Physical Chemistry Third Edition

18.3 The Ground State of the Helium Atom in Zero Order 769

function can be written as

Ψgs(0)(1, 2)Ψ(0) 1 s 1 s(1, 2)



√

1

2

[ψ100,1/ 2 (1)ψ100,− 1 / 2 (2)−ψ100,− 1 / 2 (1)ψ100,1/ 2 (2)] (18.3-1)

or

Ψ(0) 1 s 1 s(1, 2)

√

1

2

[ψ 100 (1)α(1)ψ 100 (2)β(2)−ψ 100 (1)β(1)ψ 100 (2)α(2)]



√

1

2

ψ 100 (1)ψ 100 (2)[α(1)β(2)−β(1)α(2)] (18.3-2)

where theψ 100 orbitals are hydrogen-like orbitals withZ2.

EXAMPLE18.1

Show that the function in Eq. (18.3-2) is normalized. Both the space and the spin coordinates

must be integrated.

Solution

∫

Ψ(0)∗(1, 2)Ψ(0)(1, 2)d^3 r 1 ds(1)d^3 r 2 ds(2)



1

2

∫

ψ∗ 100 (1)ψ 100 ∗ (2)[α∗(1)β∗(2)−β∗(1)α∗(2)]

×ψ 100 (1)ψ 100 (2)[α(1)β(2)−β(1)α(2)]d^3 r 1 ds(1)d^3 r 2 ds(2)



1

2

∫

ψ∗ 100 (1)ψ 100 ∗ (2)ψ 100 (1)ψ 100 (2)d^3 r 1 d^3 r 2

×

∫

[α∗(1)β∗(2)−β∗(1)α∗(2)][α(1)β(2)−β(1)α(2)]ds(1)ds(2) (18.3-3)

The integral over the space coordinates is equal to unity if the space orbitals are normalized.

The integral over the spin coordinates is equal to

∫

[α∗(1)α(1)β∗(2)β∗(2)−α∗(1)β(1)β∗(2)α(2)

−β∗(1)α(1)α∗(2)β(2)+β∗(1)β(1)α∗(2)α(2)]ds(1)ds(2) (18.3-4)

The first term and the last term equal unity after integration. The second and third terms

vanish after integration, so that this integral is equal to 2, which cancels the factor of 1/2 and

the function is normalized.

Probability Densities for Two Particles

The probability of finding electron 1 in volume elementd^3 r 1 and electron 2 in volume

elementd^3 r 2 , irrespective of the spins of the electrons, is obtained by integrating

the square of the wave function over the spin coordinates of both electrons. For the