21.2 The BH 3 Molecule and thesp^2 Hybrid Orbitals 871
hydrogen 1sorbital have the same symmetry around the bond axis. The LCAOMOs
have coefficients of larger magnitude for the hydrogen atoms than for the beryllium
atom, corresponding to the greater electronegativity of the hydrogen atoms.
There is also a procedure in which linear combinations of the canonical orbitals
are formed such that the energy of repulsion between electrons in different orbitals is
minimized. The resulting orbitals are calledenergy-localizedorbitals.^1 For the BeH 2
molecule, the energy-localized orbitals are very similar to the localized orbitals in
Eqs. (21.1-1) and (21.1-2).
From our experience with LiH and BeH 2 , we now assert that to some usable approx-
imation we can construct localized LCAOMOs for various molecules using hybrid
orbitals where appropriate and including only two atomic orbitals in our LCAOMOs.
We use the same rules that we applied to LiH and BeH 2 : Each bonding LCAOMO will
be constructed of two atomic orbitals centered on different atoms such that the orbitals
(1) have orbital energies that are fairly close together in value, (2) overlap significantly,
and (3) have the same symmetry about the bond axis.PROBLEMS
Section 21.1: The BeH 2 Molecule and the
spHybrid Orbitals
21.1 Construct an accurate graph of the berylliumψ 2 sp(1)
hybrid orbital along thezaxis, usingz/aas the
independent variable. Use a value ofZeffequal to 2.776
(1.00 greater than the value ofZefffor Li in Section 19.4).
21.2 a.The ionization potential of a sodium atom is 5.139 eV.
Find the effective number of protons seen by the 3s
electron in a sodium atom as was done in
Example 19.4.
b. Construct an accurate graph of the sodiumψ 3 sp(1)
hybrid orbital along thezaxis, usingz/aas the
independent variable. Since the 3sorbital is positive in
the bonding region while thespzorbital is negative in
the bonding region, we take
ψ 3 sp(1)√
1 /2(ψ 3 s−ψ 3 pz)instead of using the signs that were used forψ 2 sin
ψ 2 ps(1). Use the value ofZefffrom part a.c.Sketch the orbital region of theψ 3 sp(1)hybrid
orbital.21.3 Describe the bonding in the B 2 molecule using LCAOMOs
made from 2sphybrid orbitals. What is the bond order?
Compare your description with that of Table 20.2.
Which seems more reasonable according to the VSEPR
theory?21.4Assume that BH 2 has a linear equilibrium conformation.
Compare its bonding with that of BeH 2. What orbital does
the unpaired electron occupy?21.2 The BH 3 Molecule and thesp^2 Hybrid
Orbitals
The BH 3 molecule is known from experiment to have an equilibrium nuclear
conformation that is trigonal planar (planar with three 120◦bond angles). We assume
the Born–Oppenheimer approximation and use this conformation. We place the B atom(^1) P.-O. Löwdin, ed.,Quantum Theory of Atoms,Molecules,and the Solid State, Academic Press,
San Diego, 1966, pp. 263–280.