Module:32,Huckel Molecular Orbital theory- Application Part-II PAPER: 2, PHYSICAL CHEMISTRY-I QUANTUM CHEMISTRY

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1 Subject PHYSICAL Paper No and Title TOPIC Sub-Topic (if any) 2, PHYSICAL -I QUANTUM Hückel Molecular orbital Theory Application PART II Module No. 32

2 TABLE OF CONTENTS 1. Learning outcomes 2. Hückel Molecular Orbital (HMO) Theory 3. Application of HMO theory 3.1 Butadiene 4. Summary

3 1. Learning Outcomes After studying this module, you shall be able to Find the π-electron energy and wave function for butadiene molecule Find the resonance energy, electronic charge and bond order for butadiene Understand the basis of molecular orbital diagram for π-electron systems 2. Hückel Molecular orbital theory HMO theory is an approximate method which simplifies variation method to treat planar conjugated hydrocarbons. This theory treats the π electrons separately from σ electrons. Properties of the conjugated molecules are primarily determined by π-electrons. The consideration of σ-π electron separation in a multi-electron molecule in HMO theory reduces the problem to the study of only π electrons.hmo calculations are carried out using variation method and LCAO(π)-MO approximation. According to LCAO-MO approximation, the MO is written as, φ a = n i=1 c i ψ 2pz i -(1) For a planar conjugated hydrocarbon, the only atomic orbitals of π symmetry are the 2pπ orbtials on carbon. In this module, we have consistently assumed the plane of the molecule as x-y plane with π orbital in the z axis, perpendicular to the molecular plane. And the approximate energy is given by, E a = φ a H φ a dτ φ a φ a dτ -(2) The Hamiltonian Ĥ incorporates the effect of the interaction of π electron with the rest of the molecule (nuclei, inner electrons, σ bonds) in an average way. In HMO method, π

4 electrons are assumed to be moving in a potential generated by the nuclei and σ electrons of the molecule. The Secular determinant for the two π electron system is of the form, [ H 11 ES 11 H 12 ES 12 H 21 ES 21 H 22 ES 22 ] [ c 1 c 2 ] = 0 -(3) To solve the Secular determinant for an n-π electron system, Hückel treated the Hii, Hij, Sij and Sij integrals as parameters that can be evaluated empirically by fitting the theory to experimental results. 1. H ij = ψ i H ψ j dτ α (i = j) H ij = { β (i j) Coulomb integral Resonance integral 2. S ij = ψ i ψ j dτ 1 (i = j) S ij = { 0 (i j) Overlap integral Taking into account the assumptions of HMO theory, the secular determinant reduces to, α-e β β α-e = 0 -(4) In this manner, Hückel determinant can be generated for an n-π electron system The expansion of an n x nhückel determinant yields a polynomial equation which has n real roots giving n energy levels and n molecular orbitals for then-π electron system. The energy of any a th molecular orbital (MO) is given by E a = α + x a β, where x a is the a th root of the polynomial -(5)

5 3. Application of HMO theory In this section, we shall apply HMO theory to simple conjugated molecule viz., butadiene. 3.1 Butadiene We consider here the case of 1,3-butadiene. 1,3-butadiene molecule is a linear sequence of four carbon atoms where each carbon 2pz orbital contributes 1 electron to the HMO π-electron system. Or simply, one can say that HMO theory treats 1,3-butadieneas a four π-electron system. Note: 1,3-butadiene molecule exists in trans and cis configuration but HMO theory does not disingush between the two and consider butadiene as a linear combination of four carbon 2pz orbitals respectively where lies one π electron in each pz orbital perpendicular to the molecular plane. The four atomic orbitals (AOs) combine to form molecular orbitals (MOs). Labeling the four carbons as 1, 2, 3 and 4,

