Chem 373- Lecture 25: Introduction to Molecular Orbital Theory

8/3/2019 Chem 373- Lecture 25: Introduction to Molecular Orbital Theory

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Lecture 25: Introduction to Molecular Orbital

Theory

The material in this lecture covers the following in Atkins.

14 Molecular structure

Molecular Orbital Theory

(a) Linear combinations of atomic orbitals

(b) bonding orbitals

(c) anti-bonding orbitals

Lecture on-line

Introduction to Molecular Orbital theory (PowerPoint)Introduction to Molecular Orbital Theory (PDF)

Handout for this lecture


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Molecular Orbital Theory H2+

We shall now discuss ways toapproximately solve :

H r R r R E (R r Re e N e N e N e N( , ) ( , ) ) ( , ) =

and represent the many - electronwave - function r Re N( , )

Here

H = T V + V + Ve e Ne ee NN +

We shall this time use :


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1 = [A(1) A(2)B B( ) ( )]

[ ( ) ( ) ( ) ( )]

2 1

1 2 1 2

+

BA

In valence bond theory we localizedone electron to each atom

el 1. el 2.

or BA

In molecular orbital theory each electron movesover the whole molecule

BA

el 1.el 2.

BAand

el 2.el 1.

The two theories are limiting cases (see later) of a more exact theory

Both electrons can be on the same nuclei


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We used the orbitals of theone - electron hydrogen to build up

wavefunctions for many - electron

atoms

We shall use the orbitals of

the one - electron H molecule

to describe many electron

diatomic molecules

2+


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The

molecule

hamiltonian for the H2+

:is given by

H =

h

22

2me e

Kinetic energy of electron

e

ro A

2

14

1

Attraction of el. 1 by A

e

ro B

2

14

1

Attraction of el. 1 by B

+e

Ro

2

4

1

rA1

A B

1

rB1

R

Repulsion between A and B


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We R R E R Rcould now solve : H(r (r (r1 1 1

, ) , ) ( ) , ) =With

Ve

r r Ro A B

= + 2

1 14

1 1 1

[

Where

H

m

V r R

e

e

= +h

22

1

2

( , )

To this end we write (r as a linear combinationof atomic orbitals (LCAO)

1 , )R

This is possible but tedious

We shall instead find approximate soluions

The atomic orbitals are in general those centered on

the atoms of our molecule

Was first done by a Danish astronomer in 1926


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1 1 1 3 3 2

1

s r A r ea

A A A

r

o

A ao( ) ( )

( )

/

/= =


For H that is :2+

1 1 1 3 3 2

1

s r A r ea

A B B

r

o

B ao

( ) ( )( )

/

/= =

Where r and r are related by :1A 1B

r r R r RB A A1 1 2 2 12= + cos

rA1

A B

1

rB1

R

We now assume the electron to be equally

likely to be on each nuclei and write the

molecular orbitals as :+ = +( ) ( ) ( )1 1 1A B = ( ) ( ) ( )1 1 1A B

Note : we have as many linear combinations

as we have atomic orbitals


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+

= +( ) ( ) ( )1 1 1A B

= ( ) ( ) ( )1 1 1A B

We can normalize our orbitals for easy energy

calculation and use in probability density

+ + = + + =( ) ( ) ( ( ) ( ))( ( ) ( ))1 1 1 1 1 1 12dv N A B A B dv

+ + = + + =

( ) ( ) ( ( ) ( ) ( ( ) ( )

( ( ) ( )

1 1 1 1 1 1

2 1 1 1

2 2

2

dv N A A dv N B B dv

N A B dv

A normalized B normalized overlap S > 1

+ + +

( ) ( )1 1 1 2dv N= +1 N

+2S N = 1

2

2

NS= +

12 1( )


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+ = +( ) ( ) ( )1 1 1A B = ( ) ( ) ( )1 1 1A B

After normalization

+

=+

+( )( )

[ ( ) ( )]11

2 11 1

SA B

=

( )( )

[ ( ) ( )]11

2 11 1

SA B

The electron probability density :

+ + = + +( ) ( ) ( ) [ ( ) ( )]1 1

1

2 1 1 1

2

S A B

=+

++

++

1

2 11

1

2 11 2

1

2 112 2

( )( )

( )( )

( )( )

SA

SB

SAB

Density at A

(Reduced)

Density at B

(Reduced)

Density between A and B

(Increased)


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(a)The amplitude of

the bonding molecularorbital in a hydrogen

molecule-ion in a plane

containing the two nuclei and

(b) a contour

representation

of the amplitude.


