INTRODUCTION:
A popular theoretical technique for examining the electrical structure of conjugated systems in organic chemistry is the Huckel method. The linear combination of atomic orbitals (LCAO) may explain the electrons in the conjugated system, a presumption made by this technique. The LCAO method creates molecular orbitals by combining the atomic orbitals of nearby atoms, which may then be utilized to characterize the electrical characteristics of the molecule4. The stability, reactivity, optical, and magnetic characteristics of conjugated systems may all be predicted using the Huckel technique5,6. Examples are the conjugated chemical compounds benzene, cyclopentadienyl, cyclobutadiene, and furan7,8. The design and synthesis of organic semiconductors for electrical and optoelectronic devices are only two examples of how the study of conjugated systems has significant applications in materials science. The wave function of molecular orbital is written as
Where:
Φj represents the 2pz orbital of the jth carbon atom.
n is the number of molecular orbitals.
Ci,j are the coefficients of the i and j values, where i can take values of 1, 2, 3, ..., n9,10.
An effective theoretical method for examining the electrical structure of conjugated cyclic compounds is the Huckel molecular orbital theory. This hypothesis is comparable to the homonuclear diatomic molecules' molecular orbital theory. The 2pz atomic orbitals are employed as the fundamental functions in the Huckel molecular orbital theory, while the 1s atomic orbitals are the basic functions in the H2 molecule. Huckel made several presumptions about the Coulomb integral, exchange integral, and overlap integral in his approach11. The Coulomb integral in this context refers to the energy of an electron in the 2pz orbital on the ith carbon atom, while the overlap integral refers to the amount of overlap between the 2pz atomic orbitals12,13. The exchange or resonance integral represents the energy of interaction between two atomic orbitals. A molecule must be planar, cyclic, and have a cyclic electron cloud (system) to be classified as aromatic. The molecule must also adhere to Huckel's rule, which specifies that the number of electrons in the molecule must equal (4n+2), where n is an integer. It is important to remember that the number of p orbitals involved in the pi system need not be the same as the number of pi electrons. As a result, the Huckel rule may be used to treat anions, cations, and radicals as aromatic compounds. The literature has further information on the Huckel molecular orbital theory14,15.
A conjugated cyclic molecule with six carbon atoms and six hydrogen atoms, the benzene system has been extensively studied using the Huckel molecular orbital theory. To meet the Huckel rule (4n+2) for aromaticity, the Huckel theory postulates that the pi electrons in the benzene system are delocalized and create a cyclic cloud of electron density. The theory makes assumptions about Coulomb integrals, exchange integrals, and overlap integrals and employs the 2pz atomic orbitals of the carbon atoms in the benzene ring as its fundamental units. Important benzene molecular characteristics, such as the orbital energies, wave functions, electron density, and charge density, may be calculated using the Huckel theory16. According to the hypothesis, alternative double bonds and a planar molecule exist in the completely conjugated system of the benzene ring. The carbon atoms in the ring may form sigma bonds via the axial overlap of half-filled atomic orbitals and pi bonds through the lateral overlap of half-filled atomic orbitals thanks to sp2 hybridization. The molecular plane is perpendicular to each 2pz orbital. The study of substituted benzene systems, where substituents on the ring may change the electron density and impact the stability of the molecule, has been expanded by the Huckel theory17. Cyclobutene and naphthalene are two examples of additional conjugated cyclic compounds to which the idea has been applied. The Huckel molecular orbital theory, extensively used in organic chemistry, is a potent tool for comprehending conjugated cyclic compounds' electronic structure and stability18.
Figure 1: Benzene
Removing a hydride anion from one of the sp3 hybridised carbon atoms in the benzene ring framework (figs 1 and 2) creates an sp2 hybridised carbon atom, producing the cyclic molecule known as benzene cation. As shown in Figure 2, the molecule, as a result, becomes planar and has a cyclic electron cloud. The Huckel rule states that a molecule must contain a cyclic electron cloud with (4n+2) electrons to be classified as aromatic. On the other hand, the benzene cation only has six electrons, which deviates from the Huckel rule. According to the Huckel rule, the benzene cation is not considered aromatic. However, because of the delocalization of electrons in the system, it still shows certain aromatic characteristics. The benzene cation becomes more stable due to electrons' delocalisation, making it a highly reactive species in organic chemistry8.
Benzene cation [C6H6] +
Benzene radical [C6H6].
