subject physical chemistry topic quantum chemistry...

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CHEMISTRY

PAPER:2, PHYSICAL CHEMISTRY-I

MODULE: 31, Hückel Molecular orbital Theory – Application PART I

Subject PHYSICAL CHEMISTRY

Paper No and Title 2, PHYSICAL CHEMISTRY-II

TOPIC QUANTUM CHEMISTRY

Sub-Topic (if any) Hückel Molecular orbital Theory – Application PART I

Module No. CHE_P2_M31

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TABLE OF CONTENTS

1. Learning outcomes

2. Hückel Molecular Orbital (HMO) Theory

3. Application of HMO theory

3.1 Ethylene

4. Summary

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1. Learning Outcomes

After studying this module, you shall be able to

Appreciate the simplification introduced by E Hückel for studying organic conjugated

molecules.

Find the π-electron energy and wave function for ethylene molecule

Understand the basis of molecular orbital diagram for π-electron systems

2. Hückel Molecular orbital (HMO) 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 electro 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,

n

i

ipia zc

1

2

HMO theory approximates the π molecular orbitals as linear combination of atomic

orbitals. 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.

For a two π electron system φa becomes,

-(1)

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2pz22c

1pz21c

a

And the approximate energy is given by,

d

dHE

aa

aa

a *

^*

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

electrons are assumed to be moving in a potential generated by the nuclei and σ electrons

of the molecule.

The Secular determinant obtained for two π electron system can be written as,

02

1

22222121

12121111

c

c

ESHESH

ESHESH

In order 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. dHH jpipij zz 2

^*

2

)(

)(

ji

jiH ij

2. dS jpipij zz 2

*

2

)(0

)(1

ji

jiSij

Taking into account the assumptions of HMO theory, the secular determinant reduces to,

Coulomb integral

Resonance integral

Overlap integral

-(4)

-(2)

-(3)

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In this manner, Hückel determinant can be generated for an n-π electron system

The expansion of an n x n Hückel determinant yields a polynomial equation that has n

real roots for n-π electron system leading to n energy levels and n molecular orbitals.

The energy of any ath molecular orbital (MO) is given by

aa xE , where xais the ath root of the polynomial.

The values of the coulomb integral α and the resonance integral β are always negative. If

the root xa is positive, then the energy level corresponds to a more negative value and is

more stable (Bonding molecular orbital) while a negative value of root gives antibonding

molecular orbital.

3. Application of HMO theory

In this section, we shall apply HMO theory to ethylene having 2 π electrons with one

double bond.

3.1 Ethylene

We consider here the case of ethylene, C2H4.

Ethylene is a 16 electron system but HMO theory reduces this to a two π electron system.

0

E

E

-(6)

-(5)

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HMO theory treats ethylene as a two electron problem, with one π electron on each

carbon atom in p-orbital, perpendicular to the molecular plane. These two atomic orbitals

(AOs) combine to form molecular orbitals (MOs).

Labeling the two carbons as 1 and 2,

The Secular determinant obtained for ethylene molecule is of the form,

222121 pzpza cc

02

1

22222121

12121111

c

c

ESHESH

ESHESH

Taking into account the assumptions of HMO theory, the secular determinant transforms

into Hückel determinant as,

01 21122211

21122211

SSSS

HHHH

02

1

21

121

c

c

E

E

-(7)

-(8)

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002

1

E

E

c

c

Let,

E

This reduces the Hückel determinant as,

which on expansion gives,

1

012

So, the energies of the molecular orbitals are,

)(,1

)(,1

ABMOOrbitalMoleculargAntibondinEIf

BMOOrbitalMolecularBondingEIf

0

E

E

01

1

The number of molecular orbitals that are generated using LCAO approximation are equal to

the number of combining atomic orbitals.

-(9)

-(10)

-(11)

-(12)

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HMO energy level diagram for ethylene

Total (π bond) energy = 2(α + β)

[As there are two electrons in the orbital with energy α+β]

Using λ as

E

,

the secular equations are obtained as

01

1

2

1

c

c

0

0

21

21

cc

cc

-(13)

-(14)

-(15)

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21

21

,1

,1

ccIf

ccIf

ccc 21

Now, if we apply the normalization condition,

1* daa

1)( 2

222121 dcc pzpz

1)( 2

2212

2 dc pzpz

1)2( 2212

2

22

2

12

2 dc pzpzpzpz

1]011[2 c

2

1c

Molecular

orbital

λ E c1 c2 Number of

nodes

BMO -1 α+β

2

1c

2

1c

0

ABMO 1 α-β

2

1c

2

1c

1

If i = j, ψi2 = 1

If i ≠ j, ψi2 = 0

The sum of the squares of the coefficients is always unity.

-(16)

-(17)

-(18)

-(19)

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With this, one can now write the two normalized wavefunctions corresponding to two

Hückel molecular orbitals for ethylene as,

)(2

12212 pzpzBMO

)(2

12212 pzpzABMO

The pictorial representation of the two Hückel molecular orbitals viz., BMO and ABMO

for ethylene is shown below.

-(20)

-(21)

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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 ethylene, the total π electron energy Eπ is

given by

22 E

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, so

we can consider the energy of two electrons, each one in isolated and non-interacting

atomic orbitals as 2α, then the π bond formation energy becomes,

isolatedformationbond EEE )(

In general,

moleculetheinatomsCofnumbernwherenEE formationbond ,)(

For ethylene,

2222)( formationbondE

2β is the total π bonding energy on formation of the ethylene molecule.

-(22)

-(23)

-(24)

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4. Summary

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.

HMO calculations are carried out using variation method and LCAO(π)-MO

approximation

Application of HMO theory to ethylene molecule

Ethylene is a 16 electron system but HMO theory reduces this to a two π electron

system.

)(2

12212 pzpzBMO

)(2

12212 pzpzABMO

22 E

Molecular

orbital

λ E c1 c2 Number of

nodes

BMO -1 α+β

2

1c

2

1c

0

ABMO 1 α-β

2

1c

2

1c

1

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