developed. Among them we can mention some refined NICS measures (such as the
NICS profiles [6, 7] or NICS πzz values [8]) and new defined indices of aromaticity
based on electron delocalization measures such as PDI [9], FLU [10], I ring [11], and
MCI [12] and the normalized versions of the last two, I NG and I NB [13].
Many of the novel aromatic compounds found present high symmetry. Symmetry
is one of the usual features of aromatic compounds. Although not all aromatic
species are symmetric, the most archetypal aromatic compounds are highly symmetric and possess degenerate highest-occupied molecular orbitals. These orbitals
can be fully occupied resulting in a closed-shell structure or can be same-spin
half-filled. This is the case of paradigmatic aromatic species, namely, benzene,
B 6 H 6
2− closo borane cluster, C 60
10+ , Al 4
2− , but also of triplet C 5 H 5
+ or C 60
1− with a
spin of 11/2. The closed-shell or same-spin half-filled electronic structure is the
origin of several rules of aromaticity such as the 4n + 2 Hückel [14], 4n Baird [15],
2n + 2 Wade-Mingos [16, 17], 2(n + 1)
2 Hirsch [18] or the 2n
2 + 2n + 1 [19] rules.
In this chapter we review the results of our recent investigations on the
abovementioned rules of aromaticity. It is worth noting that extension of these rules
to aromatic polycyclic hydrocarbons (PAH) has led to the Clar π-sextet rule [20, 21]
and the Glidewell-Lloyd [22] extension of Clar’s rule. These extensions will not be
discussed in the present chapter. Neither we will comment on the 4n rule for
Möbius aromaticity [23, 24] nor on its extension by Rzepa in what is known as the
linking number rule [25].
12.2 Hückel’s 4n + 2 Rule
The classification of molecules into groups based on similar molecular properties,
structure, or reactivity has been one of the goals of chemistry from the very
beginning. Benzene and related compounds attracted a lot of attention because of
their peculiar stability and reactivity and were gathered to form the group of aromatic molecules. Understanding the peculiarities of aromatic molecules and the
features that a molecule should display to join the group of aromatic compounds
became a goal of chemistry. Since aromaticity is not directly observable, theoretical
calculations have played a key role in classifying molecules as aromatic. In 1931,
Hückel put forward the importance of delocalized electrons (π-electrons) to
rationalize molecular properties of conjugated molecules [26]. According to Hückel
molecular orbital (HMO) theory, the topology of the molecule and the number of πelectrons determines the stability of the molecule. An outcome of the HMO theory
applied to cyclic conjugated hydrocarbons was the well-known 4n + 2 stability rule
[14]. Its connection to the concepts of aromaticity and antiaromaticity has played a
central role in organic chemistry.
Hückel’s rule of aromaticity states that monocyclic conjugated hydrocarbons with
4n + 2 π-electrons are aromatic, whereas systems with 4n π-electrons are antiaromatic. Consequently, the number of π-electrons is crucial to determine stability,
structure, and reactivity of aromatic and antiaromatic systems. In 1954, Doering and
322
F. Feixas et al.
NICS profiles [6, 7] or NICS πzz values [8]) and new defined indices of aromaticity
based on electron delocalization measures such as PDI [9], FLU [10], I ring [11], and
MCI [12] and the normalized versions of the last two, I NG and I NB [13].
Many of the novel aromatic compounds found present high symmetry. Symmetry
is one of the usual features of aromatic compounds. Although not all aromatic
species are symmetric, the most archetypal aromatic compounds are highly symmetric and possess degenerate highest-occupied molecular orbitals. These orbitals
can be fully occupied resulting in a closed-shell structure or can be same-spin
half-filled. This is the case of paradigmatic aromatic species, namely, benzene,
B 6 H 6
2− closo borane cluster, C 60
10+ , Al 4
2− , but also of triplet C 5 H 5
+ or C 60
1− with a
spin of 11/2. The closed-shell or same-spin half-filled electronic structure is the
origin of several rules of aromaticity such as the 4n + 2 Hückel [14], 4n Baird [15],
2n + 2 Wade-Mingos [16, 17], 2(n + 1)
2 Hirsch [18] or the 2n
2 + 2n + 1 [19] rules.
In this chapter we review the results of our recent investigations on the
abovementioned rules of aromaticity. It is worth noting that extension of these rules
to aromatic polycyclic hydrocarbons (PAH) has led to the Clar π-sextet rule [20, 21]
and the Glidewell-Lloyd [22] extension of Clar’s rule. These extensions will not be
discussed in the present chapter. Neither we will comment on the 4n rule for
Möbius aromaticity [23, 24] nor on its extension by Rzepa in what is known as the
linking number rule [25].
12.2 Hückel’s 4n + 2 Rule
The classification of molecules into groups based on similar molecular properties,
structure, or reactivity has been one of the goals of chemistry from the very
beginning. Benzene and related compounds attracted a lot of attention because of
their peculiar stability and reactivity and were gathered to form the group of aromatic molecules. Understanding the peculiarities of aromatic molecules and the
features that a molecule should display to join the group of aromatic compounds
became a goal of chemistry. Since aromaticity is not directly observable, theoretical
calculations have played a key role in classifying molecules as aromatic. In 1931,
Hückel put forward the importance of delocalized electrons (π-electrons) to
rationalize molecular properties of conjugated molecules [26]. According to Hückel
molecular orbital (HMO) theory, the topology of the molecule and the number of πelectrons determines the stability of the molecule. An outcome of the HMO theory
applied to cyclic conjugated hydrocarbons was the well-known 4n + 2 stability rule
[14]. Its connection to the concepts of aromaticity and antiaromaticity has played a
central role in organic chemistry.
Hückel’s rule of aromaticity states that monocyclic conjugated hydrocarbons with
4n + 2 π-electrons are aromatic, whereas systems with 4n π-electrons are antiaromatic. Consequently, the number of π-electrons is crucial to determine stability,
structure, and reactivity of aromatic and antiaromatic systems. In 1954, Doering and
322
F. Feixas et al.