Benzene anion [C6H6]-
Figure 2: Benzene system
As shown in Figure 2, a Benzene radical is created when a hydrogen atom is extracted from the sp3 hybridised carbon atom of the Benzene ring. As a consequence of this procedure, carbon hybridization becomes sp2. The planar benzene radical possesses a cyclic electron cloud. In contrast to the Benzene cation, the Benzene radical does not follow Huckel's rule since it includes seven electrons instead of the 6n+2 electrons needed for aromaticity. As a result, it is not regarded as aromatic. Figure 3 illustrates the creation of the Benzene anion by the abstraction of a proton from the sp3 hybrid carbon ring in Benzene. The carbon atom's sp3 hybridization is transformed into an sp2 hybridization. The cyclic Benzene anion that results from this process is planar and features a cyclic electron cloud, which complies with the Huckel rule. As a result, the aromatic species of the benzene anion are relatively stable (fig 3). Unlike the Benzene cation and the radical, it is simple to obtain by straightforward treatment. This is because the benzene cation and radical are very erratic and reactive species, and their synthesis calls for unique experimental circumstances.
All resonance and overlap integrals are considered in contemporary computational approaches, resulting in precise predictions of a molecular system's features. However, the Huckel approximation may still help indicate the molecule's properties. Despite its value, the Huckel technique has certain drawbacks because of its strict presumptions. More complex theories have been created to get beyond these restrictions and provide us with a better knowledge of the characteristics of cyclic molecular systems. These theories can forecast a molecule's structure or geometry, its energies, electron delocalization energy, and charge density. The stability of a cyclic chemical system may be anticipated by considering these factors. These ideas have made it possible to create new molecular systems with specific features for various chemistry, physics, and materials science applications8,10.
THEORETICAL METHOD:
a) Formation of the Benzene system
Figure 3: Benzene system, i.e., formation of cation, anion and radical species
b) Secular determinant for Benzene system
C6H6+(cation), C6H6- (anion), and C6H6. (radical). The number of carbon atoms is the same in the three systems, so the secular determinant is unchanged.
|
|A| |
= |
|
c) Secular equation: (unchanged for the system)
The secular equation determines the molecular orbitals and their energy in the Huckel molecular orbital theory. In a cyclic conjugated system, imagine a carbon atom (C1) next to atoms C6 and C2. The formula denotes the related secular equation for this carbon atom: XC1 + C2 + C6 = 0, where X is the eigenvalue or energy of the molecular orbital centred on carbon atom C1, C2, and C6 are the coefficients or weights of the atomic orbitals on atoms C2 and C6, respectively, and X is the value of the orbital. It is essential to notice that the Huckel technique has three non-zero elements in each secular equation. The exact molecular structure may be found in all six secular equations for a cyclic conjugated system by permuting the indices. Solving for the eigenvalues and coefficients of the molecular orbitals is made more accessible as a result. Trigonometric functions may be used to solve the secular equation. The secular equation's solutions determine the energy and forms of the molecular orbitals. The circle technique is a simple geometric structure that may be used to depict trigonometric answers. By creating a circle with a radius equal to the total of the coefficients of the nearby atoms, the eigenvalues are then ascertained by locating the locations where the circle intersects the x-axis. The intersection points along the x-axis are located according to the coefficients of the nearby atoms, and the quantity of intersection points reflects the amount of molecular orbitals. The three non-zero components in the secular equation for a carbon atom in a cyclic conjugated system may be solved using trigonometric functions. The circle technique, a simple geometric design, may be used to depict the solutions since they provide the energies and forms of the molecular orbitals8-9.
Similarly, for C2, C3, C4, C5and C6 the secular equation as follows,
XC1 + C2 + C6 = 0, C1 + XC2 + C3 = 0, C2 + XC3 + C4 = 0,
C3 + XC4 + C5 = 0, C4 + XC5 + C6 = 0
C1 + C5 + XC6 = 0
d) X values:(Unchanged for the system):
The X values can be obtained by solving the secular determinant,
X1 = 1, X2 = 1 , X3
= - 1 , X4 = -1, X5 = -2, X6 = 2. Energy levels, X = 
e) Total π energy:
Eπ = 
iEi= n1E1 + n2E2 + n3E3
+ n4E4 + n5E5 + n6E6
Where ni = a number of electrons in the ith energy state, Ei = ith energy equation or value.
f) Delocalization Energy (DE):
D.E = (π electron energy of the system) – (π electron energy of an equivalent number of isolated double bonds).
