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Quim. Nova, Vol. XY, No. 00, 1-10, 200_ http://dx.doi.org/10.21577/0100-4042.20230017
*e-mail: im_sutjahja@itb.ac.id
AROMATICITY OF AZA AROMATIC MOLECULES: PREDICTION FROM HÜCKEL THEORY WITH
MODIFIED PARAMETERS
Inge M. Sutjahjaa,*,, Yuanita P. D. Sudarsob, Sho Dhiya ‘Ulhaqa and Erik Bekti Yutomoa
aDepartment of Physics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, 40132 Bandung, Indonesia
bDepartment of Physics, Faculty of Information Technology and Science, Parahyangan Catholic University, 40141 Bandung,
Indonesia
Recebido em 14/10/2022; aceito em 11/11/2022; publicado na web 27/01/2023
Hückel theory is a simple and powerful method for predicting the molecular orbital and the energy of conjugated molecules. However,
the presence of nitrogen atoms in aza aromatic molecules alters the Coulomb and resonance integrals owing to the difference in
electronegativity between nitrogen and carbon atoms. In this study, we focus on acridine and phenazine. Further correction is
implemented based on the ring current model, thus revealing the change in resonance integral for the carbon–carbon bond along
the bridge of the molecule. The Hamiltonian of the π–electron system in the Hückel method is solved using the HuLiS software.
Various geometry-based aromaticity indices are used to obtain the aromaticity indices of the two non-equivalent rings. For further
evaluation, the results for bond lengths are used to calculate the associated bond energy. Considering the carbon–hydrogen (CH)
bonds, the total molecular energy is compared with the experimental heats of formation for a number of benzenoid hydrocarbons
and aza aromatics, in addition to the two studied molecules. Finally, the correlation between the nitrogen atom on the aromaticity
index and the ring energy content is evaluated to determine to which extent the Hückel model agrees with previous experimental and
advanced computational studies.
Keywords: Hückel theory; aza aromatic molecule; HuLiS software; geometry-based aromaticity; bond energy.
INTRODUCTION
Polycyclic π-conjugated carbon-based molecules have numerous
potential applications in organic electronics; therefore, a detailed
understanding of their fundamental properties is crucial.1 Doping
heteroatoms into carbon-based molecules is an effective strategy to
adjust their physical and chemical properties of the material, thereby
improving their performance in electronic, photonic, optoelectronic,
and spintronic applications.2-11 Nitrogen is the most commonly used
dopant as it has a similar covalent radius as carbon while providing
one extra electron. A fundamental property of heteroatom-doped
carbon-based molecules is their stability with respect to the control
of energy levels, which can be measured by the energy gap (Egap).
Aromaticity is related to stability and Egap, and it has been a
central concept in organic chemistry for over a century.12 Among
the numerous aromaticity indices,13,14 a geometry-based aromaticity
index is the most simple, successful, and widely used to quantitatively
describe the π−electron delocalization in homo- and hetero-
atomic molecules. Starting from the harmonic oscillator model of
aromaticity (HOMA),15-17 the model was developed for the harmonic
oscillator model of electron delocalization (HOMED)18,19 and the
harmonic oscillator model of heterocyclic electron delocalization
(HOMHED).20 These aromaticity indices use bond length data from
molecular geometry, and the uniformity of π–electron distribution
in the molecule is associated with the equalization of bond lengths.
To calculate the aromaticity index, each of the three models uses
some reference molecules or their related hybridizations to measure
the optimal bond length from the single bond, Rs, and the double
bond, Rd.
Hückel molecular orbital (HMO) theory is a well-known, simple
theory, easily understood by undergraduate students, to predict
Egap and the behavior of the π–electron system.21-23 The HMO
theory uses only the topology of the molecule, is independent of
structural parameters, such as bond lengths and bond angles, and
ignores the strain energy. For the heteroatoms, the Coulomb and
resonance integral parameters can be adjusted according to the atom
type and coordination number. These parameterizations are due to
the different core energies of the heteroatom and the change in the
effective electronegativity, compared with carbon, of the remaining
non-bonded p orbitals at the center. Although the parameterizations
are standardized in numerous chemistry and physics textbooks, the
importance of distinguishing the Coulomb parameter of carbon atoms
adjacent to heteroatoms is rarely emphasized. This is so particular
for nitrogen-containing molecules, due to the large electronegativity
of nitrogen.24,25 This parameterization should also depend on the
molecule, which would require an additional parameter to measure
the accuracy of the parameterization.
