survey, the conjugated π-electron systems considered (mainly benzenoid hydrocarbons) belong to the class of species whose Kekulé structures have equal “parity”.
Another reason for the increase of interest to Kekulé–structure–based studies
was the recognition that in mathematics there exists a fully equivalent notion,
namely the concept of perfect matching (or, in an earlier terminology, of 1-factor)
[26–28]. This soon resulted in various mathematics–oriented researches on Kekulé
structures, see e.g. [29, 30] (which are not the concern of the present survey).
Anyway, beginning with the 1970s, a number of approaches was put forward, all
based on counting or examination of Kekulé structures (and only Kekulé structures!), all offering reasonably good quantitative prediction of various molecular
properties. We provide a short survey of these methods in the subsequent section.
11.2 Kekulé–Structure–Based Approaches: Success
11.2.1 Resonance Theory
In 1973 William Herndon put forward a resonance–theoretical model, according to
which he was able to calculate the resonance energies of benzenoid hydrocarbons
with accuracy tantamount to the best (in that time) molecular–orbital theories [31].
Let it be mentioned that similar ideas were proposed by Simpson in 1953 [32], but
in that time had little impact.
In Herndon’s resonance theory, the basic assumption is that Kekulé structures
alone suffice to describe the π-electron configuration of the ground state of a
conjugated hydrocarbon. The respective wave function is of the form
jWi ¼
1
ffiffiffiffi
K
p
X K
i¼1
jk i i
ð 11:2Þ
where jk i i is the wave function associated to the Kekulé structure
k i ; i ¼ 1; 2; . . .; K. The resonance energy is then calculated as
RE ¼
2
K
X
1 i\j K
hk i j ^
Hjk j i
ð 11:3Þ
where ^
H is a formal Hamiltonian operator and hk i j ^
Hjk j i are the corresponding
matrix elements. In Herndon’s model
hk i j ^
Hjk j i ¼ c 1 ¼ 0:841 eV
if the Kekulé structures k i and k j differ in the position of exactly three double bonds,
300
I. Gutman and S. Radenković
Another reason for the increase of interest to Kekulé–structure–based studies
was the recognition that in mathematics there exists a fully equivalent notion,
namely the concept of perfect matching (or, in an earlier terminology, of 1-factor)
[26–28]. This soon resulted in various mathematics–oriented researches on Kekulé
structures, see e.g. [29, 30] (which are not the concern of the present survey).
Anyway, beginning with the 1970s, a number of approaches was put forward, all
based on counting or examination of Kekulé structures (and only Kekulé structures!), all offering reasonably good quantitative prediction of various molecular
properties. We provide a short survey of these methods in the subsequent section.
11.2 Kekulé–Structure–Based Approaches: Success
11.2.1 Resonance Theory
In 1973 William Herndon put forward a resonance–theoretical model, according to
which he was able to calculate the resonance energies of benzenoid hydrocarbons
with accuracy tantamount to the best (in that time) molecular–orbital theories [31].
Let it be mentioned that similar ideas were proposed by Simpson in 1953 [32], but
in that time had little impact.
In Herndon’s resonance theory, the basic assumption is that Kekulé structures
alone suffice to describe the π-electron configuration of the ground state of a
conjugated hydrocarbon. The respective wave function is of the form
jWi ¼
1
ffiffiffiffi
K
p
X K
i¼1
jk i i
ð 11:2Þ
where jk i i is the wave function associated to the Kekulé structure
k i ; i ¼ 1; 2; . . .; K. The resonance energy is then calculated as
RE ¼
2
K
X
1 i\j K
hk i j ^
Hjk j i
ð 11:3Þ
where ^
H is a formal Hamiltonian operator and hk i j ^
Hjk j i are the corresponding
matrix elements. In Herndon’s model
hk i j ^
Hjk j i ¼ c 1 ¼ 0:841 eV
if the Kekulé structures k i and k j differ in the position of exactly three double bonds,
300
I. Gutman and S. Radenković