Computational Chemistry

a, thecoulomb integral

Z

fiH^fidv¼Hii¼a i:e:basis functions on the same atom; '(4: 61 a)

b, the bond integral or resonance integral

Z

fiH^fjdv¼Hij¼

Z

fjH^fidv¼Hji¼b '(4: 61 b)

for basis functions on adjacent atoms;
Z
fiH^fjdv¼Hij¼

Z

fjH^fidv¼Hji¼ 0 '(4: 61 c)

for basis functions neither on the same or on adjacent atoms.
To give these approximations some physical significance, we must realize that in
quantum mechanical calculations the zero of energy is normally taken as
corresponding to infinite separation of the particles of a system. In the simplest
view,a, the coulomb integral, is the energy of the molecule relative to a zero of
energy taken as the electron and basis function (i.e. AO; in the simple H€uckel
method,fis usually a carbonpAO) at infinite separation. Since the energy of the
system actually decreases as the electron falls from infinity into the orbital,ais
negative (Fig.4.13). The negative ofa, in this view, is the ionization energy (a
positive quantity) of the orbital (the ionization energy of the orbital is defined as the
energy needed to remove an electron from the orbital to an infinite distance away).

C

C C

a = Hii < 0 kJ mol–1

b = Hij < 0 kJ mol–1

0

Energy

electron falls from infinite e–

distance into a p AO on C electron falls from infinite

distance into a MO formed

by overlap of two p AOs on

adjacent carbons

Fig. 4.13 The coulomb integralais most simply (but not too accurately) viewed as the energy of
an electron in a carbon 2porbital, relative to its energy an infinite distance away. The bond integral
(resonance integral)bis most simply (but not too accurately) viewed as the energy of an electron
in an MO formed by adjacent 2porbitals, relative to its energy an infinite distance away

4.3 The Application of the Schr€odinger Equation to Chemistry by H€uckel 127