Physical Chemistry , 1st ed.

nodal surface between the nuclei; the electron’s probability of being at that

point is exactly zero. If all atomic orbitals were assumed to combine to make

molecular orbitals, half of the molecular orbitals would be bonding and half

would be antibonding orbitals. (There are also nonbonding orbitalsthat do not

contribute to molecular bonding, but they will not be considered at this point.)

One useful definition is bond order. If the number of electrons in bonding

orbitals were nbondand the number of electrons in antibonding orbitals were

nantibond, the bond order nis

n

nbond

2

nantibond

 (12.48)

Bond order is qualitatively related to the strength and number (that is, single,

double, triple) of bonds between atoms in a molecule.

It has been assumed that our system and our orbitals are cylindrically

symmetric. This makes sense, because our molecular orbitals were spherical

to start with and the combination of two spheres yields a shape that is cylin-

drical about the line connecting the two nuclei. A cylindrical wavefunction

has a magnitude that is symmetric about some axis, in this case defined by

the line drawn directly between the two nuclei. Such a line is often used to

indicate a bond. Any orbital whose behavior or magnitude is cylindrical

about the bond between the two atoms is called a sigma() orbital. The com-

bination of the two atomic wavefunctions in H 2 therefore yields one bond-

ing sigma orbital (denoted ) and one antibonding sigma orbital (denoted

*). Figure 12.19 shows a labeled molecular orbital diagram for the two mol-

ecular orbitals of H 2 .

Since the orbital for H 2 has a lower energy than the two individual or-

bitals of the separated H atoms, if an electron were residing in that orbital, the

overall energy of the molecular system would be lowered. Lower energies are

more stable, and so the H 2 system would be considered energetically stable

in its ground state. Although it requires special conditions to generate, H 2 is

a stable species relative to separated H H. Because there is one electron in

a bonding orbital and none in an antibonding orbital, the bond order of H 2 

is ^12 . This also implies that a bond exists and that the species would be stable.

If, however, the electron in H 2 were to absorb energy and be promoted into

the antibonding orbital, repulsion between nuclei would be increased and the

molecule should break apart into the stabler H Hspecies. This is indeed

what happens experimentally.

12.13 Molecular Orbitals of Other Diatomic Molecules

The concept of molecular orbitals can be extended to diatomic molecules

larger than the H 2 system. By including a second electron, we can consider

the neutral hydrogen molecule, H 2. For the ground electronic state, we can

borrow the molecular orbital diagram for H 2 , which has a single electron in

the bonding orbital. The second electron in H 2 also resides in this orbital,

but its spin must be opposite the spin of the first electron to satisfy the Pauli

exclusion principle. The MO diagram for H 2 is shown in Figure 12.20.

The approximate wavefunction for H 2 is similar to that for the He atom in

that there are now two electrons that need spatial functions, and the overall

410 CHAPTER 12 Atoms and Molecules

Energy

H 2 +

(^)
H1
1 s 1 s
H2
Figure 12.19 A molecular orbital diagram for
H 2 , showing the and  labels for the molec-
ular orbitals. (Compare this to Figure 12.17.) In
the ground state, the single electron occupies the
lowest-energy molecular orbital. Since this repre-
sents a lowering of the energy with respect to the
energy of the atoms, the molecule is more stable
than the separated atoms.
(^) *
H1
1 s 1 s
Energy
H2
H 2
Figure 12.20 A qualitative molecular orbital
diagram for H 2 is very similar to that of H 2 ,ex-
cept for the presence of the second electron with
a spin opposite that of the first.