lecture 23: introduction to valence bond theory

8/3/2019 Lecture 23: Introduction to Valence Bond Theory

1/18

Lecture 23: Introduction to Valence Bond

TheoryThe material in this lecture covers the following in Atkins.

14 Molecular structure

Valence-bond theory14.1 The hydrogen molecule

(a) The spatial wavefunction

(b) The role of the electron spin

Lecture on-line

Introduction to Valence Bond Theory (PowerPoint)

Introduction to Valence Bond Theory (PDF)Handout for this lecture


2/18

Valence Bond Theory Basic Theory

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 start with the :


3/18

Valence Bond Theory

H HBH H

H

CH

H

H

HH Cl OH

H

NH

H

H

In valence bond theory we startby writing down the Lewis structure

of our molecule

Subsequentlyas

r ri i

we write r Rthe product of electron pair

functions as

e N

i

( , )

( , )2 1 2

( , ) ( , ) ( , )

.. ( , ) ( , ).. ( , )

r Re N 1 2i

=

r r r r

r r r r r ri i j j j n n n1 2 3 4

2 1 2 2 1 2 2 1 2

Pair 1

Pair 2

Pair i Pair j

Pair n

Basic Theory


4/18

Valence Bond Theory

H HWe shall now illustrate this simple theory for H2

We have two well separated hydrogen

atoms A and B

each with one electronA

B

1

rA1

2

rB2

We can describe hydrogen A

by 1s 1sHA HA

( ) ( ); ( ) ( )r rA A1 1

1 1 We can describe hydrogen B

by 1s 1sHB HB

( ) ( ); ( ) ( )r rB B2 2

2 2

Or in shortA(1) A(1) ( ); ( )1 1 B(2) B(2) ( ); ( )2 2

We now bring the two hydrogens together to form H2

B

2

rB2

A

1

rA1

Basic Theory


5/18

Valence Bond Theory

A

1

rA1

B

2

rB2

The hamiltonian of theH molecule is :2

H me

ro A= h

2

12

2

12 4

1

Hamiltonian of HA

h

2

22

2

22 4

1

m

e

re o B Hamiltonian of HB

e

ro B

2

14

1

rB1

Attraction between el. 1 and H - atom B

e

ro A

2

24

1

Attraction between el. 2 and H - atom B

rA2

+ ero

2

124

1

Repulsion between el. 2 and el. 1

r12

+e

Ro

2

4

1

R

Repulsion between A and B

Basic Theory


6/18

Valence Bond Theory

A

1

rA1

B

2

rB2rB1

rA2

r12

R

W

Hm m

e

RV

o

e can thus write the Hamiltonian as

= + +h h

2

12

2

12

2

2 2 4

1

where

Ve

r

e

r

e

r

e

r

e

ro A o A o A o B o= +

2

1

2

2

2

2

2

1

2

124

1

4

1

4

1

4

1

4

1

Subsequently

as

r ri i

we write r R

the product of electron pair

functions as

e N

i

( , )

( , )2 1 2

A(1) A(1) ( ); ( )1 1 B(2) B(2) ( ); ( )2 2

( , , ) ( , )r r1 2 1R r r= 1 2

We r rshall further write as linear

combinations of product between

functions on A and B

1 ( , )1 2

Basic Theory


7/18

Valence Bond Theory

A

1

rA1

B

2

rB2rB1

rA2

r12

R

A(1) A(1) ( ); ( )1 1 B(2) B(2) ( ); ( )2 2From the functions

We can construct

the products :A(1)

A(1)

A(1)

A(1)

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1 2 2

1 2 2

1 2 2

1 2 2

B

B

B

B

Allowing for interchange of

the two affords :

A(2)

A(2)

A(2)A(2)

( ) ( ) ( )

( ) ( ) ( )

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

2 1 1

2 1 1

2 1 12 1 1

B

B

BB

We

R r r

have further that

r r1 2 1 ( , , ) ( , )= 1 2

Must be anti symmetric

r r r r1 2 1 2 1 1 ( , , ) ( , ) ( , , ) ( , )R r r R r r= = = 1 2 2 1

Basic Theory


8/18

Valence Bond Theory

A(1)

