Appendix E: Orbital Localization and Orthogonality
231
The interactions between electrons placed in the ϕ a and ϕ b orbitals are represented
by the Coulomb and exchange integrals
ϕ a ϕ b
r
−1
12
ϕ a ϕ b
and
ϕ a ϕ b
r
−1
12
ϕ b ϕ a
.
(E.6)
Figure E.2 shows the dependence of S ab , ε and the two-electron integrals on the
internuclear distance. As the internuclear distance increases the overlaps between
the χ α functions and between the charge distributions ρ α = −χ
2
α tend to zero. So,
the queue of each ϕ α orbital on the other nucleus, that is proportional to ε, tends
to vanish and ϕ α → χ α . Therefore, when R is large enough the Coulomb integral
approaches the interaction of two nonoverlapping spherical charge distributions, i.e.,
tends to 1/R as shown in Fig. E.2 (note that this interaction is perfectly balanced
by the nucleus–electron attractions plus the nucleus–nucleus repulsion). At the same
time the exchange integral decreases much faster, because the integrand contains
two factors ϕ a (r i )ϕ b (r i ), with i = 1, 2, that are everywhere small when the two
orbitals are not overlapping. The dependence of the exchange integral on R beyond an
intermediate distance (say 2.5 bohr) can be approximated by an exponential function,
as shown in Fig. E.2.
References
1. Lowdin, P.-O.: On the Nonorthogonality problem connected with the use of atomic
wavefunctions in the theory of molecules and crystals. J. Chem. Phys. 18, 365–375
(1950)
2. Mayer, I.: On Löwdin’s method of symmetric orthogonalization. Int. J. Quantum
Chem. 90, 63–65 (2002)
231
The interactions between electrons placed in the ϕ a and ϕ b orbitals are represented
by the Coulomb and exchange integrals
ϕ a ϕ b
r
−1
12
ϕ a ϕ b
and
ϕ a ϕ b
r
−1
12
ϕ b ϕ a
.
(E.6)
Figure E.2 shows the dependence of S ab , ε and the two-electron integrals on the
internuclear distance. As the internuclear distance increases the overlaps between
the χ α functions and between the charge distributions ρ α = −χ
2
α tend to zero. So,
the queue of each ϕ α orbital on the other nucleus, that is proportional to ε, tends
to vanish and ϕ α → χ α . Therefore, when R is large enough the Coulomb integral
approaches the interaction of two nonoverlapping spherical charge distributions, i.e.,
tends to 1/R as shown in Fig. E.2 (note that this interaction is perfectly balanced
by the nucleus–electron attractions plus the nucleus–nucleus repulsion). At the same
time the exchange integral decreases much faster, because the integrand contains
two factors ϕ a (r i )ϕ b (r i ), with i = 1, 2, that are everywhere small when the two
orbitals are not overlapping. The dependence of the exchange integral on R beyond an
intermediate distance (say 2.5 bohr) can be approximated by an exponential function,
as shown in Fig. E.2.
References
1. Lowdin, P.-O.: On the Nonorthogonality problem connected with the use of atomic
wavefunctions in the theory of molecules and crystals. J. Chem. Phys. 18, 365–375
(1950)
2. Mayer, I.: On Löwdin’s method of symmetric orthogonalization. Int. J. Quantum
Chem. 90, 63–65 (2002)