An odd number of carbon atoms in a conjugated system, then the comparison energy is taken to be that of an equivalent number of isolated double bonds plus alpha for the energy of the odd electron.
g) Wave functions:
The general wave function equation is,
Ψi = a1 P1 + a2 P2 + a3 P3 + a4 P4 + a5 P5 + a6 P6
Solving the wave functions for the Benzene system in terms of coefficients (a) by using the normalized and orthogonalized condition as
I ψj
dX =1--------(i = j, Normalized condition)
I ψj
dX =0 ---------- (i 
j, Orthogonalized condition)
The wave function for the system is,
Ψ1 = 
(P1 + P2 + P3 +
P4 + P5 + P6)
Ψ2 = 
(P2 + P3 – P5 –
P6)
Ψ3 = 
(P1 + 0.5P2 - 0.5P3
- P4 – 0.5P5 + 0.5P6)
Ψ4 = (P2 - P3 + P5 –
P6)
Ψ5 = (P1 - 0.5P2 - 0.5P3
+ P4 – 0.5P5 - 0.5P6)
Ψ6 = 
(P1 - P2 + P3 -
P4 + P5 - P6)
h) Electron density
The total electron density at an atom r is defined as the sum of electron densities contributed by different electrons in each HMO as,
Electron Density (qr)
= 
niai2
Where ai is the coefficient of the atom r in the ith HMO and ni is the number of electrons in that HMO. (Values of n = 0, 1, or 2).
i) Charge Density
In the π system, a neutral carbon is associated with an electron density of 1.0, and so the equation defines the net charge density, Charge density = 1-qr
RESULTS AND DISCUSSION
A conjugated cyclic system has numerous resonance configurations because the electrons are delocalized across the whole system. Additionally, due to this delocalization, the system's single bonds behave like double bonds even if no true double bonds are present. The Huckel approximation theory often investigates the electrical structure and characteristics of conjugated cyclic systems. By adopting a variety of assumptions, such as utilizing 2pz atomic orbitals as the fundamental functions and neglecting specific overlap integrals, this theory streamlines the difficult computations needed in molecular orbital theory. Calculating molecular properties like the number of molecular orbitals, their energies, and the coefficients of each atomic orbital is possible using the Huckel approximation. According to Huckel's rule of 4n+2, where n is an integer, electrons in a cyclic system may also be used to identify whether it is aromatic.
One secular equation may be derived from the Huckel theory for each atomic orbital in the cyclic system. These equations describe the linear combinations of atomic orbitals that comprise the system's molecular orbitals. Each secular equation has three non-zero terms and is trigonometrically solvable. It is possible to derive the six secular equations for a conjugated cyclic system from one another by permuting the indices, as they all have the same chemical structure. The circle approach is a simple geometric structure that may be used to depict the trigonometric solutions to secular equations. To estimate the energy levels and coefficients of the molecular orbitals utilizing this approach, plot the three coefficients of each atomic orbital on a circle and then locate the locations where the two circles cross. This method visually depicts the molecular orbitals and their energies, which may help comprehend the conjugated cyclic system's electrical characteristics.
a) Total π energy:
= 6α + 8β
[C6H6]. radical:
Eπ = 2(α + 2β) + 4(α + β) + (α - β)
= 7α + 7β
[C6H6]– anion:
Eπ = 2(α + 2β) + 4(α + β) + 2(α - β)
= 8α + 6β
According to the value of total π energy of a system in which anion species shows high energy indicates more
stability.
b) Delocalization energy (DE):
[C6H6] + cation:
EDE = 6α + 8β - {2(α + β) + 4(α + β)}
= 2β
[C6H6]. radical:
EDE = 7α + 7β - {2(α + β) + 4(α +β) + (α - β)}
= 2β
[C6H6]– anion:
EDE = 8α + 6β - {2(α + β) + 4(α +β) + 2(α - β)}
= 2β
The delocalization energy of a Benzene system remains unchanged.
c) Electron and charge density
at C1, qr = n1a12 + n2a22 + n3a32
= 2 x (
)2 + 2 x ()2 + 2 x ()2
= 1
at C2, qr = n1a12+ n2a22+ n3a32
= 2 x (0)2 + 2 x (
)2 + 2 x ()2
= 1
at C3, qr = n1a12 + n2a22 + n3a32
= 2 x (
)2 + 2 x (
)2 + 2 x (
)2
= 1
at C4, qr = n1a12 + n2a22 + n3a32
= 2 x (0)2+ 2 x ()2 + 2 x (
)2
= 1
at C5, qr = n1a12 + n2a22 + n3a32
= 2 x ()2 + 2 x ()2 + 2 x ()2
= 1
at C6, qr = n1a12 + n2a22 + n3a32
= 2 x ()2 + 2 x (
)2 + 2 x ()2
= 1
Charge density = 1- qr
At, C1 = 0, C2 = 0, C3 = 0, C4 = 0, C5 = 0, C5 = 0.
