Physical Chemistry Third Edition

768 18 The Electronic States of Atoms. II. The Zero-Order Approximation for Multielectron Atoms

where theψfunctions are spin orbitals and whereCis a constant that can be chosen to

normalize the function. By writing Eq. (18.2-6) we haveantisymmetrizedthe wave

function. We do not now specify which electron occupies which spin orbital.

Exercise 18.1

By explicit manipulation, show that the function of Eq. (18.2-6) obeys Eq. (18.2-5).

There is an important fact about electrons that we can see in Eq. (18.2-6). If the spin

orbitalsψ 1 andψ 2 are the same function, the antisymmetric wave function is the differ-

ence of two identical terms and vanishes. Therefore, a given spin orbital cannot occur

more than once in any term of an antisymmetrized two-electron orbital wave func-

tion. This is an example of thePauli exclusion principle, which applies to orbital wave

functions for any number of electrons:In an antisymmetrized orbital wave function,

the same spin orbital cannot occur more than once in each term.Another statement of

the Pauli exclusion principle is:In an antisymmetrized orbital wave function, no two

electrons can occupy the same spin orbital.

The Pauli exclusion principle is named
for Wolfgang Pauli, 1900–1958, an
Italian-American physicist who received
the 1945 Nobel Prize in physics for his
contributions to quantum mechanics.
The probability density for the antisymmetrized wave function of Eq. (18.2-6) is

Ψ(1, 2)∗Ψ(1, 2)|C|^2

[

|ψ 1 (1)|^2 |ψ 2 (2)|^2 −ψ 1 (1)∗ψ 2 (2)∗ψ 1 (2)

−ψ 2 (1)∗ψ 1 (1)ψ 1 (2)∗ψ 2 (2)+|ψ 2 (1)|^2 |ψ 1 (2)|^2

]

(18.2-7)

To normalize the wave function, we integrate the function of Eq. (18.2-7) over the

space coordinates and spin coordinates of both particles and set the result equal to

unity. If the orbitals are normalized the first term and the fourth term both give unity

after integration. Because the hydrogen-like spin orbitals are orthogonal to each other,

the second and third terms give zero after integration. The result is

1 |C|^2 [1+1] 2 |C|^2 (18.2-8)

IfCis taken to be real and positive,

C

√

1

2

(18.2-9)

PROBLEMS

Section 18.2: The Indistinguishability of Electrons and
the Pauli Exclusion Principle

18.3 Pretend that electrons are bosons with zero spin. Describe
how the ground state of the helium atom would differ from
the actual ground state in the orbital approximation.

18.4 Pretend that electrons are bosons with zero spin.

Describe how the excited states of the helium atom

would differ from the actual excited states in the orbital

approximation.

18.3 The Ground State of the Helium Atom in Zero Order

The lowest-energy state of a system is called itsground state. The lowest zero-order

energy of the helium atom corresponds to both electrons occupying 1shydrogen-

like orbitals with different spins. The antisymmetrized zero-order ground-state wave