A(1)

A(1)

A(1)

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1 2 2

1 2 2

1 2 2

1 2 2

B

B

B

B

A(2)

A(2)

A(2)

A(2)

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

2 1 1

2 1 1

2 1 1

2 1 1

B

B

B

B

From the product functions

We can construct the

anti - symmetric linear

combinations : 1 = [A(1) A(1)B B( ) ( )]

[ ( ) ( ) ( ) ( )]

2 2

1 2 1 2

+

- Symmetric in space

- anti - symmetric in spin

< S singlet2 >= 0 :

2

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

[ ( ) ( ) ( ) ( )]

2 2

1 2 1 2

+

- Anti - symmetric in space

- symmetric in spin

< S triplet2 >= 2 2h :

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

4

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

ms = 0

ms = 1ms = 1

Basic Theory


9/18

Valence Bond Theory

1 1= C [A(1) A(1)BB BB ( ) ( )]

[ ( ) ( ) ( ) ( )]2 2

1 2 1 2+

2 2= C [A(1) A(1)BB BB ( ) ( )][ ( ) ( ) ( ) ( )]2 21 2 1 2 +

3 3= C [A(1) A(1)BB BB ( ) ( )] ( ) ( )2 2 1 2 4 4= C [A(1) A(1)BB BB ( ) ( )] ( ) ( )2 2 1 2

Singlet

triplet

Electron

n n

dr dr dr dspinn

density :

(r1 ) ( , ,.. ) ( , ,.. )

..

*

= 1 2 1 2

2 3probability of finding el.

no matter where other el.are

Singlet density :

sin ( )( ) ( )

( ) ( )

gHA HB

S SA B

S

11

1

1

12 1 1

1

2 2

2

=+

++

+ +

A B BA

positive overlap density

Density build up between nuclei

A(1) B(1)

S s s dA B= 1 1 1 1( ) ( )

Overlap between

1s orbitals on A and B

Basic Theory

CCi insures normalization


10/18

BA

Valence Bond Theory

1 1= C [A(1) A(1)BB BB ( ) ( )]

[ ( ) ( ) ( ) ( )]2 2

1 2 1 2+

2 2= C [A(1) A(1)BB BB ( ) ( )][ ( ) ( ) ( ) ( )]2 21 2 1 2 +

3 3= C [A(1) A(1)BB BB ( ) ( )] ( ) ( )2 2 1 2 4 4= C [A(1) A(1)BB BB ( ) ( )] ( ) ( )2 2 1 2

Singlet

triplet

Triplet density :

sin ( )( ) ( )

( ) ( )

gHA HB

S SA B

S

11

1

1

12 1 1

1

2 2

2

=

+

A B

negative overlap density

Density reduced between nuclei

Basic Theory


11/18

Valence Bond Theory

1 1= C [A(1) A(1)BB BB ( ) ( )]( ) ( ) ( ) ( )]

2 21 2 1 2

+

2 2= C [A(1) A(1)BB BB ( ) ( )]

[ ( ) ( ) ( ) ( )]2 2

1 2 1 2

+

3 3= C [A(1) A(1)BB BB ( ) ( )] ( ) ( )2 2 1 2

4 4= C [A(1) A(1)BB BB ( ) ( )] ( ) ( )2 2 1 2

Singlet :

E E J KS

eR

Ho

= + ++

+21 42

2

Triplet :E E

J K

S

e

RH

o

= +

+2

1 42

2

Energy hydrogen atom

A(1) B(1)

S s s dA B= 1 1 1 1( ) ( )

Overlap between

1s orbitals on A and B

Basic Theory


12/18

Valence Bond Theory

Singlet :

E EJ K

S

e

RH

o

= ++

++2

1 42

2

Triplet :

E EJ K

S

e

RH

o

= +

+21 42

2

J

e

A A r dvo B=

2

114 1 1

1

( ) ( )

A

1

rA1

B

2

rB2rB1

rA2

r12

R

e

B B

r

dv

o A

2

2

2

4

2 21

( ) ( )

eB B

r

A A dv dv

o

2

12

2 1

4

2 21

1 1

( ) ( ) ( ) ( )

