The Chemistry Maths Book, Second Edition

302 Chapter 10Functions in 3 dimensions

Integrals over all space

When the region Vis the whole three-dimensional space, the volume integral is

(10.11)

(10.12)

These integrals are important, for example, in quantum chemistry for the evaluation

of atomic and molecular properties from wave functions obtained as solutions of the

Schrödinger equation.

EXAMPLE 10.8The integral over all space of the electron probability density of

the 1 sorbital of the hydrogen atom, (see Table 10.1), is

The integral over the angles

(10.13)

is the complete solid angle around a point. Then, making use of the standard integral

This result is consistent with the interpretation of|ψ|

2

dvas the probability of

finding the electron in elementdv. The total probability, the probability of finding

the electron somewhere, is the integral over all space, and must by unity. In fact, the

coefficients of the orbitals in Table 10.1 have been chosen for this to be true. The

orbitals are said to be normalized (to unity).

0 Exercises 20–22

Zψ

1

2

0

3

0

3

12

2

41

s

d

aa

v=× ×=

π

π

() 2

Z

0

1

∞

erdrna

−+ar n n

= !, 2

ZZ

00

2

4

ππ

sinθθddφ= π

=

−

1

0

3

0

2 2

00

2

0

π

ππ

a

erdr d d

ra

ZZZ

∞

2

sinθθ φ

ZZZZψφθθ

1

2

0

3

0

2

00

2 2

1

0

s

ra

d

a

v= er drdd

−

π

ππ

∞

2

sin

ψ

1

2

0

3

2

1

0

s

ra

= ae

−

() 2

2

π

=,ZZZ ,

0

2

00

2

ππ

∞

fr r()sinθθθφφdrd d in spherical poolars

Z ZZZ

V

f dv=,f x y z dx dy dz, in cartes

−

+

−

+

−

+

∞

∞

∞

∞

∞

∞

() iians