Computational Chemistry

Diagonalization of the three-basis function matrix of Eq.4.64gives

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

v 1 v 2 v 3

C

e 1 ; 0 ; 0

0 ; e 2 ; 0

0 ; 0 ; e 3

e C"^1

(4.68)

The energy levels and MOs corresponding to these results are shown in
Fig.4.16.
Diagonalization of the four-basis-function matrix of Eq.4.65gives

+

• 
  

+

• 
  

+

• 
  

C

C

C

+

• 
  

• 
  

+ –

• 
  

C

C

C

+

• 
  
  
  
  • 
    
    
    
    • C
      C
    
    
    
    
  
  
  
  

C

energy

e = 1.414 antibonding MO

e = 0 nonbonding MO

a-b

a+b

a

y 3 = 0.500 1 – 0.707 2 + 500 (^3)
y 2 = 0.700 1 + 0.000 2 – 0.700 (^3)
f
fff
e = –1.414 bonding MO
y 3 = 0.500 fff 1 + 0.707 2 + 0.500 (^3)
f f
Fig. 4.16Thepmolecular orbitals andpenergy levels for an acyclic three-p-orbital system in the
simple H€uckel method. The MOs are composed of the basis functions (threepAOs) and
the eigenvectors (thec’s), while the energies of the MOs follow from the eigenvalues
(Eq.4.68). In the drawings of the MOs, the relative sizes of the AOs in each MO suggest the
relative contribution of each AO to that MO. This diagram is for the propenyl radical. Thepaired
arrowsrepresent a pair of electrons of opposite spin, in the fully-occupied lowest MO,c 1 , and the
single arrowrepresents an unpaired electron in the nonbonding MO,c 2 ; the highestpMO,c 3 , is
empty in the radical
4.3 The Application of the Schr€odinger Equation to Chemistry by H€uckel 131