Physical Chemistry Third Edition

764 18 The Electronic States of Atoms. II. The Zero-Order Approximation for Multielectron Atoms

Nucleus ( 1 Ze)

Electron 1

2 e

2 e

z

x

y

Electron 2

 1


 2

r 1 r^12

r 2

 2

 1

Figure 18.1 The Helium Atom System.

18.1 The Helium-Like Atom

The helium atom contains two electrons and a nucleus containing two protons. We

define ahelium-like atomto have two electrons and a nucleus withZprotons, so that

Z2 represents the He atom,Z3 represents the Li+ion, and so on. The helium-

like atom is shown in Figure 18.1. Our treatment will apply to any value ofZ. With the

hydrogen-like atom, the motion of the electron relative to the nucleus was equivalent to

the motion of a fictitious particle with mass equal to the reduced mass of the electron and

the nucleus. Replacing the reduced mass by the mass of the electron in our formulas was

equivalent to assuming a stationary nucleus. This was a good approximation, because

the mass of the nucleus is large compared to mass of the electron, which means that

the nucleus remains close to the center of mass.

The helium nucleus is even more massive than the hydrogen nucleus, and we will

assume that the nucleus is stationary while studying the motion of the electrons. Of

course, the nucleus can move, but as it moves, the electrons follow along, adapting to

a new location of the nucleus almost as though it had always been there. This is similar

to ignoring the motion of the earth around the sun when we describe the motion of the

moon relative to the earth.

Assuming a stationary nucleus, the classical Hamiltonian function of the electrons

in a helium-like atom withZprotons in the nucleus is

Hcl

1

2 m

p^21 +

1

2 m

p^22 +

1

4 πε 0

(

−

Ze^2

r 2

−

Ze^2

r 1

+

e^2

r 12

)

(18.1-1)

wheremis the mass of an electron,−eis the charge of the electron,p^21 is the square

of the momentum of electron 1,p^22 is the square of the momentum of electron 2, and

the distances are as labeled in Figure 18.1. The Hamiltonian operator is obtained by

making the replacements shown in Eq. (16.3-8) and its analogues:

Ĥ−h ̄

2

2 m

(∇^21 +∇ 22 )+

1

4 πε 0

(

−

Ze^2

r 2

−

Ze^2

r 1

+

e^2

r 12

)

(18.1-2)

where∇ 12 and∇ 22 are the Laplacian operators for the coordinates of electrons 1 and 2

as defined in Eq. (15.2-26) and Eq. (B-45).

The Hamiltonian operator of Eq. (18.1-2) gives a Schrödinger equation that cannot

be solved exactly. No three-body system can be solved exactly, either classically or