Cyclopropene System: A Theoretical Investigation into Secular Determinants, π-Energy, Delocalization Energy, Wave Functions, Electron Density and Charge Density

 

K. B. Handage, S. S. Jadhav, P. R. More, A. D. Matere, P. R. Pote, V. R. Jadhav*

Department of Chemistry, K. K. Wagh Art’s, Commerce and Science College,

Pimpalgaon (B), 422209, Tal Niphad, Nashik, Maharashtra (India)

*Corresponding Author E-mail: mevikramjadhav@gmail.com

 

 

INTRODUCTION:

In 1931, Erich Hückel conducted a ground breaking study on the characteristics of conjugated aromatic cyclic compounds. These compounds include well-known molecules like Cyclopropene, benzene, cyclobutene, and naphthalene¹. Later on, C. Coulson, H. C. Longuet Higgins, and R. Hoffmann further expanded on Hückel work. A crucial characteristic of conjugated systems is planarity, which denotes that the molecules' alternating double bonds are completely conjugated. A molecule containing "n" carbon atoms goes through sp² hybridization and develops a planar shape².

 

The 2p_z orbitals, which are aligned in the plane of the molecule, are involved in the sp² hybridization. In a conjugated system, two nearby atoms with half-filled atomic orbitals might overlap laterally or sideways to form a structure known as a π bond. The conjugated system has unique electrical characteristics because of the delocalization of electrons caused by this kind of link. A sigma bond is formed when two nearby atoms have half-filled atomic orbitals that overlap axially or end-to-end. Single bonds between atoms frequently contain sigma bonds, which are more confined³. The Linear Combination of Atomic Orbitals (LCAO) approximation may be used to derive the molecular orbital wave function since the molecule's plane is perpendicular to each 2p_z orbital. Using this approximation, atomic orbitals may be combined to form molecular orbitals, illuminating how the electrons are distributed throughout the molecule⁴.

 

In organic chemistry, the Hückel method is a popular theoretical tool for examining the electrical structure of conjugated compounds. It calculates the distribution of electrons in the conjugated system using the linear combination of atomic orbitals (LCAO) idea. The basis for this method is the assumption that molecular orbitals may be described as linear combinations of atomic orbitals from nearby atoms. The LCAO technique forms molecular orbitals by combining the atomic orbitals of adjacent atoms. Understanding the electrical properties of the conjugated molecule, including its electronic structure, bonding, and stability, depends heavily on these molecular orbitals⁵. The Hückel technique is an extensively used in organic chemistry to investigate the behavior of conjugated aromatic cyclic compounds and other similar systems. The Hückel method may predict conjugated systems' stability, reactivity, and optical and magnetic properties. Examples are the conjugated chemical substances Cyclopropene, cyclopentadienyl, cyclobutadiene, and furan. The study of conjugated systems has essential applications in materials science, including designing and manufacturing organic semiconductors for electrical and optoelectronic devices. The 2p_z orbital of the j^th carbon atom is represented by the symbol "j" in the wave function of a molecular orbital^{6, 7}.

 

 

Where:

Φ_j represents the 2p_z orbital of the j^th carbon atom.

n is the number of molecular orbitals.

C_i,j are the coefficients of the i and j values, where i can take values of 1, 2, 3, ..., n.

 

In the molecular orbital theory of homonuclear diatomic compounds, the 1s atomic orbitals are the primary functions in the H₂ molecule, and the 2p_z atomic orbitals are used as the fundamental functions in the Hückel molecular orbital theory⁸. Regarding the Coulomb integral, exchange integral, and overlap integral, Hückel strategy included several assumptions. In this context, the overlap integral and the Coulomb integral relate to the amount of overlap between the 2p_z atomic orbitals and the energy of an electron in the i^th carbon atom 2p_z orbital, respectively. The exchange or resonance integral represents the energy of interaction between two atomic orbitals. A molecule must be planar, cyclic, and possess a cyclic electron cloud (system) to be categorized as aromatic. Hückel rule stipulates that the number of electrons in the molecule must equal (4n + 2) π, where n is an integer, which also applies to the molecule. It's crucial to remember that the p orbitals and π electrons need not be equal in the π system. The Hückel rule may thus be applied to anions, cations, and radicals to consider them aromatic compounds. The Hückel molecular orbital hypothesis is covered in further detail in the literature^9-13.

