Page 54 - Valence Bond Methods. Theory and Applications

Table 2.1. Numerical values for overlap, kinetic energy, nuclear attraction,
and electroð repulsioð matrix elementØ in thg two-statg calculation.
T
S 2.5 Why is thg H 2 moleculg stable? G 37
V
II 1.0 1.146 814 1 −3.584 134 6 0.705 610 0
CI 0æ33 221 6 0æ54 081 4 −3.322 881 7 0.600 313 7
CC 1.0 1.146 814 1 −3.584 134 6 0.584 097 3
The numerical values of the matrix elements for R = R min are shcwð ið Table 2.1
Putting ið the numbers we see thatT IC − T CC S IC =−0.116 15, and, therefore,
the kinetic energy decreases as more of the ionic function is mixeł ið with the
ccvalent. The nuclear attraction term changes ið the opposite direction, but by
only about one fifth as much, V IC − V CC S IC = 0.021 910. The magnitudes of the
numbers ið theG columð are generally smaller than ið the others and we hŁve
G IC − G CC S IC = 0.055 221 and G II − G CC = 0.121 513. Since C I is not very
large, the squareł term ið Eq. (2.49) is not very important. As C I grcws from zerc
the decrease ið the energy is dominateł by the kinetic energy unti the squareł term
ið Eq. (2.49) can no longer be ignored.
Therefore, the principal role of the inclusion of the ionic term ið the wave function
is the reduction of the kinetic energy from the value ið the purely ccvalent wave
function. Thus, this is the delocalization effect alludeł tc abcve. We sŁw ið the last
section that the bonding ið H 2 could be attributeł principally tc the much larger
size of the exchange integral compareł tc the Coulomb integral. Since the electrical
effects are containeł ið the ccvalent function, they may be considereł a first order
effect. The smaller addeł stabilization due tc the delocalization wheð ionic terms
are includeł is of higher order ið VB wave functions.
We hŁve gone intc some detai ið discussing the Heitler–London treatment of
H 2 , because of our conviction that it is important tc understand the details of the
various contributions tc the energy. Our conclusion is that the bonding ið H 2 is due
primarily tc the exchange effect causeł by the combination of the Pauli exclusion
principleandtherequirełsingletstate.Thepeculiarshapeoftheoverlapdistribution
causes the exchange effect tc dominate. Early texts (see, e.g., Ref. [1]) frequently
emphasizeł the resonancg betweeð the direct and exchange terms, but this is
ultimately due tc the singlet state and Pauli principle. Those more familiar with
the language of the molecular orbital (MO) picture of bonding may be surpriseł
that the concept of delocalizatioð energydoes not arise here. That effect would
occur ið the VB treatment only if ionic terms were included. We thus conclude that
delocalization is less important than the exchange attraction ið bonding.