20.2 LCAOMOs. Approximate Molecular Orbitals That Are Linear Combinations of Atomic Orbitals 835
Overlap
region(a)(b)(c)Figure 20.7 The Orbital Region for theσg 1 sandσ∗u 1 sLCAO Molecular Orbitals.(a) The overlapping orbital regions of the 1sA and
1 sB atomic orbitals. (b) The orbital region of theσg 1 sLCAOMO. (c) The orbital region of theσ∗u 1 sLCAOMO. In parts b and c, positive
regions are in black and negative regions are in gray.
EBO/eVrAB/10–10m(^) g 1 s
energy
*u 1 senergy
Born–Oppenheimer
energy of exact
orbitals
1
4
23
2
0
2
4
6
Figure 20.8 The Orbital Energies for theσg1sandσ∗u1sLCAO Molecular Orbitals.
The 2pxand 2pyatomic orbitals are not included in Eq. (20.2-8) because they have
different symmetry about the bond axis than does the exact ground-state orbital. If they
were included with nonzero coefficients, the LCAOMO would not be an eigenfunc-
tion of the same symmetry operators as the exact orbitals. If the variation energy is
minimized with respect to theccoefficients, the function of Eq. (20.2-8) would give a
better (lower) value than does theσg 1 sorbital.
Exercise 20.6
Argue that the 2pxand 2pyatomic orbitals are eigenfunctions of thêC 2 zoperator with eigenvalue
−1, while the 2pzorbital is an eigenfunction with eigenvalue+1. Argue that a linear combination
of all three of these orbitals is not an eigenfunction of thêC 2 zoperator.