Pauling26 established a formula relating bond length to bond
order and an empirical rule for the relation between bond order
and bond energy;27 subsequently, Krygowski developed a model to
estimate bond energy from bond length data.28 Summation of the bond
energy over all bonds of a cyclic molecule provides the ring energy
content (REC), and summation over the entire molecule provides
the molecule energy content (MEC).29 Considering the CH bond
energy, one may obtain the total molecular energy and compare it
with the experimental data for the heat of formation.28 Krygowski28
and Cyrański30 reported that the local and global aromaticity index of
the HOMA correlates well with REC and MEC, with a higher mutual
correlation for angular polyacenes compared with linear systems.
However, the analyses so far are limited to benzenoid hydrocarbon
molecules, leaving some questions regarding heteroatom molecules,
including aza aromatic molecules.
This study applied a variation in the Hückel parameters of
the aza aromatic molecules, acridine and phenazine. Previous
theoretical studies have shown that the carbon atom adjacent to
nitrogen should have a different Coulomb parameter owing to
Artigo
Sutjahja et al.
2Quim. Nova
the large electronegativity of nitrogen.24,25 However, the variation
applied is not systematic, and generalization for its dependence on
molecules is unclear.31 A further correction was performed based on
the ring current model, which describes the ow of the π−electron
system along the molecule’s perimeter due to delocalization,32,33 as
in the case of aromatic hydrocarbon molecules.34,35 The latter was
implemented by assigning different resonance integral parameters
for carbon–carbon atoms along the bridge of the molecule.36 For this
purpose, we used the HuLiS software,37 which provides a facility for
the proposed variations. The bond length data were further analyzed
to predict the geometry-based aromaticity index based on the HOMA,
HOMED, and HOMHED models. The results were used to examine
the effect of each modied Hückel parameter on the molecular energy
level and aromaticity of the two inequivalent rings. To estimate the
bond energy from the bond length and its relation to the aromaticity
index, the bond lengths of the single and double bonds were varied
based on the values proposed in the three structure-based aromaticity
models mentioned above.
Hückel molecular orbital theory
Parametrization of the Hamiltonian matrix element according to
Hückel theory is:21-23
(1)
where pi is the 2pz atomic orbital of the ith order of the C atom; α is the
Coulomb integral; and β is the resonance integral of the π-electron
system. In addition, the overlap integral was diagonal.
(2)
For a molecule with a heteroatom, the Coulomb integral (hX)
and resonance integral (kXY) of the π–electron system depends on the
core energy and the change in the effective electronegativity of the
remaining non-bonded p orbitals at the center. In this case, the values
of α and β in Equation (1) can be written as21,22
(3a)
(3b)
Table 1 lists the commonly accepted values of hX and kXY for C
and N atoms.22
According to Hückel theory, the molecular orbital ψ is formed
from atomic orbital pi through a linear combination of atomic orbitals
(LCAO) model, and the coefcients of atomic orbital {cni} can be
further used to obtain the π-electron bond order:
(4)
with νn is the number of π-electrons occupying the corresponding
energy level.
The energy gap is the difference between highest occupied
molecular orbital (HOMO) level and the lowest unoccupied molecular
orbital (LUMO) level,
(5)
In addition, the delocalization energy is the difference between the
total energies of the π–electron system and the simple isolated system.
(6)
The isolated state of acridine (C13H9N) consists of six ethylene
(C2H2) and one methane imine (CH3N), whereas phenazine (C12H8N2)
consists of ve ethylene and two methane imines.38 The energies
of one molecule of ethylene and methane imine are 2(α + β) and
2α+2.61β, respectively.