Int. el.1 with Nuc. B

Int. el.2 with Nuc. A

el 1 with el 2

2EH

2EH+

e2

4oR

2EH+

e2

4 oR+

J

1 S2

E

Triplet2EH

+e2

4 oR+

J

1+ S2

singlet

Basic Theory


13/18

Valence Bond Theory

Singlet :

E EJ K

S

e

RH

o

= ++

+

+21 42

2

Triplet :

E E J KS

eR

Ho

= + +2

1 42

2

Ke

A B

r

dv

o B

= 2

1

1

4

1 11

( ) ( )

e

A B

r

dv

o A

2

2

2

4

2 21

( ) ( )

+ e

A B

r

A B dv dv

o

2

12

2 1

4

2 21

1 1

( ) ( ) ( ) ( )

Int. overlap dens. with Nuc. B

Int. overlap dens with Nuc. A

int. overlap dens with itself

BA

BA

BA

K is negative for singlet

since overlap density

positive

K enters with oppositesign for triplet since

overlap density

negative

Basic Theory


14/18

Valence Bond Theory

Singlet :

E EJ K

S

e

RH

o

= ++

+

+21 42

2

Triplet :

E EJ K

S

e

RH

o

= +

+21 42

2

2EH

2EH+ e2

4oR

2EH+ e

2

4oR+ J

1 S2

E

Triplet

2EH+ e

2

4oR+ J

1+ S2

singlet

singlet

Triplet

E = 2EH +

J + K

1+ S2 +

e2

4oR

E = 2EH +J K

1 S2+ e

2

4oR

Positive overlap density

makes singlet more

stable than triplet

and separate hydrogen

atomsorigin of chemical

bond

Basic Theory


15/18


H H

It is very difficult to represent valence-bond

wavefunctions because they refer to twoelectrons simultaneously. However, this

illustration is an attempt. The atomic orbital

for electron 1 is represented by the blackcontours, and that of electron 2 is

represented by the green contours. The

top illustration represents A(1)B(2), and

the middle illustration represents thecontribution A(2)B(1). When the two

contributions are superimposed, there

is interference between the black

contributions and between the greencontributions, resulting in an enhanced

(two-electron) density in the internuclear

region.


16/18

he molecular potential energy curve for theydrogen molecule showing the variation of

he energy of the molecule as the bond length

s changed. The calculated curve refers to the

alence-bond model.


H H


17/18

You should know that :

In valence bond theory we start by writing down theLewis structure of our molecule Subsequently wewrite r R as the product of electron pair functions

as r Re N

i e N 1 2

i

( , )( , ) ( , ) ( , ) ( , )

.. ( , ) ( , ).. ( , )

rr rr rr rr rr rr

rr rr rr rr rr rr ii ii ii ii jj jj jj nn nn nn

2 1 2 1 2 3 4

2 1 2 2 1 2 2 1 2

=

You should know that the singlet function= C [A(1) A(1)

is more stable than the triplet (e.i. =C [A(1) A(1)

1 1

22

BB BB

BB BB

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

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

2 2 1 2 1 2

2 2 1 2 1 2

+

+

What you should learn from this lecture

You

S S

A B

SHowever

gHA HB

are not asked to derive the expressionfor the density of th singlet .

you should know that densityis increased in the bonding region and thatcontribute to the stability of the singlet

sin ( )( ) ( ) ( ) ( )

,

1 1

1

1

1

2 1 1

12 2 2=

++

++

+


18/18

What you should learn from this lecture

You

S S

A B

SHowever

tripHA HB

are not asked to derive the expression

for the density of the triplet .

you should know that density is decreased inthe bonding region and that contribute to the higherenergy of the triplet

( )( ) ( ) ( ) ( )

,

11

1

1

1

2 1 1

12 2 2=

+

You

E EJ K

S

e

Rand

E E J KS

eR

However

g Ho

triplet Ho

will not be required to derive the energy

expression for the singlet

you should know that the (negative) exchangeintegral K is responsible for the lower energy of the singlet.

It is related to the buildup of charge in the bonding region.

the tripletsin = ++

++

= +

+

21 4

21 4

2

2

2

2

lecture 23: introduction to valence bond theory

Documents