Physical Chemistry Third Edition

770 18 The Electronic States of Atoms. II. The Zero-Order Approximation for Multielectron Atoms

zero-order wave function of a helium atom in its ground state

Probability

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)

ψ 100 ∗(1)ψ 100 ∗(2)ψ 100 (1)ψ 100 (2)d^3 r 1 d^3 r 2 (18.3-5)

The second equality follows from Example 18.1. The two-electron probability density

irrespective of spin is

(

Probability

density

)

ψ 100 ∗(1)ψ 100 (1)ψ 100 ∗(2)ψ 100 (2)

|ψ 100 (r 1 ,θ 1 ,φ 1 )|^2 |ψ 100 (r 2 ,θ 2 ,φ 2 )|^2 (18.3-6)

The probability of finding electron 1 in the volume elementd^3 r 1 is given by inte-

grating the two-electron probability density over all positions of particle 2:

(

Probability of finding

particle 1 ind^3 r 1

)

[

∫

|ψ 100 (1)|^2 ψ 100 (2)|^2 d^3 r 2 ]d^3 r 1

|ψ 100 (r 1 ,θ 1 ,φ 1 )|^2 d^3 r 1

|ψ 100 (r 1 ,θ 1 ,φ 1 )|^2 r^21 sin(θ 1 )dr 1 dθ 1 dφ 1 (18.3-7)

The probability for the second electron is the same function. The total density of

electrons is

(Total electron density) 2 |ψ 100 (r,θ,φ)|^2 (18.3-8)

The density of electric charge due to the electrons is

(Total electron charge density)− 2 e|ψ 100 (r,θ,φ)|^2 (18.3-9)

where−eis the electron’s charge.

Now consider the spins of the electrons. We take the wave function in Eq. (18.3-5)

before the spin integrations were carried out. Multiplying out the spin factors, we obtain

Probability

1

2

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

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

−α∗(1)β(1)β∗(2)α(2)+β∗(1)β(1)α∗(2)α(2)] (18.3-10)

The first term in the sum of products of spin functions corresponds to electron 1 with

spin up and electron 2 with spin down. The fourth term corresponds to electron 1 with

spin down and electron 2 with spin up. The second and third terms areexchange terms

with no classical interpretation.

We now integrate this function over the space and spin coordinates of particle 2

to find the probability of finding particle 1. The space integration yields unity by

the normalization of the space orbitalψ 100 (2). Integrating the spin coordinates of

particle 2 gives zero for the second and third terms in the sum of spin functions by the