Bond order and bond length correlation
Pauling established a well-known formula relating the bond length
R(p) to its bond order p as26
(7)
where c is an empirical constant and R(1) is the standard single bond
length. The value of c can be calculated by taking the bond lengths
for the typical single (p = l) and double (p = 2) bonds as follows:
(8)
Using Equation (7), it is also possible to calculate the bond
number, p, for any bond length, R(p), as follows:
(9)
Focusing to Huckel theory that deals with π-electron system,
the bond length can be calculated from the π-electron bond order
following Gordy’s formula:39,40
(10)
where a and b are the modied Gordy values, equal to 7.33 and 2.09
for CC bonding, and 6.52 and 2.03 for CN bonding, respectively;40
and N = 1 + ρij is the total bond order.39
Geometry-based aromaticity index
The geometry-based aromaticity index of the HOMA13-17
is simple, successful, and widely used,41 both for homo- and
heteroatomic systems. Subsequently, the HOMED18,19 and HOMHED
were developed.20 For each of these models, the bond length values for
the single bond, Rs, and the double bond, Rd, were proposed according
to the reference molecules.
In general the HOMA index can be calculated as13-17,42,43
(11)
where n is the number of bonds taken into summation; and Rj,i is the
experimental or computed bond length of the system for a certain
type of bond (j). In Equation (11), the summation is also held for
Table 1. Values of Hückel parametrization22
Element hXkXY
ChC = 0.00 kCC = 1.00
N2* hN = 0.51 kCN = 1.02
N3** hN = 1.37 kCN = 0.89
*Dicoordinated. **Tricoordinated, planar geometry.
Aromaticity of aza aromatic molecules 3Vol. XY, No. 00
all types of bonds that are present in the ring of a molecule, that is,
carbon–carbon (CC), carbon–nitrogen (CN), and nitrogen–nitrogen
(NN) for the aza molecule. The optimal bond lengths Ro and a were
calculated using the following formula.
(12)
(13)
where w is the ratio of the stretching force constants for pure single
(Rs) and double (Rd) bonds, and the common value is equal to 2 for CC
and CX bonds.44 Table 2 presents the Rs, Rd, Ro, α, and c values for CC,
CN, and NN bonds, and the reference molecule for each type of bond.
In Equation (11), it is clear that the decreased aromaticity in the
π–electron system can be described by two different and independent
mechanisms, namely, an increase in the bond length alternation
(GEO) and an extension of the mean bond length (bond elongation)
(EN); the latter is due to a decrease in the resonance energy. These
two dearomatization terms, i.e., geometric GEO and energetic EN
contributions, are calculated by transforming the bond order of CX or
XY bonds. This is derived from Equation (9) considering the virtual
CC bonds according to the Pauli formula (Equation (7)),
(14)
such that
where rav represents the average value of ri.
The HOMED index was proposed by Raczyńska et al.18 in 2010,
primarily for molecules that contain heteroatoms. The HOMED index
is formally the same as that of the HOMA, Equation (11) but differs
for the reference molecules. Herein, quantum-chemical methods were
used to estimate the bond lengths using simple saturated systems for
single bonds and simple unsaturated systems for double bonds. The
optimal bond length, Ro, was chosen from various simple molecules
for which equalization of the bonds occurs. Table 3 presents the
Rs, Rd, and Ro values for the HOMED calculations, along with the
reference molecules.
The α calculation follows the following rules. For molecules
with even numbers of bonds (2i), α can be calculated using Equation
(13). In contrast, for molecules with odd numbers of bonds (2i + 1),
α can be calculated using Equations (15) and (16), each of them for
(i + 1) single bonds and (i) double bonds, and (i) single bonds, and
(i + 1) double bonds.
(15)
(16)
The HOMHED model proposed by Frizzo et al.20 in 2012 is
based on the average experimental (X-ray diffraction and neutron
diffraction) data of several reference molecules. Table 4 presents
the Rs, Rd, Ro, and α values for the HOMHED calculation, along
with the hybridization of the reference molecules for a particular
type of bonding. The HOMHED index was calculated using
Equation (11).