+ =

+

+( )

( )

[ ( ) ( )]11

2 1

1 1

S

A B


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A representation of theconstructive interference

that occurs when two H 1s

orbitals overlap and forma bonding orbital.

Compare this illustration

with Fig.14.14.

+ =

+

+( )

( )

[ ( ) ( )]11

2 1

1 1

S

A B


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The boundary surfaceof a orbital enclosesthe region where the

electrons that occupy

the orbital are

most likely to be found.

Note that the orbital has

cylindrical symmetry.

( ) ( ) ( ) ( ) ( ) ( ) ( )11

2 1 1

1

2 1 1 2

1

2 1 1

2 2

= + + + + +S A S B S AB


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The electron density calculated

by forming the square of the

wavefunction used

to construct Fig.14.14.Note the accumulation

of electron density in the

internuclear region.

( )( )

( )( )

( )( )

( )11

2 11

1

2 11 2

1

2 112 2=

++

++

+SA

SB

SAB


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Hm

er r Re

eo A B

= + h2 2 21 12 4

1 1 1

[ ] + = + +( )

( )[ ( ) ( )]1 1

2 11 1

SA B

We H dvcan now evaluate the energy : E =+ + + ( ) ( ) ( )1 1 1

E =+1

2 1

1 1 1 1 1

( )

[ ( ) ( )] ( )[ ( ) ( )]

+

+ +

S

A B H A B dv

=1

2 11

2 4

11

22

2

1( )] ( )[ ] ( )

+

SA

m

e

rA dv

ee

o A

h

+ 12 1

12 4

1 12

22

1( )] ( )[ ] ( )

+ SB

me

rB dv

ee

o Bh

+1

2 1

1

4

11

2

1( )

] ( )[ ] ( )

+

S

Ae

r

A dv

o B+

1

2 11

4

11

2

1( )] ( )[ ] ( )

+

SB

e

rB dv

o A

+ 22 1

12 4

1 1 14

1

2

2

2

1 1

2

( )] ( )[ [ ]] ( )

+ + +SA

me

r rB dv e

Ree

o A B o

h

rA1

A B

1

rB1

R


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E =+

1

2 11 1 1 1 1

( )[ ( ) ( )] ( )[ ( ) ( )]

++

+

SA B H A B dv

=1

2 11

2 4

11

22

2

1( )] ( )[ ] ( )

+

SA

m

e

rA dv

ee

o A

h

+ 12 1

12 4

1 12

22

1( )] ( )[ ] ( )

+

SB

me

rB dv

ee

o B

h

+1

2 1

1

4

11

2

1( )

] ( )[ ] ( )

+

S

Ae

r

A dv

o B+

1

2 11

4

11

2

1( )] ( )[ ] ( )

+

SB

e

rB dv

o A

+ 22 1

12 4

1 1 12 2 2

1 1( )] ( )[ [ ]] ( )

+ +

SA

me

r rB dv

ee

o A B

h

rA1

1

rB1

+ e Ro

2

41

E

2(1+ S)1sH

E2(1+ S)

1sH

J

2(1+ S)J

2(1+ S)

K'(1+S)


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- =J

S SA A

e

rdv

o B2 1

1

2 11 1

4

12

1( ) ( )] ( ) ( )[ ]

+ +

- J =2 1

12 1

1 14

12

1( ) ( )] ( ) ( [ ])

+ +

S SB B e

rdv

o A

A B

rB1

R

1

2(1+ S)] A(1)A(1)dv

Interaction (attraction) between nucleus B

and the charge cloud A(1)A(1)/2 1( )+ S

Interaction (attraction) between nucleus A

and the charge cloud B(1)B(1)/2 1( )+ S


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K' =2

2 11

2 4

1 11

22

2

1 1( )] ( )[ [ ]] ( )

+ +

SA

m

e

r rB dv

ee

o A B

h

rA1

A B

1

rB1

R

=2

2 11

2 4

11

2

2 11 1

4

1

22

2

1

2

1

( )] ( )[ ]] ( )

( )] ( ) ( )[ ]

+

+

SA

m

e

rB dv

SA B

e

rdv

ee

o B

o A

h

SE SsH1 1/( )+

K

Interaction (attraction) between nucleus Aand the charge cloud A(1)B(1)/( )1+ S


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+e

Ro

2

4

1

E+ =E

2(1+ S)

1sH +E

2(1+ S)

1sH J

2(1+ S)

J

2(1+ S)

K-SE

(1+S)

1sH

ES e

Ro+=

+ +

E E

(1+S)

J+K

(1+S)

1sH 1sH2

4

1

We

E

e

Ro

get in a similar way that

= E

J - K

(1- S)1sH +

2

4

1

Ee

Ro

+ = +EJ+K

(1+S)1sH

2

4

1


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The calculatedand experimental molecular

potential energy curves for

a hydrogen molecule-ion.

