PAPER:2, PHYSICAL CHEMISTRY-I QUANTUM CHEMISTRY. Module No. 34. Hückel Molecular orbital Theory Application PART IV

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1 Subject PHYSICAL Paper No and Title TOPIC Sub-Topic (if any), PHYSICAL -II QUANTUM Hückel Molecular orbital Theory Module No. 34

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

3 1. Learning Outcomes After studying this module, you shall be able to Find the π-electron energy and wavefunction for cyclobutadiene Understand the basis of molecular orbital diagram for π-electron systems. Hückel Molecular orbital theory HMO theory is an approximate theory that gives a quick picture of the molecular orbital energy diagram of organic conjugated molecules. The Hückel theory treats only π- electrons in a planar conjugated molecule. HMO calculations are carried out using variation method and LCAO(π)-MO approximation. The basis set for MO approximation consists of one pπ-orbital on each atom. The σ skeleton of the conjugated molecule is assumed frozen and no σ-π interactions are considered in this study. In addition, electronelectron repulsions are neglected in π-hamiltonian of conjugated molecule. Note: 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. According to LCAO-MO approximation, the MO is written as, φ a = n i=1 c i ψ pz i -(1) And the approximate energy is given by,

4 E a = φ a * Ĥφ a dτ φ a * φ a dτ E o -() A trial function that depends linearly on the variational parameters leads to a secular determinant which gives secular equation as an approximation to the energy. H 11 ES 11. H 1n ES 1n... = 0 H n1 ES n1. H nn ES nn 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 -(3) -(3). S ij = ψ i ψ j dτ 1 (i = j) S ij = { 0 (i j) Overlap integral -(4) The expansion of an n x n Hückel determinant yields a polynomial equation which has n real roots which n-π electron system has n energy levels and n molecular orbitals. 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) 3. Application of HMO theory In this section, we shall apply HMO theory to a cyclic conjugated molecule viz., cyclobutadiene.

5 3.1 Cyclobutadiene Now, we will consider the treatment of cyclobutadiene molecule using HMO theory in detail. Cyclobutadiene is a 4 π-electron cyclic system where the carbon atoms are adjacent with each carbon pz orbital contributes 1 electron to the HMO π-electron system. The four atomic orbitals (AOs) combine to form molecular orbitals (MOs). Labeling the four carbons as 1,, 3 and 4, The Hückel molecular orbital wavefunction for this system becomes φ a = c 1 ψ pz1 + c ψ pz + c 3 ψ pz3 + c 4 ψ pz4 -(6) The Secular equations obtained for cyclobutadiene molecule are of the form: (H 11 ES 11 )c 1 + (H 1 ES 1 )c + (H 13 ES 13 )c 3 + (H 14 ES 14 )c 4 = 0 (H 1 ES 1 )c 1 + (H ES )c + (H 3 ES 3 )c 3 + (H 4 ES 4 )c 4 = 0

6 (H 31 ES 31 )c 1 + (H 3 ES 3 )c + (H 33 ES 33 )c 3 + (H 34 ES 34 )c 4 = 0 (H 41 ES 41 )c 1 + (H 4 ES 4 )c + (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 1 ES 1 H 13 ES 13 H 14 ES 14 c 1 H [ 1 ES 1 H ES H 3 ES 3 H 4 ES 4 c ] [ H 31 ES 31 H 3 ES 3 H 33 ES 33 H 34 ES 34 c ] = 0 3 H 41 ES 41 H 4 ES 4 H 43 ES 43 H 44 ES 44 c 4 -(7) In this case, carbon atom 1 is connected to carbon atom 4, i.e., the C1 and C4 are neighbors. Taking into account the assumptions of HMO theory, the secular determinant transforms into Hückel determinant as, H 11 = H = H 33 = H 44 = α H 1 = H 1 = H 3 = H 3 = H 34 = H 43 = H 14 = H 41 = β H 13 = H 31 = H 4 = H 4 = 0 S 11 = S = S 33 = S 44 = 1 S 1 = S 1 = S 13 = S 31 = S 14 = S 41 = S 3 = S 3 = S 4 = S 4 = S 34 = S 43 = 0 c 1 α E β 0 β β α E β 0 c [ ] [ 0 β α E β c ] = 0 3 β 0 β α E c 4 -(8) c 1 c c 3 α E β 0 β β α E β 0 [ ] 0 [ ] = 0 0 β α E β c 4 β 0 β α E