Table 2. Values of Rs, Rd, Ro, α and c for the HOMA index along with reference molecules15,43
Type of bond Rs (Å) Rd (Å) Ro (Å) αcReference molecule
CC 1.467 1.349 1.388 257.7 0.1702 1,3-butadiene, CH2=CH−CH=CH2
CN 1.465 1.269 1.334 93.52 0.2828 Methylamine, H2N−CH3 and methylene imine,
HN=CH2
NN 1.420 1.254 1.309 130.33 0.2395 (CH3)2C=N–N(CH3)2 and H3C–N=N–CH3
Table 3. Values of Rs, Rd, and Ro for the HOMED index along with the reference molecules18,19
Type of bond Rs (Å) Molecule reference Rd (Å) Molecule reference Ro (Å) Reference molecule
CC 1.5300 H3C−CH31.3288 H2C=CH21.3943
CN 1.4658 H3C−NH21.2670 H2C=NH 1.3342
NN 1.4742
1.2348
1.3193
Table 4. Values of Rs, Rd, Ro, and α for the HOMHED index along with the hybridization of the reference molecules20
Type of bond Rs (Å) Rd (Å) Ro (Å) αHybridization of the reference molecules
CC 1.530 1.316 1.387 78.6 Csp3−Csp3; Csp2−Csp2
CN 1.474 1.271 1.339 87.4 C−Nsp3
NN 1.454 1.240 1.311 78.6 Nsp3−Nsp3; C–N=N–C
Sutjahja et al.
4Quim. Nova
Bond energy from bond length
Bond energy is related to geometry-based aromaticity indices
and is estimated from the bond length. For benzenoid hydrocarbon
molecules, Krygowsky et al.28 has developed a model that relates
the bond energy, E(i), calculated directly from the bond length as:
(17)
where R(1) and E(1) are the bond lengths of a single bond and its
associated energy, respectively; R(i) and E(i) are the investigated bond
lengths in the molecule and its related energy, respectively; and α′ is
an empirical constant that can be calculated according to the reference
bond lengths and bond energies that correspond to the single and
double bonds. Generalization of the above formula to aza aromatic
molecules was performed by considering the CN and NN bonds in
addition to CC bonding. The REC can be obtained by summing E(i)
for all bonds in a ring of the molecule, as follows.
(18)
whereas the MEC can be obtained by summing over the molecule.
Considering the carbon and hydrogen bonds in the molecule, one can
obtain the total bond energy of the molecule:
(19)
This value can be compared with the experimental data for the
heat of formation from atoms (HtFfA),28
(20)
where ∆Hf,molecule is the heat of formation of the corresponding
molecule; and HA is the heat of atomization; additionally, summation
(k) is performed for all atoms that make up the molecule.
METHODS
HuLiS software
HuLiS37 is a software package for molecular electronic structure
calculations based on the Hückel method. Figure 1 shows the HuLiS
software interface. The input parameter used to solve the Hamiltonian
system is the molecular drawing. This software allows for a change
in the Coulomb integral of any atom (α) or resonance integral (β) by
clicking on a certain atom or bonding.
The output consists of the energy level and the coefcients of the
linear combination of molecular orbitals. Additionally, this software
can display the electron distribution prole for each energy level.
Parametrization of Hückel parameters
For various N-doped molecules of pyridine, quinoline, iso-
quinoline, and acridine, Longuet-Higgins and Coulson in 194624
proposed the same correction to the Coulomb integrals of nitrogen
and carbon atoms that directly bonded to nitrogen, whereas Dasgupta
in 196425 proposed different parameter values that depend on the
molecular type, as presented in Table 5. In this table, α is the Coulomb
integral for all other carbon atoms and β is the resonance integral of
the CC bonding.
The data presented in Table 5 reveal a signicant difference in the
parameter values, whereas the commonly accepted values are listed
in Table 1, without the need to give the difference for the carbon
atom close to nitrogen.
Herein, we performed a systematic study for the parametrization
of the Hückel parameters of acridine and phenazine, which consist
of the Coulomb integral of carbon adjacent to nitrogen (αCN) and
the resonance integral for CC bonding (βCC) along the bridge of the
molecule, expressed in terms of resonance energy,36
βCC = γ β (21)
The values of αCN are 0, 0.025, 0.050, 0.075, 0.0825, 0.100, and
0.250; and 1.0 ≤ γ ≤ 1.5 with steps of 0.1.
Bond energy and heat of atomization
To compare the total bond energy and heat of formation from
atoms, Table 6 lists the experimental data for the heat of atomization
Figure 1. HuliS software interface
Table 5. Different Coulomb and resonance parameters of aza aromatic mo-
lecules from previous studies24,25
Molecule αNαCN Reference
Pyridine α + 2β α + 0.25β 24
α + 0.2β α + 0.025β 25
Quinoline α + 0.2β 25
Iso- Quinoline α + 0.2β 25
Acridine α + 2β α + 0.25β 24
α + 0.66β α + 0.0825β 25
Aromaticity of aza aromatic molecules 5Vol. XY, No. 00
of C, H, and N from the gas phase and the bond dissociation energies
of C−H, C−C, C=C, C−N, and C=N.45 Table 7 lists the experimental
data for the enthalpy of formation of the investigated molecules.
