The Chemistry Maths Book, Second Edition

15.2 Orthogonal expansions 417

and that, when the value of the variable is restricted to the interval− 11 ≤ 1 x 1 ≤ 1 + 1 ,the

orthogonality of the Legendre polynomials can be used to derive a general formula

for the coefficientsc

l

in (15.3) in terms of the coefficientsa

l

in (15.1). The Legendre

polynomials were discussed in Section 13.4, and the first few are

P

0

(x) = 11 P

1

(x)= 1 x

(15.4)

The function P

l

(x)is a polynomial of degree lin x, and it is possible to express

every powerx

l

as a linear combination of Legendre polynomials of degree up to l.

For example, it follows from (15.4) that

x

0

= 1 P

0

x

1

= 1 P

1

(15.5)

The power series (15.1) can therefore be written (using only terms from (15.5)),

(15.6)

and this has the required form (15.3). The Legendre polynomials occur in the physical

sciences with the variablex 1 = 1 cos 1 θ, so that we are interested only in the interval

− 11 ≤ 1 x 1 ≤ 1 +1. The polynomials are orthogonal in this interval (see equations (13.25)

and (13.27)),

(15.7)

Z

−

+

,

=

• 
  

=

≠

• 
  

=

1

1

2

21

0

2

21

PxPxdx

l

kl

l

k

kl kl

() () δ

if

if ll











++

















++

















• 
  

8

35

8

63

4

4

5

5

a

P

a

P

+++

















+++

















2

3

4

7

3

5

4

9

24

2

35

aa

P

aa

P

33

= +++

















++++











a

aa

Pa

aa

0

24

01

35

35

3

5

3

7









P

1

+++++++

a

PPP

a

PPP

4

420

5

531

35

8207

63

()( ) 82827 

fx aP aP

a

PP

a

()=++( ++) (PP+)

00 11

2

20

3

31

3

2

5

23

xPPP

5

531

1

63

=++() 82827

xPPP

4

420

1

35

=++() 8207

xPP

3

31

1

5

=+() 23

xPP

2

20

1

3

=+() 2

Px x x x

5

53

1

8

() (=−+63 70 15)

Px x x

4

42

1

8

() (=−+35 30 3)

Px x x

3

3

1

2

() (=− 53 )

Px x

2

2

1

2

() (=− 31 )