Cyclobutadiene System (C₄H₃⁺, C₄H₃^-, and C₄H₃^.): A Theoretical Study for Solving Secular Determinant, Delocalization Energy, Electron Density, and Charge Density

Vikram R. Jadhav^1*, J. S. Aher², A. M. Bhagare³, M. S. Jamdhade⁴, and P. B. Wadhawane⁵

^1,3,4,5Department of Chemistry, K. K. Wagh Art’s, Science, and Commerce College, Pimpalgaon (B),

422209, Tal Niphad, Nashik, Maharashtra (India).

²Professor and HOD, Department of Chemistry, K.T. H. M College, Gangapur Road,

Nashik, 422002, SPPU, MS (India).

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

ABSTRACT:

In this theoretical study, we have mainly focus on the Hückel approximation method. Nowadays many modern methods like computational method, are utilized to understanding the molecular parameters but it has some difficulties such as not easily understood, and non-availability for everywhere, students want to know the molecular parameters then the theoretical methods or technique are preferable and it is conceivable to get secular parameters, π energy, wave functions, electron density, and charge density, as an account of cyclobutadiene system i.e. C₄H₃⁺ (cation), C₄H₃^- (anion), and C₄H₃^. (radical). Here, it has presented the secular determinant, and secular equation of the Hückel approximation technique and applied to cyclobutadiene system to communicate their delocalization energies, wave functions, and also its electron, and charge density at each carbon atom in terms of understanding the stable molecular configuration of cyclobutadiene system. It is settled by the Hückel approximation method using assumptions or characteristics such as coulomb integrals, exchange integrals, and overlap integrals. This is a simple way theoretical method, which will be preferable to graduate and post-graduate understudies to understanding the molecular parameters and to investigate the stable configuration of a cyclobutadiene system.

KEYWORDS: Charge density, Cyclobutadiene system, Hückel theory, Secular equation, Wave function.

INTRODUCTION:

Hückel molecular orbital (HMO) theory of conjugated systems for linear polyenes such as allyl system, 1, 3 butadiene, ethylene, etc., and cyclic polyenes such as benzene, cyclopentadiene, cyclobutadiene, naphthalene, etc., were investigated and studied by E. Hückel during 1930, afterward his method successfully extended by C. Coulson, H. C. Longuet-Higgins, and R. Hoffman¹.

In unsaturated molecules with an alternate single and double bond, in which conjugation occurs, molecule generally planer, in n-carbon atoms, which is sp² hybridized and has a 2p_z orbital centered on it, electrons in sp² orbital consider as sigma electrons, which is participated in the formation of a single bond and contained in the molecular plane. Pi electrons in the 2p₂ orbital participated in double bond formation and exactly perpendicular to the molecular plane. In a molecular system all the carbon atoms are treated equally, so the coulomb integrals α for the atomic orbitals that contribute to the pi orbitals are set equal, for example, in cyclobutadiene system⁷, we consider the two sigma bonds as fixed, and concentrate on finding the energies of the double pi bond. In a modern computation, all the overlap integrals and resonance integrals may be considered but for the molecular orbital energy level⁹, some of the additional assumptions of the Hückel approximations can be included in which all overlap integrals, resonance integrals between non-neighbors are set equal to zero, and remaining resonance integrals are set equal^1,6,10. Other information regarding matrix formulation, wave functions, energies, electron density, and charge density has been given in elsewhere^1,3,4,6. The most notable example, cyclobutadiene^1,2,8,11 as shown in figure 1 and discussion about its system has mentioned here.

Figure 1. Cyclobutadiene

Cyclobutadiene cation is formed from cyclobutadiene molecule, it’ formation, as shown in figure 5, when the hydride (H^-) ion is removed from the sp² hybridized⁵ carbon atom in the cyclobutadiene ring⁴, the hybridization of the carbon atom, is remained unchanged, however, the cyclobutadiene cation is planar, two π electron cloud, and not obeys Hückel rule, therefore it is an antiaromatic compound^7,11 as shown in figure 2.

Figure 2. Cyclobutadiene cation [C₄H₃] ⁺

The abstraction of hydrogen radicals into the cyclobutadiene molecule at the sp³ hybridized⁵ carbon atom which is produced the cyclobutadiene radical as shown in figure 5. Like the cyclobutadiene cation, the hybridization of the carbon remains unchanged, the cyclobutadiene radical has five π electron cloud, planer, and not obeys Hückel rule, therefore, it is nonaromatic as shown in figure 3.

Figure 3. Cyclobutadiene radical [C₄H₃]^.

The abstraction of hydrogen using a strong base from the sp² hybridized⁵ carbon atom into the cyclobutadiene, which is produced the cyclobutadiene anion as shown in figure 4. Like, cyclobutadiene cation, and radical the carbon atom shows sp² hybridization. Cyclobutadiene anion is a planer, six π electron cloud, and obeys Hückel rule. Therefore, cyclobutadiene anion is a stable aromatic species. Hence, the cyclobutadiene cation and the radicals are unstable and highly reactive species.

