Physical Chemistry Third Edition

766 18 The Electronic States of Atoms. II. The Zero-Order Approximation for Multielectron Atoms

These are two hydrogen-like Schrödinger equations. The eigenvaluesE 1 andE 2 are

hydrogen-like orbital energies. The total electronic energy in the zero-order approxi-

mation is

E(0)n 1 n 2 En 1 (HL)+En 2 (HL)−(13.60 eV)

[

Z^2

n^21

+

Z^2

n^22

]

(18.1-12)

wheren 1 andn 2 are two values of the principal quantum number for a hydrogen-like

atom.

The orbitalsψ 1 (1) andψ 2 (2) are hydrogen-like orbitals:

Ψ(0)(1, 2)ψ 1 (1)ψ 2 (2)ψn 1 l 1 m 1 (1)ψn 2 l 2 m 2 (2) (18.1-13)

The function in Eq. (18.1-13) satisfies the zero-order Schrödinger equation, but does

not include the spin angular momentum, which can be included by replacing the space

orbitals by spin orbitals:

Ψ(0)(1, 2)ψ 1 (1)ψ 2 (2)ψn 1 l 1 m 1 ,ms 1 (1)ψn 2 l 2 m 2 ,ms 2 (2) (18.1-14)

We can also represent the spin orbitals as products of space orbitals and spin functions.

One of several possible orbital wave functions would then be

Ψ(0)(1, 2)ψ 1 (1)ψ 2 (2)ψn 1 l 1 m 1 (1)α(1)ψn 2 l 2 m 2 (2)β(2) (18.1-15)

PROBLEMS

Section 18.1: The Helium-Like Atom

18.1 Assume that each of the two electrons in a helium atom is
50.0 pm (5. 00 × 10 −^11 m) from the nucleus.

a.If the two electrons lie on a line passing through the

nucleus and are on opposite sides of the nucleus,

calculate the potential energy of the electron–electron

repulsion. Compare it with the potential energy of the

electron–nucleus attraction.

b.Repeat the calculation of part a assuming that the

two electrons and the nucleus are at the corners

of an equilateral triangle, 50.0 pm from each

other.

18.2 It has been proposed that much of the error due to

assuming a stationary nucleus for the helium atom can be

removed by replacing the mass of each electron by the

reduced mass of an electron and the helium nucleus.

Assume that the helium nucleus is a^4 He nucleus with a

mass of 4.00141 atomic mass units, and calculate the

reduced mass. Compare it with the mass of an electron.

Calculate the effect that inclusion of this would have on

the zero-order ground-state energy of a helium atom.

18.2 The Indistinguishability of Electrons and

the Pauli Exclusion Principle

If a two-electron wave function is denoted byΨ(1, 2), the probability of finding electron

1 in the volume elementd^3 r 1 and simultaneously finding electron 2 in the volume

elementd^3 r 2 is given by

(Probability)|Ψ(1, 2)|^2 d^3 r 1 d^3 r 2 Ψ∗(1, 2)Ψ(1, 2)d^3 r 1 d^3 r 2 (18.2-1)