spin-orbital wavefunction must be antisymmetric with respect to exchange of
the two electrons. Recall that the wavefunction for the electron in the bonding
orbital of H 2 , from equation 12.43, is
H 2 ,1
2 1
2S 12
(H(1)H(2))
The other wavefunction from equation 12.43 is for the antibonding orbital.
Since both electrons can be described with this spatial wavefunction, the spa-
tial wavefunction for the H 2 molecule is the product of two such ’s:
H 2
2
1
2 S 12
[H(1)(el.1)H(2)(el.1)][H(1)(el.2)H(2)(el.2)]
(12.49)where each linear combination has been labeled as referring to electron 1 or
electron 2 (el.1 or 2). This spatial wavefunction must be multiplied by the anti-
symmetric spin function 1/ 2 [ (1) (2) (2) (1)] to get the complete
wavefunction that satisfies the Pauli principle (that is, is antisymmetric). The
complete wavefunction is
H 2
1
2
2 1
2 S 12
[H(1)(el.1) H(2)(el.1)] (12.50)
[(H(1)(el.2) H(2)(el.2)][ (1) (2) (2) (1)]
The average energy of this wavefunction can also be evaluated versus R
under the Born-Oppenheimer approximation. We can calculate that, with re-
spect to two separated H atoms, the decrease in energy upon bonding is
4.32 10 ^19 J for the hydrogen molecule (compared to 7.59 10 ^19 J ex-
perimentally), at a minimum-energy Rof 0.85 Å (compared to 0.74 Å experi-
mentally). The calculated bond order of H 2 is 1, corresponding to the existence
of a single bond.
Because both electrons reside in the bonding orbital, an “electron config-
uration” of^2 can be used to describe the ground electronic state of H 2.
(Because H 2 is a homonuclear diatomic molecule, we add a subscript “g” to the
label—g^2 —to indicate the orbital’s symmetry property with respect to the
center of the molecule. Electrons in the antibonding orbital are labeled u,
the “u” also referring to the orbital’s symmetry properties.†Symmetry will be
discussed in the next chapter.) To emphasize that the electrons in the orbital
derive from 1selectrons from H atoms, the more detailed (g 1 s)^2 label can also
be used.
Larger atoms have more occupied atomic orbitals that can combine into
molecular orbitals. It is common to use the second row of atoms, Li through
Ne, to illustrate the principles. For diatomics having electrons originating from
different atomic electronic shells, we adopt the approximation that only atomic
orbitals of similar energies will combine to make molecular orbitals.
Thus, for Li 2 , the 1sorbital from one Li atom will interact with the 1sor-
bital from the other Li atom, as we saw occurring in H 2. Additionally, the 2s
orbital of the first Li atom will interact with the 2sorbital of the second Li
atom, creating another bonding and antibonding pair of molecular orbitals.
The four 1selectrons will fill the g 1 sand g 1 smolecular orbitals, and the two
12.13 Molecular Orbitals of Other Diatomic Molecules 411†The labels “g” and “u” stand for the German words geradeand ungerade,respectively.