7 α E β 0 β β α E β 0 = 0 0 β α E β β 0 β α E -(9) Let, λ = α E β This reduces the Hückel determinant as, -(10) λ λ 1 0 = λ λ -(11) The Hückel determinant leads to a polynomial equation, x 4 4x = 0 Thus, we have the roots of the polynomial equation as, λ 1 = ; λ = 0; λ 3 = 0; λ 4 = -(1) -(13) So, the energies of the molecular orbitals of cyclobutadiene are of the form, λ 1 = E 1 = α + β λ = λ 3 = 0 E = E 3 = α λ 4 = E 4 = α β -(14) -(15) -(16) The two energy levels viz., E and E3 are degenerate (doubly degenerate non-bonding levels). The Hückel energy level diagram for cyclobutadiene is shown below:

8 Total π energy (E π ) = (α + β) + α = 4α + 4β E D.E. = E π E ethylene = 4α + 4β 4α 4β = 0 -(17) -(18) Delocalization energy for cyclobutadiene is zero which indicates that there is no additional stability in the molecule by delocalization. (The ring is small and is under strain which is not compensated by delocalization making it difficult to synthesize) Now, we solve for HMO coefficients, In terms of λ which is taken as λ = α E, the secular equations for cyclobutadiene are as follows, β c 1 λ λ 1 0 c [ ] [ 0 1 λ 1 c ] = λ c 4 -(19)

9 λc 1 + c c 4 = 0 c 1 + λc + c = c + λc 3 + c 4 = 0 -(0) c c 3 + λc 4 = 0 We now solve the above set of equations by the method of elimination. For, λ =, the secular equations become c 1 + c c 4 = 0 c 1 c + c = c c 3 + c 4 = 0 c c 3 c 4 = 0 -(1) -() -(3) -(4) Subtracting equation (3) from equation (1) gives, c 1 + c 3 = 0 -(5) c 1 = c 3 -(6) Multiplying equation (4) by and then subtracting it from equation (5) gives, 4c 1 + 4c 4 = 0 -(7) c 1 = c 4 -(8)

10 Now, one can eliminate c4 from equation (1) by multiplying it by 4 and then subtracting equation (7) from it, 4c 1 + 4c = 0 c 1 = c -(9) -(30) The sum of squares of coefficients is always unity, i.e., from normalization condition we have, c 1 + c + c 3 + c 4 = 1 -(31) Now, using equations (6), (8) and (30), we get normalization condition for λ = as, 4c 1 = 1 or c 1 = c = c 3 = c 4 = 1 = 1 4 -(3) So, we get the wavefunction corresponding to λ = as, φ 1 = 1 (ψ p z1 + ψ pz + ψ pz3 + ψ pz4 ) -(33) For, λ = 0 (corresponding to degenerate energy levels viz., E and E3), the secular equations become 0c 1 + c c 4 = 0 c 1 + 0c + c = c + 0c 3 + c 4 = 0 -(34) c c 3 + 0c 4 = 0 The value of the coefficients corresponding to λ = 0 cannot be determined using the above set of equations alone. We know that the energy levels corresponding to λ =

11 0 (E and E 3 ) are degenerate. And in case of degenerate orbitals, HMO method cannot determine the coefficients uniquely. One can choose any value for c1, c, c3 and c4 provided that they satisfy the conditions of normalization and orthogonality as well as the values of c1, c, c3 and c4 must satisfy the above set of equations. A simple method to satisfy the above mentioned three conditions is to set any one of the coefficients equal to zero. Let c = 0, then we have from equation (34), c 4 = 0 c 1 = c 3 -(35) From normalization condition, c 1 + c + c 3 + c 4 = 1 c 1 + c 3 = 1 c 1 = 1 c 1 = 1 -(36) c 3 = 1 -(37) So, we get the wavefunction corresponding to λ = 0 as, φ = 1 (ψ pz1 ψ pz3 ) -(38) We arbitrarily chose c = 0 or in other words φ but we cannot repeat the same process for determining φ3. This is because φ3 must be orthogonal to φ1, φ and φ4. φ 1 φ 3 dτ = 0 -(39)

12 φ φ 3 dτ = 0 -(40) φ 3 φ 4 dτ = 0 -(41) Note: We can choose any equation (from 39, 40 & 41) for finding φ3 as HMO method cannot determine the coefficients uniquely. For instance, taking equation (40), we get [ 1 (ψ pz1 ψ pz3 )] [c 1 ψ pz1 + c ψ pz + c 3 ψ pz3 + c 4 ψ pz4 ]dτ = 0 -(4) On expanding the equation (4) we get, [ 1 c 1 ψ pz1 + 1 c ψ pz1 ψ pz + 1 c ψ pz1 ψ pz3 + 1 c ψ pz1 ψ pz4 1 c 1 ψ pz1 ψ pz3 1 c ψ pz ψ pz3 1 c 3 ψ pz3 1 c 4 ψ pz4 ψ pz3 ] dτ = 0 [ 1 c 1 ψ pz1 1 c 3 ψ pz3 ] dτ = 0 1 c 1 1 c 3 = 0

