Computational Chemistry

The analogous procedure, beginning with Eq.4.45and differentiating with
respect toc 2 leads to

ðH 12 "ES 12 Þc 1 þðH 22 "ES 22 Þc 2 ¼ 0 (4.47)

Equation4.47can be written as Eq.4.48

ðH 21 "ES 21 Þc 1 þðH 22 "ES 22 Þc 2 ¼ 0 (4.48)

since as shown in Eqs.4.44H 12 ¼H 21 andS 12 ¼S 21 , and the form used in Eq.4.48
is preferable because it makes it easy to remember the pattern for the two-basis

+

• 
  
  
  
  • 
    
    
    
    • 
      
    
    
    
    
  
  
  
  

+

• 
  

+

• MO 2

antibonding MO

MO 1

AO 1 AO 2

+

• 
  

+

• 
  

bonding MO

Two p AOs

two pi-type MOs

+

• 
  

+

• 
  

Three p AOs

AO 1 AO 2

+

• 
  

AO 3

MO 1

bonding MO

MO 3

antibonding MO

MO 2

nonbonding MO

energy

+

f 1 f 2

f 1 f 2

f 1 f 2 f 3

C 11 f 1 + C 21 f 2

C 12 f 2 + C 22 f 2

C 11 f 1 + C 21 f 2

C 13 f 1 + C 23 f 2 + C 33 f 3

C 12 f 1 + C 22 f 2 + C 32 f 3

C 11 f 1 + C 21 f 2 + C 31 f 3

C 12 f 1 + C 22 f 2

y 2

y 1

y 2

y 1

y 3

y 2

y 1

+

+ +

+ –

energy

Two s AOs

coefficient of basis

function 1 in MO 1

coefficient of basis

function 2 in MO 1

MO 1

AO 1 AO 2 MO 2

bonding MO

antibonding MO

two sigma-type MOs

a node (AOs change sign here)

+

• 
  

+

• 
  

+

• 
  

+

+ –

• 
  

+

• 
  

+

• 
  
  
  
  • 
    
    
    
    • 
      
    
    
    
    
  
  
  
  

three pi-type MOs

energy

node

C 22 = 0

nodes

Fig. 4.11 Linear combination ofnatomic orbitals (or, more generally, basis functions) givesn
MOs. The coefficientscare weighting factors that determine the magnitude and the sign of the
contribution from each basis function. The functions contributing to the MO change sign at a node
(actually anodal plane) and the energy of the MOs increases with the number of nodes

122 4 Introduction to Quantum Mechanics in Computational Chemistry