Computational Chemistry

0 " 10 " 1

" 10 " 10

0 " 10 " 1

" 10 " 10

0

B

B

B

@

1

C

C

C

A

¼

0 :500 0:500 0:500 0: 500

0 : 500 " 0 :500 0: 500 " 0 : 500

0 : 500 " 0 : 500 " 0 :500 0: 500

0 :500 0: 500 " 0 : 500 " 0 : 500

0

B

B

B@

1

C

C

CA

" 2000

0000

0000

0002

0

B

B

B@

1

C

C

CA

0 :500 0:500 0:500 0: 500

0 : 500 " 0 : 500 " 0 :500 0: 500

0 :500 0: 500 " 0 : 500 " 0 : 500

0 : 500 " 0 :500 0: 500 " 0 : 500

0

B

B

B@

1

C

C

CA

v 1 v 2 v 3 v 4

e 1 000

0 e 2 00

00 e 3 0

000 e 4

C e C"^1 ð 4 : 69 Þ

The energy levels and MOs from these results are shown in Fig.4.17. Note that all
these matrix diagonalizations yield orthonormal eigenvectors:vi+vi¼1 andvi+vj¼0,
as required the fact that the Fock matrices are symmetric (see the discussion of matrix
diagonalization inSection 4.3.3).

+

• 
  

+

• 
  

+

• 
  

+

• C C

C C

+

+ –

• 
  

+

+ –

• C C

C C

+

• 
  

+

• 
  
  
  
  • 
    
  
  
  
  

+

• 
  

C C

C C

+

+ –

• 
  

+

+ –

• 
  

C C

C C

energy

+

bonding MO

antibonding MO

nonbonding MOs

e = –2

e = 2

e = 0 e = 0

y 4 = 0.500 j 1 – 0.500 j 2 + 0.500 j 3 – 0.500 j 4

y 3 = 0.500 j 1 + 0.500 j 2 – 0.500 j 3 – 0.500 j 4

y 2 = 0.500 j 1 – 0.500 j 2 – 0.500 j 3 + 0.500 j 4

y 1 = 0.500 j 1 + 0.500 j 2 + 0.500 j 3 + 0.500 j 4

a - 2 b

a + 2 b

a - b

a + b

a

Fig. 4.17 Thepmolecular orbitals andpenergy levels for a cyclic four-p-orbital system in the
simple H€uckel method. The MOs are composed of the basis functions (fourpAOs) and the
eigenvectors, while the energies of the MOs follow from the eigenvalues (Eq.4.69). This particular
diagram is for the square cyclobutadiene molecule. Thepaired arrowsrepresent a pair of electrons
of opposite spin, in the fully-occupied lowest MO,c 1 , and thesingle arrowsrepresents unpaired
electrons of the same spin, one in each of the two nonbonding MOs,c 2 andc 3 ; the highestpMO,
c 4 , is empty in the neutral molecule

132 4 Introduction to Quantum Mechanics in Computational Chemistry