacts with maleic anhydride affording a Diels-Alder adduct while
the other is unreactive. Can you make an educated guess?
Two isomeric tribenzoperylenes

cis and trans-dibenzapentalene
Clararomatic PAH
With the Clar structures drawn it is clear that isomer 1 is the one that reacts
with maleic anhydride affording a Diels-Alder adduct while 2 is unreactive.

I. Gutman et al. Chem. Phys. Lett. 2004, 397, 412

A
A
A
A A
A
A A
A A
B
B B
B B
B
B B
B
B
C
C
C
C C
C C
D
D
E
D E F
Mol. 1
Mol. 2 Mol. 3
Mol. 4
Mol. 5
Mol. 6
Mol. 7
Mol. 8
Mol. 9 Mol. 10
Molecule
Param. Ring 1 2 3 4 5 6 7 8 9 10
A 0.080 0.069 0.086 0.084 0.083 0.080 0.084 0.068 0.058 0.071
B 0.047 0.043 0.026 0.034 0.041 0.051 0.034 0.044 0.051 0.045
C 0.044 0.068 0.059 0.031 0.045 0.082 0.031
D 0.073 0.045 0.033
E 0.071 0.029
PDI
F 0.055
A 0.856 0.834 0.889 0.811 0.872 0.847 0.807 0.840 0.749 0.721
B 0.435 0.553 -0.030 0.383 0.356 0.520 -0.518 0.550 0.686 0.477
C 0.518 0.788 0.640 -0.496 0.570 0.853 -0.511
D 0.838 0.479 0.341
E 0.723 0.510
HOMA
F 0.732
A -10.06 -12.74 -8.63 -9.44 -9.93 -9.80 -9.30 -12.63 -10.97 -12.22
B -6.82 -5.07 -1.18 -4.13 -5.38 -7.15 -3.74 -4.82 -9.33 -5.88
C -5.47 -11.27 -8.35 -2.29 -13.04 -9.77 -1.83
D -11.58 -6.04 -5.86
E -12.38 0.05
NICS
F -11.39
PAHs with a unique Clar structure

Mol. 11
Mol. 12
Mol. 13
a a
a
b b
b
c
c
A A
A
B
B
Mol. 14 Mol. 15
Mol. 16
Mol. 17
a
a
a
a
b
b
b
b
c
c
A
A
A
A
B
B
B
B
C
C
C
C
D
D
D
E
E F
Molecule
Param. Ring 11 12 13 14 15 16 17
A 0.076 0.066 0.080 0.069 0.079 0.036 0.083
B 0.066 0.053 0.066 0.057 0.048 0.040
C 0.038 0.031 0.064 0.067
D 0.084 0.085 0.047
E 0.085 0.054
PDI
F 0.079
A 0.769 0.619 0.829 0.697 0.749 0.447 0.877
B 0.696 0.542 0.730 0.305 0.666 0.333
C 0.266 -0.097 0.783 0.783
D 0.883 0.820 0.464
E 0.883 0.568
HOMA
F 0.824
A -9.98 -8.84 -9.94 -9.30 -10.19 -1.90 -9.83
B -12.60 -7.69 -11.69 -7.68 -10.06 -5.06
C -4.58 -3.91 -12.70 -11.36
D -9.81 -9.55 -6.42
E -8.99 -7.81
NICS
F -9.84
G. Portella, J. Poater and M. Solà J. Phys. Org. Chem. 2005, 18, 785.

3. Glidewell-Lloyd extension of Clar’s p-sextet rule
The total p-electron population in polycyclic systems tends to form the
smallest 4n+2 groups and to avoid the formation of the smallest 4n groups
C. Glidewell and D. Lloyd Tetrahedron 1984, 40, 4455.
For benzenoid systems this reduces to the Clar’s p-sextet rule
Examples:
YES NO YES NO
YES NO NOYES NO

4. 4n lowest-lying triplet excited state Baird’s rule
5/2/2015
It strictly holds for monocyclic systems like
benzene and cyclooctatetraene
Dnh
Stability comes from the half-full shell electronic configuration
even n