6 φ a = c 1 ψ 2pz1 + c 2 ψ 2pz2 + c 3 ψ 2pz3 + c 4 ψ 2pz4 -(6) The Secular equations obtained for butadiene molecule are of the form: (H 11 ES 11 )c 1 + (H 12 ES 12 )c 2 + (H 13 ES 13 )c 3 + (H 14 ES 14 )c 4 = 0 (H 21 ES 21 )c 1 + (H 22 ES 22 )c 2 + (H 23 ES 23 )c 3 + (H 24 ES 24 )c 4 = 0 (H 31 ES 31 )c 1 + (H 32 ES 32 )c 2 + (H 33 ES 33 )c 3 + (H 34 ES 34 )c 4 = 0 (H 41 ES 41 )c 1 + (H 42 ES 42 )c 2 + (H 43 ES 43 )c 3 + (H 44 ES 44 )c 4 = 0 which can be written in the form of secular determinant as, H 11 -ES 11 H 12 -ES 12 H 13 -ES 13 H 14 -ES 14 c 1 c 2 H 21 -ES 21 H 22 -ES 22 H 23 -ES 23 H 24 -ES 24 [ H 31 -ES 31 H 32 -ES 32 H 33 -ES 33 H 34 -ES c ] = c 4 [ H 41 -ES 41 H 42 -ES 42 H 43 -ES 43 H 44 -ES 44 ] -(7) Taking into account the assumptions of HMO theory, the secular determinant transforms into Hückel determinant as, H 11 = H 22 = H 33 = H 44 = α H 12 = H 21 = H 23 = H 32 = H 34 = H 43 = β H 13 = H 31 = H 14 = H 41 = 0 S 11 = S 22 = S 33 = S 44 = 1 S 12 = S 21 = S 13 = S 31 = S 14 = S 41 = S 23 = S 32 = S 24 = S 42 = S 34 = S 43 = 0

7 c 1 α E β 0 0 β α E β 0 c 2 [ ] [ 0 β α E β c ] = β α E c 4 c 1 c 2 c 3 α E β 0 0 β α E β 0 [ ] 0 [ ] = 0 0 β α E β c β α E α E β 0 0 β α E β 0 = 0 0 β α E β 0 0 β α E -(8) Let, λ = α E β This reduces the Hückel determinant as, λ λ 1 0 = λ λ -(9) which on expansion gives, λ λ 1 λ λ 1 = λ 0 1 λ λ[λ(λ 2 1) 1(λ) + 0] 1(λ 2 1) = 0 which gives, λ 4 3λ = 0 Let λ 2 = z -(10)

8 z 2 3z = 0 z = 3 ± λ 2 = z = 3 ± 5 2 λ = ± ( 3 ± ) 2 λ = ± ( ) 2 = ±1.618 ; ± ( ) 2 = ± We assumed earlier λ = α E β -(11) So, the energies of the molecular orbitals of butadiene are obtained as, If λ = , E = α β (Bonding Molecular Orbital BMO) If λ = 1.618, E = α 1.618β (Antibonding Molecular Orbital ABMO) If λ = , E = α β (Bonding Molecular Orbital BMO) If λ = 0.618, E = α 0.618β (Antibonding Molecular Orbital ABMO) The number of molecular orbitals that are generated using LCAO approximation are equal to the number of combining atomic orbitals. The Hückel energy level diagram for butadiene is shown below:

9 HMO energy level diagram for butadiene Total (π bond)energy = 2(α β) + 2(α β) -(12) Total (π bond)energy = 4α β Using λ asλ = α E β the secular equations are obtained as c 1 λ λ 1 0 c 2 [ ] [ 0 1 λ 1 c ] = λ c 4 -(13)

10 λc 1 + c = 0 c 1 + λc 2 + c = c 2 + λc 3 + c 4 = 0 -(14) c 3 + λc 4 = 0 If λ = c 2 = 1.618c 1 ; c 3 = c 1 ; c 4 = c 1 If λ = c 2 = 0.618c 1 ; c 3 = 0.618c 1 ; c 4 = c 1 If λ = c 2 = 0.618c 1 ; c 3 = 0.618c 1 ; c 4 = c 1 If λ = c 2 = 1.618c 1 ; c 3 = 1.618c 1 ; c 4 = c 1 -(15) -(16) -(17) -(18) Now, we know that the sum of squares of coefficients is always unity, c c c c 4 2 = 1 -(19) Putting equations (15, 16, 17 & 18) respectively in equation (19) gives the value of c1 and so forth the values of all the other coefficients in each respective case. λ = c 1 = 0.372; c 2 = 0.602; c 3 = 0.602; c 4 = λ = c 1 = 0.602; c 2 = 0.372; c 3 = 0.372; c 4 = λ = c 1 = 0.602; c 2 = 0.372; c 3 = 0.372; c 4 = λ = c 1 = 0.372; c 2 = 0.602; c 3 = 0.602; c 4 = Substituting the values of coefficients in equation (6) gives the four molecular orbitals (2 BMOs & 2 ABMOs) of butadiene, φ 1 = 0.372ψ 2pz ψ 2pz ψ 2pz ψ 2pz4

