Physical Chemistry Third Edition

784 18 The Electronic States of Atoms. II. The Zero-Order Approximation for Multielectron Atoms

PROBLEMS

Section 18.6: The Lithium Atom

18.13Show that the wave function of Eq. (18.6-5) is
normalized if the orbitals are normalized and orthogonal
to each other. The normalization integral is an integral
over the coordinates of all three electrons. There will be
36 terms in the integrand. Look for a way to write down
the result of integrating each term without having to write
all of the integrands, using the orthogonality and
normalization of the orbitals.

18.14Show that the following determinant vanishes:

∣∣

∣

∣∣

∣

123

123

214

∣∣

∣

∣∣

∣

18.15Show that the following determinant vanishes:

∣

∣

∣∣

∣

∣

121

222

313

∣

∣

∣∣

∣

∣

18.16Show that the following determinants are equal to the

negative of each other:

∣∣

∣

∣∣

∣

121

212

313

∣∣

∣

∣∣

∣

and

∣∣

∣

∣∣

∣

212

121

313

∣∣

∣

∣∣

∣

18.17Find and explain the relationship between the two

determinants:

∣

∣

∣∣

∣

∣

121

212

310

∣

∣

∣∣

∣

∣

and

∣

∣

∣∣

∣

∣

363

212

310

∣

∣

∣∣

∣

∣

18.18Find the possible term symbols for the excited

configuration (1s)^2 (2p).

18.19Find the possible term symbols for the excited

configuration (1s)(2s)(2p).

18.20Consider the excited-state configuration (1s)(2s)(3s) for a

lithium atom. Write a table analogous to Table 18.1.

a.Show that quartet states withS 3 /2 can occur.

b.Write the term symbols for all terms that occur.

c.Find the zero-order energy eigenvalue for this

configuration.

18.7 Atoms with More Than Three Electrons

The treatment of other atoms in zero order is analogous to the helium and lithium treat-

ments. For an atom with atomic numberZ(Zprotons in the nucleus andZelectrons),

the stationary-nucleus Hamiltonian operator is

Ĥ−h ̄

2

2 m

∑Z

i 1

∇i^2 −

Ze^2

4 πε 0

∑Z

i 1

1

r 1

+

e^2

4 πε 0

∑Z

i 2

∑i−^1

j 1

1

rij

(18.6-9)

whereriis the distance from the nucleus to theith electron andrijis the distance from

theith electron to thejth electron.

The first two sums in Eq. (18.6-9) are a sum of hydrogen-like one-electron

Hamiltonian operators. In zero order, we ignore the double sum representing the

electron–electron repulsions and obtain the Hamiltonian operator:

Ĥ(0)

∑Z

i 1

ĤHL(i) (18.6-10)