5/2/2015
N-2(t) N-2(s) N N+2(s) N+2(t)
p 2.543 2.644 3.359 3.466 3.451
(1-2) 1.356 1.251 1.397 1.349 1.368
p
(1-2) 0.401 0.287 0.422 0.382 0.409
(1-3) 0.082 0.125 0.067 0.119 0.085
p
(1-3) 0.051 0.088 0.036 0.082 0.056
(1-4) 0.109 0.064 0.102 0.062 0.092
p
(1-4) 0.099 0.054 0.093 0.052 0.082
FLU 0.009 0.029 0.000 0.024 0.004
MCI 0.162 -0.001 0.078 0.003 0.095
AROMATIC
4N (triplet)
4N2 (singlet)
ANTIAROMATIC
4N (singlet)
(1-2)  (1-3)  (1-4) 
1-2
1-31-4
F. Feixas, E. Matito, M. Solà and J. Poater Phys. Chem. Chem. Phys. 2010, 12, 7126

Excited state intramolecular proton transfer (ESIPT) mechanism

5Ta3
-
MCI 0.776
MCI
orb 0.362
MCIp
orb 0.178
MCI
orb 0.235
Aromaticity p
(2e’)2(3a1’)2(1e’’)2(1e1’)4(1a2’’)2(2a1’)2(1a1’)2
5Ta3
-
3Hf3
 p  p 
F. Feixas, E. Matito, M. Duran, J. Poater and M. Solà Theor. Chem. Acc. 2011, 128, 419
 and p Baird aromatic and  Hückel aromatic

5. Soncini and Fowler generalized form of Baird’s rule
5/2/2015
The lowest-lying electronic states with even spin
(singlet, quintet,. . .) of rings with (4n+2)p-electrons and
the lowest-lying states with odd spin (triplet, septet,. . .)
of monocycles with 4np-electrons are aromatic
Ground state, S = 0 Quintet state, S = 2
A. Soncini and P. W. Fowler, Chem. Phys. Lett., 2008, 450, 431

5/2/2015
CASSCF(6,6)/6-311++G(d,p)
F. Feixas, J. Vandenbussche, P. Bultinck, E. Matito, and M. Solà, Phys. Chem. Chem. Phys., 2011, 13, 20690.
5. Soncini and Fowler generalized form of Baird’s rule

Möbius Aromaticity
M. Rosenberg, C. Dahlstrand, K. Kilså and H. Ottosson, Chem. Rev. 2014, 114, 5379.

6. 2(n+1)2 Hirsch’s rule
Rotor rigid solution
𝐻Ψ 𝜃, 𝜑 = 𝐸Ψ 𝜃, 𝜑
𝐻 = −
1
2𝑚
𝛻2
= −
ℏ2
2𝐼
1
𝑠𝑖𝑛𝜃
𝜕
𝜕𝜃
𝑠𝑖𝑛𝜃
𝜕
𝜕𝜃
+
1
𝑠𝑖𝑛2 𝜃
𝜕2
𝜕2 𝜑
𝐻 𝑌𝑙
𝑚
𝜃, 𝜑 =
ℏ2
2𝐼
𝑙 𝑙 + 1 𝑌𝑙
𝑚
𝜃, 𝜑
2, 8, 18, 32, 50, 72, 98… = 2(n+1)2
2
8
18
32
n = 0
n = 1
n = 2
n = 3
The 2(n+1)2 Hirsch rule is a rule for spherical aromaticity

6. 2(n+1)2 Hirsch’s rule
Z. Cheng and R. B. Bruce Chem. Rev., 2005, 105, 3613

2(n+1)2 Hirsch’s rule
4n lowest-lying
triplet excited state
Baird’s rule
4n+2 Hückel’s rule
?OPEN-SHELL
CLOSED-SHELL
PLANAR SPHERICAL

7. 2n2+2n+1 (S=N+½) rule
The 2n2+2n+1 (S=N+½) rule is a rule for open-shell spherical aromaticity
2
8
18
32
1, 5, 13, 25, 41, 61, 75… = 2n2+2n+1 (S=N+½)
n = 0
n = 1
n = 2
n = 3
J. Poater and M. Solà Chem. Commun., 2011, 47, 11647

7. 2n2+2n+1 (S=N+½) rule
1, 5, 13, 25, 41, 61, 75… = 2n2+2n+1 (S=n+1/2)
Systems symm. NICS(1)zz MCI r(C,C) Spin
C20
2+ Ih -7.4 0.020 1.447 S = 0
C20
7+ Ih -4.0 0.035 1.494 S = 5/2
C20
5- Ih -18.0 0.024 1.508 S = 7/2
2, 8, 18, 32, 50, 72, 98… = 2(n+1)2

