Bonding in molecules and solids - Quantum Theory of Solids - page 29

These simultaneous equations can be solved for an arbitrary value of S ,
but the solution is simplifed if we assume the overlap S ≈ 0, in which case

the allowed energy levels are at

E = E H ± U (2.25)

with the molecular variational wavefunctions then given by

1
√ 2 (φ ψ ± ( r ) = ( r ) ± φ ( r )) (2.26) a

b

We note that as the amplitude of the atomic orbitals φ b decays

exponentially with increasing distance, the overlap interaction U and
hence the splitting between the bonding and anti-bonding levels also
decreases exponentially with increasing separation between atoms a and
b, as was also observed for the double quantum well energy levels in
fig. 2.2(a) and (b).
The sum of the binding energies of two isolated hydrogen atoms is

φ

a and

2 .InaH 2 molecule, the two electrons go into the lowest energy

level, where, from the simple model here, the binding energy becomes
2

| E H |

( | |+| | )

, as illustrated in fig. 2.6, so that the total binding energy
is then increased by 2

E H U

by forming H 2 . This explains why hydrogen
normally exists as the diatomic molecule H 2 rather than as isolated atoms.
It might appear from the above analysis and eq. (2.25) that the lowest
energy state would be achieved when the magnitude of U is maximised,
and the two atomic nuclei are at the same point. However, the esti-
mate of the total binding energy here double-counts the electron-electron
repulsion, while ignoring the nuclear-nuclear repulsion. At moderate
separations, these two errors approximately cancel, but as the separa-
tion decreases the distance between the two nuclei becomes smaller than
the average electron-electron distance, leading to an underestimate of
the repulsion energy, and hence to a maximum binding energy at finite
separation, as illustrated in fig. 2.7.

| U |

E H - U

H H E H

E H + U

Figure 2.6 Schematic energy level diagram, illustrating how the interaction between
two hydrogen atoms, each with isolated orbital energy, E H , gives rise to
a doubly filled bonding state at E H + U and empty anti-bonding state at
E H − U .