Because the molecular orbitals must be normalized, we can determine ex-
pressions for c 1 and c 2. Normalizing the first equation in 12.40:
H 2 ,1H 2 ,1d 1 c^21 (H(1)H(1) 2 H(1)H(2)H(2)H(2))d
1 c^21 (2 2 *H(1)H(2)d)
where the fact that the atomic orbitals on the same atom are normalized has
been used for simplification. The integral *H(1)H(2)dinvolves a wave-
function from each atomic center and, as discussed above, cannot be assumed
to be identically zero. This is an example of an overlap integral and is usually
given the abbreviation S 12. (We defined overlap integrals in our previous dis-
cussion of linear variation theory.) The normalization of the molecular wave-
function proceeds as
1 c^21 (2 2 S 12 )c 1
2 1
2 S 12
(12.41)
By performing a similar normalization, the coefficient for the second molecu-
lar orbital can be shown as
c 2
2 1
2 S 12
(12.42)
The two coefficients are not the same as long as S 12 is not zero! The complete
wavefunctions are
H 2 ,1
2 1
2 S 12
(H(1)H(2))
H 2 ,2
2 1
2 S 12
(H(1)H(2))
(12.43)
Now we will evaluate the average energies of these two molecular orbitals for
H 2 . Using the first wavefunction and assuming the purely electronic
Hamiltonian where the nuclei are separated at some distance R:
E 1 c^21 (H(1)HˆH(1)H(1)HˆH(2)H(2)HˆH(1)H(2)HˆH(2))d
(Note the subscripts on each of the ’s.) We substitute the following defini-
tions from linear variation theory into the above equation:
H 11 H 22 H(1)HˆH(1)dH(2)HˆH(2)d
H 21 H 12 H(1)HˆH(2)dH(2)HˆH(1)d
(12.44)
These integrals are very similar to those in equation 12.28, except that now
wavefunctions from different atoms can interact mathematically.H 11 and H 22
are simply the energies of the atomic orbitals.H 12 and H 21 ,however,represent
a sort of energy of mixing of wavefunctions from two different atoms. These in-
tegrals are called resonance integrals.Integrals of this sort—where one comes
from one atomic center and the other comes from another atomic center—
are not predicted by classical mechanics and are of purely quantum-mechanical
origin. The equalities in equation 12.44 arise from the fact that both atoms are
hydrogens. If this were a heteronuclear diatomic system, each resonance inte-
gral H 11 ,H 12 ,H 21 , and H 22 would have its own independent definition.
12.11 Introduction to LCAO-MO Theory 407