RESULTS AND DISCUSSION
The results of Hückel method
The molecular structures of acridine and phenazine are shown in
Figure 2. The direct application of the Hückel method to acridine and
phenazine by applying the Coulomb integral and resonance integral
parameters listed in Table 1 gave the energy level diagram shown in
the above panel of Figure 2. This gure shows that for both molecules,
the energy levels of the HOMO-1 and LUMO+1 states consist of
degenerate states. However, group theory predicted no degeneracy
for the two molecules in the C2v or D2h point groups.51-53
As shown in Figure 2, the aromaticity indices for the two non-
equivalent rings from the previous structural data of acridine54 and
phenazine53 clearly demonstrate that the ring containing the nitrogen
atom (B) is more aromatic than the benzene ring (A).55,56 Furthermore,
for ring-B of the two molecules, the index value of EN is comparable
to GEO. We observe that, particularly for acridine, this variation in
the aromaticity index cannot be qualitatively obtained from the direct
application of the Hückel method.
Correction to Hückel parameters
Figure 3 illustrates the results of the HOMA aromaticity index
of the two non-equivalent molecular rings of (a) acridine and (b)
phenazine for the variation of αCN (i) and βCC (ii). Evidently, different
models of the aromaticity index (HOMED and HOMHED) exhibited
similar proles, although with different absolute values. As shown in
this gure, the standard deviation of the bond length was compared
with the corresponding values from Phillips54 for acridine and
Wozniak et al.53 for phenazine.
Evidently from Figure 3(i), upon varying αCN, the aromaticity
index of the ring-A is smaller than the aromaticity index of ring-B that
contains nitrogen for αCN = 0.1 for acridine, whereas for phenazine, it
occurred for all values of αCN and a minimum in the standard deviation
value of the bond length was observed for αCN = 0.1. However, using
different parameters for carbon atoms close to nitrogen removed the
degeneracy in the energy levels of acridine and phenazine, which is
in good agreement with a previous semi-empirical self-consistent
eld study.24 When varying βCC, a minimum in the standard deviation
value of bond length occurred at βCC = 1.1 for acridine and 1.2
for phenazine. Thus, we concluded that the best modied Hückel
parameters for αCN and βCC are 0.1 and 1.1 for acridine and 0.1 and
1.2 for phenazine, respectively. In addition, set values of aromaticity
indices for acridine are {H = 0.805, G = 0.088, E = 0.107} for ring-A
and {H = 0.837, G=0.044, E = 0.119} for ring-B. The same indices
for phenazine are {H = 0.791, G = 0.104, E = 0.105} for ring-A and
{H = 0.883, G=0.057, E = 0.060} for ring-B. By modifying the
Hückel parameters, it can be seen that the aromaticity indices of
the two molecules qualitatively agree with the experimental results
presented previously.
Table 6. Heat of atomization, bond dissociation energy45
∆Hf (kcal mol-1)Eb (kcal mol-1)
C(g) 171.2 C−H98.3
H(g) 52.1 C−C82.6
N(g) 112.9 C=C 144.0
C−N72.8
C=N 147.0
Table 7. Enthalpy of formation of molecule
Molecule Formula ∆Hf
(kJ mol-1)
∆Hf
(kcal mol-1)Reference
Benzene (l) C6H649 11.70 46
Naphthalene (s) C10H878 18.63 46
Anthracene (s) C14H10 127 30.33 46
Phenanthrene (s) C14H10 110 26.27 46
Pyridine (l) C5H5N 100.02 23.89 47
Acridine (g) C13H9N 273.9 65.42 48,49
Phenazine (g) C12H8N2338.3 80.80 50
Figure 2. Molecular structure with atomic numbering and energy level diagram of (a) acridine and (b) phenazine. The numbers in the rings represent the index
values of HOMA (H), GEO (G), and EN (E) calculated from experimental geometric data53,54
Sutjahja et al.
