Physical Chemistry Third Edition

20.1 The Born–Oppenheimer Approximation and the Hydrogen Molecule Ion 831

Solution

The function is

ψ 1 s

√

1

π

(

Z

a

) 3 / 2

e−Zr/a

√

1

π

(

Z

a

) 3 / 2

e−Z(x

(^2) +y (^2) +z (^2) ) 1 / (^2) /a
Whenxis replaced by−x,yis replaced by−y, andzis replaced by−z, the value ofris
unchanged so that the function is unchanged.
̂iψ 1 sψ 1 s
Theψ 1 sfunction is an eigenfunction of the inversion operator with eigenvalue 1.

EXAMPLE20.4

Determine the spherical polar coordinates of̂iPand̂σhPifPrepresents a point whose

spherical polar coordinates are (r,θ,φ).

Solution

̂iP̂i(r,θ,φ)(r, 180◦−θ, 180◦+φ)(r,π−θ,π+φ)

̂σhP̂σh(r,θ,φ)(r, 180◦−θ,φ)(r,π−θ,φ)

Exercise 20.3

Show that theψ 2 pzhydrogen-like orbital is an eigenfunction of thêσhoperator with eigen-

value−1.

The electronic energy eigenfunctions of a molecule can be eigenfunctions of the

symmetry operators that commute with the electronic Hamiltonian. The symmetry

operators that commute with the electronic Hamiltonian are said to “belong” to the

molecule. In order for a symmetry operator to commute with the electronic Hamiltonian,

it must leave the potential energy unchanged when applied to the electron’s coordinates.

Otherwise, a different result would occur if the symmetry operator were applied to an

electronic wave function after application of the Hamiltonian than if the operators were

applied in the other order.

In the Born–Oppenheimer approximation a symmetry operator operates on all of the

electron coordinates but does not affect the nuclear coordinates. If a symmetry operator

leaves the potential energy unchanged, it must move every electron to a position where

it either is at the same distance from each nucleus as in its original position, or is at the

same distance from another nucleus of the same kind. We can test whether a symmetry

operator belongs to a molecule by applying it to all electrons and seeing if each electron

is now at the same distance from each nucleus as it originally was from a nucleus of

the same kind.

There is a second test to determine whether a symmetry operator belongs to a

molecule in a particular nuclear conformation. We apply it to all of the nuclei of the

molecule instead of applying it to the electrons. If a symmetry operator moves every

nucleus to a location previously occupied by that nucleus or by a nucleus of the same