11 φ 2 = 0.602ψ 2pz ψ 2pz ψ 2pz ψ 2pz4 φ 3 = 0.602ψ 2pz ψ 2pz ψ 2pz ψ 2pz4 φ 4 = 0.372ψ 2pz ψ 2pz ψ 2pz ψ 2pz4 The pictorial representation of the four Hückel molecular orbitals viz., 2 BMOs and 2 ABMOs for butadiene is shown below. Nodes (or nodal planes) are regions where the probability of finding electrons is zero. The ground state of butadiene has no nodes. The number of nodes in an MO increases with increase in energy. Thus φ 2 has one node (shown by a dot in the figure), φ 3 has two nodes and φ 4 has three nodes. The total π electron energy (or π electron binding energy) Eπ is taken as the sum of the energies corresponding to each π electron. For butadiene, the total π electron energy Eπ is given by E π = 4α β -(20)

12 The energy of two π electrons in ethylene is E ethylene = 2α + 2β Resonance energy is defined as the difference in the energy of π electrons in a molecule and the sum of energies of isolated double bond E π(resonance energy) = E π E ethylene = 4α β 2(2α + 2β) = 0.472β Resonance energy is the measure of the stability of a molecule. Hence butadiene molecule is more stable than two ethylene molecules by a factor of 0.472β. Another related term is π bond formation energy which is the energy released when a π bond is formed. Since the contribution of α is same in the molecules as in the atoms,

13 so we can consider the energy of four electrons, each one in isolated and noninteracting atomic orbitals as 4α, then the π bond formation energy becomes, E π(bond formation) = E π E isolated -(21) E π(bond formation) = 4α β 4α = 4.472β 4.472β is the total π bonding energy of formation of the butadiene molecule. Total π-electronic charge on n th carbon atom is given by For butadiene, 2 q n = n i c in, n i is the number of electrons in ith MO i q 1 = 2c c = 2(0.372) 2 + 2(0.602) 2 = 1 The value unity indicates that the π electrons in butadiene are uniformly distributed over the molecule. One can try and find out q2, q3 and q4 for which the value will come out to be unity. Π-bond order between adjacent carbon atoms is given by BO π ab = n i c ia c ib i where n i is the number of π electrons in ith MO c ia c ib is the π electorn charge in ith MO between adjacent carbon atoms a and b For butadiene, BO π 12 = 2c 11 c c 21 c c 31 c c 41 c 42 BO π 12 = 2. (0.372). (0.602) + 2. (0.602). (0.372) BO π 12 = BO 12 π = BO π 34, by symmetry

14 BO π 23 = 2c 12 c c 22 c 23 BO π 23 = 2. (0.602). (0.602) + 2. (0.372). ( 0.372) BO π 23 = There is a σ bond between each carbon atom which is taken into account while reporting the total bond order. The total bond order is given by BO total π ab = 1 + BO ab For butadiene, total bond order comes out to be, BO total 12 = BO total 34 = BO total 23 = The bond order values for butadiene are in very good agreement with the experimental results.

15 4. Summary HMO theory is an approximate method which simplifies variation method to treat planar conjugated hydrocarbons. Properties of the conjugated molecules are primarily determined by π-electrons. HMO calculations are carried out using variation method and LCAO(π)-MO approximation Application of HMO theory to butadiene molecule 1,3-butadiene is a four π-electron system E = α ± 1.618β; α ± 0.618β φ 1 = 0.372ψ 2pz ψ 2pz ψ 2pz ψ 2pz4 φ 2 = 0.602ψ 2pz ψ 2pz ψ 2pz ψ 2pz4 φ 3 = 0.602ψ 2pz ψ 2pz ψ 2pz ψ 2pz4 φ 4 = 0.372ψ 2pz ψ 2pz ψ 2pz ψ 2pz4 Total (π bond)energy = 4α β E π(resonance energy) = 0.472β E π(bond formation) = 4.472β Total π-electronic charge on n th carbon atom of butadiene molecule is unity which indicates that the π electrons in butadiene are uniformly distributed over the molecule. Π-bond order between adjacent carbon atoms BO π 12 = BO π 34 = BO π 23 = 0.448