6Quim. Nova
Bond length and molecular orbital description
Figure 4 shows the calculated bond lengths for some unique bonds
of the two molecules for the best parameters compared with previous
theoretical and experimental studies. To study the effect of nitrogen,
we plotted anthracene data from a previous study using the same
method with the best correction to the Hückel parameters.36 Figure
4 shows that the results are in good agreement with previous studies.
For the best corrected Hückel parameters, the standard deviation
values of the bond length compared with the experimental data53,54
were 1.08 % for acridine and 1.55 % for phenazine.
Figure 4 shows that for both molecules the CC bond distance of
the benzene ring (ring A) was 1.34–1.44 Å, whereas the CN distance
of the nitrogen-containing ring (ring B) was approximately 1.34 Å,
both consistent with a delocalized scheme. The bonding properties
of the outer rings of acridine and phenazine were similar to each
other and to those of anthracene, differing only in the middle ring.
This result is in good agreement with previous studies51,57 and is
supported by the fact that the force constants of the outer rings of
the three molecules are similar, differing in the middle ring due to
the presence of the nitrogen atom.57
Further, we noted a mistake in the bonding labels in Figure 6(a)
in Sudarso et al.36 with reference to the numbering of the carbon
atoms of anthracene in Figure 2(a) of the same reference; the correct
bonding labels should be 1-2, 1-11, 9-11, 2-3, 11-12.
The molecular energy level diagram for the best modified
Figure 3. Results of aromaticity index of HOMA of the two non-equivalent rings (A and B) of (a) acridine and (b) phenazine, for the variation of (i) αCN (for
βCC= 1.0) and (ii) βCC (for αCN = 0.1). For all graphs, the right axis shows the standard deviation of bond length
Figure 4. Bond length of (a) acridine and (b) phenazine from the present work with the best result for the modied Hückel parameters, compared with the data
for anthracene36 and experimental and computational data from previous studies.53,54,57-60 Shading denotes CN bonding
Aromaticity of aza aromatic molecules 7Vol. XY, No. 00
Hückel parameter and the associated HOMO and LUMO obtained
from LCAO model are summarized in Table 8 for both acridine and
phenazine. Data for anthracene were obtained from our previous
study.36
Evident from the molecular energy level diagram, degeneracy in
the energy levels of acridine and phenazine was removed, which is
consistent with a previous semi-empirical self-consistent eld-LCAO-
MO study.24 The HOMO and LUMO plots obtained resembled those
obtained from the density functional theory (DFT) studies of Adad61
for acridine and Zendaoui59 for phenazine. The HOMO and LUMO
plots for acridine and phenazine were similar to those of anthracene.
In particular, the HOMO is localized primarily on the two C6 rings
but extends to a certain degree to the central ring. In contrast, the
LUMO is localized primarily on the central ring and presents a non-
negligible contribution from the C6 rings. For acridine and phenazine,
the essential contributor of the central ring is from the nitrogen atoms.
Table 9 compares some of the energy characteristics of the two
studied molecules with those of anthracene, compared with previous
experimental and advanced calculation studies. This correction
shows that the HOMO-LUMO gap energy was 0.84β for acridine
and 0.77β for phenazine. In addition, the delocalization energy was
5.72β for acridine and 6.07β for phenazine. The higher gap energy
of acridine relative to phenazine agrees with previous experiments
and advanced computational studies. Compared with anthracene, the
trend of the gap energy was consistent with the experimental optical
band-gap data, and the gap energy decreased with increasing nitrogen
to carbon ratio, which is in good agreement with a previous study.62
The delocalization energy of phenazine was found to be larger than
acridine, which is in good agreement with the increased molecular
stability when increasing the number of nitrogen atoms, but still
smaller than the parent molecule anthracene.
Based on the result of the HOMO energy, with β < 0, one
might expect a substantial shift to higher ionization energies from
anthracene to acridine and phenazine, which is consistent with
previous computational studies.67,68 In contrast, the opposite trend
was observed for the LUMO energy.