Figure 4. Cyclobutadiene anion [C₄H₃]^-

In modern computation chemistry, the molecular parameters have been obtained by using sophisticated technology along with Hückel approximation method¹⁰ to understand the molecular parameters as geometries, π energies, delocalization energy, charge density, electron density, and consequently, to predict the stability nature of a cyclic molecular system for further applications and uses.

THEORETICAL METHOD:

Formation of cyclobutadiene system (cation, radical, and anion):

Figure 5. Cyclobutadiene system

The secular determinant for cyclobutadiene system:

C₄H₃⁺(cation), C₄H₃^- (anion), and C₄H₃^. (radical) the total number of carbon atoms are the same in the three species, so the secular determinant has remained the same,

Secular equation:

Figure 6. Cyclobutadiene system (mentioned the appropriate position of carbon atoms)

The carbon atom (C₁), is next to atoms C₄, and C₂ as shown in figure 6, then the following conditions for the secular equation are arises,

XC₁ + C₂ + C₄ = 0

v Each secular equation has 3-digit terms. 

v All the four secular equations have the same molecular structure, and they can be obtained from each other by permuting the indices. 

v It can be solved by using trigonometric functions.

v This trigonometric solution can be done by using a simple Circle method.

A similar method is used for C₂, C_3, and C₄ carbon atoms, the following secular equation can be getting, C₁ + XC₂ + C₃ = 0, C₂ + XC₃ + C₄ = 0, C₁ + C₃ + XC₄ = 0

X values:

The following ‘X’ values can be obtained by solving the above secular determinant of the cyclobutadiene system, 

X₁ = - 2.000, X₂ = X₃ = 0, and X₄ = 2, the Energy level equation as, X = (α - E) / β

Total π energy:

E_π = _iE_i = n₁E₁ + n₂E₂ + n₃E₃ + n₄E₄, Where n_i = number of electrons in the i^th energy level, E_i = i^th energy level equation.

Delocalization energy (DE):

D.E = (π electron energy of a species) – (π electron energy of an equivalent number of isolated double bonds + energy of an odd number of electrons).

Wave functions:

The wave function equation of cyclobutadiene system as follows,

Ψ_i = a₁ P₁ + a₂ P₂ + a₃ P₃ + a₄ P₄

Solving the above wave function equation for the cyclobutadiene system using the terms coefficients (a) including the normalized, and orthogonalized conditions,

_i ψ_j dX = 1 --------------- (i = j, normalized condition)

_i ψ_j dX = 0 ---------- (i  j, orthogonalized condition)

The complete wave function of a cyclobutadiene system as follows,

Ψ₁ = 0.5 (P₁ + P₂ + P₃ + P₄)

Ψ₂ = 0.7071 P₁ – 0.7071 P₃

Ψ₃ = 0.7071P₂ - 0.7071P₄

Ψ₄ = 0.5P₁ – 0.5P₂ + 0.5P₃ - 0.5 P₄

Electron density:

The total electron density at each carbon atom (r) is the sum of electron densities contributed by different electrons in each Hückel molecular orbital as therefor the Electron Density (q_r) = n_ia_i²

Where a_i is the coefficients of the carbon atom (r) in the i^th molecular orbital, and n_i is the total number of electrons at an energy level.

Charge Density:

In the π system, a neutral carbon atom in a cyclobutadiene system shows 1.0 electron density and so the net charge density can be calculated using the equation, Charge density = 1 - q_r

RESULT AND DISCUSSION:

In a cyclic conjugated system, the π electrons are delocalized in a cyclic system and the single bonds which are present between, which can participate in delocalization and shows double bond character. Theoretical Hückel approximation method can be used to solve the following molecular parameters,

Total π energy:

v [C₄H₃] ⁺ cation: E_π = 2(α + 2β) = 2α + 4β

v [C₄H₃]^. Radical: E_π = 2(α + 2β) +3α = 5α + 4β

v [C₄H₃] ^– anion: E_π = 2(α + 2β) + 4α = 6α + 4β

According to the total π energy of a cyclobutadiene species (cation, radical, and anion) out of this, the anion species show higher energy i.e. more stability.