 

The Hückel molecular orbital theory has been extensively used to examine the cyclic molecule composed of three carbon atoms and three hydrogen atoms that comprise the Cyclopropene system. Applying the Hückel rule (4n + 2) for aromaticity to this system, it is hypothesized that the delocalization of the π electrons in Cyclopropene results in the development of a cyclic cloud of electron density. The Hückel theory uses the 2p_z atomic orbitals of the carbon atoms in the Cyclopropene ring as its basic building blocks. It makes specific assumptions concerning Coulomb integrals, exchange integrals, and overlap integrals¹⁴. This theory allows for calculating many significant molecular properties of Cyclopropene, such as orbital energies, wave functions, electron densities, and charge densities. The Hückel hypothesis proposes the presence of alternating double bonds, resulting in a planar molecule, in the fully conjugated system of the Cyclopropene ring. Due to sp² hybridization, the carbon atoms in the Cyclopropene ring may form sigma bonds via the axial and lateral overlap of their half-filled atomic orbitals, respectively, and π bonds. Bonds are arranged so that a stable, planar molecular structure is formed¹⁵.

 

The Cyclopropene system (fig 2) has a perpendicular molecular plane, a defining trait of molecules with sp² hybridization. The Hückel theory has also been extended to the analysis of modified Cyclopropene systems, in which substituents on the ring may change the electron density and affect the molecule's stability. This enables a clearer understanding of how different functional groups affect the characteristics of the Cyclopropene system. The Hückel theory has been used to study additional conjugated cyclic compounds, cyclobutene and naphthalene, and Cyclopropene (fig 1). Understanding the electronic structure and stability of conjugated cyclic compounds may be done with the help of the Hückel molecular orbital theory, which is still a powerful and popular technique in organic chemistry. It is an essential approach for comprehending the behavior of these compounds because of its capacity to provide information about orbital interactions and electron delocalization^16-17.

 

Figure 1: Cyclopropene

 

The cyclic molecule known as Cyclopropene cation is produced when a hydride anion is removed from one of the sp³ hybridized carbon atoms in the Cyclopropene ring (fig 3). Figure 3 shows how this process makes the molecule flat and formed a cyclic electron cloud. The Hückel rule states that an aromatic molecule must have a cyclic electron cloud that contains (4n + 2) π electrons, where "n" is an integer. The (4n + 2) π rule is obey the Cyclopropene cation, which possesses two electrons, the Cyclopropene cation would be regarded as aromatic when using the Hückel method. The delocalization of its electrons in the solution allows the Cyclopropene cation to display specific aromatic properties while entirely obeying the Hückel rule. Stability improvement brought about by electron delocalization is a crucial characteristic of aromatic compounds¹⁸.

 

Figure 2: Cyclopropene system

 

In contrast, the considerable delocalization of the π electrons across the Cyclopropene ring makes the Cyclopropene cation more stable. The Cyclopropene cation is a highly reactive species in organic chemistry because it may easily take part in various chemical reactions due to its wide delocalization. The Cyclopropene cation is an essential species in organic chemistry because of its delocalized electron cloud, high stability, and reactivity. In conjugated cyclic compounds, studying these aromatic properties and the delocalization of electrons offers essential insights into their behavior and reactivity. Figure 3 shows how particular procedures involving the removal of hydrogen atoms from the Cyclopropene ring result in the formation of a Cyclopropene radical and a Cyclopropene anion. The Cyclopropene radical is formed when a hydrogen atom is removed from the sp³ hybridized carbon atom in the Cyclopropene ring. As the hybridization of the carbon atom shifts from sp³ to sp², a planar Cyclopropene radical with a cyclic electron cloud is produced. The Cyclopropene radical, however, deviates from the Hückel rule since it has three electrons instead of the necessary (4n + 2) π electrons for aromaticity. It is not categorized as an aromatic species as a result. The radical Cyclopropene is very unstable and reactive. The Cyclopropene anion is produced when a proton is removed from the sp³ hybridized carbon atom in the Cyclopropene ring, causing the hybridization to change into sp². Because the planar Cyclopropene anion has the necessary (4n) π electrons for anti-aromaticity, it has a cyclic electron cloud that fits the anti-aromatic rule. The Cyclopropene anion is unstable and is easily accessible via basic processing¹⁸.

