Computational Chemistry

Care eigenvectors, and the diagonal elements of«are eigenvalues; cf. Eq.4.38and
the associated discussion of eigenfunctions and eigenvalues. The result of
Eq. (4.66) is readily checked by actually multiplying the matrices (multiplication
here is aided by knowing that an analytical rather than numerical diagonalization
shows that,0.707 are approximations to 1/

pffi
2). Note thatCC"^1 ¼ 1 , and thatC"^1
is the transpose ofC. The first eigenvector ofC, the left-hand column, corresponds
to the first eigenvalue of«, the top left element; the second eigenvector corresponds
to the second eigenvalue. The individual eigenvectors,v 1 and v 2 , are column
matrices:

0 : 707

0 : 707



" 1 and

0 : 707

" 0 : 707



 1

v 1 v 2

(4.67)

Figure4.15shows a common way of depicting the results for this two-orbital
calculation. Since the coefficients are weighting factors for the contributions of the
basis functions to the MOs (Fig.4.11 and associated discussion), thec’s of
eigenvectorv 1 combine with the basis functions to give MO 1 (c 1 ) and thec’s of
eigenvectorv 2 combine with these same basis functions to give MO 2 (c 2 ). MOs
belowaare bonding and MOs aboveaare antibonding. The«matrix translates into
an energy level diagram withc 1 of energya+bandc 2 of energya"b, i.e. the
MOs lie one |b| unit below and one |b| above the nonbondingalevel. Sinceb, likea,
is negative, thea+banda"blevels are of lower and higher energy, respectively,
than the nonbondingalevel.

C C

+

+

+

• 
  
  
  
  • 
    
    
    
    • C C
    
    
    
    
  
  
  
  

nonbonding level

bonding MO

antibonding MO

energy

e = 1

e = –1

y 2 = 0.707 1 – 0.707 (^2)
a - b
a + b
a
ff
y 2 = 0.707 ff 1 + 0.707 (^2)
Fig. 4.15 Thepmolecular orbitals andpenergy levels for a two-p-orbital system in the simple
H€uckel method. The MOs are composed of the basis functions (twopAOs) and the eigenvectors,
while the energies of the MOs follow from the eigenvalues (Eq.4.66). The paired arrows represent
a pair of electrons of opposite spin (in the electronic ground state of the neutral ethene moleculec 1
is occupied andc 2 is empty)
130 4 Introduction to Quantum Mechanics in Computational Chemistry