Computational Chemistry

The quantityb, the bond integral or resonance integral, is, in the simplest view,
the energy of an electron in the overlap region (roughly, a two-center MO) of
adjacentporbitals relative to a zero of energy taken as the electron and two-center
MO at infinite separation. Likea,bis a negative energy quantity. A rough, naive
estimate of the value ofbwould be the average of the ionization energies (a positive
quantity) of the two adjacent AOs, multiplied by some fraction to allow for the fact
that the two orbitals do not coincide but are actually separated. These views ofaand
bare oversimplifications [ 30 ].
We derived the 2&2 matrices of Eqs.4.55starting with a two-orbital system.
These results can be generalized to n orbitals:

H¼

H 11 H 12 ... H 1 n

H 21 H 22 ... H 2 n

... ... ... ...

Hn 1 Hn 2 ... Hnn

0

B

BB

B

@

1

C

CC

C

A

(4.62)

TheHelements of Eq.4.62becomea,b, or 0 according to the rules of Eqs. 4.61.
This will be clear from the examples in Fig.4.14.

C C

H

H

H

H

1 2

*

* = + or. or –

1

2

3

1

3 2

4

a

b

c

a a a a a a a a

b b a

bb

b b

b b

b

b b

b

b

b

0

0

0

0

0

0

Fig. 4.14 Some conjugated molecules, theirporbital arrays, simplified representations of the
molecules, and their simple H€uckel Fock matrices. Same-atom interactions area, adjacent-atom
interactions areb, and all other interactions are 0. To diagonalize the matrices, we usea¼0 and
b¼" 1

128 4 Introduction to Quantum Mechanics in Computational Chemistry