 

The Cyclopropene cation is less reactive and more stable than the radical and the anion. Modern computational methods consider all resonance and overlap integrals, allowing for exact forecasts of different chemical system characteristics. The Hückel approximation may provide helpful insights into the characteristics of specific molecules notwithstanding this development. However, the Hückel method has several drawbacks because of its rigid assumptions. More intricate theories have been formed to get over these restrictions and provide a better understanding of the properties of cyclic molecular systems. These cutting-edge theories can forecast the geometry, energy, electron delocalization, and charge density of molecules, which may be used to determine how stable cyclic chemical systems are. With the help of these discoveries, novel molecular systems with particular properties have been designed and created, making them helpful in various applications in chemistry, physics, and materials science^19-20.

 

THEORETICAL METHOD:

1)    Formation of the Cyclopropene system:

 

Figure 3: Cyclopropene system, i.e., formation of cation, anion and radical species

 

2)    Secular determinant for Cyclopropene system:

C₃H₃⁺ (cation), C₃H₃^- (anion), and C₃H₃^* (Radical). The number of carbon atoms is the same in the three systems, so the secular determinant is unchanged.

 

 

 

 

3)    Secular equation: (unchanged for the system):

For carbon atom C₂ in a cyclic conjugated system, the related secular equation is:

 

XC₁ + C₂ + C₃ = 0

 

Here, X is the eigenvalue or energy of the molecular orbital centred on carbon atom C₂, C₁ and C₃ are the coefficients or weights of the atomic orbitals on atoms C₁ and C₃, respectively. Similarly, for carbon atom C₃, the secular equation is:

XC₁ + C₂ + C₃ = 0

 

In these secular equations, the Hückel technique still has three non-zero elements, and they can be solved using trigonometric functions. The circle technique, a geometric approach, can also be employed to visualize and depict the solutions. By creating a circle with a radius equal to the sum of the coefficients of the nearby atoms (for each carbon atom), the eigenvalues can be determined by identifying the points where the circle intersects the x-axis. The intersection points along the x-axis correspond to the eigenvalues or energies of the molecular orbitals. The number of intersection points represents the number of molecular orbitals in the system. Overall, solving the secular equations for carbon atoms in a cyclic conjugated system allows for the determination of the energies and shapes of the molecular orbitals. The Hückel molecular orbital theory, with its simplified approach, is valuable for gaining insights into the electronic structure and stability of conjugated cyclic compounds. The use of trigonometric functions and the circle technique offers a straightforward method for solving these equations and understanding the behaviour of molecular orbitals in such systems^7-8.

 

C₁ + XC₂ + C₃ = 0,

C₁ + C₂ + XC₃ = 0

 

 

4)    X values:(Unchanged for the system)

The X values can be obtained by solving the secular determinant,

X₁ = 1, X₂ = 1, X₃ = - 2

Energy levels, X =

 

 

5)    Total π energy

E_π = _I E_i

       = n₁E₁ + n₂E₂ + n₃E₃

Where,

n_i = a number of electrons in the i^th energy state,

E_i = i^th energy equation or value.

 

6)    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.

 

7)    Wave functions:

The general wave function equation is,

Ψ_i = a₁ P₁ + a₂ P₂ + a₃ P₃

 

Solving the wave functions for the Cyclopropene system in terms of coefficients (a) by using the normalized and orthogonalized condition as,

 

 ------------ (i = j, Normalized condition)

 ------------ (i  j, Orthogonalized condition)

 

The wave function for the system as shown in the fig 4,

 

 

Figure 4: Wave function of Cyclopropene system

 

8)    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 (q_r) = n_ia_i²

Where, a_i is the coefficient of the atom r in the i^th HMO and n_i is the number of electrons in that HMO. (Values of n = 0, 1, or 2).

