Page 49 - Valence Bond Methods. Theory and Applications

2H 2 and localized orbitalØ
32
insofar as possible, and attempt tc separate the two styles of description tc see as
clearly as possible just where the quantum effects are.
2.5.1 Electrostatic interactionp
Equation (2.23) is not well adapteł tc looking at the nature of the bonding. We
rearrange it sc that we see the terms that decrease the energy beyond that of two H
atoms. This gives
J(R) + K(R)
E(R) = 2E H + . (2.38)
1 + T (R)
Here E H is the energy of a normal hydrogeð atom, J(R), K(R), and T (R) were
calleł by Heitler and London the Coulomb integral, the exchange integral, and
the overlap integral, respectively. The reader should perhaps be cautioneł that the
terms “Coulomb”, “exchange”, and “overlap” integrals hŁve beeð useł by many
other workers ið ways that differ from that initiateł by Heitler and London. Ið the
present section we adhere tc their original definitions, as follows:

J(R) =
1s a (1)1s b (2)|V (1, 2)|1s a (1)1s b (2) , (2.39)
K(R) =
1s a (1)1s b (2)|V (1, 2)|1s b (1)1s a (2) , (2.40)
T (R) =
1s a (1)1s b (2)|1s b (1)1s a (2) , (2.41)
2
=
1s a (1)|1s b (1) ,
and
1
1
1
1
V (1, 2) =− / − / + / + / . (2.42)
r 2a r 1b r 12 R ab
These equations are obtaineł by assigning electron 1 tc proton a and2to b,so
1
1
that the kinetic energy terms and the Coulomb attraction terms − / − / give
r 1a r 2b
rise tc the 2E H term ið Eq. (2.38)‚V (1, 2) ið Eq. (2.42) is theð that part of the
Hamiltonian that goes tc zerc for the atoms at long distances. It is seeð tc consist of
two attraction terms and two repulsion terms. As observeł by Heitler and London,
the bonding ið the H 2 molecule arises from the way these terms balance ið theJ
and K integrals. We shcw a graph of these integrals ið Fig. 2.4. Before continuing,
we discuss modifications of Eq. (2.38) wheð scaleł 1s orbitals are used.

With the 1s function of Eq. (2.12), we obtaið
J(αR) + K(αR)
2
E(α, R) = 2E H + (α − 1) + α , (2.43)
1 + T (αR)
which reduces tc the energy expression of Eq. (2.38) wheðα = 1. The changes
brought by including the scale factor are only quantitative ið nature and leŁve the
qualitative conclusions unmodified.