Bond energy and heat of formation from atoms
Figure 5 shows a comparison of the total molecular energy with
the experimental HtFfA for various benzenoid hydrocarbons (a)
and aza aromatic molecules (b). For the benzenoid hydrocarbons
anthracene and phenanthrene, we used the best modied Hückel
parameters, as reported previously.36 For both groups of molecules,
increasing the number of rings increased HtFfA, and the HtFfA value
of the angular molecule, phenantrene was larger than that of the
linear molecule, anthracene. For aza aromatic molecules, increasing
the number of nitrogen atoms from acridine to phenazine decreased
HtFfA. In addition, the dependence of HtFfA on the number of ring
molecules and nitrogen atoms is shown by the smaller corresponding
values of pyridine compared to those of benzene and the larger
corresponding values of acridine compared with those of anthracene.
In general, the variation in the total molecular energy related
to the three aromaticity indices followed the experimental HtFfA
value. However, a smaller deviation was observed for the bond
energies derived from the bond lengths based on the HOMED,
Table 8. Energy level diagram and plots of HOMO and LUMO of acridine and phenazine from the present work compared with those of anthracene. Different
colors of red and grey circles indicate the positive and negative coefcients of linear combinations, respectively
Anthracene Acridine Phenazine
MO Energy level diagram
HOMO
LUMO
Sutjahja et al.
8Quim. Nova
followed by those derived from the HOMHED and HOMA. In
particular, all models predicted a larger total molecular energy
value for phenanthrene compared with anthracene, which is in
agreement with the experimental HtFfA. Thus, the modied Hückel
parameters yielded total molecular energy values that matched the
experimental data. Compared with benzenoid hydrocarbon molecules,
a larger deviation between the calculated total energy molecular and
experimental HtFfA values was found for the aza aromatic molecules.
To further correlate the bond energy with the aromaticity index,
Table 10 presents the corresponding aromaticity HOMED index with
the REC divided by the number of CC and CN bonds (n).
Evidently, aromaticity varied with an increase in the number
of rings and the number of nitrogen atoms in the molecule. For
anthracene, the aromaticity of the central benzene ring was higher
than that of the outer benzene ring. The opposite trend was observed
for phenanthrene, which is in good agreement with our previous
study36 using the HOMA aromaticity index. With an increased
number of rings, the aromaticity index generally decreased, as
reported previously30,55. Upon replacement of carbon with nitrogen,
the aromaticity index of nitrogen-containing rings increased (for
example, 0.994 for acridine as compared with 0.972 for anthracene),
whereas the aromaticity index of the benzene ring decreased (0.952
for acridine as compared with 0.961 for anthracene). This result is
consistent with previous studies.55,56 For acridine and phenazine, the
aromaticity index of the nitrogen-containing ring was higher than that
of the benzene ring, which was consistent with previous studies using
geometric experimental data.55,56 However, the smaller aromaticity
index for the benzene- and nitrogen-containing rings in phenazine
compared with those of acridine indicates that further increase in the
number of nitrogen atoms decreased the aromaticity index.
Interestingly, for both groups of molecules, the variation in REC/n
from the present study generally followed the corresponding values
from experimental geometry data.30 The results of the present study
show that the REC/n value generally decreased as the number of
rings increased. In addition, upon replacing carbon with nitrogen, the
REC/n value generally decreased for both single-ring molecules and
for benzene- and nitrogen-containing rings. The results presented in
this study should be supported by further analysis based on advanced
calculations studies, in particular, studies on the relationship between
aromaticity and the number of nitrogen atoms and NN bonding, as
well as the topological environment of the ring.32,55
CONCLUSIONS
This study presented a correction to the Hückel parameters of
the aza aromatic molecules acridine and phenazine. The parameter
correction consists of the Coulomb integral of the carbon atom
adjacent to nitrogen (αCN) and the resonance integral of the
carbon–carbon atoms (βCC). The latter is based on the ring current
model, which describes the delocalization of π-electrons along the
perimeter of the molecule. The calculation was performed using
HuLiS software. The resulting bond order was transformed into a
bond length based on Gordy’s formula with modied Gordy values.
The validity of the results was examined using the structurally based
aromaticity indices of the HOMA, HOMED, and HOMHED. For the
three aromaticity indices, the experimental data of the two molecules
revealed that the nitrogen-containing ring is more aromatic than the
benzene ring. Additionally, both αCN and βCC parameters were found to
be responsible for the excessive degeneracy in the molecular orbitals.