9)    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- q_r

 

RESULTS AND DISCUSSION:

Electrons are delocalized across a conjugated cyclic system, producing a variety of resonance configurations. Despite the absence of actual double bonds, this delocalization makes the single bonds in the system behave like double bonds. It is usual practice to use the Hückel approximation theory to analyse the electrical properties and structure of these conjugated cyclic systems. The Hückel theory streamlines the challenging calculations necessary in molecular orbital theory by assuming certain assumptions, such as utilizing 2p_z atomic orbitals as the basic functions and ignoring particular overlap integrals. The number of molecular orbitals, their energies, and the coefficients of each atomic orbital may all be calculated through this simplification. The number of electrons in a cyclic system may be used to assess if the system is aromatic in accordance with Hückel rule, often known as the (4n + 2) π rule, where "n" is an integer. The system is regarded as aromatic if it has (4n + 2) π electrons. One secular equation derived from the Hückel theory corresponds to each atomic orbital in the cyclic system. The linear combinations of atomic orbitals that make up the system's molecular orbitals are described by these secular equations. Three non-zero terms make up each secular equation, which may be resolved trigonometrically. It is significant to note that, because to their same chemical structure, the three secular equations for a conjugated cyclic system may be obtained from one another by permuting the index values. The circle method is a straightforward geometric technique for representing the trigonometric answers to the secular equations. The three coefficients of each atomic orbital are plotted on a circle, and the points where the two circles meet are identified in order to estimate the energy levels and coefficients of the molecular orbitals using this method. The molecular orbitals and their energies are better understood using graphical method, which also sheds light on the electrical properties of the conjugated cyclic system. The study of conjugated cyclic systems is made simpler by the Hückel approximation theory, enabling the identification of molecular characteristics and the use of Hückel rule to determine aromaticity. The comprehension of the system's electrical structure and behaviour is aided by the use of trigonometric solutions and the circle method, which gives molecular orbitals and their energies a visual representation.

 

a)    Total π energy:

According to the value of total π energy of a system in which cation species shows high energy indicates more

Stability.

 

b)   Delocalization energy (DE):

 

According to the delocalization energy, Cyclopropene anion shows zero value, therefore it is highly unstable. The increasing order of stability of the Cyclopropene system: Cyclopropene cation (Aromatic) > Cyclopropene Anion (Anti-aromatic) > Cyclopropene anion (Anti-aromatic)

 

c)    Electron and charge density:

 

The stability and aromaticity of many species, including the Cyclopropene cation, radical, and anion, are significantly influenced by the electron density (fig. 5) in the Cyclopropene system.

 

Cyclopropene Cation (Aromatic):

The electron density distribution in the Cyclopropene cation indicates that each carbon atom contributes differentially to the overall electron density. Carbon atom C₂, with a coefficient of 1.000, has the highest electron density contribution, followed by C₁ (0.6667) and C₃ (0.3333). This distribution is in line with the Cyclopropene cation being classified as aromatic. The higher electron density at C₂ is consistent with its role as a hub of electron delocalization in the aromatic system. The presence of aromaticity is associated with enhanced stability due to the resonance of π electrons, which in turn is reflected in the electron density distribution.

 

Cyclopropene Radical (Non-Aromatic):

The electron density distribution in the Cyclopropene radical demonstrates a significant increase in electron density compared to the cationic system. Carbon atom C₁ now has the highest electron density contribution (1.3333), followed by C₂ (1.000) and C₃ (0.6667). Despite the increased electron density, the Cyclopropene radical is classified as non-aromatic. This may be attributed to the fact that the electron distribution is uneven and not evenly spread across the system, leading to a lack of complete resonance stabilization, which is characteristic of aromatic compounds.

 

Figure 5: Graphical method of electron density for Cyclopropene system

 

Cyclopropene Radical (Non-Aromatic):

The electron density distribution in the Cyclopropene anion displays balanced contributions from all carbon atoms. Both C₁ and C₂ have equal coefficients of 1.000, while C₃ has a coefficient of 0.5000. This symmetrical distribution is typical of anti-aromatic systems. Anti-aromatic compounds, unlike aromatic ones, have destabilizing effects due to the presence of anti-resonance, which disrupts electron delocalization. This phenomenon is evident in the even distribution of electron density among all three carbon atoms. The electron density distributions in the Cyclopropene system showcase the distinctive characteristics of its cationic, radical, and anionic forms. The variations in electron density among the carbon atoms correspond to the aromatic, non-aromatic, and anti-aromatic nature of each species, respectively. These electron density patterns provide insights into the stability and reactivity of these species, making them valuable tools for understanding the behavior of conjugated cyclic systems.

