Computational Chemistry

"

p

2 bc 11 þbc 21 ¼ 0 ; so c 21 =c 11 ¼

p

2

(Recall thecijnotation;c 11 is the coefficient for atom 1 inc 1 ,c 21 is the
coefficient for atom 2 inc 1 , etc.). SubstitutingE¼a+

pffi
2 binto the second
secular equation we get

"bc 11 þbc 31 ¼ 0 ; so c 11 ¼c 31

We now have the relative values of thec’s:

c 11 =c 11 ¼ 1 ;c 21 =c 11 ¼

p

2 ;c 31 =c 11 ¼ 1 (4.84)

To find the actual values of thec’s, we utilize the fact that the MO (we are
talking about MO level 1,c 1 ) must be normalized:

Z

c^21 dv¼ 1 (4.85)

Now, from the LCAO method

c 1 ¼c 11 f 1 þc 21 f 2 þc 31 f 3 (4.86)

Therefore

c^21 ¼c^211 f^21 þc^221 f^22 þc^231 f^23 þ 2 c 11 c 21 f 1 f 2

þ 2 c 11 c 31 f 1 f 3 þ 2 c 21 c 31 f 2 f 3 (4.87)

So from Eq.4.87, and recalling that in the SHM we pretend that the basis
functionsfare orthonormal, i.e. thatSij¼dij, we get

Z

c^21 dv¼c^211 þc^221 þc^231 ¼ 1 (4.88)

Using the ratios of thec’s from Eq.4.84:

c^211

c^211

þ

c^221

c^211

þ

c^231

c^211

¼

1

c^211

i.e.

12 þ

ffiffiffi

2

p 2

þ 12 ¼

1

c^211

and so

c 11 ¼

1

2

4.3 The Application of the Schr€odinger Equation to Chemistry by H€uckel 151