Computational Chemistry

Kij¼

Z

c$ið 1 Þc$jð 2 Þ

1

r 12



cið 2 Þcjð 1 Þdv 1 dv 2 ð 5 : 22 Þ

Kis called an exchange integral; mathematically, it arises from Slater determinant
expansion terms that differ only in exchange of electrons. Note that the terms on
either side of 1/r 12 differ by exchange of electrons. It is often said to have no simple
physical interpretation, and even to represent an “exchange force”, but looking at
Eq.5.17, we see we can regardKas a kind of correction toJ, reducing the effect ofJ
(bothJandKare positive, withKsmaller), i.e. reducing the electrostatic potential
energy due to the mutualci,cjcharge cloud repulsion referred to above in connec-
tion withJandK. This reduction in repulsion arises because as particles with an
antisymmetric wavefunction, two electrons can’t occupy the same spin orbital
(roughly, can’t be at the same point in space), and can occupy the same spatial
orbital only if they have opposite spins. Thus two electronsof the same spinavoid
each other more assiduously than expected only from the coulombic repulsion that is
taken into account byJ. We could consider the summed 2J#Kterms of Eq.5.17to
be the true coulombic repulsion (within the charge cloud model), corrected for
electron spin, i.e. corrected for the Pauli exclusion principle effect. TheJandK
interactions are shown in Fig.5.4for a four-electron molecule, the smallest closed-
shell system in whichKintegrals arise. A detailed exposition of the significance of
the Hartree– Fock integrals is given by Dewar [ 15 ]. Note that outside the nucleus the
only significant forces in atoms and molecules are electrostatic; there are no vague
“quantum-mechanical forces” in chemistry [ 16 ]. Chemical reactions involve the
shuffling of atomic nuclei under the influence of the electromagnetic force.

4 J integrals

(between electrons in different

spatial MOs)

2 K integrals

(between electrons of the same spin)

Fig. 5.4 TheJintegrals represent interactions between electrons in different spatial orbitals; the
Kintegrals represent interactions between electrons of the same spin

188 5 Ab initio Calculations