 

CONCLUSION:

Theoretical insights have been gained from the study of the electrical and molecular properties of the Cyclopropene system. C₃H₃⁺, C₃H₃^-, and C₃H₃^* were the three species taken into consideration. Even while each species had the different delocalization energy, they each had unique characteristics, notably in terms of their stability and aromaticity. The findings showed that the lowest π energy species, the aromatic Cyclopropene cation (C₃H₃⁺), was the most stable species. The Cyclopropene anion (C₃H₃^-) was thought to be non-aromatic, whereas the Cyclopropene radical (C₃H₃^*) was thought to be anti-aromatic. The finding highlighted how important electron and charge density are in defining the electrical properties of the Cyclopropene system. Each carbon atom on the Cyclopropene anion had a greater electron density, which suggests a better likelihood of finding electrons nearby. This discovery is crucial to comprehending the Cyclopropene system's molecular behavior. Our grasp of the electrical and molecular characteristics of the Cyclopropene system has generally improved as a result of this theoretical investigation. Students may use the results as a useful tool to learn more about this important compound's behavior and to spark further investigation into its properties and possible uses. Researchers may better understand the complexity of conjugated cyclic compounds and their wide range of features by learning more about the stability and aromaticity of various species in the Cyclopropene system. This information may open up new directions for the research and use of these substances in a number of disciplines, including organic chemistry, materials science, and drug development.

 

CONFLICT OF INTEREST:

The authors have no conflicts of interest regarding this investigation.

 

ACKNOWLEDGMENTS:

I want to express my sincere gratitude to Dr. D. T. Dhage and Dr A. M. Bhagare (HOD, Chemistry Dept.) and Dr A. C. 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.

 

REFERENCES:

1.     Hückel, E. Quantum-theoretical contributions to the benzene problem. I. The electron configuration of benzene and related compounds. Z. Physic. 1931; 70(3-4): 204-286.

2.     Longuet-Higgins, H. C., McDowell, C. A., Bawn, C. E. H., Lawrence, A. S. C., and Mills, O. S. General and physical chemistry. Annual Reports on the Progress of Chemistry. 1951; 48:7-86. https://doi.org/10.1039/AR9514800007

3.     Kutzelnigg, W. What I like about Hückel theory. Journal of Computational Chemistry. 2007; 28(1): 25-34. https://doi.org/10.1002/jcc.20470

4.     Kopf, A., and Saalfrank, P. Electron transport through molecules treated by LCAO-MO Green’s functions with absorbing boundaries. Chemical Physics Letters. 2004; 386(1-3): 17-24. https://doi.org/10.1016/j.cplett.2003.12.118

5.     Hoffmann, R. (1963). An extended Hückel theory. I. hydrocarbons. The Journal of Chemical Physics. 1963; 39(6): 1397-1412. https://doi.org/10.1063/1.1734456

6.     Jug, K., Hiberty, P. C., Shaik, S. σ− π Energy separation in modern electronic theory for ground states of conjugated systems. Chemical Reviews. 2005; 101(5): 1477-1500. https://doi.org/10.1021/cr990328e

7.     Jadhav, V. R., Aher, J. S., Bhagare, A. M., Jamdhade, M. S., and Wadhawane, P. B. Cyclobutadiene System (C4H3+, C4H3-, and C4H3): A Theoretical Study for Solving Secular Determinant, Delocalization Energy, Electron Density, and Charge Density. 2020. 10.5958/0974-4150.2020.00076.0 

8.     Jadhav, Vikram, Theoretical Approach to Understanding an Electron Density, Charge Density and Most Stable Configuration of H3 System (March 15, 2020). International Journal of Research and Review. 2020; 7(3): SSRN: https://ssrn.com/abstract=3684255