7. 2n2+2n+1 (S=N+½) rule
Systems symm Ring NICS(1)zz MCI BLA Spin
C60 Ih 6-MR 0.8 0.018 0.058 S = 0
5-MR 21.5 0.011
C60
1- Ih 6-MR -1.4 0.017 0.002 S = 11/2
5-MR -19.9 0.049
C60
19+ Ih 6-MR -14.9 0.019 0.013 S = 9/2
5-MR -25.3 0.041
C60
10+ Ih 6-MR -18.6 0.011 0.030 S = 0
5-MR -29.5 0.017
1, 5, 13, 25, 41, 61, 75… = 2n2+2n+1 (S=n+1/2)
2, 8, 18, 32, 50, 72, 98… = 2(n+1)2

7. 2n2+2n+1 (S=N+½) rule
1, 5, 13, 25, 41, 61, 75… = 2n2+2n+1 (S=n+1/2)
2, 8, 18, 32, 50, 72, 98… = 2(n+1)2
Systems symm. Ring NICS(1)zz MCI BLA Spin
C80 S6 5-MR 10.7 0.019 S = 0
6-MR -5.2 0.012 0.025
5-MR 26.3 0.018
6-MR 11.3 0.014 0.001
6-MR -5.1 0.012 0.025
C80
8+ Ih 6-MR -7.2 0.011 0.015 S = 0
5-MR -4.0 0.017
C80
5- Ih 6-MR -20.8 0.019 0.012 S = 13/2
5-MR -5.5 0.034

7. 2n2+2n+1 (S=N+½) rule
1, 5, 13, 25, 41, 61, 75… = 2n2+2n+1 (S=n+1/2)
2, 8, 18, 32, 50, 72, 98… = 2(n+1)2
Systems symm. NICS(1)zz MCI Spin
Ge12
2- Ih -5.5 0.049 S = 0
Ge12
1- Ih -405.9 0.113 S = 5/2
Ge12
4+ Ih -69.0 0.088 S = 0

8. The 2n+2 o 4n+2 Wade-Mingos rule
• Closo, [BnHn]2-, n+1 cage electron pairs
• Nido, BnHn+4, n+2
• Arachno, BnHn+6, n+3

Closo, [BnHn]2- boron clusters
• They are very stable. For instance, Li2[B12H12] do not
decompose below 600 ºC.
• [BnHn]2- boron clusters (especially for n = 6, 7, 10 and 12)
tend to retain their molecular structure.
• Lipscomb suggested that [BnHn]2- boron clusters are aromatic.
• Rules: 2n+2 Wade’s rule and 4n+2 Mingos’ rule. Both are
equivalent because Wade’s rule refers to the skeletal electron
pairs and Mingos’ rule incorporates the exo electron pairs of the
n B-H bonds (2n electrons).

Closo [B6H6]2- boron cluster
26 VE, 13 e-pairs, 6 e-pairs for B–H bonding
7 e-pairs for the cage

Hydrocarbon vs borohydride analogy
-TO DEFINE THE MODEL ORGANIC COMPOUND
-TO DRAW THE MODEL ORGANIC COMPOUND
-TO DEFINE THE CONFINED SPACE
-TO TRANSMUTE THE C ELEMENTS INTO B ELEMENTS + 1 e-
- FOR EACH B–B BOND ONE CAN ADD A SACRIFICIAL ATOM (H+ or BH4+)
-TO OBEY THE 8 ELECTRONS RULE
-TO GENERATE THE NEW STRUCTURE BY STRUCTURE RELAXATION
Procedure:
ABBREVIATIONS USED
cS CONFINED SPACE
eT ELECTRONIC TRANSMUTATION
sA SACRIFICIAL ATOM
sR STRUCTURAL RELAXATION

Examples:
J. Poater, M. Solà, C. Viñas and F. Teixidor
Chem. Eur. J., 2013, 19, 4169

TNVE=22

J. Poater, M. Solà, C. Viñas and F. Teixidor Angew. Chem. Int. Ed., 2014, 53, 12191
4n+2 Wade-Mingos’ rule (3D aromaticity) is connected to the 4n+2
Hückel rule (2D aromaticity) through the hydrocarbon-borohydride
analogy. Total number of valence electrons remain the same.
TNVE=26 TNVE=30

27/3/2008
MolInfBioInf2008 Budapest
THE AROMATICITY PUZZLE 51
http://iqcc.udg.edu

EIGHT RULES OF AROMATICITY

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