Page 65 - Valence Bond Methods. Theory and Applications

3 H 2 and delocalized orbitalØ
48
wher S = 1s a |1s b , and th signs ar appropriat for
tioð gives us two new functions (see Eq. (1.48))
(3.3)
|A = P|1s a + Q|1s b ,
|B = Q|1s a + P|1s b , S > 0. This orthogonaliza-
(3.4)
wher
1 1
P = √ + √ , (3.5)
2 1 + S 2 1 − S
1 1
Q = √ − √ . (3.6)
2 1 + S 2 1 − S
We us thes ið a single Heitler–Londoð covalent configuration,
orth = A(1)B(2) + B(1)A(2),

and calculat th energy. When R →∞ w obtaið E =−1 au, just as w should. At
❛
R = 0.741 A, howver, wher w hŁve seen that th energy should b a minimum,
w obtaið E =−0.6091 hartree, much higher than th correct valu of –1.1744
hartree. Th result for this orthogonalized basis, which represents not only no
binding but actual repulsion, could hardly b worse.
It is interesting to consider this functioð ið terms of th covalent and ionic func-
tions of Chapter 2. If th |A and |B functions ið terms of th basic AOs ar
substituted into orth and th result normalized, one sees that
√
1 + S 2
orth = (ψ C − Sψ I ),
1 − S 2
where, as always, S is th orbital overlap. Thus, this is exactly th symmetrically
orthogonalized functioð closer to ψ C discussed above, and its vector representatioð
ið Fig. 2.à is clearly a considerable distanc from th optimum eigenvector. Thus w
should not b surprised at th poor valu for th variational energy corresponding
to orth .
Th early workers do not comment particularly oð this result, but, ið light of
present understanding, w may say that th symmetric orthogonalizatioð gives very
clos to th poorest possible linear combinatioð for determining th lowest energy.
This results from th added kinetic energy of th orbitals produced by a node that is
not needed. Alternatively, one could say that th symmetric orthogonalizatioð yields
antibonding orbitals wher bonding orbitals ar needed. This is a gooł example of
how th orthogonalizatioð between different centers can hŁve serious consequences
for obtaining gooł energies and wave functions. We shall see shortly that ther are,
howver, linear combinations determined ið other ways that work quit well.