20.1 The Born–Oppenheimer Approximation and the Hydrogen Molecule Ion 827
antibonding orbital, which has a shorter de Broglie wavelength because of the node
between the nuclei.Angular Momentum Properties of Molecular Orbitals
In Chapter 15 we asserted that a set of commuting operators can have a set of com-
mon eigenfunctions. For example, the hydrogen atom spin orbitals can be chosen to
be eigenfunctions ofĤ,̂L^2 ,̂Lz, and̂Sz. ThêL^2 operator does not commute with the
electronic Hamiltonian of the H+ 2 ion. The physical reason for this is that all directions
are not equivalent as they are with atoms, because of the presence of the two fixed
nuclei. The operator̂Lzdoes commute with the electronic Hamiltonian operator if the
internuclear axis is chosen as thezaxis. The molecular orbitals can be eigenfunctions
of̂Lz:̂Lzψhmψ ̄ (20.1-8)wheremis an integer. There is no quantum numberlso that the values ofmare not
limited to a specific range. We define a non-negative quantum numberλ, equal to the
magnitude ofm:λ|m| (20.1-9)The exact Born–Oppenheimer molecular orbitalsψ 1 andψ 2 both correspond toλ0.
A nonzero value ofλcorresponds to two states becausemcan be positive or negative.
Sincelis not a good quantum number, we use the quantum numberλto classify
molecular orbitals, using the following Greek-letter designations:Value ofλ Symbol
0 σ
1 π
2 δ
3 φ
etc.Note the similarity with the letterss,p,d, andfin the atom case. Both the ground state
orbital and the first excited-state orbital of H+ 2 areσ(sigma) orbitals. Sigma orbitals of
diatomic molecules contain the factorΦ 0 and have orbital regions that are cylindrically
symmetric about the bond axis. The other orbitals can be chosen to include either the
realΦfunctions or the complexΦfunctions, and have nodal planes containing the
internuclear axis.
The spin operator̂Szcommutes with the H+ 2 Hamiltonian operator, so we can have
molecular spin orbitals that are eigenfunctions of̂Sz. The spin eigenvalues are equal
tomsh ̄±(1/2)h ̄, the same as with the hydrogen atom. We can represent the spin
orbitals by a space orbital multiplied by the spin functionαor the spin functionβ,as
with atomic orbitals.Symmetry Properties of the Molecular Orbitals
There is another class of operators that can commute with the electronic Hamiltonian
operator of the H+ 2 ion. These operators aresymmetry operators. Before we can define
how these operators operate on functions, we define how they operate on a point in