9.     Nagaoka, S. I., Yamasaki, Y., Teramae, H., Nagashima, U., and Kokubo, T. Addition to Practical Training in Simple Hückel Theory: Matrix Diagonalization via Tridiagonalization, Cyclobutadiene, and Visualization of Molecular Orbitals. Journal of Chemical Education. 2020; 97(8): 2373-2374. https://doi.org/10.1021/acs.jchemed.0c00857

10.  McKee, W. C., Wu, J. I., Rzepa, H. S., Schleyer, P. V. R. (2013). A Hückel Theory Perspective on Möbius Aromaticity. Organic letters. 2013; 15(13): 3432-3435. https://doi.org/10.1021/ol401491s

11.  Pedersen, K. G., Borges, A., Hedegård, P., Solomon, G. C., and Strange, M. Illusory connection between cross-conjugation and quantum interference. The Journal of Physical Chemistry C.n 2015; 119(48): 26919-26924. https://doi.org/10.1021/acs.jpcc.5b10407

12.  El Bakouri, O., Szczepanik, D. W., Jorner, K., Ayub, R., Bultinck, P., Solà, M., and Ottosson, H. Three-Dimensional Fully π-Conjugated Macrocycles: When 3D-Aromatic and When 2D-Aromatic-in-3D?. Journal of the American Chemical Society. 2022; 144(19): 8560-8575. https://doi.org/10.1021/jacs.1c13478

13.  Wypych, K., Dimitrova, M., Sundholm, D., Pawlicki, M. Diagnosing Ring Current (s) in Figure-Eight Skeletons: A 3D Through-Space Conjugation in the Two-Loops Crossing. Organic Letters. 2022; 24(27): 4876-4880. https://doi.org/10.1021/acs.orglett.2c01625

14.  Jadhav, V. R., Madhuri, J., Pooja, W., Baste, Y. R. Cyclopentadienyl System: Solving the Secular Determinant, π Energy, Delocalization Energy, Wave Functions, Electron Density and Charge Density. Journal of Chemistry, Environmental Sciences and its Applications. 2020; 6(2): 21-26. https://doi.org/10.15415/jce.2020.62002

15.  Kuldeep B. Handage, Snehal S. Jadhav, Vikram R. Jadhav. Benzene System: A Theoretical Exploration of its Secular Determinant, π-Energy, Delocalization Energy, Wave Functions, Electron Density, and Charge Density. Asian Journal of Research in Chemistry. 2023; 16(4):265-0. doi: 10.52711/0974-4150.2023.00044.

16.  Zhu, Z. B., Wei, Y., Shi, M. Recent developments of cyclopropene chemistry. Chemical Society Reviews. 2011; 40(11): 5534-5563. https://doi.org/10.1039/C1CS15074J

17.  Tapia, O., Andres, J., Aullo, J. M., Cardenas, R. Electronic aspects of the hydride transfer mechanism: Part 2. Ab-initio analytical gradient studies of the pyridinium cation/1, 4-dihydropyridine, cyclopropenyl-cation/cyclopropene and formaldehyde/methanolate model reactant systems. Journal of Molecular Structure: Theochem. 1988; 167(3-4): 395-412. https://doi.org/10.1016/0166-1280(88)80240-9

18.  Allen, Annette D., and Thomas T. Tidwell. Antiaromaticity in open-shell cyclopropenyl to cycloheptatrienyl cations, anions, free radicals, and radical ions. Chemical Reviews. 2001; 101(5): 1333-1348. https://doi.org/10.1021/cr990316t

19.  Glukhovtsev, M. N., Laiter, S., Pross, A. Thermochemical assessment of the aromatic and antiaromatic characters of the cyclopropenyl cation, cyclopropenyl anion, and cyclopropenyl radical: A high-level computational study. The Journal of Physical Chemistry. 1996; 100(45): 17801-17806. https://doi.org/10.1021/jp961882k

20.  D'yakonov, I. A., Kostikov, R. R. The cyclopropenyl cation. Russian Chemical Reviews. 1967; 36(8): 557. 10.1070/RC1967v036n08ABEH001671

 

 

Received on 09.08.2023                    Modified on 11.09.2023

Accepted on 27.09.2023                   ©AJRC All right reserved