Computational Chemistry

A 11 x 1 þA 12 x 2 ¼b 1

A 21 x 1 þA 22 x 2 ¼b 2

Using determinants:

x 1 ¼

b 1 A 12

b 2 A 22

D

x 2 ¼

A 11 b 1

A 21 b 2

(^)
(^)

D

D¼

A 11 A 12

A 21 A 22

whereDis thedeterminant of the system.
Ifb 1 ¼b 2 ¼0 (the situation in the secular equations), then in the equations
forx 1 andx 2 the numerator is zero, and sox 1 ¼0/Dandx 2 ¼0/D. The only way
thatx 1 andx 2 can be nonzero in this case is that the determinant of the system be
zero, i.e.

D¼ 0

for thenx 1 ¼0/0 andx 2 ¼0/0, and 0/0 can have any finite value; mathematicians
call it indeterminate. This is easy to see:
Let

0

0

¼a

then

a& 0 ¼ 0

which is true for any finite value ofa.
So for the secular equations the requirement that thec’s be nonzero is that the
determinant of the system be zero:

D¼

H 11 "ES 11 H 12 "ES 12

H 21 "ES 21 H 22 "ES 22

¼ 0 (4.72)

4.3 The Application of the Schr€odinger Equation to Chemistry by H€uckel 147