13 c 1 = c 3 -(43) But c 1 + c 3 = 0, from equation (34) c 1 = c 3 = 0 Since c + c 4 = 0 c = c 4 From normalization condition, c 1 + c + c 3 + c 4 = 1 c + c 4 = 1 c = 1 c = 1 -(44) c 4 = 1 -(45) So, we get the wavefunction corresponding to λ = 0 (degenerate with φ) as, φ 3 = 1 (ψ pz ψ pz4 ) -(46) For, λ =, the secular equations become c 1 + c c 4 = 0 c 1 + c + c = c + c 3 + c 4 = 0 -(47) -(48) -(49) c c 3 + c 4 = 0 -(50)

14 Subtracting equation (49) from equation (47) gives, c 1 c 3 = 0 -(51) c 1 = c 3 -(5) Multiplying equation (50) by and then adding it to equation (51) gives, 4c 1 + 4c 4 = 0 -(53) c 1 = c 4 -(54) Now, one can eliminate c4 from equation (47) by multiplying it by 4 and then subtracting equation (53) from it, 4c 1 + 4c = 0 c 1 = c -(55) -(56) From normalization condition, c 1 + c + c 3 + c 4 = 1 Now, using equations (5), (54) and (56), we get normalization condition for λ = as, 4c 1 = 1 or c 1 = 1 = 1 4 -(57) c 3 = 1 c = c 4 = 1 -(58) -(59) So, we get the wavefunction corresponding to λ = as,

15 φ 4 = 1 (ψ p z1 ψ pz + ψ pz3 ψ pz4 ) -(60) The pictorial representation of the four Hückel molecular orbitals for cyclobutadiene is shown below: Electron density: The total electron density is taken as the sum of electron densities contributed by different electron in each HMO q n = n i c in i n i is the number of electrons in ith HMO (0,1 or ) In case of cyclobutadiene, two π-electrons are in energy state E1 while the third and fourth electrons singly occupy degenerate energy states E and E3. q 1 = ( 1 ) + 1 ( 1 ) ( 1 ) = 1

16 q = ( 1 ) ( 1 ) + 0 ( 1 ) = 1 q 3 = ( 1 ) + 1 ( 1 ) ( 1 ) = 1 q 4 = ( 1 ) ( 1 ) + 0 ( 1 ) = 1 Charge density In a conjugated molecule, a neutral carbon is associated with an electron density of 1.0 and the net charge density is defined as ε n = 1 q n For cyclobutadiene, ε 1 = 1 1 = 0 = ε = ε 3 = ε 4 Π-bond order between adjacent carbon atoms is given by BO π ab = n i c ia c ib 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 i For cyclobutadiene, BO π 1 = ( 1 1 ) + (1 1 0) + (1 0 1 ) BO π 1 = 0.5 BO π 3 = ( 1 1 ) + (1 0 1 ) + (1 1 0) BO π 3 = 0.5

17 BO π 34 = ( 1 1 ) + (1 1 0) + (1 0 1 ) BO π 31 = 0.5 BO π 41 = ( 1 1 ) + (1 0 1 ) + (1 1 0) BO π 41 = 0.5 This implies that all the bonds in cyclobutadiene are neither single bonds nor double bonds. All the bonds have character intermediate between single and double bonds. 4. Summary HMO theory is an approximate theory that gives a quick picture of the molecular orbital energy diagram of organic conjugated molecules. The Hückel theory treats only π- electrons in a planar conjugated molecule. HMO calculations are carried out using variation method and LCAO(π)-MO approximation. The basis set for MO approximation consists of one pπ-orbital on each atom. Application of HMO theory to cyclobutadiene Cyclobutadiene is a four electron system E 1 = α + β E = E 3 = α E 4 = α β The two energy levels viz., E and E3 are degenerate. E π = 4α + 4β E D.E. = 0 Delocalization energy for cyclobutadiene is zero which indicates that there is no additional stability in the molecule by delocalization. (The ring is small and is under strain which is not compensated by delocalization making it difficult to synthesize)

18 φ 1 = 1 (ψ p z1 + ψ pz + ψ pz3 + ψ pz4 ) φ = 1 (ψ pz1 ψ pz3 ) φ 3 = 1 (ψ pz ψ pz4 ) φ 4 = 1 (ψ p z1 ψ pz + ψ pz3 ψ pz4 ) All the bonds in cyclobutadiene are neither single bonds nor double bonds. All the bonds have character intermediate between single and double bonds.