Page 736 - Advanced Organic Chemistry Part A - Structure and Mechanisms, 5th ed (2007) - Carey _ Sundberg

both bond length alternation and bond stretching into consideration. The aromaticity 719
index HOMA is formulated as
SECTION 8.1

2 2 Criteria of Aromaticity
HOMA = 1−
R opt − R av + R av − R i (8.1)
n
where R is average bond length, R is the optimum bond length, R is an individual
av opt i
bond length, n is the number of bonds, and is a constant such that HOMA is equal
to zero for a single localized structure. The HOMA equation can be rewritten as
2
HOMA = 1− R opt −R i (8.2)
n
The R opt values are derived from butadiene, which is thereby the implicit standard of
the definition. For C−C bonds, = 257.7 Å −2 and R opt = 1 388 Å. Further definition
of bond lengths is required for structures with heteroatoms.
In the formulation in Equation (8.1), the first term, the deviation from the optimum
length, accounts for the energy associated with bond length effects, and the second term,
which reflects deviation from the average, accounts for bond alternation. The HOMA
energy and alternation terms have been calculated for many aromatic compounds. 15
The values for benzene, naphthalene, anthracene, and phenanthrene are shown below. 16
Note the decrease as the molecules get larger. HOMA indices can also be assigned
to the individual rings in polycyclic structures. A relatively low HOMA is assigned
to the center ring in phenanthrene. We will see later that this is in accord with other
properties of phenanthrene. In contrast, the center ring in anthracene has a slightly
higher index than the terminal rings.

0.870
0.991 0.787 0.458
0.723 0.632
0.991 0.811 0.718 0.741

Another aromaticity index based on bond lengths was devised by Bird. 17 The
input into the index are bond order values derived from bond lengths. (See Section
1.4.6 to review the relationship between bond length and bond order.) The formulation
of the index is

V 100 N −N 2 a
I = 100 1− V = and N = −b (8.3)
V N n R
k 2
where N is the arithmetic mean of bond orders. V is the value of V for the corre-
k
sponding localized structures, and a and b are constants for each bond type.
The index I is assigned as 100 for benzene. The values are scaled for ring size and
A
ring fusions, and increase with ring size so that naphthalene (142) and anthracene (206)
have higher indices than benzene. This method is particularly useful for comparing
heterocyclic compounds with hydrocarbons (see Section 8.6).

15 T. M. Krygowski and M. Cyranski, Tetrahedron, 52, 1713 (1996).
16 M. K. Cyranski, B. T. Stepien, and T. M. Krygowski, Tetrahedron, 56, 9663 (2000).
17
C. W. Bird, Tetrahedron, 41, 1409 (1985); C. W. Bird, Tetrahedron, 48, 335 (1992); C. W. Bird,
Tetrahedron, 52, 9945 (1996); C. W. Bird, Tetrahedron, 54, 4641 (1998).