Computational Chemistry

Chapter 4, Harder Questions, Answers

Q3

Suppose we tried to simplify the SHM even further, by ignoringallinteractionsi,j;
i 6 ¼j(ignoring adjacent interactions instead of setting them¼b). What effect would
this have on energy levels? Can you see the answer without looking at a matrix or
determinant?
Setting all adjacent orbital interactions equal to zero removes all connectivity
information. It dissociates the molecule into isolated atoms! This follows because in
the SHM the sole structural information about a molecule is provided by whichi,j
pairs areband which are zero: two atoms are connected if and only their interaction
is represented byb; they are not connected if and only their interaction is repre-
sented by 0.
A look at Fock matrices may make this more concrete. Diagonalization of the
standard SHM matrix for the propenyl system gives

0 # 10

# 10 # 1

0 # 10

0

B

@

1

C

A¼

0 :500 0:707 0: 500

0 : 707 0 # 0 : 707

0 : 500 # 0 :707 0: 500

0

B

@

1

C

A

# 1 :414 0 0

000

001 : 414

0

B

@

1

C

A

0 :500 0:707 0: 500

0 : 707 0 # 0 : 707

0 : 500 # 0 :707 0: 500

0

B

@

1

C

A

(1)

+

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    • 
      
    
    
    
    
  
  
  
  

+

• 
  

+

+

• 
  


• 
  

+

• 
  

+ +

• 
  


• 
  

0

-1

1

-2

2

-1.414

1.414

0

Three molecular orbitals with different energies and p-atomic-orbital contributions.

Diagonalization of the no-adjacent-interaction matrix gives

612 Answers