Mathematical Foundations - Models for Bonding in Chemistry - page 22

the number of electrons in the H 2 molecule, and:
ð drP exch ov <

ð r ; r Þ

S 2 ð dr n a

Þ o Þ þ b

S

Sa 2 Sb 2

¼ ð r Þ b ð r Þ ð r ð r Þ a ð r Þ ð r

ð 1 : 108 Þ

1 þ

:

S

¼

S 2 ð

2S 2S Þ¼ 0

1 þ

in agreement with Equations (1.92). However, the energy changes asso-
ciated with the quantum mechanical exchange-overlap component
(Equation 1.106) of the interaction energy are the greatest contributors
to the energy of the chemical bond (see Table 1.2).
Equations (1.105) and (1.106) are the Heitler-London counterpart of
the corresponding quantities (Equations 3.4 and 3.5 on page 340 of
Ruedenberg's paper (1962), which refers to a LCAO-MOwave function.
Ruedenberg calls Equation (1.106) 'themodification of the quasi-classical
density due to the interference effect', while we, more literally, speak of
exchange

½ ð Þ ð Þ ½ ð Þ ð Þ ½

Sa 2

ð Þ ½

Sb 2

ð Þ

a r b r , b r a r and overlap r , r densities.

Finally, it is worth noting that, while:

q A cl q b cl

¼ ¼ 1 ð 1 : 109 Þ

is the classical electron charge on separate Aand B (one electron on eachH
atom),

S

exch ov exch ov

q ¼ q ¼

S 2 >

0 ð 1 : 110 Þ

AB BA

1 þ

is the fraction of electronic charge transferred in the bond region, due towhat
Ruedenberg calls the 'constructive interference', and which means bonding.

10 3 E h Þ

Table 1.2 Optimized bond energies and their components ð for ground

state H 2

1

S g

D E exch ov

R/a 0 D E cb D E ð Þ

1 15.85 104.43 88.58

1.2 9.93 119.03 128.96

1.4 19.42 119.63 139.05

1.6 21.83 112.54 134.37

1.8 21.08 101.60 122.68

2 18.99 89.02 108.01

4 1.68 9.68 11.36

6 0.06 0.45 0.51

8 0.00 2 0.01 5 0.01 7