The Chemistry Maths Book, Second Edition

486 Chapter 17Determinants

The general condition that a determinant be zero is discussed in Section 17.5. Systems

of linear equations are discussed in greater detail as matrix equations in Chapter 19.

0 Exercise 18

Secular equations

A number of problems in the physical sciences give rise to systems of equations of the

form

a

11

x

1

1 + 1 a

12

x

2

1 + 1 a

13

x

3

1 + 1  1 + 1 a

1 n

x

n

1 = 1 λx

1

a

21

x

1

1 + 1 a

22

x

2

1 + 1 a

23

x

3

1 + 1  1 + 1 a

2 n

x

n

1 = 1 λx

2

a

31

x

1

1 + 1 a

32

x

2

1 + 1 a

33

x

3

1 + 1  1 + 1 a

3 n

x

n

1 = 1 λx

3

(17.37)

a

n 1

x

1

1 + 1 a

n 2

x

2

1 + 1 a

n 3

x

3

1 + 1  1 + 1 a

nn

x

n

1 = 1 λx

n

where λis a parameter to be determined. For example, in molecular-orbital theory,

the Schrödinger equation is replaced by such a set of linear equations in which the

quantitiesx

1

, x

2

, =, x

n

represents an orbital and λthe corresponding orbital energy.

The equations can be written as

(a

11

1 − 1 λ)x

1

1 + a

12

x

2

• a

13

x

3

• 1  1 + a

1n

x

n

= 10

a

21

x

1

• 1 (a

22

1 − 1 λ)x

2

1 + a

23

x

3

• 1  1 + a

2 n

x

n

= 10

a

31

x

1

• a

32

x

2

• 1 (a

33

1 − 1 λ)x

3

• 1  1 + a

3 n

x

n

= 10 (17.38)

a

n 1

x

1

• a

n 2

x

2

• a

n 3

x

3

• 1  1 + 1 (a

nn

1 − 1 λ)x

n

= 10

These homogeneous equations, called secular equations, have non-trivial solution

only if the determinant of the coefficients is zero,

(17.39)

The determinant is called a secular determinantin this context. It is zero only for

some values of the parameter λand these are obtained by solving equation (17.39).

Because the expansion of the secular determinant is a polynomial of degree nin λ, the

required values of λare the nroots of the polynomial.

()

()

(

aaa a

aa a a

aaa

n

n

11 12 13 1

21 22

23 2

31 32 3

−

−

λ

λ





33

3

1

23

0

−

−

=

λ

λ

)

()





a

aaa a

n

n

nn

nn