844 20 The Electronic States of Diatomic Molecules
In order to discuss molecules beyond Be 2 , we construct additional LCAOMOs,
following the same policies that we applied earlier: (1) Each LCAOMO is a combi-
nation of two atomic orbitals centered on different nuclei; (2) each LCAOMO is an
eigenfunction of the symmetry operators belonging to the molecule. From each pair of
atomic orbitals, a bonding orbital and an antibonding orbital can be constructed. We
construct six space orbitals from the real atomic orbitals of the 2psubshells:ψσg 2 pzC(ψ 2 pzA−ψ 2 pzB) (20.3-17a)
ψσ∗u 2 pzC(ψ 2 pzA+ψ 2 pzB) (20.3-17b)
ψπu 2 pxC(ψ 2 pxA+ψ 2 pxB) (20.3-17c)
ψπ∗g 2 pxC(ψ 2 pA−ψ 2 pxB) (20.3-17d)
ψπu 2 pyC(ψ 2 pyA+ψ 2 pyB) (20.3-17e)
ψπ∗g 2 pyC(ψ 2 pyA−ψ 2 pyB) (20.3-17f )The normalizing constantccan have a different value in each case. The complex 2p
atomic orbitals could be used as basis functions instead of the real 2porbitals in order
to have a definite value ofLzfor each orbital.
Figure 20.11 shows the orbital regions of the real LCAOMOs constructed from 2p
atomic orbitals. The 2pzatomic orbitals produce sigma molecular orbitals since they
correspond tom0. Theπu 2 pxandπu 2 pyorbitals in Eq. (20.3-17) are called pi
orbitals since the 2pxand 2pyatomic orbitals are linear combinations of the 2p1 and
2 p,−1 orbitals. If complex LCAOMOs were constructed from the 2p1 and 2p,− 1
orbitals, the bonding orbitals would be calledπu 2 p1 andπu 2 p,−1 and the antibond-
ing orbitals would be calledπ∗g 2 p1 andπ∗g 2 p,−1. These orbitals would be eigen-
functions of thêLzoperator withm1 andm−1. The bonding pi orbitals are
“u” instead of “g” because the two lobes of the 2patomic orbitals have opposite
signs, making the bonding pi orbitals eigenfunctions of the inversion operator with
eigenvalue−1.
Figure 20.12 shows acorrelation diagramin which the energies of the atomic
orbitals and the LCAO molecular orbitals are shown schematically with line segments
connecting the LCAOMOs and the atomic orbitals from which they were constructed.
When using these orbitals for a particular diatomic molecule, the atomic orbitals with(a) (b) (c) (d)Figure 20.11 Orbital Regions for LCAO Molecular Orbitals Made from 2pAtomic Orbitals.(a) Theσg 2 pzLCAOMO. (b) Theσ∗u 2 py
LCAOMO. (c) Theπu 2 pyLCAOMO. (d) Theπ∗g 2 pyLCAOMO. Positive regions are in black and negative regions are in gray.