Delocalization energy (DE):

v [C₄H₃] ⁺ cation: E_DE = 2α + 4β - {2(α + β)} = 2β

v [C₄H₃]^. Radical: E_DE = 5α + 4β – {2(α + β) + 3α} = 2β

v [C₄H₃] ^– anion: E_DE = 6α + 4β - {2(α + β) + 4α} = 2β

The delocalization energy of a cyclobutadiene system remains same i.e. 2β

Electron and charge density:

v [C₄H₃] ⁺ cation:

Electron Density (q_r),

at C₁, q_r = n₁a₁² + n₂a₂² + n₃a₃² + n₄a₄² = 2 x (0.5)² + 0 x (0.5)² x 0 x (0.5)² + 0 x (0.5)² = 0.50

at C₂, q_r = n₁a₁² + n₃a₃² = 2 x (0.7071)² + 0 x (0.7071)² = 1.00

at C_3, q_r = n₂a₂² + n₄a₄² = 0 x (0.7071)² + 0 x (0.7071)² = 0.00

at C₄, q_r = n₁a₁² + n₂a₂² + n₃a₃² + n₄a₄² = 2 x (0.5)² = 0.50

Charge density = 1- q_r

At, C₁ = 0.500, C₂ = 0.00, C₃ = 1.00, and C₄ = 0.50

v  [C₄H₃]^. Radical:

Electron Density (q_r),

at C₁, q_r = n₁a₁² + n₂a₂² + n₃a₃² + n₄a₄² = 2 x (0.5)² + 2 x (0.5)² + 1 x (0.5)² = 1.250

at C₂ q_r = n₁a₁² + n₃a₃² = 2 x (0.7071)² + 1 x (- 0.7071)² = 1.499

at C₃, q_r = n₂a₂² + n₄a₄² =2 x (0.7071)² + 1 x (- 0.7071)² = 1.499

at C₄, q_r = n₁a₁² + n₂a₂² + n₃a₃² + n₄a₄² = 2 x (0.5)² + 2 x (-0.5)² + 1 x (0.5)² = 1.250

Charge density = 1- q_r

At, C₁ = - 0.250, C₂ = - 0.499, C₃ = -0.499, C₄ = - 0.250

v [C₄H₃]^- (Anion):

Electron Density (q_r) 

at C₁, q_r = n₁a₁² + n₂a₂² + n₃a₃² + n₄a₄²= 2 x (0.5)² + 2 x (0.5)² + 2 x (0.5)² = 1.500

at C₂, q_r = n₁a₁² + n₃a₃² = 2(0.7071)² + 2(-0.7071)² = 2.000

at C₃, q_r = n₂a₂² + n₄a₄² = 2(0.7071)² +2(-0.7071)² = 2.000

at C₄, q_r = n₁a₁² + n₂a₂² + n₃a₃² + n₄a₄² = 2(0.5)² + 2(-0.5)² + 2(0.5)² + 0 (0.5)² = 1.500

Charge density = 1 - q_r

At, C₁ = -0.500, C₂ = -1.00, C₃ = -1.00, C₄ = -0.500

Figure 7. 3D Surface graph of electron density at each carbon atom into the cyclobutadiene system

Figure 8. 3D Surface graph of Charge density at each carbon atom in the cyclobutadiene system

According to the above data, the three-dimensional surface graph of the electron density, and charge density at each carbon atom of the cyclobutadiene system i.e. C₄H₃⁺ (cation), C₄H₃^- (anion), and C₄H₃^. (radical) has shown in figure 7 and 8, which is indicate that the maximum surface electron density as well as charge density at each carbon atom of the cyclobutadiene anion as compared to the cyclobutadiene cation and radical, because of maximum probability of finding an electron near at the carbon atom. 

CONCLUSION:

In this study, theoretical results of the cyclobutadiene system i.e. C₄H₃⁺ (cation), C₄H₃^- (anion), and C₄H₃^. (radical) using the Hückel approximation method implies that the species of cyclobutadiene system shows the same delocalization energy i.e. 2β, but the pi energy of a species is different, according to this the stability order as, C₄H₃⁺ (cation, antiaromatic) < C₄H₃^. (radical, nonaromatic) < C₄H₃^- (anion, aromatic). The electron and charge density can reveal the electronic properties of a cyclobutadiene system as shown in graphical method, in which concluded that the high electron density of a cyclobutadiene anion at each carbon atom as compared with cyclobutadiene cation, and cyclobutadiene anion, it means the high probability of a finding the electron is more around the carbon atoms. accordingly, graduate, and post-graduate students easily understand an idea regarding the whole molecular parameters of a cyclobutadiene system.  

ACKNOWLEDGEMENTS:

The authors are thankful to Principal, HOD, and Chemistry faculty of  M. V. P. Samaj’s, K. K. W. Arts, Science, and Commerce college Pimpalgaon (B), Nashik for the support and encouragement of this work.

CONFLICT OF INTEREST:

The authors declare no conflict of interest.

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Received on 26.07.2020                    Modified on 17.08.2020

Accepted on 14.09.2020                   ©AJRC All right reserved

Asian J. Research Chem. 2020; 13(6):419-423.

DOI: 10.5958/0974-4150.2020.00076.0