The Chemistry Maths Book, Second Edition

578 Chapter 20Numerical methods

Formula (20.37) is used in the following examples to derive two results that are

important in statistical thermodynamics.

EXAMPLE 20.13The rotational partition function of the linear rigid rotor

The energy levels of the linear rigid rotor are

with degeneraciesg

J

1 = 12 J 1 + 11 , where Jis the rotational quantum number and Iis the

moment of inertia. The rotational partition function is then

whereθ 1 = 1 h

2

28 π

2

Ikis called the rotational temperature.

Putf(J) 1 = 1 (2J 1 + 1 1)e

−J(J+1)θ 2 T

in the Euler–MacLaurin formula (20.37), witha 1 = 10 ,

b 1 → 1 ∞. Then

The integral is evaluated by means of the substitutionu 1 = 1 J(J 1 + 1 1). Then

Also,f(J)and all its derivative go to zero asJ 1 → 1 ∞, and the values atJ 1 = 10 can be

obtained from the expansion off(J)as a power series in J,

Thenf(0) 1 = 11 ,f′(0) 1 = 121 − 1 θ 2 T, f′′(0) 1 = 1 − 12 θ 2 T 1 + 112 θ

2

2 T

2

1 − 1 θ

3

2 T

3

, and all higher

derivatives atJ 1 = 10 are of order(θ 2 T)

2

or higher. Then

q (20.38)

T

TT

=++













• 
  













θ

1 θθ

3

1

15

2

terms in gand higher

fJ

T

J

T

T

()=+ − J













+− +

















12 +−

3

2

2

2

2

2

θθθ θ

TT

TT

+− J

















• 
  

2

6

2

2

3

3

3

θθ



ZZ

0

1

0

21

∞∞

()

()

Je dJ e du

T

JJ T u T

+==

−+/θθ−/

θ

+′−′−′′′ − ′′′ +

1

12

0

1

720

[ ( ) ( )]ff f f∞∞[ ( ) ( )] 0 

qfJ fJdJff

J

== ++

=

∑

0 0

1

2

0

∞

∞

()Z () [() ()]∞

qge Je

J

J

EkT

J

JJ T

J

==+

=

−

=

−+

∑∑

00

1

21

∞∞

()

()θ

E

JJ h

I

J

J

=

• 
  

,=,,,,

()1

8

0123

2

2

π

...