[C6H6]. radical:
Electron Density (qr),
at C1, qr = n1a12 + n2a22 + n3a32 + n4a42
= 2 x ()2 + 2 x ()2 + 2 x ()2 + 1 x ()2
= 1.1667
at C2, qr = n1a12 + n2a22 + n3a32 + n4a42
= 2 x (0)2 + 2 x ()2 + 2 x ()2 + 1 x (0)2
= 1
at C3, qr = n1a12 + n2a22 + n3a32 + n4a42
= 2 x ()2+ 2 x ()2 + 2 x ()2+ 1 x (
)2
= 1.3333
at C4, qr = n1a12 + n2a22 + n3a32 + n4a42
= 2 x (0)2 + 2 x ()2 + 2 x ()2 + 1 x (0)2
= 1
at C5, qr = n1a12 + n2a22 + n3a32 + n4a42
= 2 x ()2 + 2 x ()2 + 2 x ()2 + 1 x ()2
= 1.3333
at C6, qr = n1a12 + n2a22 + n3a32 + n4a42
= 2 x ()2+ 2 x ()2 + 2 x ()2+ 1 x ()2
= 1.1667
Charge density = 1- qr
At, C1 = -0.1667, C2 = 0, C3 = -0.3333, C4 = 0, C5 = -0.3333, C6 = - 0.1667
[C6H6]-(Anion)
Electron Density (qr),
at C1, qr = n1a12 + n2a22 + n3a32 + n4a42
= 2 x ()2 + 2 x ()2 + 2 x ()2 + 2 x ()2
= 1.3333
at C2, qr = n1a12 + n2a22 + n3a32 + n4a42
= 2 x (0)2 + 2 x ()2 + 2 x ()2 + 2 x (0)2
= 1
at C3, qr = n1a12 + n2a22 + n3a32 + n4a42
= 2 x ()2 + 2 x ()2 + 2 x ()2 + 2 x ()2
= 1.6667
at C4, qr = n1a12 + n2a22 + n3a32 + n4a42
= 2 x (0)2 + 2 x ()2+ 2 x ()2 + 2 x (0)2
= 1
at C5, qr = n1a12 + n2a22 + n3a32 + n4a42
= 2 x ()2 + 2 x ()2 + 2 x ()2 + 2 x ()2
= 1.6667
at C6, qr = n1a12 + n2a22 + n3a32 + n4a42
= 2 x ()2 + 2 x ()2 + 2 x ()2 + 2 x ()2
= 1.3333
Charge density = 1- qr
At, C1 = -0.3333, C2 = 0, C3 = -0.6667, C4 = 0, C5 = - 0.6667, C6 = - 0.3333
Figure 4: Graphical method of electron density for Benzene system
CONCLUSION:
This research has offered insightful theoretical understandings of the electrical and molecular characteristics of the Benzene system. The findings for C6H6+, C6H6-, and C6H6.. Although all three species have the same delocalization energy (2), only the benzene anion has the lowest pi energy. The stability trend follows Anion C6H6- (antiaromatic), radical C6H6. (nonaromatic), and cation C6H6+ (aromatic) (fig 4). The research also emphasises how electron and charge density play a crucial role in establishing the electrical characteristics of the Benzene system. The Benzene anion has a larger electron density at each carbon atom, as seen by the graphical depiction of electron density in Figure 4, suggesting a better likelihood of finding electrons nearby. Understanding the molecular characteristics of the Benzene system depends heavily on this discovery. Overall, this study's theoretical approach has given us an important new understanding of the electrical and molecular characteristics of the Benzene system. Students may learn more about this significant compound's behaviour using the findings, which may inspire the additional study of its characteristics and uses.
CONFLICT OF INTEREST:
The authors have no conflicts of interest regarding this investigation.
ACKNOWLEDGMENTS:
I want to express my sincere gratitude to Pr. (Dr) D. B Shinde and Dr Arun. M. Bhagare (HOD, Chemistry Dept) and Dr Akshay Dhaygude for their invaluable support and encouragement throughout this study. Their guidance and expertise have been instrumental in completing this work. I am also thankful to the Chemistry Department of our institute for providing the necessary resources and facilities for this research. The support and cooperation of the staff members have been greatly appreciated.
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Received on 26.04.2023 Modified on 30.05.2023
Accepted on 11.07.2023 ©AJRC All right reserved
Asian J. Research Chem. 2023; 16(4):265-270.
DOI: 10.52711/0974-4150.2023.00044