E

e

Ro+ = + +E

J+K

(1+S)1sH

2

4

1

Ee

Ro

+= EJ - K

(1- S)

1sH

2

4

1


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A representation of thedestructive interference

that occurs when two H1s

orbitals overlap and form an

antibonding * orbital.Compare this illustration

with Fig.14.20.

= ( )

( )[ ( ) ( )]1 1

2 11 1

SA B


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(a) The amplitude of the

antibonding molecular

orbital in a

hydrogen molecule-ion in

a plane containing the two

nuclei and

(b) a contour representation

of the amplitude. Note the

internuclear node.

=

( )( )

[ ( ) ( )]11

2 11 1

SA B


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The electron density calculated

by forming the square of the

wavefunction used to constructFig.14.20. Note the elimination

of electron density from the

internuclear region.

= +

( ) ( ) ( )

( )( )

( )( )

11

2 1 1

1

2 11 2

1

2 11

2

2

S A

SB

SAB


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(b) In an antibonding

orbital, the nuclei are

attracted to an accumulation

of electron density outside

the internuclear region.

A partial explanation of the origin of bondingand antibonding effects.

(a) In a bonding orbital,

the nuclei are attracted to

the accumulation of electron

density in the internuclear

region.

+ = ++( )

( )[ ( ) ( )]1

1

2 11 1

SA B =

( )( )

[ ( ) ( )]11

2 11 1

SA B


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A molecular orbital energylevel diagram for orbitals

constructed from the overlap

of H1s orbitals; the separation

of the levels corresponds tothat found at the equilibrium

bond length. The ground

electronic configuration ofH2 is obtained by

accommodating the two

electrons in the lowest

available orbital(the bonding orbital).



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rA1

A B

1

rB1

R

What you should learn from this lecture

Be

Hm

e

r

e

r

e

Ree

o A o B o

able to construct the Hamiltonian

= +h

22

2

1

2

1

2

2 4

1

4

1

4

1

Understand the justification for constructingthe two molecular orbitals as linear combinationsof atomic orbitals

and

be able to evaluate the normalizationconstants from the overlap integral

S = A(1)B(1)dv

+ + = + =

( ) ( ( ) ( )) ( ) ( ( ) ( ))1 1 1 1 1 1N A B N A B

and

You

E eR

or Ee

R

o

o

are not asked to derive the energy expressions

for E J + K(1+S)

EJ - K

(1-S)

+ 1sH

- 1sH

:

:

+

= +

= +

2

2

41

4

1


26/27


However you should understand the

meening of

J = and

K = and undertand how K

stabilize and destabalizes .

A Ae

rdv

A Be

rdv

o B

o A

( ) ( )[ ]

( ) ( )[ ]

1 14

1

1 14

1

2

1

2

1

+

rA1

A B

1

rB1

R

It

SA

SB

SA B

You

A B A B

S

A B

is expected that you can derive the expressionfor the density due to as :

shoule further understand the meening of

and Also you should understand

how influence the energy E as well

as the features of Fig 14.16

+

+

( )( )

( )( )

( )( )

( ) ( )

( ) , ( ) , ( ) ( ).

( )

( ) ( )

11

2 11

1

2 11 2

1

2 11 1

1 1 1 1

21

2 1

1 1

2 2

2 2

=+

++

++

+

+


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It

SA

SB

SA B

You

A B A B

SA B

is expected that you can derive the expression

for the density due to as :

shoule further understand the meening of

and Also you should understand

how influences the energy E as well

as the features of Fig 14.21

-

-

( )

( )( )

( )( )

( )( ) ( )

( ) , ( ) , ( ) ( ).

( )( ) ( )

11

2 11

1

2 11 2

1

2 11 1

1 1 1 1

21

2 11 1

2 2

2 2

=

+

Unders dfor N A B N A B

respectively

tan( ) ( ( ) ( )) ( ) ( ( ) ( )),

.

the features of the plots in Fig. 14.17 and 14.19and + + = + = 1 1 1 1 1 1

Make note of the potential energy curve in Fig 14.18 andobserve that E is lowered less in energy (compared to

E E is increased (see also Fig 14.24+

1s -than

Chem 373- Lecture 25: Introduction to Molecular Orbital Theory

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