The Chemistry Maths Book, Second Edition

20.5 Numerical integration 579

The largest value of θfor a molecule is θ 1 = 1 87.5 1 K for H

2

but most molecules

have rotational temperatures close to 0 K. In that case, T 2 θ 1 >> 11 except at very low

temperatures, so that the first term on the right side of (20.38) is much the largest, and

is the rotational partition function normally used in chemistry (after correction for

symmetry).

EXAMPLE 20.14Stirling’s approximation for ln 1 n!

The quantity ln 1 n! for integer noccurs in probability theory and enters statistical

thermodynamics with nequal to the number of particles in a mole; that is, with

n 1 ≈ 110

23

. Stirling’s approximation for the logarithm of the factorial of a very large

number,ln 1 n! 1 ≈ 1 n 1 ln 1 n 1 − 1 n, can be derived from the Euler–MacLaurin formula. We

putf(x) 1 = 1 ln 1 x. Then

ln 1 n! 1 = 1 ln 111 + 1 ln 121 +1-1+ 1 ln 1 n

≈ 1 n 1 ln 1 n 1 − 1 n when n 1 >> 11

(see also Section 21.6). A more complete treatment gives

0 Exercise 26

Gaussian quadratures

All numerical integration formulas have the form

(20.39)

and involve the evaluation of the integrand at npoints,x

1

, x

2

, =, x

n

, with coefficients,

or Gaussian weights,w

i

determined by the form of the interpolation formula. In the

Newton–Cotes formulas discussed above the points are equally spaced. In Gaussian

quadrature formulas the points are chosen to give the greatest possible accuracy for a

given interpolation formula, and are not equally spaced.

7

Z

a

b

i

n

ii

fxdx() ≈ fx( )

=

∑

1

w

ln! lnnnnn ln n

n

nn

≈−+ +− +

1

2

2

1

12

1

360

1

1260

35

π

≈−













+++−













xxx n

n

n

ln (ln ln )

1

1

2

1

1

12

1

1

=+++′−′+Z

1

1

2

1

1

12

1

n

lnxdx [ ( ) ( )]f n f [ ( ) ( )]f n f 

7

Gauss presented his new method in Methodus nova integralium valores per approximationem inveniendiin

1816.