Page 66 - Valence Bond Methods. Theory and Applications

3.n Optimal delocalized orbitalØ
3.2 Optimal delocalized orbitals
We now iðvestigat orbitals that rang over both centers with linear combinations
that minimiz th calculated energy. For this simple two-electroð system thes may
all b viwed as extensions of th Coulso[Fisher approach w describ next. We 49
us th basis of Table 2.à and compar thes results with th appropriat full MCVB
calculations of Sectioð 2.8À
3.2.1 The method of Coulson and Fisher[15]
Th first calculatioð of th energy of H 2 for optimal delocalized orbitals used

A = 1s a + λ1s b , (3.7)
B = λ1s a + 1s b , (3.8)
and, using th “covalent” function, A(1)B(2) + B(1)A(2)¤ ið th Rayleigh quotient,
adjusted th valu of λ to minimiz th energy. We will not duplicat this calculatioð
here, but bring this up, becaus th methods w will discuss ar generalizations of
th Coulso[Fisher approach wher w us ið th orbitals all of th functions of
our basis with th appropriat symmetry.

3.2.2 Complementary orbitals
It will b observed that th Coulso[Fisher functions satisfy th relations

σ h A = B (3.9)

and

σ h B = A, (3.10)
wher σ h is th operatioð of D ∞h that reflects th molecule end for end. If A and B
1
1
ar also of σ symmetry, th “covalent” functioð A(1)B(2) + B(1)A(2) is of +
g
symmetry. Thus, th overall stat symmetry is correct, although th orbitals do not
belong to a single irreducible representation. For our first calculatioð w take all of
th σ-typ AOs of th basis and form (th unnormalized)
A = 1s a + a1s b + b2s a + c2s b + dp zà + ep zŁ (3.11)

and
B = σ h A, (3.12)

1 Th reader should not carefully th two different uses of th symboo σ here. One is a group operation, th other
th stat designatioð of an orbital.