The best values of αCN and βCC were 0.1 and 1.1 for acridine and 0.1 and
1.2 for phenazine, respectively. Comparison of the calculated bond
lengths with previous experimental and advanced calculation studies
revealed a delocalized scheme of CC and CN bonding. Compared with
anthracene, the bonding properties of the outer rings of acridine and
phenazine were similar to each other and also similar to anthracene,
differing only in the middle ring owing to the presence of nitrogen
Table 9. HOMO–LUMO gap energy, delocalization energy, HOMO and LUMO energies, and optical band gap of phenazine and acridine from the present work
(p.w) and previous experimental and advanced calculation studies, compared with those of anthracene.
Molecule HOMO–LUMO
gap energy Edeloc EHOMO ELUMO
HOMO–LUMO
gap energy (eV) Eg,opt (eV)
Anthracene 0.82β36 8.08β36 0.43β36 -0.39β36 3.5863 3.20*63
Acridine 0.84β (p.w) 5.72β (p.w) 0.52β (p.w) -0.32β (p.w) 3.764
3.70161 3.22*65
Phenazine 0.77β (p.w) 6.07β (p.w) 0.59β (p.w) -0.18β (p.w) 2.3759 3.0766
*Estimated from the absorption spectra by determining the wavelength of absorption onset and converting it from nm to eV.62
Figure 5. Total molecular energy based on HOMA, HOMED and HOMHED calculations compared with the experimental HtFfA for (a) benzenoid hydrocarbon
and (b) aza aromatic molecules. The numbers on (a) are Emolecule of anthracene and phenantrene calculated based on HOMED
Aromaticity of aza aromatic molecules 9Vol. XY, No. 00
atoms. The HOMO-LUMO energy gap was 0.84β for acridine and
0.77β for phenazine, which is in good agreement with the smaller
energy gap of phenazine compared to acridine from previous DFT and
experimental studies. The delocalization energy of phenazine (6.07β)
was larger than that of acridine (5.72β), which is in good agreement
with the trend of increasing molecular stability with an increasing
number of nitrogen atoms, but they are still smaller than anthracene.
To further validate the aromaticity index, we calculated the
molecular bond energy from the bond length for several benzenoid
hydrocarbons and aza aromatic molecules, including acridine and
phenazine. Considering the energy related to the CH bond, the total
molecular energy data were compared with the experimental data
for the heat of formation from atoms (HtFfA). The comparison
revealed that in general, modied Hückel parameters with calculated
bond energies derived from the three aromaticity indices based on
each single and double bond length as the reference resemble the
experimental HtFfA. The smallest deviation was observed for the
bond energy derived from the HOMED model. However, a larger
difference between the two values for aza aromatic molecules may
indicate some limitations of the existing model. A correlation study
between the HOMED index and REC divided by the number of CC
and CN bonds revealed a different role between the number of rings
and nitrogen atoms. Remarkably, the results from the simple Hückel
theory with modied parameters can resemble the experimental
results, from the viewpoints of the aromaticity index and energy
related to the bond length. However, advanced computational
studies are needed to conrm these results and to achieve a better
understanding of the relationship between the aromaticity index and
molecular energy from bond length and the bonding properties of a
particular molecule.
Table 10. Aromaticity indices of HOMED and REC/n for the individual rings. The values in the parentheses are calculated from experimental geometry data
listed in the reference
Group Molecule Molecule Ring HOMED index REC/n Reference
Benzenoid
hydrocarbon
Benzene
A 0.997 122.3 (118.8) 69
Naphthalene*
A 0.967 119.1 (116.6) 70
Anthracene
A 0.961 118.1 (116.7)
71
B 0.972 115.5 (115.3)
Phenanthrene
A 0.986 119.5 (117.0)
71
B 0.922 114.9 (111.3)
Aza aromatic
Pyridine
A 0.999 118.7 (113.8) 72
Acridine
A 0.952 115.7 (116.1)
54
B 0.994 112.7 (114.1)
Phenazine
A 0.948 115.8 (118.4)
53
B 0.970 112.1 (110.8)
*The modied Hückel parameters of a = 0.5, γ = 1.5.
ACKNOWLEDGEMENTS
This study is the output of the P2MI ITB 2023 research scheme.
It is dedicated to the late Professor Pantur Silaban for his memorable
contributions to theoretical physics at the Physics Department,
Faculty of Mathematics and Natural Sciences, Institut Teknologi
Bandung.
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