Physical Chemistry Third Edition

832 20 The Electronic States of Diatomic Molecules

kind (same isotope of the same element) it belongs to the molecule. For a multielectron

molecule, this test is generally easier to apply than the test in which the electrons are

moved.

The operation of the inversion operator on the electron of an H+ 2 molecule ion is

illustrated in Figure 20.6. This motion brings the electron to the same distance from

nucleus A as it originally was from nucleus B and vice versa, and does not change

the potential energy. The inversion operator belongs to the H+ 2 molecule ion. We can

also apply the inversion operator to the nuclei and see that it simply exchanges the

nuclei.

Position

of the

electron

after the

inversion

operation

Nucleus B

Original

position

of the

electron

Nucleus A

x

z

y

Figure 20.6 The Effect of the Sym-
metry Operator̂i on the Electron of
the H+ 2 Ion.

Exercise 20.4

a.Show that the symmetry operatorŝσh,̂Cnz, and̂C 2 abelong to the H+ 2 molecule ion, where

nis any positive integer and whereastands for any axis in thexyplane. Show this first by

applying the symmetry operators to the electron’s position with fixed nuclei. Then apply them

to the nuclei with a fixed electron.

b.ThêC∞zoperator corresponds to an infinitesimal rotation about thezaxis, and infinitely

many applications of this operator can correspond to rotation by any angle. Show that̂C∞z

belongs to the H+ 2 ion.

c.There are infinitely manŷσvoperators. Show that all of them belong to the H+ 2 ion.

The ground-state molecular orbitalψ 1 is an eigenfunction of each of the symmetry

operators in Exercise 20.4, and each eigenvalue is equal to+1. The first excited-state

orbitalψ 2 is also an eigenfunction of these operators, but the eigenvalues of̂i,̂σh, and

̂C 2 aare equal to−1. An eigenfunction having an eigenvalue of̂iequal to+1 is denoted

by a subscript g (from the Germangerade, meaning “even”) and an eigenvalue of̂i

equal to−1 is denoted by a subscript u (from the Germanungerade, meaning “odd”).

An eigenfunction having an eigenvalue of̂σhequal to−1 is denoted by an asterisk

(*). Orbitals with asterisks are antibonding since they have a nodal plane through the

origin perpendicular to the bond axis. No superscript or subscript is used to denote an

eigenvalue of̂σhequal to+1. We now use the notationsψ 1 σgandψ 2 σ∗ufor our first

two H+ 2 molecular orbitals.

PROBLEMS

Section 20.1: The Born–Oppenheimer Approximation
and the Hydrogen Molecule Ion

20.1 What is the symmetry operation that is equivalent to the
product̂îσh? Is this the same as the product̂σĥi?

20.2 What is the symmetry operation that is equivalent to the
product̂C 2 ẑi? Is it the same as the product̂îC 2 z?

20.3 Write the function that is equal tôC 4 zψ 2 px.

20.4 Write the function that is equal tôC 2 zψ 2 px.

20.5 a.Draw a sketch of the orbital region of the 3dx (^2) −y 2
hydrogen-like orbital.
b. List the symmetry operators of which this orbital
is an eigenfunction. Give the eigenvalue of each
operator for the 3dx (^2) −y 2 hydrogen-like orbital.
20.6.The symbol (called a column vector)
⎡
⎣
a
b
c
⎤
⎦is used to
represent the point given byxa,yb, andzc. The
symbol (called a row vector) [abc] is equivalent in some
ways.